Differential and Integral Calculus

TH E M OD ERN M ATH EM ATICAL SERIES

LUCIEN AUGUSTUS W AIT GE NERAL E D ITOR (SENIOR PROFESSOR OF M ATH E MATICS IN CORNELL UN IVERSITY) THE MODERN MATHEMATICAL SE IE R S.

LUC IEN AUGU T S US WAIT,

Semor Pro ess M athe ‘ ( f or of mati cs 1 7: Cornell Um versityJ

G ENE A ED R L ITOR.

This s eries in cludes the follo wing works

ANA YTIC E M ETR B . H L G O Y . J . T an d JO PH AL y ANNER SE LEN . DIFFERE TIA N L CALCULUS . B J M M C M H O an d V S Y D y A ES A N IRGIL N ER . INTEGRAL CALC L B A. U US . D . M y URRAY . D IFFERE TIA A D I N L N NTEGRAL CALCULUS . B V S Y D n y IRGIL N ER a d J . I H H UTC INSON .

L L H . . A EB RA B J. T E EM ENTARY G . y ANNER

B J M M C M H . E LEM ENTARY GEOMETRY . y A ES A ON

The An al tic Geom et r D iffere tial Calculus and In t e ral Calculus ub y y, n , g (p “ lished in S ept ember of 1898) were writt en prim arily t o m eet t he n eeds of college s tuden t s urs uin c ou rs es in E n in ee rin an d Archit ect ure accordin l rae p g g g ; g y, p t ical roblem s in illus t ratio O f e n eral ri ci les u der discus sion la an p , n g p n p n , p y rt in h o important p a e ac b ok . Thes e t hree boo s t reatin t heir s ub ec ts in a wa that is s im le an d r i k , g j y p p act al et t ho ro u hl ri o ro us an d att rac tive t o bot h t eac her an d s t ud n c , y g y g , e t , re ceived s uch en e ral an d hea rt a ro val of t eachers an d have be en s o idel ado t ed g y pp , w y p in th e bes t colle es a n d u n ivers it ies o f t h e c o un t r t hat other boo s ritt e on g y, k , w n t he s ame en eral lan a re bei added t o t he s eries . g p , ng Th e D ifferent ial an d Int egral Calc ulus in on e v olum e was writt en especially for thos e ins t it ut ion s where t he time given t o t hes e s ubject s is n ot s ufﬁcient t o u s e ad va t n ageo usly t he t wo s eparat e books . The m ore elem en t a ry books o f this s eries are d es ign ed t o implant the s pirit of t h h Thi il m a e t h e or from t he e ot er boo s int o t he s econ dar s chools . s l k y w k w k , s chools u t hro u t he u n iv ers it con tin uous an d harm on ious and free from p gh y, , t he abrupt t rans ition which t h e s t udent s o oft en expe riences in ch an gin g from i r r h s p epa at ory t o his college m athem at ics . DIFFERENTIAL AND INTEGRAL CALCULUS

D ER - H D V R G L SN Y P . . E I I , (GOTTING N)

AN D

JOH N IRW IN (CH ICAGO) O F C O RNELL UNIVERSIT Y

NEW YORK CINCINNATI CH ICAGO A M E R I C A N B O O K C O M PA N Y RYR I CO GH T, 1 902, B Y

I I Y D E R AN D J H N I H H I V RG L SN O . UTC NSON

’ E D S H D NTERE AT STATIONER ALL, LON ON . PREFACE

TH E favorable reception acc orded the two v olum es on the Calcu lu s in this s eries s hows that they have been s erviceable

n a dem an d has ar s en in s u p plyi n g a real n eed . A ge er l i for

S m a m n the s u c s in r m s u a a i ilar tre t e t of bje t b iefer for , it ble A d for u s e in shorte r an d m ore ele m en tary cou rs es . ccor

in l in s n s e n u m e u s u s s an d s u s n s the g y, re po to ro req e t gge tio ,

pre s en t volum e has been prepared . The part o n t he D iffe ren tial C al cu lu s is of e ss en tially the s am e characte r as the form er s eparate volu me (whi ch will

h x bu n be refe rre d to in t e te t as D . t the ra ge of topics is res tri cte d ; various theorem s have bee n pu t in les s ah

s rac m an d e e a ern a o s ha e een n . t t for , f w r lt tive pro f v b give The chapter on the expan s ion of fun ction s has been s o arran ged that the rem ai n der theorem m ay be omitted W ith

m n h n n u h In h ea m en out arri g t e c o ti ity of t e s u bj e ct . t e tr t t of fun ction s of two in depen den t variables n o us e is m ade of

x a a an au iliary v ri ble .

Th h r e c a acteris tic featu res of t he larger b o ok are retain ed . Som e of the s e are as follo ws :

1 . The de a i n riv tive s pres e ted rigorou s ly as a li m it .

Th r f 2. e p o ces s o differen ti ation is s o arran ged as to give t hex-d a e a u n c n u in h ch u is a un c n eriv tiv of f tio of , w i f tio of x ; the res ultin g typ e form s b ein g prin te d in full-face ‘ letters in the text an d c olle cte d for refe ren ce at the e n d of the cha pter . vi PR E FACE

x m n d n 3 . M a i a a m i im a are dis cu s s ed as the tu rn in g

a u s in the a a n o f a u n c n h c m e a h v l e v ri tio f tio , wit o pl te gr p i cal repres en tation .

4 The n t on s of a e an d d . o i r t s ifferen tials are s o pres en ted as t o ro n a u ra ou t o f the de Of a de a v an d ar g w t lly i a riv ti e , e n ot in trodu ce d u n til the s tu den t has be com e familiar with the proces s of ﬁ n di n g the de rivative an d with it s u s e in

u d n the ar a on of a u n i s t yi g v i ti f ct on .

5 . The re a e d he r es of in ﬁex ion s cu a u an d l t t o i , rv t re ,

s m es ece e d rec an d m r h n a y ptot r iv i t c o p e e s ive treatm en t . The part on t he In tegral C alcu lus has been written en l tire y an ew . The ﬁ rs t ﬁ v e chapte rs dis cu s s the ordi n ary m ethods of

n ra The aim ha n t r h i teg tion . s b ee o m ake c le a t e ra tion ale o f e ach r ces s an d e n c u a the s u den s t o ec m p o , to o r ge t t b o e

n de n d n m i pe e t O f for ulas . The m ethod of redu ction has been put in the s imples t pos s ible form ; in the s ol u tion of proble m s s tu d en ts n e ed m a e n o u s e o f rm u as o f e du c ion k fo l r t .

In the s o u on Of at n a ac n s n s m e n s re l ti r io l fr tio i to i pl r o e , care has been taken to show the logical bas is of the u s u al a u m n ss ptio s . The ration aliz ation Of a diﬁ eren tial c on tain in g the s qu are root of a qu adratic e xpres s i on has been treated m u ch m ore

h Th o em is n e d e m e r ca fully t an u s u al . e pr bl i t rprete g o t i lly as e qu ival en t t o the ration al e xpres s ion o f the c oOrdin at es of a variable poi n t on a c on ic in te rm s of a varyin g param eter . This m akes clear ho w t he requ ired tran s form ation s are s ug gested an d pu ts the s ubje ct in a m ore attractive form . Spe cial care has been taken in pres e n tin g the su bje ct of i n tegration regarded as a s u mm ation s o as to combin e rigor

d n n u he an d s m c . The d n a ca es sc i pli ity or i ry s of i o ti ity, eit r PR E FACE

n an d t he a n n are ih of the i tegr or of vari ble of i tegratio , c lu ded in t he dis cus s ion .

de r n the o m u a n h are the d n n In ivi g f r l for le gt of , eﬁ itio

f u ch en h is en as the m the s um ch ds a o s l gt giv li it of of or , d n on h ch e ad ex s s es s b the u s e o f the eﬁ iti w i r ily pre it elf, y

- m ean a ue he em in t he m a de n e n a . v l t or , for of ﬁ it i tegr l

The exe c s s h ch are n ew h u h u the are r i e ; w i t ro g o t book ,

d d Nu m u us ra e exam carefu llygra e . ero s ill t tiv ples are worked ou t in the ex an d are ac com an e d b v a us s u st n s t t , p i y rio gge io an d m a e a n h r n re rks r l ti g to both t eo y a d practice . The au thors grate fu lly ack n owledge their i n debtedn es s to

he c a u e P ess r Jam s M cM ahon for erm s s n t ir olle g , rof o e , p i io ’ to m ake fre e u s e of M cM ahon an d Sn yder s D iffere n tial

Ca cu us a n u m e a u a e s u s n s an d as l l , for b r of v l bl gge tio , for s is t an ce in ad n n s the m n u t an d re i g portio of a s crip proof .

CONTENTS

D IFFEREN TIAL CALCULUS

CHAPTER I

FUN D AMENTAL PRINCIPLE S ARTICLE m r ition s 1 . Ele enta y deﬁn

2 . I nﬁ n ites im als an d in ﬁ n ite s

u dam e t a t e orem s co cer i in fi n ite sim als an d im i 3 . F n n l h n n ng l ts in general Comparison Of variables in fi n im l n m ris o of ites a s a d of in fi n i . rd 5 . Co pa n te s O ers of m ag n it u de Useful illu strations of in fi n it e sim als of different o rders Continu ity Of f unction s Comparison of sim ult aneou s infinite sim al increm ent s of two related v ariables D e itio f deri tive 9 . fin n o a va om e tri a i u str tio s of a derivative 10. Ge c l ll a n The o ratio f dif ere tiatio 1 1 . pe n o f n n I re i n r i u c ti 1 2 . nc as ng a d dec e as ng f n on s A ebr ic te st of the i terva s of i crea i an d decre a i 1 3 . lg a n l n s ng s ng f i of a u ti 14. Differentiation o a funct on f nc on

CHAPTER II

D IFFERENTIATION O F TH E ELEMENTAR Y FORMS

D ifferent iation of th e produ ct o f a cons tant a n d a v ariable D iffere n tiation Of a s u m D ifferent iat io n of a prod uct D iffere ntiat ion of a qu ot ie n t D ifferentiation o f a com m en su rable power o f a fu nction Elem e ntary t ran s cende nt al fu nction s D iffere tiatio of o as an d lo a n n l go, g,

1X X C ON TE N TS ARTICLE D i fere i i f th s im e ex o e tia u ti 22 . f nt at on o e pl p n n l f nc on i re t i i h e era e x o e tia u ti 23 . D ffe n at o n of t e g n l p n n l f nc on Diﬁe re n t iation of an incom m e nsurable powe r i f in u 25 . D iffe rentiat on o s e e i tio Of co u 26 . D iff r nt a n s iff tiat i O f t an u 27 . D eren o n i iati Of c t u 28 . D ffere nt on o

i ti f. s ec u 29 . D iffere nt a o n o i i f s c u 30. D ffere ntiat o n o c er u 3 1 . D iffe re ntiatio n of v s

“ 1 i r iatio o f s in u 32 . D ffe e nt n D iﬁ r n i ti n of t he rem ai i i r 33 . e e t a o n ng nve s e t rigon om etric form s r 34. Table of fu ndam en t al fo m s

CHAPTE R III S UCCESSIV E D IF FERENTIATION

i f he n th derivati e 35 . D eﬁn tio n o t v Expres sion for the nth de rivative in cert ain cases

CHAPTER IV E X PANSION O F FUNCTIONS

Convergence an d diverge nce of s erie s General te st for interv al of convergen ce Re m ainder after n t erm s ’ M aclaurin s expan sion of a fu nction in power-serie s ’ Taylor s se ries ’ R olle s theore m ’ F orm of re mainde r in M aclau rin s series Another e xpress ion for rem ainder Theorem of m ean value

CHAPTER V IND ETERMINATE FORMS

D eﬁnition of an inde term in at e form Indet erm in ate fo rm s m ay h ave dete rminate value s Evalu atio n by de v elopm e nt Evalu ation by differe ntiation CON TEN TS

ARTICLE a u ti of the i det ermi at e orm 50. Ev l a on n n f E a u ati Of the orm 00 0 51 . v l on f

E a u atio of t he orm 00 co 52 . v l n f E u atio of t he orm 53 . val n f E u t i of the orm s 0 54. val a on f 00

C H APTER VI M OD E O F VARIATION O F FUNCTIONS O F ON E VARIAB LE

RevI ew of increasing an d de cre asing fu nctions Turning values of a fu nctio n Critical valu es of t he variable ’ M eth od of dete rm ining wh et her d) (23) c h anges it s sign in passing th rou gh z ero o r inﬁnity Second m ethod of de term ining whet her (33) c hanges Sign in pass ing t hrou gh z ero ’ Conditions for m axim a an d m inim a derived from Taylor s t heorem The m ax im a an d m ini m a of any contin u ou s functio n occur alt ern ately Simpliﬁcations t h at do n ot alte r crit ical values Ge ometric problem s in m axim a an d m inima

CHAPTER VII

RATES AN D D IFFERENTIALS

R te . T 64 . a s im e as indepe nde nt variable

6 5 . Abbreviate d notatio n for rat es

6 6 . D iffere ntials oft e n substitu t ed for rate s

CHAPTER VIII D IFF E RE NTIATION O F FUNCTIONS O F Two VARIAB LE S

D eﬁnition of continuity Part ial differentiatio n Total diffe rential Langu age o f differe ntials D ifferent iation Of implicit fu nctions S uccess ive partial differentiation Order of differentiatio n in diﬁ e rent C ON TE N TS

CHAPTER IX

CH ANGE O F T H E VARIAB LE ARTICLE I rcha e of de e de t an d i de e de t variab 74 . nt e ng p n n n p n n les f the de e de t variab e 75 . Change o p n n l e O th i de e de t variab e 76 . Chang f e n p n n l

APPLI CATI ON S TO GE OME TR Y CHAPTER X TANGENTS A N D N O R M ALS i Geo m etric m ea i of — n ng iiZ Equ ation of t ange nt an d norm al at a given point Le th of ta e t orm a subt a e t s ub orm a ng ng n , n l, ng n , n l

POLAR COOR D INATES i f 80. M e an ng O p gg dy do R t i b t ee an d 81 . ela on e w n p dx dp m o ar u b e t n 82. Le t o f ta e t or a s t a a d o ar sub ng h ng n , n l, p l ng n , p l norm al

C HAPTER XI

D E V V E O F AR C A A N D S E RI ATI AN , REA , VOLUME , URFAC O F REV OLUTION

D erivative of an arc

Tri o om etri c mea i of g n n ng ’ dx dy D erivative of the volu m e of a s olid of revolution D erivat ive of a s urface of rev olu tion D erivat ive of are in polar coordinates D erivative of area in polar coCrdin ates

CH AP ’ I‘ ER XII

AS Y MPTOTES

rbo i an d arabo ic ra ches 89. Hype l c p l b n D itio of a recti i e ar as m tote 90. eﬁn n l n y p C ON TE N TS x iii

ARTICLE D E TERMINATION O F AS Y MPTOTES 1 M et od of im itin i t erce t s 9 . h l g n p

92 . M et hod of i s ect io . I ite ordi ate s as m tote s ara e n p n nﬁn n , y p p ll l t o ax es

M t od Of subs titu t io . Ob i u e as m tot e s 93 . e h n l q y p N u mber of as m totes 94. y p

M ethod of ex a sio . Ex icit u ctions 95 . p n n pl f n

CHAPTER XIII

D IRE CTION O F B END ING . POINTS O F INFL EX ION

Concavity u pward an d downward Algebraic t est for pos itive an d n egativ e bending i h Analytical derivat on of t e test for t he directio n of be n ding . Concavity an d convexity towards the axis

CHAPTER XIV

CONTACT AN D CURV ATURE

Order of contact N umber Of condition s implied by cont act Contact of o dd an d of even order Circle of cu rv at ure Length of radiu s of curvatu re ; coordin ate s of center of curvat ure D irection of radiu s of curvature Tot al cu rvature of a give n arc ; average curvature M easu re of curvat ure at a give n point Cu rvatu re Of oscu lat ing circle D irect derivation of the expressio n for K an d R in polar

EVOLUTE S AN D INV OLUTES

1 10. D eﬁnition of an evolute

1 11 . Properties of the evolute

CHAPTER XV

SINGULAR . POIN TS 1 2 D 1 . eﬁnition of a singular point

1 1 3 . D ete rm ination of singul ar point s of algebraic cu rves xiv C ON TE N TS ARTICLE 4 M u t i e oi 1 1 . l pl p nts

u s s . 1 1 5 . C p u ate oi t s 1 16 . Conj g p n

CHAPTER XVI EN V ELOPES m i 1 17 . Fa ly Of cu rves

1 1 8 . Env elo pe of a family of cu rves The e u e r 1 1 9. nv elope to ch s ev e y curve of the fam ily e f rm a s f 120. Envelop o n o l o a given c urve

21 . Two aramete rs on e e uatio of co ditio £ p , q n n n

IN TEGRAL CALCULUS

CHAPTER I

GENERAL PRINCIPLES O F INTEGRATION

The fu ndam ent al problem Integration by inspect ion The fu ndam ent al form ulas of integration Cert ain ge neral principles Inte gration by part s Integratio n by s ubs titu tion Additional s tandard form s

CHAPTER II

RE D UCTION FORMULAS

CHAPTE R III

INTEGRATION O F RATIONAL FRACTIONS

D m s itio of ratio a ract io s . 1 32. eco po n n l f n t d 1 3 . E I . actors of the rst de ree one re ea e 3 CAS F ﬁ g , n p C ON TEN TS XV

m II. actors of the rs t de ree s o e re e ate d 1 34. E CAS F ﬁ g , p f u d r t i a t rs II . Occu rre ce o a a c c o o e re eated 1 35 . C I ASE n q f , n n p

IV . O ccu rre ce of u adratic actors som e re ated 1 36 . C e ASE n q f , p e r theorem on the i te ratio Of ratio a ra ti s 1 37 . G ne al n g n n l f c o n

CHAPTER IV

INTEGRATION B Y RATIONALIZ ATION / Integration of function s cont aining the irrat ionality Q ax b

2 Integratio n O f expre ss io n s co ntaining \/ax bx c Ge neral theore m on the integration ofirrational fu nct ions

CHAPTER V

INTEGRATION O F TRIGONOMETRIC A N D O TH ER TRAN S C E N D E N TAL FUNC TIONS

Integration by substit ution

2 z I te ratio of s ec “ dx cosec n x dx n g n f x , f

m ? " 1 m ? " 1 I te ratio of s ec a: t an dx co se c a: cot dx n g n j x , I x

‘ I t e ratio of t a " dx cot n dx n g n f n x , [ x

m " Inte gration o f j sin x cos x dx

‘ dx I te rat io of n g n ’ j a b co s x

w Inte gration of fe s in n x dx

CHAPTE R VI

INTE GRATION AS A SUMMATION

148. The de ite i te r ﬁn n g al . 248 149 . Geom e trical inte rpret atio n of the deﬁnite int egral as an area 253

1 50. Ge era iz at io of h r r n l n t e a ea fo m ula . Pos itive an d negative

1 51 . Cert ain prope rtie s of deﬁnite integrals

152. D e itio of the fin n definite integral whe n f(x) becom e s infinite . Infinite limits CON TEN TS

CHAPTER VII

GEOM E TRICAL A LICATIONS ARTICLE PP A n r co rdi 1 3 r s . Re t a a o ates 5 . ea c gul n r t d 1 54. A e as . S econd m e ho t r e n 1 55 . Precau ions t o be obs e v d i e valu ating deﬁnite int egrals

’ r rdi t 1 56 . Areas . Pola coo na es

. Re ct r r in t 1 57 Length of curves . angula coci d a es r r in t 1 58 . Length of cu rves . Pola coC d a es 15 M e asu rem e t of arcs b t he aid of aram etric re re se t ti n 9 . n y p p n a o r r f o uti 1 60. A ea of su face o rev l on V um e f so id of revo utio 1 6 1 . ol o l l n l ne us a ications 1 62. Misce la o ppl

CHAPTER VIII

SUCCESSIV E INTE G RATION

S ucce ssive integration of fu nction s of a s ingle v ariable Int egrat ion of fu nct ions of s everal variables Integratio n of a total differential M ultiple integrals D eﬁnite m u ltiple integrals Plane areas by dou ble integration Volume s

2 I FFER E N TI A CA C U Us GIL I D L L L [ .

x m e a ples of limit s m et with in elem en tary geom etry are u s u a the n d h lly of s e c o kin d ; i . e . t e variable does n ot reach h im t e l it . The limitin g val u e s of algebraic expres s ion s are

m o re re u e n h r f q tly of t e ﬁ s t kin d .

E . . the u ctio has the im it 1 e b m g , f n n l wh n 35 eco es z ero ; it has x2 1

the im it 0 e 3: b m i l wh n e co e s nﬁnite . The fu nction sin x has the limit 0

e 3: becomes z ero t an has the limit 1 e be me wh n x wh n x co s 2.

EXERC IS ES

— 1 . Let x Ax B C S ow t at 0 x 0 ¢ ( , y) + y ; h h y) , ) are the e u atio s f t o er i q n O w p pe nd cular lines .

2 2 . If x 2 e 1 33 S o t at s in = in x f( ) , h w h f( g) s x 1 m u 3 , If ( p sho w that . bo) , x + 1 1 + z y

4 If = z ~ . f(x) log il zS how t hat f(x) + f(y) f <ﬁ >

2 2 5 . Give x V I x ﬁn d x . n f( ) , )

6 . If x : x f( y) f( ) prove that f(1) 0 .

7 . Give x S o t at 0 0 an d t at n f( y) h w h f( ) , h 23 bei n x a os itive i t e er. pf( ) f p ) , p ng y p n g

8 . Us i t he s am e otatio as in the ast ex am e rove t at ng n n l pl , p h mx m x m b i e an ratio a racti . f( ) f( ) , ng y n l f on

2 . Infi n i A ha h t es im als and infi nites . variable t t approac es z e ro m i as a s an in n it es im a l . In her rds an in fin i li it fi ot wo , t es im al is a v ari able that bec om es s m aller than an y n um ber tha n t ca be assign ed . The re cip ro cal of an i n fin ites im al is then a vari able that

e c om s ar e han an n u m er ha c an be as s n ed an d b e l g r t y b t t ig , i s calle d an inﬁ n it e v ari abl e .

" E . . the u mbe r is an i itesim a e n is ta e ar er an d g , n Q ) nﬁn l wh n k n l g ar i 2" i l ger ; an d it s rec procal s an in ﬁnite variable . 1 F UN DAM E N TAL PRI N CIPLE S 3

“ Fro m the deﬁn ition s of the words limit an d in ﬁn it es i ” m al the o n us ul c a es are mm d a e n n ces foll wi g ef oroll ri i e i t i fere .

1 . The d ff n c e n a e an d i m Cor. i ere betwee a vari bl t s li it i n m s an i ﬁn ites i al variable .

. 2 . C on s the diﬁ eren ce n a n an Cor ver ely, if betwee c o s t t an d a a a e be an n n s m a h n th c n s n is the v ri bl i ﬁ ite i l , t e e o ta t

h r limit of t e va iable .

c n en n c the s m 5 u s d n d ca e For o v ie e , y bol will be e to i i t that a variable approaches a con stan t as a lim it ; thu s the s ymbolic form a; 5 a is to be read the variable 3: approaches t he con s tan t a as a lim it . “ 5 . 00 1 m The s pe cial form a: is read 1 : beco es in ﬁn ite . The corollarie s j us t m en tion ed m ay accordin gly be s ym bolically s tated thu s :

é = a 5 1 . If x a h n x a h n 04 0 , t e + , w erei ;

= a _ h éz I a n d n . 2 . f x + , a a t e x a

h h 1 i It will appear that t e c ief u s e of C or . s to c on vert

en m re a n s n the rm rd n a u a n s giv li it l tio i to fo of o i ry eq tio , s o that they m ay be c ombin e d or tran s form e d by the laws

n d h r 2 il gover in g the equ ality of n u mbers ; an t en C o . w l s erve to ex r s s the resu in the o n a o rm a m a n p e lt rigi l f of li it rel tio .

In all cas e s h h a r a ac u a ec m s u a , w et er va i ble t lly b o e eq l to it s m or n ot t he m an r e is ha he d ffe li it , i port t p op rty t t t ir i r en i n An n n m a i n ot n ce s a in ﬁ n itesim al . i ﬁ ites i l s e ces s arily

m I n in all s tage s of its hi s t ory a s m all n u ber . t s es s e ce lies in it s e d c as n n um e r ca ha n z for it s pow r of e re i g i lly, vi g ero

mi an d n ot in the s m a n ss an the c n s an val li t, ll e of y of o t t u I u n de n d a an es it m ay pas s through . t is freq e tly ﬁ e s “ n n e s m a u an but hi s x e s s n sh u d be i ﬁ it ly ll q tity, t e pr io o l

n e e d in the a e s n s . Thu s a c n s n n um i t rpret bov e e o ta t ber,

h s ma m a be is n ot an n n s ma . owever ll it y , i ﬁ ite i l 4 I FFE R E N TIA CA C U Us OIL I D L L L [ .

3 . Fun damental theorems con cern ing inﬁnites imals and

h n h limits in general . T e f ollowi g t e o re m s are us e f ul in the p ro ces s es o f t he calcul us ; the ﬁrs t thre e relate to in

ﬁ n it im l h u t o m in es a s t e as t s e n e . , l fo r li it g ral

Th du o f n Theorem 1 . e pro ct an i fin ites imal a by an y fi n it e c on s tan t k is an in fi n ites im al

i . e . a 5 : 0 , if ,

h a t en le 0.

c an as s n d n um r. Th n b h h s s a For , let be y ig e be e , y ypot e i , 0 can c m es s han hen c e lea can c m e e ss han 0 the be o e l t ; be o l t , 70

r r n d n um er n d i h r n a ass a s e e n s m . arbit y, ig e b , , t e for , i ﬁ ite i al

Theorem 2 . The algebrai c s u m of an y fi n ite n umber n of in fin it es im als is an i n fin ites imal

5 0 5 0 t . . 05 e , if , 6 ,

h n a 8 t e + , +

Fo r the s u m of the n variables does n ot at an y s tage

m r a e xc d n t m e the ar s o f hem bu t h s n u e ic lly ee i s l ge t t , t i produ ct is an in ﬁn ites im al by the ore m 1 ; hen ce t he s um

of t he n vari ables is either an i n ﬁ n itesim al or ze ro .

N h n i ite u mber f in ﬁn it es im als m a be OTE . T e sum of a nﬁn n o y

i nitesim al ite or i ite accordi t o circum st a ces . nfi , fin , nfin , ng n

b a ite co sta t an d if n be a variable t at becom es E . . if a e g , fin n n , h a fl 9 _ _ are all in ite sim a variab es but in ite ; t e , , ’ ; fin h n 2 fin l l 11 n

£ 3 t simal to n terms which is in i e , 5 3 ﬁn n n

whi e 9 9 t o n term s a hich is ite l , w ﬁn , n n

t o n terms which is inﬁnite . 3 F UN DAMEN TAL PRIN CIPLE S 5

h Theorem 3 . T e produ ct of two or m ore in ﬁn it es im als

n n is a i ﬁ nites imal .

Theo m 4 . If a a es x a a s e u a an d re two v ri bl , y be lw y q l ,

o n e of t hem a a oach a m a t h n t he o he a if , , ppr li it , e t r p

roache th m p s e s a e limit .

I Theorem 5 . f the s um of a ﬁ n ite n u mber of vari ables be v r a e hen t he m he s um is u a t he s um a i bl , t li it of t ir eq l to of their lim its ;

t . e . lim as , ( + y +

let x é a é b For, , y ,

Th n = m A . 2 e a r 1 . x a t C or . + , , [ , — h n a i O i O w erei , B , hen c e x + y + b or 0 T ut h . 2 . 3 + , [ 2 2 r . hen ce b Art . C o , y , ,

lim x + li1n y + ~

h T eorem 6 . If t he produ ct of a ﬁ n ite n u mber o f vari a

es a a e hen t he m h du c is e u a bl be v ri bl , t li it of t eir pro t q l to

h r d t e p o uct of their limits .

Th o m 7 . If the u n a es x e re q otie t of two v riabl , y be

a a e h n t he m he u n is e u a th v ri bl , t e li it of t ir q otie t q l to e

u n he m s d d h m s n ot th q otie t of t ir li it , provi e t es e li it are bo

n n n ot h z . i ﬁ ite , or bot ero

4 . m h n Comparis on of variables . So e of t e pri cipl es j us t es tablished will n ow be u s ed in comparin g vari ables with

h h Th r m r n o r h e ac ot er . e elative i po ta ce f two va i ables t at are approachi n g lim its is m eas u red by t he limit of their ratio . 6 IFFE R E N TIA CA C U Us O IL I D L L L [ .

D EFI N ITI O N . O n e v ar a e a is s aid t o be n n es m a i bl i ﬁ it i l ,

n n e or n in c om aris n it h an o her ar a e T hen i ﬁ it , ﬁ ite , p o w t v i bl w

h h r r io a a: i z n n t e n t e m e a s e o e . li it of t i t r , i fi i , or fi it “ In t he s t wo cas es the hras e n n it es im a o r n n e fir t , p i fi l i fi it in c om p aris on with is s om etim es replac ed by the les s pre “ ” hr n n m a n n r h cis e p as e i fi itely s ller or i fi itely l a ge r t an .

In the h rd cas e the ar a es s d the s m e t i , v i bl will be ai to be of a d m or er of agn itu de . The follo win g theorem an d c orollary are u s efu l in com parin g two v ariables

Theorem 8 . The limit of the qu otien t of an y two varia

es x is n ot a er d b add n hem an n u m ers bl , y lt e y i g to t y two b

a h ch are es ec e n n s m a in c m s n h , B , w i r p tiv ly i ﬁ ite i l o pari o wit thes e variables ;

i . e . ,

provided

x a x For s in c e

s b h m s 4 6 ha it follow , y t eore , , t t

x a x li m li m A B a

but b he m s 7 5 an d h hes s , y t ore , , ypot i ,

h o t eref re , 4 F UN DAME N TAL PRIN CI PLE S 7

If the diﬁ e ren ce e n v a s x be Cor. betw e two riable , y

n n t es m a as her the m h is 1 an d i ﬁ i i l to eit , li it of t eir ratio , con vers ely ; x ’ 9 0 h n i . e . , if , t e y 9

s n c For, i e

5 i C or. 2 . 1 0 Art . 2 h n ce an d 1 . e , g [ ,

x y é C n e s 1 h n 0. o v r ely, if , t e ;

For b Art . 2 , y ,

” “ y i . e . , a

m n . 5 . ris on of inﬁ n ites i als a d of inﬁni s d r of Compa , te Or e s

I ha a read e en a e d ha magnitude . t s l y b s t t t t an y two variable s are s aid to be of the s am e order of m agn itu de when the lim it

he ra is a n e n um r ha is s a is n h r of t ir tio ﬁ it be ; t t to y, eit e

n n e n or z e . In s s r c s e an ua v a a s i ﬁ it ro le p e i l g ge , two ri ble

‘ are of the s am e orde r of m agn itu de whe n on e vari able is

n e he r n n ar er n or n n e m a r h th h it i ﬁ itely l g i ﬁ it ly s lle t an e ot er .

n s an ce 9 is t he s m e rder as hen k is an For i t , M of a o B w y ﬁ n ite n u m ber ; thu s a ﬁ n ite m u ltiplier or divis or does n ot

af ec t he de m a n u d of an a a e he h r f t or r of g it e y v ri bl , w t e

n n es m a n e or n n i fi it i l , fi it , i fi ite .

In a m n n in fin it esim als an on e of h m a proble i volvi g , y t e , , m ay be chos en as a s tan dard of com paris on as t o m agn i tu de then a is called t he prin cipal in fin ites im al Of the firs t

‘ 1 o d an d a is ca ed t he n c a n n the rs r er, ll pri ip l i ﬁ ite of ﬁ t

order . 8 I FFE R E N TIA CA C U US OIL I D L L L [ .

T o tes t for the o rde r n o f an y giv en in ﬁn ite s im al B with

e e re n ce t he i n c i a n n s m a a on h ch de r f to pr p l i ﬁ ite i l w i it pen ds , it is n e ces s ary to s ele ct an ex pon en t n s u ch that

lim 3 «w on

h i e e n s n e c on s an n ot z . w r i k a ﬁ it t t , ero

he n n is n e a ve 8 is n n e de — W g ti , , i ﬁ it of or r n . An

n n es m a or n n e of de z e is a n e n u m i fi it i l, i fi it or r ro , fi it ber .

4 3 to ﬁ n d the order of the variab e 3 x 4 x it re ere c l , w h f n e to a: s the ri i i it im a a p nc pal nﬁn es l . 2 3 4 Com ari it x x E in su ccess io p ng w h , , , n

4 3 I n 3 a: 4 x lim 4 x : 0 n n ) , ot ite ; x é o ﬁn

4 3 lim 3 33 — 4 33 lim —4 ﬁm te é ’ ; x O x3

8 lim 3 x 4 9: lim n co , ot ﬁ n ite ;

4 3 3 hence 3 x 4 x is an inﬁnite sim al of the s am e order of s m allne ss as 33 ;

t at is of the t ird order. h , h

The o rde r of large n es s of an i n ﬁn ite variable can be teste d

b e h r in s m a wa . For n s an ce e a n as t e n c a a i il r y i t , if x t k p i ip l

n n e le t be re u e d ﬁ n d the d t he a a e i ﬁ it , it q ir to or er of v ri bl 4 3 3 4 m n h x an d x 3 x 4 x . Co pari g wit

° t ' 1 m S ca ll m f 3 x x é oo 3 x z oo ( x

- = x . cc 4 x oo 23

4 3 hen ce 3 x 4 x is an i n ﬁn ite of t he s ame orde r of largen es s

i s that is O f t he u h o d . , fo rt r er The p roces s o f ﬁ n din g the limit o f the ratio Of two in

im l is ac t a e d b the o o n r n c as e d ﬁn it e s a s f ili t y f ll wi g p i iple , b

1 0 IFFE REN TIA CA C U Us CH I D L L L [ . .

B d d n m y ivi i g each e mber of thes e in e qu aliti es by s in 9

9 s ec ,

bu t s e c 9 1 hen 9 5 0 , w ,

lim 9 lim Sin 9 hen c 1 an d e Gi o , 9 5 0 s in 9 9

S m ar b diIfidin the n e u a es b i il ly, y g i q liti y

9 cos 9 t an 9

lim 0 lim t an 9 h n c 1 an d e e ei o , t an 9 9

Th 9 . 9 C r 1 . e n um rs s in t an 9 in o be , , are ﬁn it es im als of the s am e order .

h x 9 . 9 or. 2 T e e ss n s s in t an 9 are in n i i C pre io , 9 ﬁ t es 9 m al as t o .

Th orem 2 . If on e an e 9 a h an e an e gl , of rig t tri gl , be

n n e s m a t he rs rde hen the h n us e r an d i ﬁ it i l of ﬁ t o r , t ypote the adjacen t s ide a: are e ither both

n or he are in fi n it es im als Of fi ite , t y the s am e orde r ; an d the oppos ite s ide y is an i n fin ites im al o f orde r

FIG . 2 . n on e higher than that o f r a d x . m For cos 9 h ch a r ach s the a u e 1 as 9 i 0 , w i pp o e v l ; 7’ hen ce x 7' are in ﬁn it es im als the s am d h ch m a , of e or er ; w i y

h r be t e orde z e ro .

A s T s in 9 l o y , an d s in 9 is of order 1 ; therefore y is of order on e higher

han T b h em 1 Art . 5 . t , y t eor , FUN DAME N TAL PR I N CIPL E S 1 1

I h am 9 the r de r an r. n t e s e cas e s d Co , if be of ﬁ t or ,

T an d a: be t he rd r n h n t he d f ere n ce e e en if of o e , t e i f b tw

n d i n r r 2 T a x s an i ﬁ n ites im al Of o de n + . 2 m 2 9 z 2 2 2 2 7 8 T — x = = T s i11 9 T — x y , ; T a:

2 bu t the ders s in 9 T E are re s e c i 2 n 2 n or of , + , p t vely , , ;

h e b h m 2 Art . 5 T is ord t erefor y t eore , , x of er — 2 n + 2 n = n + 2

Theorem 3 . The differen ce be twe en the le n gth of an in ﬁ n ites im al arc of a circle an d it s chord is of at le as t the

h rd d h n h i h r t rd t i or er w e t e arc s t e ﬁ s o e r .

For let CD be t he arc an d CB D B an en s at it s , , , , t g t

T r m ex tremities . hen by ele m en ta y ge o etry

r chord CD a c CD D B B C.

Let the an gle B OD 9 be take n as the prin cipal in ﬁn i

t es im al Th n s n ce arc . e , i

CD 2 T9 an d T is n e , ﬁ it ,

urc i of d 1 . CD s or er . A a n s n ce AD is g i , i of

order 1 ( Th . an d an gle AD B 9 is of or

der 1 h n ce D B is FI G . 3 . , e of

1 n d B D A i T 2 r d a D s d 3 h . C o . he e or er , of or er ( , ) t refor

ch d D is d r 3 (D B B C) or C of or e .

r D ch rd CD is d h en ce a c C at eas . H o of or er, l t , t ree

Theorem 4 . The differen ce b etwe en the len gth of an y

n n e s m a arc n cu a u re an d it s ch rd is an i ﬁ it i l (of ﬁ ite rv t ) o ,

h r n n s m a at eas the d d . i ﬁ ite i l of, l t , t ir o er

h imi i ratio of NOTE . The c urvature is s aid t o be ﬁnite whe n t e l t ng the length of a s m all chord to the acute angle betwee n the t angent s at its ex r mi i t e t es is ite an d n ot z ero . ﬁn , 1 2 IFFE RE N TIA CAL C UL US C I D L [ m .

If P be s u ch an arc the ch d P an d the an Q , or O gl e TSP

e are b h h s s in ﬁn it es im als of the a ' , y ypot e i , s me o rderﬁ Le t the an gle TS P be t he

n Th c a n n es m a . en pri ip l i ﬁ it i l , s in ce TS P S R S Q RP ,

it follows that the gre ater Of

the a an es s a S R l tter two gl , y Q ,

is the s rder h e the of ﬁr t o , w il othe r m ay be of the ﬁ rs t or

h h d r h a e o e . A s t e ig r r l o , gre ate r of the two s egm en ts

R P R s a t he a e r is of Q , , y l tt ,

t he s rde hi e R m a ﬁr t o r , w l Q y

be Of the rs o r h h rd r ﬁ t ig e r o e .

A n b t he m 2 B S are of the s am e rd an d gai , y ore , Q , O o er,

r f h PR , PS a e O t e s am e o rde r .

arc P cho rd P S S P P e m . Q Q O Q , [g o

( QS QR ) ( S P RP) ;

z bu t si n ce OS QR QS ( 1 c o s B) 2 QS s in g-

z an d s m SP R P 2 SP s n - , , i ilarly, i g

an d s n c ach hes e d uc s is at as of the h d , i e e of t pro t , le t , t ir

de he n ce are P ch rd P is at eas t he h rd or r, Q o O of, l t , t i orde r.

* If TS P were of i er order t a P the curvat ure ould be z ero h gh h n Q , w ; if of lo er order the curvat ure ould be i ite the ormer is the case at w , w nﬁn f

an in ﬁexion the latt er at a cus . , p 6— E 1 8 7. J FUN DAM N TAL PRIN CI PL ES

EXERC IS ES

1 ria e av i a ri t n e at C dra CD er . Let AB C be a t ngl h ng gh a gl ; w p e n dic ular t o AB D E e r e dicu ar to CB E F er e dic u ar t o D B p , p p n l , p p n l , ’ F G perpe ndicular t o E B ; le t t he angle B AC be an inﬁ n ites im al of the

rst order AB rem ai i ite . Pro ve t at ﬁ , n ng ﬁn h

re d CD , CB a of or er 1

E r f r D B , D a e o o der 2

E B E F CB CD are of order 3 , , ( ) ;

FB PC D B D E are of order 4 . , , ( )

2 f t r r i h re a h t ri ? B D ? C D E . O wh a o de s t e a of t e angle AB C C

3 A i i f t two re ct an ar . s tra t l e o co st a t e t s ides be ee ul gh n , n n l ng h , l w n g ' ’ ’ ’ h i . strai t i es CAA CB B . Let AB A B be two os ition s of t e e gh l n , , , p l n S ow t at in the imit e the t wo os itio s coi cide h h , l , wh n p n n , AA’ CB B B ’ CA

. h n n n d n den a a e 7 Continuity of fun ction s . W e a i epe t v ri bl

x in as s n r m a 6 as s es h u h e n m ed a , p i g f o to , p t ro g ev ry i ter i te

i con t in u ou a u s ca e d s . v l e , it ll A fu n cti on f(x) of an i n dep en den t v ariable a: is s aid to

c n n u ous at an a u e x he n x is n e a an d be o ti y v l 1 w f( 1) ﬁ it , re l ,

d m n a an d s uch ha in ha e r wa a: a ach $ eter i te , t t w t ve y ppro 1 , $3 ,

h h ~ i in c a s n d n den f th l w ach . w i f( 1) i epe t o e a of appro From t he d eﬁn ition of a lim it it follows that c o rres pon din g to a s mall i n crem en t of t he vari able the in crem en t of the

un c n is a s s m an d h c s n d n an n u m f tio l o all , t at orre po i g to y ber

6 us ss n d an h n um 8 can d m n d , previo ly a ig e , ot er ber be eter i e , s uch that when it remain s n um e rically les s than 8 the

diﬁ eren ce

is nu m c l eri al y les s than e . 1 4 IFFE R E N TIA C IL I D L AL C UL US [O .

2 E . . the u ctio x x 3 g , f n n f( ) x 2

is co ti u ous at the n n valu e x 1 .

” 1 = 6 1 k : 6 5 k f( ) , f( ) k ,

2 1 h 5 11. 71 : lz 5 f( ) ( h) .

If the differe ce 1 h 1 is t o be es s t a s a n f( ) f( ) l h n , y, m , only nece ss ary th at 1 h (5 1 01000000

If 8 the n for every value of k s uch th at

h 8,

it is e vide t t at 1 11 1 is ess t a n h f( ) f( ) l h n rm .

Whe n a fu n c tion is con tin u ou s at e very value o f a: within the n t e rv a o m a b is s a d be n n u u within i l fr to , it i to co ti o s l th a t in t erva .

he n a v a u e re e x s s at h h an n of the r ce d n W l l i t w ic y o e p e i g c n d n s is n ot u e d a ven u n c on ( T the o itio f lﬁll for gi f ti M ) ,

a d o d is con tin u ou s = un c n is s t be at £13 x . f tio i 1

be i E . . the u ctio m a com e i te as e 2 g , f n n y nﬁn , , wh n x ; 2 a x

2 2 t he u ctio m a be im a i ar as 9 x e x 9 f n n y g n y, , wh n ; 1 the u ctio m a be i determi ate as s in e a: i 0 f n n y n n , 5, wh n ;

a the va u e of the u ctio m a de e d u o the m a er ﬁn lly, l f n n y p n p n nn ic the variab e a roac es the va u e as as in the u ctio wh h l pp h l ] , f n n

I _ z 2 3 o l 1 31

= — — — é whe n x _ l when x h k 5 2 as k O. + ; , f( )

A c on tin u o u s fu n c tion actu ally attai n s it s limit for

u of h h n c n n u val e t e variable within t e regio of o ti ity, h t e variable m ay be s ubs titu ted di re ctly. - 7 s . ] F UN DAM E N TAL PR I N CIPLE S 1 5

1 h n n m It m ay be shown as on p . 4 t at a y p oly o ial

” "" 1 ax bx [n a pos itive i n t eger .

n n u r n is co ti u o s for e ve y ﬁ ite val ue o f re . The ordin ary fu n ction s i n volvi n g radicals an d ratios are

n in u n c o t ou s o ly for ce rtain in tervals . The trigon o m etric fun ction s s in a: an d cos x are c on ti n u ou s for all real ﬁn ite valu es of x ; the other trigon om et ric fu n c

r r n r i rm in n d n ti on s a e ati o ally ex p es s ible n te s of s e a c os i e .

Show th at t an x is discontinu ous whe n x Ji m

” The ex p on en tial fu n ction a an d t he logarithm ic fu n ction

o a: are each c n n u u s the rm er for all n i e v a u es l g o ti o , fo ﬁ t l l f D C . . x the a e a l n e s v a ue s o a: . of , l tt r for ﬁ it po iti e v l [ , p

8 . Comparis on of s imultaneous inﬁ n it es imal in crement s of

h f r i r n rn d t wo related variables . T e las t e w a t cles we e c o ce e with the prin ciples t o be u s ed in com pari n g an y two in ﬁn i

I h n h l h h t es im als . n t e u s a n s e t e aw b ch ac ill tr tio giv , y w i e

ar a e a oached z er was as s n e d or e s e the t wo v ari v i bl ppr o ig , l ables we re c o n n ected by a fix e d relati on ; an d t he obj e ct was m i to fin d the lim it of their rati o . The val ue o f this li t gav e th m l e relati ve im portan ce of the in fi n it es i a s . In the pres en t arti cle t he parti cular in fin it es im als c om

are d are n ot the r n c a a a es as hem s e ves bu t p p i ip l v ri bl , y t l , s m u an e ou s n crem e n s h of hes e v a a e s as he s a i lt i t , k t ri bl , t y t rt ou t r m n a ues x an d ar in an as s n e d m an n er f o give v l 1 , 9 1 v y ig , as in the familiar in s tan ce of the abs ci s s a an d ordin ate Of a

n give cu rve .

The v ar a es x are hen t o be e ac e d b he e u iv a i bl , y t r pl y t ir q

en s 93 k k in h ch the n cre m e n s ll are hem l t 1 + , y1 + , w i i t , k t s e s ar a s an d can d s re d be o h m ade t o a r ach lve v i ble , , if e i , b t pp o z ero as a lim it ; for s i n ce y is su ppos ed to be a c on tin u ou s 1 6 IFFE RE N TIA CA C CA I D L L UL US ( . .

u n c n 33 it s n c e m e n c an be m ade as s m a a d re d f tio of , i r t ll s esi b a n t he n crem en of x s u f c i n t m y t ki g i t ﬁ e ly s all .

The de e rm n a on o f t he im it o f the ra k t o h as it t i ti l tio of , a r aches z er s u e c an as s n ed re a on e t e e n a: pp o o , bj t to ig l ti b w an d is t he un dam e n t a o em of the D if e ren t ia y, f l pr bl f l

u C al c lu s . h E . . t e a n be g , let rel tio

2 a x

x be s m u an e o us v a ue s of the a a es x an d let 1 , 9 1 i lt l v ri bl , y ;

h n as c han s t he v a u e 23 - 71 le t c han e t he w e ge to l 1 4 4 y g to — h u e k . T en val yl l

7 2 h n 2 $ 2. R . e ce k 1

Thi is a a n c n n e c n t he n c m en s k Is . s rel tio o ti g i re t , H ere it is t o be ob s erv e d t hat t he rel ati on b etw een the in fin ites im als h is n ot d rec v n bu t has rs t o be , k i tly gi e , fi t de ri ve d from t he kn ow n relation betwe en T an d y. Let it n ex t be re qu ire d to c omp are the s e s im ultan e ou s i n cre me n ts by fi n din g t he limit of their rati o when they

z r approach t he limit e o .

B d s n y ivi io , k “ 2 23 h i: 1

m hen c 2 93 . e , bli 0 % 1

This res ult m ay be e xp re s s e d in fam iliar lan guage by

ha hen n cre as es h ou h t he v a u e x hen s ayin g t t w x i t r g l 1 , t y

s 293 m es as m u ch as x an d hus hen It c n n u s in creas e 1 ti ; t w o ti e

1 8 “ IFFE R E N TIA CA C U US OH I D L L L [ . .

Thu th s e ratio o f c orres pon din g in crem en ts takes the

s uc ces s e a u es m iv v l , an d can

be r u h as n ea 8 as d es red b a n it s m a en u h b o g t r to i y t ki g ll o g .

' As a ot er ex am e let t he re atio obetw n h pl , l n een a: an d y be

2 3 y 33 .

? 3 T en 56 h y] 1 ,

x (91 + ( i

he ce b ex an sion an d s u bt ract io n , y p n,

2 2 2 le [C 3 $ 11 3 $ E yl 1 t ,

2 lc 2 k h 3 x 3 x h ( y, ) ( , + l

1c 3 x 2 + 3 x h + h2 l l . k 2 Ic y, + I T f i M herefore lim l m , as h 2 k y,

an d b Art . 4 t heorem 8 , y , , a 2 m g2 Si l—0 h 2 yl

“ The n a a u s x h n i iti l v l e of , y, ave bee written with

s u s c s Sho ha n the n cr m n s h It a b ript to w t t o ly i e e t , v ry

du n the a e ra c ces s an d a s o t o e m ha z h ri g lg b i pro , l p s i e t e fact that the lim it of t he ratio of t he s im ultan e ou s i n cre m e n ts depen ds on the parti cular valu es through which the

ar a s are as s n he n he are s u os d v i ble p i g, w t y pp e to take h h d thes e in cre m en ts . Wit t is un ers tan din g t he s ubs cripts

d M r n m h re a be om e . eo v e r the c n s it la will e fter itt o , i re e t ,

for a d s n c n s s be d n e d b the s m o s will , gre ter i ti t e , e ot y y b l “ ” “ Ax A ad n cr m en as n c m n . , y, re i e t of , i re e t of y

2 2 = 2 — Ex . 1 . If x G ﬁ n d im . Let the i itia va u es Of the y , l ii n l l variab es be de oted b x an d let the variab es take the res ect ive l n y , y, l p i crem e ts Ax A s o t at t eir n ew va ue s a Ax A s a sti n n , y, h h l , y y h ll ll re i T s at isfy the given lat o n . hen

2 “ (as A5 (y Aw a . s A I I 1 9 . ) F UN D ME N TAL PR N C PLE S

B ex ans io an d s ubtractio y p n , n,

- 2 2 x Ax + (Ax) + 2 y o Ay +

e ce AT 2 Ax —A 2 A h n ( x ) y ( y y) ,

lim Ay lim 2 Ax ‘ x Therefore _ Ax i O Ax 5 0 E 2 y + Ay

The negative Sign indicat es t hat wh e n Ax an d the rat io x z are os itive A is y p , y

e ative t at is an i creas e in roduces n g ; h , n x p in T is m be i strat e a decre ase y. h ay llu d geom et rically by drawing the circle W hose 2 2 2 e u ati is x Fi q on y a ( g .

2 2 — Ex . 2 . If x = 2 x + y y ,

lim Ay 2 x + 2 prove Ax é 0 Fm 5 . K5} 2 y 1 .

S m a h n t he e a n een a: an d is n i il rly, w e r l tio betw y give t he e xplicit fun c tion al form

A x Ax y + y ¢< + ) .

AA TO: Ax) TO) hen c e Ax Ax

h n the m < ) is n the m h s a o can be W e for of 1 give , li it of t i r ti

e u a d an d x s s d as a u n c n x . This un c on val te , e pre e f tio of f ti is then calle d the d eriva t ive of the fun c tion with

regard to the in depen den t variable x .

T m h va o f a u n c n h he . for al deﬁn itio n of t e deri tive f tio wit

regard to its variable is given in the n ext arti cle . 20 I FFE R E N TI A C A C U US OIL I D L L L [ .

If 9 . D ﬁ ni i n f i i e t o o a der vat ve . to a variable a s m all

n crem en be ve n an d the corres on d n n c em e n i t gi , if p i g i r t of a con n u us un c on of t he ar a e be de e m n e d h n ti o f ti v i bl t r i , t e t he limit Of the ratio of the in cre m en t of the fu n ction to t he in c em en of t he ar a e he n the a e r n m r t v i bl , w l tt i cre en t a oaches t he im i z er is ca e d t he de va th ppr l t o , ll ri tive of e fu n ctio n as to the variable .

If be a n an d c n n u u s u n c n x an d m ﬁ ite o ti o f tio of , A a s m a n cr m en en t o a he n the de r at v e ( T as ll i e t giv , t iv i of M ) t o x is lim 9X3: Are 9606 2 lim Ax i O Ax é O Ace Are

It is im portan t to dis ti n guish between lim an d

lim Ax Th inﬁ n ites im als an d t he ratio of thei r limits . e latter is in determin ate of the form 8an d m ay have an y valu e ; bu t t he m has u s u a a d m n a a u e as u s a d in for er lly eter i te v l , ill tr te h t he e xamples of t e las t articl e .

EXERC IS ES

2 2 Find t he derivative of x x as to x .

2 Find the d erivative of 3 x 4 a: 3 as to x.

Find t he derivative of as to x. 2}x

4 3 derivative of x 2 as to x . 4 . ind the F 3 a:

m on 1 ons f d ri i e . S c 0 . Geometrical illus trati o a e vat v o e cept ion of the m e an in g an d u se of a derivative will be

afforded by on e or two geometrical illus tration s . Let be a fu n ction of a: that rem ain s ﬁn ite an d c o n tinuou s for all val ues of as between certain as signed con F UN DAME N TAL PR IN CI PLE S 21 ’ s t an t s a an d b an d t he a a es x be a n as the ; let v ri bl , y t ke

e c an u r rd m n Th h r r t g la coO in at es Of a ovin g poi t . en t e el a

n e we en a: an d is e s n e d ra h ca h n the tio b t y r pre e t g p i lly, wit i as s n ed un ds c n n u b the cu e h s e u a on 1 s ig bo of o ti ity, y rv w o eq ti

a Mos)

x x t he c oo d n a es t wo o n s P Let ( 1 , yl ) , ( 2, y2) be r i t of p i t

P on h s cu . Then is e d n ha the ra 2, t i rve it vi e t t t tio

9 2 9 1 x x 2 1 is e u a t an a he n a is the n c n a on an e the q l to , w rei i li ti gl of

n n - L t m s ca t e P P the x ax s . e P be o d n e e an d e li 1 2 to i 2 ve ar r n h 5 i Th a e t o c o n c d n ce h P s o a x x . en e r r i i e wit I, t t 2 1 , y2 y] t he s e can n e P P a ach s n eare an d n eare c oin ci t li 1 2 ppro e r r to d n ce h t he t an en n d n at the n P an d e wit g t li e raw poi t I, the i n clin ation -an gle a o f the s e can t approaches as a m t he n c n a n P li it i li tio a n ama) an gle 4) of the tan gen t

n li e .

“v" y en ce ” H ,

t an a é. t n a cf) .

192 yl Thu s t an x x 2 l

FIG . 6 . hen x x = w 2 1 , y2

It m a be o s e ve d h 50 be ut d c e u a t o E y b r t at if 2 p ire tly q l l , an d the ra o on t he e ou d in en era as s u m e yz to 9 1 , ti l ft w l , g l , the n d e rm n a e orm 9 as in h r c as s n d n the i et i t f , ot e e of fi i g 0 limit of the ratio of t wo in fin it es im als ; bu t it has j us t b een s h n ha t he ra of the in fin it e sim als x 93 has ow t t tio yz yl , 2 1 , n h e e s s a de e m n a e m viz . t an vert le , t r i t li it, , 22 IFFE RE N TI ’ A CA OH . 1 C U US . D L L L I

They are thu s in ﬁn it es im als of t he s ame order e x cept

I . when ‘I’ is 0 or 2 ‘

If the d ffe n ces x —rr be den e d b AT A i re z l , yz 9 1 ot y , g,

hen x Ax A t 1 + , + y ;

bu t s n c ( ax , i e 9 M ) ,

s ha = ( a = it follow t t y1 M l ) , y2

hen c e the ratio of the s im u ltan e ou s in crem en ts m ay be written in the variou s forms

A ( 5 ( 5 A1” 5 9 9 2 9 1 90 2) 90 1 ) (90 1) Ace 23 — 93 93 — 33 Ax 2 1 2 1

In the las t form x is regarde d as the i n dep en de n t v ariable an d Are as it s i n depe n den t in crem en t ; the n u m erator is the

n c e m e n of the un c n ( E cau s ed b t he chan e O f a: i r t f tio M ) , y g

h a u h a u Th m h r m t e e 23 t e e x Ax . e s f o v l 1 to v l , + li it of t i

a o as Ace 0 is the a u t he der the un c n r ti , , v l e of ivative of f tio

x hen a: has the a u 95 . e re x s an ds an ¢( ) w v l e 1 H 1 t for y

o Thu h d as s ign ed v alu e f x . s t e erivativ e of an y c on ti n u ou s fu n ction is an othe r fun ction of x which m easu res the s lope of the tan gen t to the cu rve d rawn at t he poin t whos e abs ci s s a is 93 .

i the s e of t he t a e t i e t o the curve at t he Ex . F nd lop ng n l n y oi t 1 p n ( , 2

2 lim (96 A9 0 tan d) Ax 0 A1: F UN DAME N TAL PR IN CIPLE S 23

e ce t an 4 e x 1 an d t he e u at io of the t a e t i e at H n (f) , wh n z q n ng n l n the oi t 1 2 is 2 4 x p n ( , ) y (

As an o her u s ra on t he c di n a s Of P 23 t ill t ti , if oor te be ( , y) ,

h o f x h an d s e Q, x + A , + A , n t o ( y y) t e Y = = n d PS = R = N PR Ax a A . M , Q y If t he are a OAPM be den ote d by

hen 2 is e v de n s m e u n c n 2 , t i tly o f tio of the abs cis s a x ; als o if area CAON

deno e d b Ax hen the be t y z , t area MN OP is Ax ; it is the in cre

FIG . 7 . m n t a e n b t he u n c n 2 hen e t k y f tio , w

B u e n h x takes the in crem en t Ax . t M N QP li s betwee t e

c n s MR M hen ce re ta gle , Q ;

Ax Az A Ax y (y y) ,

Thus the d n a e an d the area ach x es s d as a , if or i t be e e pr e

u n c n o f t he a s c s s a the de r e of t he ar a un c n f tio b i , ivativ e f tio

h r ard t o t he a s c a i e ua the rd n a u n n wit eg b iss s q l to o i te f ctio .

Ex . If t he area i c u ded bet ee a curv e t he ax is of x an d the n l w n , , ordi at e ose ab scis sa is x be ive b t he e u at io n wh , g n y q n

x3 z ,

ﬁ n d t he e u atio of t he q n curve .

i Ax Ax O Ax

ir 2 2 3 33 3 [ + x . Ada-2 0 24 IFFE R E N TIA CA C n U . I D L L L Us (o .

1 1 . The ti n f iff opera o o d erentiation . It has been s ee n in a n um be r of ex amples that whe n t he operati on in di cat ed by

lim (9017 “ I“ A33) Ax " o Ax is p erform e d on a give n fu n ction the re s ult of the o ra on i an o he u n c n h pe ti s t r f tio Of x . T e lat te r fu n ction m ay have properties s im ilar t o thos e of or it m ay be of an e n tirely diffe re n t clas s . The o pe ration abov e i n dicated is for b re vity de n Ct ed by da m) t he sym b ol an d the res ultin g de rivative fu n ction by dx

hu s t ,

dc a) im A 5 11m IE Ax> ili Alﬁ e 22 3 96 4 A

The proc es s of p erfo rm in g this i n dicated op eration is

’ d i n t ia t ion x h c alled t he ﬁ ere of d) ( ) with regard to x . T e

' s m o he n s o e n s e ara e is ca ed the d i er y b l 3 w p k of p t ly, ll ﬂ £x

' en tia tin o era t or an d ex res ses ha an u n c o n e n g p , p t t y f ti writt a t e is t o d f e n a ed h e a d x u s as the f r it be i f re ti t wit r g r to , j t symbol cos preﬁxed to ¢(x) in dicates that the l atter is to ha a ce r a n a n m ed u n n am e ha ve t i oper tio perfor po it, ly, t t

n of ﬁ n di n g it s cos i e . The pro ces s of differen tiati ng ¢(x) c on s is ts o f the follow in g s teps

G a m a n c m n the a a . 1 . ive s ll i re e t to v ri ble

m u t he e u n n e m en the un c n . 2 . C o p te r s lti g i cr t of f tio

D de the n c em en t he u n c n b t he n c m n 3 . i v i i r t of f tio y i re e t o f the variable .

4 O a n the m h s u n as the n c m n . bt i li it of t i q otie t i re e t of h the variable approac e s z ero .

is mbol i s ometim s re laced b the si le letter D Th s y s e p y ng .

26 D IFFE R E N TI A CA U C US CH . I L L L [ .

u ctio i e x is e ative an d an i creas i u ctio i e f n n wh l n g , n ng f n n wh l x is posi 2 tive . T is is e s o b the ocus of the e u atio = x 4 h w ll h wn y l q n y + (Fig.

FIG . 8 .

A ai the orm of the cu rve s t at is a decreasi u c g n, f y hows h ng f n 5 x

' t io as x ass e s ro m co t o 0 -an d a so a decre as i u ctio as x n , p f , l ng f n n,

ass es rom 0 t o 00 . e x as s es t ro u 0 t he u ctio c a es p f Wh n p h gh , f n n h ng r m dis continu ou sly f o the value co to the valu e 00 (Fig .

M os t fu n ction s are in cre as in g fun ction s for s om e v alu es of the

ar a e an d decreas v i bl , in g fu n cti on s for

others .

2 is an E . . V2 Tx x g , FIG . 10 . incre a sing fu nction fro m x 0 to x T an d a decreas i u ctio ro m x T t o x 2 T Fi . , ng f n n f z ( g

A fu n ction is s aid t o be an in cre as in g or de creas i n g fu n c tion in the vicin ity of a given valu e of x ac cordin g as it l in creas es or decre as es as x in creas es throu gh a s mall in terva in clu din g thi s valu e . FUN DAME N TAL PR I N CIPLE S 27

1 Al br ic t e f e in s f incr in nd 3 . ge a s t o th terval o eas g a de

in Let x be a u n c on of x an d let be rea creas g . y d( ) f ti , it l , con n uo u s an d diffe e n a e for all a u e x rom a t o6 s . ti , r ti bl v l of f Then by deﬁ n ition y is in creas in g or de creas in g at a poin t x x acc d n as 1 , or i g ( 5 Ax 9 3 90 1 ) (70 1 )

is s or n a e h re Ax is a s m a s n u m . po itive eg tiv , w e ll po itive ber The s ign of this expres s ion is n ot chan ged if it be divided b Ax n o m a how s m a Ax m a he n ce x is an y , tter ll y be ; ¢( )

n creas n dec eas n u n c on at the a u as acc d i i g or a r i g f ti v l e ] , or in g as d im ( A ( _ f 1 9 1 Ax é O dx Ax

is pos itive o r n egative . Thu s the in te rvals in whi ch qS(x) is an in c reas in g fu n c tion

r th m i h i a e e s a e as the in tervals n w ich (x) s pos iti ve .

i d h a s in hich the u ctio Ex . F n t e interv l w f n n

9 x2 + 12 x —6

The de ri ative is is increasin g or decreasing. v

6 x2 18 x 12 6 (x l ) (x

' e ce as x asse s rom co t o 1 t he derive d u ctio b x is ositive h n , p f , f n n q ( ) , p — an d increase s fro m cc ) to — i. s . rom ( ) co t o 1 as x asses , f I ; p ’ rom 1 t o 2 S x is e ative an d x f , q ( ) n g , ¢ ( ) m de creas es ro m S 1 t o i. s . ro f q ( ) , f — 2 an d as x s ses ro m 2 t 00 1 to ; pa f o + , ’ r x is ositive an d i cre ase s rom p ( ) p , n f

t o i. e . rom 2 to 00 . , f The loc us of the e qu ation is

1 1 . At oi ts ere sh own in Fig. p n wh M x 0 the u ctio S x is eit er 4 ( ) , f n n q ( ) n h At u oi ts incre as ing n o r de creasing. s ch p n h x f x the t angent is parallel t o t e a is o . T u s in t is il ustratio at x 1 x 2 h h l n , , , - the ta e t is ara e to the x ax is . FI 1 1 ng n p ll l G . , 28 IFFE R E N TI A CA I I D L L C UL Us [C L .

EXERC IS ES

1 i . F nd the inte rvals of incre as ing an d decreas ing for the fun c tion

x x3 2 d> ( ) 5 2 x x 4 .

Here — The function in cre ases fro m x 00 t o x 1 ; decrease s fro m x to x i cre ase s ro m x t o x 00 n f .

2 i d the i terv a s of i cre i n e i . F n n l n as ng a d d cre as ng for the fu nctio n

= x3 2 x2 x —4 y + ,

n s re the urve is r h - a d how whe c pa allel t o t e x ax is .

i t h 3 . At how m any po n s can t e slope of the t angent to the curve

8 2 y 2 x 3 x 1 i be 1 ? 1 Find the po nt s .

om u te the a e at ic the ol i u r e r 4 . C p ngl wh h f low ng c v s inte sect

= 2 — = 2 3 x 1 2 x 3 . y , y +

1 4 Diff r n i ion of func ion of func i n . Su s . e e t at a t a t o ppo e

hat n s ad O f e n en d ec as a u n c o n x t y, i te b i g giv ir tly f ti of , i ex es s e d as a un c n an o h a a e a h ch is s pr f tio of t er v ri bl , w i

L u r its elf e xpres s ed as a fu n cti on of x . et it be re q i ed to ﬁn d the de rivative Of y with regard to the in depen den t variable x . h Let in whi ch a is a fu n ction of x . W en x chan es t o the v a u e x Ax let u an d u n de r the en g l + , y, giv -- a n s c han e t he v a u es u Au A . Then rel tio , g to l l , y+ y — A Au u + Au ) u Au ( ) . __ y AA f f( ,

Ax Au. Ax Au Ax he n c e u a n m s e , q ti g li it ,

d d da dfﬁ l da y _ y ) . dx da dx da dx 13 F UN DAME N TAL PR I N CI PLE S 29

This res ult m ay be s tated as follows

The deriva tive of a fu n ction of u with regard to the p rodu ct of the deriva tive of the fu n ction with

t o an d the deriva tive of u wi h regard t x .

EXERC IS ES

= 2 y 1 . Give n y 3 u ﬁ n d Zx dy d a " : da 81

d“ - 2 _ 35 36 x (3 “3 + 4 dx du It; D = 2 = 3 — 2 . Given 3 u 4 u + 5 u 2 x 5 s a y , ; n g 1 —- 2 —y 3 . Give n u = 5 x 2 4 y , x + ; ﬁ n d u s27

1 3 d 2 y _ 4 . Gi ven 3 u + u ﬁ n d y 2 , 3 u 3 dx CH APTER II

DIFFER ENTIATION OF TH E ELEMENTARY FOR MS

In cen ar c es the m e an n the s m re t ti l , i g of y bol was ex plain ed an d illu s trate d ; an d a m ethod of exp res s in g its

a u as a un c on x was x em e d in cas es in h ch v l e , f ti , e pliﬁ , w i y was s m e a e ra c un c n x b d c us e the a i pl lg b i f tio of , y ire t of d Th m hod i n a a th m o eﬁ n ition . is et s ot lw ys e s t c on ven ien t on in th d ffe ren on of m m ca e d u n e e i tiati ore c o pli t f cti on s . The pres en t chapte r will be devote d t o t he es tablishm en t

s m e e n a ru es o f d fferen a n h ch in m an of o g er l l i ti tio w i will , y

cases s e the r u e in ac t he de n n . , av t o bl of go g b k to ﬁ itio The n ex t ﬁ ve articles tre at of the diffe ren tiation of alge brai e fun ction s an d of algebraic combin atio n s of other diffe r

i l n on en t ab e fu cti s .

n 1 5 . Different iation of the product of a const ant a d a variable .

Let y cx .

Th n A c x Ax e y y ( ) ,

c x Ax ex c A3] ( ) ,

Ar Ax

therefore c . I E TI A TI ON OF E E E N TA R Y F OR S 81 C H I . IFFE R N . D L M M

If = cu he e u is a u n c on x h n b Cor. y , w r f ti of , t e , y 1 4 rt . A , (1 ) d er; d x

The derivative of the p rodu ct of a con s tan t an d a varia ble is equ a l to the con stan t mu ltip lied by the derivative of the variable .

Differentiat ion of a s um .

= r y f( ) we ) .

A = x Ax x Ax Ax r y f< ) ¢< > + w ) . — Ar fez + Ax> f< x> ¢< x Ax ) 5G) Ax Ax Ax

Th e b e u a n the m s h m em ers er fore , y q ti g li it of bot b , l ’ h 33 3 T . 5 . f ( , Eli )

f h h . I in c u v r u n n r. 1 w a e c s Co w i , , f tio o f x hen , t d d u d v d i v u v w 2 d x d ac d oe d ae

The derivative of the s u m of a ﬁn ite n umber of fun ction s is equ al to the s um of their derivatives .

Cor. 2 . If u c c e n a co n s an h n y , b i g t t , t e

hen c A Au e , y ,

dy du

The l as t e quation as s erts that all fu n ction s which diffe r from each other on ly by an additive con s tan t have the s am e

der a iv tive . 82 I FFE R E N TIA CA C D L L UL US [Cm II .

Ge m r ca the add i n a con s an h o et i lly, it o of t t as t he eﬁ e ct of m ovin g the c u rve y u (x) parallel to the y axis ; this opera tion will o bviou s ly n ot chan ge t he s lope at poin ts that have

the s am e x . Fm m C2 dx dx dx bu t om the u h u a n a e fr fo rt eq tio bov ,

dz da

do he n ce o ws ha , it foll t t dx

va tive o a cons tan t is z The deri f ero.

If the u mber of u ctio s be i ite t e r m o e 5 of Art . 3 m a n ot n f n n nfin , h y a t at is t he im it of t he su m m a n ot be e u a t o t he s um of the pply; h , l y q l im its an d e ce the derivative of the s u m m a n ot be e u al t o the s u m l , h n y q T u the de ri ati e of n i ite serie c a of the deriva tive s . h s v v a nfin s nnot always be found by difi ere ntiatin g it term by t erm .

i n f uc . 1 7 . Different atio o a prod t

Let r

Ar f< fv Ax) Then Ax Ax

B y s u btractin g an d addin g in t he n u m e r

h s res u m a be re arran e d hu s ator, t i lt y g t — Ar f( r + Arv) f(w) 503 ) ¢< x Ax) + f< x> Ax Ax Ax

’ 5 6 a r a h z r u s n Art . 3 he re m s N ow let Ax pp o c e o , i g , t o , , an d n oti n g that the ﬁ rs t factor (1) (x Ax) appro aches ¢(x)

Th n s in ce by hypothes is is con ti n u o us ( Art . e gi

34 N cm 11 DIFFERE TIAL CALCUL US [ .

f(x Aw) f (x) Av ¢< w Arc ) M r ) Ax Ax

( x Ax) Ax) Aw Aw)

B su rac n an d add n x x in the n u m e a t r y bt ti g i g ¢( )f( ) r o , t his ex pres s ion m ay be writte n

x + Ax x 4 03) Ag Ax Ax)

e n ce b u a n m i s H , y eq ti g li t , ’ dy ¢ ( x) f ( x) _ Tb Art . 3 s . 6 7 . 2 [ , , dz [M 9 01 Thi s res ult m ay be written in the brie fer form d u d v

d u d oc d ac (5) v

The derivative o a ra ction the u otien t o two un ction s is f f , q f f , equ al to the den omin ator mu ltip lied by the derivative of the n u mera tor min u s the n u mera tor mu ltip lied by the deriva tive o the den omin ator divided b the s uare o the den omin a tor . f , y q f

n f c mm ns u b o of unc ion . 1 9 . Differentiatio o a o e ra le p wer a f t

” Let u in h ch u is un c n x . Th n h are y , w i a f tio of e t ere thre e cas e s to c on s ider .

1 n a s n e . . po itive i t ger

n a e in e e . 2 . n a eg tiv t g r

a o mm en s u a e rac on . 3 . n c r bl f ti

a s v n e er . 1 . n po iti e i t g

the ac ors u v w This is a particu lar cas e o f f t , , ,

T hu s bein g equ al . du n —l dx 18 DIFFERENTIATION OF ELEMENTARY FORMS 35

2 a n e a n er . . n g tive i teg

m i n h ch m is a s n e . Le t n , w i po itive i t ger

' m Then u u

” H d“ by an d dx

- m - l da mu dx

” “ l hen ce gig n u Z:

m m e n s u a e rac on . 3 . n a co r bl f ti

Let n 2 h e are h n e rs h ch 5, w er p , q bot i teg , w i either p os itive or n egative .

" Then y u

‘ n 1 up he ce y ,

i p

cl du y p d _ P“ dx dx

v n the e u e d de a v e Sol i g for r q ir riv ti ,

- dy p gl du u ’ dx q dx

n d u n —l d u hen c e n u (6 ) d x d x

The derivative of an y commen s u rable p ower of a fun ction is equal to the exp on en t of the p ower mu ltip lied by the p ower with its ex on en t dimin ished b u n it mu lti lied b the derivative p y y, p y

of the fun ction .

l i d eri tive are ide i If two u ctio s be ide tica the r va s t cal. f n n n , n 36 11 D IFFE R E N TIAL CALCUL US [cm .

Thes e theorem s will be fo u n d s ufﬁ cie n t for t he differen tia tion of an y f un c tion t hat in vol ves on ly the Operat ion s of

dd t n s u ract o n m u t ic at on d vis on an d in o a a i io , bt i , l ipl i , i i , v l t i on in which the ex pon en t is an in tege r or com m e n s u rable fracti on . The followin g exam ples will s erve to illu strate the the o

e m s an d s h w the c m n e d a ca n the en a r , will o o bi ppli tio of g er l fo rm s (1 ) to

| LLUS T RAT IVE EXAM PLES

2 — 3 x 2 dy . x + l dx

— — 2 (33 + 2) (3 x 1)

2 —i < 3 x (2) dx

6 x.

dx _ 03 1 1 . $3 dz T Subst it ute t e se re s u t s i n t he ex re ssio for he n h l p n 573

— 2 — 2 dy ( x + 1 )6 x (3 x 2 2 3 x 4 6 x + 2 dx (x (x

u 2 z = 3 8 ﬁ n d — . u ( + Zs 7 d 93! 2 (3 3 + by (3) (is

i i 2 (1 5 Ar - (1 5 3 ) by (6 ) d

5 8

5 s 2

2 = (3 3 + 2) 6 8 . DIFFEREN TIA TI ON OF ELEM ENTA R Y FOR M S 37

S ubstitu te th e se valu es in the expres s io n for Then

d“ ? i L J 5 -i 6 s \/ l 5 3 ds

3 . y

irst a s a u o tie t F , q n ,

2 — 2 2 ( V1 x x )

dx dx dx

m 7 i , (1

2 by (2)

Sim i ar for th i l ly e ot her t erm s . Combin ng the results

Ex . 3 m a a s o be or ed b rst ratio a iz i the e m y l w k y ﬁ n l ng d no in ator.

EXE RC IS ES — Find the x derivatives of the following fu nct ion s

1 o . g l .

“ ' 8 2 : . y 1 x + 3 . 3 _ . y c x

1 0 .

V5 3 38 C II D IFFERENTIAL CALCUL US [ m .

3 3 33 + 2

2 2 y 3 (x x

2 y 3 u 7 .

3 2 z 4 u — y 6 14 + 12 u 3 .

— 2 y 3 u + 6

z y ux.

2 2 4 y u + 3 xu + x y 1 1 < a + xr m a x n 2 3 < + 1 (m e y u x w.

2 5 5 4 a 2 2 3 4 5 8 . Given ( a x) a 5 a x 10 a r 1 0 a x 5 ax x ; ﬁn d 4 (a x) by diﬁ eren t iat ion .

3 2 9 . Sho w t hat t he slope of the t angent t o the cu rv e y x is never

e at ive . S o w ere t he s o e i n g h wh l p ncrease s or decre ases .

z 2 2 2 2 2 ' 3 0 . Give b x a a b ﬁn d 1 b di ere tiat i as to x V n y , ( ) y ff n ng ; jix (2) by differentiat ing as t o y ; (3) by s olv ing for y an d differentiating

l r t h re u t s of h . as t o x . Co mpa e e s l t e t hree m et hods

3 1 . S o t at orm . 31 is a s e cia case of h w h f p , p l

2 a 3 2 . At at oi t of the cu rve ax i t he s o e O? 1 ? 1 ? .1 wh p n y s l p

~ 3 3 3 2 . Tr e t he curve x 3 x 7 ac y z x 1 . d = 2 — y 3 4 . y an d u 5 x 1 ; ﬁ n d dx

2 2 2 3 A a e h x 3 : O 7 5 . t wh t angl do t e cu rv es y 12 x an d y x 6 6 inte rsect ?

E m 20 . n le e tary tran s cendent al fun ctions .

The followin g fu n ction s are call e d t ra n s cen d en t a l fun c tion s

S im le ex on en tial u n c n s c n s s n a c n s an p p f tio , o i ti g of o t t

x n n is v ar a e n u mb e r rais ed t o a power who se e po e t i bl , ” as a ; 19- 21 ] DIFFERENTIATION OF ELEMEN TAR Y FORMS 39

en era l ex o en tia l u n c ion s in v o v n a ar a e a s e d g p n f t , l i g v i bl r i s m “ t o a o e r hos e e x on en t is v ar a e as x p w w p i bl ,

9“ the lo arithmic u n c on s as o x lo u g f ti , l ga , gb ;

” " the in commen su ra ble owers of a v a a e as x u p ri bl , , ;

the tri on omet ric u n c tion s as s in u cos u g f , , ;

“ 1 - 1 he in vers e tri on om etric u n ct n s as s in t n t u a x . g f io , ,

Th re are s he an s ce n de n a u n c on s but h e till ot r tr t l f ti , t ey

n d d i h will n ot be co s i ere n t is b ook .

The n x ur ar c es a of the o ar hm c the e t fo ti l tre t l g it i , two

d h n m m x n en a u n c on s an t e co en s u a e er . e po ti l f ti , i r bl pow

1 Diff en on of lo x and lo u . 2 . er tiati ga g“,

Let lo x . y g,

Then A lo x Ax y y g , ( ) ,

A o x Ax o x y l ga ( ) l ga Ax Ax

1 A}?

c n en n c n h for Ax an d ea an n o v ie e writi g , r rr gi g,

Ay 1 x lo 1 ga Ax x h

dv 1 lim hen ce lo w i g dx x k 0

Th m r l lo arit mic u ctio lo u is n ot clas s iﬁed s e arat el e ore gene a g h f n n g, p y,

logo, u as it can be redu ced t o th e quot ient 40 OH 11 . DIFFERENTIAL CALCUL US [ .

To e a u a e the ex s s n 1 hen h 0 x an d v l t pre io w , e p it b the n o m a he rem s u os n t o a a e os y bi i l t o , pp i g be l rg p itive 77: i n teger m . The e xpan s i on m ay be written

1 2 m2 1 2 3 whi ch can be put in t he form

1 2 N o w as m ec m e s v e r ar e the e rm s e c m e y g , ’ ’ b o b o l t m m v r s m a an d he n m 00 the s er es a roache s the im e y ll, w i pp l it 1 1 1

The n u m erical valu e of this lim it can be readily cal cu late d

x m n Th s n u m e is an m or an to an y de sire d appro i atio . i b r i p t t c n an h ch is de n ot e d b the e e e an d is e u a o s t t , w i y l tt r , q l to m ; thus

e 2 71 8281 8 m i co

The n u m ber e is kn own as the n atu ral or N aperi an base ;

n d ar hm s h s as ar ca d n a u a or N a erian a log it to t i b e e ll e t r l . p

m N a u r l ogarith s . t al l ogarithm s will be writte n withou t a

it This m ethod of obt aining e is rather t oo brief t o be rigorous it assumes

is a os itive i te er but t at is e u ivale t t o restricti Ax t o p n g , h q n ng f; a roac z ero in a art icular wa It also a lies t he t eorem s of limits t o pp h p y. pp h The roo is com let ed the sum an d p rodu ct of an inﬁ n it e n umber of t erm s . p f p ’ ” on 1 5 of M cM aho n an d S der s D i fere tial C alculus . p . 3 ny f n

42 DIFFERENTIAL CALCUL US

The deriva tive of an exp on en tia l fu n ction with a con s tan t bas e is e u a l to the rodu ct o the u n ction the n atu ra l lo ar t q p f f , g i hm of th ba a d the derivative o t e e s e n h ex on en t . , f p

23 . Diff n i t ion of the n l x n n ere t a ge era e po e tial function .

” u y , in h h ar h u n n c u v e c o s of x . w i , bot f ti

T the ari hm o f o h d n d d f r a e s es a e en a . Th k log t b t i , i f ti te en

: v lo u log y g ,

gi log u by form s (8 )

q dv v 10 g dx dx u d

” r - 1 h i t lo u c u 1 1 t erefore g g; 3; ( )

The derivative of an exp on en tia l fu n ction in which the bas e is a ls o a variable is obtain ed b rs t d erentia tin re ardin y ﬁ iff g , g g the bas e a s con s tan t an d a ain re a rdin the ex on en t as , g , g g p

con stan t an d a ddin the res u lts . , g In the diffe re n tiat ion of an y given fu n cti on of this form it is u su ally bett er n o t t o s u bs titu te in the form u l a dire ctly

h m h d in n but t o a t e e o us u s e d de v i. e . pply t j t ri i g , to diffe ren tiate the logarithm of the fun ction by the precedin g rul e s .

2 “L 5 : 4 ﬁn d Ex . x y ( , gi

2 2 log y (2 V x 5) log (4 x

1 d x 8 x y 2 lo 4 x x2 5 g ( ) 2 - vx2 _ 5 4 x 7

d VT 4 x2 y - . x 5 x [ 22 DIFFER ENTIATIO N OF ELEMENTAR Y FORMS 43

n . 24 . Differentiation of an incomme s urable power

" Let M y ,

h m n r Th n in whic n is an in co m e s u able con s tan t . e

o n lo u l g y g ,

1 dy n da ’ y dx u dx

dy da __ _ n M y dx u dx

d du n _ 1 ° dx

This has the s am e form as s o that the q ualifyi n g m 1 m 9 n n o d . c om en s urable of Art . ca w be o itte

EXE RC IS ES

Find the x derivatives of the followin g f u nction s

1 : . y log (33 + a) . 2 e. 2 y 3 = lo 4 x ’ . y g ( l + o”

1 + x “ 3 4 y = log 3/ 6 (1 95 ) — ex _ e x 1 x2 y 5 = IO ° x - x . y g e + e 1 i x2

6 = x lo x . y g

n z x lo x 7 . y g n “ z x a n m z x lo x y 8 . y g

9 . \/a 1 0 : x/x . y 1 o 2 1 1 . lo 3 x x . y g , ( ) log x 2 2 1 2 . lo x 7 x . = y g10 ( + ) c (loe r) 1 3 . o a . lo lo x . y l gz y g ( g )

x“ " 1 4 2 . e , y y 5L .

1 5 log 2 . y 2/ a . 44 D on n IFFERENTIAL CAL C UL US ( . .

The following f unctio ns c an be eas ily differe nt iate d by ﬁrst t aking t be r f t he e u he logarithm s of both m em s o q at ion s .

x x 5 3 2 3 2 . x a 3 x a 2 ( ( y ( ) ( x) .

= 30 . y x 1 — Arti cles 25 31 will tre at of the diﬁ ere n tiation of the

T m r un n rigo n o et ic F ctio s .

i i t . 25 . Differentiation of s n

Le t y s in u .

A s in ( u Au ) s in u Au The n y Au Ax

1 2 c os — 2 u Au s in —— Au Au 5( ) 2 Au Ax

s in 1 Au Au cos i t I. Au ( 2 ) W Ax w1 B u t h n Au i 0 c os u Au é cos u an d 1 , w e , ( % ) , Au b Art . 6 h n ce as s m t he i m i y ; e , p g to l t ,

d d a sin u cos i t ° ( 1 2) d oc 3 5

The derivative of the s in e of a fu n ction is equ al to the p rod u ct of the cosin e of the fun ction an d the derivative of the fu n ction .

u 26 . Differentiation of cos .

Let y c os u s in

cos it s in u 24 DIFFERENTIA TION OF ELEMENTAR Y FORMS 45

1 The de riva tive of the cosin e of a fu n ction is equ a l to min u s

” the p rodu ct of the s in e of the fun ction an d the deriva tive of the fu n ction .

27 . Differentiation of t an u .

y t an u

d c os a — s i n u s i n u

Then 2 c os u

2 h tan u sec it t at is ,

The derivative of the tangen t of a fu n ction is equ al to the p rodu ct of the s qu are of the s ecan t of the fun ction an d the derivative of the fun ction .

iff r n i tion of cot u . 28 . D e e t a

y : c o t u

d 1 d y 8 O t a n u b 1 4 z ( ) ’ ( ) dx t an u dx

2 cot u csc it

The deriva tive of the cotangen t of a fu n ction is equa l to min u s the p rod uct of the s qu a re of the cosecan t of the fun ction a n d the deriva tive of the fu n ction . 46 L c 11 DIFFERENTIAL CA CUL US [ a .

Differ n 29 . n of s c e tiatio e u .

Let y s e c u

dy 1 d Th n c os u = e z + dx cos u Et

sec u z tan u sec u

The derivative of the s ecan t of a fu n ction is

rodu ct o the s ecan t o the un ction t he tan en t p f f f , g

tion a n d the eriva tive o the un ction . , d f f

Diff r n i t on of cs c it . 3 0 . e e t a i 1 cs c u Let y . s m u

l 1 d Th n El — e 2 dx Si n u dx

d “ csc u cs c u cot u . d x

The derivative of the cos ecan t of a fun ction is equ a l to min u s the roduct o the cos ecan t o the un ction the cotan ent o the p f f f , g f

un ction an d the deriva tive o the un ction . f , f f

D ff ent i n of s . 3 1 . i er iat o ver it

= Let y v e rs u 1 cos u .

dy d fhen cos u . dx dx

d “ vers u _ sin u . (1 8) d x

The derivative of the versed-sin e of a fu n ction is equ a l to the produ ct of the sin e of the fun ction an d the derivative of the fu n ction . 219 DIFFERENTIA TION OF ELEMENTAR Y FORM S 47

EXE RC IS ES

Find t he x derivatives of the following fu nctions

= 1 n . 1 . . y si 7 x 8 co s (a == 2 s 5 . . y co x 1 9 . 2 n 3 : s in x cos mx . y = — 4 = sin 2 20 . x + lo cos x . y x cos x. y g ( g) 5 = 3 . y s in x . = = 2 v b g 6 . 8 1f1 5 $ y . 2 7 . = sin 7 x. y n sin u y s i ( ) .

8 . “2 “ y s in e . = 3 9 . sin x cos x . “ y sin e o l y og x.

1 0 . t au x see x . y v

2 2 2 1 1 . sin 1 2 x . , y ( ) s in z 4 z y x . 1 2 t an 3 . y ( = 2 y csc 4 x. 2 2 1 3 t an x lo sec x . . y g ( ) y = s ec (4 x

t 1 4 . lo tan x n . y g (5 i ) 3 T z y v ers x . 1 V 5 . lo sin 5 . y g = 2 / Vi cot x + sec x x. 1 y } = in x 1 6 . t an a . s 4. y y y

” : sin nx sin x.

W — l Diff sin u . 32 . erentiation of

“ 1 Let y s in u .

Th n sin u e y ,

an d b diﬁ eren tiatin h m m ers h s den , y g bot e b of t i i tity, ig glib cos g dx dx

1 du

z dx cos dx \/ 1 s in dx y\ i y

t i t i .

’ l The ambigu ity of s ign ac co rds with t he fact that s in u

is m an - a ued u n c n it s n ce an u u be a y v l f tio of , i , for y val e of 48 CAL C L US GIL 11 DIFFERENTIAL U [ .

e e n 1 an d 1 t he re is a s e ri e s o f an e s hos n e is u tw , gl w e s i an d he n u rece iv e s an n c re as e s om e o f he s e an le s in , w i , t g

' l d s in u r a n d m d re h n fo r o c e s e a s o e ec as e ; e ce , s om e f t he m , du

n r I n t h t is s e a d fo s o m e n e a e . t be s ee a po itiv , g tiv will ,

‘ l he n s n u ie s in t he rs t o r ou r h u art e r it n c reas es w i l ﬁ f t q , i

h u an d hen in the s e con d o r h rd it de creas es as 11 wit , , w t i ,

f r he n o f h r an d o u rth n c re as es . en ce o t a es t e s t i H , gl ﬁ f

u a e rs q rt ,

at . l a 1 1 d a 8 1 11 u s in u : (1 9) 2 d x d“ u

In the other q u arte rs t he m m us s ig n is t o be u s e d be fo re the radi cal .

ff i f h m in in in r i onomet ic 3 3 . Di erent ation o t e re a g ve s e tr g r forms . The de ri vative s of the othe r i n ve rs e t rigo n om e t rl c fu n c tion s can be e as ily obtain e d by t he m e tho d e mploye d in the

Th u r o o s las t arti cl e . e res lt s a e as f ll w

‘ - 1 l r 20 cos n ( in l s t an d 2d qu art e s) . ( ) d ; 2 Z; VI — u

—1 1 21 mn it ( in all qu arters ) . ( ) da 1 + 15n

“ 1 2 cot u in all u art e rs . ( 2) 2 ( q ) ( lb: 1 I10 1 d - 1 23 sco u ( in 1 s t an d 3d q u art e rs ) . ( ) d x d i xi x/ “ 2 1

“ l d u 24 csc u (in 1 s t an d 3d qu art ers ) . ( ) die 2 d x u v; _ 1

m 25 vers ( in l s t an d 2d qu arte rs ) . ( ) di “m d x

25 re eive the The radical s in fo rm s ( ) p h n the an es are a en in the o ppo s ite s ign s res pe ctiv ely w e gl t k

qu arters othe r than thos e s tated .

50 CAL L H 11 D I FFE R E N TI AL C U US [O . .

sin u cos i t

cos it

tan u

d u s ec u d x

i t (380 u csc u cot u fl - d x

VGI‘ S u

‘ l sin u

- l cos n

- 1 cot u

‘ l sec u

1 d u ‘ l csc u

“ l vers i t DIFFER ENTIATION OF ELEMENTAR Y FORMS 51

EXE RC IS ES O N C H APT ER II

Find t he x derivatives of t he following f unctions

l 3 $2 5 x3 7 y 1 5 - I / + . x a tan \ y z ( ) VE ax. 3 5 1 a 2 _ . y 7 + x x 7 1 1 — 6 y 0 0 t 3 .

4 — 4 1 7 . t an x 2 , q y

58 lo y l og s m x . — — 1 8 . l 1 ﬁ y og ( x) . 1 i n

" 1 1 9 CO M . . y S 5 3 cos x y 1 x i 1 1 0 _ -— ° g log tan x ( — x ) 2 VZ LU e u lo sin x. x , g “ y 1 x 2 — 2 1 . y log ( x a ) s ee a 3/ 2 log “ ” 2 2 . e u lo x. 7 y z , g 1 f

Q ’ 2 z -- = 2 3 . lo S e s sec x . 1 —x y g I ,

\/ 1 2 4 3 3 — = x 2 . x 3 + + y axy 0.

x y e cos x. 5 2 2 3 3 2 . x x y y 0.

- 1 l 3 3 = o = vers 6 x x . y 2 . + y + y (x ) 2 2 — 2 7 . x x x . £ y + y + y - i s l 1 4 = t n l . 3j a 2 = = 2 i 2 u . 3 + 5 co s x 8 . y s n ( u log x

2 9 . For wh at valu e s of x is the fu nction an increas ing f unc tion ?

1 c os x 3 1 - “ 1 1 n ti . Sho w t hat t he x deriv ative of t an 8 ot a f unc on 1 + co s x of x . 2 2 3 2 £ ff . Find at what point s of the ellips e z 1 the t ange nt cu t s o 5 a e u a i t e r q l n cept s on the ax e s . 3 3 . Find t he point s at which the slope o f t he cu rve y t an x is twice t at of t h i z: h e l ne y x .

3 4 : Find the angle which the curves y sin x an d y co s x m ake it eac ot er at t e ir oi t of i terse c tio w h h h h p n n n . C H APTER I II

S UCCES SIVE DIFFER ENTIATION

3 5 . D ﬁ ni i n f n h d i i hen a e t o o t er vat ve . W give n f un ction y ¢(x ) is diffe re n t iat e d with r egard t o x by the rules of

Ch I hen t he res u apter , t lt

is a n ew fun ctio n o f x which m ay its elf be diﬁ e ren tiat ed by

the s am u s . Thus e r le ,

d d 4) ( x) dx 512

The left-han d m e mbe r is u s u ally abbreviated to ’ the h -h n d m e m r t o c) x ha is rig t a be l ( ) ; t t ,

D iﬁ ren t iatin a a n an d us n a m n a n e g g i i g si ilar ot tio ,

3 4 as 3 c ( ) ,

n d n f r a s o o o an y n um b e r of differen tiatio ns . Thus the z d u s m l e x ss es h is t o be d e n a d h y bo 2 pre t at y iff re ti te wit dx

a d t o x an d h the es u n de vat is th n be reg r , t at r lti g ri ive e to d3 y . m d f e re n a d . S m a i n d ca es the e o an ce i f ti te i il rly, 3 i t p rf r of 011 52 11 111 S CCE S S I E D IFFE REN TIA TI ON 53 C . . U V

d the e ra n h e t m es In e n a the op tio d t r e i , g er l , n a gl s m o m e an s that 1 s t o be di ffe ren ti ate d n t l m es l n y b l ’' y dx h s ucces s io n wit regard to x .

= 4 Ex . 1 . x s in 2 x If y + ,

dy 3 2 4 x + 2 co s x, dx

2 2 12 x 4 s in 2 x ng ,

3 d y 2 4 x 8 cos , dx3

4 d y 24 + l 6 s in 2 x. dx“

If an im plimt equ ation bet we en x an d y be given an d the d a es o f h e ard x are e u re d is n ot eriv tiv y wit r g to r q i , it n e ces s ary to s ol ve t he equ ation for e ither variable before p erform i ng the diffe re n tiation . 2 C 4 4 2 = —3 E 2 . i e x 4 a x ﬁn d x . G v n + y + y 0 ; 4 (ix

2 x4 4 4 a fe 0 ( + 9 + y) , dx d d d 4 2 x 4 62 x 0 + y + y ’ dx dx dx

dv dv 4 3 4 3 ' 2 2 x y + 4 a x + 4 a y r: 0 dx dx

The last e qu atio n is n ow to be solved for

3 f dy x a y 2 dx y3 a x D iffere tiat in a ain n g g , 54 A H 111 C L C O . DIFFERENTIAL UL US [ .

2 The value of fro m (1 ) is n o w to be s ubstit u ted in the last equ a dx

t io an d t he re su ti ex res sio sim i ed . The a orm m a be n , l ng p n pl fi fin l f y writte n fi 2 3 3 4 4 “ 2 2 4 “ d2 2 a x 1 0 a x a r/c 3 x x y y y ( y ) y ( .+ y )

In like m ann er higher derivat ives m ay be fou nd .

0 0 0 O

n f r the n t h der1vat1 ve 1n c ai n c s es . For 3 6 . Ex pres s w o ert a

r n u n c io n s a n e a e x res s m n fo r t he n th de a c e tai f t , ge r l p riv tive can be readily obtain ed in term s of n .

" Ex . 1 . If c t e y z , h n

“ “ i an o sitive i te er. If e W e where n s y p n g y ,

i 2 . If s n x Ex . y ,

— z z in x -- Zg cos x s ( I g),

cos (x s in (x

J : sin dx" ( 2

1 " s in ax 51 3 : a sin ax -- n y , I dx” ( g)

EXE RC IS ES O N C H APT ER III

B y 4 2 z = 3 x 5 x 3 x 9 ﬁ n d z c lo x ﬁ n d y + + ; s y g ; 2 i (133

a ? d y “ d y 2 5 ﬁ d x2 10 x ﬂn d : 2 x + 3 x + ; y e , y n dx3 dx2

s 1 d y d f! . 2 i s ec 1: ﬁn d . ’ 3 y 5 dx dx3

_ ﬁ n d y _ 10 g s in x ; ﬁ n d

g : s in x co s x ﬁ n d 1 0 . y ; ﬁn jis 35 S UCUE S S I VE D IFFE REN TIA TI ON

4 d }, ﬁ n d 1 9 cos mx ﬁ n d “ y ; dx

5 d y 4 2 . —— (I) lo ﬁ n d ° y z g x , 5 Clx

12 9 _ ﬁ n d y s i n x ; 237 5 d 3 —r = l c ﬁ n d “ ° y og ( + e 3 dx

d3 2 2 z y y = (x 3 x + 3) e ; ﬁ n d dx3

4 y : x log x ; ﬁ n d

—— y : ﬁ n d xy; ﬁ n d ZZn

2 01 31 2 6 _ 1 y — . y xe ; ﬁ n d dxz

z d ]! dy Z 0 2 7 _ m x r v = . e s o e 2 2 . y , p + y 0 dxz dx

2 d ? C z / h! 2 z x r e x _ — 8 . y ax s in ; p ov 2 x dx2 dx

d 2 1 2 y 2 9 y a x"+ prove x w n (n l ) y

2 ’ - d y ay 1 2 . 2 3 in x rove 1 x x 0 . s ( ) , p ( dx

d y 2 —1 __ — 3 3 n prov e 1 y . y z x log x ; ﬁ n d 27x

1 / 3 2 = . y ; 4 x2 _ 1

5 2 3 v x 3 . y x e ; pro e 2 e

z 3 6 n . y cos x ; ﬁ d

3 7 r h . F o m t e relat io n prove th at C HAPTER IV

EXPANSION OF FUNCTIONS

It is s om etim es n eces s ary to e xpan d a given fu n ction in a

h n s e es e s t e de e n d n a a e . n s an ce ri of pow r of i p e t v ri bl For i t , in o rder to compu t e an d tabulate the s ucc es s i ve n um e rical

a u es s in x for d f eren a u s x is con n en v l of i f t v l e of , it ve i t to have s in x de v elope d in a s eries of po we rs of x with coefﬁ i d c en t s i n depen en t of x . Simple cas e s of s u ch developm en t have been met with in

a e a . e x am e b t he n m a he m lg br For pl , y bi o i l t ore , — - 1 ”( f l ) ( a x) n a n n an x an i z an d a a n b d n a d s n g i , y or i ry ivi io ,

( 2) 1 x

It is O s r ed h ev r ha the s s is to be b e v , ow e , t t erie a proper repres e n tative of the fu n c tion on ly for value s of x within a

d n in 1 ho ds r a n n e a . n s an ce the e ce t i i t rv l For i t , i tity ( ) l o n ly for valu es of x betwe en a an d a when n is n ot a pos itive i n teger ; an d the iden tity in ( 2) holds o n ly for

1 In ach o f hes ex valu es of x betwee n 1 an d . e t e am s a n e v a u e u s de of the s a d m s be n ple , if ﬁ it l o t i t te li it give

x the s u m an n n e n u m e e m s the s e r e s to , of i fi it b r of t r of i will be n n e h t he un c n in t he s m m i fi it , w ile f tio fir t e ber will be fi n ite .

58 L CA L C L n US o . IV DIFFERENTIA U ( .

h s e comm on ra o is 3 x is c on v e r e n d w o ti , g t or i verge n t

acc rd n a s 3 x is n u m e rl call es s o r ea e r han u n o i g y l . gr t t ity.

The c n di i n for c on er n ce is 1 3 x 1 an d h n o t o v ge < , e ce

the n e r a c n e e n ce is e e n n d i t v l of o v rg b tw e ga 4.

38 . G n er l s for int r l f n The e a te t e va o co vergence . follow ing s u m m ary o f algebrai c p rin ciples leads u p to a t es t that is s u fﬁ cien t to ﬁn d t he in terval of co n vergen ce for a s e ries o f

the m s u s u a n d ha is a s r s c n s s n o f s i e o t l ki , t t , e ie o i ti g po tiv " n e ra o e rs x in h ch the c f c en of x is a n n i t g l p w of , w i oe ﬁ i t k ow

fun cti on o f n .

If i h n n u n h 1 . s s a v c a c e ases n but ,, ariable t at o ti lly i r wit ,

all va ues n em a n s es s han s m e x e d n u m e r h for l of r i l t o ﬁ b ,

h n 8 a oaches s m e d e n e m n o t rea e r han h. t e , ppr o ﬁ it li it g t t [Thi s foll ows fro m t he de ﬁn iti on of a limit ]

2 If n s e r es o f os v e e m s is n n t o be c n . o e i p iti t r k ow o ver

e n an d t he e rm s o f an o he s e s be s v e an d e s s g t, if t t r rie po iti l

h h o es o n d n e rms o f the s s er es he n the t an t e c rr p i g t ﬁr t i , t latte r s eries is co n ve rge n t . [ Us e

m th rm s es m 3 . If a e a e n er e s , ft r giv t , te of a eri for a

e om et r c r e s s on h n dec re asi ng g i p ogr i , t e ( a ) The s ucces s ive te rm s approach n earer an d n eare r to z e ro as a lim it ; (6) The s u m o f all the term s approaches s om e ﬁx ed con

s e m e h d as a c e . s tan t as a lim it . [ U t o of l t rti l ]

h m a s e r es be os an d a er a e n 4 . If t e ter s of i p itive , if ft giv te rm t he ratio o f each te rm to t he p rece di n g be les s than a

xe d ro e r rac o n t he s e r e s is c n e n . Us e 2 an d ﬁ p p f ti , i o v rge t [

If he re be a s e r1 es A c on s s n o f an n n e n u m e 5 . t i ti g i ﬁ it b r

h an d n e a i e m s an d an he r s e es B o f bot pos itive g t ve t r , if ot ri ,

ll the e m s s v is obtain e d the refrom by m akin g a t r po iti e ,

be c n e r e n he n the s es A is c n ve r en . k n own to o v g t , t eri o g t 37 EXPANSION OF FUNCTIONS 59

the s ve e rm s o f A m us orm a c o n r en s er es For po iti t t f ve g t i , othe rwis e t he s e ries B c ou ld n ot be c o n v e rge n t ; s imilarly

h n e a e e rm s o f A m u s o m a c n e r e n ri t e g tiv t t f r o v g t s e es .

h u s o f hes e c n ve r e n s e r e be an d L Le t t e s m t o g t i s u v . e t the ﬁrs t n te rm s o f s erie s A c o n t ai n m po s itiv e te rm s an d p

n e a e e rm s . Let E T S de n o the s u m o f t he g tiv t m, p , , te

s e rm s t he s um o f t he n e a i e rm s an d t he s um po itiv te , g t v te ,

Th n = ll erm s es e c e . e S 2 N a n T . o w of t r p tiv ly n m p

he n n a roaches n n m a n d a s o a r ach n n w pp i ﬁ ity, p l pp o i ﬁ ity an d he n ce i lim S l m T — v. n 17 5 00 p ,

r i n r The re fore t he se ies A s co ve gen t .

The a bs olu t e va lu e r D E FIN ITIO N S . of a eal n u mbe r x is

n m r a a u e a e n os e an d i n its u e ic l v l t k p itiv ly, s writte Ix I. The e qu ati o n | x | a l in dicates that the abs ol u te val u e o f x is

e u a t o the a s o u e a ue a . hen ho e r x an d a q l b l t v l of W , w ve ,

r ace d b on e r e x re s s i n s is c n e n n a e repl y l g p o , it o v ie t to write the a i n in t he o m x a in h ch t he s m rel t o f r l , w i y bol is “ ” ead e u a s in a s u e a u . In m an n r the s m r q l b ol t v l e like e , y bols will be u s ed to i n dicate that t he e xpr es s io n o n t he e has es c e a s m a or a e n um e ica a u l ft r pe tiv ly ller, l rg r, r l v l e than the on e o n the right . An y s e ries o f term s is s aid t o be a bs olu t ely o r a n con d ition a lly co n ve rgen t when the s eries form ed by thei r abs o

h n e es i on r n but u e a ue i c n e e n . e a s s c e e l t v l s s o v rg t W ri v g t,

m n the s eries form ed by m akin g each . ter pos iti ve is ot c n e e n t he s s s is s d be con d it ion a ll o v rg t , ﬁr t erie ai to y c on ve rge n t .

The appropriat enes s ofthis terminologyis due t o th e fact that the t erms Of an abs olute l co ver e t s eries can be rearra ed in an wa it out alt eri y n g n ng y y, w h ng the limit of the s u m o f the s eries ; and that this is n ot t rue of a conditionally 6 0 I CAL C L US CH Iv D FFERENTIAL U [ . .

E . . the s eries is absolute l co ver e t bu t the g , y n g n ;

s e ries 1 is co ditio al c r } 4 5 n n ly onve ge nt .

I 6 . f there be a n y s erie s of te rm s in which aft er s om e ﬁxed t e rm the rat i o of e ach te rm t o the p recedin g is n u m erically

e ss t han a xe d ro e rac on hen l ﬁ p p r f ti ; t , ( a) the s u c ces s i v e te rm s of t he s eries app roach n earer an d n ea re r t o z e ro as a lim it ;

’ (b) the s um o f all the t erm s appfoache s s om e ﬁx e d c on s an as a m an d the s e r e s is a s o u e t t li it ; i b l t ly con vergen t . Us e 3 4 [ , ,

1 . i Ex . F nd the inte rval o f converge nce of the s eries - 2 - 3 - 4 1 + 2 o 2 x + 3 4 x + 4 8 x + 5 l 6 x + — 1 ‘ 4 e re the n th t erm a is n 2n xn an d t he n 1 st term a is H n , ( + ) n. “ " n 1 2n x e ce ( ) , h n . " (n 1 ) 27‘x (n 11: n

the re ore e n n 00 f wh ,

It follo ws by (6 ) th at the s eries is absolutely c onvergent whe n 1 2 x 1 an d t at the i t e rva of co ver e c e is be t ee an d < < , h n l n g n w n 5 h ri i i e t i 5. T e se es s ev d n ly n ot convergent whe n x has e ther of the extre m e val ues .

Ex . 2 i d the i t er a of ve r e ce o f the erie . F n n v l con g n s s

x x3 x5 x7 2 - 1 (2 n _ 1) 3 n

2 2 2 l "+ 1 2 n 1 an u n + 1 n x

2 u x e ce e n h n 2 wh n u 3 n

1 1 1 co ver e t s er1es T us th e u meri cal value o f th e s er1es — -- n g n . h n I IE ﬁ gé is independent of th e order or grou ping but the value of the s eries 1 4 i can be m ade e qu al t o an y num be r what ever by s uitable

m l o d . 43 rearra em e t . For a s i e roo s ee Os o ng n [ p p f, g , pp , - 38 39 . EXPANS ION OF FUNCTIONS 6 1

2 — t u s the s e ries is abs o u te co ve r e t en 1 i. e . e 3 x 3 h l ly n g n wh < , , wh n , g2 T m an d the inte rval o f co nve rgen ce is fro m 3 t o 3 . he ex t re e value s

o f x in the rese t case re de r the s eries co ditio a l co ver e t . , p n , n n n l y n g n

1 x 1 x 3 1 x 5 l x 7

Ex 3 . Show t hat the s er1 es + 3 2 3 A5 2 3 7 2 3 has the s am e inte rval of conve rge n ce as the last ; bu t th at the ex trem e value s of x render the s e ries abs ol ut ely co nvergent .

Th r d 3 9 . Remain der after n t erm s . e l as t article t eate of the i n terv al of c on v e rge n ce of a give n s erie s withou t re fe ren ce t o the q u e s t1 on whethe r o r n ot it was t he develop

men o f an n o n u n c on . O n the o he han d t he s e e s t y k w f ti t r , ri that pres e n t them s elve s in this chapte r are the de velopm en ts of ve n un c ion s an d the rs u e s o n ha a s s is gi f t , ﬁ t q ti t t ri e c on cern i n g thos e valu es o f x for whi ch t he fu n ction is e qu ivalen t t o it s de velopm en t .

he n a s e r e s has s uch a en era n un c on t he d ffe W i g ti g f ti , i r e n ce b etwe en the valu e Of the fu n cti on an d t he s u m o f t he

ﬁ rs t n t erm s o f it s dev elopm en t is calle d t he r em a in d e r T h h a fte r n t e r m s . hu s x be t e u n c o n , S x t e if f( ) f ti n ( ) s u m o f t he rs n e m s the s e r es an d R x the ﬁ t t r of i , n ( )

e m a n de o t a n e d b s u ac n S x om x h n r i r b i y btr ti g n ( ) fr f( ) , t e

in h ch S x R x are u n c on s of n as as o f x . w i n ( ) , n ( ) f ti well

“m m 0 h n R x , t e S x $ 00 , ( ) ”1 m n ( )

hu s the m the s e es S x is the e n er n un c n t li it of ri n ( ) g ati g f tio

hen th m of the m a n d r 1 o e u e n t his w e li it re i e s z er . Fr q ly t 1 m i n h s a s uf c e e s t for t e c on ve n ce a s e r e s S x . ﬁ i t t rge of i 1 m n ( )

If a s e r es roceed in n e a rs o f x a t he re i p i t gr l powe , p cedi n g c on ditio n s are to be m odiﬁe d by s u bs titu ti n g x a

’ for h w1 x ; o t e r s e e ach crite rion is t o b e applie d as b efore . 6 2 D CAL C UL US Cm IV IFFERENTIAL [ .

’ - "e 40 . M aclaurin s ex n s ion of fun ction in ower s eri pa a a p es . It will n o w be s ho wn that all the de vel opm en ts of fu n ction s in powe r-s eries given in algebra an d trigon om etry are bu t s ec a cas es on e e n a o m u a x an p i l of g er l f r l of e p s ion .

It is r s d ﬁn d a o m u fo r the x n p opo e to f r la e pa s io n , in as ce n d n os n e a ers o f x a an s s n d i g p itive i t gr l pow , of y a ig e

u n c n h ch h its s ucces s d a s i n nu u f tio w i , wit ive eriv tive , s c o ti o s i t h h n e vi ci n ity of t e val ue x a . The p relimi n ary in ves tigati on will proceed on the hypothe s is ha t he as s i n e d un c n x has s u ch a d o m n t t g f tio f( ) evel p e t, an d that the latte r can be treate d as iden tical with the form er for all val u es of x within a certai n i n te rval of e quiva h h F h h h le n ce t at in cl udes t e val u e x a . rom t i s yp ot es is the coefﬁ cie n ts of the differen t powe rs of x a will be de

I h n e m a n e t h a d o f the te rm in e d . t will t e r i to t s t e v li ity r u b ﬁn din the c n d on s ha m us be u d in es lt y . g o iti t t t f lﬁlle , orde r that t he s eries s o obtain ed m ay be a proper rep res en ta tion of the ge n eratin g fu n ction . Let t he as s u m ed iden tity be

2 3 f(x) 3- A E ( x a) + 0 (x a ) D (x a ) 4 E x a m 1 ( ) , ( ) in h ch A B 0 are un de m n e d c f c n s in de e n w i , , ter i oe ﬁ ie t p den t o f x . Su cces s ive diﬁ eren tiation with regard to x s upplies t he

o n add on a de n t es on the h hes s ha the foll wi g iti l i iti , ypot i t t derivati v e of e ach s eri e s can be obtain e d by diﬁ eren tiatin g

e rm b e m an d ha has s m n a u a n c it t y t r , t t it o e i terv l of eq iv le e with it s c orres pon din g fun cti on

* N amed after Colin M aclaurin (1 698 wh o publis hed it in his “ ” b Treat is e on Fluxions but he dis tinctly s ays it was known y — ff rentialis S tirling ( 16 90 1 77 who als o published it in his M eth odus D i e l Art an d by Tay or (s ee . EXPANSION OF FUNCTION S 6 3

— 2 a ) + 3 D (x 7 a ) + 4 E (x ” - — 3 — z f ( x ) 2 0 + 3 2 D (x a) 4 3 E ( x a ) + m

. - - — 3 2 D + 4 3 2 E ( x a )

If n ow the s c a a u a be v en t o x the n , , pe i l v l e gi , followi g equati on s will be obtain ed :

a = A B 2 3 f< > , . 0 ,

n ce He , 0 f

Thu s the c e fﬁ c e n s in 1 are de e rm n ed an d h r o i t ( ) t i , t e e qu ire d de velopm e n t is

3 f < w> (w (w a >

(n ) f ( a ) ” (w a ) n !

’ Th s s e r es is n o n as M a cla u rin s s er es an d the t he o i i k w i ,

’ m u a i s ca e d M cl u rin h re m e xp re s s e d in the fo r l ll a a s t eore m .

1 in o ers of x a . Ex . . Expand log x p w

’ ” ere x lo x x x H f< > g , f ( ) i, f ( )

” e ce a a H n , f ( ) f ( )

an d b the re u ired develo m e t is , y q p n

1 — log x : lo a + -- x a ( x g ( ) 3 a 3 a 6 4 Us Cm IV DIFFERENTIAL CALCUL [ .

The condition for the conv ergence of this s eries is

“ 1 n (n 1 ) (x n a

9” a |< 1 .

O< x < 2 a .

- ” This series m ay be called t he de velop men t of log x in the vicin ity 01

e in t e v ici it f x z x z a . It s deve lopm nt h n y o 1 has t he s impler form — log x : x 1

r b which holds fo valu es of x etween 0 an d 2 .

2 . t h m n Ex . Show t hat e develop ent of ii powers of x a is 1 1 1 1 1 — — 2 - — 3 (x a ) (x a ) + x a p a

z an d th at t he serie s is conv ergent from x 0 t o x z 2 a .

m 3 v Ex . . D e elop e in powers of x 2

3 2 4 D v x 2 x n r x . Ex . . e elop 5 x 7 i pow e s of 1

? 5 . . Ex . D evelop 3 y 14 y 7 in powers of y 3

The e xpan sion of a fu n c ti on f( x) in a s eries of as cen ding powe rs o f x can be obtai n e d at o n ce from form u la ( 2) by

The s e r s hen c m e s givi n g a t he p articular v al ue z ero . ie t be o

H (n ) n f 0 9” -+ (3) 723

in wers f x n ﬁ nd t he i te rva of co ver 6 . Ex a d s in x o o a d Ex . p n p , n l n gence o f t he s erie s . H ere f(x)

’ x cos x f ( ) ,

6 6 On I V Us . DIFFERENTIAL CALCUL ( .

As i a s ec a cas e ut a e . p l , p

T e lo a lo e 1 h n g g ,

x 2 3 n

i . 0 ex : 1 0 0 i 0 0 . n f

T ese se ries are conve r e t for e ve r ite value of h g n y ﬁn x.

’ 41 . n Taylor s s eries . If a fu ction of the s u m of two n u m

ers a an d x be n a x is u e n des i a e t o b give , f( ) , it freq tly r bl

x n d t he u n c n in r n h m a e s o e e s a x . e p f tio pow of of t , y

In the u n c n a x a is re a d d as c n s an f tio f( ) , to be g r e o t t , s o ha c n s de e d as a un c n x m a be x an ded b t t, o i r f tio of , it y e p y

3 h d n In h orm a h rm u a t e c a c . a u t e fo l ( ) of pre e i g rti le t t f l , c on s tan t te rm in t he expan s ion is the v alu e whi ch t he f u n c

on has he n x is m ade e u a z e h n ce t he rs t e rm ti w q l to ro , e ﬁ t

In h in t he ex pan s io n of f( a x) m ay be written f( a ) . t e s am e m an n er the coefﬁ cien ts of the s u cces s ive powe rs of x are t he co s n d n der a e s a x as x in rre po i g iv tiv of f( + ) to , which x is pu t e q ual to z ero after t he differen tiatio n has Th bee n perform ed . e ex pan s io n m ay the re fo re be writte n

u (n )

Th s s e es ro m the n am e it s d s c o v e er is n o n as i ri , f of i r , k w

’ T l r s eries an d the h em e x r s s d b t he m u a i s a y o s , t eor p e e y for l ’ m kn own as Taylo r s theore .

E n in ers f x . Ex . xpand si (a x) pow o

e re a x s in a x H f( ) ( ) , hence f( a)

’ an d a cos a f ( ) ,

i a os a - s n 2 c 3 H en ce sin (a + x) sin a + cos a x x x + 2 ! 3 , ” “ M EXPANSION OF FUNCTIONS 6 7

13y J “ y EX ERC IS ES

1 Ex a d t an x in owers o f x. . p n p

2 om are the ex a s io of t an x it the u otie t de rive d . C p p n n w h q n dividing the s eries fo r s in x by t hat for cos x.

E n Art . 4 . See x s . 6 a d 7 , 0

2 4 s i u x x x 3 Prove lo ﬁ . g x 6 1 80 — $8 3 4 . Prove x 2 -4 5

2 2 x“ 6 -O72 x8 5 Prove lo co s x V . g 4 ! 8 1

b di isio m a i u s e of t he ex o e tia s eries . 6 . Ex a d v p n y n , k ng p n n l 1 f:x

5 f x l 1 x to t he term in v o vi x b 7 . d the ex a s io o e o Fin p n n g ( ) l ng , y ac m ultiplying t ogethe r a s ufficient nu mber o f t erm s of the series for e an d for log (1 x) .

2 n rs . V 8 . Expand x i po we of x

of . V 9 . Expand in powers x i x

4 x 3 2 1 0 . Arrange (3 x) ) x) (3 x) 2 in powers

o f x .

4 2 . A n e cess ary restriction impos ed u pon the s eries s o that it m ay be a corre ct repres en tative of the gen e ratin g

“ u n c i n i s ha t he m a n d a r n e m s m a m ad f t o , t t re i er fte t r y be e

m e h n n n n um e b a n i a n h s all r t a a y giv e b r y t ki g t l rge e oug . B efo re de rivi n g the gen eral form for this rem ain der it is

n eces s ary t o pro v e t he foll owi n g theo rem .

’ R olle s t r If x an d it s s der a ve ar n heo em . f( ) ﬁr t iv ti e co

tin u ou s for all a u s x e e e n a an d b an d v l e of b tw , if ’ h an sh hen x n sh for s om a u x be bot v i , t f ( ) will va i e v l e of e en a tw an d 6 . B y s uppos iti on f( x) can n ot be c om e i n ﬁ n ite fo r an y v alu e h h ’ x s u c a a x b. If x does n ot v an s h m us of , t t f ( ) i , it t a wa s os or a a s n e a i v hen ce x m us l y be p itive lw y be g t e ; , f( ) t 68 Us CE W DIFFERENTIAL CALCUL [ .

co n tin u ally in creas e or co n tin ually de creas e as x in creas es

fro m a t o 6 ( Art .

Th s is im os s e s n c b h h s s a 0 n i p ibl , i e y ypot e e f( ) a d

b = 0 hen ce at s om e n x ee n a an d 6 x m us f( ) ; , poi t betw , f( ) t

c eas e t o n c e as e an d n d c eas e ceas e d c i r begi to e r , or to e reas e

n an d begin to i cre as e .

h i d n e d b t he e u n x T is po n t x is eﬁ y q atio f ( ) 0.

To r t he s am e h n e m e r ca let = x be p ove t i g g o t i lly, y f( ) t he e quation of a c on tin u ou s

cu e h ch c s es th x-ax rv , w i ro s e is

at d s an c s x = x x = b m the i t e , fro T o rigin . he n at s om e poin t P b e twee n a an d b the tan gen t to the cu rve is parallel to the FI G . 12 . x-ax s s n c b s u os n i , i e y pp itio

h is n o d s c n n u in th h n e t ere i o ti ity e s lope of t e tan gen t . He c at t he poin t P r QL—. — f ( rv) 0 ~ dx

’ 43 . n the Form of remai der in M aclaurin s s eries . Let

re m a n de r a e r it erm s be d n e d b R x a h ch is i ft t e ot y n ( , ) , w i

x n d in ce e a h t he a f un cti on of a a as well as of n . S c of s u cc ed n e m s is d s e b x R m a con e i g t r ivi ibl y ( n y be v en ie n tly written in the form

R a a . m. ) w , )

r m is n ow d e m n b x a s o h the The p oble to et r i e g ( , ) t at rel ati o n ? 3 foe) = f< a > a ) (a: a) 42 EXPANSION OF FUNCTIONS 6 9 m a be an a e a c de n in h ch the r ht -han d m em y lg br i i tity, w i ig be r c o n a n s on t he rs it t m s of t he s e r es h t he t i ly ﬁ t er i , wit r d a t e r n m Th n m ai n e s . u s b ran s s e er f t r , y t po i g, — f< x> f< a > a > a )

— — ” (”f a ) (93 4 5 0 {Zj iiz

a n ew u n c n F z be de n e d as o s Let f tio , ( ) , ﬁ foll w

— 2 F e) E ﬂ x) f< z > z) ( x z >

M 950“ a ) ( x z ) ( x ( 3) ( ii

This fun ction F ( z ) van is hes as is s een by

n s ec n an d a s o an shes s n ce he n i p tio , it l v i i it t

ec m es d n ca h t he e -ban ber of he n ce b o i e ti l wit l ft , ’ ’ b Ro e s h m it s de r v a F z an shes s m e y ll t eore , i tive ( ) v i for o

a u 2: e e n x an d a s a z . B u t v l e of b tw e , y l

we z ) z )

— x a n 1 ¢ ( , ) — z ) + (r z>

Thes e te rm s can cel e ach othe r in p airs e x cept the las t two ; hen ce ( F < z> = fj }; [w a >

" S n c I z an s hes hen z s ha i e ( ) v i w z l , it follow t t

a w . )

In h s x res s on 2 s e e e n x an d a an d t i e p i 1 lie b tw , be repre s en ted by a 9 x a ( ) , 70 C m IV DIFFERENTIAL CALCUL Us [ .

he re 0 is a os iv e ro e r ra i n r m 4 w p it p p f ct on . H e ce f o ( )

a 9 a w , ) c n,

n ) e R x a M 3: a n j n ( , ) M ( >

The c om plete form of the ex pan s ion of f(x) is then

£ 2 1 2 a ) (a: a ) 2 1

— ” < w a > W (a: in h h i an n r The s er es m a r w ic n s y pos iti v e i tege . i y be ca

ed an des re d n u m e o f e m s b n creas n n an d the ri to y i b r t r y i i g , las t term in (5) give s t he rem ain d e r ( o r error) after t he ﬁ rs t

e m the r The m o 9 x a n d i t t r s of s e ie s . sy b l ( ) ) i i c ate s that f(x) is to be differen tiate d n tim e s with regard to x an d ha x is hen t o e ac d b a 9 x , t t t be r pl e y ( a)

44 . n In s ead o f ut An other ex pres s ion for the remai der. t p

n R x a in the m ti g n ( , ) for

it !

is s m m s con en en it , o eti e v i t to write it

R x a x a « f x a . , ( , ) ( ) I ( , ) " P c e d n as re t he x ss n for E z ro e i g befo , e pre io ( ) will be

’ M x x « r x a . E ( ) ( l ( , )

— — in h ch a 0 x — a x —z x d l 9 w i + ( ) , 1 ( ) ( )

who This form of the rem ainder was found by L agrange ( 1 736 ’ u blis ed it in h M xn r d A démie des Scie ces a B erli 1 772 . p h t e é oi es e l ca n n , EXPANSION OF FUNCTIONS 71

“) f a 9 013 a — — i ( n l 1 en ce a 1 9 x d n H ) ( ) ( o , ( n 1 ) l

n “ 0 n —1 an d R x a 1 0 M x a ,, ( , ) ( ) ( ) (Z —

An example o f the u s e of this fo rm of rem ain der is fur h is he d b the s e r es for lo x in e s o f x a he n x a y i g pow r , w

“ is n e a an d a s in the ex an s n o f a x g tive , l o p io ( )

1 . Find the interval of equ iv ale nce for the developme nt of log x in

o ers of x a e a is a ositive u mber. p w , wh n p n

r m Art . 4 ere o 0 Ex . 1 H , f , ,

n —l i n ———L x ’ xn

" hence ) a 9 7 “ ﬂ ( 0 D b lﬁ ’

x —a n _ £_ a,n d b Art 43 R a 2 . , y , n (x’ ) l x a) :i

irst l i T e en i i F et x a be pos tive . h n wh t l es betwee n 0 an d a it is u m erical e ss t a a 0 x a s i ce 0 is a os it ive ro er racti n ly l h n ( ) , n p p p f on ; he nce whe n n i 00

A ai e x a is e ative an d u m e ricall les s t ha a t he s eco d g n , wh n n g n y n , n orm of h i d t e rem a er mu st be e m o e d . As be ore f n pl y f ,

M — f( a G ag a

x _ l a e ce R x a 1 0 n h n n ( , ) | K ) m

— e u 1 ) — [a 0 (a M A 0 M “ l — — ° — f a 0 (a x) a - t9(a av)

This form of the rem ainder was fou nd by Cauchy ( 1 789 an d ﬁrst ” ublis d in hi l in ﬁn i im al 1 26 e s Le on s s ur le calcu t és 8 . p h c , 2 IV 7 DIFFERENTIAL CALCUL US [Cm .

The act or it i the brac ets is u m erica es s t a 1 e ce the f w h n k n lly l h n , h n n 1 st o wer can be m ade es s t a an ive n u mber b ta i ( ) p l h n y g n , y k ng

i i true f r all va ues of x be t e n d . n large e no ugh . Th s s o l w e n 0 a a T e re o re lo x an d its deve o m e t in o ers of x a a re e u iva h f , g l p n p w q e t it i t he i terva o f co ver e ce of the se rie s t at is for all l n w h n n l n g n , h , n d 2 values of x be twe e n 0 a a .

' 2 o t at t he de elo m e t f x i i i i Ex . . Sh w h v p n o n po s t ve powers of x a o ds for all value s of x t at m a e the serie s co ve r e t t at is e h l h k n g n ; h , wh n x lies between 0 an d 2 a .

If the u n c on is x n d d in s o f x t he c m f ti e pa e power , o plete form will be O M ) ? as

” f0 9x + ”g

t he s o m em a n d r an d ﬁr t f r of r i e ,

f(rv)

for the s econ d form of rem ain der . ’ m ar the c m e e m Ta s s e es Art . 41 Si il ly, o pl t for of ylor ri ( ) be co m es

ﬂ “ 50 )

r h r m m n de an d fo t e ﬁ s t for of re ai r,

f “) 2 f( a 93) = f( a i x

( 1 w” < 4)

for t he s e con d form of remain der .

74 A CI-I IV DIFFERENTIAL C LCUL US [ . .

The act or 1 is e ua t o or reater t a u it but in f q l g h n n y, n )

t he latter cas e beco mes m ore an d m ore ear u it a n i e n ly n y s ncr ases . n The factor I S n um ericallyles s th an 1 for any value of n when a i 9x

x iv n va ue bet ee z er n T is g e n a y l w n o a d a . he product is therefore re at er t a u it for va ues o f it es s t a som e n ite umber N but g h n n y l l h n ﬁ n ,

le s s t a u it for va ues of n reater t a N . T e in the s ame wa h n n y l g h n h n, y as for t he recedi case p ng ,

k B N+ I RN,

k k2R RN+ 2 RN+ l N9

i i it mber an d k a r c R s a e u r ti . wh h N ﬁn n p ope frac on Hence

“m R (1 0 m, n i ce )

for ever va ue of m rovided x ie s bet ee 0 an d a . y l , p l w n i e t h i t rv f co ver i x a S c e e a o e ce s b Art . 38 rom x a t o n n l n g n , y , f , it re m ai s to ex am i e the va ue of R x a e x ies bet ee 0 n n l n ( , ) wh n l w n orm of an d a . For this purpos e it is necess ary t o u se the s econd f rem ai der ic m a be ritte n , wh h y w n ,

" t R x a ﬁ — x a n ( , ) W (

1 “ e B ut e _x_ is e ative an d ess t a 1 the ex ressio is a wh n n g l h n , p n a x 1 + 0 a

ro er ractio he ce its n l st o er a roaches ze ro as a im it . p p f n , n ( ) p w pp l - 1 M M ic The expre s sion is m ade up of n 1 factors (n 1 ) l (a y m k x — eac of ic is ite an d e k is su fﬁ cie t of the form h wh h ﬁn , wh n n ly k a

ar e is um e rica ess t a 1 e ce the im it of the e tire roduct l g , n lly l h n ; h n l n p T e or is z ero . her f e R x a i 0 e n i 00 n ( , ) , wh n ,

if x lies betwee n a an d a . “ T ere ore for va u es ofx it i t is i terva the u ctio a x an d h f , l w h n h n l f n n ( ) it s e xpans io n are e quivale nt . EXPANSION OF FUNCTIONS 75

L 45 . Theorem of mean value . e t f(x) be a c on tin u ou s

n n o f h h ha a der It an h n r r fu ctio x w ic s i vative . c t e be ep e s e n ted by the ordin ates of a c u rve whos e e qu ation is y

1 3 let In Fi . g ,

x GIV, x h OR ,

f(x) = Nﬂ lv -l B K — Then f(x + k) f(x) M K an d

FI G . 13 . t an M H If, h H M

B u t at s ome poi n t S b etw e e n H an d K t he tan ge n t to the c u rve is parallel to t he s ecan t H K Si n c e t he abs cis s a of S is greate r than x an d les s than x b it m ay be repres en te d

b x 9k m h ch 9 is a s n u m e es s han u n . y , w i po itive b r l t ity The s lope of the tan ge n t at S is then expres s ed by hen ce

from which ' for: h) = f< x> hf (fv on

The theorem e xpres s ed by this form ula is kn own as the th eorem o m e n l f a va u e .

’ The th eore m o f mean value can als o be established from Taylor s

t eorem b . h y putting it 1 in e quation (3) of Art . 44

EX ERC IS ES O N C HAPT ER IV

1 . Ex a d cos x in o er f p n ( h) p w s o k .

2 . Ex a d t n i p n a (x h) n powers ofx .

3 . B ex a di cos x h in o e rs of on e of its variab es rove P y p n ng ( + ) p w l , p t he t eo m h re co s (x h) co s x cos b s in x s in h.

’ 4 . Ex a p nd log(x h) by Taylor s t heore m in powers of h.

5 5 . Ex a d x in o e r f p n ( y) p w s o y. 4 IV 5 . DIFFERENTIAL CAL C UL US [Cm .

x x2 x4 6 Prove lo 1 e” lo 2 . g( ) g 23 -4 1

x l l 1 I 7 r ve lo . P o g — - - 2 x l x l 2 (x l ) 3 (x

2 14 8 . Prove + R . {2

9 . If x rove t at the ex a s io o f x in o ers of x ) , p h p n n f( ) p w 2 i co t ai o eve o wers of x as cos x V 1 x if x — —x w ll n n nly n p , , ; f( ) f( ) , the ex a s io of x i i vo ve o odd o e rs of x as s in x t an x p n n f( ) w ll n l nly p w , , , x

1 x2

1 0 . S o t at in the ex a s io for s in x in o e rs of x h w h p n n p w ,

lirn

0. C HAPTER V

INDETER MINATE FORMS

46 . H he the va u e s a v n u n c on x c o it rto l of gi e f ti f( ) , rre

on din as s n d a u s the a a e x ha n s p g to ig e v l e of v ri bl , ve bee l a n e d b d e c s u s u on . The u n c n m a h obt i y ir t b tit ti f tio y, ow e e r n o v e the ar e in s u ch a wa h for ce a n v , i v l v iabl y t at rt i value s of t he latte r the co rres pon din g valu es of the fu n ctio n

d m r c an n ot be foun by e e s ubs tit ution .

Fo r exam e t he un c n pl , f tio

x -x e e

s in x for the a u e x = 0 as s u m es t he m an d the c s n d v l , for 8, orre po

‘ in g val ue of the fun cti on is thu s n ot dire ctly dete rmin ed . In suc h a cas e t he e xp re s s i on for the fun c tio n is s aid to ass u me an in d et erm in a te form for t he as s ign ed valu e of the variable . The example j us t give n ill ustrates the i n determi n aten es s

m s u e n c cu e n ce n m ha in h ch t he of o t freq t o rr ; a ely, t t w i give n fun ction is the qu otie n t of two other fun ction s that v n s h th m h a i for e s a e val ue of t e variable .

Thus if f( x)

an d hen x a s the s c a va u a the un c n s x if, w t ke pe i l l e , f tio ¢( )

an d h an s h t h n bot v i , e M a ) 0 f( a M a ) 0

is n de m n a e in m an d can n o n d e d d e m n e i ter i t for , t be re er et r i at

h u u wit o t f rther trans fo rm atio n . 77 E A C m 78 DIFF RENTIAL C LCUL Us [ v .

Ind min t fo ms ma h d rmin 4 . e u 7 eter a e r y ave et ate val es .

h d n d r 9 i h A c as e as alre a y bee n otice (A t . ) n whic an ex 0 res s on ha as s um es the o m for a ce a n a u e o f it s p i t t f r 0 rt i v l

ar a e a e s a de n e v a u de n de n u on t he law v i bl t k ﬁ it l e , pe t p of v ari ation of t he fun cti on in the vicin ity of the as sign ed v alu e of t he v ariable .

As an o he ex am le co n s de the u n c n t r p , i r f tio

2 2 x a

x — a

If this relation be twe en x an d y be written in the form s

z 2 x — a = z — a x — a — x i < > , < > < y

e e n h can be re e s en ed ra h ca as in it will b s e t at it pr t g p i lly,

the a r n e s t he ﬁgu re ( Fig . by p i of li

— z x a 0,

— — = 0 y x a .

H e n ce when x has the valu e o f a there is an i n deﬁn ite n u mb e r of corre s pon di ng

on the ocu s all s u a ed on the poin ts l , it t lin e x a ; an d acc ordin gly for this

m ha e an a u e ha an d valu e of x the fun c tion y ay v y v l w tever, is therefo re in dete rmi n ate .

u e d f e en rom a the c r s n d n When x has an y v al i f r t f , o re po i g

d r m the e u a on x a . N ow v alu e of y is det ermi n e f o q ti y , d n of t he i n ﬁ n ite n u m b er of differen t v alu es of y c orres pon i g

= a h e is on e a cu ar a ue AP h c h is con to x , t er p rti l v l w i

h n x a es t in u ou s with the s erie s of val ue s tak en by y w e t k

h m a s u cce s s iv e v al ue s in the v ici n ity of x a . T is y be

. = It is ob calle d the d et erm in a t e va lu e of y when x a .

= in the u n x a an d is t ain e d by pu tt in g x a eq atio y + , therefore y 2 a . INDETERMINA TE FORMS 79

This res u lt m ay be s tate d withou t referen ce to a lo cus as

: hen x a the u n c n follows W , f tio

2 2 x — a

x — a i n de e m n a e an d has an n n e n u m r f d e re n s i t r i t , i ﬁ it be O iff t valu es ; bu t am on g thes e valu es there is on e de term in at e valu e whi ch is co n tin u o u s with the s e rie s of v alu es taken by the fu n ction as x in creas es thro ugh the valu e a ; this deter m in ate o r s i n gular value m ay the n be deﬁ n ed by

z — z lim x a

In evalu atin g this lim it the in ﬁ n ites im al facto r x a m ay be re m o ve d rom n u m e at or an d de n m n a or s in ce h s ac or f r o i t , t i f t is n o t z ero hi e x is d ffe re n ro m a h n c the det e rmi , w l i t f ; e e n ate valu e of the fu n cti on is

lim x + a _ 2 a ‘ x i a 1

Ex . 1 . i d t he d et erm i at e va u e e x l of t he u ctio F n n l , wh n , f n n

x3 2 x2 3 x ’ 3 x3 — 3 x2 — x + 1

ic at t he im it t a es the orm wh h , l , k f 8

This expre ssio n m ay be written in the form

(x2 + 3 x) (x 1 )

2 x + 3 x x

ic reduces to . W e x 1 t i s becom e s wh h h n , h 3 x2 1

E 2 x . . Evaluate the express ion

8 2 2 8 x ax a x a 2 2 2 x3 x6 ax ab e x wh n a . 80 on DIFFE RENTIAL CALCUL US [ v .

3 . Ex . D e term ine the value of

x3 — 7 x2 + 3 x + 14

x3 + 3 x2 — 1 7 x + 14 - when x _ 2 .

Ex . 4 . Ev te i alua wh en x 0.

‘ M u ti bot umerato r and de omi at r b \/ 2 2 ( l ply h n n n o y a a x . )

E u i n b d m n . In s m e c as es the 48 . val at o y evelop e t o com m on van is hin g fact or can be bes t re moved aft e r exp an s ion i s n eries .

Ex. 1 . Co s ider the u ctio m e tio ed in Art . 46 n f n n n n ,

e * x

s in x

e u m erator an d de om i ator are de ve o ed in o ers of x the Wh n n n n l p p w , express ion becom es 2 3 1 — x + 2! 3 !

c) 2 l 3 4 -1 2 x + r + ~ 2 . + 3 1 s

x 3 x‘ + 3 1 6

i l z r hic has the dete rm ate value 2 he x t a es the va ue e o . w h n , w n k

2 As ot er exam e evaluate he x i O the u ctio Ex . . a n h pl , , w n , f n n

- 1 x s in x

s in 3x B y development it becom es

1 x3 t 5 5 +

x3 3

R movi the co mm o actor an d t e utti x 0 the resu t is . e ng n f , h n p ng , l 3

82 C V DIFFERENTIAL CALCUL US [ a .

B d d n b x a an d hen e n x é a y ivi i g y t l tti g , it follows that l lim f( x) f at ) ﬁ x a ﬂ fv) PT“)

’ The un c on s b a in en e ra h b f ti q ( ) will g l bot e ﬁn ite .

“ f I 0 1 ﬂ ) I a ( ) a 0 h n 0. f ( ) , I ( ) , t e (Ma )

’ If a a = 0 f ( ) , (M

’ ’ If a an d c a are h z e the m n a u e of f ( ) t ( ) bot ro , li iti g v l

’ iggis to be Obtain e d by carryin g Taylor s de velopm en t 2 on e m a h em n t he c mm n ac x a an d ter f rt er, r ovi g o o f tor ( ) , n h i then le ttin g x app roach a . T e res u lt s £ 623 ’ ” S m a a a a ) a all i il rly, if f( ) , f ( ) , f ( ) ; 4 ( ) an sh is d in the s am m an n ha v i , it prove e er t t

lim f( fv) x a 9506 )

on u n a res ul t is a n e d ha is n ot n d m n an d s o , til obt i t t i eter i ate in form . H en ce the rule

To eva lu ate an ex res sion o the orm d eren tia te n umer p f f 8, iﬁ ator an d den omin a tor s ep arately ; s ubstitu te the critica l va lu e

th ir deriva tives an d e uate the uotien t o the deriva of x in e , q q f tives to the in determin ate form .

1 cos 0 1 E u t e e 0. Ex . . val a wh n 0 92

= 1 — cos f (9) 0,

’ ’ sin t 0 2 0 f (0) 0, q ( ) ,

’ 0 0 e 0. f ( ) , m ) IND ETERMINA TE FORMS 83

0 0 3 0, 2,

1 , 2:

lim 1 cos 0 1 he nce 9 5. 0 92 2

11 6 ” e ‘ “ 2 co s x 4

Ex . 2 . i d F n $ 3 0 x4 — lim ex + e —4 lim ex _ e x Q Sin x - (t i O x4

lim e” + e x 5 0

1 11 x s m x cos x Ex . 3 . i d 5- F n 96 0 x3

lim x5 —2 x3 —4 x2 + 9 x —4 4 Fi d Ex . . n x4 —2 x3 + 2 x —1

In this exam e s o that x 1 is a actor n um erator pl , h w f m i deno nator . s lim 3 t au x — 3 x —a

EX . 5 . Fin d [ L x i o x5

In n h s r c s s ar cu a m s the applyi g t i p o e to p ti l r proble , work can o ften be shorten ed by e val uatin g a n o n -van ishi n g factor in e ither n u m erator or den omi n ator before performin g the ff di ere n ti ation .

a: i 0

The given expre s sion m ay be writt en

lim t au x lim lim tan x

x x 84 DIFFERENTIAL CALCULUS

In n e a x = an d 0 ge r l , if f( ) if , S a 0 t hen q ( ) ,

11 111 f(z ) a x X( ) a (13 013)

lim ll m _ x x .. = ) x ¢( x) x a

li n in x cos z 7 s x Ex i d i ﬂ F n x 3 (2 x

x 1 z 8 1 11 (x 1)

EX ERC IS ES

Evaluate the following e xpre ss io ns

1 _ C O S x

0 . e x e x 0. , wh n wh n s 1 n x

W e n x o. h t au x s x cos x s iu x e = ;p wh n x 0 . x3 _ 1 whe n 1 . I 1 -1 S in 17 h O a z _ 1 a n x : 0 whe . bac 1 2 e“ s iu x x x s in ax he x : 0. e x 0 w n wh n . x2 + x lo 1 —x S i n bx g ( ) 1

n _ — (1 + x) 1 tau x s iu x = e x 0 whe n x 0. wh n x x3 0 There are other i n determin ate form s than They are 0 0 co 00 , 00

50 . Evaluation of the indetermin at e form

f( Z 00 Let t he fu n cti on be come whe n x a . It is re 0°

hm f< 2: ui d ﬁn d q re to x _

86 D V IFFERENTIAL CALCUL Us ( C a .

(X) h ch has the m h n x = a an d has th d t m i n w i for 5 w e , e e er ate

a u 0 h ch is n ot z . en ce b 2 v l e , w i ero H y ( )

’ lim f(a?) GP( in) f ( a ) f

Th b s u c n c erefore , y btra ti g ,

' fee) f o) ,

lim fCB ) lim 4606 ) If 00 th n 0 wh ch can be t e d r e a , e “ a , i r ate ches) f(w)

h u as t e previo s cas e .

51 . 00 Evaluat ion of the form 0. Let the fun ction be ¢ ( x) s u ch that ¢( a)

1 r 1 ( a) 0. 0 Th s m a n h ch a e t h m h n i y be writte w i t k s e for 0 w e

( 93) a is s u s u d x an d h c m s un d th v b tit te for , t erefore o e er e abo e

Art . r ule . (

co 00 52 . Evaluation of the form .

The m 00 co m a be n z n n for y ﬁ ite , ero , or i ﬁ ite . / 2 For n s an c c n s der \ x ax x t he a u x 00 . i t e , o i for v l e

It is th e rm 00 00 but b m u n an d d d n b of fo , y ltiplyi g ivi i g y ax s ax x it be com es which has the form

h n x 00 w e

A a n b d d n h m s b x a s the m g i , y ivi i g bot ter y , it t ke for

0 whi ch becomes when x 0 . 9 1 x

The re is here n o gen eral rule of procedu re as in t he

e u s cas s but b m an s n s rm a n s an d pr vio e , y e of tra fo tio proper si c INDETERMINA TE FORMS 87 grou pin g of terms it is often po s s ible to brin g it in to on e 9 99 the m s . u n a u n c n h ch c m e of for , Freq e tly f tio w i be o s 0 co co 00 for a critical valu e of x can be put in the form

u t

v w

n h ch v w c m z e . Th s can be du ed to i w i , be o e ro i re c

uw vt which is then of the form 3

1 “

Ex . 1 . i d sec x tan x F n x 2, ( ) .

This ex res sio assum es t he form 00 00 but can be written p n ,

1 sin x 1 sin x

co s x cos x cos x

whic is of the orm an d ives z ero hen evalu ated. h f 8, g w

i i x tan x e ce i sec ) 0 . H n x -gg (

11m n n Ex . 2 . Prove s ee x t an x 00 1 0 accordin as g ( ) , , , g 2 n > , 2, < 2.

EX ERC IS ES

Evalu ate the following expressions

5 5 0 . (a l ) x when x 0 . log sin x

—lOg—x sea hen x = 0 . 2 . w 6 . when x cot x sec 5 x 2

n i 2 2 Lx 3 . e x 00 . 7 x tan he x a . wh n . (a ) w n ex 2 a

t an x 1 tan x s ec 2 x e x 4 when x 8 . ( ) wh n tan 5 x 88 n v DIFFERENTIAL C ALCUL Us [O .

—l ﬂ 9 . e x 0 . e x wh n z wh n 1 . } e

lo x 1 3 2 . c sc x e x 0. g 0 \I wh n wh en x 0 . x7l 1 en 3? 1 ° wh en x z 1 . wh log x x 1

E u t on of the form 53 . val a i

Let the f u n c tio n u as s um e the form whe n

x a .

In rde ev a u a e h s ex es s n a e the o a hm o r to l t t i pr io , t k l g rit of

Th n both s ides . e iI i —& log u W e ) log ere ) 1

W e ) 0 Thl S x r s s1 u as s um s the m he n x = a an d can e p e o e for w , 0 4 be e valuated by the m ethod of Art . 9 .

If t he du ce d a ue h s ac n d n e d b m re v l of t i fr tio be e ot y , m then l og u m an d u e

° m 1 is n t i det ermi te bu t is e u a t o 1 . N O T . The or o a E f n n , q l ° For let ass um e the orm 1 e x (1 . , f wh n

Put it

Then log u d4x) 10 s whic h e quals z ero when x a ;

° e ce lo u 0 u e 1 . h n g ,

f m 5 4 . Evaluation of the or s ° c Le t be com e c whe n x a .

Put u

Then log u TC”) log 9°C”)

90 U n s o . v 54 DIFFERENTIAL CALCUL ( . .

8 m 3° 8 . cot x e x ( ) wh n O.

co I}x when x 0 wh n = e x 0.

2 1 o = W he n x 1 0 2 — x 1 x _ 1

se c x w e x I ” h n 1 2 . 2 s in e x 0 wh n 0 . C HAPTER VI

-MODE OF VAR IATION OF UNCTIONS OF ONE VARIAB LE _ F

h ha e m e h ds e xh n h m h r 55 . In t is c pt r t o of ibiti g t e arc o m ode a n u n c on s as t he ar a e a es all of v riatio of f ti , v i bl t k

i u s ion r m — oo 00 d a ues n s cc s be d s cus s . v l e f o to + , will i e

m h e n e n in 1 3 th u Sim ple exa ples ave b e giv Art . of e s e that can be made of the derivative fu n ction ( x) for this

u p rpos e . The fu n dam en tal prin ciple employed is that when x in c as es h u h t he a u e a n c eas es hr u h the re t ro g v l , i r t o g

a u e a if is s ve an d ha de creas es v l ¢ ( ) po iti , t t ’ h h throu gh t he valu e if d) ( a) is n egati ve . T u s t e qu es tion o f ﬁn din g whethe r d( x) in creas es or de creas es throu gh an assign ed valu e is redu ced to determ in i n g th e s ign of ( a) .

1 ind - et r . F wh he the function

¢ (x) x2 i crease s or de creas es t rou t he va ue s 5 2 = n h gh l q ( ) 1, t 1 10 an d s t ate at at va u e of x the u ct io ceases to i crease q ) , wh l f n n n

an d be i s t o decre ase or co vers e . g n , n ly

56 . Turn in lu f func n It s ha t he g v a es o a tio . follow t t val u es o f x at which ¢ (x) ceas es t o in creas e an d begin s to ’ de creas e are thos e at which gb ( x) chan ges s ign from p os itive to n egative ; an d that t he valu es o f x at which gb(x) c eas es to decreas e an d begin s to i n creas e are thos e at which (x) chan es it s s i n r m In t he m g g f o n egative to p os itiv e . for er cas e ( ) x is s a d t o as s hr u h a m ax im u m in the a e , I ( ) i p t o g , l tt r, a in i m m u m valu e . on v 1 92 DIFFERENTIAL CALCUL US ( . .

he t ur i v ue s f the u ti . 1 i d t a o c o Ex . F n n ng l f n n

2 3 3 x2 12 x 4 x + ,

an d ex hibit the m ode ofvariation ofthe fu nction bysketch

in g the c u rve y oS(x) .

2 Here 6 x 6 x 12 6 (x + 1 ) (x

’ e ce t x is e at ive e x ies bet ee 1 an d 2 h n q ( ) n g wh n l w n + , T an d po sitive for all othe r value s o fx. hu s gb(x) increases

ro m x co t o x 1 decreas e s rom x 1 to x 2 f , f z ,

: : a n d incre ase s fro m x 2 t o x 00 . Hence 1 ) is a

m ax im u m va u e of < ) x an d < ) 2 a m i im u m . l 1 ( ) , 1 ( ) n

FI G . f h ur The general form o t e c ve (Fig. 1 5) m ay be i e rred ro m the ast s tate m e t an d rom the o o i s im u a eo nf f l n , f f ll w ng lt n us value s o f x a n d y

x — oo — 2 3 , 2 , l , , ,

— cc 0 - 9 —1 6 y , , , ,

Ex 2 . Ex ibit he riat i f he . h t va on o t fu nc tio n

‘ e : x l i 9 ev) ( ) + . e e i sp c ally it s t u rning value s . 2 1 Since (x) ’ 5 1 (96

e ce x c a es si at x 1 bei h n ( ) h ng gn , ng e at ive e x l i it e e n x l n g wh n , nﬁn wh , T an d po s it ive whe n x 1 . h u s 2 FI G . 16 . is a m inim u m t u rning valu e of

The ra of the u ct io is as s o in Fi . 1 6 it a vertica ta ent g ph f n n h wn g , w h l ng at the o i t 1 p n ( ,

x . 3 Ex m r xim a an d m i im a t he u ctio E . a ine fo m a n f n n

— 1 i ) + 1 . 1 ’ 1 ere qS (x) i H 3 (x

e ce eve r c a e s s i but is h n n h ng gn , T r i i n o always po sitive . he e s accord ngly The ur e x ha a t u rning valu e . c v y c]; ( ) s ve rtical t a e t at the oi t 1 s i ce ng n p n ( , n ﬁ i i it e e = Fi . m s w x 1 . g F , 17 . nﬁn h n ( (g

94 D IFFERENTIAL CALCUL Us [Cm vi .

u a an d as s s h u h m ax m um u val e , p e t ro g a i val e ’ S m a ( a h is n a an d a h os i il rly, if t ( ) eg tive ( ) p itive, the n pas s es thro u gh a m in imum valu e

If a h an d h ha the s am s n ho e e ( ) ) ve e ig , w v r

s m a h m a he n a is n ot u n n a u b x . ll y be , t ¢( ) a t r i g v l e of q ( )

E i d he e h u x. F n t turning valu s oft e f nction

a Me ) : (as W e l ) 2a l ) (x so 1m 1) a (x 1) (x 1) 2(5 x

f e ce ( x becom e s z ero at x 1 and 1 it does not becdme H n I ( ) , fi, ; infinite for an y finite value of x.

Thus the critica va ues are 1 1 . , l l , g,

FIG . 18 .

e n x 1 h the t ree actors of t a e the s i n s Wh , h f k g an d he x 1 h t e becom e w n , h y t hus does n ot change s ign as x in creases t hrou gh l ; oS( 0 is n ot a t urning value of e x h the t re e actors of ave si s Wh n g , h f h gn an d e x h t e becom e wh n , h y t h us changes sign fro m t o as x increas es through 1 1 1052 is a m aximu m value of 58 VARIA TION OF FUNCTIONS 95

’ i all he x : l h the t ree actors of S x ave the s i s F n y, w n , h f q ( ) h gn an d whe n x 1 h they becom e t u s c a es s i rom to as x i creases t rou 1 an d h h ng gn f n h gh ,

0 is a minim um valu e of qS(x) . The deportm e nt of t he fu nctio n an d it s ﬁrst derivative in the v icinity

o f the critica va ue s m a be t abul ated as o o s m ic in c . dec . s ta d l l y f ll w , wh h , n for i creas i decreas i re s ective n ng, ng, p ly

x — 1 — h — 1 — 1 + h — h h 1 _ h 1 s s + ‘ ’ d) (x) 0 0 0 -I

n c . n . m ax . dec . c . . n ¢ (x) i i c de m in i c .

The general m arch of the function m ay be e x hibite d graphically by i t h oi re u t n t racing the curv e y ¢ (x) (Fig . u s ng e foreg ng s l a d observin g the following s im ultaneous valu es of x an d y :

z — oo _ x , 2,

00 , 9,

59 Second method of determining whether chan ges

n n n hr u h z r The n m h d m a be s ig i pas s i g t o g e o. followi g et o y e m ploye d whe n t he fu n c tion an d it s de ri v atives are c on tin u

o u s in the v i ci nity o f t he c riti cal val ue x a . ’ Su os h n x in c e as es hr u h the a u e a ha b x pp e , w e r t o g v l , t t g ( )

n r hr u h z r I chan ges s ig f om pos it i ve t o g e o to n egative . t s

chan e o m s v t o z e ro is a de creas e an d s o is t he g fr po iti e , ’ chan ge fro m z e ro t o n e gati ve ; thu s qS ( x) is a dec reas in g

u n c n at x = a an d hen ce it s der v a " x is n e a f tio , i tive 4 ( ) g

ti v e at x a . ’ O n the her han d if ) x chan e s s n rom n e a v e ot , d ( ) g ig f g ti

h u h z e r t o s v e is an n creas n un c n an d t ro g o po iti , it i i g f tio ( x) is po s itive at x a ; hen ce

’ The fu n ction ¢( x) ha s a maximu m va lu e when d) ( a ) 0 ” an d ct ( a ) is n egative ha s a min im u m va lu e when ’ ’ 0 an d gb ( a) is p ositive . n v 1 DIFFERENTIAL CALCUL Us (O .

It m a ha en howe v e r ha a is a z y pp , , t t ( ) ls o ero .

In h s cas e de erm n e het her x has a u n n t i , to t i w ¢( ) t r i g

a u e is n eces s ar roce e d t he h h d r e e a e . v l , it y to p to ig r iv ti v s ’ If x is a m ax m u m x is n e a ve u s e o re an sh qb( ) i , gb ( ) g ti J t b f v i

in an d n e a u s a e r the reas on ven a ov e but g, g tive j t ft , for gi b ;

he chan r m n a e t o z e o is an n creas e an d the t t ge f o eg tiv r i , ” n ge from z ero t o n egative is a decreas e ; thus cf) ( x)

h n r m n n t d c a ges f o i c re as i g o e creas in g as x pas s es throu gh a . He n ce (x) chan ges S ign fro m pos itive thro ugh z ero t o W n a e an d s as e e ha it s de va v c x eg tiv , it follow , b for , t t ri ti e t ( )

is n egativ e . Thus gS( a) is a m ax im u m valu e of ¢(x) if

0 0 n e a v e . Sim i ar a is , , g ti l ly, ¢( ) a m in im u m valu e of ¢(x ) if

an d pos itive . If it happ en that it is n ec es s ary to pro ceed to

h h d a es t o es u rn n u Th s till ig e r eriv tiv t t for t i g val es . e res u lt m ay then be ge n e raliz e d as follows

The fu n ction ¢( x) has a maximum ( or minimum) va lu e a t ’ ” m x a on e or m ore o the deriva tives ) a ) a ( a if f d ( ) , d ( ) , p ( )

van is h an d the rs t on e that does n ot van is h is o even order if ﬁ f ,

an d n ega tive ( or p ositive) .

n the critica va ues in the ex am e of Art . 58 b the eco d Ex . Fi d l l pl y s n

m eth od . (x 1 ) 2 (5 x 1 ) + 2(x 1) (x 1 ) (5 x 5 (x 1) (x ° 2 4 (5 x + 3 x 3 x ” c l 1 6 e ce is a m i imu m va ue of x h ( ) , h n n l qb( ) ,

t 0 e ce it is eces sar t o ﬁn d q , h n n y

2 x 2 x

S 24 e ce 1 is n eit er a m aximum n or a m i im u m q , h n ) h n v alue of

is m axim u m A ai 5 is n e ative , hen ce ( a g n , (i) g g . pg) value o f

98 C C H v 1 . DIFFERENTIAL ALCUL US [ .

’ In cas e gb ( a ) 0 the pre ce din g e quation s becom e

‘ ﬂ—ﬁ a ég r A gat .

01 2 ) 3 ¢< a h) (Pal ) h _ $h +

an d h c an be taken s o s m all that the ﬁrs t te rm on t he right

is n um e r ca ar er han e her Of the s econ d e rm s hen ce i lly l g t it t , ¢( a h) ¢( a ) an d ( a h) ¢( a) are b oth n egative when

( ) a is n e a e an d o h s tive hen a i I ( ) g tiv , b t po i w ( ) s p os itive . Thu s ¢( a) is a m axim u m ( or m in im u m ) valu e of qb(x) ’ hen a is z er an d a is n e a ve o r w ( ) o qb ( ) g ti ( pos itive) .

If s hou d ha e n ha a is a s o z er hen it l pp t t ( ) l o , t

V — —M 3 —fL—a l ¢< a h> h + ﬁ m

an d b the s am e e as n n as e re o s ha a y r o i g b fo , it f llow t t for m axim u m ( or m in im u m ) the re are the fu rther con dition s W ha a e u a s z er an d ha a is n e a v e o r t t ( ) q l o , t t ¢ ( ) g ti (

positiv e) .

Pr c e d n in h s wa t he en era c n c u s on s at e d in o e i g t i y, g l o l i t t he last article is e vide n t .

i of the recedi ex am e s can be s o ved b the e era 1 . W c Ex . h h p ng pl l y g n l rule here referre d t o ?

2 re stri ti im s ed u o x t at it s ou d Ex . . W hy was th e c on po p n ( ) h h l c a e si n b assi t rou z ero rat e r t a b assi t rou h ng g y p ng h gh , h h n y p ng h gh inﬁ n ity

The m x im and minim of an con inuous func ion 6 1 . a a a y t t

It has e n s e en t ha the m ax m u m an d occur alt ern ately. b e t i 60 VARIA TI ON OF FUNCTIONS 99 m in im u m valu es of a ration al polyn omial o ccu r altern ately

h n th ar a e is c n nu a n c eas ed or d m n she d . w e e v i bl o ti lly i r , i i i This prin ciple is als o tru e in the cas e o f e ve ry c on tin u ou s

un c n o f a s n e v ar1able . let b t wo f tio i gl For , ¢ ( ) be m ax im um valu es of in which a is s u pp os e d les s than

Th n h n a h th u n n i d ea n h n 6 . e e x e c s ec s , w , f tio r i g ; w e — x z b h the fu n ct l on 1 s 1 n creas in h e n a n s u fﬁ , g, b i g t ke

B u in n om a decr a cien tly s m all an d pos itive . t pas s i g fr e s in an n c e as n s a e a con n u u s u n c on m u s at g to i r i g t t , ti o f ti t , s m e n erm e d a e a u x chan e om de cr as n o i t i t v l e of , g fr e i g to

m m n n cre as n ha is m u s as s hr u h a n m u . ce i i g, t t , t p t o g i i He , b etween two m axim a there m u s t be at leas t on e mi n im u m . It can be s im ilarly prove d that betwe en two m in im a the re m n m x m u m ust be at le as t o e a i .

im ﬁ c on h d not c it c l lu s . 6 2 . S pli ati s t at o alter r i a va e The

o rk n din t he c t ca v a u es of the a a e in t he w of ﬁ g ri i l l v ri bl , cas e an en u n c n m a en s m d b m an s of y giv f tio , y oft be i pliﬁe y e f h - O t e followin g s elf e vide n t p rin ciples .

1 . hen 0 is n de en den x an a u e Of x ha es W i p t of , y v l t t giv a tu rn in g v alu e t o 0 4> ( x) gives als o a turn in g valu e t o

Th n an d con ve rs ely. es e two tu r in g values are of the s am e or Oppos ite kin d accordin g as c is pos itive or n egative .

n 2 . A y valu e of x that gives a tu rn in g valu e to c qS(x) giv es als o a t u rn l n g v alu e of the s ame kin d t o an d c on vers ely.

3 . hen n is n d en den x an a u of x ha s W i ep t of , y v l e t t give a tu rn in g valu e t o [95 gives als o a tu rn in g v alu e to

an d c n r h h r h u rn n a u re o v e s ely. W e t e t es e t i g v l es a

the s am e or os e n d d n ds on the s n n an d of opp it ki epe ig of , als o on the s ign of [qt 1 00 D IFFERENTIAL CALCUL US

EX ERC IS ES

i d the crit ica va ues of x in t he o o i u ctio s dete rmi e the F n l l f ll w ng f n n , n at ure f th u tio e n btai t he r h o e c at ac a d O a of t e u ction . n f n n h , n g ph f n

J 2 — x x 6 . ( 5 .

3 — 2 — — 2 - 3 2 = z . u 2 x 1 5 x + 36 x 4. 7 . a 5 + 1 2 x 3 x 2 x .

3 u = x — 1 3 x . ( ) ( J 4 . u siu x + cos x. ? 8 9 . u sin x cos x. 3

a — 2 x

1 o t at a u adratic i t e r u i a h n 0 . Sh w h q n g al f nct on alw ys as o e m axi

m um or o n e m i im um but ever bot . , n , n h

u ic i s in r 1 1 . Sh ow t hat a c b ntegral function ha ge ne al both a m ax i

m u m an d a mi im u m va ue bu t m a have n eit er. n l , y h

1 2 ow t at he unctio x % has n either a maximum n or a . Sh h t f n ( b) m inim um value .

m The r c r bl m in m im n min m . 6 3 . Geo et i p o e s ax a a d i a the ory of the tu rn i n g valu es Of a fu n ction has importan t application s in s olvin g problem s con ce rn in g geom etric

m x m a m n m a i . e . the d m n a n the ar es a i or i i , , eter i tio of l g t or the s m alle s t valu e a m agn itu de m ay have while s atis fyin g

r certain stated geomet i c con dition s . The ﬁrs t s tep is to expres s the m agn itu de in qu e stion

» algebraically. If the res u ltin g expre s sion c on tai n s m ore

han on e a the s d c n d n s l u n sh n u h t v riable , tate o itio wi l f r i e o g

e a n s n h s a a s s o ha the h s r l tio betwee t e e v ri ble , t t all ot er m x d r e The ex ss n t o ay be e pre s s e in te m s of on e . pre io

m x m z d m n m z d n hu s m ad a un c be a i i e or i i i e , bei g t e f

n a s n can a d b the ced tio of i gle variable , be tre te y pre in g rules .

1 02 A On v 1 DIFFERENTIAL C LCUL US [ .

Ex . 3 . Find the area of the greate st rectangle th at can be ins cribe d

n i n e li i a g ve l pse . An ins cribe d re ctangle will evide ntly be s ym met ric with re gard t o the principal axes of the e i ll ps e . Let a 6 de ote the , n lengths of the s emi-ax es

0 A OB Fi . let 2 x , ( g , 2 31 be the dim en sions o f an i scribed rect a n ngle . The n the area is

FI G . 20 . 4 x 1 u 9 ) ( ) in ic the variab es x m a be re arded as the coordi ates of the wh h l , y y g n vertex P an d are t ere ore s u b ect t o the e u ation of the el i se , h f j q l p

x2 2 2 2 1 . ( ) 52 55

It is geom et rically evident th at there is s om e pos ition of P for which the i scribe d rect a e is a m axim u m n ngl . The elimin ation of y from by m ean s of gives the fu nction of b x to e m axim ized,

B Art . 6 2 the critica va ues of x are n ot a tered if this u ctio be y , l l l f n n 9 41 divided b th e co st a t an d t e s u are d . e ce the va ues of x y n n , h n q H n , l a hic re der u a m ax im u m ive a so a m axim um va u e to the u ctio w h n , g l l f n n

2 2 2 2 4 (I) (x) x (a x a x x .

2 3 2 2 ’ x 2 a x 4 x 2 x a 2 x qS ( ) ( ) ,

2 2 2 a 12 x ;

e ce b t he u s ua te s ts the critica valu es x re der an d h n , y l , l i : n f2 di alu es Of are t ere ore the area u a m axim u m . The corres o v h f , p n ng y give n by an d the vertex P m ay be at an y of the four poin ts denoted by VARIATI ON OF FUNCTIONS 1 03

ivi in eac case the s ame m aximu m i s cribed rect a e ose g ng h n ngl , wh dim e s io s are a \/2 b\/2 an d ose area is 2 ah or al t at o f the n n , , wh , h f h r ci cu m scribe d rectangle .

x 4 i th r t i r i i E . . F nd e g e a est cyl nder th at can be cu t f o m a g ve n r ght

co e ose ei t is h an d the radius of hose bas e is a . n , wh h gh , w Le t the cone be gene rat ed by t he revolu tion of t he t riangle OAB an d th i s rib c i d r (Fig . e n c ed yl n e be ge nerated by the revolution of the rectangle AP. A AB n d le the L et 0 h, a , a t

coOrdin at es Of P be x . T e the ( , y) h n - 2 - fu nction t o be m axim iz e d is 71 y (h x) y a s ubj ect to the relation _ x h This expression becom es 2

The critica va u e of x is h an d V l l s ,

EX ERC IS ES O N C HAPT ER VI

1 Thr u a i i it i an a e draw a s trai t i i . o gh g ve n po nt w h n ngl gh l ne wh ch u t e t is r b e m b t he m eth shall c off a m inim u m t riangle . Solv h p o l y od of

the ca cu u s an d a so b eom etr . l l , l y g y [Take given lines as coordinat e ax es ]

2 . The vo u m e o f a c i der bei consta t ﬁ n d it s orm e the l yl n ng n , f wh n r e ntire su face is a m inim um .

2 3 A n l r c r b u i o tai . recta u a ou t is t o e b t so as to c a ive are a 0 g , l n n g n , r i i f r n e f it s id i d its an d a wall al e ady c on stru ct ed s ava lable o o o s es . F n dimen s ion s s o t hat the expe n se incu rred m ay be the least pos s ible .

4 ube is ive . H ow do . The su m of t he su rfaces of a s phere an d a c g n their volum es compare wh e n the s um Of t heir volum es is a m inim u m

5 axim u m arabo a ic . What is the length of the axis of the m p l wh h can be cu t rom a ive ri t circu ar co e i ve t at the are a of a f g n gh l n , g n h parabola is e qu al to t wo t hirds of the pro duct of it s base an d altit u de

6 an be i scribed in a iven . D et erm ine the greate st rect angle which c n g triangle who se bas e is 2 b and whos e altitude is a. n v 1 6 3 04 O . . 1 DIFFERENTIAL CALCUL US ( .

he m f i ir f cir e 7 . T fla e o a candle s d ectly over the cent er o a cl whos e u f he ﬂa me ab e t h radius is 5 inches . What o ght to be t he height o t ov e plane o f t he c ircle s o as to illum inate the circ umference as m uc h a s pos s ib e su os i the i te sit o f t he i t to var direct as the s i e o f l , pp ng n n y l gh y ly n t he a e u de r ic it s tri e s the i lu mi at ed s ur ace an d i verse ngl n wh h k l n f , n ly as t he square o f its dis tance fro m t he illuminated point

8 A u 1 of a teb ard 3 i c e n an d 1 4 i c e . rect ang lar p ece p s o 0 n h s lo g n h s r i d t he ide of t is s u are wide has a s quare c ut ou t at e ach corne . F n s h q so that the re m ainder m ay form a box o f m aximu m conte nt s .

in t he a titu de o f th r t i der of reates t vo um e in 9 . F d l e igh cyl n g l s cribed in a sphere wh ose radiu s is r.

1 0 . T rou t he oi t a b a i e is dra su c t at the art i t er h gh p n ( , ) l n wn h h p n

’ i d its cepted betwee n t he rectangular cobrdin ate axes is a m inim um . F n le ngth .

1 i t he t i r 1 . G ven slan he ght a of a ight cone ; ﬁ n d it s altitu de when the volu m e is a m ax im um .

1 2 i r h r d u of ircu ie f . I . T e a s a c la p ce o paper is r Find the arc of the s ector whic h m us t be c ut fro m it so th at the re m aining sector m ay form t he co nvex s urface of a pone of m ax im u m volu m e .

3 r r 1 . Find relation betwee n lengt h of circula a c an d radiu s in orde r he r a o f a ir u ar s ect or o f i e erim ete r u d be x t h at t a e c c l g v n p s ho l a m a imu m .

1 4 m h r f . O n the i e o m t e ce te s of two s eres o rad1 1 r R ﬁ n d l n j g n ph , , the dist an ce of the point from t he cent er of the ﬁrs t sphere fro m which the maximu m of s phe rical surface is vis ible .

1 5 D s ribe a circ e i e t r n i ir l th . e c l w th it s c n e o a g ven c c e s o at the h t r e t h len gth of t e arc in e c pted wi hin t e given circle sh all be a m aximu m .

1 06 on vn DIFFERENTIAL CALCUL US [ .

an o her u n c n o f the m e . For n s an ce le t th or y t f tio ti i t , e

u an of an e e c r c c u rren be 0 at the m e t an d 0 A q tity l t i t ti , + 0

Th n t he v ra a t at t he tim e t At . e a e ge r e of change of ou r A0 ren t in the in te rv al At is ; this is the average in creas e 2 7

- An d in c urren t u n its p er s econ d . the a ctu al ra te of change at the in stan t den oted by t is

At é O At di

This is the n umbe r o f c u rren t -u n its that wou ld be gain ed in the n e xt s e con d if the rate of gain were u n iform from the m h m t t e t 1 . S n ce b Art . 14 ti e to ti e + i , y ,

, dx dt IE hen ce m easu res the ratio of the rates of change of g an d

O f x .

It follows that the res u lt of diﬁ eren tiatin g

m ay be written in e ither of the forms

293 f < x> . dt dt

The e m is O e n c on n n an d m a a s latt r for ft ve ie t, y l o be Obtain ed directly from ( 1 ) by differe ntiating both s ides

I m h chan e with re gard to t . t ay be read : t e rate of g of ’ h g is f ( x) times the rate of c an ge of x . n n t he us ra n a m n n P let it s Retu r i g to ill t tio of ovi g poi t , dx h n m as u s the c oOrdin at es at tim e t be x an d g . T e e re dt - r rate of chan ge of the x coo dm at e . Sin ce velocity has been deﬁn ed as the rate at which a poin t RATE S AN D DIFFEREN TI AL S 1 07

2 is m n the a 1 3 m a be ca d the c t h ch the ovi g, r te y lle velo i y w i dt

n has in the d c n the x- x s or m r e poi t P ire tio of a i , , o e bri fly, th - m n en the c e x co po t of velo ity of P .

h n 1 5 h h t n It was s ow on p . 0 t at t e actu al velocity a a y in s tan t t is equ al to the s pace that wou ld be pas s ed over in a u n m ided the oc e e u n m du r n it of ti e , prov vel ity w r ifor i g

h u n A d n h - a . cc t e x c m o n en of e c t t it or i gly, o p t v lo ity fi:

h i 22 h ch P m ay be repres en ted by t e dis tan ce PA ( F g . ) w i wou ld pas s over in the dire ction of the x -ax is duri ng a u n it

O f tim e if the v elocity re main ed un iform . dv S m ar is the c m n n the e c P an d i il ly g o po e t of v lo ity of , dt m ay be repres en te d by the dis tan c e PB . ll The velocity ﬁ; of P alon g the c u rve c an be r epre s en te d b the d s an c P 0 m a u r d on the n en n e the y i t e , e s e ta g t li to cu I i rve at P . t s eviden t that PC is the diagon al O f the rec

an e PA PB t gl ,

2 S n ce P0 PA2 PB 2 i , it follows that

(4) FI G . 22.

Ex . 1 . If a oi t describe the s trai t i e 3 x 4 5 an d p n gh l n g , i crease h u its er s eco d ﬁ nd the rates of i crease of an d of s . n n p n , n g

y 2 i x:

3 dx hence dt 4 di

dx When dl 1 08 D IFFE RE N TI AL CAL H V C L C . II U US [ .

E 2 2 x . . A i po nt des cribes the parabola y z 1 2 x in su c h a way that wh e n x 3 t he abs cis s a is in cre asing at the rate of 2 fe et per s econd ; at at rate is t e i cre asi i d also the rate of i cre se f wh y h n n ng F n n a o 3 .

Si ce 12 x n ,

d dx t e 2 y 12 h n g , dt di

( dx 6 dx 1g 6 , ’ dt g dt vm dt

e ce e x z : 3 an d 2 it o o s t a t i : , w , , f w j : 2 . h n h n Z? ll h tll

, C 3 E - i i/ 9- e ce eet er seco d . I , h n f p n dt (d l Y (d l Y dt

3 A . er i Ex . p son s walking t oward the foot o f a to we r o n a horiz ontal plane at t he rat e of 5 m ile s per h ou r ; at wh at rate is he approaching the to ic is 60 eet i e he is 80 ee t rom t he bott om ? p, wh h f h gh , wh n f f Let x be the dist a ce rom t he oot of t he t o er at t im e t an d the n f f w , y , t r h dis ance f om t e top at the sam e t im e. Then

2 2 272 6 0 y , dx gg an d g dl dx W e x is 80 eet is 100 e et e ce if _ _ is 5 mi es er our h n f , g f ; h n l p h , dt Ch is 4 m iles per h ou r .

Abb d n ion for s . hen as in the a 6 5 . o reviate tat rate W , bove e x am s a m de r a ve is a ac r of each m em er of an ple , ti e iv ti f to b e u n is u s u a c n en e n t o r e in s ead of the q atio , it lly o v i t w it , t dx ly s m s g— t he a ev a on s dx an d d for the a es y bol , bbr i ti g, r t dt dt Thus the es u o f of chan ge of t he variables x an d g . r lt differen tiatin g (1 ) m ay be written in e ither of the fo rm s

g 1 4 x f ( ) . A.»

d x y ’ é—- x s

1 1 0 D IFFE R E N TI A AL n II L C C L O . V U US ( .

2 A Ex . . r i ve t cal wheel of radiu s 10 ft . is m aking 5 revolution s per e b ut x i i d t h r z s cond a o a ﬁ ed ax s . F n e ho i ontal an d vert ical velocit ie s Of ° a oi t o n t he circ um ere ce s it u at ed 30 rom the riz p n f n f ho ontal.

Si ce x 1 0 cos 0 10 s in 9 n , g ,

t e dx 10 s in 0d9 d 1 0 cos Od . h n , g o

B ut dd 1 0 7; radia s er seco d n p n ,

e ce dx s in eet e r seco d h n 0 f p n ,

n a d dg cos 0 feet per second .

3 . T t h Ex . race e c hanges in the horiz ontal an d vertical velocity in a m t e re o ut io co ple v l n .

. Th 6 6 Differentials oft en s ubs titut ed for rates . e symbols

dx d hav e e en de n e d a as the s chan x , g b ﬁ bove rate of ge of

d n d an g per s ec o .

So m e m es h h m a c n n n d ti , owever, t ey y o ve ie tly be allowe

s an d for an n um ers ar e or s m a ha are to t y two b , l g ll , t t pro

or n a t o h s e ra s the u a n s e n ho m en e u s p tio l t e te ; eq tio , b i g og o

in hem n ot be affe c d . It is us ua in s u ch cas s t o t , will te l e s pe ak of t he n um bers dx an d dg by the m ore gen eral n am e ‘ ‘ ; of d iﬁ eren tials ; they m ay the n be either the rates them

s e ves o r an t wo n u m s in the s am a . l , y ber e r tio This will be es pe cially c on ven ie n t in p roblem s in which m the tim e variable is n ot expli citly en tion ed .

° ° 1 e x i creases rom 45 to 45 ﬁ n d the i crease of Ex . . Wh n n f n lo s in x as s u m i t at t he ratio of the rate s O f c a e of t he u c tio g10 , ng h h ng f n n

an d t he variable rem ains s ensibly co nst ant th rou ghou t the s hort in terval.

x . 4343 dx. Here dg logwe cot xdx . 4343 cot dx

' 4363 radia s. Let dx 15 .00 n

T (l . 001895 he n g ,

xim ate i crem e t of lo sin x. which is the appro n n g10

° lo sin 45 lo 2 . 1 50515 B ut g10 5 g ,

° ’ lo sin 45 15 .148620. therefore g10 6 5 R A TE S AN D D IFFE R E N TI ALS 1 1 1

2 ’ Ex . 2 . Ex a di lo sin x h as far as h b Ta or s t eorem p n ng g10 ( ) y yl h ,

an d t e u tti x . 785398 h . 004363 S ow at is t he error m h n p ng , , h wh ade b e e ti th t ird te c e rm as was do e in Ex . 1 . y n gl ng h , n

° ° Ex . 3 . e x varies rom 6 0 t o 6 0 ﬁ n d the i crease in sin Wh n f n x .

Ex . 4 . S o t at lo x i cre ase s m ore s o t a x e x lo e h w h glo n l wly h n , wh n g10 ,

t at is x .4343 . h ,

Ex . 5 . Two sides a b of a tria e are m easured an d a so the iii , ngl , l clude d a ngle 0 ; ﬁ n d t he e rror in t he com pu t ed lengt h of the t hird side

c du e t o a s m a error in t he obs er ed a ll v ngle 0 . 2 2 2 D iffere tiate the e u atio c O b 2 a b co s 0 re rdi a b as [ n q n , ga ng , constant ]

. A e e i i i r Ex 6 . v ss s s a n o t e t at the rate f 10 m i es er u r l l ng hw s o l p ho . At what rat e is s he m aking north latit u de ?

2 7 I he arabo a 12 x ﬁ n d the oi t ic t he r i t Ex . . n t at o d a e p l y , p n wh h n

an d abscis sa are increas ing e qu ally.

t f the rst uadra t d es t he a e i re 8 . At hat ar o c as e Ex . w p ﬁ q n o ngl n twice as fas t as its s ine ?

9 i d the rate of c a e in the area of a s u are e the side Ex . . F n h ng q wh n

r s ec d . 6 is increas ing at a ft . pe on

3 1 In the u ctio 2 x 6 w at is the va ue of x at the Ex 0 . . f n n y + , h l point where g increases 24 tim es as fas t a s x ?

/ 1 1 A circu ar at e of m et a ex a ds b e at s o t at it s diam et er 0 Ex . . l pl l p n y h h increases u niformly at the rat e of 2 inch es p er s econd ; at wh at rate is the surface in cre asing when the diam et er is 5 inch es ?

8 2 1 2 at i the va u e of x at the oi t at ich x 5 x 1 7 x Ex . . Wh s l p n wh 8 an d x 3 x ch ange at the sam e rate ?

1 3 i the rate of c an e of t he ordi ate . i h oi t s t c Ex . F nd t e p n a wh h h g n 2 g x3 6 x 3 x 5 is e qu al to the rate of change of the s lope of t he t n h a gent to t e curve .

Ex . 1 4 . The relatio betwee 3 the s ace t rou ic a bod a s n n , p h gh wh h y f ll , 2 an d t the tim e of a i is s 1 6 I s o t at t he velocit is e ua , f ll ng, ; h w h y q l t o 32 t .

The rat e of c hange of velocity is called acceleration ; show that the acce eratio l n of the falling body is a constant . 1 1 2 D I FFE R E N TIAL AL on V II 6 C C L U . 6 U S [ .

’ 1 5 E . A b d m s cordi t o t he 4 x . o o ve ac law 3 y ng cos (n t e) . Show t h at its acceleratio n is proportio nal t o the space throu gh which it has

m ove d . V 1 b r 6 . If od be ro ected u ds in a u Ex . a y p j pwa a v c u m with an initial ve ocit v t o at e i t i it rise an d at i be t he t im e of l y 0, wh h gh w ll , wh w ll as cent ?

‘/ 1 7 . A bod is ro ected u ards ith a ve ocit of a eet er Ex . y p j pw w l y f p A t er at tim e i l it ret ur se cond . f wh w l n

1 8 I A be t he area of a circle of radiu s x s o t h th i Ex . . f at e c rcum , h w

5 i m tri fere nce is Interpret t h s fact geo e cally. ?x

2 2 1 9 A oi t de scribi the circ e x 25 . asses t r u Ex . p n ng l y p h o gh

3 4 it a ve o cit o f 20 eet er s eco d . in d its com o e t ve oci ( , ) w h l y f p n F p n n l ties parallel to t he axes .

1 14 D IFFE RE N TIAL CALC L US On v r U [ n .

6 8 . P I artial differentiation . f in the fu n ction

= x z f( , y) a x d u n hen ﬁ e val e 21 be give to g , t

x f( ’ 9 1) is a un c n of x n an d the ae of h n in f tio o ly, r t c a ge z caus ed by a chan ge in x is expres s e d by

dz ( 1 ) dz

dz in h ch i s n d on the s u s n ha i n w i obtai e ppo itio t t g s co stan t . dx

To i n dicate this fact withou t the qu alifyin g v e rbal s tate m n u n 1 n in the m e t , eq atio ( ) will be writte for

( 2) dx

? The s ymbol rep res en ts the re s ult Obtai n ed by diffe r 3x e n t iatin x h re ard x the v a r a e e n re a d as g wit g to , i bl y b i g t te a con stan t ; it is called t he p a rtial d eriva tive o f z with regard to x . 1 1 h l Fr m the d n n d ffe en a n Art . t e a o eﬁ itio of i r ti tio , , parti derivative is the re s ult O f t he i n dicated operation

— Ax é o ex A23 9 S m a the s m 3 e res n s the su O a n e d i il rly, y bol r p e t re lt bt i b diﬁ ere n t iatin 2 h e a d the ar a le x e n y g wit r g r to g, v i b b i g treated as a con s tan t ; it is called the p arti al derivativ e of 2 with regard to g . The partial de rivative of 2: with regard to g is accordi n gly t he res ul t o f the i n dicated operatio n F UN C TI ON S OF T W O VAR IAB LE S 1 1 5

“ — dz lim 93 “ A x f( s z I Q) f( 9 3) M O By Ar

' ' ﬁ dx 1s ca d t he a rtia l x -d i eren t ia l z an d lle p ﬁ of , 6x

z ' ‘ 1 h - dg s calle d t e p a rt ia l y d iﬁ eren tial of z . 53;

m c the u a n Geo etri ally, two eq tio s

il = K e y) , y y1 d n the cu r e s c n the s u ac x m ade eﬁ e v of e tio of rf e z f( , g) z h f b the an . The de a e deﬁn e t e s e O y pl e g gl riv tiv s lop 6x n h r t he tan gen t li e t o t is cu ve .

S m a hen x has a v e n c n s an a u x x the i il rly, w gi o t t v l e, 1 , 5—92 partial deriv ati ve is t he expres s ion for the mpe o f t he 39

an n the c u e cu t m t he s u c x b t ge t to rv fro rfa e z f( , g) y = h lan x x . t e p e 1 The e qu ation s of thes e two tan gen t lin es at the poin t

x 2 are ( 1 , 2 1 , 1)

z = z — z —l w—x v vp 1 ( l ) . 8x 1

az $ = x Z — z ——l 1 , 1 { y 9 1) a 9 1 an d hen ce the e qu ation of the plan e con tain in g in ters ectin g lin es is 6 z — a ﬁ — i e O v1) ;1

The plan e is called the tan gen t plan e to the s u rface

x at the n x z . z f( , g) poi t ( 1 , gl , l ) 1 1 6 Us Ca vn r DIFFERENTIAL CALCUL [ .

EXE RC IS ES

N/ 1 4 2 2 3 Q au . Give x 3 x 7 x rove t hat x ﬁ — n u y , p + y 4 a . 6 x dy

- 1 3 Q —au 2 . Give t an s o t at x 2 n u , h w h y O. x 6 x By

t u y i - B 3 u e ) ; ﬁ n d i 6 27 By

6 u 6 “ 4 sin x __ . u y; ﬁ n d + am By

2 § 5 l V ﬁ . . u og (x + x ﬁ n d n + y 6 90 By

u log (t an x t an y t an z ) ; show t hat

§ﬂ 523 s sin 2 x + sin 2 y s in ? z ax 831 dz

Q9 123 2 u log (x y) ; s how t hat 6 27 631 6 “ Jr

6 9 . T iff n If h x an d otal d ere tial . bot y be allowed to vary

in the un c on 2 x the s u s on ha n u f ti f( , y) , ﬁr t q e ti t t at rally

ar s s is d m n the m ean n th d f n i e to eter i e i g of e i fere tial of z .

x Let f( 1 ,

an d 2 A2 x Ax A 1 f( 1 , y, g)

be two val ues of t he fu n c tion co rres pon din g to the two pairs

r a Ax a u es o f t he a es x an d x A . of v l v i bl 1 , 9 1 1 , 9 1 y The diffe re n ce

AZ = x A5” A — x f( 1 , 3h y) f( 1 , 591)

m a a d d as c m s d a s the s a y be reg r e o po e of two p rt , ﬁr t p rt

th n e m n h ch s h n x chan s m x b e in g e i cr e t w i z take w e ge fro 1

x Ax h m a n s c n s an an d the s ec to 1 , w ile y re i o t t (y yl ) , on d part being t he addition al in crem en t which 2 takes when

1 1 8 on . VIII DIFFERENTIAL CALCUL US ( .

Equation (1 ) m ay be written in t he form

62 az d2 — dx — d + y’ 6 x ay from whi ch the followin g the orem can be s tate d : the tota l diﬁereu tia l of a fun ction of two variables is equ al to the su m of the p a rtia l diﬁeren tial The s am e m e tho d can be directly applied to fu n ction s of

h e e or m e a s . Thu s a un n h t r or v riable , if z be f ctio of t e

ar a s x u v i ble , y, , x u ¢( , ) ,

ab ’ c i a x -—ﬁ d Qé i y + du . dx ay au

2 2 3 EX . 1 . Give ax bx 0 x e n z y y y,

2 2 2 t dz u 2 bx 3 cx dx 2 ax x hen ( p/ y ) ( y 6 e) dy. — y y l !’ Ex . 2 . Give x t e d z x dx an d d z x lo x d . n z , h n z y , y g y

y- l ?! H ence dz yx dx x log x dy.

x d dx - 1 g y EX . 3 . Give u t an s o t at da n , h w h z 2 x x y

i the c ar ct eris t i e u tio of er e 4 . Assu m a c a a ct as v t Ex . R ng h q n p f g , p , in ic v is vo u m e res su re t abs o u te t em erature and R a co st a t wh h l , p p , l p , n n ,

e x res s eac of the di ere tia s d v d dl in term s of t he ot er two . p h ff n l , p , , h

2 ? 2 5 A art ic e m ove s on t he s erica su r ace x z Ex . . p l ph l f y ° a vert ic al m eridian plane inclined at an angle of 6 0 t o the zx plane . If t he x-com o e t of its velocit be 1 a er s eco d he x a ﬁ n d p n n y 1 6 p n w n % , - - t he y component an d the z compo ne nt velocity.

Since t he n

B ut si ce dx 1 a an d the e u atio of the ive m eridia a e is n 1 6 , q n g n n pl n

x tan e ce d dam/5 V5 an d T ere ore y h n y , y h f

d”; £3 gﬁ ' n feet per second . 2 1 5 6 9 FUNCTIONS OF TWO VARIAB LES 1 1 9

n of differ n i l . The s u s the c d 7 0 . La guage e t a s re lt of pre e ing articles m ay be s tate d thu s The parti al z -differen ti al du e to a chan ge in x is e qu al to

- h r - the x diffe ren tial m u ltiplied by t e pa tial x de rivative . The p artial z-differen tial du e to a chan ge in y is e qu al t o

- n a m u e d b the a a -d r the y diffe re ti l ltipli y p rti l y e ivative . The total z -differen tial is equ al to the s u m of t he p artial

- z differe n tials . O n e advan tage of writin g the equ ation in the diffe ren tial form is that it m ay be divided when n eces s ary by the dif f r n t ial an he a a 8 h ch x an d m a be e e of y ot r v ri ble , to w i y y

a d an d h n em e m e n ha t he a d f e rel te , t e , r b ri g t t r tio of two i f r en t ials a s m a x s s ed as d t he (or r te ) y be e pre a erivative , equation would bec om e

dz az dx az + dg ds 6x ds 6g ds

In a cu a be n ot n d n den bu t is u n c n p rti l r, if y i epe t , a f tio of x h n 3 m a be ch s n as x s e an d the c d n u a , t e y o e it lf, pre e i g eq tion becom es d z 6 z z Q__ dg . dx 6 x ay dx

If the u n c n a n e n x an d n f tio al rel tio b twee y be give ,

e a M ) , d z h n the s m su u d n d h h e t e a e re lt wo l be obtai e , w et er b dx de erm n ed b the es n m h d s m n a e d t i y pr e t et o , or y be ﬁr t eli i t from the relation fv f( , y) , an d the u n u n ff Th res lti g e q atio be di eren tiated as to x . e m e h d t o of this article frequ en tly s horten s the pro ces s . 6 z d z I i _ _ t s here well to n ote the differen c e between an . dx dx The former is the partial derivative of the fun ction al ex 1 20 US C r] . V III DIFFERENTIAL CALCUL [ .

es s n for z h re a d x on t h pr io wit g r to , e s u ppos ition that y

Th a h is c o n stan t . e l tte r IS t e total derivati v e o f z with

e a d x hen acc u n is e n th r g r to , w o t tak of e fac t that y is its elf a fun ction of x

2 2 z Ex . 1 . Give z x lo x ﬁ n d n + 31 , y g ; Zx

( lz

he nce

Ex . 2 . If z 1 sho t at , w h

h 1 Diff en i ion of im lici func ion . If in t e e a on 7 . er t at p t t s r l ti

x z as s um e d t o be c n s an hen f( , y) , be o t t, t

hen ce

Qf. 8x from W t h

In all s u ch cas es either variable is an impli cit fun ction of the h an d hus the as u a n u n s hes a ru e ot er, t l t eq tio f r i l for

fi n din g the derivative of an implicit fu n ction . d 3 3 a . Ex . 1 . Give x 3 ax z e fi n d n + y + y , dx 2 d x a 2 2 y y Si ce 3 x 3 a 3 3 a x — 0 _ — n ( y) + ( g ) ’ 2 gZ dx y ax t 2 — - Ex . . Z x y E 3 I 2 2 fi n d — X. . f a x + 2 hxy + by dx

4 4 is . Ex . 4 . Give x 0 ﬁ n d n y , dx

1 22 OH V III. DIFFERENTIAL CALCUL US [ .

c n s an . The n b t he the orem of m r o t t y ean valu e (A t . the in crem e n t of the fun ction is e qu al to the in cremen t of the variable m u ltipli ed by the de riv ative taken for s ome

a ue n rm e d a e e en x an d x h ha is v l i te i t betw ; t t ,

x h — x : li — x 0h f( + , y) f( , y) f( + , y) . ei

N x le t chan e k x m a n n c n s an an d e t y g to y , re i i g o t t, h Th take the in crem en t of the fun ction on t e left . e n by the he m of m ean a u e a ed — x 9h as a t ore v l ppli to f( , y)

un c n h the n c em n k f tio of y wit i r e t ,

h k — w fv h [f , y ) f( , y [f( , y) fee 20] a a— Mh f(w + 0h y + 0 k ay 5x Th N ow let thes e in crem en ts be given in revers ed order . en — -f( rv + h y) ] f(x w] i k i x e i 0 7c M f< + y + 3 ) , dx dy hen ce 8 6 h 0 k = 0 h 0 h: , y + ) f + 2 , y + 3 g; %

This relation is tru e for an y val u es of h an d k for which all the fun cti on s m en tion ed are c on tin u o us .

hen h 79 a ach z W , ppro ero ,

x 9h 9 k an d x 0 h 0 70 , y 1 , 2 , y 3 a ach x an d ppro , y,

93 all 9 k x 9 k f( , y 1 ) , f< 2 , 3/ a ach x an d s m a the d i s h n c ppro f( , y) ; i il rly for er vative ; e e d 6 “ x $ f( , y) f( 9 y) , (Ty 5; a ay

s n c x : z or, i e f( , y) , z z a z a z dy dx dx 6g FUNCTIONS OF T W O VARIAB LES 1 23

r It s d r ct ha u n d c res n d n con Co . follow i e ly t t er or po i g dition s t he orde r of differen tiation in the higher partial derivatives is in diffe ren t .

3 3 s 8 2 6 z a z 2 dx dy dx sas sy By 6x

EX ERC IS ES

2 2 3 Verify t hat whe n u x y 623;

3 g 8 u d u V 2 a 2 . Veri t at , w e u x x . fy h 2 h n y y dx 69 dx

z (Va d u Veri t at e u lo 1 x fy h wh n y g ( y) . dx 63; dy 6 3:

4 . In Ex . 3 are t ere an exce tio al va u es of x for hic the p h y p n l , y w h relatio n is n ot t ru e ?

2 ﬁ 5 . Give u x veri t he orm u a n ( . + y ﬁ, fy f l 2 2 d a 8 u (Wu 2 + 2 ” x 2 y 2 6 93 6 95 0e dy

3 6 Gi e u x s o t at the ex re s sio in t he e t m ember . v n ( h w h p n l f 1 5 is u £ 4 of the diﬁ erential e qu ation in Ex . eq al to

2 2 2 l 7 i rove t at l— - — , p h ig gjénZ

2“ 2“ ' 8 2 6 u x t n ro e t t 8 . Give sec y + a) a y p v a a n ( ( h 2 dx dy

4 4 4 V 6 a 8 21 d a 9 1 n CO erI t h . Gwen s x S y v f at y 3 2 2 6 x dx By 6 96 c9x (99 2 1 t 0 . Given u (4 ab prove t hat (i £1 d C H APTER IX

CH ANGE OF VARIAB LE

4 . n 7 Interchange of depe dent and independent v ariables .

If be a c n n u us un c o n x d n d b the u n y o ti o f ti of , eﬁ e y eq atio 3 x : 0 the s m r s n s the d f( , y) , y bol epre e t erivative of y 3x

h a d x h n on e x s s . If x be e a ded as a wit reg r to , w e e i t r g r dx un c on de n e d b the s am u a n t he s m f ti of y, ﬁ y e eq tio , y bol dv e s en s t he d x h a d h n on e r pre t erivative of wit reg r to y, w e 6 3 I ﬁn d h n 1 e xists . t is requ ired to t e relation betwee dx dx

x chan m t he n a u es x the u s Let , y ge fro i iti l val 1 , y, to val e

h : x Ax A s u c t e n x 0. 1 , y, y, bje t to relatio f( , y) Th n s n c e , i e Ag 1 , Ax Ev Av

It s b n t he m t ha follow , y taki g li it , t do 1 dx dx

n ce i an d x be con n ected b a un ction a l relation the He , f y y f h deriva tive derivative of y with regard to x is the recip roca l of t e

of x with regard to y. i This pro ce s s is kn own as chan gin g the i n depen den t var a

The c s on d n a n s for the h h r ble from x to y. orre p i g rel tio ig e 1 24

1 26 D U Cm x IFFERENTIAL CALCUL S [ I .

i s

Q 2 £ 3 u r } Hen ce : ct ( z + ) z z ) 4 ( ) 2 dx dx

The highe r x-de rivatives o f y can be s imilarly in m -d r te r s of x e ivatives o f z .

Ch n of the in n 6 n . 7 . a ge depe de t variable Let y be a

u n c n x an d h x an d be u n n f tio of , let bot y f ctio s of a n ew ( I v ari able t . t is requ ired to e xpres s 21 in term s of fig 2 61 3, dy 9113 an d in ms an d . z ter of dx dt diz

B Art . 1 4 y ,

hen ce

In practical ex amples it is us u ally better to work by the m e thods here illu s trate d than to u se the res ultin g form ulas .

EXE RC IS ES

1 a e the i de e de t variab e rom x t o z in the e u ation . Ch ng n p n n l f q di d 2 v y — z x m 0 he x e . + H , , w n dx2 dx

dv dz dx d z di 4 d y —2 8 y —2 8 513, 8 8 2 dx2 d z d z 2 dzu dy d y en ce x x 0 becom es 0 H 2 y ? y dx dx d z 75 . CH ANCE OF VARIAB LE 1 27

Interchange the fu nctio n an d the variable in the equ ation 2 2 2 + 2 y dx2 (dx )

Interchan ge x an d y in the equ atio n i iu (ft) I

e the i de e de t variab e rom x t o in the e u atio 4 . Chang n p n n l f y q n

dy (ﬂy ﬂ 3 2 ( 11: dx dx (dx y

Change the depe ndent variable from y t o z in the e qu ation

dx2

6 C a e the i de e de t variab e rom x t o in the e u at io . h ng n p n n l f y q n

2 — x x u : 0 e z lo x ; + , wh n y g

7 I is a u ctio of x an d x a u ctio of t he tim e t ex ress . f y f n n , f n n , p - - - acce e ratio in t erm s of the x acce eratio an d the x ve ocit . y l n l n, l y

d ( _ v _ ﬂ, Since ’ d l a x dl

d ll fl d i é fi 433 . fl . hence di2 dx dt2 di dz(dx )

2 i a d dx d z/ p : _ _ _ d rcdx ) dx dx dt dx2 d l

2 1 9 Q 41 3 23 . dl2 dx d i2 dx2 dt

In the abbreviate d ot atio fo r t-derivatives n n ,

2 d3/ d r/ _ 2 2 d x + dx . 2 ( ) dx (71

in th u tio 8 . C an e t he i de e de t ariab e ro m x t o u e e a A, h g n p n n v l f q n 2 9 d v dy 0 e x z t an u . , wh n 1 28 n IX US o . ; 76 DIFFERENTIAL CALCUL ( .

9 a e h i de e de t riab e rom L . Ch ng t e n p n n va l f x to t in the equ ation

2 — 2 —3/ — 1 x x 0 en x = cos t. ( ) Z , wh gx %

1 0 S t t h e uatio . how h a t e q n

1 re m ain s unc hange d in form by the s u bstit u tio n x . z

1 1 I t erc a e t he variab e an d t he u ctio in the e u atio . n h ng l f n n q n

2 . . 2 2 3 2 dx2 (dx ) (dx)

2 n he de e de t variab e rom to z in the e uatio 1 . Cha ge t p n n l f y q n d d - - f 2 1 : 0 hen = z . I w H , w y dig ( ag

1 30 Cm x DIFFERENTIAL CALCUL US [ .

n m u 79 . Len h of t n n or l s b n n ubn o m l gt a ge t , a , ta ge t , s r a .

The s egm e n ts of t he tan gen t an d n o rm al in t ercepted be

w en the n an en c an d the ax s 0X are ca ed t e poi t of t g y i ll ,

es e c e the t a n en t len th an d t he n or m a l len th r p tiv ly, g g g ,

’ an d t hei r p rojection s on 0X are calle d t he s u bta ng en t an d the b o s u n rm a l .

N V t M V N M

FI G . 23 FI G . 23 a .

l he an en an d n o rm a t o the cu r Thu s in Fi . 23 et t , g , t g t l ve

OK in T an d IV an d le t MP be the P 0 at P m e e t the ax is ,

Then ordin ate of P .

i h an en en h TP s t e t g t l gt ,

h n orm a en h PN t e l l gt ,

he s u an en TM t bt g t,

MN t he s u bn orm al .

d n o e d res e c v e b t n 7 v. Thes e will be e t , p ti ly, y , , ,

X TP de n d b ( ) an d r an ) Let the an gle be ote y f, w ite t d

Th n (I) e 2 \/ 1 + 7n

91>

dx d ] — = 0 0 t = - v = t an = T 9 1 ¢ y1 ; y1 ¢ 9 1 dyi in 612 TAN GE N TS AN D NORMALS 1 31

The su btan gen t is m eas ured fro m the in ters ection of the tan gen t to the foot of t he ordin ate ; it is therefore pos itive when the foot of the o rdin ate is t o the right of the in t ers ec

n Th u n rm a i m r d r tion of tan ge t . e s b o l s eas u e f om t he fo ot

the rd n a e the n e rs ec on n m a an d is s of o i t to i t ti of or l , po itive when t he n o rm al cu ts OK t o t he right of the fo ot of the

B o h ar t he r re r n rd n a e . e os e n e a e cc d o i t t efo p itiv or g tiv , a o i g

r as g!) is acu te o obtu s e .

The e x res s on s for 7 v m a a s a n e d b n d n p i , y l o be obt i y ﬁ i g

m e u a on s Art . 78 the n erc s m ad b the fro q ti , i t ept e y T t an gen t an d n orm al on the ax is 0X . he in te rce pt of the

an en s u rac e d r m x v s 7 an d x s u ac e d m t g t bt t f o 1 gi e , 1 btr t fro

v the i n tercept o f the n orm al giv es .

i d the i t erce t s ade u o he ax es b t he t a e t at t he Ex . F n n p m p n t y ng n oi t x on t he curve x/x V an d s o t at t eir s um is p n ( l , + ? h w h h c on st ant . Dif ere tiati the e u atio of the curve f n ng q n ,

1 Cl —-y : O. _ dx \/x Vy

Hence the e qu ation ofthe t angent is

x m . y ( l )

The x i terce t 1 s x \/ x an d the i terce t is V x e ce n p l l 1 , y n p y, l , 1 , h n t heir su m is

If a s eries of line s be drawn su c h t hat the su m of t he intercept s of e ac is the s am e co st a t acco u t bei t a e of t he si s th e o rm h n n , n ng k n gn , f r of the parabola t o whic h they are all t ange nt can be eadily se en .

EXE RC IS ES

1 i h t an d the orm a t o the e i . F nd t e e qu ation s of the t ange n n l ll pse 2 m 1 at the oi t x . Co m are the roces s it t at e o ed 2 p n ( 1 , yl ) p p w h h pl y a

in analytic ge o m e t ry t o obtain the sam e re sults . 1 32 DIFFEREN TIAL CALCUL US [Cm x J 2 2 2 d i z — . Fin the e qu at on of the t angent to t he cu rve x (x + y) a (x y)

at t he origin .

3 . i d t he e u atio s of the ta e t an d orm a at t he oi t 1 3 J F n q n ng n n l p n ( , ) 2 3 z on t he cu rve y 9 x .

h u a i th t a 4 . Find t e eq t o ns of e ngent an d norm al to each of the following curves at the point indicated

3 8 a h i f r (a ) v at t e po nt o which x 2 a . T , x,

2 2 3 2 x x at the oi t s for hic x 1 . y , p n w h 2 z: 4 x at the oi t 2 . (y) y p , p n (p , p ) 2 2 = 5 i d t he v a u e of the subt a e t of z 3 x 12 at x 4. . F n l ng n y

Compare the proces s with t h at alre ady given in analytic geom etry.

2 6 h t h n . . Find t e leng h of t e t a gent to the curve y z 2 x at x 8

—L ‘ 7 . i d the oints at ic the t a e t is ara e to the ax is of x , F n p wh h ng n p ll l , an d a t which it is perpendicular t o the x axis for eac h of the following CIII‘ VCS 2 2 2 (a ) a x hxy by 1 . 3 ”3 “8 (B) y ax

3 2 (r) 51 x (2 a x) .

‘ ’ h ditio th h n 2 2 = 2 2 = 8 . i d t e co at t e co Ics ax b 1 a x b 1 q F n n n + y , + y

s hall cu t at right angle s .

2 2 2 2 i d the a e at ic x 5 i tersect s x 1 8 9 . 4 7 F n ngl wh h y n 8 y 1 4. Compare wit h Ex . 8 .

2 1 0 w t hat in the e ui at era erb a 2 x a . Sho q l l hyp ol v the area of the t riangle form ed by a variable t angen t an d the coordin ate ax es is const ant 2 an d e qu al t o C .

2 2 2 1 1 . At what angle doe s y 8 x int ersect 4 x 2 y 48 ?

V " 2 " n 1 1 . D et erm ine t he subnorm al to the cu rve y a x .

1 3 i d th u t th c . F n e val e s of x for which t he tangent o e urve s 2 a (x a) (x 0 )

is parallel to the axis of x .

2 1 4 a u ta e t of the h erbo a x = a is e u a to . Show th t the s b ng n yp l y q l

the absciss a of the oi t of ta e c bu t o os ite in s i . p n ng n y, pp gn

2 5 t e a bo x h s c nstant subnorma . 1 . Prove that h p ra la y 4 a a a o l

1 34 R S C IL x DIFFE ENTIAL CALCUL U [ .

2 B ut 1 c os A0 2 s in A0 p( ) p 5 ,

h n ce t an « f e i Ag? 0 1 p 8 111 AO 2 —AO AO

1 . l S l O S n c — — l the c d n e u a n du c s t o i e ? , pre e i g q tio re e Al 0 gzﬁ A if t an xjr p dp dp as

1 A Ex . . i t cribes i po n des a c rcle of radi u s p . Prove t hat at a ny in st ant t he arc velocity is p time s t he a e ve ocit ngl l y, ds d0

i.e . , p cit dt

i 2 . i ribes a e Ex . W he n a po nt desc g v n c rve rove t at at an i s ta t the ve ocit u , p h y n n l y S —dp has a radiu s com ponent and a com dt dt

pone nt perpendicu lar to the radiu s vector

an d e ce t at p g, h n h d d0 = i = cos ¢ g in ¢ p FI G . 26 . d

d y ”d o 1 n and 8 . Relation betwee d x d p If the in itial li n e be take n as the axis

x the an en n P m s an of , t g t li e at ake

an gle d) with this lin e .

Hen ce 0 + 3b = qb ;

" " FIG ‘ 27 ° 1 1 0 t an tan i . e . , 80 TANGEN TS AN D NORMALS 1 35

8 2 . L n h of n nt n o m l o ub n n e gt ta ge , r a , p lar s ta ge t , and polar

bn m The or on s of the an en an d n rm s u or al . p ti t g t o al in t er cept e d b etween t he poin t of tan gen cy P an d t he li n e throu gh the o e er en d cu ar t o t he rad u s v e c r OP are c a p l p p i l i to , lle d

the p ola r t a ng en t length an d the p ola r n orm a l length ; the i r p roj e cti on s on t his perpen dicu lar are calle d the p ola r

u bt a n en t an d ola r s u bn orm l s g p a .

FI G . 28 a . FIG . 28 b.

Thu s let the an n an d n orm a at P m e the er en , t ge t l et p p dicular t o OP in the porn t s N an d M Then

PN is t he o ar an en n h p l t g t le gt ,

PM is the o ar n rm a n h p l o l le gt ,

ON is the o ar s u an en p l bt g t , h OM is t e polar s u bn ormal .

They are all s e en to be in depen den t of t he dire c tion of

Th n h of hes e n es n o be the i n itial lin e . e le gt s t li will w

dete rm in e d .

= Sin ce PN OP s ec OPN : p s ec X!”

hen ce polar tan ge n t len gth p 1 36 CAL L c a DIFFERENTIAL C U US [ x .

9 2 ‘7l A ain ON : OP t an OPN _ t an 1 , = g , p 1 p clp d0 hen ce polar s ubtan gen t _ p dp

PM : OP cs c

hen ce polar n orm al len gth

i 0M : OP c ot OPN = i i d0 ,

dP _ hen ce s u n m al ° polar b or d0

The s ign s of the polar tan gen t len gth an d polar n ormal

n h are am u u on ac ou n h d a Th le gt big o s c t of t e ra ic l . e dire c f 2 ti on of t he s ubtan gen t is determi n ed by the s ign of p i i P he n g is s the d s an c ON sh u d m as u e d W po itive , i t e o l be e r gP the h an d hen n e a the an s to rig t , w g tive , to left of ob erver placed at O an d looki n g alon g OP ; for when 0 in creas es

h is s e Art . an d is an acu e an as wit p, g po itiv ( t gle ( P l i 2 b h n 0 d a n a s — 1 s n in F . 8 c s s as s g ) w e e re e p i cre e , egative , filg i F 2 an d xjr s obtu s e ( ig. 8 a) .

EXERC IS ES

1 In the curve a sin ﬁnd . . p 0, a

2 In the s ira of Arc imede s a0 s how t at t au 0 an d ﬁn d . p l h p , h gh the o ar subta e t o ar orm a an d o ar s ub orm a . Trace the p l ng n , p l n l, p l n l

c u rve .

2 2 3 i r th c urve a 0 0 8 2 the va ues of all the ex res sio s . F nd fo e p 0 l p n t reate d in this article .

4 t at in the cu rve a the o ar subta e t is of co stant . Show h p0 p l ng n n

T ace t he c urve . length . r

CHAPTER XI

DER IVATIVE OF AN AR C AR EA VOLUME A D , , , N SUR FACE OF REVOLUTION

. Th n h 83 Derivative of an arc. e le gt 8 of the arc AP o f a v en cu x m eas u d m a x d n A t o an gi rve y f( ) , re fro ﬁ e poi t y

n P is a un c on t he a s c s s a x the a t er n poi t , f ti of b i of l t poi t,

m x d b a e at n the m 8 an d ay be e pres s e y r l io of for qS(x) . The determi n ation of t he fun ctio n (I) when t he fo rm of

is n n is an m r an an d s m m es d f cu ro em f k ow , i po t t o eti i ﬁ lt p bl Th e in it s o u in the In tegral Calculu s . e ﬁrst s t p s l ti on is

9 ’ t o d m n the m the de a un c n i S x eter i e for of riv tive f tio o ( ) , dx which is eas ily don e by the m ethods of the D iffe ren ti al

C al culu s .

P n s on the c u v e Fi . let x Let O be two poi t r ( g , y

the coOrdin at e s of P x Ax be ; , y Ay t hos e of O; s the le n gth of the arc AP ; 8 As that of

D r th d n the arc AO. aw e or i ates

MP N an d dra PR ara e , O; w p ll l

Th PR = Ax R = A M N . n to e , O y;

: n 2 are P As . He ce mg , 9 , O

Chord P O (Ay)

P Q 1 Ax

éﬁ AS P Q AS Therefo re ' Ax R O Ax P O 1 38 on . x1 . 83 R V O F AR C AR EA E TC . 1 39 DE I A TIVES , ,

T akin g t he lim it of bot h m em bers as Ax approaches z e ro

21 4 d n 1 Ar an . Ar . 4 u t b t 6 Th . an d t p t i g 2 , y , , 4 0 235 T h 8 r. o t t h . Co s a , , it f llow

ds Similarly do

Mo re ove r,from Art . 6 5

or in the diffe re n tial n otati o n

2 2 2 ds dx dy .

as a—s ' ” " 8 4 . Tri gonomet rl c meani n g of d x (zy Ax Ax P O S"1 0 8 c os RP 2 As P O As it follows by taki ng the limit that

dx cos ds

her n ( ) n the m the an e RP is t he an w ei I , bei g li it of gl O, gle

n m h - h ch the an n at the x a es h t e x ax s . w i t ge t poi t ( , y) k wit i l i S m ar 3 sin c hen ce Ef s ec ii cs c i il ly, p; w $8 dx dy Usi n g the idea of a rate or dif fe ren tial all hes e a n s m a , t rel tio y b c n n Fi e o ve ien tly e xhibite d by g .

80 .

Thes e res u lts m ay al s o be de 0 rived from e quati on s ( 2) of 313 Art . 83 b u n t an , y p tti g dx 140 D IFFE RE N TIAL CALC L US Cm x1 U [ .

8 5 . D i ti e of the o um of s o d of r o u i L er va v v l e a li ev l t on . et t he cu rve AP re o e a o u the x -ax s an d hu s e n e r O v lv b t i , t g ate a s u rface of re v ol ution ; let V be the volum e i n cl ude d betwe en

h s s u r ace the an e e n era t i f , pl g ted

b the xed o rd n a e at A an d y ﬁ i t , the plan e gen erated by any ordi

n ate M P . Let AV be t he v olum e gen er

FIG . 30 a . a d b the area M te y P N O . Then AV lies b etwe en the v ol um es of the cylin de rs gen erated b the c n e s PMNR an d S M N ha is y re ta gl O ; t t , 2 2 w Ax A u A y V < (y p) Ax .

D d n b Ax an d a n m it s ivi i g y t ki g li , d V

L 86 . r c of r o ution et S be the De ivative of a s urfa e ev l .

h u r a n r d the arc AP Fi an d are a of t e s f ce ge e ate by ( g .

h n AS that gen erate d by the arc P O w os e le gth is As . ' ’ D a P P ara e t o OX r w O , O p ll l

n h r an d e qu al in le gth t o t e a c P O. Then it m ay be as s u m e d as an axiom that the area ge n erate d by P O lies betwe en the are as gen ’ ’ erat ed b P an d P i . e . y O O; ,

2 n As AS 2 7r A As . FIG 31 y (y y) . .

D din b As an d as s n t o t he m t ivi g y p i g li i , dS 2 Tr y, ds

dS dS cls _ __ dx ds dx

x1. 142 D IFFE RE N TIAL CAL C UL US [Cm 88 .

. L 88 Derivat ive of area in polar coordin ates . et A be the a rea of OK P m eas u re d from a ﬁ xe d radiu s ve cto r OK to an y other radiu s v e cto r OP

h r N le t AA be t e a e a of OP O. D raw arcs PM ON, wit h O as

a ce n e . Then the a e a P O t r r O, I lies be twee n t he are as of the

FIG . 33 .

s ect rs OPM an d ON i. e . o O; , 2 0 2 0 p A AA 300 Ap) A .

D d n b A0 an d as s n t o the m t h n A0 0 it ivi i g y p i g li i , w e , follows that

For the de rivati ve of t he are a of a c u rv e in rect an gular

6 1 1 Ordin c o 1 0 . T i at es s e e Art . he es u s , r lt 2x

EX ERC IS ES O N C H APT ER XI

2 ds (IA dS 1 . In the arabo a 4 ax d p l y , ﬁn dx dx dx dx d 1( 3 2 2 = 2 2 i d an f r he i . . F n d o t c rcle x + y a dx do

3 i i f r ” . F nd o the c urve 6 cos x 1 . i17

4 - e e erated b . Find the x de rivative of t he volum e of the con g n y i re volving the l n e y ax about the axis of x.

5 i d the x-deriv tive h f the e i soid of revo utio . f t e vo u me o F n a o l ll p l n , 2 x_ form ed b revo vi g 1 about its or axis . 4 y l n 2 a

1 o 0 3 _ 6 . In the curve p a ﬁ n d d0 J C HAPT ER XII

ASYMPTOTES

n . h n 89 . H yperbolic and parabolic bra ches W e a c urve has a ran ch e x n d n n n t t he an e n t s d ra n at b te i g to i ﬁ i y, t g w s u cces s i ve poi n t s of this bran ch m ay ten d t o c oin cide wit h

d n t e x d n as in the am ar cas e o f the h a eﬁ i ﬁ e li e f ili yperbol a .

O n t he o the hand the s cc s s an n s m a m o v e ar he r r , u e ive t ge t y f t

n d he o ut the e d as in t he cas o f h a fart r of fi l e t e parabola . The s e t wo ki n ds of i n fin ite bran ches m ay be calle d hyp erbolic an d p a ra bolic . The c haract e r of each of the i n fi n ite bran ches of a c u rve can

d m h n h n h always be e te r i n ed w e t e equ ati o of t e c u rve is k n own .

If h n 90 . Defi nition of a rectilinear as ympt ote . t e ta gen ts at s u cces si ve poin ts o f a cu rve appro ach a fi xed s t raight li n e as a lim itin g pos ition whe n t he poi n t of c on tac t m o v es farthe r an d a t he r a o n an n n t e ran ch of t he ven c u rve f r l g y i fi i b gi , then t he fi xed li n e is calle d an a sym p tote o f t he cu rve . This defi n it ion m ay be s t ate d m ore b riefly bu t les s pre cis ely as follows : An asymptote to a cu rv e is a tan gen t

hos e n c n -act is at n n bu t h ch is n ot s e w poi t of o t i ﬁ ity, w i it lf

n n n e tirely at i ﬁ ity.

D ETERM INATION OF ASYM PTOTES

1 . Th on h 9 Method of limit in g intercepts . e equ ati of t e

an e n at an n x e n t g t y poi t ( 1 , p1) b i g d — 3 — 55 0 0 1 0 0 1 672 1 43 1 44 D I F T L u x11 F E RE N IA CAL C UL US [C .

the in tercepts m ade by this lin e on the c o o rdin ate ax es are d x ' l 0 y ’ 0 1 l dz 1

dx x y 1 1 1 d yl — Suppos e the cu rve has a b ran ch 0 11 which x . cc an d

i 0 Th 1 p 0 . en from ( ) t he lim its can be fou n d t o which t he n erc e t s x a roach as t he c oOrdin at e s x of the i t p 0, yo pp 1 , 0 1

o n n n I h im p i t of co tact te d t o be com e i n ﬁ n ite . f t es e l its be d n d b a b the u a n ofthe c s n d n as m o e is e ote y , , eq tio orre po i g y pt t

‘’ i + a b

Except in s pe cial cas es this m ethod is u s ually t oo compli c at ed to be of practical u s e in determin i n g the e qu atio n s of

Th r thr e o he the as ym ptotes of a give n cu rve . e re a e e t r

r n c a m h ds h ch a a s s u f ce de erm n e the p i ip l et o , w i will lw y ﬁ to t i as ym ptotes of cu r ves whos e e qu ati on s in volve on ly algebraic

Th m c d t h m e h d o f n ec n u n c on s . s e a be a e e s s f ti e y ll t o i p tio ,

d x n n o f s u s u n an e a s . b tit tio , of p io

92 . M th d of ins c ion . Inﬁ ni ordin s m o s e o pe t te ates , a y pt te

he n raic e ua on in t wo c o parallel t o ax es . W an algeb q ti

rd n a es x an d is a on a z e d c e a ed o f rac on s an d o i t y r ti li , l r f ti , a an e d accord n s of on e the c oOrdin at es s a rr g i g to power of , y

a e s the m y, it t k for

" 2 a bx 0193 ex u 0 s ( ( wa y . . in h ch u is a n m a of the de ree n in e rm s o f t he w i n poly o i l g t 1 her coOrdin at e x an d u is o f de re e n . ot , n _ l g

hen an u s e n x t he u a n d e rm n e s n W y val e i giv to , eq tio et i valu e s for y. Let it be requ ired to ﬁn d for what valu e of x the corre spon ding ordi n ate y has an in ﬁ n ite val u e .

1 46 D IFFE RE N TI AL CAL C L OIL xn U US [ .

t wo v a u s x ha m a e n n e n am e h s v a u es l e of t t k y i ﬁ it ; ly, t o e l of 2 x ha m a e dx ex = 0 an d the e u a n s the n n t t k + + f , q tio of i ﬁ ite

rd n a es are ou n d b ac r n h s as u a n n d o i t f y f to i g t i l t eq tio ; a s o on .

S m ar b arran n the ua on of the cu acc d i il ly, y gi g eq ti rve or in t o o e s x is as ﬁn d ha u g p w r of , it e y to w t val es of y give an i n ﬁn ite v alu e to x .

Ex . 1 . In the curve

3 2 2 2 — 2 2 x + x y + xy = x y

of t he i ite ordi at e an d dete rm i e the ite oi t nfin n , n fin p n in which this line m eets t he c urve . T i is a ubic e uatio in ic the c cie t of 3 is z er h s c q n wh h oeffi n y o . Arranged in pow ers of y it is

2 3 (33 + 1 ) + yx + (2 x 0 .

t he e qu at io n for y becom es

2 O ° 2 2 0 y + y + 1 the t wo root s of c are z 00 2 e ce t he e u atio of the whi h y , y ; h n q n infinite ordin at e is x infinit e ordinat e m eets the cu rve again in ere are n o infinite valu e s of x for i fin te v alue s of y.

2 h i es x a an d O are as m totes to the Ex . . S o t at t e h w h l n , y y p 2 2 34 curve a x y (x a ) (Fig. )

2 2 3 3 i d the as m t ot es of the c urve x a x C Ex . . F n y p (y ) y 92 AS YM P TO TE S 1 4 7

M ho of ubs ut ion . b i u m . he 93 et d s tit O l q e as y ptot es . T as ym ptotes that are n o t parallel to either axis can be fou n d b the m h d o f s u s i u io n h ch is a ica e t o ll y et o b t t t , w i ppl bl a

‘ a e a c cu r s an d is o f s c u h n t he u a i lg br i ve , e pe ial val e w e eq t o n 1 8 g1 ve n In t he 1 mplicit form

f( a v) 0.

Co n sider the s traight li n e

mx b 2 y , ( ) and let it be requ ired to de termi n e m an d b s o that this lin e

h r s ha an as m o e t e c u ve x 0 . ll be y pt t to f( , y) S in ce an as ymptote is t he - limiting p os itio n of a li ne that m ee t s t he c u r ve in two poin ts that t e n d to c oincide at

n n he n b m a in 1 an d 2 s m u an us t he i ﬁ ity, t , y k g ( ) ( ) i lt eo ,

s u n ua n in x re lti g eq tio ,

x mx b 0 f( , ) ,

n Th r is to have two of it s roots i ﬁ nite . is requ i e s that the h h h t f h coefﬁ cie n ts of t e two ig es powe rs o x s all v an is h .

Th s c ef c n s e u a e d z r u rn s h e u a n s e e o ﬁ ie t , q t to e o , f i two q tio , fro m whi ch the requ i red val u e s of m an d b can be dete r

m n d . Th s a u s s u s u d in i e t he i e e e v l e , b tit te w ll giv

n m equ atio n of a asy ptote .

3 2 4 . i d t he as m totes t o t he c urve x 2 a x . Ex. F n y p y ( )

In t he rs t ace t e re are e vide t n o as m tot es ara e t o eit e r ﬁ pl , h n ly y p p ll l h

o f the Coordi at e axes . To dete rm i e the ob i ue as m totes m a e t he n n l q y p , k

e uation of t he curve s im u t a eou s it mx b an d e im i ate . q l n w h y , l n y Then 2 mx b 8 x 2 a x ( ) ( ) , or arra ed in o e rs of x , ng p w ,

2 2 2 ( 1 m3 ) x3 (3 m b 2 d ) x + 3 b mx

3 = 2 — = 171 + 1 0 an d 3 m b 2 a 0. 148 D IFFE RE N TIAL CAL C L US Ca xn U [ .

The n m

he nce y x

is the e uatio of an as m t o te q n y p . The t ird i t ers e tio f t h n c n o his line with t he given cubic fou nd

2 3 2 a t he e u atio 3 mb x O : 0 e q n , wh nce x

T is is the o l ob i ue as m tote as the h n y l q y p , other root s of t he equ ation for e i i r m ar m ag n a y.

5 . 2 2 2 Ex . i d the as m totes to the curve a x F n y p y( ) a (a x) .

er th i n d e 2 . T ﬁ H e l ne y 0 is a horiz o ntal asymptote by Art . 9 o the ob i ue a m t te s o s ut mx b. l q y p , p y

1 50 D IFFER EN TIAL AL C x11 C C UL US [ m .

ho f x n i n x A M d o . E ici n h h . s o unc ou 9 5 et e pa pl t f tio s . lt g the t wo fo regoin g m ethods are in all cas es s ufﬁ cien t t o ﬁ n d

t he as m o s a e ra c c u es et in c er a n a y pt te of lg b i rv , y t i s peci l c as e s t he o bliqu e as ymptotes are m os t c on ve n ie n tly fou n d

th h d x n n in d n di I i by e m et o of e pa s io es ce n g powers . t s bas ed on the p rin ciple that a s traight lin e will be an as ymp

‘ tote to a c urve when t he differen ce betwe en the ordi n ate s

the c u e an d of the n e c or es on d n t o a c om m on of , rv li , r p i g

h z ro a the a c a e om n n a s c s s a a ac s s s ss c es e . b i , ppro e e b i b i ﬁ it It will appe ar fro m the proce s s of applyi n gthis p ri n ciple that a li n e an s werin g the co n dition j u s t s tated will als o

s atis fy the origin al de ﬁn ition of an asym ptote . The p ri n cip al valu e of the m ethod of e xpan s ion is that it exhibits t he m an n er in which e ach in ﬁn ite bra n ch ap

proaches it s asymptote .

h Ex . Find the a symptote s of t e curve

g whe n ce y : i x (1 _ )(1

H O

— — l — :t x 1 + 1 + ( x 2 x2

th bli u e as m to te s are x 1 Fi . Hence e o q y p y ; t ( ) ( g The s i oft he ex t t erm s o ws t at e x co t he c urve is above gn n h h wh n , the ﬁ rst asym pt o te a n d below the s econd ; an d vice vers a when x co . AS YM PTO TE S 1 51

The sam e m ethod m ay be applied to cases in which x 18 an explicit fu nctio n of y This m ethod can als o be e x te nded so a s t o apply to cu rves de ﬁn e d by ’ an im ici io e M M h n n e r s D i fer t e uat x 0 . S e c a o a d S d pl q n , f( , y) [ ny f ” e n l a tia C cu us . l l , p

FI G . 37

EX ERC IS ES O N C H APT ER XII

Find the asymptote s of each of the following curve s

, I£ 2 2 if 2 2 1 . a x : b 2 x c . 7 . x a z y( ) ( ) ( )y (y b) x .

L 8 .

2 x 2 ax 2 2 3 t 9 . xy x y O .

1 2 2 2 2 2 2 3 3 . x 1 0 . x 3 a x y a (x y( ) . 1 3 3 V 1 . x 3 axy + y O 4 = C . y a + 3 3 3 7 1 2 x 2: I . y (1 . 5 2 V ° 27 W x) 1 3 4 2 2 2 2 4 . x t x y a x b O.

2 2 4 4 2 ' 6 . 1 1 4 x x . . x x . V y ( ) y a y 3 2 — 2 1 5 . x 2 x x + y y 1 . C HAPTER XIII

DIRECTION OF B ENDING. POINTS OF INFLEXION

96 . onc i u r nd n A cu i C av ty pwa d a dow ward . rve s s aid to

con ca ve d ow n wa rd in t he c n o f n P h n be vi i ity a poi t w e ,

a n d s an ce on ch s d P the u r e i for ﬁ ite i t ea i e of , c v s s itu ated

FI G . 38.

the an n d n at h n as in the a cs AD below t ge t raw t at poi t , r , FE It is con c a ve u p w a rd whe n the cu rve lies above t he

n n as in the a cs D 17 H E ta ge t , r ,

B y drawin g s u cc es s iv e tan gen ts to t he cu rve, as in the

u is e as s n ha the n c n ac ad an c s ﬁg re , it ily ee t t if poi t of o t t v e

the h t he an en s n s in t he s d ec n of to rig t , t g t wi g po itive ir tio

n hen t he c n ca is u a d an d in the n e a v e rotatio w o vity pw r , g ti

h n n e u a d dire ction when t e c o cavity is do wn ward . H e c pw r c n a m a e ca e d a s v e en d n t he cu an d o c vity y b ll po iti b i g of rve ,

d n a d c n cav n e a ve n d n . ow w r o ity, g ti be i g A p oin t at whi ch the direction of b e n din g chan ges con e tin uou s l m os v e t o n e a v e o r vice vers a as at F or y fro p iti g ti , ,

1 54 D IFFE REN TIAL CAL C Cm III UL US [ X .

u n c n x acc rd n as the c n ca is u d or d f tio of , o i g o vity pwar own

a d an d he n ce ha it s x -de at ve is os e n e a w r , t t riv i p itiv or g tive . Thus the ben di n g of the c u rve is in the pos itive or n ega 2 d y d re c n of a n acc d n as the u n n tive i tio rot tio , or i g f ctio dx2 pos itive or n egativ e . dy At a poin t of in ﬁexion t he s lope is a m axim u m or dx z d v m n m u m an d here o e it s de e chan s s n om i i , t f r rivativ 2 ge ig fr dx

n e e r m n Th po sitive to gativ or f o egativ e to pos itive . is latter con ditio n is e v iden tly both n e cess ary an d s u fﬁcien t in

de ha the n x m a a n in ﬂexion on or r t t poi t ( , y) y be poi t of the given cu rve

n c the c d n s the n s in ﬁexion on the He e , oor i ate of poi t of c u r ve y f(x) m ay be foun d by s olvin g the e qu ation s

’ an d then tes tin g whether f ( x) chan ges it s s ign as x pass es

T. n r a through t he criti cal valu e s thu s obtain e d . o a y c itic l

th s c s n ds the n valu e a that s atis ﬁes e te t , orre po poi t of in ﬁ xion a e ( ,

h c urve Ex . 1 . For t e 2 y (06

i t s of in ﬂexion an d s o the mode of variatio of the s o e ﬁ n d the po n , h w n l p an d o f the ordinate . dy 4 x x2 ere ( H dx

3 324 x2

= It i be see hence the critical values for inﬂexion s are x i f . w ll n i3

he seco d derivativbc a es si rom t at as x i crease s t rou , t g g f h n h gh $3 n h n n

iti e to e ative e ce t ese is an inﬂexion at ic the co cavit pos v n g , h n h wh h n y

J n d to do ard . Simi ar at x the co changes from u pwar wnw l ly, V3 D IRE CTI ON OF B EN D IN G 1 55

ca i r v ty change s from downwa d to u pward . The following num erical t able will help to s how the m ode of variatio n of the ordinate an d of t he h s o e an d t e directio of be di . l p , n n ng 1 As x increases from co to 3 the be di is ositive an d the s o e n ng p , l p co ntinuallyincre ases from 00 t hrou gh

zero to a m ax im um va ue ﬁ ic l , wh h

is the slope of the stat io nary t angent

dra at the oi t é 4 wn p n , 9 ) As x co ntinue s t o increase from l 1 to - t he be nding is nega V3 tive an d the s om , slope decrease fr 8 throu z er a m i im um _ gh o to n

8 o o 0 1 4 value whIch IS the 8 10 pe of the s tat lon ary tan gent at V3 9 1 i a as x i creases rom - to co the be di 1 8 F n lly, n f , n ng vs an d the slope increase s from the value 8 r 00 th ough z ero to .

The va ue s x —1 0 1 at l , , + , w ic the s o e ass es throu zero h h l p p gh , correspond to turn ing values of t he

ordinate .

Ex . 2 . Examine for inﬁexions the

23! l (x 4 dx 3

d - i v _ 2 (x 4) 2 dx2 9

en ce at the oi t 4 H , p n , Si 2 n d — r i 4 a a e ite . e x 252 nﬁn Wh n , ﬂ 18 ositive an d e x 4 is e ative . p , wh n , n g d 2x 1 56 D I FFE R EN TIAL CAL C L US Cm III U [ X .

Thus t here is a oi t of in ﬁexion at 4 at ic the s o e i p n , wh h l p s in it e an d t he be di c a e s rom the os itive t o the e ative ﬁn , n ng h ng f p n g ir ti d ec on .

4 EX . 3 . Co sider th cur e = n e v y x . d 1 3 i -Z 2 93 4 . 1 x , 2 13 . dx dx2 d2 At 0 —3 is z ero but the ( , , dx2 2 d 2 c urve has n o In ﬁeXIO n for — ever , n dx2 i change s S gn (Fig. FI G . 41 .

98 . An alyt ical derivation of the t es t for t he direction of

L h u on of a c u r e be n d l bending . et t e e q ati v y a et

P x be a o n u on . Then the e u a n the , ( 1 , yl ) , p i t p it q tio of tan gen t at P is 23 y 0 1 1)

Su s ha hen x chan es m x x h the ppo e t t w g fro 1 to 1 , n ate of the tan gen t chan ge from

t o ha t he cu y1 t t of rve ” m hen is fro yl to y ; t it pro pos ed to de term in e t he s ign of ’ ’ the diﬂeren ce of ordin ates y y

(x ' h) 47 c orre spon din g to the s am e ab r .

FIG‘ 42 °

s a x h. s cis 1 ’ B Ta s h em y ylor t eor ,

, = ‘ ‘ b 2 2 f( 2’ I f ( 1) f ( 0 + 1 1 2

u a n of the an en an d from the ab o ve e q tio t g t,

, 23 “ “ h 2 y 01 ( 1 I 1 ) A H en ce a y1 + r e a) . an d it follows that

” ' ’ v v gf l H

1 58 D I FFE RE N TI AL CAL C L S OIL III 99 U U [ X . .

I n the vicin it o a iven oin t x the c y f g p ( , y) u rve is con vex or

- con ca ve to the x axis a ccordin a s the rodu ct , g p y is p os itive or n ega tive .

EX ERC IS ES O N C H APT ER X III

1 . Ex am i e the curv e 2 3 x fo r oi t s of in ﬁ eXi n y ( p n on .

2 2 2 2 . S o t at the c h w h urv e a y x (a x ) has a point of in ﬁexion r i t h e o ig n . s 8 a 3 i d t he oi t . F n p n s of in ﬁex ion on the curve y 2 2 x 4 a m

71 4 . In the cu rve a x rov e t at the ori i i y , p h g n is a po nt o f in ﬁexion re o itive odd if m an d n a p s int egers .

5 t th u r z i . Sh ow hat e c ve y c s n has an inﬁnite nu mber of point s of in x i tr i t li ﬂe ion ly ng on a s a gh ne .

2 2 = 6 . Sh ow th at the cu rv e y(x + a ) x has t hre e point s of inﬂexion lying on a s traight line ; ﬁ n d the e qu at ion of t he line

2 7 . If x be the e u atio of a curv e rove t at the abscis sas of y f( ) q n , p h it s point s of in ﬁex ion satis fy the e qu ation

21 158 )

2 2 3 8 D r th f h 2 a e ar it i t of . aw e part o t e cu rve a y g ax n s po n

in ﬂexion an d ﬁ n d the e u atio of the statio ar ta e t . , q n n y ng n CH APTER XIV

CONTACT AND CURVATUR E

n . The n s n s c n the 1 00 . Order of co t act poi t of i ter e tio of two cu rves ( ) x “f x y l ( ) , y I ( ) are foun d by m aki n g the two e qu atio n s s im u ltan eou s ; that is b n d n h s e u es o f x h ch , y ﬁ i g t o val for w i

9 (x) «1» ( 2 )

u = a is on e a u e ha a h u S ppos e x v l t t s tisﬁes t is e q ati on .

Then the n t x = a is c m m n t the poi , o o o

u c r ves .

If m re e the two cu es hav the s am e an en a , o ov r, rv e t g t t

h s n he are s a d t o ou ch e ach her or t o av e t i poi t, t y i t ot , h co r rd h h h The a u f n ta ct ofthe ﬁ s t o er wit e ac ot e r . v l es o y an d o f are thu s the s am e for both cu rves at the poin t in

u s n h ch e u es ha q e tio , w i r q ir t t

( ) a ‘ ’ a I ( ) f ( ) , 9T“) WOO 2 d g If in add n the a u e be t he s am ch , itio , v l of e for ea dx2 c u t he n hen rve at poi t, t

a 4 ( ) , an d the cu rves are s aid to have a c on tact of the s e con d

rd r h h o e wit eac othe r .

“ If ¢ ( a ) an d all the de rivative s up to t he n th

d n c u s e e u a each h the c u es are s a d or er i l iv be q l to ot er, rv i 1 59 1 6 0 D I FFER E N TI AL CAL C L US cm XIV U [ .

T t o have c on tact o f the n t h o rder . his is s een t o re qu ire

1 d n h n + c o n iti on s . H e ce if t e e qu ation of the cu rve

be v n an d the e ua on on gi e , if q ti of a s ec d curve be written in t he fo rm in which «b(x) pro ceeds in ers of x h u n de erm n e d coe ﬁ c e n s hen n 1 pow wit t i f i t , t + of thes e c oe fﬁ cien ts co uld be determ in ed by requI rI n g the s eco n d cu rve t o hav e con tact o f t he n th orde r with the given c u rve at a giv en poin t .

1 1 . 0 Number of condition s implied by cont act . A s traight

n e has ar i rar c n s t an s h ch can be d e m n ed li two b t y o t , w i et r i by two co n dition s acc o rdi n gly a s t raight lin e can be d rawn i which tou ches a giv en cu rve at an y s pe c ﬁ ed poi n t . For if the u a n n e be r en mx b h n eq tio of a li w itt g , t e

2 dx dx hen c h u h an ar rar oin x = a o n a e n cu r e e , t ro g y bit y p t giv v

— h n t ac the r g j a lin e can be drawn w i ch has c o t of ﬁ s t

r th the cu r e bu t h ch has n ot in en e ra c on ac o rde wi v , w i g l t t - o f the s econ d order ; for the two c on di ti on s for ﬁrs t order c on tact are

whi ch are j u s t s ufﬁcien t t o dete rm in e m an d b. In gen eral n o lin e can be drawn havin g c on tact of an order higher than the ﬁ rs t with a given cu rve : but there m h h h s can be don e . ex a are ce rtai n poin ts at w i c t i For ple ,

- d on ac is O a the additi o n al c on dition for s e con d o r er c t t ( ) ,

‘ whi ch is s atis fie d when the poin t x a is a p oin t of in flexion o n t he given c u rve y = Thu s . the tan ge n t at a poin t rde of in fi exion on a cu rve has con tact of the s econ d o r h with t e cu rve .

1 6 2 E T “ A m XIv D I FFE R N I AL C L C UL US [C .

Sin ce by hyp othes is the two c u rves hav e c on tact of the n th d r at the o n hos e abs cl s s a i s a hen ce or e p i t w ,

1> a f a « 9 ( ) I ( ) , N O WOO,

hn + l an d 9 1

bu t h s ex es s n h n h is s uf c en d m n sh d has t i pr io , w e ﬁ i tly i i i e , t he s am e s ign a s hn -I-l

n ce n be odd d es n ot chan S n h n h is He , if , yl y2 o ge ig w e c han e d n o h an d hu s the cu r es do n ot c os s each g i t , t two v r

h h n d if he t the com m n o n . O n the o e a n ot r a o p i t t r , be e ven chan es S n h h an d he re re hen the , y1 y2 g ig wit ; t fo w v c o n tact is of e ven order the c u rves c ros s each other at their c omm on poin t .

exam e the an e n n e u su a es en re on on e For pl , t g t li lly li ti ly

d the cu r e but at a oin in ﬁex ion the an e n s i e of v , p t of t g t cros s es the cu rv e .

A a n the Ci rc e of s econ d -o rder c on ac c r s s es the g i , l t t o

h a o n n o ed a r in h ch cu rve ex cept at t e s pe ci l p i ts t l te , w i t he circle has c on tac t of the third orde r .

EX ERC IS ES

1 i d the order of co t act of the curves . F n n

2 z n z — 4 y x a d y x l .

2 i d the order of c o t act of the curve s . F n n

3 x y an d x y 1

the order of co t act of the curves 3 . Fin d n

= 2 — 2 — = — 2 4 y x 4 an d x 2 y 3 y .

- 4 term i e t he arabo a havi it s axis ara e t o the ax is, . D e n p l ng p ll l y 3 t t he which has the clo se st pos sible co nt act wit h t he cu rve (127; x a

oi t a a . p n ( , ) 102. CON TAC T AN D C UR VA T UR E 1 6 3

5 D te rmi e a strai t i e ic has o tact of the second order . e n gh l n wh h c n wit h the cu rve = 2 — 2 — y x 3 x 9 x + 9 .

6 i d the rd r f . F n o e o contact of

2 — y = log (x - 1 ) an d x 6 x + 2 y +

at the oi t 2 O . p n ( , )

7 hat m u t . W s be the value of a in order that the cu rves — an d xy = 3 x 1

n tact of the secon d order ?

1 03 . Th c e h h Circle of curvature . e ircl t at as c on tact of

' t he clos es t o rde r with a given cu rve at a s pe ciﬁed poin t is called the os cu la ting circle or circle of c u rvature of the

n n Th cu rve at the gi ve p oi t . e radiu s of this circle is called the ad u s of cu r a u re an d it s cen er s ca d th en e r i v t , t i lle e c t r

u a ur at t he a n d of c rv t e s sig e poin t .

1 04 . Len gth of radius of curv ature ; coordin at es of center

Le h u of curvature . t t e eq ati on of a ci rcle be (X ( 1 ) in h ch R is the ad us an d a 8 are the c o d n s the w i r i , , , o r i ate of c n r the c u rren c oOrdin ates e n de n e d b X Y e te , t b i g ot y , to distin gu is h them from the c obrdin at es of a poin t on the given cu rve .

It is e u r d de m n R a 8 S O ha h s c c r q i e to ter i e , , , , t t t i ir le m ay ha ve con tact of the s e con d orde r with the given cu rve h at t e n x . poi t ( , y)

F m b su cc s s e d ff n a on s h ro y e iv i ere ti ti , it follow t at d Y (X

d 2 Y 1 Y —8 — + + ( 1 ) ZZ X 16 4 D IFFE R E N TI L CAL C OIL XIV A UL US [ .

If the circle ( 1 ) has c on tact of t he s e con d order at the

o n x h the en cu rve hen hen X = x p i t ( , g) wit giv , t w it is n e ces s ary that Y y, 2 d Y dg 02 V ﬁg . , 2 2 dX dx dX dx

S u bs titu tin g thes e ex pre s s i on s in

(w — B

— 1 + + (y B)

when ce

an d n ll b s u s u n in ﬁ a y, y b tit tio

1 5 . 0 D c ion of d us of cu u . S n c at an n ire t ra i rvat re i e , y poi t

P o n the en cu e the a ue is the s am the giv rv , v l of e for

cu e an d the s cu a n c c e for hat n s ha rv o l ti g ir l t poi t , it follow t t

he ha the s am an n an d n rm a at P an d h n c t y ve e t ge t o l , e e

that the radi us of cu rvatu re coin cides with the n orm al . 2 A a n s n c the a u $3 is the s am e fo r o h cu s g i , i e v l e of b t rve at

h t he hav e the s am e d ec n P ws r m Art . 97 a , it follo f o , t t y ir tio

xrv 1 6 E N TI C L n . 6 D IFF R E AL A C UL US (o .

1 06 . T ur ur f i n otal c vat e o a g ve arc ; average curvature .

h o f an arc Fi 4 in hi h h T e total curvatu re P O ( g . 6 ) w c t e ben din g is c on tin u ou s an d in on e dirco

n is t he n hr u h h ch the tio , a gle t o g w i t an gen t s wi n gs as the p oin t of c on tact m o ve s from the in itial p oin t P t o the

m n a n in h ds ter i l poi t O; or, ot er wor , it is the an gle between the tan gen ts at FIG . 46 . P an d m eas u e d r m the m O, r f o for er to th hu h cu r a u re o f en re e latter . T s t e total v t a giv a is pos i tive or n egative acc ordi n g as the ben din g is in the pos itive or n egative direction of r otati on . The t otal cu rvatu re o f an are divided by the le n gth of the

’ r i d h u r ur h r Th if h a c s ca t e c v a e t e a c . us t e lle a verage t of ,

e n h the arc P As cen m e s an d it s a l gt of O be ti et r , if tot l cu u A ) d n s h n it cur u is rvat re be 4 ra ia , t e s average v at re 8 d n s c n m ra ia per e ti eter .

1 07 . The m ea su re Meas ure of curv ature at a given point . Of the c u rva t u re of a given cu rve at a given poin t P is the limit which the average cu rvatu re of t he arc P O approaches whe n the poin t O approaches c oin ciden ce with P .

Sin ce the ave rage cu rvatu re of the arc P O is t he m easu re of t he cu rvat u re at the poin t P is

x : r: A8 = O , AS 618 an d m ay be regarded as the rate of deﬂection of the are from the an n m d r u n n h n as the t ge t es ti ate pe it of le gt ; or agai , ratio of the an gu lar velo city of the tan gen t to the lin ear velocity of the poin t of c on tact . 106—108 ] CON TACT AN D C UR VATURE 1 6 7

x to in m s of x an d h de s s e To e press ter , g, t eir rivative , ob rve

—1 13 When ce (b t an

r 3 the e fore [Art . 8

1 08 . Curv at ure of os culatin g circle . A cu rve an d it s os c u latin g circle at P have the s am e m eas u r e of cu rvatu re at

ha n t t poi t . ’ For let it E be h s c e m e asu es of cu a u , , t eir re pe tiv r rv t re at

he n Th r m r 1 A t . 0 t n ac x . en o 7 poi t of co t t ( , g) f ,

div

B ut this is the re ciprocal of the expre s s ion for the radiu s

cu va u E h n of r t re ( q . p . e ce

1c : 1 6 8 D I FFE E N I A on I R T L . X CAL C UL US ( V .

That IS : the m ea su re of curva tu re te at a p oin t P is the reci rocal o the radiu s o curva tu e p f f r R for t hat p oin t . Sin ce a curve an d it s os cu latin g c ircle have the s am e radiu s at

he n c n ac s rom h r u h t ir poi t of o t t , it follow f t is es lt t at the m e as u e cu a u r is a s the s am h r of rv t e l o e for bot . It is on accou n t of this property that the os culatin g circle

h r m is c alle d t e ci cle of c u rvatu re . This is s o etim es u s e d as

h d n n ro h f u u r Th d u t e eﬁ i g p perty of t e circle o c rvat e . e ra i s of curvatu re at P wo uld then be deﬁ n e d as the radiu s of the circle whos e me as u re of cu rvatu re is the s am e as that o f the

I u u n d m r en cu v at t he n P . t s a e as o A t . giv r e poi t v l , f fro 1 h Art . 04 . 1 06 an d Art . 1 07 cc ds t ha v n in , a or wit t gi e

EX ERC IS ES

2 i 1 f curvat ure of the curve 4 a x at the ori . . Find the radius o y g n

2 2 i d the radiu s of curvat ur e of the c urve c . F n v at the origin .

3 s 2 3 in d the radiu s of cur ature of the curve a bx ox at t he . F v y y i orig n .

. Find t he radius of curv at ure for e ach of the following curve s

2 4 x m . Re ct a ul ar erbo a . . y ng hyp l

— 1 . e rb l 7 Hyp o a .

n ‘ l ” 6 a y x . General parabola.

f / V x x . 7 5: O Parab a . . y ol

2 2‘ f i . 8 x o . oc c o d . y Hyp y l

2 9 issoid . . y C “N — 5 E 1 . ate ar . 0 . y 309 e ) C n y

1 70 D I FFE RE N TIAL CAL OI XIV C UL US [ L .

1 hen u i s a n as de en den a i a e h x W t ke p t v r bl , t e e pres p Sl on for 1c ass um e s the s 1 mpler form

S n c n in ﬂex ron 1c an shes an d chan s s n i e at a poi t of v i ge ig , hen c the c n d n n of in ﬁexion ex r s s e d in e o itio for a poi t , p e 2 d a a cobrdin at es is h u s h l n sh an d chan pol r , t at a l va i ge d02 Si gn . EX ERC IS ES

Fin d the radiu s of curv ature for e ach of the followin g curves " — 2 = 2 1 . z 3 . = 2 a cos a . 5 . cos 2 a . p a . p 0 p 0 9 2 2 — = t = = 2 2 . 4 os . 6 . 2 a 1 cos 0 . . p a cos 0 . p c §0 a p ( ) 7 . p0 a .

EVOLUTE S AND INVOLUTES

h n P m e s l . h n t e 1 1 0 . Deﬁ nition of an evo ute W e poi t ov

n t he v e n c u ve the c en t er of cu a u r 0 des cr es alo g gi r , rv t e ib

r an other cu r ve whi ch is called the evolu te of the ﬁ s t .

x 0 be the e u n the en cu r e . Then Let f( , y) q atio of giv v the equ ation of the lo cu s des cribed by the poin t 0 is fou n d by elim in atin g x an d y from the three e qu ation s

dx 109 CON TAC T AN D O UR VA TURE 1 71

an d hu s a n n a n n a 3 the c d n a es t obt i i g relatio betwee , , , oor i t

u u of the c en ter of c rvat re . N o gen eral proces s of elimin ation can be give n ; the m ethod to be adopted depen ds u pon the form of t he given

u a n x 0. eq tio f( , y)

2 1 h u t e f the bo 4 . i d t e evo o ara a x. Ex . F n l p l y p d d2 1 Si ce = 2 ixi ix 2 n y p , 52 p ,

— - — 2 2 4 g1 x a p x (1 + 173 ) 217 il3 2 6”

“ 2 2 2 2 2 2 31 B (1 I n ) 219 56 209 96 P 95 ) ;

- a = 2 3 x = _ s ixi p + , p p .

The res ult of elimin ating x between the last two e quation s is i s o 3: ( p er.

2 2 3 27 0 1 0 1 3 . 1 72 D IFFE RE N TI AL CAL C UL US

ic is the e u atio of the e vo ute of the arabola a wh h q n l p , , ,8 cu rre t r n coo din at es .

2 EX. . Find t he evolut e of t he ellipse

x2 2 y _ RI F

d y . _ y + y {L 2 2 2 ’ 5 b dx dx a y

da x — 2 y 2 2 2 (11 7 b iix - b b x — b2 2; 2 2 2 2 _ a ’ 2 2 2 2 y 4 3 ( y dx 5 y a y ( FF) a y whe n ce 4 2 2 2 2 e (0 3 911 x a y y y : 1 B 2 “ , 4 2 a b ( b a b b

2 a _ 62 Therefore

Si m Ilarl y, a4

E im i ati x bet ee the e uatio of the locus l n ng , y w n q n de scribed b a 8 is y ( , , ) i f s i i 52 (mo (in (a ba (F g. )

Th ha t o im 1 1 1 Pro er i s of the e o u . e e o u e s w . p t e v l te v l t

an o t s h n h d port t pr pe r ie t at will o w be establis e .

I a t t Th o t t o . . The n orm l he cu rve is tangen t to he ev lu e e p re a on s con n c n the c oOrdin at es a 8 o f the ce n er o f l ti e ti g ( , , ) t c u a u h the cobrdin at es x o f the c es nd n rv t re wit ( , y) orr po i g

n n r 1 4 o the cu a e b Art . 0 poi t rve , y ,

— x —a 8 O + (y , ) ,

B d f e e n a in 1 as x c n s der n a as y i f r ti t g ( ) to , o i i g . B , 9

un c n s x f tio of ,

2 ' d d a d . y d _— O. (2 B> 2 dx dx dx (ix

1 74 D CAL C L Us Cm XI V IFFERENTIAL U [ .

A a n m 1 an d g i , fro ( ) dB

dx — — x a 9 8

en ce ch h s ac n H , ea of t e e fr tio s is equ al t o

( 9)

in h ch a is the arc the u e . C m a A w i of evol t ( o p re rt .

N x m u n n u m era an d den m n at r e t, ltiplyi g tor o i o of the ﬁrs t m em e of 8 b x a an d hos e the s e c n d m em e b r ( ) y , t of o b r by

an d c m n n n e w n u m e a ors an d den m n a r g B , o bi i g r t o i to s , it follows that each of the fracti on s i n ( 8) is equal to

— (iv a ) + < y

h ch u s b 7 an d w i eq al 15? y ( )

B y c om bin ing with

i ” dx has

that is, dx

Th ' R c n s an 1 0 erefore 0 j : o t t, ( )

wherein a is m easu red from a ﬁx ed poin t A on the evolu te .

u u re n s Now 0 0 be the cen s . c the , let 1 , 2 ter of rvat for poi t CONTACT A N D CURVA TURE 1 75

P P on the v n cu le t P 0 P 0 R an d I, 2 gi e rve ; 1 1 2 2 2 ;

h n d 0 0 let t e a cs A 0 A0 be de e b . Th n r 1 , 2 ot y 1 , 2 e

a“ R 6 R b 1 j ; I 2 j : 2, y

h is a 0 R R t at , , 2 1: ( 2 1) ;

hen ce arc 0 R R . , 2 2 I Th 49 us in Fi . , g , P 0 0 0 P 0 1 1 1 2 2 2, P 0 0 0 P 0 et c . 2 2 2 3 3 3 ,

n ce a h ead be ra e d He , if t r w pp ar un d t he e u e an d he n o vol t , t be u n u n d the en d of can wo , free it be m ade race ou t the n a cu to t origi l rve . From this prop erty the lo cu s o f the FIG . 49 . cen ters of cu rv atu re of a given cu rv e i s c a e d the evolu t e of ha cu r e an d the a er is ca d ll t t v , l tt lle the in l t t h m r vo u e of e fo r e .

hen the s n is u n ou n d e ach o n of des cr s a W tri g w , p i t it ibe d f e en n u e hen ce an cu r e has an n n n u m e i f r t i vol t ; , y v i ﬁ ite b r

' f n o v o u es bu t on l on e e o u e . i l t , y v l t An y two of thes e in volu tes in tercept a c on s tan t dis tan ce on he c mm n n rm a an d are ca ed a ra llel cu r es on t ir o o o l , ll p v acc u n h o t of t is property.

E i h X . F nd the length of that part of the evolute of t e parabola which lies i side t he r n cu ve . rom F h i er e h F ig. 47 the requ ired length is t wice t e d ff ence betw e n t e

ta e t s 0 P an d P C bot of ic are orm a s to the arabola . ng n 3 3 O O, h wh h n l p To ﬁ n d the coordi at es of t he o i t P rite the e u at io of the t an n p n 3 , w q n e t t o the evo u t e at C an d ﬁ n d t he ot er oi t at ic it i tersects g n l 3 , h p n wh h n the r b pa a ola .

The coordi ates of 0 the oi t of i t erse ctio of the t wo curves are n 3 , p n n n ,

8 4 an d the e u atio of t he 't an en t at C is ( p ; q n g 3

‘ — = 2 x 8 p 0. ‘

1 76 N on , XIV DIFFERE TIAL CALCUL Us ( .

T is t a e t i t erse ct s the arabo a at t he oi t 2 h ng n n p l p n ( p ,

ic is P . wh h 3

The v a ue of the radius of cu rvatu re is M he ce P 0 2 l , n 0 0 p ,

P 0 e ce t he arc 0 0 is 2 an d the re u ire 3 3 p , h n 0 3 p q d length of the evolu t e is t herefore 4 p

EXERC lS ES

Find the coordinate s of the ce nte r of cu rvat ure for each of the follow in g cu rves l 2 2 2 l ~ = . x + y a

3 2 f " —Cl—L—L —L — — ﬂ z l V 4 . 2 x o . . a g ﬂ g 2/

Find the e qu ation s of the evolutes of the following curve s

2 2 2 2 ? 2 2 % § % 1 5 . x 6 . a 6 x a 6 . x a . y a . g

r f n e li se is a mi im u m at the en d 8 . Sho w th at the cu rvatu e o a l p n of the m i or axis an d t at the os cu ati circ e at t is oin t has con n , h l ng l h p tact of the third order with the curve .

FIG . 50.

Fi . This circle of curvat ure m u st be e ntirely o ut s ide the ellipse ( g the o sider t wo oi ts P P o n e on , e ac side of B the end of For, c n p n I, 2, h ,

1v C x . 1 1 1 78 DIFFERENTIAL CALCUL Us [ m 1 .

the cente r cu rv ature the v ertex A (Fig. prove 2 t at C E : a e in ic 6 h , wh h is t he e ccent ricity of t he e i se an d e ce t at C D ll p ; h n h , A C F C E orm a eo C , , f g m etric series whos e co m S o a so m on rat io is e . h w l t at D A AF FE orm a h , , f

s im ilar s erie s .

1 4 If H be t he ce ter of . n curvatu re at B s o t at , h w h the point H is without o r it i t he e i se accordi w h n ll p , ng / or a cord as a or l) x 5, c 2 in g as e o r

b i s ectio 1 5 . Show y n p n of t he ﬁgure t hat fou r real norm als c an be d rawn t o t he ellipse from an y poin t

it i the evo ute . FIG . 52 . w h n l C HAPTER XV

SINGULAR POINTS

in l . h 1 1 2 . Deﬁ nition of a s gu ar point If t e equ ation ( y x 0 e s n e d b a c u r the d a e L f( , g) be r pre e t y ve , eriv tiv ’ dx

h n has a de ermn a e a u e m easu the th w e it t i t v l , res s lope of e

an n at the n x . Ther m a be c er a n o n s t ge t poi t ( , g) e y t i p i t on the c u ve ho e er at h ch the ex res s on for the r , w v , w i p i d r v a e as s u m e s an u s or n d m n a e rm an d e i tiv ill ory i eter i t fo ; , in c n s e u en ce the s10 e the an n at s u ch a n can o q , p of t ge t poi t

d m n d b h m h d f 1 0 h n d r e e t e e o Art . u ot be i e ctly et r i y t o . S c v a u e s of x are ca e d s in u la r va lu es an d the c rre l , g ll g , o

n h c u r e are c ed s in u la r oin t s spon din g p oin ts o t e v all g p .

l 1 D min on of s in ul r oin s of br ic curv s . 1 3 . eter ati g a p t a ge a e When the e quation of t he cu rve is rat ion aliz e d an d cleare d

ac n s a the m x 0. of fr tio , let it t ke for f( , g)

Th s es b d ffe ren a n h e a d x as in i giv , y i ti tio wit r g r to ,

Ar t . 71 , (if Gf clg dx 6g dx Bf dx when ce 6f ar

In o d r ha ’ m a ec m e us r it is h e r e t t ZZ y b o ill o y, t erefor neces s ary that ( 2) 1 80 CAL C on x v DIFFERENTIAL UL us ( . .

Thus t o de m n e h h a n cu x 0 ter i w et er give rve f( , g) 6f has s n u a o n s ut an d ach e u al i g l r p i t , p e q to z ero an d 76 :

s e hes e u a n s x an d olv t e q tio for g .

If an a of v a u s of x an d s o u n d s s the y p ir l e g, fo , ati fy

e u a n x 0 the n d e m n d b hem is a q tio f ( , g) , poi t et r i e y t

n u a o n on h s i g l r p i t t e cu rve . To dete rm i n e the appearan ce of the cu rve in the vicin ity

s n u ar n x e a u a t he n d e m n m of a i g l poi t ( 1 , gl ) , v l te i et r i ate for 3f dg 6 x 0 , dx 57 0 by ﬁn din g the limit approached c on tin u ous ly by the slope é é the an n he n x x . of t ge t w l , y gl

Hen ce

2 A s 49 7 . [ rt ,

at the o n x . p i t ( 1 , gl )

Th u on c ea d ac n s s d m n the is e q ati l re of fr tio give , to eter i e s o x the u d a c l pe at ( 1 , gl ) , q a r ti (i W f 8 z 2 ( ) ag 8x 6g 6x

The This qu adrat ic e qu ation has in gen eral two roots .

n xc n s ccu h n s m u an u s at the n in o ly e eptio o r w e i lt eo ly, poi t

u s n q e tio ,

1 82 N US C m x v DIFFERE TIAL CALCUL [ .

— - ere z ﬁ x —ii x —x —4 —2 2 H gé g l ﬂ y 4 y .

[ The v a u e s x 0 0 i s atis t e se three e uatio l , g w ll fy h q ns, hence O 0 i i r s a s u a oi t . ( , ) ng l p n 82 — S1n ce 6 + 6 x = 6 at a£ 6 27 6 x 6g — 48 y z 4 at

e ce the e uatio determi in the slo e is ro m h n q n n g p , f r “ 4 " — PG 0, Ga; (a; )

of hich the root s are 1 an d . It ol o s t at 0 0 is a doub e w g f l w h ( , ) l -

oi t at ic t he t an e t s ave the s o es 1 . p n wh h g n h l p , g

' Cus s : 1 . N x let H O. Th n 1 5 p . e t e two ta gen ts are then

de n n h are d r h c n c a d cas s c n s e . If t e oi i t, t ere two e to o i c urve recedes from t he tan gen t in both directi on s f rom t he

n of an en c the s n u a o n is ca ed a t a cn ocle . poi t t g y, i g l r p i t ll

‘ T wo dis tin c t bran ches of the cu rv e tou ch e ach other at

i . this poin t . ( See F g If both b ran ches of the cu rve rece de from the tan gen t in

n n d r c on r m the n an en c the n is o ly o e i e ti f o poi t of t g y, poi t

d a c c alle u sp . 1 11 SINGULAR POI NTS 1 83

re a a n he e are a h If He g i t r two c s e s to be dis tin gu is e d . the bran ches rec e de fro m t he p oin t on Oppos ite s ide s of the

d u e an en t he cu s is s a d t o be of t he s n d o bl t g t, p i ﬁr t ki ; if

he reced on the s am e s de is ca ed cu s the s c n d t y e i , it ll a p of e o

n d ki . The method of in ves tigation will be illus trated by a few

x am e ples .

Ex. 1 .

af _ 2 3 5 if 4 4 a x 6 x 2 a . + , y dx 63/

The oi t i s atis x : 0 0 — 0 he ce it is a p n w ll fy f( , y) , Egg , g ; n

si u ar oin t . Pro ceedi to the s eco d derivatives ng l p ng n ,

2 6 f _ 2 2 4 — 12 a x + 30 x 0 at 0 0 2 ( ) r 70 6

at ”

ﬁ af 4 2 a .

The t wo va ues of are the re ore coi cide t an d each e u al t o zero . l f n n , q

rom the orm of the e u at io the cu rve is evide t s m m etrica it F f q n , n ly y l w h

re ard t o bot ax es e ce t he oi t 0 0 is a t ac ode . g h ; h n p n ( , ) n N 0 part of the cu rve can be at a gre at er dist ance from the g-ax is t han — i a at ich oi ts is i it e . The m ax imu m va u e of corre , wh p n ZZ nﬁn l g

s on ds to x z B e e e x : 0 x : t ere is a oi t of p i a . tw n , h p n in ﬂe i x on (Fig.

2 3 EX . 2 . x = x : 0 f( , g) g ;

2 3 x z: 2 , y 55 55

e ce the oi t 0 0 is a sin u H n p n ( , ) g

lar po int . 6 2 f _ — urt er “ —6 x 0 at 0 F h 2 ( a 1 676 ” 6 21 dx ay S on xv DIFFERENTIAL CALCUL U [ .

y T ere ore the two root s of the u adratic e u ati de i — h f q q on ﬁn ng are bo th Zx

e ua to z ero . So far t is cas e is ex act i e the ast o n e but ere n o q l , h ly l k l , h

art of t he cu rve ie s t o t he e t of t he ax is . O n the ri t s ide the p l l f y gh , - cu rve is s m m et ric it re ard t o t he x ax is . As x i creases i creas e s y w h g n , y n ; r r m x im a n or m i i n t e e a e n o a m a a d n o inﬂexion s Fi . h n , ( g

3 4 2 2 2 2 E . x . x x 2 ax ax a 0 . f( , y) z y g

The oi t 0 0 is a si u ar oi and t he root s ofthe u adratic de in p n ( , ) ng l p nt, q ﬁn g

( a are bot e ua t o z er h q l o . dx

Let a be os itive . S o vi the e u atio for p l ng q n y,

e x is e ativ e is im a l n ar e x 0 0 he x is Wh n n g , g g y; wh n , g ; w n ositive but e ss t a a has two ositive va u es t ere ore two bra c e s p , l h n , 3; p l , h f n h

FI G . 55 . FIG. 56 .

are a - b ch b com e s i ite havi bove t he x ax is . x a on e ra e When , n nﬁn , ng

th a m t t r r h r i ate a . The ori i e sy p o e x a ; t he ot he b anc h has t e o d n 31 g n is there fore a cu sp of t he seco nd kind (Fig.

1 1 6 H n e a . In . on u int s . as let C j gate po L tly, be g tive this cas e there are n o real tan gen ts ; hen c e n o poin ts in the im me diate v i c in ity of the given p oi n t s atis fy the e q uation of t he cu rve .

in t S uch an i s olated poin t is called a conj uga t e p o .

1 86 A C R xv 1 6 1 . DIFFERENTIAL C LCUL Us [ .

EX ERC IS ES O N C H APT ER XV

Exam ine e ach of t he following cu rv e s fo r m u ltiple points an d ﬁn d the e q uat ion s of t he t ange nt s at e ac h s uch point

2 2 z a I 1 . a x b y

3 75

2 a x

3 i % i . x y a .

4 ? 4 . a ) x . 2 g 5 . z a x bx x g + + i c .

W he n a c urv e has two parallel asym ptote s it is s aid to h ave a n ode at

h A EX . 6 . inﬁnity in the directio n of t e parallel asympto t es . pply to

? 6 (xk - T) 4 y + y = 0

2 4 1 0 . y x

1 1 o t t the c ur e = x lo x h a t erm i ati oi t at the . Sh w ha v g g as n ng p n o rigin . C HAPTER XVI

ENVELOPES

Th cu s . e e u a n a cu 1 1 7 . Family of rve q tio of rve ,

x 0 f( , ? u u n s s d s the v ar a e s x an d ce a n c oefﬁ s ally i volve , be i e i bl g, rt i

i n t ha s er e t o ﬁx the s z e s ha e an d os n the c e s t t v i , p , p itio of

The c ef c en s are ca e d co n s an s h e ren c e cu rve . o ﬁ i t ll t t wit r fe

o t he var a es x an d bu t has e e n s e en in re v ou s t i bl g, it b p i chapte rs that they m ay take differen t val ues in diﬁ ere n t

r em s h e the orm t he u a on is res erve d . Let p obl , w il f of eq ti p “ 9 , h n a en a s er a be on e of thes e c on s tan ts . T e if be giv ies

n um e a u es an d the cu s of the e ua n corre of rical v l , if lo q tio , s on din t o each s e c a a u of a be race d a s e r e s p g p i l v l e t , i of cu r e s is o a n e d all ha n the s am e en e a cha ac e v bt i , vi g g r l r t r , bu t d ffer n s om e ha r m e ach her in s z e sha e i i g w t f o ot i , p , or

A m u s s o a n d is ca d a a m il pos ition . s ys te of c rve obt i e lle f y

of cu rves .

exam e h k be x ed an d a ra the e u a For pl , if , ﬁ , p be rbit ry, q 2 n k 2 x h re res n s a am o f ara o as tio (g ) p ( ) p e t f ily p b l ,

ach cu ve of h ch has the s am er ex h k an d t he e r w i e v t ( , ) ,

s am e ax s : h but a d ffe ren a u s re c um . A a n k i g , i t l t t g i , if

be the a rar con s an h s e u a n e res n s a am rbit y t t, t i q tio r p e t f ily

a a as hav n a a e ax e s the s am e a u s c u m an d of p r bol i g p r ll l , l t re t ,

havin g the i r ve rtices 0 11 the s am e lin e x h. The pres en ce of an arbitraryc on s tan t a in the e qu ation of a cu rve is in dicate d in fu n ction al n otation by writin g the 187 1 88 R A US C a XVI DIFFE ENTIAL C LCUL [ .

: u a on in the o m x a 0 . The u an t or h ch eq ti f r f( , g , ) q ti y , w i is con s tan t fo r the s am e c u rve bu t diffe re n t for differen t

u e is ca e d t he a ra m e ter o f h m h c s t e a . T e u rv , ll p f ily eq a tion s of t wo n eighbo ri n g c u rves are then written

x a : 0 x a h : 0 f( , g, ) , f( , g , ) ,

i m n m Th in whi ch h s a s all i cre e n t of a . es e c u rves can be

u h as n ea c n c den ce as des r d b d m n h n bro g t r to oi i i e y i i is i g h.

1 1 8 . A n Envelope of a family of curves . poi t of in te r s e ctio n o f t wo n eighbo rin g cu rves of the fam ily te n ds toward

n h h n d Th a limitin g pos itio as t e cu rves approac c oi ci en ce . e lo cu s of s u ch limitin g poin ts of in ters e ction is called the m en velop e of the fa ily.

x a . 0 x a h O 1 Let f( , g , ) T , f( , g, ) ( )

B th he rem m a u be two c urves of the fam ily. y e t o of e n val e 4 (Art . 5) 8 a a 9h 2 g, g, g , + ) , ( )

h ch on acc u n ua n du c s w i , o t of eq tio re e to af e : x 04 h 0 . ( , + ) 3a

n c s ha in the m h n h =- O H e e , it follow t t li it, w e , dj x a : 0 k , y, ) da is the equ ation of a cu rve pas s i n g throu gh the lim itin g

om n e r n the cu v x a = 0 h it s p t s of i t s ectio of r e f( , g , ) wit

Th s de m n e s an ass n d a u con s ecu tiv e c urve . i ter i for y ig e v l e

t he c r of a a deﬁ n ite limitin g poi n t of i n ters e ctio n on or e

The o cus all s u ch s pon din g m emb e r of the family. l of

1 90 C v DIFFERENTIAL CALCUL US [ m x 1 . 8 a1 bu t 0 hen c e t he m e of the an en h , p t g t to t e 5g 8a e n o e at the n x is en b vel p poi t ( , g) giv y

6 3: 6g dx

B ut this eq uation deﬁn es t he direct ion of the tan gen t to t he c urv e x a : 0 at the s am e n an d here o e a f( , g , ) poi t, t f r limitin g p oi n t of in te rs ection on an y m emb er of the fam ily is a n c n ac h s cu h h n poi t of o t t of t i rve wit t e e velope .

E i th x . F nd e e nvelope of t he family of lin es

y z mx + £a obt ained by varying m . D i fere t ia te 1 as t o m f n ( ) , (2)

To e imi ate m m u ti 2 b m an d s u are s u are 1 an d s ub l n , l ply ( ) y q ; q ( ) i t ract the ﬁrs t from the s eco nd . The e nvelope s fou nd to be the parabola

2 y 4p x .

1 2 f i The u e 0 . f n m l o n cu Envelope o or a s a g ve rve . evol t

1 1 0 was d e n ed as t he o cus of t he cen ers of cu a ( Art . ) ﬁ l t rv

The c en r c u r a u e was sh n t o the n t u re . te of v t r ow be poi t of

h n i n te rs ection of c on s e cu tiv e n orm als ( Art . w e ce by

1 8 t he en e t he n m a s is the o u . Art . 1 v lope of or l ev l te

2 E i d t he e ve o e of the orm a s t o the arabo a 4 x. x . F n n l p n l p l y p Th e u atio of the orm a at x is ' e q n n l ( l ,

1 — 73 — m y y ( l) a l g§

2 or e imin ati x b m ean s of the e u at io 4 33 , l ng l y q n g1 P I, 1 - 121 E N E L OPE S 1 9 . 3 V 1 9 1

The e ve o e of t is i e e t a es all va u e s is re u ired . n l p h l n , wh n 9 1 k l , q D ifferentiating as to 3 2 x g1 , 2 8 p 2 p

4 9 2 1 y x 2 ° 1 3 ( P)

Substitu ti t is va ue for in the result ng h l gl ,

3 4 7: 2 0 1 0 ,

is the equation of the requ ired e volute .

1 21 m on u n . Two r s e o of cond on In m an pa a eter , eq ati iti . y cas es a fam ily of c urves m ay have two param ete rs which are

in con n e cted by an e qu ation . Eor s t an cey the equ ati on of

the n rm a a en cu e c on a n s a am e s x o l to giv rv t i two p r ter 1 , 9 1

’ h c are c n n ec d b the e u a o n t he r I h w i h o te y q ti of cu ve . n s u c cas es on e param ete r m ay be elimin ate d by m ean s of the g1 ven

‘

e a on an d the other e a e d as e re . r l ti , tr t b fo

hen the e m n a n is d f cu r o m h u a W li i tio i ﬁ lt to pe f r , bot eq 6 n s m a be d ffe ren a e d as on e o f the a am e ers tio y i ti t to p r t a ,

d n t he he a am e 8 a a n e s u c n a . This r gar i g ot r p r et r , f tio of as e s u u a n s r m h ch a 3 an d _ m a e m giv fo r eq tio f o w i , , y be li

in at ed the sul n u a n n h the d s d , re ti g eq tio bei g t at of e ire

en velope .

. 1 i d h e f t h Ex . F n t e e nvelop o e line

; 1 _ + 1 ’ a b

he m of i i t r e i i co t t su ts n e c pts rem a n ng ns ant . The t wo e qu at ions are 192 CH X VI DIFFERENTIAL CALCUL US [ . .

D iffe rentiate both equ atio ns as to a ;

x 1 db 2 2 (1 b da

db 1 + da

' a b

da x 1 ic re du ce s t o , wh h a2 222

x y x y a b a b 1 whence a a b a + b c / Therefore x x V; V5

i . X is the e qu at o n of the des ired e nvelope [Com pare E . p .

. 2 i t he e h m i of o x ia e i se s EX . F nd nvelope of t e fa ly c a l ll p r cons tant a ea . 2 l Here + z l ;

FIG . 59 .

1 94 c XV I 1 2 1 m . DIFFERENTIAL CALCUL Us [ .

3 . E i se s are describe d it comm o ce t ers an d ax es an d avi ll p w h n n , h ng he s u of m i- s u t m the se axe e a to c . i d t ei e v e q l F n h r n lope .

4 V . Find the e nvelope of the straight lines having the product of their i te rce ts on the co ordi ate x e e n p n a s qu al to W.

l 2 5 . i d t he e ve o e of t he i es 8 m x a r V I m m F n n l p l n g , ( ) ,

be ing a variable param eter.

6 A ir m o c t r . c cle ve s w ith it s en e on a parabola whos e equ ation is 2 4 ax an d as ses throu the vert ex of the arabo a . i d its y , p gh p l F n e nvelope .

7 f di u r h n h r . Find the envelope o a perpen c la t o t e orm alto t e pa abola ? — 4 ax dra t rou the i t ers ect io of the orm a it t he x axis . y , wn h gh n n n l w h

h u i 8 . Show that the curve s deﬁned by t e eq at ons 0° B E

in hic a an d 8 are aram eters all as s t rou ou r xed oints ﬁ n d w h , p , p h gh f ﬁ p ;

them .

“ 2 2 3 3 : 8 at 9 . In the oda am i 2 a x a x s o t n l f ly (3; ) ( ) y , h w h t he u su a roces s ives for e ve o e a co m os it e ocu s m ade u of the l p g n l p p l , p - node locus (a line) an d the e nvelope proper (an ellipse) . INTEGRAL CALCULUS

CHAPTER I

GENERAL PR INCIPLES OF INTEGRATION

m The fun damental problem . The fu n da e n tal prob lem the D f e en a C a cu u s as ex a n e d in t he e ce d of i f r ti l l l , pl i pr in a e s is h s g p g , t i

Given a u n ction x o an in de en den t va riable x to f f ( ) , f p , determin e its deriva tive f (x) It is n o w ro os e d t o c n s de t he n s e r em v iz p p o i r i ver p obl , ’ Given an un ction x to determin e the u n ction x y f f ( ) , f f( ) t having f (x) for its deriva ive . The s tu dy of this i n vers e problem is on e of the obje cts of the In tegral C alcu l us . ’ The en u n c n x is ca e d the in t e ra n d the giv f tio f ( ) ll g ,

un c n x h ch is t o ou n d is ca d the in t e ra l an d f tio f( ) w i be f lle g , the pro ce s s gon e throu gh in orde r t o obtain the u n k n own fun ction f(x) is calle d in t egra tion . The operation an d res ult o f diffe ren tiation are s ym boliz e d by t he form ula d x f ) f ( ) , dx

n in the n n d ff e n a s or, writte otatio of i er ti l ,

are ) f (x) dx 195 1 96 IN TE R AL CAL L n I C U . G US (o .

The op e ration of in tegrati on is in dicated by preﬁxin g t he

s m o the u n c on o r d fferen a h s e n e ra y b l f to f ti , i ti l , w o i t g l it d A ord n h m is e u ire d t o ﬁn . cc t e u a n e ra n r q i gly, for l of i t g tio is written thus : x f( ) dx .

o n n s a sh d u s a the d ff e n l h Foll wi g lo g e t bli e ge , i er tia , rat er

han the de a e of the un n o n un ct on x is e n t riv tiv , k w f i f( ) writt u n d th n n a on O n e of the advan a es s o er e s ig of i tegr ti . t g of d n is ha t he a a h s c t o h ch the n oi g t t v ri ble , wit re pe t w i i te

ration is rm ed is x c m en on ed . Th s is g perfo , e pli itly ti i , of c u s n ot n eces s a hen n on e ar a e is n ed o r e , ry w o ly v i bl i volv , bu t is es s en tial when s everal variables e n te r i n to the in te

ran d o r a chan e is m ad du n t he r cess g , ge of variabl e ri g p o

r n of i n teg atio .

Int e tion b in s c ion . The m s v ou s aid 1 23 . gra y pe t o t ob i to the probl em of i n tegration is a kn owle dge of t he ru les an d

d f e n a n It e ue n ha en s ha the res ults of i fe r ti tio . fr q tly pp t t requ ire d fu n ction f( x) can be dete rm in e d at on ce by re col le ctin g the res u lt obtai n ed in s om e previo u s differen tiati on .

x m s u s be e u d ﬁn d For e a ple , ppo e it to r q ire to

h x dx the d ff en a s in x It will be recalled t at c os is i er ti l of , an d thu s the an s wer to the propos ed i n tegration is directly

a n d . Th is obt i e at ,

in fc os x dx s x .

A a n s u s is e u d n e ate g i , ppo e it r q ire to i t gr

1 98 IN TE RAL AL I G C C UL US [cm .

1 24 f n . The und m n o mu s of i t i n h n the f a e tal r la egrat o . W e — orm u as diffe re n a ion 1 . 49 50 are rn e in f l of ti t ( ) pp , bo m n d the m e h d n s ec on re err d t o in the reced n i , t o of i p ti f e p i g

n the o n u n d m arti cle leads at o c e to foll wi g f a en tal i n tegrals . Upon thes e s oon er or l ater ev ery i n tegration m us t be m ade to depen d .

u n d u I . f

' I

fa u d u

u = “ fe d u c .

cos u d u z sin u . v. I

— sin u au cos a . VI . f

Q z u fsec u d u tan .

‘z - jcosec u d u cot u .

u fsec u tan u d u sec .

scc u cot u d u cosec u . X . fco

—1 ‘ l sin n, or cos u .

“ “ l l u tan u , or cot .

“ " 1 l sec u , or coss e u . 124 GE N E RAL PR IN CIPLE S OF IN TE GR ATI ON 1 99

I n h n l . n a t e a e 1 25 . Certain general pri cip es pplyi g bo v form u las o f in tegration ce rtain p rin ciples which follow from

n m n t he ru l es of differen tiation s hou ld be bor e in i d .

( a) The in tegra l of the s um of a ﬁn ite n umber of fu n ction s is equ a l to the s um of the in tegrals of each fun ction taken s ep a rately.

h m Ar . 1 T is follows fro t 6 .

xam For e ple ,

g l — —c x i fn dx = fx dx fi g

(b) A con stan t fa ctor mag be removed from on e side of the

Sign of in tegra tion to the other .

s n c For, i e d c u = c du ( ) , it follows that

z fc d u c d u cu .

To u s a u d n ill tr te , let it be req ire to i tegrate

The n u m erical factor 5 is ﬁrs t placed ou ts ide t he s ign of

n r n h h rm u a I i d n a a e c s a d . Acc i teg tio , ft r w i fo l pplie or i gly,

A a n s u s the n ra g i , ppo e i teg l

z fx ﬁ—l is b I i d n h he to e fou n d . t s rea ily oti ced t at except for t con st an t fac to r 2 t he n um erator of t he in tegran d is the ex act de a o f the d n m n a an d o mu a II u d riv tive e o i tor , f r l wo l be 200 IN TE R AL CAL C L US cm I G U [ .

a ca . All ha is e u e d th n in rd ppli ble t t r q ir , e , o er to reduce

the 1v en 1 n t e ral a n n rm is t o m u ti n g g to k ow fo , l ply i s ide the

s n n e a n b 2 an d ou ig of i t gr tio y ts ide by 5. This giv es

2 x dx 2 x dx d x - 1 ( 1

I h n t is c on n ection it m u s t n ot be forgott en that an exp res s ion con tain ing the variable of in tegration can n ot be removed

rom on e s ide o the si n o in te ratio to t oth f f g f g n he er . ( c) An arbitrary con stan t m ay be a dded to the resu lt of i in tegra t on .

For t he d r a of a c n s an is z o an d hen c , e iv tive o t t er , e

du d a c ( ) , from which follo ws

du = d u c f f ( ) u c .

Th s c n s an is ca e d t he con s t a n t o in te ra ti n i o t t ll f g o . It will be s een from this that t he res ult o f i n tegration is n ot u n u e but ha an n u m Of u n c n s d ffer n r m iq , t t y ber f tio ( i i g f o e ach o h h e n b an add v c n s an c an be t er, owev r, o ly y iti e o t t)

h h h h m n x n d foun d w ic ave t e s a e give e pres s io for eri vative . 1 C m a Art . 6 C o r. [ o p re , 2 2 3 2 Thu s an on e the u n c n s x —1 x 1 x a , y of f tio , + , + , th m f x a x a et c . s e e as a s u n e e o ( ) ( ) , , will rv ol tio of probl in tegratin g [ 2 x dx . It ofte n happ en s that diffe ren t m ethods of i n te grati on lead

ll u ch d ffe re n ces ho e ver can o ccu d ffe n e u . A s t o i re t r s lts i , w , r o nly in t he con s ta n t term s .

Fo r ex a m e pl ,

2 [ 3 ( x dx 8f( x + 1 ) d(x (x

3 4 2 x 3 x + 3 x + 1 .

202 I N TE RAL CAL C L US I G U [ CE .

2 1 0 8 x co s x dx dx f 5 3 ).

s i m n x x f n ( ) d .

2 ’ sin x o x ax f .

3 2 2 3 . co s x dx = co s x 1 s in x x f ( f ( ) d ).

2 2 5 . s i x f n dx.

2 2 t an x dx z s ec x 1 f [ f( ) dx] .

2 2 2 t a x sec x dx. 2 8 . oo sec ax x f n f ( b) d .

. L 2 9 x/ 2 . cot x cos ec x dx . \ f

’ 2 dx sec x dx j . in ) s x cos x t an x

i’ 3 1 . e u fs c t an u da .

' t u d a = co se c u co 3 2 . oot u du f ( J cosec u

‘ t an u d u

J s ec u 125- 12a] G EN E RAL PR IN CIPL E S OF I N TE GR A TI ON 203

I Integration by parts . f u an d v are fu n ction s for differen tiati n g a p ro du ct gi ves the form u la

h n ce b n e a n an d ran s os n m s w e , y i t gr ti g t p i g ter ,

‘ n d v u v — v d u I J .

This form u l a affo rds a m os t valu able m ethod of in tegra

B i u on n n as n e ra on b a s . t s s e a v en ti , k ow i t g ti y p rt y gi

n e a is m ade de en d on an o h n e a h ch in i t gr l to p t er i t gr l , w i m an y im portan t c as e s is of s im pler form an d m ore readily

r h i n teg able t an the origin al on e .

Ex . 1 .

Assu m e a lo x dv dx. g , ﬂ T e v x . h n , x

B s ubstituti in the orm u a for i te ratio b art y ng f l n g n y p s,

’ flog x ax x log x

x log x x x (log x 1)

x (log x log e) x log :

EX. 2 .

z = : u x, do e dx.

” du dx, v e ,

z m ” fxe dx xc“ fe dx e (x

Suppos e t hat a diﬁ eren t choice had been m ade for u an d re se t rob em s a p n p l , y ” u e dv x dx . z , 204 IN TE RAL CAL C L US I On . G U ( .

Fro m t his wo uld follow

u z d e dx, v

a It will be observed t hat the n ew integral fgez dx is le ss s imple in

orm t a t he ori i a on e an d he ce t he res e t choice of u an d d o f h n g n l , n p n n is n ot a fortu nate o e . o e era ru e can be aid do f r t h N g n l l l wn o e selection of u an d dv. S e ra t ria s m a be e ce s sar be ore a su it ab n e n ve l l y n y f le o ca be found . It is t o be rem ar ed ho eve r t at as far as ossib e d v s ou d be k , w , h p l h l c os e in su c a wa t at its i te ra m a be as s im e as oss ib e h n h y h n g l y pl p l , while u s hou ld be s o c ho se n t hat in differentiat ing it a m aterial s im

liﬁ cation is brou t abou t . T u s in Ex . 1 b t a i u lo x t he p gh h , y k ng g , r de t a u ctio is m ade to dis e r b if i t In t ansce n n l f n n app a y d fere nt a ion . ” EX . 2 the re se ce of eit er x or e reve t s direct i te ratio . The , p n h p n n g n rs t ac tor x can be rem oved b differe tiatio an d t u s the choice ﬁ f y n n, h u x is natu rally sugges ted .

x ﬂ a dx. Ex . 3 . [

Fro m the prece ding rem ark it is evide nt t hat the only c hoice which will s im plify t he int egral is

u x

e ce da 2 x dx v H n ,

2 x az 2 ﬁ z a n d x a dx log a log a

the sam e m et od t o the n ew i te ra assu min Apply h n g l, g

x z u x, dv a dx,

m a d a dx v whe nce , log a

sea” 1

log a log a

xa "c a“ 2 log a (log a )

t i in the re ce di ormu a substitu ng p ng f l ,

x 2 a 2 2 x f a’ dx [x

206 1 on . I N TE GR AL CAL C UL US ( .

2 2 m a e the seco d acto r the diff re tia f T i i f h k n f e n l o a x . h nk ng o t e atte r as a n e w variab e the i te ra d co t ai s t is variab e aﬁ e cte d b l l , n g n n n h l y an e x o e t an d m ulti ied b the dif ere tial of the varia b e in p n n pl y f n l , i r wh c h cas e fo m u la I can be applied .

2 w EX . . dx . f x Assum e

Then and 2 2

Here again it is n ot necess ary to write ou t the details of the s ubstitu tio as it is e a s t o t i of lo x as a n e w i de e de t variab e an d to n , y h nk g n p n n l I i perform the inte grat ion with re spe ct t o that . t s th en readily s ee n that the expres s ion t o be integrat ed cons is ts of the v ariable log x m ul tiplied by its differential an d th at the integration is accordingly x

u n i i h re d ce d t o a im m ediate appl c at on of t e ﬁrs t form u la of inte gration . Thus 2 flog x d (log x)

1 dx . E1 3 . e 1 + x2

“ l T Think of t an x as a n ew variable an d apply formula IV. hus

1 dx 1 - ' 1 u m W d(tan x) c . 1 x2

‘ l s in x dx 4 x . E . f x2

- l Think of s in x as a n ew v ariable an d as the differential of

A u . th at variable . pply form la 1

2 5 x 2 x 3 x 1 dx. Ex . . f( ) ( )

Th the rm M ultiply an d divide by 2 . e integral the n t akes fo

2 %j (x + 2 x 3) (2 x + 2) dx.

2 Observ in t at 2 x 2 dx is the dif ere tia of x 2 x 3 an d t i g h ( ) f n l , h nk in of t he atte r ex res s io as a n ew variab e it is s ee t at orm u a I is g l p n l , n h f l direct a licab e eadi t o the resu t ly pp l , l ng l }(x2 2 x GE N E RAL PR IN CI PLE S OF IN TE GRA TI ON 207

2 2 6 1 sin x 1 x dx. EX . o cos x . fl g ( ) ( )

M ake the s ubstitution

1 z .

The given integral t akes t he form

o cos s in { fl g z z dz.

M a e a seco d cha e of variab e k n ng l ,

cos z y.

Then sin z dz The tran s form ed integral is

2 lo 7 d / .% f g .

Art . 1 26 can be at t o i the r sult f Ex . 1 o ce a i c e o e d . wh h , , n ppl It will be observed t hat two su bs tit u tions which natu rally su ggest t e m se v e r m the orm of the i te ra d are m ade in s u cces s io Th h l s f o f n g n n . e two to et er are obviou s e u iva e t to the on e tra sform atio g h ly q l n n n ,

2 cos (x 1) g.

Eit er u t x az or els e divide u m erator an d de omi ator b a an d h p , n n n y , write in the form

R e ardi f as a n ew variab e t is com es u der an d ive s the g ng l , h n XI g a resu lt

— 8 1 . In a s imilar m anner tre at Ex s . 0

Ex 8 . .

E X . 9 .

Try also the s ubstit ution x 208 IN TE RAL CAL C L Us CH 1 G U ( . .

EX . 1 0 .

Tr als o t he s u bstitution x y a z.

E 1 1 X . .

M ake the tran s form ation

rom t his o o s b diﬁ ere ntiat ion F f ll w , y ,

t at is h ,

dx dz dz

2 2 2 s i a x i a + x z

EX . 1 2 . I ? 2 x a

x —a 1 = + z Assu m e z ; that is , x a x + a l - z

The re as o ns for the cho ice o f s ubstitut ion m ade in this an d the pre

b m 1 33 an d 1 . ceding ex ample will e ade clear in Arts . 39

x . 1 3 cos ec x dx. E . I

i n divide b s It l b r M ult ply a d y co ec x cot x. wi l e eadily s ee n the integral the n t akes the form If An other m et hod would be to u se the trigonom etric formula

x x s1 n x = 2 sm — cos 2 2 whe nce fcosec x dx f z f 2 s in cos 2 2

1 4 x . Ex . . fse c dx = Put x z , an d us e Ex. 13 .

210 IN TEGRA AL L C C L on . I U Us ( .

d “ 1 VII X 2 2 u a a.

d u 1 u a VIII _ 10 X 2 2 g _ a 2 a u + a

1 —1 XIX . sec or cosec

— 1 ° vers 9 a

XXI . tan u d u lo cos u lo sec u f g g .

u u . XXII . f oot d log sin u

7 III se u d u u u XX . c log (scc tan ) log tan + { (5 Ti )

t - XXIV. cosec u d u log (cosec u cot u ) log tan f 22

Int ra1§ of the fo m 1 29 . eg r

In tegrals of this form are of s u ch frequ en t o ccu rren ce as

m n Th n r n i r i to des erve s pe ci al e tion . e i teg atio s e ad ly effe cted by the s u bs titu tion of a n e w variable whi ch re du ces the

r T r t o d r d r adical t o a s imple r fo m . wo cas es a e be c on s i e e

i n acc ordin g as a s p os itive or egative . h v . In a d d n h CAS E I . a p ositi e t is c s e by iv i i g out t e 2 c oefﬁcien t of x t he radi cal m ay be written — 128 129 ] GE N ERAL PRIN CI PL E S OF I N TE GR ATI ON 21 1

The given in tegral then take s the form

z + B (Ax + B ) dx x= x +

2 aB bA

2 a \/ a

2 aB — bA 10 g z +

2 — aB bA b 2 b 10 g x + - x + 2 a \/ a 2 a

h A ive . n d C SE II . a n e at a is n a e b d n g W e eg tiv , y ivi i g out the pos itive n u m ber a the radical be com e s

an d in c on s e qu en ce the in tegral takes the form 1 (Ax B ) dx

2 a B — bA j f 2 a x/ d 21 2 I N TE RAL CAL C L US on . 1 G U ( .

O n div idi u m ng n erato r an d deno minato r by V3 the inte gral reduce s to (a s ) I m ’

i b m e f the s u bstit tio wh ch y an s o u n x g, z can be written

—z + V3 dz ’ ) (35 1 4 d J (h mfl fi j m fi éfi

- log (z + m

‘ r t r n d d om i V2 Th D ivide n u me a o a en nator by . e in te gral becom es

2 x 1

(z x

” 1 s i n (4 x

1 3 0 of the fo . Int egrals rm

In te grals of this type can be redu ced t o the form given in the precedin g article by m ean s of the reciprocal s u bs titu tion

1 Ax + B Z

IN TE G RAL CAL C UL US

dx I 2 x (log x) x

sec x 2

a — b t an x)

d dd 3 3 - l — . fvers i a 1 cos 0 Vx

' t an x dx

2 a b t an x

‘ dx [S ubstitute x j 1 cot x

dx 2 ? Pu t 1 cot x 3 5 o 511 V 33 O fl g ( ) 3 x {A C HAPTE R II

R EDUCTION FOR MULAS

1 h In A s . 1 29 30 t e n a n c a n s m e ex 1 3 1 . rt , i tegr tio of ert i i pl

2 pres s ion s con tain in g an irration ality of the form Vax bx c A h has e n x a n ed . s w as s n in Art . 1 29 the ad ca b e e pl i ow , r i l

2 2 can be redu ced to the form j: x j: a by a chan ge of vari

I em a n s h how h n a n able . t r i s to ow t e i tegr tio can be per

m ed in s u ch cas s as x m e for e , for e a pl ,

2 2 a j: x i a dx f ,

n bein g an y in teger . For this pu rpos e it is c on ven ien t to con s ider a m ore gen eral

n a h ch h r d n r iz t e e ce a e s e c a cas e s v . type of i tegr l of w i p i g p i l , ,

m ” x a bx p dx 1 f ( ) , ( ) in h ch m n are an n u m e s ha e n e ra or rac w i , , p y b r w tev r, i t g l f

i n l t o a s e o r n e a . , po itiv g tive

It is t o be r m a e d in the s ac ha n can h u e rk ﬁr t pl e t t , wit o t

os s of en e a b r rd d Fo r r e e a e as os v e . n e e l g r lity, g p iti , if w l n e a e s a n n t he n e ran d c u d be r en g tiv , y , i t g o l w itt

m m x ” p x ax ) .

m ex r ss on h ch is the s am e e as x a is p e i , w i of typ (

ha the ex n en of x n de h r n h t t po t i s i t e p a e t es is is pos itive . 21 5 21 6 IN TE R AL CAL OIL II G C UL US [ .

It will n ow be p roved that an in tegral of the typ e ( 1 ) can in gen era l be redu ced to on e of the four in tegra ls

” a bx p dx b A bx” pdx ( ) ) , ( ) ) ,

m ( c) A fx ( a (d) p lus an algebraic term of the form

" B x ( a

A B A u. are c a n c n s an s h ch be d Here , , , , ert i o t t w i will eter

m n d e n i e pr s e tly. Obs e rve that ih e ach of the fou r cas es the i n tegral t o which ( 1 ) is red uce d is of t he s am e type as bu t that

h n h i x n z ce a n c a s av a en ace n the e on e s v i . rt i ge e t k pl p t , , the expon e n t m o f t he m on omial factor is in cre as ed or d m n s h d b n i i i e y ,

the x n en t he n m a is n cre as e d or d m n or, e po t p of bi o i l i i i ishe d by un ity. The valu e s of A an d u are de term in ed by the followin g r ule Comp are the exp on en ts of the monomial factors in the given

S e ect in tegra l an d in the in tegra l to which it is to be redu ced . l

he the les s of the two n umbers a n d in crea se it by u n ity. T

m are the ex onen ts resu lt is the valu e o A. I n like man n er co f , p p

t b ia to th two in te ra ls s elect the less an d of he in om l fa c rs in e g , ,

in crea s e b u n it . This ives u . y y g ,

Thu s is des e d t o re du ce the v en n a to , if it ir gi i tegr l

m ” A x bx p dx f ) ,

ﬁrs t write down the formu la

m " " ’ " fx ( a bx ) p dx Af bx ) 1 dx B x ( a

21 8 I TE RAL 11 N G CAL C UL US [Cm .

Th e is on e cas e h e er in hi ch h du i er , ow v , w t is re ction s

m s viz . h n i po s ible , , w e m n 1 0 p ,

in h as e A an d B m i n 4 for a c eco e n e . S ee Ex . . t t b ﬁ it [ , p In a s im ilar m an n er the three followi n g form u lae m ay be derived

m ” fx ( a bx ) p dx b< m + n + np + 1 ) x ( a + bx ) d + a (m + 1 ) a (m + l )

m ” fx ( a bx ) p dx

m 1 x ( a dx

m ” fx ( a bx ) p dx

m+ l n + l x a bx p n + 1 < ) ( a bx ) p dx [D] (“ KP 1 ) au (p 1 )

The cas es in which t he above redu ction s are imp os s ible are , For form ulae [A] an d whe n m np 1 0 ;

for form u la [B ] whe n m 1 0 1 for fo rm ul a [D] when p 0.

3 2 2 1 x Va x dx. Ex . . f

3 h m o omia actor ere x i ste ad of x the i te ratio could If t e n l f w n , n g n

z e ec ted b u s i orm u a I . Si ce in the res e t case m 3 e asily b e ff y ng f l n p n , n 2 o rm u a A ic dimi is es m b n i l reduce t he above , f l [ ] , wh h n h y , w l integral t o o n e th at can be directly int egrate d .

I s te ad of s ubstit u tin in A a s m i t re adi be do e it is best t o n g [ ] , gh ly n , w u sed apply t o partic ular problem s the s am e m ode o f proce dure th at as in de rivi T r are two adv a t a es in t is. ng the gen eral form ula . h e e n g h

irs t it m e the t ude t in de e de t of the ormu as an d seco d, F , ak s s n p n n f l , n m de in the sam e rob em t he or whe n several redu ctions h av e t o be a p l , w k

EX . is generally sh orter. [See R ED UC TI ON F ORM ULAS 21 9

Accordingly assum e

l 3 2 2 2 2 2 % 2 2 fx (a x ) dx Aj x (a x ) dx B x (a the values of A an d pt having bee n determ ined by the previously given

rule . 2 D i fere tiate an d divide the resu ti e u atio b x a 2 Thi f n , l ng q n y ( x3 . s gives 2 2 2 x : A B 2 a 5 x + ( ) ,

rom hich b e uati coe ﬁcie ts of i e o ers of x f w , y q ng f n l k p w ,

2 2 a A B T , i, an d e ce h n , f Q ? 2 % 2 2 3 v 2 x2 dx — x x dx x a 2 f z a gfw ) }; ( x3

2 2 2 (1 3 x ) (a

B o o i the su estio s of Art . 129 this i te ra can be reduce y f ll w ng gg n , n g l d to the form

in ich wh z x 1 . As sum e - 2 % 2 f(z 4)t Af (A 4) dz Ez (z In determi i A otice t at m 0 in both i te ra s s o that n ng n h n g l , A= O A so + l ,

h 2 m a be o o ed . A ther m ethod T e m ode of proce dure of EX . y f ll w no can a s o be u se d as o ows l , f ll On writing in the form

x2 2 a x 2dx f ( ) , an d observing th at ‘ ‘ _ x dx ‘ ' 1 x 2 2 a x 2dx vers ( ) , 5 2 j a 2 ax x

it will be see n that the integration m ay be effected in the present case b T is is os sib e y reducing e ach of the expon ent s m an d p by u nity. h p l assum e s in ce n 1 an d m can accordingly be diminished by 1 . Hen ce

. 2 ' ’ i ’ i Iw (2 a xﬁ dx A fx (2 a xﬁ dx E x (2 a 220 IN TE R AL CALC L Us C II G U [ m .

The ex o e t f h bi p n n o t e no mial in the n ew int egral m ay be re duced in t urn by ass uming

‘ ” “ ” x 2 2 a x 2dx A x 2 2 a 56 d B x2 2 2 f ( ) f ( ) ( a x) .

When this expre s s ion is s u bs tit ut e d for the integral in the second

member of the recedi e u atio the resu t t e s the r p ng q n, l ak fo m

2 l’ B x (2 a xﬁ Cx (2 a

’ ’ ” ’ " ' in ic A B C are ritte for bre vit in the ace of A A A B B wh h , , w n y pl , , ' A B are ca cu ated i re s ective . The va ues of C n the u sua m a e r p ly l , , l l l nn b diﬁ ere n tiatin s im i i an d e u ati coe cie ts of li e ers y g, pl fy ng, q ng fﬁ n k pow

of x .

The m et od u st ive re u ires two reductio s an d he ce is ess h j g n q n , n l

suitab e t a t at em o ed in EX . 2 ic re u ires but on e reductio . l h n h pl y , wh h q n

The u de e m n n the a u s A an d u. m a n ow r le for t r i i g v l e of , y

be ad an a e us a re v e d . m be the ex on en s v t g o ly bb iat Let , p p t ' ’ the ac s in t he en n ra an d m t he c e of two f tor giv i teg l , , p orr

h n f h s pon din g e xp on en ts in t e n e w i tegral . O t es e t wo ’ ' a rs m an d m on e the n u m e s in the on e a is p i , , p , p , of b r p ir

han the c rr s n d n n u m in t he o h Th less t o e po i g ber t er pair . is fact will be expres s ed brie ﬂy by s ayin g that the on e pair is

h h h h u n d n n h les s than t e ot e r p air . Wit t is ers ta di g t e pre cedin g ru le m ay be ex pres s ed as follows

S elect the less o the two airs o ex on en ts m an d f p f p , p ’ ’ m I n crea s e ea ch n u mber in the a ir s elected b unit . , p . p y y

This ives the air o ex on en ts A 1 . g p f p , 1

4 $ 613; 4 Ex . . I 2 % a )

As sum e su cces sively

- . 4 2 2 2 ’ 5 2 2 h dx A' x x a dx B x x a f ( ) ( ) ,

- — 4 2 2 2 " 2 2 ” 3 2 x (x a ) dx A a ) dx B x (x — — 2 2 2 2 m z 2 2 il/ 2 x (x a ) dx A fcc a ) dx B x(x

IN TE RAL CAL C L US C a II. 13 1 G U [ .

EX . 6 1 3 . j am

EX . 7

Ex . 8

EX . 1 5

9 X .

EX . 1 6 i 1 dx Ex . 1 0 $ . j 2 2 2 x \/ a x

dx E 1 1 EX . 1 7 x . J f (M ar 1 2 2 % EX . 1 2 . a x f( ) dx.

EX . 1 9 t . Show h at

” dx 1 + 2 n - 3 2 ( ) 5(x + c) ” 2 c (n 1 ) CH APTER III

INTEGRATION OF R ATIONAL FRACTIONS

1 3 2 . n fr c ns The c the Decompos ition of ratio al a tio . obje t of p res en t chapter is to s how how to in tegrate fraction s of the form M , f (x)

n ar o n i wherein ¢(x) a d «f( x) e p ly omials n x . The des ired resu lt is accom plis he d by t he m ethod of s epa ratin g t he given fraction in to a su m of term s of a s im ple r

n d an d n e a n e rm b e m . ki , i t gr ti g t y t r Ifthe degre e of the n u m erator is equ al to or greater than the de re the de n m n a r the n d ca e d d s n can g e of o i to , i i t ivi io be carried o ut u n til a rem ai n der is o btain ed which is of lower d h h H n th c n an egree t an t e den omin ator . e ce e fra tio c be redu ced to the form

in h ch th de r e o f i h n h w i e g e f(x) s les s t a t at of d (x) . ( x) As to the i n tegration of the rem ain der fraction f it is f ix) to be rem arked in the ﬁ rs t place that the m e tho ds of the precedin g articles are s u fﬁcien t to effect the in tegration of s u ch s imple fraction s as

' ' A A M x --n M x -—Z W Px l I + Q 1 ’ ’ ’ ) — — 2 z z z z ? 2 ( x a (x a) x i a ( x zt a ) x + mx + n

N ow the su m of s ev eral s u ch fraction s is a frac tion of the

n d un d i n ho n um r is c on s d a n v z . o e s o ki er i er tio , , w e e at r of 223 224 I N TE R AL CAL C L US Cm II I G U [ .

o e de h n it d n m n or Th l w r gree t a s e o i at . e qu es tion n at urally

ar s s as h h the c on e s e is s s e ha is : can ever i e to w et er v r po ibl , t t y

fraction be s ep arated in to a su m offractions of a s s imp le x typ es as thos e given in

The an s er is es . w , y Sin c e the s um of s everal fraction s has for it s den omin a

t he as c mm on m u e the s e a d n m n a rs tor le t o ltipl of ver l e o i to , i —z it follo ws that if can be s eparated i n to a s um of {Fzx3 s m ac n s the d n om n a rs hes e ac on s m u i pler fr tio , e i to of t fr ti s t

N o i be divis o rs of xp(x) . w it s k n own from Algebra that every p olyn omia l 11r (:v) having rea l coefiicien ts ( an d on ly thos e havi n g real coefficien ts are to be c on s idered in what follo ws) can be sep ara ted in to fa ctors of either the first or the s econ d

de ree the coe icien ts o ea ch a ctor bein real. g , ﬁ f f g This fact n aturally leads to t he dis c us s ion of fou r differen t

0 3 8 8 8 .

' h c n be a a d n a c t rs t he I . W en « f(x) a s ep r te i to re l fa o of

s d n o a e . ﬁr t egree , two lik — — — b x c . E . . d x g , ) ( ) ( )

h n the ea c s are the rs d s m II . W e r l fa tor all of ﬁ t egree , o e d of which are repeate .

2 3 x b x c . E . . a x g , ( ) ( ) ( )

h n s m e t he ac s are n cess the III . W e o of f tor e arily of

e c n d de bu t n o su ch are a e . s o gree , two lik

2 2 2 2 1 b x a E . . « x x a x x x g , f( ) ( ) ( ) ( ) ( )

n n d de c s ccu s m h ch IV . Whe s eco gree fa tor o r . o e of w i are re peate d . — 1 x b . E . . x g , + ) ( )

TE GRAL CAL C UL US

Ex . 2 2 x2 + 3 x —2

EX . 3

EX . 4 . 2 — 2 — 3 x + 4 x + 3 2 x x x 3 “3 dx 1 3 Ex . 4 EX . 5 . 2 L K) x + 7 x + 1 2

dx " x dx L EX . 1 4

EX . 6 . M2 2 —52 x2 —4 x + 1

Ex . 7 . 2 (x 4) (4 x2 i — Se ar t M 2 cx + ac a b+ bc [ p a e into par X 8 . — E . 3 x + 2 tia / (x a) (x b) (x 0) l v 2 4’ M Ex 9 . EX . 1 6 . 5 2 (x +

C AS E II . F c o s f the ﬁ s d e om r 1 34 . a t r o r t egr e , s epeate

As sum e

' 5 x2 3 x + 1 A B C D (1) I I I x (x x III - 1 (“f

To u sti t is as s u m tio observe that j fy h p n, a In addi t he ractio s in t he ri t-ha d member the east com ( ) ng f n gh n , l m on m ultiple of t he de nom in ators will be x (x which is identical - wit h the denom inator in t he left hand m ember. b urt er t he ex res sio s x x 1 x x are the o l ( ) F h , p n , , ( ( n y o es ic can be assu m ed as de om i ators of the artia ract io s n wh h n n p l f n , s ince thes e are the only divisors of x (x

c e e u atio 1 is c e ared of ractio s an d the coef cie ts of ( ) Wh n q n ( ) l f n , ﬁ n

i e o ers of x in bot m em bers are e u ate d ou r e uatio s are ob l k p w h q , f q n t ain ed ic is ex act the ri t um ber rom hic to determ i e the , wh h ly gh n f w h n o ur u o B D co st a ts A C . f nkn wn n n , , , Inst e ad of the m et hod j ust indicated in (c) for calculating the coefﬁ

cien t s a m ore ra id rocess ould be a s o o s : , p p w f ll w B c e ari of ractio s the ide tit 1 m a be ritte n y l ng f n , n y ( ) y w

2 — 5 x 3 x + 1 = A (x Cx (x 133 . INTEGRATION OF RA TI ON AL FR ACTION S 227

Putting x 1 gives at once

3 D .

Substit ute for D the va ue ust ou d an d tra s ose the corres o d l j f n , n p p n in g T is ive s t erm . h g ' 5 x2 C x (x

It can be s ee n by ins pection t hat the right -h and m ember of the re s ult As t is re at i is n i is divis ib e b x 1 . o a de t it it o o s t at the l y h l n n y, f ll w h - i i i ib e b x 1 e this ac r is r left hand m ember s also d v s l y . Wh n f to e m oved rom bot m embers the e u atio redu ces to f h , q n

5 x —1 = A x ( 0 93.

N o u : 1 T e w p t x . h n

0 4.

i t u e ud for tr s o e an di i Subs t t ut e he va o C a s d v de b x 1 . l f n , n p , y The result IS x 1 B x. 1 A. ( )

B 1 v1n x the va u es 0 an d 1 in s u ccess io it is ou d t at y g g l n, f n h

A 1 , B 1 . Accordi ngly,

2 — . (5 x 3 x + 1 2dx l 1 4 3 f = _ + — + x(x J( x x l (x

x —l 8 x — 5 log x 2 (x

Ex . f(x

x2 —l 1 x - 26 dx B ( 4 2 1 d(I: 3 . V Q — L EX . 1 0 f 2 x a 3 2 ( ) [ x + 3 x x dx 1 2 a “ 1 x . 1 1 d x bx 2 2 ? E dx. f(x a ) f( )

V Ex . x2(x V8) ?

5 x 5 x 3 2 111130 Part l al f$ 62;

x4 2 d2x2 a4

Ex . 1 3

— [Substit ute x a z z 228 CAL C L 1 INTEGRAL U US [ on 11 .

1 35 . CASE III . Occurrence of u dr q a atic fact ors , none

repeated .

(x2 l ) (x2 2 x + 2) As sum e 2 4 x + 5 x + 4 Ax + B C x + D 1 + ( ) 2 x + 1 x2 + 2 x + 2 Then

2 2 (2) 4 x + 5 x + 4 = (Ax E) (x + 2 35 + D ) (x2 +

B y e qu ating coefﬁcients of like powers of x

= 4 Q E + D , from which

He nce the give n inte gral becom es 2 x dx _ 1 _1 x + 1 Q t an 95 + t an (H 1 l l x2 + l j x2 + 2 x + 2

To m ake clear the reason s for the as sum ptio n whic h was m ade c on cer i the orm of e uatio I observe t at s i ce t he actors of the n ng f q n ( ) , h n f 2 2 de om i at or in the e t m ember are x 1 an d x 2 x 2 t ese m u s t n n l f , h

ece s s aril be the de o mi at ors in the ri t m e mber. A so s i ce t he n y n n gh l , n um erat or of t he ive ractio is of o er de re e t a its de om i ator n g n f n l w g h n n n , the nu m erator O f e ach partial fractio n m u s t be of lower degree than its

de om i ato r. As the atte r is of the se co d de ree in e ac case the n n l n g h , m os t gen eral form for a n u m erator fu lfilling t his requirem e nt to be oflower degree th an it s denominator) is an expre s sion of the fi rst degree ' s uc as Ax B or Cx D . h , N otice besides t at in e uati the coe cie t s of i e o ers of x in , , h q ng ffi n l k p w oppos ite m embe rs of e qu atio n fou r e qu ations are obt ained which exact su ce to determi e the our u o coe c ie ts A B C D . ly ffi n f nkn wn ffi n , , ,

4 dx

EX . 2 . 2 x4f + 2 x

’ ‘ “ I” x dx 3 7 EX . EX j 2 j 4 2 (x + 1 ) (x + 1) x + x + l

’ dx x dx Ex 4 Ex 8 8 j + a (x m ' . Ex. 5 . Ex . 9 . 2 2 2 j z i(x a ) (x (x i ) (z + 2 93 +

230 CAL m III — C L Us C . 136 1 37 INTEGRAL U [ .

The ﬁrst term becom es

9 u x dx dx 1 — 2 — -1 log (x 2) - tan j x2 2 5x2 2

The s eco d i te rated b the m et d f r o o eductio C a . II i e s n , n g y h n ( h p ) , g v

x 1 x t an “ l 2 x + 2 « 5 V2

i a b a i orm u a I the as t te rm i F n lly, y pply ng f l l ntegrates imm ediately ence H i 5 — 4 -- 3 2 x x i 8 x -4- 4 x 2 dx = 1og (x2 (x2 +

— 2 — x l g 3 x + 2) dx Ex . 2 . Ex , 5 , ! 2 f 2 2 2 x + 1 x (x + 1)

2 2 ’3 2 2 x x - a - a 2 x -l—2 a x —x Ex s i _ )— . ax Ex 6 f 2 “ 2 2 2 (x + a > (a: + a )

2 x dx x5 dx

Ex 7 . f 2 3 (1 + x )

The prin ciples u s e d in the pre cedin g cas es ih -the assum p tion of the partial fraction s m ay be s u mm ed u p as follows :

E ach of the den omin ators of the p artia l fra ction s con ta in s

o e on e an d on ly n e p rime fa ctor of the given den omin a tor . Wh n

a re eated rim e a ctor occu rs all o its d eren t owers must p p f , f iﬁ p

o be u sed a s den omin at rs of the p artia l fra ction s . The n u merator of ea ch of the a s su m ed fra ction s is of degree on e lower than the degree of the p rime factor occurring in the

corresp on ding den omin ator .

h m . S n c n a rac n c an 1 3 7 . General t eore i e every ratio l f tio

be n e a d b rs s e a a n n ces sa n s m er i t gr te y ﬁ t p r ti g, if e ry, i to i pl fractio n s in acc ordan ce with s om e on e of t he cas es c on s idered

a e the m r an c n c u s n is at on ce d du c bov , i po t t o l io e ible

ra tion a l ra ction can be oun d an d is The in tegral of every f f , - ex res s ible in term s o a l ebraic lo arithmic an d in vers e tri o p f g , g , g

n ometric fu n ction s . C HAPTER IV

INTEGR ATION B Y RATIONALIZ ATION

At the en d of t he pre cedi n g chapte r it was rem arked that

ra u n c on can n Th eve ry ration al algeb ic f ti be i tegrated . e qu es tion as to t he p os sibility of in tegratin g i rration al fu n c

n d Th s h a r tion s has n ex t to be c o s idere . i as l eady been

' he d u on in Cha er II he a ce a n r a to uc p pt , w re rt i type of i r f tion al fu n cti on s was t reated by the m ethod o redu ction . In the pres en t chapte r it is propos ed to co n s ider the

m les t cas es of r a o n a un c n s viz . h s e con a n n p i r ti l f tio , , t o t i i g

Of a n a z a n s u ch u n c on can n a e d . r tio li tio , every f ti be i tegr t

1 3 8 . Integration of functions contain in g the irration ality

t he n th r an ex s s n the s d ree in x bu t n o oot of pre io of ﬁr t eg ,

he a n a c an be re u ce d a n a m b ot r irr tio lity, it d to ratio l for y mean s of the su bs titu tion

= Vax + b z .

2 2 2 x + 3 z .

dx z z dz , IV INTEGRAL OAL O UL vs [cm .

x§ + x

It would appear at ﬁrst s ight that this integrand cont ain s s everal r i i V7 / e i rat o a ties viz . 53 x x It IS re adi s ee o ev er t at t n l , , . ly n, h w , h h y

are all o e rs of V5 an d he ce the substit utio {721} i ratio a iz p w , n n z w ll n l e the ex res i t p s on o be in tegrated . 7 dx i d” UK. Ex u 3 h M m e . f / — x \ x + l (x 2

dx dx Ex . 7 Ex . 4 _ . — — ( x a a M

Ex . i f2 v x —1 + x x§ + xlg

he n ra n a s the m V ax b c d W two ir tio litie of for + , +

ccu in the n e an d the s ad ca can m ad t o dis o r i t gr , ﬁr t r i l be e appear by the s u bs t itu tion

The s e con d radical then redu ces to

b d ) + .

h x r n an d he m ethod of t e n e t a ticle ca be applied.

2 1 39 . Int ion of ex r s s ion s con ai nin x/ egrat p e t g a w bx + 0 , 2 E ex s s n c n a n n V ax bx c bu t n o he very pre io o t i i g + + , ot r

a n a can be a n a z d b a s u s u n . irr tio lity, r tio li e y proper b tit tio

In de t o m a e the n ec s s a s e s c ea e r a m ca or r k e ry t p l r , geo etri l

n n h o em r u e i te rpret atio of t e pr bl will be ve y s ful . To thi s en d let the given radical be repres en ted by g ;

h is let t at , ( 1 )

234 INTEGRAL CAL C UL US [GEL IV

n s n e s e c n . The he c rr s n d n h poi t of i t r tio ot r root , o e po i g to t e

n P is variable poi t ,

From this follo ws

— 2 = z x —l = x g ( ) 2 z — 1

2 z — 1

Two particu lar cas es of the m ethod given above des erve

n to be oti ced .

a hen the con ic in ter ect t - ( ) W s s he x axis .

2 In this cas e the quadratic expres s ion ax bx c has real

ac s s a f tor , y, — x) ( x

The co n ic (1 ) i n te rs e cts the x-axis in the two poin ts

a O an d her on e h ch m a c n en n ( , ) eit of w i y be o v ie tly s ele cte d for t he poin t Q . The e qu ation of an y lin e QP through the ﬁrs t poin t is

z x a {9 ( ) , an d the u a n an n h ou h the s c n d n eq tio of y li e t r g e o poi t,

’ a z (w B) (A )

E h on e h s e u a n s c m n d h it er of t e e q tio , o bi e wit will h effect t e des ired ration aliz atio n .

(b) When the con ic is an hyp erbola .

2 This cas e occu rs when the coefﬁcien t of x is pos itive .

Th cu x n d n n in d n d c on s e rve e te s to i ﬁ ity two iffere t ire ti , h n am the d re c n s the as m s . If on e t e ely, i tio of y ptote of

n s at n n on t he cu be a e n the n t he poi t i ﬁ ity rve t k for poi t Q, lin e s QP pas s in g throu gh this poin t are p arallel to that INTEGRATION B Y RATI ON ALIZ ATI ON 235

h as ymptote which tou ches the curve at Q . T e eq uation s of the asym ptotes are

Ac cordingly the lin es paralle l to the on e asymptote are

V5 x z g , an d thos e parallel to t he other

' g = (B )

Either of thes e equ ation s u s ed in place of (2) will s erve 2 equ ally well in expre s s in g x an d g( = V ax bx c) ration

ally ih term s of a n ew variable z .

2 The conic y \/x 2 x 1 is an hyperbola an d formula (B ) can be T is i s applied . h g ve

e ce b s u ari an d s o vi for x wh n y q ng l ng ,

2 z + 1 x ’ z) an d accordin gly 2 z 2 z 1 2 z )

2 (l z)

he t ese ex res io s are substit uted in the ive i te ra it becom es W n h p s n g n n g l,

2 z 2 1 4 2 z d + d 2 z % 2(1 + z) f( 1 + z 2 - z + 4 log (1 + z) + 1 H ] 1

cou d be u h i i l sed for t e purpose of rat on aliz at on . 236 L GIL I V INTEGRAL CA CUL Us [ .

2 Ex . .

The de omi ator be i ratio a iz ed the i te ra d ta es the orm n n ng n l , n g n k f

“ (1 x) The conic

i te rs ect s t he x-ax is in two oi t s 1 n p n ( A; , If t he oi t 1 0 be c os e for t he e u atio of an i e assi p n ( , ) h n Q , q n y l n p ng t hrough t his poin t is y

The s im ult aneou s s olutio n of these two equ at io ns gives

2 z 1 2 x z 2 2 2 1 z 1

2 2 VI x 2 z dz en ce dx wh 2 2 (1 x) z 1

“ 1 2 ( z t an z )

- l t ail x —l x — l

Ex . 3 .

1 40 From ha ce des c m n d h the h m . w t pre , o bi e wit t eore of

1 87 s ha er a n a un c on de en d n rt . A , it follow t t ev y r tio l f ti p i g on ly on x an d the s qu are root of a polyn om ial of n o t highe r

n h e c n d de e e in x can be n e ra ed an d the s u tha t e s o gr i t g t , re lt

n n n expres s ed in te rm s of k ow fu ction s .

EX ERC IS ES O N C H APT ER IV

2 , — § 2 (x a ﬁ (x a )

CH APTER V

INTEGR ATION OF TRIGONOM ETR IC AND OTH ER TRAN S CENDENTAL FUNCTIONS

In e ard the n a n r n m 1 41 . r g to i tegr tio of t igo o etric fun c

n s is t o be m d in the s ace ha tio , it re arke ﬁr t pl t t every ration al trigon om etric fu n ction can be ration ally expre ss ed i m n n n n t er s of s i e a d cos i e . It is accordin gly eviden t that s uch fun ction s can be in te grated by m ean s of the s ubs titution

s iu x 3 .

A t h u u n ha e n ff c d the n n fte r e s bs tit tio s b e e e te , i tegra d m ay in volve t he irration ality

— 2 z ( cos x) .

Th can m d b n a z a n as x n d in the is be re ove y ratio li tio , e plai e

ced n ch the m h d du c n m a be pre i g apter, or et o of re tio y employed .

The s ubs titution cos x 2 will s erve equ ally well .

It is u s u a as h n the n m c lly e ier, owever, to i tegrate trigo o etri form s withou t an y s u ch previous tran s form ation to algebraic

The f llo in es the c s s m s fun c tion s . o vV g articl treat of a e of o t fre qu en t occurren ce .

i‘ n zn 1 42 . x d x cosec x d x . fsec , f

In this cas e n is s uppos ed to be a pos itive in teger .

zn If s e c x dx be writte n in the form

’ i’H z 1 t an x s ec x s ec x clx ( ( ) , 238 H V l 41—l 43 C . . . ] TR I G ON OM E TR I C F UN C TI ON S the ﬁrs t in tegral be com es

t z f( an x x) .

If be ex pan ded by the bin omial form u la an d n a d m b m t he u d es u is ad i tegr te ter y ter , req ire r lt re ily

n obtai ed .

In m an n like er,

i ‘ z z fcosec n x cos e c x dx

2 f( cot x ( cot x) .

Th s as m can n a d as in the ced n c s i l t for be i tegr te , pre i g a e ,

h n m i by expan din g t e bi o i al n the in tegran d . The s ame m ethod will eviden tly apply to in tegrals of th form

m zn m i n t an x s e c x dx oot x cos ec x dx f , f , i in which m s any n umber .

EX ERC IS ES

dx 4 4 4 4 s in 4x cos x (cos x s in x)

dx - 4 z 3 t an x 8 6 0 23 dx) . 8 ( Sl n x COS x

I 2 ﬁ cos x dx g sin ﬁx

m 2 1 m 2 l 1 43 . scc x tan x d x oosec x cot x d x . f , f

In h n d i a s e n e er or z e r s o ha t es e i n tegra s n s po itiv i t g , o , t t

2 n 1 i an s e odd n e r h e m is u n r s c d s y po itiv i t ge , w il e tri te 240 R OAL OUL US on . v INTEG AL ( .

The ﬁrs t in tegral m ay be written in t he form

m -1 2n fs ee x t an x s ec x t an x dx

m ‘ l z ” s ec x s ec x 1 d s ec x f ( ) ( ) ,

h h z w ic can be in tegrated afte r e xpan din g (se c x by the b n m m i o ial for ula .

S m a i il rly,

m 2n+ 1 m ‘ l zn fc os ec x c ot x dx fcos e c x cot x cos e c x cot x dx

- m " 1 z ” c os ec x c os e c x 1 o f ( ) d( c s e c x) .

EX ERC IS ES

z a 5 1 . e t an x dx. 5 s c x . t an x [ f dx.

’ dx 2 3 5 " 3 3 . cos ec x cot x dx. 6 . z sec x tan x x f j d ).

7 . t I an x dx.

4 3 . s i x x . cot dx 8 . o x f n f o t dx.

” tan av d x ” d M 44 . oot x x . I , f

The ﬁrs t i n tegral can be treated thu s

n z ft an x dx t an x dx

z ftan x ( s ec x 1 ) dx

-l t an n x n ' f t an x dx . 1 ‘ f

W h n n is a s v e n e e r t he ex on n tan x m a e po iti i t g , p e t of y be dim in ished by s u cces s ive application s of this fo rm ula u n e c m s z hen n is e n or on e h n n til it b o e ero (w ve ) , (w e

is odd) .

242 INTEGRAL CALCUL US [Ca V .

m n 1 4 sin ac cos x d x . 5 . f

a Ei h m or n os i ( ) t er a p tive odd int eger.

If on e t he ex o nen s fo r ex am e m is a os e odd of p t , pl , p itiv

n e e r the en n e ra m a be r en i t g , giv i t g l y w itt

" 2 2 ” x c os x s in x dx i cos x o f( ) c s x d ( c os x) .

m 1 S n ce m is odd m 1 is e en an d h e o e i , v , t er f r

e n c th n m n x n d d n pos itive i n teger . H e e bi o ial ca be e pa e i to a n e n u m e erm s an d hu s the n e ra n can ﬁ it b r of t , t i t g tio be m d e as ily c o ple te .

5 1 . s i c o s x dx. Ex . f n

According to the m eth od j u st indic ated this integral can be redu ced to

“ 2 2 ' % f sin x v cos x d (cos x ) f(l cos x) (cos x) d (oos x)

1 1 ’ c g s} 2 2 os x 3 co x 1 1 0 0 5 x .

s 2 s in x x. Ex . 5 Ex . . f d

4 3 i 3 x s x dx Ex . . s co . f n 3 sin x dx

4 Ex . . f D

(b) m + n an even n egative integer.

In this cas e the i n tegral m ay be put in the form

m” m c os x dx t an x dx f ,

h h can be n e ra e d b Art . 1 42 s n ce t he ex on en w ic i t g t y , i p t — (m n ) of s e c x is an e ve n pos itive i n tege r . TRIG ON OM E TR I C FUNCTIONS 243

7 Ex . . f COSg(I?

The int egration is effected in the following s teps

i‘ 4 t an x s ec x dx

i 2 t an x (t au x 1 ) d (t a n

’ % n 2 2 t an x (g 4t a x) .

Ex . 1 1 Ex . 8

Ex . 1 2 Ex . 9

1 Ex . 1 3 Ex . 0

( 0 ) M ultiple an gles .

hen m an d n are h en s e n e s n e a on W bot ev po itiv i t ger , i t gr ti

h m u n Th m ay be effecte d by t e u s e of ltipl e a gles . e trigo n o m e t ric form u las u s e d for this p u rpos e are \

1 co s 2 x

1 2 z cos x c os x 2

s in 2 x $1 11 93 C O S 2

4 1 4 ia c s x x . . fs o d

2 4 2 fs in x cos x dx f( s 1n x co s x) co s dx

2 si u 2 x 1 cos 2 x 4 2

— 2 sm 2 2 x d 1 Si n 2 x co 2 d x % x + 1 3 s x ( ) f \ ; 2

o s 4 x

— 1 3 x in 4 x sin 2 x. ﬁ gﬁ s + 1 5 244 U Cm v INTEGRAL CALCUL S [ .

2 z f‘ 4 Ex . 1 5 . in . 1 cos x s x dx E x . 7 . sin I f x cos x dx.

. ﬁ 6 ‘ 4 4 Ex . 1 6 . s iu x cos x dx. Ex . 1 8 . siu x x J( cos ) dx. t/

I te rat e the two o o i b the aid of m u ti e n g f ll w ng y l pl angles .

d dx 1 Q Ex . 9 . 2 Ex . 0 . 4 ‘ 3 I s in x cos x Is 1 n x co s x sin 3 x cos x

Iptegrat e t he following by an y of the preceding m et hods

’ s in ‘ x — LL1 M 2 2 Ex . 2 1 . dx dx 2 s ee x 2 ( co s x ) dx. j l2 I 2 f cos x cos x

2 / 2 2 Ex . 2 4 . \ . 2 2 . x Ex j f a x dx.

2 5 J 3 EX . Ex . 2 . :1 2 I I nch / 5 a

[S ubstitu te x [ Substit ut e x a s ec

d x 1 46 j a + b cos x a + b sin w

Write a bc os x in the fo rm

2 x 2 x — 2 x os —+ s m c os s 1n 2 2 2 2

dx Then — fa + b cos x a b a + b

a - b

246 E R U US on v INT G AL CALC L ( . .

dx a s in x b cos x) 2

1 cos2 x

a “ w” 1 47 . e sin u md x e cos u md x . f , f

m” In a e s in n x dx b s ass u m n tegr te f y part , i g

” in u s nx, an d do e dx . This gives

w” “ w” e s in n x in c s 1 f dx e s nx fe o nx dx. ( )

In the s am ex s s on a n as su m n th s m tegrate e pre i ag i , i g i ti e

” i u e , do s n n x dx . Then a w” ” ” 2 dx o x n d . fe s i n n x e c s n ﬁfe c os x x ( )

M u 1 n d 2 an Th n a ltiply ( ) by ga ( ) by g d add . e i tegr ls

the h m m s are m n a ed an d the s u is rig t e ber eli i t , re lt

B y s ubtractin g ( 1) from the form ula

is obtain ed .

EX ERC IS ES O N C H APT ER V 1 h at . S ow th

"+2 n n (1 n ) fsec x dx t an x sec x n fsec x dx.

” 2 I te rat e b art s t a in u sec x do see x dx . n g y p , k g , 146 TR I G ON OM E TRI C F UN C TI ON S 247

2 t t . Show h a

"+2 1 n cos ec x dx cot x cos ecn x n cosec” x ( ) j f dx. 5

s in x cos x

1 dx 5 Q e cos dx. 8 . 5 2 j 2 (1 x) l x

' s in 2 x [Pu t x cos dx. j ex

‘ s in x dx m’ i ii 5 1 0 e s n x dx. j cos x , f

ez in 2 1 s x s in x dx. 1 , f

SU GG T 2 = [ E S I O N . 2 s in x s in x cos x cos 3 1 2 h h . S ow t at

2 a b 2 a 6 ( , ) ( ) Use the t rigonom etric form u la

sin a s in 3 [cos ( a 3 ) cos (a , p ,

1 3 . Show that w M . f s in ax cos bx dz 2 (a b) 2 (a b)

Show that sin (a b)x s in (a b)x f oos ax cos bx dx — . 2 (a b) 2 (a + b)

” 3 6 s i 1 7 . t n x cot x dx. f n x 0 0 3 x dx . f ( a ) I f C HAPTER VI

INTEGR ATION AS A SUMMATION

1 4 I h h 8 . n t e preceding ﬁve c apters variou s m e thods of

n x Th n i tegration have be en e plain ed . e ﬁ al object in every cas e has be en to de term in e a fu n ction F(x) s u ch that its

d h a e n un derivative should be i en tical wit giv f ction f(x) .

It i n o w o ed t o an a z e h s d a m re u s prop s ly t i i e a little o f lly, an d to show that it re adily leads to a v iew of i n t egration which is of the highes t in te res t an d impo rtan ce . In orde r t o obtai n the derivat ive o f a fu n cti on F( x) it is n eces sary in the ﬁrs t plac e t o det e rmi n e the in cre m en t

' F( x Ax ) E( x) ( 1) which the fu n ction E(x) takes when the in depen den t vari h m able x t ake s t e i n cre en t Ax . The ex pres s ion (1 ) can be pu t in a form more c on v en ien t

s n u r s s it as s um e d h for pre e t p po e , if be t at for all valu es of x u n der c on s ideratio n F( x Ax) can be expan ded by m ean s Of ’ h 41 T o r or m Art . a s e . yl t e [ , p Thi s expan s ion is

F(x Ax) u F ” ’ ) 2 E 3 F( x) F (x) Ax g (Ax) gg (Ax)

F m h s b n s s n F x the n c m n 1 is ob ro t i , y tra po i g ( ) , i re e t ( ) t ain e d in the m a i s es z . for of eri , v ,

F( x Ax) F( x) ﬁ ’ l i fl ? F ( x ) Ax Ax + ggmx)

f(x) Ax qS( x) Ax .

0 C vi 25 INTEGRAL CALCUL US [ m .

an d s u s u e in t he rs erm of the re e d n b tit t ﬁ t t p c i g equ ation . The additio n of the ab ov e n e qu ation s then giv es

F( b) -F( a )

Ax [f( a ) + f( a Ax) +f( a 2 Ax) +f( a + n 1 Ax)]

Ax 1

This e x res s on for F b F a h e de en d n on the p i ( ) ( ) , w il p i g

v en u n ct o n x c n a n s a s a s er s s ucces s e gi f i f( ) , o t i l o ie of iv

v a u es v iz . l of ,

Ax) + qb( a n 1

This lat te r can be gotten rid of by takin g it s limit as Ax approaches z ero .

Fo r s n ce , i N E C”) 2 c< x> Ax + < Ax> + 2 3 , it follows that m O 4 Ai; 0 ( )

n d hen c (I) de n o e t he n u m e ca a es m of the a e , if t ri lly gre t t ter s eries 9601 A51 0+ then Ax ¢( a Ax) i Ax [ < I) (I) ( n terms ) ]

[éAx n CID (b

B u s n ce on accou n t i , t of

l r l (I) : 0 Ax i o ’ it foll ows that

Ax + Ax A32O we ) w ) 1 an d hen ce F(b) F( a ) l ” 5 a + a Ax f( a n 1 Ax) Ax . ( ) Al; 0 M ) f< ) ] INTEGRATION AS A SUMMA TION 251

The s econ d member of (5) is den oted for brevity by the

x dx f( ) , an d is calle d the d eﬁ n ite in t egra l off(x) betwe en the limits

a an d 6 .

S u os e on e the m s s a the u m 6 is pp of li it , y pper li it ,

d r h e the o he r ha x d e ar ed as a a e s a e v a ue . r g v i bl , w il t ﬁ l To em phas iz e this as s u m pti on c on cern in g t he variability of

6 let e ac d b t he t x . Then e u on 5 , it be r pl e y let er q ati ( ) m ay be written 3 F a a Ax a n 1 A Ax e ) A320 [f( ) +f( ) + f< v]

F( a ) . ( 6 )

re t he e m F a has a x ed a h u h a i rar a ue He t r ( ) ﬁ , lt o g rb t y, v l depen di n g on t he part i cu lar choic e that is m ade for the c on I m n a . r rd d on n o n r s ta t t ay be ega e as a c s ta t f i teg ation . Form u la ( 6 ) exp res s es in two s tep s the s olu tion of the problem of determi n in g the fu n ction F( x) :

( 1 ) Fin d the su m of the s eries of n terms

a Ax a 2 An a n foo. f( ) , f( a f( ( m e thes e being the va lues of the given fu n ction f(x) corresp on ding to the n e u idis tan t va lu es o x q f ,

2 m — a a Ax a Ax a n 1 Ax . , + , + , , + ( )

2 Fin d the limit o the rodu ct o this s um b Ax as Ax ( ) f p f g , a roa ches zero while n in creas es to in n it s ub ect to the con pp ﬁ y, j i d tion n Ax x a .

The addition of an arbitrary c on stan t of in tegratio n m akes t he o u h s l ti on t e m os t gen eral pos s ibl e . The m e thod j u s t form u late d fo r de te rm in i n g the i n tegral E(x) of a giv en fu n ction f( x) is n ot s u itable for the ac tual 252 US Cm v i INTEGRAL CALCUL [ .

n e n s n ce h f f x a ew exce n c . E 1 s s . work of i t gr tio , i , wit ptio ( ,

2 e o the s u m m a on of the s er s in the h -h n d b l w) , ti ie rig t a

en n u a ﬁ m emb er of ( 6 ) pres ts i s per ble dif cu lties .

O n the her han d m u a 5 adm s a s m e ot , for l ( ) it of very i pl geom etrical or phys ical in terp re tation in m os t of the applica

n s of the ca c u us an d he e n es on e it s ch m e s . tio l l , r i li of ief rit It places before on e a very con ve n ien t an d u s eful form ulati on of m an the em s e m e r m chan cs h s cs y of probl of g o t y, e i , p y i ,

et c . t he n s u n o f h ch is m s ad ff c d b , ﬁ al ol tio w i o t re ily e e te y the evalu ation of the deﬁn ite in tegral

i n h un n x in the followin g m an n e r . F rs t obtai t e f ctio F ( ) by in tegratin g f (x) dx ac cordin g to the methods already h r explai n ed in the pre ce din g c apte s . D eterm i n e F ( b) an d F ( a) by s u bs titu tin g the limits I) an d

F n u ac F a m E h . a in the res u lt . i ally s btr t ( ) fro ( ) gives fe m M ) E ca)

as the valu e of the deﬁn ite in tegral .

x b the m ethod of su m m at io . x 1 . Give x e ﬁ n d F x E . n f( ) , ( ) y n e orm u la 6 ives For t he sake of brevity write Ax h. Th n f ( ) g

i lD — a a +h a+2h a + (n l )k F x e e e e h F a ° ( ) h l-O[ ] ( )

The su m in the right m embe r m ay be writ ten

" 2 h “ e“ [1 e e c

(by the form ula for summ ing a geom etric series) - 1 _ ez a — e“h nh z x u a b e, — ea ( ez a l )

‘ 254 CAL GUL US Cm v1 INTEGRAL [ .

A s er es r c an e s PA P A is hu s m d e ach i of e t gl l , 1 2 , t for e ,

h h s e n ire h n the en r h of w i c lie t ly wit i giv a ea . T e s e will be

t h i t rior n B refe rre d to as e n e r ecta gles . y producin g t he

n es a read d ra n a s er es of e c an les SA S A is li l y w , i r t g I , l z,

o m ed h ch be ca ed the exterior re n I f r w i will ll cta gl es . t is cle ar that the val ue of the giv en area will always lie betwe e n the s um the n r r an d the su m the ex e rec of i te io , of t rior

an es or x res s d in m u a t gl , , e p e a for l ,

PA P A P B a a B SA S A 1 1 2 nﬂ re AP Q 1 I 2 ( 7)

The diﬁ eren ce between the s u m of the ex terior an d the su m of the in terior re ctan gles is

S R S R S R rec an e S T T Ax . I I 2 n _ 1 ,, t gl n_ 1 Q

If the fu n cti on f(x) does n ot becom e i n ﬁn ite as x varies fro m a b T be n e an d hen ce T -Ax a roach z e o to , Q will ﬁ it Q will pp r

n h m of h u m of th s im ult an e ou s ly wit h Ax . H e ce t e li it t e s e exterior r e ctan gl es e qu al s t he lim it of the s u m o f the in te

Fr m ha the a a is e u a rior re ctan gles . o ( 7) it follows t t re q l

m n m h m to the c om o li it of t es e t wo s u s . To determ in e this s um obs erve that

R c n APR A AA a Ax . e ta gle I 1 AP 1 f( )

S m a A P R A a Ax Ax i il rly 1 1 2 2 f( ) ,

= 1 f( a n Ax) Ax. Add n i g, s u m of re ctan gles

[f( a ) + f< a Ax) + f< a m Ao JAx .

H en ce b e u i i n Ax a ach z r , y r q r g to ppro e o ,

Area AP QB

1 a + a n 1 Ax Ax . A520 [f( ) f( ) ] 149 INTEGRATION As A SUMMATION 255

The ex pres s i on j u s t obtai n e d for the are a is iden tical wi th that o ccu rri n g in t he right -han d m em ber of an d affords on e of the s im ples t an d m os t in teres ti n g of the geom etri cal Th in te rpretation s of that formula . u s

fa w n: f ( 9)

1 G n r iz ion of the r fo mu . Pos i and 50 . e e al at a ea r la tive

In ad n the m the s u m n egative area . ste of taki g li it of of

h n r o x c an es a m n e a t e i te i r (or e terior) re t gl , ore ge r l pro

r d a e a s s n t m e d a e e c an cedu e woul be to t k erie of i er i t r t gles .

33 A 1 I

x an a u x e een a an d a Ax x an a u e Let 1 be y v l e of b tw , 2 y v l

n Ax an d a Q Th n x Ax ou a + Ax et c . e d betwee , f( 1 ) w l

h n m d a t e are a a c an e K LA Fi . 62 t e e be of re t gl IA ( g ) i r e i t

n PA an d S A ha is betwee 1 I ; t t ,

PA x Ax S A . 1 < f( 1 ) < 1

s P A x Ax S A et c . Likewi e 1 2 < f( 2) < I 2, 256 U on v1 INTEGRAL CALC LUS ( . .

< s u m ext e ri ec an es of or r t gl , an d thhrefore ( cf. Fig . “ A ea B 1 0 r AP Q AL; 0 ( )

If the a a be u n d is e n re a o e the x-ax s t he re to fo ti ly b v i , l If o rdin ates are al pos itive . at the s am e tim e Ax be taken

s ha is b a o m u a 8 1 0 e po itive (t t , if ) , f r l ( ) or ( ) giv s a posi

s n t he a ea . O n t he h han d the a is n a tive ig to r ot er , rea eg - tive if below the x axis . If the cu rve g = f(x) is p artly abo ve an d partly below the

-ax s the a u the de n n a 8 re re x i , v l e of ﬁ ite i tegr l ( ) will be p s en ted by t he algeb raic s um of the pos itive an d n egative areas limite d by this cu rv e .

1 C r in o er i s of d fini in s . m the 1 5 . e ta pr p t e e te tegral Fro b defin itio n of the de fin ite in t egral f f( x) dx as the lim it of a e parti cu lar s u m [form ul a p . c rtain importan t d properties m ay be dedu ce .

( a ) In terchanging the limits a an d 6 changes the sign of the deﬁn ite in tegral.

Fo r if x s tarts at t he u pper limit 6 an d dimi n ishes by the

n e a n c m en s Ax ch n e addition o f s u c ce ss ive g tive i re t ) , a a g of s ign will occu r in form ula givin g

F( a) F( b) ﬂwx) dx .

“ He n c e f( x) dx dx .

b a n u mber between a an d b a c < b then ( 6) If e e ( ) ,

c = fﬂ e ) dx ff( e ) dx dx .

258 US Cm v r INTEGRAL CALCUL [ .

' a e for u mi a a u e x = x h ch is es s h n t k pper li t v l , w i l t a b, ’ a < x b. Then acc d n the ced n s ul s , or i g to pre i g re t ,

' E(x ) F( a) = fﬂe > de

’ N let x n c as e an d a ach 6 as m ow i re ppro li it . If at the s am e tim e the in tegral t ﬁ f(x) dx ( 14)

aches a d n n e m ha m appro efi ite , fi it li it , t t li it will be defin ed as the valu e of the in tegral

in the c s un d c n sde a n ha a e er o i r tio ; t t is,

' x < b.

O n the other han d the in tegral ( 1 4) m ay in creas e withou t

m . he n h ha en s the n a ll be s a d ha li it W t at pp , i tegr l wi i to ve an n n u i ﬁ ite val e, or

b In s m a m n n e a 00 the u x dx a i il r a r, if f( ) , val e of ff( ) will be deﬁn ed to be the lim it of the i n tegral

' a < x < b

’ h m as x dimin ishes an d approac es a as li it .

F n a c 00 he e c is an n u m e n a an d i lly, if f( ) , w r y b r betwee b is n c ss d e m n the m an n , it e e ary to et r i e e i g of

x dx an d f x dx fiffi( ) , fif( ) b the m h d u s s u e s d an d hen add the su s y et o j t gg te , t two re lt in accordan c e with form ula INTEGRATION As A SUMM ATION 259

the m s a b ha e n a s um d e n Heretofore li it , v bee s e to b ﬁ ite .

The cas in h ch on e of the m s s a b is n n e is e w i li it , y , i fi it , readily dispos ed of by in tegratin g from a t o a fi n ite u pper ’ m x an d h n c n s der n the m h ch th li it , t e o i i g li it w i e in tegral ’ h n a ac s as x c s e s n n t . Th s m hen on ppro e i rea to i fi i y i li it, w e

x s s ll d n d as the a u the n ra s o ha e i t , wi be eﬁ e v l e of i teg l , t t

{ x dx f( ) f( x) dx .

An ex actly s im ilar m o de of procedu re i s to be followed if the m a is 00 h m s n lower li it , or if bot li it are i ﬁn ite .

EX ERC IS ES

E 1 . ithou er r i he i ti x. Prove, W t fo m t te ra o , p ng n g —n da 2 xeﬂc dx

1 x2 . . a

2 . h ut i e r t Ex . Wit o nt g a ing s how that

?“ ’ ' 2 x a x “ x dx _ I 2 2 j 2 2 a x + a a x + a

3 If x an d x r Ex . . y ¢ ( ) g ll/ ( ) a e the e qu ation s of two curves ic are co ti u ous bet ee x a an d x b an d su c t at to eac wh h n n w n , h h h va ue of x co rre s o ds but on e va ue of rove t at the l p n l g, p h are a bou ded b the se c urves an d the two ordi ates x a x b is n y n , n u m erically e qu al to (93) “O ldﬁ

2 k 2 4 . Pr e t at the area of the circ e x h Ex . ov h l ( ) (g ) equal t o h+ r

5 Eva uate Ex . . l

6 . E u ate Ex . val

7 . e Ex . Evalu at C H APTER V II

GEOMETR ICAL APPLICATIONS

Ar s . R c n l c o 1 53 . e u I ea ta g ar o rdin ates . t was shown in

r 1 49 h A t . t at the area b o un de d by the cu rve g the - x ax s an d the d n a es x a x b is e s n d b i , two or i t , , r pre e te y the deﬁn ite in tegral j;91x) dx

In an ex actly s imilar m an n er it c an be s hown that t he are a

m e d b the cu e the -ax s an d the a s c s s as a li it y rv , g i , two b i g ,

8 is s n d b g , , repre e te y ( 2)

' m d th n 1 h h It was re arke at e e d of Art . 50 t at w en b is greate r than a the in tegral ( 1 ) giv es a pos itive or n egative

u c d n a the r i r h - res lt ac or i g s a e a s above o below t e x axis .

S m 8 a the n a 2 s a s i ilarly, if , > , i tegr l ( ) give po itive or n egative res ult accordin g as the area which it repres en ts is

th h th - x to e rig t or left of e g a is . When e ver it is requ ire d to determin e the area of a ﬁgu re which is partly on o n e s ide an d partly on t he o ther s ide of t he c o d n a e ax s is n e ces s a ca cu a e the s e o r i t i , it ry to l l t po itiv an d t he n e a e areas s a e an d add the esul s ach g tiv epar t ly r t , e

f x . E . 5 n h a s s n . O . take wit po itive ig [ , p

1 S econd me hod . An h m h d of d m n n 54 . t ot er et o eter i i g

2 . It he is as e d on the es u Art . 1 0 . 3 t area b r lt of , p was there shown that if 2: repres e n ts the area m easured

262 S C v n INTEGRAL CALCUL U [ m .

EX ERC IS ES

2 1 . i d the area bou ded b the arabola 4 ax h - y p y , t e x axis F n n , n h a d t e ordin ate x b. 5 2 . i d t he area of the tria e orm ed b the in e F n , ngl f y l 2 an d a i the coord nat e ax es .

- - 3 . Find the are a betwee n the x axis an d on e semi u ndu lation of the cur sin x ve g .

— i — 4 . i d the area bou ded b the mi cubi a z i F n n y se c l parabolajy n x a ml

t he lin e x 5 .

z - 5 . i d t he area bet ee the curve sin x cos x an d the x axis F n w n g , ri i from the o g n to the point at whic h x 2 7r.

FI G . 66 .

An examination of the curve will show t hat the area is partly above - an d art be o the x axis. The curv e cros ses the axis at x = an d p ly l w 3, at x 2

The rst ortio of area hich is ositive is obt ai ed b in te rat ﬁ p n , w p , n y g 77 — » in ro The resu t is . The xt t ortio s of area are g f m O t o . l i ne wo p n 2 73: S r e ative an d are ca cu ated b i te rati rom to . The resu t is n g , l l y n g ng f l 2 i

. The as t ortio ic is os it iv e is ou d b i te rati rom g l p n , wh h p , f n , y n g ng f

to 2 7r t o be e ce tot a area . , H n l g

6 h - xis n d the cur e : a sin 4 x . i d t he r e t e x a a v F n a ea betwe n y ,

from the origin t o x 7r.

3 h u ica arabo a x the -ax is 7 . i d t he ar ou ded t e c b F n ea b n by l p l y , y , an d t he 1111 8 y 8 . 1 54 GE OM E TR I CAL APPL I CA TI ON S 268

2 8 h u e the arab a . Find t e area bo nd d by p ol y x an d the line

E . . x . Cf. x 3 3, [ , p

2 i th e ar b u de h r o x an d the 9 . F nd ea o n d by t e pa ab la g two lin e s

1 i d 2 0 . F n the area bou nded by the parabola g 4 p x an d the lin e

x : a an d s o t at it is two t irds t he are a of th e c ircu m scribi , h w h h ng rec ta ngle . What is the area bounded by the cur ve an d it s latu s rectum ?

2 2 1 1 . i d the area of the circ e x 2 ax F n l g 0.

1 2 . i d the area bou ded b t he coordi ate axes the itc F n n y n , w h 3 8 a

an d t he ordi ate x x . B i creas i x wit ou t im it y , 2 2 n 1 y n ng l h l x 4 a - ﬁ n d the area betwee n the curve an d the x axis .

h i £ : 1 3 i d t e area of t he e e 1 . . F n ll ps 2 2 a £

' ?" b 1 4 i d the area of the oc c oid x a . . F n hyp y l ﬂ

” 1 5 . i d the area bou n ded b the lo arit m ic curve a t he F n y g h y , - x ax is an d the two ordi at es x x x x . S o t at the resu t is , n 1 , 2 h w h l i r i proport o nal t o the diﬁ eren ce between t he o d nat es .

Precaution s t o be obs erved in evaluating deﬁ nite

Th m th d n f r d rm integrals . e t wo e o s j u s t give o e te in in g

an a eas are s s en a a e in t he r c s s s u d pl e r e ti lly lik p o e e req ire , n am ely

(1 ) t o ﬁn d the i n tegral of t he given fu n ction f(x)

2 s u s u e for x the m n a u es a an d b ( ) to b tit t two li iti g v l ,

n r h r r r h d a d s u bt act t e ﬁ s t es ult f om t e s ec on .

Erron eou s es ul s m a be r ache d ho e r b an in r t y e , w ve , y

u u i c a ti o s appli cat on of this p roces s .

In rac ca ro b e m s the cas e re u r n s ec a car is p ti l p l , q i i g p i l e that in whi ch f x be com es in ﬁn ite for s om e valu e of x ( ) ,

n n d h n h h n a e n e e e a a b. e a a e s s c a s a b tw W t t pp , p i l i ve tig

on m u b m d h n r r 1 52 . ti s t e a e afte r t e m an e of A t . 264 CAL C L S on . vn INTEGRAL U U ( .

Ex . 1 . i d t he are bo a u ded b the curve x c th n F n n y 3; ( , e coordi ate axes an d the ordi ate x 2 n , n . A direct applicatio n of the formula gives

area, o (x

b where the s mbo is a s i of s ubstit utio i dicati that the va u es y l 1 gn n , n ng l

b a are to be i se rte d for x in the ex re s sio 1m m ed1 at el , n p n yp recedm g the

si an d the s eco d re su t subtracte r 1 gn, n l d f o m the ﬁrst . T is re su t 1 i h l s ncorrect . A glance at the e qu ation of the cu rve s hows 0 bec m r o es in ﬁn ite fo x 1 . It is accordingly

FIG . 67 .

ece ssar ﬁn d the are u de b ordi te n y to a OOFA (Fig. 67) bo n d y an na AP ’ corre s o di t a ue x h For t i rti p n ng o v al x w ich is le ss t han 1 . h s po on the area x is ite an d o sitive an d orm u a can be im m ediate f( ) ﬁn p , f l ly a ied ith the resu t ppl , w l

” m’ ’PA C C are a OC c . fm 1

’ If n o w x be m ade t o i crease an d a roac 1 as a limit the v a u e of n pp h , l the expres s ion for the are a will increase w ithout lim it . A like re sult is obtaine d for the area inclu de d be tween the ordinates

n 2 e ce the re u ired area is i ite . x 1 a d x . H n q nﬁn

8 2 2 2 = “ the Ex . 2 . i d the rea imite d b the curve x a 8 x F n a l y y ( ) ,

1 coordi ate axes an d the ordi at e x 3 a . n , n

c a VII 26 6 INTEGRAL CAL C UL Us [ .

- 1 3 r b u ded b cur e n he r Ex . . i d the a ea o v t a x t cob di nate F n n y g , ax e s an d the i e x l , l n . In t his problem we have to deal wit h a - m a va u ed u ctio of x . In act to ny l f n n f ,

e ach value of x corre sponds . an inﬁnit e — mb r f u f 1 Th nu e o val es o t an x . e problem accordingly has an in deﬁ n it en e s s which m u st be re m oved b addi

tion al as su mption .

" 1 The curve y t an x cons ist s of an in ite u m ber of bra c es corres o di ﬁn n n h , p n ng ordin ate s of which differ by integer m ulti ples of Eac h branch is continu ou s for

all ﬁnite v alues of x (s ee Fig. It is i t t ct n e f t ese FIG 69 . ev de l n e ces s ar o s e e o o . n y y l h branches for the bou ndary of the propos ed i r ll th AB is are a an d d s ca d a e ot ers . Su ose for ex am e the bra c , h pp , pl , n h

' se ected . The ordi ate to this bra c has the va u e 71 e x is z ero l n n h l wh n , —5 7 an d 1 n creases con tm u ou sly t o 7: + 2 as x 1 ncreases con t 1nu ous ly e ce the re u ired area is t o 1 . H n q

“ 1 —1 2 t an x dx [x t an x elog (55 D] ;

5 7r log 2 . 4

4 i d the area o f the ara e o ram st ri AB C 0 Fi . Ex . . F n p ll l g p ( g

x3 2 5 i d the are a bet e e t he cis soid an d its as m Ex . . F n w n y y p — 23 tote x = 2 a .

2 2 2 6 i d the area i c os ed b the curve 33 2 a 2 x an d it s Ex . . F n n l y 31 (y ) asymptote s .

2 - h re b u ded b the cu rv e a x x a the x ax is 7 . i d t e a a o Ex . F n n y y( ) , , = an d the asym pt o te x a .

8 i d the are a i c u ded bet ee the cu rve 2 Ex . . F n n l w n ( an d it s asymptote .

ictio m u st be aced on the ex o e t k in order 9 . at restr Ex . Wh n pl p n n b he curve 1 x k 1 its as m tote x 1 th at the area bou nded y t ( ) y , y p , an d the coordinate axe s m ay be ﬁnite ? 1 55 GE OME TR I CAL APPLI CATI ON S 26 7

Ar s . Pol Coo din s . P an arc 1 56 . ea ar r ate Let Q be of a cu rve whos e equ ation in p olar coordin ates is

p = f( 9) ( 3)

Let it be requ ire d to ﬁ n d the area bou n de d by this cu rv e an d the two radii OP an d OQ . D raw from the origin a s eries

ad 0P 0P at of r ii 1 , 2,

L he r e qu al an gle s AH. e t t cob din at es o f the o n s P P P p i t , 1 , 2 ,

" ‘ be “ a Q ( a (Pp (P2,

b D ra the c c ( , w ir le ’ ’ arcs PR R P R R In I I , 1 2 2 , the circu lar s ector P OR

ad u s OP a r i ,

FIG . 70 .

arc PR = a -A 1 0 ;

2 = « a e a P 0R a A0. r 1 2

’ 1= 2 S m a a ea P 01l =- o A9 i il rly r I 2 2 , 1 ,

1— 2 a a P 018 : A9 re 2 3 2 p2 ,

— 2 a a P — A9° re 2 _ 2 Pn 1

The s um o f thes e s e ctori al areas is

? 2 ace a pa a ) Ae < 4)

This is an a x m a e a u for the re u ed a a O ppro i t v l e q ir re P Q, whi ch is les s than the tru e valu e by the am ou n ts con tai n e d in the n e ec e d r an u r r n P a o s PR P R P et c . gl t t i g l p tio l l , 1 2 2,

Su s e the u re PR P re v o e d a ou t 0 u n i it o ccu es ppo ﬁg I 1 lv b t l pi ’ ’ ’ the s on P R R an d s m ar h P R Th n P e t c . e po iti I 2 i il ly wit 1 2 2, the s u m of all the parts n egle cted is e vide n tly les s than the 26 8 I R L CAL C UL us c a . VII NTEG A ( .

’ ’ s P R R P _ t he a ea h ch a roache z e r h trip l n n 1 , r of w i pp s o as t e

s ec o a an e A9 is m de a t ri l gl a to pproach z ero . i n c are a P 0 1 2 2 A6 He e 0 Al? 0 2 p1 p2

2 2p de.

An th m h d c du is 111 d in E 1 o er et o of pro e re rate x .

mm d n i e iately followi g .

2 2 Ex . 1 . i d the area f the F n o le mniscate p a cos 2 0. Le t A de note the area of the sector F OQ m eas ured from the polar ax is to

an arbitrary radius vector OQ . The dif

A 88 . 142 ferential of area is rt . ( , p )

2 ? (L4 d a cos 2 Odd 2p 9 2 ,

e ce b i te ratio wh n , y n g n,

2 A 2a f0 0 8 2 0d9 3 Fm 71 . t , 2 3s o c .

If 0 ere z ero the i e O ou d occu the i it ia os itio OP and w , l n Q w l py n l p n the are b T at a would e z ero . h is

whe n

The substitution Of this result in the preceding form ula gives = O 0 + C .

2 A g s ia

—77 In order to ﬁn d the total area of t he ﬁgure put 9 . In t his case

n f he s m met r OQ will be t angent to the lem niscat e at 0 . O accou nt o t y y o f t he curve t he resu t obtai ed wi be on e ou rt the tota area an d , l n ll f h l , t he re ore 2 f area “

2 i d the area oft he cardioid a 1 co s Ex . . F n p ( J

he are a of the t ree oo s of the c urv e a sin 3 . x 3 . in d t 9 E . F h l p p

2 0 CAL C L Ca v n 7 INTEGRAL U Us [ .

This de ﬁn iti on is im m e diately c on vertible in to a form u la s u itable for dire ct appli cati on .

For let the n t s P P be s o ch s n ha , poi I , 2 , o e t t

F R = P E I I Z

dra n ara e t o h - the n es PR et c . e n t e x ax s . li I, b i g w p ll l i

D e n o e b A the n c rem en R P of . Then t he chord t y g i t i , g

P P has the en h ,_ I , l gt

VOW? (4 2 ?

£1 It is clear that is the valu e of c orres pon din g to 2x 5?x

f Art . C . m e n the cu r e en P an d P . s o poi t of v betwe , _ 1 , [

en ce b s u s u n l n 5 an d u s n t he r n c e H , y b tit ti g ( ) i g p i ipl - m d in d r n t he area rm u a Art . 1 50 e ploye e ivi g fo l ,

d 6 are P Q y, ( )

’ ’ ” ” h ch x an d x are the c d n a es of P an d in w i ( , g ) ( , y ) oor i t

Q res p ectively. The s am e res u lt wou ld als o be obtain ed by in tegratin g the 9 1 3 . n or the de ri a e arc 1 an d . expre s s i o s f v tiv of , ( ) p

2 1 i d the e t of arc of the arabo a = 4 x m easured Ex . . F n l ng h p l g p from the vertex to on e ext remity of the lat u s rectum .

In this case

1 p he ce le th of are dx. an d n ng j: 5

’2 8 h e t of arc of the s e mi-cubica arabo a a x 2 . i d t e Ex . F n l ng h l p l y from the origin t o the point Whose abscis sa is Z . 1 57 GE OM E TR I CAL APPLI CA TI ON S 271

l 1 3 i d the e tire en t of the oc c oid x5 % a? Ex . . F n n l g h hyp y l g

2 2 f rc f the circ e x h k 2 4 i d the e th o a o r . Ex . . F n l ng l ( ) (g )

° a 5 i d the e t of arc of the cat e ar e e rom Ex . . F n l ng h n y g( ) f

he ertex t o the oint x . t v p ( l , yl)

6 d the e th of the lo arit m ic cur e lo x rom x 1 Ex . . Fin l ng g h v y g f V to x S.

7 i d the e th of arc of the evolut e ofthe e li se Ex . . F n l ng l p

i 1 = 2 t (awe) + 04 0 (a bo .

1 58 . P . h h . Length of curves olar coordinates W en t e

u a n the cu r e is en in ar c obrdin at es let the eq tio of v giv pol ,

n s P P s o ch s n ha the c n 9 poi t I, 2 4 be o e t t ve torial a gle

M I73 .

n c s s b u a n c m e n s AH in as s n r m a n P i rea e y eq l i re t p i g f o poi t ,

on t h u r n n D r h e c v e t o the n ex t s u ccee di g p oi t PM. aw t e

l n s P R n d cu ar the ad GP . Th n i e , ,+ 1 perpe i l to r ii M e R m V I INTEG AL CALCUL US [c I.

[Cf. p .

AGVP + p S in % A0

Th m the s u m o f all s u ch chords be cc rd n e li it of will , a o i g

d n n t he n h t he arc 6 . en ce to eﬁ itio , le gt of P 2 H

d9 7 arc P Q , ( )

’ ” in h ch are t he c o d n a s an d w i (p , (p , or i te of P Q re s pec tively.

“9 1 i d t he le th of arc of the o arit hm ic s i ra z e be Ex . . F g g p p n n l l ‘ t ee the t wo oi t s 9 and w n p n (pl , 1 ) (pz,

Since

d0 1 it follows that

pz ' 2 t of arc d V a 1 ° le ng h f p (p2 Pl) P]

he le t of arc of the cardioid a 1 cos 2 . i d t Ex . F n ng h p (

3 i the en t of the ciss oid 2 a t an 0 s in 0 rom 0 O Ex . . F nd l g h p f t 0 9 I . 4

i 2 a s n . 4 . i e Ex . F nd the entire len gth ofthe curv p 0

2 : a s eo bet ee the Ex . 5 . Find the le ngth of the parabola p 5 w n o i ts 9 an d p n (pl , 1 ) (P2,

f r of the e rbo ic s ira 9 a bet ee 6 . in d the e t o a c Ex . F l ng h hyp l p l p w n the oi ts 0 an d 0 p n (P1 , 1 ) (92, 2)

274 US Cm v n INTEGRAL CALCUL [ .

E 4 . i t he e t of the i o x . F nd l ng h nv lute of the circle

a; a (co s 9 Osin y a (sin ﬂ Oco s 0)

r . f om to § g‘ Ex . 5 . i d t he en t od arc of the curve s o rom a 0 F n l g h y f, f , g r ( )

t o x ’ ( v y ) l ’ 3 3 As sum e a s ec 9 a t an . x , y 9 2 2 x— ‘y Ex . 6 . i he e t of arc of t he e i s e V d t p 1 . i F n l ng h ll 2 5 a 5

Putti a: a co s ( ) b s in ( ) ng I, y I,

2 3 4 2 (177 [1 6 i 61

2 2 by expanding e co s (;b int o a ser1es an d integratin g term by term .

x . 7 i h e t of arc o f t he cur e J E . F nd t e l ng h v

x : eo si 9 6 9 0 0 s n , y 0 from 0 = 0 to

L Area of s urface of revolution . et AQ be a con tin u ous arc of a cu rve whos e e qua ti on is e xpres s ed in re ctan gu lar

I i u e d c oordin ates x an d y. t s re q ir to de ter min e a form ula for t he area of the s u rfac e gen e rated by revolvin g the arc AQ abou t the - w axis .

It has n h n in Art . 86 bee s ow ,

1 40 ha S de n s t he a p . , t t if ote rea of t he s u rface gen erated by the rotation of AP (P bein g

a a a e n h coo d n a e s x h n v ri bl poi t wit r i t ( , y», t e

028 2 7 g ds dS from which dz

as da GE OM E TRI CAL APPLICATIONS 275

H n c b n e ra n h s e ex s s n s e e , y i t g ti g t e two pre io ,

b s u rface 2 7 1:y

B 1 + d 9 l a y, ( ) the limitin g valu es of w an d y bein g the coordin ates of the

n poi ts A an d Q . That the res u lt of i n tegrati on i s to be evalu ated be tween t he m s at an d b or a an d 8 is ead s en b o n li it ( , ) r ily e y f llowi g

h u n m d in o 1 4 . n n t e s es s a e Art . 5 d the gg ti For, e oti g in deﬁ n ite in tegral 2 n

b :v 0 s n c the area is e d n l z hen as : a y gb( ) , i e vi e t y ero w when the poin t P c oin cides with A) it follows that

n whe ce 0 ct ( a) .

M h n c n c des t h t he u d s u ac is oreover, w e P oi i wi Q req ire rf e d e m n d an d h et r i e , t erefore

s u rface gen erated by AQ cf) ( 6) 0 gt (b)

B ut acc d n t o Art . 1 48 ? is the de n n or i g , (pa) 43 00 ﬁ ite i te

ra o a n e d u n 1 m g l bt i by e val ati g ( 0) between the li its at an d b. In lik e m an n er it is fou n d that t he area of the s u rfac e obtain ed by re volvin g AQ abou t t he y-ax is is

= dy 2 7rfx

In e ach of the above cas es the re i s a choice of two form u l f r h as o t e area . That on e s hou ld be s ele cted which can m s s n o t ea ily be i tegrated . ‘ 27 a fvn 6 INTEGRAL CAL C UL US Lo .

Ex 1 i d t he s ur ac e of the c te oid obtain ed b re h . . F n f a n y volving t e

“ - cate ar _ e e about the axis rom x O to x _ a . n y y g( y , f

an d e ce b us i the seco d orm u a 'of the re uired sur ace has h n , y ng n f l q f t he are a

; Tﬂ 2 (6 e ) dx.

2 he s ur ace obt ai ed b revo vi about the -axis the Ex . . Find t f n y l ng y 2 ? qu art er of t he circle x y 2 x 2 y 1 O cont ained between the

point s wh ere it t ou che s the coordina te axes .

2 3 i t he su r e e erat ed b re vi the arabo a 4 x Ex . . F nd fac g n y vol ng p l y p - abou t the x axis rom the ori in to t he oi t 2 . f g p n (p , p )

4 . i d the ur - Ex . F n s face ge nerated by the revolution about t he y axis

of the s am e are as in Ex . 3 .

E . 5 i d the su r ace e e x . F n f g n rate d by the revolution of the ellipse 32 + 1 , 2 (1 1?

(a) about its m aj or axis (t he prolat e spheroid) ; b about it s m i or axis the ob ate s r ( ) n ( l phe oid) .

6 . i d h u Ex . F n t e s rface generated by the revolution of the cardioid a l co s abou t the p ( 9) polar axis . Regarding t he ﬁgure as re ferred in the ﬁrs t place to rect an gular axes s uc t at cos 0 sin 0 we have h h x p , y p

s ur ace 2 w ds 2 ﬁzz sin 619 f fy ,

7 i he ur ce f t e con e obt ai ed b re vo vi t at ortio Ex . . F nd t s fa o h n y l ng h p n 6 f t he ine 3 z 1 ich is i t erce ted b the coordi ate axes o l wh n p y n , a — (a ) abou t the x axis ; - (B) about the y axis .

278 GAL GUL Us Cm v n INTEGRAL [ .

the s um of the n cylin ders gen erated by the in terior re ct an

s t he an e viz . gle of pl , ,

_ z -7 2 2 W AP A l A —P— Ax ( l 1 n I n 1 ) a

has m the u m ui d B ut the m h um for li it vol e req re . li it of t is s b z is b m u a . 250 the d n in t e ral w dx an d [ y for l p ] eﬁ ite g f y , hen ce volume 71

The s m s u is ead n d b n n the a e re lt r ily. obtai e y i tegrati g h x f u Art . 85 . ss n t e d e o m e . e pre io for erivativ vol [ , p The volum e gen erated by revolu tion abou t the y-axis is fou n d by a like pro cess to be ex pres s ed by t he deﬁn ite in tegral

in h ch a an d 8 t he a u es of the x m s of w i , are v l g at e tre itie

h r t e giv e n a c .

1 i e um e he b ate s eroid obt ai ed b revo vi Ex . . F nd th vol of t o l ph n y l ng 2 2

£ Q. the e i se 1 about its mi or axis . ll p 2 2 n a 6

E 2 . i d the o u m e of the s ere obtained b revo vi t e x . F n v l ph y l ng h 2 2 2 - circle x (y [a) r abou t the y axis .

g 3 The are of the erb a x i r h ertex Ex . . o x k e te d om t e v hyp l y , n ng f

i it is revo ed abou t it s as m t ot i d the o u me e erat ed . t o nﬁn y lv y p e . F n v l g n What is t he volu m e ge nerate d by revolving the sam e are about the other asymptote 4 i d t h i i Ex . . F n e ent re volum e obt ain ed by rotating the hypocyclo d

' ‘ 353 g ag about eit er x y h a is .

Ex . 5 i r of . F nd the v olu m e obt ained by the revolution of that pa t / the parabola ﬁ + V; x Zz intercepted by the coordinate axes about o ne of t os e h axe s .

f the itch E . i th r t he re o u tio o x 6 . F nd e volu m e ge ne a ed by t v l n w 3 8 a - the x aXlS . m a 16 1 GE OM E TRI CA L APPLI CATI ON S 279

7 i d the vo u me e erate d b the revo uti Ex . . F n l g n y l on of the witch about the -ax is t a i the ortio of the curve rom the vertex x 0 y , k ng p n f ( z ) t o the oi t x p n ( l , 9 1 )

‘ W at is the imit of t is vo u m e as the oi t x m oves t r h l h l p n ( 1, owa d inﬁnity

h 8 . i d t e u m e bt i d r i Ex . F n vol o a ne by evolv ng a com plet e arch of the cycloid x a (9 s in y a (1 cos 9) abou t the x

277 ‘z Vo um e dx 1 c 3 l 77 y ( os 9) d9. 0

m 9 . i the u e bt i e d b r Ex . F nd vol o a n y evolving the cardioid

z p a (1 c os 9) abou t the polar axis .

As su m e x z : cos 9 sin . p , y p 9

Then dx d (p cos 9) d [a (1 cos 9) cos 9]

a s in 1 2 c s 9 ( o 9) d9. He nce 2 3 3 Vo u m e r dx 1 m s in 9 1 cos 2 l jy ( cos 9) (19.

1 r 0 . dr t f a i e re o ves ab u i r Ex . A qu a an o c cl v l o t t s cho d . Find the volu m e of the spindle s o ge nerated . The e quation of the circle be ing t aken in the form (w an t

- - T the x axis can be assu m ed as the axis of rot ation . he ordin ates of the rot ated arc are determ ined by th e form ula

Mis c n ous ic ions . In the c d n a c e 1 6 2 . ella e appl at pre e i g rti l the v olu m e of the s olid of rev olu ti on is shown t o be the lim it of the s um of the vol um es of a s eries of cylin dri cal

The n n h n e d is h pl ates o f thickn es s Ax . otio ere i volv , wit

m d ca on s a ca e t o a a o em s . s uitabl e o iﬁ ti , ppli bl v riety of pr bl

x am ex ce n Exs . 6 9 1 0 are us The followin g e ples ( pti g , , ) ill t rat ion s of this prin ciple . 280 CAL C L us on . vu INTEGRAL U ( .

x2 Ex . 1 . i d the vo um e of t he el i soid = F n l l p + 1 . 2 5 ;e Im a in e the s olid divided i to a u mb g n n er of t hin plate s by m eans of a e s e r e dic ular t o t he -ax i pl n p p n x s an d at e qu al distances Ax from e ach

ot er. Re ard the vo u m e of eac at e h g l h pl as approxim at ely th at of an

e i tic c i der of a titude Ax. The ll p yl n l base of the cylinder will be t he e i s e in ic the e li soid is i ters ected b on e of the cu tti ll p wh h l p n y ng planes . If the e u at io of t is a e be d e ot ed b x the e q n h pl n n y A, qu ation of the e i tic bas e of the c i der is in c ordin te s ll p yl n ( y, z p a )

2 f 2 A2 2 b 0 2

2 2 ;I s 2 2 2 2 (a A) 201 A) 21

The semi-axes ofthe ellips e are b

a G

Since the area of the ellips e is the product of the semi-axes multiplied b 71 Ex . 1 3 . it o o s t at y ( , p f ll w h

7 66 “ ca

bc and vo u me of e i tic c i der — —a2 “ A l ll p yl n ( A) A, a be in u sed in ace of Ax si (AA g pl nce is A) . The res ult of s um m ing all s uch te rm s an d t aking the limit as M ap proaches z ero is e qu ivale nt t o the de ﬁnite integral

” o “ l 2 s ? w A) dx. f } f fc a ;

0 11 account of the ellipsoid being sym m etrical with respect to the la e x 0 the im it s 0 an d a i c u de on e a t he re u ired volum e p n , l n l h lf q — an d hence ins tead of u s ing lim it s a an d + a it is m ore conve nie nt to write the deﬁnit e integral in the above form .

Ex . 1 th m h e i tic 2 . d th m t o of e o u e of t e co e Ex . Fin by e e h d v l ll p n

2 2 y

m easu red rom the z— a e as bas e t o the vertex 1 O f y pl n ( , ,

- i t r i f i e n d o bas e area A. 3 . d he vo m e of a am d o a t t u d h a f Ex . F n lu py l

282 CAL GUL Us on v 11 INTEGRAL [ .

Let O Y be the initial position of the string an d AB an y inte rm ediate i h r pos ition . S nce at every ins tant t e fo ce is exerted on the weigh t B in the dire ctio of t he s tri B A the m otio of t he o i t m u n ng , n p n s t be in the sam e directio t at is the dire ctio n of the tractrix at B i n ; h , s t he s am e as t at of t he li e B A an d e ce B A is a t a e t to the u h n h n ng n c rve . The

ex ress io for the t a e t e t is Art . 79 . 130 p n ng n l ng h ( , p )

ﬂ 2 1 1 + 1 < r1x > y 1 a i( s dy dx

Solving for

I te rati it res ect to n g ng w h p y,

2 2 2 a y / 2 — 2 y dy z x a y y

The co s t a t of i te ratio is determi ed b the assu m tio t at O a n n n g n n y p n h ( , ) b tit uti t ese co rdi es in the is the st arting point of the curve . Su s ng h o nat ' above e qu at ion we ﬁ n d C O.

Ex . 7 . A oodm a e s a tree 2 eet in diam eter cuttin a a w n f ll f , g h lfw y t rou on e ac s ide . The lowe r ace of e ac cut is oriz o ta an d t he h gh h f h h n l, ° u er ace m a e s an a e of 60 it d doe s pp f k ngl w h the lower. H ow m u ch woo he c ut out ?

8 T e Ex . . h center of a s qu are m oves along a diam et er of a given circle of radius a the a e of t he s u are bei er e dicu ar to t at of t he , pl n q ng p p n l h circ e an d it s m a it u de var i in s u c a wa t at t wo o osit e vert ice s l , gn y ng h y h pp m o e on the circu m e re i u m e f the s o id v f nce of the circle . F nd t he vol o l ge ne rate d .

9 e Ex . . The equiangular sp iral is a curve s o constructed th at the angl be twee n the radiu s vect or to an y point an d the t angent at t he s am e i po nt is const ant . Find it s e qu at ion .

. 1 D i r r t at the li e dra Ex 0 . eterm ne t he curve h aving the p ope ty h n wn from the foot of an y ordin ate of t he cu rve perpendicular to t he cor res ponding tange nt is of constant length a . GE OM E TR I CAL APPLI CA TI ON S 283

If the angle whichthe t angent m akes with the x-axis be den oted by it is at once evident t hat

a 1

2 3! V 1 + t an ¢ 1

From thi s follows

x 2 log (y a ) + C . a

Whe n the t ange nt is parallel to the x-axis the ordinate it self h i is t e pe rpend cular a . If this ordinate be chos en for t he y-axis the oi t 0 a is a oi t of th p n ( , ) p n e curve an d e ce , h n

C log a .

The e quation can accordingl y Fm , 78 . be writte n

rom t is o o s b ta i the reci roca of bot members F h f ll w , y k ng p l h ,

or ratio a iz i the de omi ator , n l ng n n ,

(2) a

Addin g (1) an d (2) an d dividing by

- e Z 3/ ) ,

ic h wh h is t e equ ation of the caten ary.

E . 1 1 l 2 at the ert h x . A right circular co ne h aving the ang e 9 v ex as its vertex 0 11 the s urface of a sphere of radiu s a an d its axis passing t rou d t he o u m e of the ortio of th h gh the cent er of the sphere . Fin v l p n e s ere i i x r ph wh ch s e te ior t o the cone . 2 2 £ 2 - E 1 . x . Find the volum e of the paraboloid z cu t oﬂ by the a2 ii a e pl n z c . 284 TE GR AL GAL L CH / VII LN GU Us [ .

EX ERC IS ES O N C H APT ER VII J “ - 1 . i d t he area bou n ded b the erbo a x a the x axis an F n y hyp l y , , d

the two ordi ate s x a x n a . n , rom the resu t obtai ed rove t at the area co t ain F l n , p h n ed between an i ite bra c of t he curv e an d it s as m tote is in i nﬁn n h y p ﬁn te .

3 3 2 . i d the area co tai ed bet ee the curves x an d x Q) F n n n w n y y.

d th area f the ev olu te of t h 3 . Fin e o e ellips e

? (W (W (a

- i d e are boun ded b th r 4 . F n th a y e pa abola V5

coordinate axes .

5 he ar contain ed bet een t . Find t ea w he curve

—x 312 a + x

an d its asymptote x a .

The i t e ratio m a be i [H 1 N T . n g n y fac litated by the substitution x a cos

6 i d the are a bet een the curve z 2 - = — . F n w g (y 2) x 1 and the

codrdin ate axes .

7 in d the area co mm o to the two el i ses v . F n l p

2 2 1 — £ Iii — + “ 1 + “ 1 w m 9 ﬁ

a an d b are two airs f ues f x n th 8 . If a . 8 o va o a d e ( ) ( , , ) p l y, formula for in tegration by part s gives b ﬂ dx oc Ly a Lx dy.

er f r Interpret this result geom et r1 cally in t m s o a ea.

i d h area bou ded b the o arit m ic or e uia ular s ira 9 . F n t e n y l g h ( q ng ) p l ea o an d the two radii p pl , P2

1 0 i he f n arc of the s ira of Archim ede s a 9 . F nd t lengt h o a p l p bet ee the oi ts w n p n (pl , (p2,

1 f the ri e erated b revo vi the circ e J 1 . Find t he su rface o ng g n y l ng l z 2 - (y le) G (k > a) about the x axis .

1 2 in d the vo um e of the rin de n ed in Ex . 1 1 . . F l g ﬁ

C HAPTER VIII

SUCCES SIVE INTEGRATION

1 6 3 . Fun ctions of a s ingle v ariable . Thu s far we have c o n s idered the problem of ﬁ n di n g the fun ction y of x when 011 ‘ i It i n o ﬁn d h on ly s given . s w propos ed to y w en it s n th dx

i n derivative s give . if” 1 du e i F ﬁ n h The m ode of proce r s eviden t . irs t d t e fu n c - l n dn y d y h ch has its de a . Th n b n e ‘ l w i ” for riv tive e , y i t dxn dx n —2 d y h d m n a n t e s u an d s o on u n a e 71. r ti g re lt , eter i e n ‘ z’ til ft r g dx

h r d n d A n s u cces s i ve in tegration s t e re qui e resu lt is fou . s a arbitrary c on s tan t s hou ld be added after each in tegration in

n t he m n a s u n the un c n order to obtai os t ge er l ol tio , f tio y will

n n ar ra c n s an s . co tai 71. bit ry o t t

s d y 1— Ex . 1 . Give n 9? 58, —1 Integration of wit h respect to x gives x§ 2 d y 1 C ? l dx2 2 x2

I te ratin a seco d tim e n g g n , dy 1 C x C l 2 dx 27 1:

2 d a lo x (7 x 0 2: C . an ﬁn lly 3) l g l 2 3 b The t riple int egration required in this example will be sym boliz ed y

- wa s 11153 —1 which will be called the trip le integral of with respe ct to x. x8 286 n vn r o . . SUCCESSIVE INTEGRA TION 287

2 . D eterm i h r Ex . ne t e c u ve s h aving the property that the radiu s of curvat ure at an y point P is proportion al t o the cube of the s ecant of the e ic the t a e t t m d angl wh h ng n a P akes wit h a ﬁxe line . If a s ys tem of rect angu lar axe s be c hose n with the give n line for - x ax is it o o s rom e u atio . 1 64 an d rom Art . 1 0 t at , f ll w f q n p , f , h 12

(Pg ’ a x2 in which a is an arbitrary cons tant . s equation reduces to

z d y dx2 from which follows 2 2 9 a (dx) C x ff 2 l

C an d 0 bei co sta ts of i te ratio . e ce the re u ired curves l 2 ng n n n g n H n q - are the parabolas h aving ax es parallel to the y axis . The ex iste ce of t he two arbitrar co sta t s C 0 in t he recedi n y n n l , 2 p ng

e u atio m a es it os s ib e to im os e u rt er co ditio s . S u o se for q n k p l p f h n n pp , ex am e it be re uired to determ i e the curve avi the ro ert pl , q n h ng p p y a read s eci ed an d avi be sides a maxim u m or a m in im um oin t l y p ﬁ , h ng ( ) p at 1 ( ,

Si ce at such a oi t 31 0 it o o s that n p n , f ll w 32:

0 a (l

' w e ce C 1 . h n l

Als o b s ubstitutin 1 6 in the e uatio o ~ the curve , y g , ) q n f ,

0 a 1 C (% a) , from which

Accordingly the required curve is £1 y (x 2

3 i h u i in t a u ar coordi ates f the curves Ex . . F nd t e eq at o n ( re c ng l n ) o having the property t h at the radius of cu rvat u re is e qu al to the cu be of n the t angent le gth .

T h de aria e . [HI NT . ake y as t e indepe n nt v bl ] 288 CAL C L v INTEGRAL U Us [ Cm m .

Ex 4 . A r i m . pa t cle oves along a pat h in a plane such that the slope of the lin e t angent at the m oving point changes at a rat e proportion al t o the re i r ca f t he abscis s a of t at i c p o l o h po nt . Find the e qu ation of the

curve .

A r . 5 i e s rti r Ex . pa t cl ta ng at re st f om a point P m oves u nder the f or u t act io o a ce s c t a t he acce eratio cf. Ex . 14 . 1 1 1 at e ac n f h h l n ( , p ) h i st a t of time is ro ortio a to is k tim es the s u are ro t f th n n p p n l ( ) q o o e tim e . H ow far will t he part icle m ove in the tim e t ?

1 6 4 . In r n f fun c i n f h teg atio o t o s o s everal variables . W en

u n c n s t wo o r m re var a es are u n de c on s d a on f tio of o i bl r i er ti , the proces s of diffe ren tiation c an in gen eral be p erform ed

h s e c an on e t he a es h e the he s wit re p t to y of vari bl , w il ot r

n n r are t reated as co s ta t du in g t he diﬁ ere n tiation . A repe tition of this p roce s s gi v es ris e t o the n oti on of s ucc es s ive partial differen tiation with res pect to on e or s eve ral of the

i f r h n u n n . A a a es n e d n t e ve c C . s . 6 8 v ri bl i volv gi f tio [ t ,

The e rs c ss ead s u es s s e an d r s e n s r ve e pro e r ily gg t it lf, p e t the r em : Given a artia l rs t or hi her deriva tive o a p obl p (ﬁ , g ) f fun ction of s evera l va ria bles with resp ect to on e or m ore of these

va ria bles to n d the ori i n al u n ction . , ﬁ g f This problem is s ol ve d by m ean s of the ordi n ary pro ces s es

n e ra n but t he adde d c n s an o f n e ra n has a of i t g tio , o t t i t g tio

Th s can m de c a b an xam . n ew m ean in g . i be a le r y e ple Su ppos e a is an un kn own fun ction of x an d y s uch that

an 2 2 x + y. dx

e h s h es e c x a n e ea n at the In tegrat t i wit r p t to lo , tr ti g y

Th s s s am e tim e as thou gh it we re c on s tan t . i give

n n B ut s n ce in whi ch 4) is an adde d c on s tan t of i tegratio . i i 3) is regarde d as con s tan t du ri n g this in tegrati on the re s T d n d n othin g to preven t 96 from depen di ng on it . his epe

290 on v m GAL GUL Us . INTEGRAL ( .

wherein P an d a re u n ction s o x an d does there exist Q f f y, a fun ction u of the same varia bles having ( 1) for its tota l dilferen tial?

It is e asy to s e e that in ge n eral s u ch a fu n ction does n ot

x s . For in o de ha 1 m a a a d ffe ren i f e i t , r r t t ( ) y be tot l i t al o a

u n c n u is d n n ces s a ha an d h the f tio , it evi e tly e ry t t P Q ave form (in au 2 dx dy

h n hen m u s xis e e n an d in de W at relatio , t , t e t b tw e P Q or r that the c on dition s ( 2) m ay be s atis ﬁe d ? This is e asily fou n d as follows : D ifferen tiate the ﬁ rs t e qu ation of ( 2) with

s ec an d the s c n d h e s ec t o x . Th s s re p t to y, e o wit r p t i give

2 z GP 6 11 6 Q d u ’ ’ ay ayax dx az ay from whi ch follows aP aQ dy dx

‘ This is the relation s ou ght . h It The n ex t s tep is to ﬁ n d t e fun ction u by in tegration .

n ra on is easier to m ake this pro ces s clear by a illus t ti .

G en 2 2 2 2 x 2 d iv ( + y + + ) y,

h n d ff n ﬁn d the fu n ctio n it avi g this as it s total i ere tial .

Si nce it is foun d by diﬁ ere n t iation that

an d hen ce the n eces s ary relation ( 3) is s atis ﬁed . From ( 2) it follows that s uc cE s s I VE I N TE GR A TI ON 291

n a n h s h es c x a n I tegr ti g t i wit r pe t to lo e ,

‘ u = xy 2 x

It n ow remain s to de termin e the f un ction ¢(y) s o that

D ffe en a 4 h s e c t o a n e h n ce i r ti te ( ) wit re p t y lo , w e

da 2 ay

’ where (p(y) den otes the de rivative of qS(y) with respect to

Th a on of h r u y. e c omp ris t is es lt with ( 5) gives

2 2 or a ,

h n ce b n e ra n h es e c t o w e , y i t g ti g wit r p t y,

in which 0 is an arbitrary c on s tan t with respect to both

x an d y. H en c e

It is to be rem arke d that in i n tegrati n g ( 6) we in tegrate

exactly thos e te rm s in Q which do n ot c on tain x . H e n c e the followin g rule m ay be form ulated for in tegratin g a total differen ti al

I n te ra te with res ect to x alon e treatin a s con stan t . g P p , g y Then in tegra te with resp ect to y thos e term s of Q which con

tain but do n ot con tain x an d a dd the res u lt to ether with y , , g ' an a rbitrar con stan t C to the term s a lrea d obtain ed . y , y It is e v iden t that it wou ld be e qu ally w ell t o ﬁrs t in te

a e h es e c t o an d h n n e a h s e m s gr t Q wit r p t y, t e i t gr te t o ter of h ch c n a n x a n e h s e c x an d add the P w i o t i lo wit re p t to ,

u two res lts . A C L GUL US an . v111 INTEGRAL ( .

EX ERC IS ES

D rmi in ete ne e ach of the following cases fun ction it having the give n ex press ion for its tot al differen tial

1 . ydx x dy.

2 . s in x cos dx c x in y os s ydy. 3 . y dx x dy.

4 . my

2 2 5 . 3 x 3 a dx 3 ( y) ( g 3 ax) dy.

ydx x dy 6 ° . 2 2 2 x + y y + x2 7 — . y) dy. 4 4 2 — 8 . (x + y + x 2 xy + y

2 6 a T M ultiple int egrals . he in tegration 0 f was dx dy

c n s d d in Art . 1 64 . If E x n the ve n o i ere ( , y) be writte for gi

u n c n t he u ed n n f tio , req ir i tegratio will be repres en ted b \ the symbol u F x dx d ff ( , y) y,

an d the fun ction s ou ght will be called t he d ou ble in tegra l of

E h n x s e c x a . ( , y) wit re p t to d y

s E x x clx d dz Likewi e ff ( , y, ) y

ca ed the tri le in te ra l E x z . It s n s will be ll p g of ( , y, ) repre e t 3 6 u the fun ction u whos e third partial derivative is t he dx dy dz

e n un c n E x z . It be un d s d in ha giv f tio ( , y, ) will er too w t

s ha the o d n e a n is m h e follow t t r er of i t gr tio fro rig t to l ft,

ha is w e n e s h es c the h h n d t t , i t grate ﬁr t wit r pe t to rig t a vari

2 h n h s c an d as h s c x . able , t e wit re pe t to y, l tly wit re pe t to

d in Su ch n a s d u et c . i i tegr l ( o ble , triple , ) w ll be referre to gen eral as mu ltiple in tegrals .

294 GALGUL vs Cm V III INTEGRAL [ .

EX ERC IS ES

Evaluate the following integrals

£1 1 2 . se c (xy) dx dy.

a l “ 71 H 2 z .t f 21 511 12 2 . r in 4 . j j ? s 9d 9 dr. . If}

1 6 8 . Pl n r s b d ubl in r i n The a ea u n de d a e a ea y o e teg at o . r bo by a plan e cu rv e ( or by s everal cu rv es) can be readily e x

res s d in t he rm a de n e d u n n u p e fo of ﬁ it o ble i tegral . A ill s t r i x h at ve e ample will explai n t e m e thod .

2 x . 1 E . Find by double int egrat io n the area of the circle (x a) 2 2 (y b) r Im agine t he give n area divided into rect angles by a s eries of line s parallel t o t he y-ax is at e u a dis t a ces Ax an d q l n , a series of line s parallel to the x-axis at e qu al

dist ances Ay. The area of on e of t hes e re ctan gles is Ay T is is ca ed t h Ax . h ll e

elem e nt of area . The sum of all the rect angles int erior t o the circle will be le ss than the area requ ired by t he am ou n t cont ain ed in t he sm all su bdivisions whic h bor

FIG . 79 . der t he circu m feren ce of

d in Art . 149 it the circle . B a m et od exact a a o ou s t o t at u se y h ly n l g h , is easy t o s how t hat t he su m of th e se n eglect ed portions has a ze ro lim it whe n Ax an d Ag are bot h m ade t o approach z ero . To fi n d the v alu e of the limit of t he su m of all th e rect angles within t he circle it is conv enie nt to fi rs t add to gether all t hose whic h are con P P be on e of t e se in b o co ecu ti e ara e s . Let t a ed e tw ee n t w ns v p ll l 1 2 h - T e rem ai s co st a t parallels having the direction of the x axis . h n y n n n 16 7 S UCCE S S I VE INTEGRATION 295 l 2 1 i e x varies rom a V r -b the v a ue of the absciss a at P wh l f (y ) ( l 1) z ero of the su m of rectangles in t he st rip from P P 1 3 ev ide ntly — — a f s/ fl (v bfl 1 A im it of su m Ax Ax m r: A ( ) g [l ( )] y dx . — — (v b) 2

N ow ﬁ n d the lim it of t he su m of all su ch strips con t ain ed within the i r i i circle . This requ re s the dete m n at o n of the lim it of t he su m of term s su c h as (1 ) for the different valu e s o f y corre spo nding to the differe nt

st ri s . Si ce be i s at the o e st oi t A it the va ue b r an d p n y g n l w p n w h l , i creas e s t o b r the v a ue re ac ed at B the a ex res sio for t he n , l h , fin l p n area is — — - — (y b) 2 b+ r a I r2 (y b) 2 d — Ib—r a — b 2 Ib—r a —Vfl — —b (y ) (y fl

I te rat i rs t it res ect t o x n g ng ﬁ w h p ,

T is resu t is t e i t e rate d it re s ect t o ivi h l h n n g w h p y, g ng

If t he su m m at io n had begu n by adding t he rect angles in a st rip paral lel to the -ax is an d t e addi all of t ese s tri s t he ex res sio for y , h n ng h p , p n the area would t ake the form

lt is s ee rom t is last resu t t at the order of i t e ratio in a doub e J n f h l h n g n l integral can be c hanged if the lim its of inte gration be properly m odiﬁed

at t he sam e tim e .

2 h i e e t he t o rab Ex . . Find t e are a which is nclu ded b tw e n w pa olas

2 2 = z — y 9 x an d y 72 9 x .

3 i d the r a h t o irc es Ex . . F n a e com m on t o t e w c l

2 — 2 x 8 x + y

2 — 2 — x 8 x + y 4 y +

1 6 . Th m u n d d 9 Volumes . e vol u e bo e by on e or m ore s u rfaces can be e xpres s e d as a triple in tegral when t he e qu ation s of the bou n din g s u rfac es are given . 296 11 VIII INTEGRAL GALCUL as (0 . . Q Let it be requ ired to ﬁn d the volu me bou n ded by the

s u ac AB C Fi . 80 h s e u n is z = x an d rf e ( g ) w o eq atio f( , y) ,

h rdin by t he t ree cob at e plan es .

Im agin e t he ﬁgu re divide d in to s m all - equ al re ctan gu lar

ara e eds b m ean s o f h e e s e es an es the rs p ll lopip y t r ri of pl , ﬁ t s es a a e the z - an at ua d s n ces Ax the eri p r ll l to y pl e eq l i ta ,

- l d s n c s A an d s e con d parallel to the xz plan e at equa i ta e g, - the third parallel to the xy plan e at e qual dis tan c es A2 . i The volu m e of s uch a re ctan gular s olid is Ax AyAz ; it s

The m of the sum called the elem en t of volu m e . li it of all ' i the um e u d s uch el e m en ts co n tain ed in OAB C s vol e r q ire , ’ B i c n n u u s . provided that the boun din g s urface A O s o ti o

298 CAL n GUL O . VIII 1 9. INTEGRAL Us ( . 6

The im it of in te r i l s g at on wit h re spect to y are fou nd to be y 0 (the ? 2 va u e at t he x-ax is a n d V a x the va u l ) , y ( l e of y at t he circu m fer 2 2 2 e ce of t he c ircle a x 0 in ic t he s r i n y , wh h phe e s cut by the - xy plan e ) .

i a l the lim it i va u e s for x are z ero a nd a the latte r bei t h F n l y, ng l , ng e distance fro m the origin to t he point in which the sph ere int ersect s the - x ax is . H en ce — a Va 2 a 2 x2 y: V vo u m e of s ere 8 dx l ph ) dydz . 5;lo

I te rati rs t it h re s e ct to z n g ng ﬁ w p ,

2 — 2 — 2 V = 8 a x y dx dy;

t e it res ect t o h n w h p y,

2 a a a 2 — 2 — 2 “ dx \/a x y + s in [g 2 E

“ 3 4 7rd 3 2 _ 4 5 (a 5; 4 3

2 . d the o u m e of on e of the ed u r th i r Ex . Fin v l w ges c t f om e cyl nde 2 2 2 x y a by the plan es z 0 an d z mx.

3 d h u m e m m t o r r r J Ex . . Fin t e v ol co on two ight ci cular cylinde s of

the sam e radius a wh os e ax es inte rs ect at right angle s .

4 h o m th i der x 1 im it ed Ex . . Find t e v lu e of e cyl n ( (y l

b the a e z 0 an d the erbo ic arabo oid x z . y pl n , hyp l p l y

5 i h o u m of he e i soid Ex . . F nd t e v l e t ll p

2 2 2 x 2 + + L

E t a.

i h um e of t at ort io of the e li tic arabo oid E 6 . d t e vo x . F n l h p n l p p l

2 2

2 2 a 6

which is cut offby th e plane z 0 . ANSW ERS

2 . A r Pa ge 0 t . 9

- — 3 1 . 2 913 2. 2 . 6 x 4 . 3 . 4 . 4 x

2 A r 1 t . 1 Pa ge 5 .

2 - — - 2 2 — - 1 5 2 2. 1 4 1 4 33 1 . 3 . 1 2 u 2 . 4 . 4 2 5 . 1 . y .

Ar . 1 Pa ge 2 8 . t 3

r m t o l in rom 1 t o n d 1 . In rom 00 t o dec . o c . . 00 a c . f 51 ; f ; f ; g

4 . t a Two . i n

A rt . 1 4 Pa ge 2 9 .

1 — 2 6 u 4 6 x . 0 2 4 . 6 u 2 . ( ) 0 u2 (

Page 3 7 .

0 1 5 H 0 3 8 9 T Y — 2 2 x(l x 1- 4 ai + 3 xi 0 0

l 300 AN s WE R s

2 x3 4 x

(1 main xvi

7 u2 6 5 2 Si QE ( 23 . 0 u (l + 11 )

dx 35. At ri t a les at 3 gh ng ( , ; t

Pa 4 ge 3 .

1 n + .

n - l n -‘ I nx log x x .

r m n “ l nxn l log x mx

x2 1

xn x lo 1 ) a g a.

+ x(3 x + 2 )

AN S WE R S 303

2 x2v + 3 y2 3 x2 + l

26 . j3 y2 + 1

— - 2 1 2 xy x

x l For a l valu es .

21 . x are d et ermi ed rom , y n f x 2 d e uatio of GM ; l; b x an q n 22 . 1

curve . m “ 2 au x s ec x t an x. 1r 23 . t e — x = x1r i o 4

24 . 2 - 1 t an 2 2. ax y 34 . V

— x Page s 5 4 5 5 . E erc is e s o n C h ap t e r I II

" s . 1 9 . m co (mx n g)

m (m 1 ) l ( a x) + ”

ex ex

2 l 3 og x . 8 tan x s e c2 x (3 s ec2 x 2 2 cot x cs c x .

16 s in x cos x. 24 are

2 i i . 31 3 , (2 1 03 — 271 1 n ! 32 .

1 1 (2 x (2

2"- 1 2 M 36 . cos x (x 2 ) 304 AN S WE R S ,

4 Pa ge 6 .

3 4 (x 1 ) 2 1 + (w — 8 + 4 (y

Pa e 7 g 6 .

x3 2 1 7 — 5 — 7 1 . x + + x + x R . 3 l 5 3 1 5

x2 x3 3 x4 l l x5 6 ’ 1 + + 2 3 8 30 9 .

x2 x3 3 x5 7 a: + - + R 4 7 3 1 1 2 — 16 —41 1 0 . x x x x . 2 3 40 + +

— E x Pa ge s 7 5 7 6 . e rc is e s o n C hapt e r IV

2 3 — —h 71 1 . cos x h s m x cos x + § é

3 2 2 . + t an h) + 3 h h2 h3 h4 10 g x + + + R x 2 x2 3 x3 4 x4

5 4 1 3 2 10 2 3 5 4 5 5 . x + 5 x y + 0 x y + x y + xy + y .

— A 4 7 rt . Pa ge s 7 9 8 0 .

3 .

— 4 Pa ge s 8 3 8 . — 8 . 4 . 16 . 1 .

17 . 1

— A 2 8 7 8 8 . rt . 5

0 9 . . 12 .

O or . 10. 00 according as n O or O .

1 3 . g,

14 . 1 .

306 A N S WE R S

Ex e rc is e s o n C h a r V I Pa ge 1 1 1 . pt e I

2 . . b 00145 . 9 a 1 3 6 .

10 . i 2 . l m our iles er . i p h a 1 7 1 1 . 5 1r. 16 6 2 t 3 1 2 . . The row . : ( ) 1 . 16 12 9 4 , q feet ° 1 n 1 3 . a d er s eco d . At 6 0 5 . p n

P rt . 8 a ge 1 1 6 . A 6

a: cos . 4 . ( y) my

2 A r . 1 Pa g e 1 0 . t 7 if 4 . has by + f 213

— . 7 Pa ge s 1 2 7 1 2 8 A rt . 6

d zy 8 . 0 . ‘ y du2

R 3 . z

dgx dx ﬁ ” z 0 2 2 x dy dy 2 + (d 3 d z‘ ) y y 2 (1 7 Z 2 d z 1 2 ' — cos2 z + 2 dx2

2 z d u 2 d u 0 1 Z 2 z _ z doc dy

- 7 9 A rt . Pa ge s 1 3 1 1 3 3 .

4 . 0 y 2 £13 3 a . — (13) ill 111 xl ) ‘ i 3 y s 2 x i .

—‘ = 3 p y x . 4 / l 7 3 . 6 . \ . 2 y : 9 m 5

O. (05) Parallel at points of int ers ection with ax kg;

O. Perpendicular at point s of int ers ection with has by 3/ ’ 3 2 a a \ 0 (6) Parallel at perpendicular at a: . 2

s 2 ‘ a er e dicular at O 2 a. O . (7 ) Parallel at ; p p n ( , ( ) 3 ? ) AN S WE R S 307

I l 1 1 1 m be c o l i s . t e us t o ca h y nf . ' ' a b a b

2 1 . 9 ( p ,

1 —1 A rt . 2 Page s 3 6 3 7 . 8 ! 31 0. 2 ” Polar s ubta e t Polar ormal v a2 Polar s ub ormal ng n n p , n a 0. 1r a 3 . x1 2 0 Subt a e t cot 2 0 Ta e t 1 5 , ng n p , ng n

2 a s i n 2 0 Subnormal P

2 e 2 2 a s i g wa . . , n n 2 2 2

T e ave a comm o ta e t at th e ole els e ere 7 . h y h n ng n p wh

Pa 1 4 2 Ex rc is e s o n C h a t e r X I ge . e p

a x 2 V m 4 r 4 r 2 2 /a. am m. 1 . a 1r \ a l , , , \ a: b2 - 2 ec 5 . 7r a s 23. ( a2 2 2 ” a a" 2 m lo a . 6 . m ( g )

Ex e rc is e s o n C h a t er X II Page 1 5 1 . p

z 0 t ice n r bolic 8 . a: o e a a O a: a a: a . y , , w p

— br n °h ° a .

z 9 . 3 0 0 x 7 y x y z : a a 0 t W O i ma i ar y , y g n y im a i ar two . 10 . y as ; g y . z n = t W I ° x C Ce 1 w im i t o a ar . 1 . g n y ( l w im i a t wo im i 1 2 . a; O t o a r . y x ag nary. y g n y . 3 - 1 3 . y. a; 1 on e arabolic bra ch p . wo im a i r . n 14 . x a: t a y , y g n y

— = — = : — z — a: 2 0 1 l . 1 . x x a , y a . 5 x + y , + y , y

P h t e r X I II a ge 1 5 8 . E x e rc is e s o n C ap

An in ﬁexion 1 . at 2 x y .

—4 2 = 3 . . 6 x a y 0. 2 2 xfé x/3 . , 7 a

8 . Poi t of in ﬁexion at a . a t a e t is a: B e di c a es n ( , g ) , ng n y y n ng h ng rom e ativ t o os iti e f n g e p v . 308 AN S WE R S

P 2— e s 1 6 1 3 . ag 6 Art . 1 0 2 Fi rst . 5 . y 12 a: 10.

T e do n ot t ouc . 6 . Se co d h y h n .

ird . T . a 1 h 7 .

3 x : x( a) a (y a) .

Pa e g Art . 1 08

3 i s . ax . 5 , ( y)

l ? u ( n ) xy a x)

x/E

Pa 1 ge 7 0 . Art . 1 0 9

1 . 5 , 2 2 a a 2 3 p

9 6 cos 0

a 1 7 . P ge 6 A rt . III

: z o i 0 . 1 . 06 , ﬁ f _ g — ‘ — " 4 . a z x c e : 2 g i 5 y.

? a i 5 . (a B) ( B)

i ? ? . . t s 6 . (m as a a 1 5 y a y 9 g o ( ) ( . ’ B 2 4 6 a y 2 a 7 .

x r is e s n h Page 1 8 6 . E e c o C apt e r X V

1 . O ax b O. 6 . Two od es at i it h ( , ; l; y n nfin y t e as m totes are a: 1 y p y i . a: y i: 1 . O O us of rs t i d O 2 . c . ( , ) p fi k n , y — 7 . O a a O a 0 ( , ) ; , ) ; , ) h 3 . Four cus s of rs t i d t e ta e ts are res ectivel p fi k n ng n , p y,

4 . O c o u ate om t Wl th ( , nj g p 2 — : (x a) :l: V3 y. real com i den c t t a e ts 0 . ng n , y i 8 . a co u ate o t . , nj g p n = d 0 5 . O a a x cus o s eco 9 . O O. ( , ) ; y + ; p f n ( , x , y i 0 ki d . 10 . 0 O s a t ac od e . n ( , ) n y

31 0 A N S WE R S

Pa e s 2 0 —2 6 0 . g 9 Art . 1 2 7 - % (s m 1 x) 2 14 lo . g tan 2

z x -- 1 L -l l l _ x t an mn 1

1 — _ 1 3 x 1 " t an l s 1 g or l " 1 9 ec ’ COS a a a V I4

x a a: 2 " - 1 lo ver 1 o r 1 3 i g s , s in g a x 1 - 1 in 1 3 19 . s a: lo 96 V231 2 3g ( g ( i a ) .

20 . cos 3 2 1 x —a ( x ) log ﬁ x -f a a2 -1- bcc — a 1 10 log tan ﬁ 2 3 2 V a2 + bx + a

Pa e s 2 1 3 2 1 4 Ex r is h g . e c e s o n C apt e r I

7 . V ar

2 (x + 1 )

‘ l 4 ’ - 2=c 14 . og ( 6 + v e

V 1 V2 + V x2 + 2 lo 1 5 . g 3 3

gt an 2

§log (6 x3 1 2 2:

’i ‘ l — ‘i -- z i x t an x gﬁ x l ﬁyx -1 x log 2 tan x. AN S WE R S 31 1

l lo 2 [2 2 a: og a (96 g a) ] .

1 lo co 2 2 23 . g (a s a: b sin 9 2 (b a ) x lo s in cos 2 24 . % [ g ( a; s “a 1 . l §- x §- x% i xi

- t an 1 lo 30 . ( g 1

31 . b (a b tan x)

ﬁ az

. d z 2 V2: vers a x. 33 . a

- Ex Pa ge s 2 1 9 2 2 2 . e rc is e s o n C hapt e r II

x —a

2 2 a.

r e m e ts i an be 5 . The a ra c c used are B C B C an d ng n wh h [ ] , [ ] , [ ] , [ ] , B B [ ] [ ] ,

a: l 1 t an 4 2 6 8 —l 2

— _ 2 56 1 4 - 2 x 1 t an 1 3 06 2 V3

— 2 x ar2 x s — x2 - s §n a + 2 / z 2 § a 3 a \ x + a 3 4 2 a mz + a + -—— 8

2 — ax —x + —s in 31 2 AN S WE R S

95 1 / — — 2 -1 + l ) x 1 2 x e + s in . x/ z

Pa 2 2 . ge 6 A rt . 1 3 3 “32 § M 4 , 4 x + 10 g . 2 2 x + 1

b a: lo a: — 5 . g ( b) x(x 1 )

2 + ‘ g f log (x 2

x — c 1 x + - l — a — 1 o ] 1 3 . g

x5 1 ax — b l l0 g — 5 ar:) 2 ab ax + b

a: 1 x _ I /g “ ( s in 1 h —lo — x V 2 — 2 g a + x ) . x z

— x 2 \/ x2 + 4 2 +

A . 1 4 Page 2 2 7 . rt 3 x x + a

—5 1 3 . og z o l lo - l x ( ) x + g og x] . m_ 3 3 m 1 ﬁ a ' l' bx 1 2 — 2 . log 2 ( a x ) 2 a 2: act — 1 — - 2 log [x 2 x -l 2] V2 az (m

+ log x x 1 2 2 $2 + x

a)2 — a? cc — l — 13 . log (x a) + — 2 x)

31 4 AN S WE R S

Pa ge s 2 3 6 Ex e rc is e s o n C hapt er IV

1 3 1 x 3“ 2 . %( a) 2 2 x (0 3+ 1

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(The numbers refer t o pages)

Absolu te value 59 . Co s t a t 1 . Elli se area of 263 . , n n , p , ,

r e t 59. fact or 31 199 . le th of r o f Abs olute l co ve a c 274 . y n g n , , , ng ,

r 200 . v lu Acceleratio 1 1 1 . of i t e atio e o t e o f 178 284 . n , n g n , , ,

Actu al v elocit 105 . Co t act 159 . Elli soid v olum e 280 . y, n , p , ,

Arc len th of 269 . o f odd an d eve order E v elo e 187 . , g , n , n p , 1 E i l i l Area b d ouble i te ra 16 . c c o d e th o f 273 . , y n g p y , ng ,

Co ti u it 1 3 1 13 . E uian ular s iral 282 n n y, , q g p , ,

d erivative of 23 . Co ti uo us fu ctio 13 . 284 . , n n n n,

formu la for 255 256 . Con ve r e ce 57 . E valu atio 80 81 . , , g n , n , , E n l r oiirdin at es 26 8 . onv ex 157 . volu t e 1 70 . i po a c , C , , '

brdi h i 15 . of lli 1 in rect a ular co t o t e ax s 7 e s e 76 271 284 . ng , p , , , f r at e s 260 . Critical valu es 93 . o a abola 175 . n , , p ,

As m t ot e s 143 . Cubical arabola 262 . Ex ans ion o f fu ctio s y p , p , p n n ,

A er e cu rvatu re 166 . Cus 1 82 . 56 . v ag , p ,

C cloid le th of 273 . Ex t erior rect n l s 254 a e . y , ng , g ,

B e din direction of 152 . su rface of revolu tio n g , , n , Fam il of u r 1 y c ves , 87 . B n m i l t h or m 3 . i o a e e , 7 277 . Form ula for in t egration

b art s 203 . y p , Cardioid area o f 268 . D ecreasin fun ctio 25 . , , g n , Form u las of diﬁ eren tia Cat e ar 168 283 . D e it e in t e ral 251 . n y, , ﬁn g , n tio , 49 , 50 . le th of arc 271 . eom etric m ea i o f ng , g n ng , o f in t e rat ion 198 210 m l i g , , . volu e of revo ut o 253 . n , of redu ctio 217 218 . l n , , multi e i t e ral 293 . p n g , Fu ctio 1 . n n , Cat e oid 276 . D e e d en t v ariable 1 . n , p n , ’ Cauch s form o f rem ai D eriv ative 19 20 . y n , , erbolic bra ches 143 . Hyp n , d er 71 . f r 3 , o a c , 1 8 . s iral are o f 26 9 . p , a , f Cen t er o c u rvatu re 163 . f r 2 , o a ea , 3 , 142 . oc c loid area of 263 . Hyp y , , Ch an e of variable 1 24 . of s u rf g , ace , 140 . le th of a rc of 27 1 273 . ng , , Circle are a b d ouble in o f l m v o u e 140 . , y , v olum e of revolutio of n , t e ration 295 . D rmi et e at e value 78 . g , n , 278 .

of c u rv at u re 1 63 . D ev l m e o e t 56 80 . , p n , ,

Cis s oid 16 8 . D iffer n i e t als 1 1 m li it fu io 1 20 . 0 196 . I c ct , , , p n n ,

area o f 266 . i te rat io o f 289 Im os sibilit of redu e , n g n , . p y

Com on en t velocit 107 . t p , ot al 1 17 tio 218 . y , n ,

1 52 . Co cave D iffe re tiati o erat or In c reasi fu ct ion 25 . n , n ng p , ng n ,

t o ard axis 1 57 . 2 . 4 n 1 3 15 . w , I crem e n t , , Con ditio all co ver e t iff r D e e tiatio 24 . In d e en d en t v ariable 1 . n y n g n , n n , p ,

59 . of ele m t r f m 49 In d te rmin at e fo rm 77 e a or s . e n y , , Co ditio s for co tact D irectio of cu rvatu re In ite 2 . n n n , n , ﬁn ,

16 1 . 1 it f in t r 64 . In ﬁn ite lim s o eg a

Co u at e oi t 1 84 . i D sco ti uou s fu ction 14 . tion 257 . nj g p n , n n n , ,

Co oid 281 . D iver en t s e rie s 57 . ordin at es 145 . n , g , , 3 19 320 I N D EX

In ﬁ n it e im l s a 2 . M i im um 91 . S h ere v olum e , n , p , by t riple

I t e ral 195 . M ulti le oi t s 181 . i g , p p te rat ion 297 . n n , n g , d e it e , 25 1 . S h eroid oblat e 2 ﬁn p , 76 278 . , , d u l o b e 292 . N at ural lo arithm s 40 . rolat e 2 , g , p , 76 . - multi le 292 . n u i , No u e d erivative 25. S iral ofArchim ed 3 p n q , p , es , 1 6 . f um 1 o s 99 . rm . , N o al , 129 e u ia u lar 1 37 282 q ng , , , t ri le 286 292 N tio for r t . ota a es 108 . p , , n ,

I t e rat io 195 . h li erbo c 26 9 . n g n , yp ,

b i s ec tio 197 . Obli u e as m tot e s 147 . lo rithmi a c 272. y n p n , q y p , g ,

b art s 203 . Ord er of co t act 160 . St d r f m y p , , a a d o r s 1 98 210 . n n , ,

b ration aliz atio 231 . of differe tiat io 121 St ation r . a t an en t 153 . y n , n n , y g , i 5 b s ubs titu t o 20 238 . o f i it e s im al 8 . S t e s in s ff y n, , , p di e re tiatio nﬁn n n ,

form ulas of 198 210 . f m i u d o a t e 7 . 24 . , , gn ,

of ratio al frac tio s Os culatin circle 163 . Stirli 62 n n . g , ng , .

223 . Os ood 57 Su orm l 1 b a 30 . g , n , of t ot al diff r i l 289 e e t a . Subtan e t 130 . n , g n ,

s ucces s iv e 286 . P r b l 1 1 umm a a o a 7 . S at io 251 . , , n , m m t - su a io 248 . m i i Su rf c f r s e cub cal 262 . a e o ev olu tio 140 . n , , n , I t erior rec n l 251 P r boli br n r ta es . a a c a ches 143 . a ea of 274 . n g , , , I terval of co ver e ce Paraboloid 283. n n g n , ,

57 . Parallel cu rves 175 . Tac od e 182 . , n ,

I olut e 1 0 . 1 Tan 21 v 7 Param et er 88 . e t 129 . n , , g n , ,

f ir l 2 2 P rti l d ri i 1 Ta l r 6 2 . o c c e 74 85 . a a e v at v 1 4 . o , , e , y , ’ P ﬂ x i 1 3 Ta l r r oi t o f in e on 5 . o s s e ies 66 . n , y , ’ La ra e s form of re Polar coiirdin at es 133 . Tes t s for co v er e ce 58 . g ng , n g n ,

m ain der 70 . sub orm al 135 . Tot al cu rv at u re 166 . , n , , iff r n m n i n n 1 35 . d e e t ial 1 1 L s cat e re of 268 . s ubt e t 7 e , a a , a g , ,

Le th of arc 26 9 . Problem of differe tial Tract rix 28 1 . ng , n , f 2 of evolut e 173 . calcu lu s 16 . le th o 85 . , , ng , iirdin a su rface ofrevolu tio of olar co t es 271 . of i te ral calculus 1 95 . p , n g , n , rect n ular coﬁrdin t s a g a e , m e of r lu i f 269 . Radius o f curvature 164 . volu evo t o o , n ,

1 285 . Rat es , 05 . chan e of in de ite in Ratio al fractio s i t e Tra sce de tal fun ctio s g , ﬁn n n , n n n n n ,

i n f 223 38 . t e ral 295 . rat o o . g , g , i s Lim it s i it e fo r deﬁ Ra t io aliz atio 231 233 . Tri on om et ric fu ct o , nﬁn , n n , , g n n ,

t r t i f 238 . it e i t e ral 257 . Rect an les ex terior an d i e a o o n n g , g , n g n ,

Lo arithm derivative o f i t erio r 254 . g , , n ,

Redu ti c as es of im os Variable 1 . 39 . c o n , p , Volu m e of s olid of revolu rithmi curv e 26 3 . s ibilit o f 218 . Loga c , y , — io 277 S iral le th of arc 272. formulae 217 218 . t p , ng , , n , i l i te m in der 6 1 . Volum es b t r e Re a , y p n ’ ’ Rolle s th eorem 67 ration 295 . M aclaurin s s eries , 63 . , g ,

m . Maxim u , 91

r f 26 3 . M ea v alu e theorem 75 Si ular oi t 179 . W itch a ea o , n , , ng p n , , lum e of revolu tion l e 21 . vo 257 S op ,

of 278 279 . M easu re ofcurvature 166 . Solid of revolutio 140 . , , , n,