Differential Calculus

A TEX T -B O O K

DIFFERENTIAL CALCULUS

WITH NUMEROU S WORK ED OUT EXAMPLES

GANE S

B A A A B . . NT . (C )

E B ER F E L O ND O N E I C S CIE Y F E D E U SC E M M O TH MATH MAT AL O T , O TH T H MA E IK ER-V EREI IGU G F E CIRC E IC DI P ER E C TH MAT N N , O TH OLO MAT MAT O AL MO , T . FELLOW OF THE U NIVE RSITY OF ALLAHAB AD ’ AND P R FESS R OF E I CS I UEE S C O E G E B E RES O O MATH MAT N Q N LL , NA

E N L O N G M A N ! G R E , A N D C O

3 PAT ER TE ND 9 N O S R R OW , L O ON

NEW YORK B OMBAY AND AL UTTA , , C C 1 909

A l l r i g h t s r e s e r v e d

P E FA E R C .

IN thi s work it h as b een my aim to lay before st ud ents a

l r orou s and at th e sam e t me s m le ex osit on of stri ct y ig , i , i p p i

lculu s nd it c f lic n T th e Differential C a a s hi e app ati o s . h e p resent volu me is inten ded for b eginners and is SO designed

r u rem en s of rt f h as to m eet th e eq i t Pa I . o t e C ambridge

h m t cal Tri os Ex amin at on an d of th e E xam nations Mat e a i p i , i

f n n n r i d E. c . e rees o e s n a U v t es . for th e R A. a S d g I di i i Th e chief ch aracteristics of th e p resent work may be indicated as follows Th e fun damental p rin ciples of th e Differential C alcul u s h ave been based on a p u rely arith

met cal fou n ation . Thu s th e var ou s theorems h ave een i d , i b carefully enunciated and their p roofs have been ma de quite

n n i l nn t on i e e ent of eometr ca intu ti n . In th s co ec d p d g i o i i ,

’ I may sp ecially mention th e chapters on Rolle s Th eorem

’ and Ta lor s Theorem Max ma and M nima a nd Ind eter y , i i , minate F rms 2 Al s ar cle i f ll e o . ( ) mo t every ti s o ow d by wor e out exam les s c all su te for illu strat n th e k d p , p e i y i d i g

ar l T r i r c a ter. tic e . here a e al so n umerou s ex ercises n eve y h p (3) A sp ecial cha pter deals with cu rve-tracing and th e im

ortant r r s f h - n 4 Th e or er p p ope tie o t e best know curves . ( ) d in which th e ch apters are arran ged is intended to en able th e beginner to stu dy th e Simple geometrical applications of th e Differential C alcul u s immediately after h e h as learnt differen tia i B n t on . (5) Th e mi scellaneou s notes A and are inte ded

5 2 4 6 6 3 vi P REP ACE to give th e ambitious stud en t a glimp se of th e modern

l lc ul u s researches in th e Differentia C a . This vol ume is b ased o n my exp erience in teaching th e elements of th e Differenti al C al c u lus to a l arge n u mber of

f n element ar u l s . It is therefore throu hou t o a p pi , , g y

h racter Bu t as cer ain arts o f it ma be fou n fﬁ cu lt c a . t , p y d di

e nners th e have een m ar e with an a steris and by b gi , y b k d k

n may be omitted in a ﬁrst rea di g . n A few word s may b e said here ab out Ch apter I . I a

h m t al oo h ch ro fesses to b e r orou s in it s ma t e a ic b k , w i p ig treatmen t it is essent al that th e eﬁn t ons be carefu ll , i d i i y

i n It i r Th s h a een one n th e rese t wor . s wo ded . i s b d p k ,

be h ever o s le that for th s reason ha ter I . m a ow , p s ib , i , C p y

em m a v e foun d h eavy rea ding by some stu dent s . To th y d i c is thi s : If yo u d o not gra sp th e full mean ing of a d eﬁnition in a ﬁrst rea n l eave it and a fter rea n ha ters II . an d di g , di g C p

m h n III . c o e ac to a ter I . and then o u w ll u n ersta b k C p , y i d d th e eﬁn t n r d i io b ette .

In wr t n th e resent volume I h ave er ve mu ch hel i i g p , d i d p from two oo i m nu l alc l s v z . th e excell ent l ttle a a o o b k , i , C

’ ’ fferen z ale of Professor Ern esto Pasc al and To h unter s Di i , d

’ Treat se on th e ffer n l u l F r th e h stor cal n te i Di e tial C a c u s . o i i o s in r m l n ha te VIII . a s ell as for most of th e exa es o tha C p , w p t

’ c a ter I am n e t to Pr fess r n L r z ll h p , i d b ed o o Gi o o i a s Sp e ie e

’ al e ra sc e u nd n f g b i h tra sscend ente ebene K u rven . O th e re ma n n ex am les in th s vol ume a l ar e n u m er are c omm on i i g p i , g b t o all En l sh text- oo s on th e su ect some a re or nal an d g i b k bj , igi th e oth ers are taken from th e Tripos Examinations of recent years in th e case of th e more importan t ex amples b elonging to th e last c e r h in at o t e sou rc es are c te th e tex t . g y , i d T m o fr en a nd former u l Mr . La shm Nara an y i d p pi , k i y ,

M. A. Pro fessor of Mathemat c s at th e C entral Hin u Colle e , i d g , P REFA CE vii

r I am mu ch in e te for some valu a le su estions Bena es, d b d b gg

ance in re sin h e ro f e m l and for assist vi g t p o Sh ets. I a a so

i . m fr n Dr. . Ba B A n o e . c . a ta . LL . indebted t y i d S C g h , (C b D .

u l n Pr nc al of th e Unvers t Law olle e of alcutta (D b i ) , i ip i i y C g C ,

d u n r l t n hose crit c sms an s est o s e a to ha ters V . w i i gg i , i g C p ,

r ll n n h u fuln III . and XL h ave mate a e h a ce t e se ess of th e V , , i y d

r sen l m p e t vo u e .

G. PRASAD .

B en a/res J u l 1909 . , y

C O N T E N T S .

— N. B. Th e r ns marke an a er sk ma be m e in a ﬁr rea n [ po tio d with st i y o itt d st di g . )

R A T. 1 r a e . Va i bl 2 nc n . Fu tio 3 L m . i it Continuity ff eren a c e ﬁ en 5 . *Di ti l o f ci t Ex amples on Chapter I .

In troductory Four important limits Differentia l coeffi cien t of at" Differen tial coefficient of a ” Differential coefficient of sin 9: Differential coefficien t of cos a: Differen tial co effici en t of t an 3: Diff erential co efficien t of cot a: Differential c oeffi cient of sec 3: Differential coefficien t of cosec x Diff erential coefficient of vers a: f r n fi n x Di fe e tial coef cie t of loga - 1 Diff erentia l coefficien t of sin 9:

" 1 Diff erenti al coefﬁcient of OOS cc “1 “ 1 Differential coe fﬁcient of t an x and cot 3:

“ " 1 Differen ti al coefﬁcien t of sec 1 m and cosec 9: “ Diff erenti al co efﬁcient of vers 1 a: T able of re sults to b e committed to m emory Ex amples on Ch apter II . CONTENT S

APTER CH III .

FUND MENTAL RULE FOR DIFFERE NTIATI N A S O .

Con stant Product of constant a nd functio n Sum of two function s Product of two function s P roduct of more than two function s Quotient of two functions Function of a function Inv ers e function s T h e form I E x a mples on Ch apter II .

APTER CH IV .

TANGENT AND N R A S S O M L .

T a ngent : Deﬁnition and Cartesian equ a tion (1 2 _y_ 2 N e on th e e a i n 1 ot qu t o d a:

Norm al : D eﬁnition a nd Cartesian equ ation Cartesian subtangent an d subn orm al An e be een r P ol ar coordin ates . gl tw tangen t and radius vec to

P ar b an en and bn r a 37 . ol su t g t su o m l P er end ar r e on an en 38 . p icul f om pol t g t P ed a e a n 39 . l qu tio

n P d a r e . P r i r Inversio . e l cu v s ola rec p oc a ls n r Ex ampl e s o Ch apte IV .

APTER CH V . A S YMP TOTES . Definition of asymptote G enera l rul e for finding a symptotes from Cartesian equa tion P ara llel a symptotes Asymptote s by in spection Note on curvilinear a symptot es Genera l rule for findin g asymptotes from pol ar equ atio n Note on circul ar a symptotes E x a e n r mpl s o Ch apte V .

AP ER CH T VI .

CURVATURE .

46 Cen re r a re Deﬁn n and Car e an c r na e . t of cu v tu itio t si oo di t s Rad c r a re : D eﬁn n and formulte 47 . ius of u v tu itio CONTENTS

R A T. r e and h r c rva ure 48 . Ci cl c o d of u t E l e 49 . vo ut

C n a and c n exi . P n inﬂex ion 50 . o c vity o v ty oi t of

Ex ampl es on Chapter VI .

CHAPTER VII .

ENVELOP ES . am c r e 5 1 . F ily of u v s Deﬁn n en e e 52 . itio of v lop R e for ﬁnd n th e en e e a a Of ra ne 53 . ul i g v lop of f mily st ight li s 4 Genera r e 5 . l ul n Ex amples o Chapter VII .

* R CHAPTE VIII .

RV E RA ING PR P ER IES OF P E CIA R E L . CU T C . O T S CU V S

In troductory Rules for C artesia n equation s Not e on multiple poin ts Rul es for polar equ ation s

E e . erb a P arabol a . llips Hyp ol - Semi cubical parabol a . Cissoid . Folium

C r d . C n Lem nisc ate . a dioi o choid

C en ar . r r x Cycloid . a t y T act i

L ar m c ira . Arch im edian ra ne og ith i Sp l spi l . Si spiral Ex amples on Chapter VIII

CHAPTER IX .

UCCES IV E DIFFERENTIATI N S S O . ﬁn 63 . D e ition s n 64 . Sta d ard results 5 L ’ 6 . eibnitz s theorem

66 . R n U se of p artial fraction s . ecurre ce formul ae Ex ampl es on Chapter IX .

APTER X CH .

’ ’ R LLE THE RE AND TAYL R S THE RE O S O M O O M .

’ Th r Rolle s theorem . e th eo em of th e mean value ’ T aylor s d evelopment in ﬁnite form ’ ’ r r M l r n *T aylo s theo em . ac au i s th eorem Note on contact of curves Ex n r X ampl es o Chapte . xii CONTENTS

CHAPTER XI .

MAX IMA AND MINIMA . R A T. ﬁn n 70 . De itio s Two t heorems 2 enera r e for ﬁnd n ax a nd m n 7 . G l ul i g m im a i ima Ex e n r XI ampl s o Chapte .

II CHAPTER X .

INDE ER I A E F R S T M N T O M . n r r 73 . I t oducto y

’ rm 4 . r Th e n a en a 7 Ca uchy s theo em . fu d m t l fo

0 0 m 2 x 0 0 0 0 - 0 0 0 0 75 . Th e r O 0 Fo s , , , ,

Note on compound ind etermina te forms Note on inﬁnitesim al s and inﬁnities Ex a e on C a XII mpl s h pter .

MISCELLANEOUS NOTES.

’ A e er r n n . W i st ass S fu ctio ’ ’ Ro e e re n d a r re B . ll s th o m a T ylo S th eo m r ff ren a n C . Pa tial di e ti tio D I FFE R E N T IAL C AL C U L U S .

CHAPTER I .

D EFINITIO NS .

m h h u c ss e er Variabl e . Let a: be a s bo w c t a es s c e v ev 1 . y l i k i ly y nu merical valu e fro m a given nu mber a t o anoth er given nu mber 6 . Th en a: is c alled a va ria ble an d th e totality of th e val u es of a: t f constitu es th e doma in o x .

We w ll re resent th e doma n of ac b th e s mbo a i p i y y l ( , mb k N TE . k be a n m er W be n en en u se th e O If u b , it ill co v i t to sy ol en h r e t e a bsolu te va lue k t . e. th e a e k re a to d ot of , , v lu of without g d to its n Sig . Thus

2 . Function . B a unction of a} d eﬁn ed for a ven d oma n . y f , gi i is understoo d a quan tity w h ich h as a single and d eﬁnite valu e for ever valu e f in i d n y o a: ts omai .

We enera d not u nc t n f b u ch s mbols as a7 g e e f o s o a: s f ) , . lly i y y ( F B O) ,

A EX M PLES .

2 1 . £8 sin a: are fu nc t ons of a: wh atever be th e d oma n , i i f B ‘ 1 o cc . ut Sin 51: c annot be a f unc ti on of a: for su ch a d omai n as 2 ( ,

2 . Th e e era re c r e a t a cer a n a e = 8 1 n h t mp tu u v t i pl c is y 0 + 0 Si x . If t e e era re rec r e are all d eren th e e a nd e e era re t mp tu s o d d iff t , high st low st t mp tu s ° ° be n 90 and 80 r find h e d i es ec e t a n as. i g p tiv ly , om of 3 . For th e ma n 0 a n n ma be efine b a n do i ( , fu ctio y d d y s yi g that it is a: 1 1 z er or a ccord m as or a: n n th e 2 a a e 1 3 etc . o g , h vi g v lu s , , n n n 1 hedbrh i a: ma be efine b sa n a 4 . Edrft g an f( ) y d d y yi g th t r f(cc) is 2 or 3 according as w is ra tional or ir ational .

h ﬁn t A i d t b th e Limit . A and a be n bot e s sa o e 3 . i g i , i limit of at for az= a f for an n u mber 8 h ow ever sm a bu t ﬂ ) i , y , ll r ater th an z ero th ere ex sts a c orres ond n nu mber c > 0 g e , i p i g such th at A 8 ; a: h aving every va lue su ch th at — 0 < az a < 6 .

m N TE d ﬁn n ma be ex re e in a f eren r viz . : A is 1 . e O This itio y p ss d dif t fo ,

be th e a: for cu r -a m ff er r A b e an an said to limit of f ( ) , if f ( ) di s f om y l ss th y n h fi n a ne an e er a e a: as an a e e near a . ssig d qu tity , how v sm ll , wh y v lu suf ci tly to I m w u e th n otat on x to d en t th m t of at We ill s e i T- fi ) o e e li i fl ) T a = for m a . Lim x a be 0 0 for an i e n er N e er ar e ere f( ) is s id to , if , y posit v umb how v l g , th w a ex a c rre n n n ber e > 0 s c a x > N as a in e er a e ists o spo di g um u h th t f( ) , h v g v y v lu such that - 0 < x a < e .

Lim w = A for an n m er 8 e er ma rea er an z er f( ) , if , y u b , how v s ll but g t th o , w 00 there exists a corresponding number N 0 such that

A — x 5 f( ) ,

r n as having every value g eater th a N.

Deﬁn n s m ar th e a e for th e ca e en —Oo itio s i il to bov hold s s wh ,

lim

a: = A 0 0 etc . f( , 9, _Oo

2 h n n eren a Ca i N TE . T e n o D s a e O otio of limit , which iff ti l lculus b s d , is

not so n am ar th e be nner as h e m at ﬁr a n e . For in h is u f ili to gi ight st im gi , algebraic studies h e must h ave becom e acquai nted with this notion in connec n th e nﬁn e er es For exam e a is mean b a n tio with sums of i it s i . pl , wh t t y s yi g th at 2 is th e sum of th e series

to inﬁnity ?

N in i : 2 th e c S th e sum n erm en a oth g but th s is limit to whi h to t s , t ds s

n is made greater and greater. DEFINITIONS 3

EX AM PLES .

1 1111 2 50 is For ta e an nu mber 8 h ow ever small 1 . , k y , bu t 53 : 61

r ater th an z ero . Now f g e , i

w h ere 0 < 1 ; h ence

an d conse u entl , q y ,

Th erefore nc e , Si

2 — 50 61? < 5 m+ a < 8 2 a + 5

Th erefore f e is su c h th at , i e (2 a + 8

2 2 x —a < 8 for every valu e of a: satisfying th e condition — x a < 8 .

And su ch a valu e of e is any positive number wh ich is equ al to or less th an th e po sitive root of

a + t

for as th e roduc t of th e roots of th s e u at on is -5 a ne at , p i q i , g ive u ant t o n of th r s is o t and th e oth er n t e e oot s ve e a . q i y, p i i g ive i 2 . In th e a ex a e r e a en 8 is aken be 6 ma l st mpl , p ov th t , wh t to , y be On 1 1 taken t0 be 2 l a 10” lim 3 Pr . ove that

h In 4 . cc ex s s and is ﬁn e s a for an n m er 8 If -- f( ) i t it , it follow th t , y u b , how oc a

B 2 4 DIFFERENTIAL CALC’ UL US

e er mal rea er an z er ere exi a rre n n n m r e v s l but g t th o , th sts co spo di g u be > O such th at 8 M .) <

for e er a r x a sa n th e n n v y p i , , , tisfyi g co ditio s

— 6 m — a < 6 O < x a < O < . I , i , z

1 11 - 5 . Pr e a non exi en . Let x and at be re e e ov th t cos is st t , , sp ctiv ly £10 i

en equal to n b eing a n i nteger. Th (211 1 _1 _ l x , a) , 1 lim - e er ar e n ma be . H en e a IS non ex i how v l g y c it follows th t en . m st t

For ex ed a c r n th e re ed n ex a e , if this limit ist , c o di g to p c i g mpl , it would be possible to ﬁnd a value of n so large th at _I _

931 6 2 even when .

- r non ex en . 6 . P ove that is ist t I 2 + ex

Continuit . A fu nct on x is sa d to be c i 4 . y i f( ) i on t nu ous for li m = t is ﬁn t and e u m a f x ex s s e a s a . , i f ( ) i , i q l f ( ) arr -a

E AM PLE X S .

? ’ ? z i 1 . ac is cont nu ou s for a a . For re ex sts an d e u a i , i q l s a .

unct on x is d eﬁned b sa n th at it e 2 . A f i f( ) y yi g qu als 1 o r 1“ n i z ero or d ff r n t r m z r h ac c ord as a: s e e f o e o . T s f n e x , i g i i u c tio n lim —1“ n u s for x = 0 For e is z ero nd i n nt u o . x a s is disc o i , o t eq ua l a: 0

to f(O) .

n n s for = 0 r 3 . x be x e a O m be z er If f( ) co ti uou , p ov th t f( ) ust o , when 1 - e a: d f er n r m z r f(x) = x Sin for v alu s of if e t f o e o . a:

4 . a are th e n c n n th e n n en in Ex Wh t poi ts of dis o ti uity of fu ctio giv . 3 of

Art . 2

6 DIFFERENTIAL CALC UL US

4 A n n x eﬁne b a n a e a 0 a: or 90 acc r n . fu ctio f( ) is d d y s yi g th t it qu ls , o di g

as a: is O 0 o r 0 . Pr e a e not ex . , ov th t f do s ist

1 3 5 . n r Find th e diff erential coefﬁcie t of Efo a: O.

6 . f cc h as th e same a u e w h atever a; m a be rove th at I f( ) v l y , p f

Here — lim f(re + h ) f(cc) lim 0 h =— O n= o h lim — 0 0 ' n= o * Ex m s n Ch er a ple o apt I .

” l . 1 1 . a an r e a x e a 0 1 or If f( ) t p ov th t f( ) is qu l to ,

acc r n as a: O 0 or O. o di g is ,

2 ra ce th e c r e . T u v

1 11 3 x r e a w is O or 1 acc r n as as is O or . If f( ) p ov th t f( ) o di g ” 20 0 different from 0 .

4 r . Give th e g aph of

° " n = c c (1 sinvrx) 1

2 h u 5 < w . , r e a th e i c sin n ! as end s If f 2 p ov th t limit to wh h p( ) t t g (c -it

en th e n e er n is ma e ar er and ar er is 0 or 1 acc r n as a: is wh i t g d l g l g , o di g r n r rr na atio al o i atio l . l 6 . A function f(x) is deﬁned by saying that it equal s 0 or sin (Sin it )

acc r n as a: is or is not a m e r . n th e n s c n n o di g , , ultipl of Fi d poi t of dis o ti uity

of f(x ) . 7 . Trace th e curve lim y - n z o o

8 . A function f(cr) is deﬁned by saying that it equal s 0 or Sin

1 acc r n as a: is e er z er or a s m e or ne er ese . Pr e o di g ith o ub ultipl of , ith of th ov “" ll that there are an inﬁnite numb er of points Of discontinuity of f(w) between

a and ere a < O < . B , wh B DEFINITIONS 7

9 . w = 0 or a rd n as a: is z er or ff eren r m z er race If f( ) cco i g o di t f o o , t th e curve y

10 . Trace th e curve

2 x " + 1

ﬁnd th e a e at th e n s a: 4 1 and sc s e er is n n v lu s of y poi t , di u s wh th y co ti uous t e a th se points . M r a . s [ th T ipo ,

1 S1n 90 11 . A function f(x ) is defined by saying that it equals 0 or according log as a: z er or f eren r m z er Has re a ff eren a c e fi en for a: O ? is o di f t f o o . fi ) di ti l o f ci t

12 n . I each of th e following cases disc uss th e question of th e ex istence of th e diff erential coefficient for (I) a z 1

cc = en ccz a f( ) e wh k ,

f (x ) 0 when a: a .

- cc a cos — L en 93 4 f( ) (a; ) wh 4 , rig a

f(x ) 0 when a: a .

x x a en m> a f( ) wh , = — f(re) a a: when at: < a . l x cc —a cos — en m a f( ) ( ) wh + , E l a

f (w) 0 when x a .

C c a Un . [ al utt iv , HAPTER C II .

STAN DAR D FOR M S .

Introductor . It is th e o b ec t o f th e resent c h a ter to 6 . y j p p i nvestigate and tabu late th e resu lts of d iﬁerentiating th e simple " x e e m n tar fu n ct on s viz . as a sin re co s as tan x c ot cv sec re l e y i , , , , , , , , , — “ — ” 1 l 1 1 ” 1 c osec as vers a l o cc sin a c o s re t an re co t a" sec ze , , g, , , , , , , “ 1 “ l rs re . c o sec as, ve

It w be seen ater on th at b m ean s o f c erta n rules to be ill l , y i r nd n d o f th n r gi ven in Ch apte III . a a k ow le ge e sta d a d forms o f th e resent ch a ter mo st o f th e o rd n ar func t on s c an be eas p p , i y i ily

differen tiated . Th rou gh o u t th e b ook w e Sh all alw ays c on sid er th e i nverse fu nc tions to be so d eﬁned th at

. 7—1 o 1 7r 1 < sm a: 0 < c os x < 7r 2 2 ,

7" l 7! < tan a: 2“ 2

l 7T 1 7T 0 < sec x < 7r w < , 2 ,

“ ‘ 1 0 vers a:

- r NO TE . Th oughout this book we sha ll take for g ranted th e truth of th e l n e re : I n enera l fol owi g th o m g ,

lim (e l m l m e, ) i x i lim w i r u

a re resp ectively equal to

lim

ze = a

‘l’AM {Jfa

( ac e n a ed be e in th e h e m M ) b i g ssum to positiv c ase of t last li it .

For ex e na a e C r I see a e X I. c ptio l c s s , h pt S TANDARD FORM S 9

limi s Four im ortant t . 7 . p

The follo wing limits are important a nd w ill be frequently u sed in this ch ap ter : H 1 (li l - n wh atever i t ma be . (I) , y iii-6 y

L1m (II) t = 0

‘ Lim a —1 _1o a (III) g,

Lim Sl n t (IV) t = 0 t

We proceed now to p rove th e four results given above

AS t near z er and c n e en e an 1 w e can ex an (I) I is ly o is o s qu tly l ss th , p d

“ . (1 t) by th e Binomial Theorem . Therefore 7

H ence " — - - (1 t) 1 a ha ﬂ n (in 1 ) (n 2) t {

But th e numeric al value of — — Mf - l un n n j ) { g{ gf H } re n ﬁn an r ere re mai s less than a ite qu tity as t tends to z e o . Th fo

0 ,

and c n e en , o s qu tly, lim (1 t) ” — 1 t = 0 1 t (II) We have to prove that (1 + t) tend s to e when t t ends to z ero by a n e a e as e as en t en z er b a n ne a e ssumi g positiv v lu s , w ll wh t ds to o y ssumi g g tiv a e v lu s .

Case 1 . re n t m ai s positive .

For eac a e t we can ﬁnd n e er n n 1 c a h v lu of , two i t g s , , su h th t 10 DIFFEREN TIAL CALCUL US

> 1 + t) 1 + u + 1

“E n l n ( n r n n

1 1 + n + 1

Therefore we h ave from (1 ) 1 > 1 m : n ) ( ) 1 n + 1 m Now it is a well-known algebraical result tha t 1 tends to e a s ( m

r th e integer in becomes greater and greate . Therefore

1 - 1 4 both tend to e as th e Integer n becomes greater and greater ; ( 17 5 1 - n 1 . H nce also it is obvious that ( I d i) both t e d to e it

follows from (2) that (1 t) tends to e as t ten ds to z ero .

Case 2 . t remains negative . — Put t v. Th en

wh ere

Now en t en s z er b a n ne a e a e 1) and c n e en wh t d to o y ssumi g g tiv v lu s , , o s qu tly,

w end z er b a n e a es. ere re r C a e 1 t to o y ssumi g positiv v lu Th fo it follows f om s , 1 1

a 1 en th e m H n 1 ‘ n h ce e e . th t ( t ds to li it e. e ( t) t ds to t limit e

III B th e Ex nen a e re ( ) y po ti l Th o m , S TANDARD FORM S 1 1

Therefore Q a t l _ a + t g — f + A— ﬁ + loge { ga %

Bu t th e num eric al value of

2 3 (lo e a t , a ) + log + gl 3 !

n remains less th an a ﬁnite quantity a s t te ds to z ero .

and c nse en , o qu tly , lim a t 1 loge a . i z o t

(IV) D escribe a circle of u nit radius and con struct a s in th e adjoined ﬁ r en gu e . Th

P M P A NA .

T ere re for 0 t we a e h fo , 5, h v

Sin t € t < t an t ,

O m s t 1 1 $ 0 0 1 0 6 , ( ) sm t c os t

— —7r O > t > . 2 lim 1 But it is obviou s that 1 Hence t = 0 cos t it follows from (1 ) that l im t

t = 0 t FIG . 1 .

Diff r i l n f x ” 8 . e ent a coefﬁcie t o .

If th en an d

h : 0

Two ca r ses a ise .

Ca I se . i . 12 DIFFERENTIAL CALC UL US

d Th re ore Now a s It tend s to z ero a so ten s to z ero . e f , 2 l

But b of Art . 7 y (I) ,

’ n ‘ l Th erefore x = nx ux f ( ) .

C a se II . x = 0 .

' - en n 0 0 is non ex en for O h as no mean n . Wh , f ( ) ist t , f( ) i g Wh en n > 0 lim lim — If . _ h :=0 h h = 0

" Hence n 1 is 0 0 or non-ex en a c r n as h c an e or e no t , if , f ist t co di g h g s , do s ’ e = c an e n k n 1 0 is 1 or 0 ac r n as n 1 or 1 . h g , its Sig with ; if t , f ( ) co di g

Diff r n i l fﬁci n of a” 9 . e e t a coe e t .

If th en f(ze + a nd

“h ’ lim a f (at) 0 h

l m "—1 i , a a h = 0 h

h lim a —l

. 0 k= 0 k

Bu t b f Ar o t . y (III) 7 , h 11m —l a _l “ o g‘ n= o h

’ Th r r M ‘” efo l l o a . e e o a i . e. a f g, , , g, g

14 DIFFERENTIAL CALC UL US

i n of cos x Different al coeffic e t . 1 1 . i

1 . n th e ff er n Fi d di e tial coefﬁcient of sin 2 x . 2 2 . w sin at If f( ) , 2 — 2 lim sin sin ce

lim Sin (293 + h ) Sin h h = 0 h

sin 2x .

Di f r n i i n f 1 2 e e t a coefﬁc e o an x . . f l t t

f ze = tan x th en and I f( ) , — lim tan (ce + h ) tan a: f (cc) n= 0 h — lim sin (:e + h ) c o s se Sin (1: c os nz 0 h c o s (93 + h) c os 3:

lim Sin h 1 ? 7 h = 0 h c os c os 51: c os x

1 3 . Differentia coefﬁcient of cot x l .

P roc eed n as in th e last art c e w e ﬁnd th t i g i l , a ( L 2 ” — cosec x . x S TANDARD FORMS

i l efﬁcient of sec x . 1 4 . Different a co

f m = sec at th en I f( ) ,

SOC (13 h = 0 — - lim co s cc c os (iv Fll ) V = 0 c os 33 + c os x

h cos (cc -Mi ) c o s ce

Sl n X c o s (a: h) c os x

1 5 . Differential coefficient of cosec x .

roc d n as in th e ast art c e w e ﬁnd th at P ee i g l i l ,

x cos x

' sin2 x

~ Different al coefﬁcient of vers x . 1 6 . i

f as th en x --h and I , ( i ) ,

lim vers —vers a:

h = 0 h. — — lim { l c os (x + h)} co s as} h = 0 h — lim c o s (x + h) c os a: h = 0 h

d (c os x ) d x d (vers x ) i . e dx 16 DIFFERENTIAL CALC UL US

EX AM PLES .

h = 0 h

lim M ILL h = 0 l - (se f h) + a/tan r }

1 d (tuni c)

o d / d z: 2 e tan a:

ﬁnd 2 . If lim h

— lim ese c (x +h ) sec x h = e {

3 8 6 0 x

6 8 9 0 x x

— lim esec (x +h ) sec x _1 lim et 1 — sec ce t = 0 t

h — lim sec (513 + ) Sec x cl _(sec cc ) = h 0 h d ie

i l ffi n l D f erent a coe cie t of o x . 1 7 . i f g ,

f ac = l o 617 th en cc - i and I f( ) ga , ( H ) , — li m l o e + h o a: ’ g, ( ) l ga f (ae) h = 0 h S TANDARD FORM S 1 7

a n r Z t nd t z r Th r r te d s to z e o a so e s o e o . e efo e Nowas it , l

lim h l im 1 lo 1 i 10 1 g, g“( W h == 0 t = o l

_ o e b of Art . 7 . l ga y (II . ) l — l i . Th erefore o e e . f g, , , j

fﬁci n of M 1 1 8 . Differential coe e t r X .

_1 f x th en cc h = s n cc h an d I , f( + ) i ( ) ,

h = 0 h

Now take a c ircle of u nit ra diu s an d c o n stru c t as in th e adj oine d r Let P M an d N re resen t a: ﬁgu e . O p —1 in and m h re s ec t ve . Th en h s a; ( + ) p i ly , “ 1 and sin 93 h are re resented b R ( + ) p y Q, l r arc AP and arc AO resp ec tive y . Th e e fore

h R Q

arc P Q_P Q arc P O X 3 9 R Q P Q

Hence

2 lim P Q are P Q FIG . . X n= o R Q P Q

sec 4 F OR an d

P r: P M an d 4 QR A O ,

Th ere ore by (IV) o f Art . 7 . f 18 DIFFERENTIAL CALC UL US

L P OQ lim 2 sec L P QR x — h _0 Am sin 2 L P OQ 9 sec L P QR

1 __ FOM sec A ’ ﬂ OM x/l —a 1

E 1 In r r ee a P R tend L P M th N T . e s L O e en h as n O o d to th t Q s to , stud t o ly 7r n e a in th e OP = and a n e n P R en 1. Q , e , ( to ot th t limit § th t co s qu tly Q t ds to M L OP R which is equ al to A P O .

N TE For an er me d ﬁ nd n th e feren a c e ﬁ en O 2 . oth tho of i g dif ti l o f ci t of “ 1 r n me r n d h e er n er e ca c n see Ex . 1 Art 22 in a: an t . . s oth i v s t igo o t i l fu tio s , ,

- 1 1 9 Differen ia coe fic en of cos x . . t l f i t

* 1 —1 cc cos re en a: h on h a nd If f( ) , th f( ) cos ( ) ,

h = 0 h

nd M re re en a: a nd re e e en 2 l N a O . In . et O h Fig , p s t sp ctiv ly Th

* 1 ‘ and C I 90 h are re resen e b NM a rc A and arc AP res e c os a: OS ( + ) p t d y , O p c l ere re tive y . Th fo

-1 — " 1 COS (a: h) C OS a: are AP arc AO NM

arc P Q R P

g are P Q P x RP P Q

— x cosec 1. P QR

ere re as in th e rece n ar e Th fo , p di g ticl , lim f cosec L P QR x h O

cosec I. P OM 1 1

/ z P M A l - x S TANDARD FORMS 19

“ 1 “ 1 ient of tan n x Dif erential coeffic x a d cot . 2 0 . f n If re, th e and

’ f (xl h = 0 h

Now take a circ le of unit radiu s an d construct as in adj oin ed ur Let AM and N re resent a: ﬁg e . A p

‘ " 1 d r t e Th n h t n a: an £13 h es ec v . e a + p i ly , “ ‘ 1 and t an (a + h) are represented by r AP and a r A r s t MN, a c c O e pec ively . Th erefore

h MN 9 P e £ 1 £ x MN P g P Q

F G I . 3 .

1 M nd A fro m th e similar triangles RN a ON. M N 0 . O

Th erefore lim 1 I Q arc P O x _i x

1 1 2 ? OM 1 as

S m ar i il ly 2 dx 1 x

- 1 - 1 2 1 Differen ia oe ﬁ f sec x and cosec x . . t l c f cient o

" l v 1 cc e as en a: h sce a: h and If f( ) s c , th f( ) ( ) , —1 11m sec —sec :e f (a?) h = 0 h 1 I — 4 n n . 2 l et nd r nd h re ec e . e h Fig , a rep e sent £1: a x + sp tiv ly Th , OM ON

“ se 1 “ 1 A c 33 and see (a h) are represented by a rc AP and arc O res ec e r p tiv ly . T h e efore 20 DIFFE REN TIAL CALC UL US

l i m a rc P Q h = 0 NM ON OM

arc P Q P Q ON OM P Q NM

/ ‘z 93 e w 1

* ‘ l l oc = verS ac en cc h ==vers x h and If f( ) , th f( ) ( ) ,

' l — ‘ l lim verS (cc -r h ) vers a:

In . 2 l et 1 OM and 1 ON re re en a: and a: h re e e . T en h Fig , p s t sp ctiv ly h , " " er cv a nd er ﬁe h are re re en e b NM arc AP and are A re e ve . v s v s ( ) p s t d y , O sp cti ly Therefore lim arc P Q h = 0 NM

l im P at e P —Q Q ' nz o { IVM P Q

l Th r n a fﬁc nt of Sin a: c an a o b ou nd as 1 . e diffe e ti l coe ie ls e f follow s

- I nd r Let u and U represen t sin re a espec tively .

Th en

ce = in u ce h = sin U s , + ; and h enc e — h = si n U sin u

U—u 8 111 C OS 2

UL US 22 DIFFERENTIAL CALC

“ 1 ’ cot w x f(x ) , f ( l

' = sec x iv f(ce) , f ( )

1 " 1 ’ = x w a: cosec , f ( ) f( 2 tux/x 4

“ ’ 1 = 1 w ac vers x , ( ) f( f ? V ar —r

h r Exampl es on C ap te II .

ﬁc en eac th e in Find from ﬁrst p rinciples th e differential coef i t of h of follow g func tions

° V 2 . 1 e rees . . ce (d g )

z z i - 5 . n re c s 1 4 A/x r a . si h whi h . [

‘ i 6. a: c s cosh [whi h

V sin a 9 . e . 8 . a: sin 90 .

l i —x — 12 . sin x . 1 1 . log . log 1

2 —1 \ l ac 16 . tan 1 5 . i o ) in l o ze. s n 14 . s ( g g “ 1 t an a . 17 . log S TANDARD FORMS 23

2 ze 2a t ere ce a t ﬁnd 18 . If f( ) wh , f

x = a sin t ere ce = a cos t If f( ) wh ,

— ’ If a z a l cos t ere x = a t sin t ﬁnd ce . f( ) ( ) wh ( + ) , f ( ) HAPT R C E III .

F N NTA U L E S F R IFFE E NT IAT I N U DAME L R O D R O .

2 4 . Constant .

I = nst n t Ru e . w a co a l I f f( ) , f u n t t h h d Fo r a c on stant is a q a i y w ic o es n ot vary . r r h as th e sa me va u e h r a b Th e efo e ﬂ at) l w ateve a: m y e . Ar Th erefore b Ex . 6 t . 5 , y , ,

f nd 2 5 . Product o Constant a Function.

Ru I wher is a c n a nt then e . e a o st l II f , M t? )

= < Th r r For f(a: h) a p(a3 h) . e efo e

Sum of Two nu 2 6 . 5 a ;

I then Ru le III . ff(cc) (x ) f

Th r r For f(cc h) a(ce h) Mae h) . e efo e m ’ — li o f (50 11 :

h = 0 h

73 SW?) W0 ) .

m x n It is clear th at Ru le III . ay be e te d ed to th e c a se of any n ﬁnite nu mber of fu nc tio s . FUNDAMENTAL R ULES F OR DIFFERENTIA TION 25

Th u f s, i Na?)

f N ib )

A EX M PLES .

1 = 1 n h a b l f i ze w e ve R . . x 0 s u e I f( ) ( ), y II

d sin

x c os e . d a" - h b a Ru . 2 . f e w e ve e I + , y l III f

3 . r e n th e fferen a efﬁc en a: sin a: 10 cos x W it dow di ti l co i ts of , 1 -l “ ‘ tan w+ Sec at sm x + 2 COS as. log § , _$ 932 5113 4 . n th e f eren a e ﬁc en 1 + Fi d di f ti l co f i t of I 2 3

f i 2 7 . Product o Two Funct ons.

u I The di erentia l coe icient o th e rod uc o o R le V . fi fi f p t f tw functions is i ” . o (First Func tion) (D fi Coefi. f Second)

”

i . o F r Second Function x D . Coe i st ( ) ( ﬁ ﬁ f ) ,

or tated in s mbols , s y , f (x) Wh ere f(x )

F Th r r or f(se h) h) . e efo e lim gt (x + h) H(m+ h) lim { ¢ h = 0

2 8 n e th e e a n . Dividi g both Sid s of qu tio

’ f (fv) were) b x we a e y f( ) , h v f it?) 26 DIFFERENTIAL CALC UL US

Hence it is clear that an equation similar to th e above holds for th e product of any ﬁnite numb er of func tions. F x a e l et en en n b x or e mpl , Th . d oti g y f ( ) . w e have f(x) Therefore

I 1 / 63 i T1 ( ) l (w) f(x) m

in e But Mr ) be g s ens ) . I W“) ad) Therefore I 1 1 r “r + y (2 (” f(x) rev) We

nd nse en a , co qu tly ,

' f (ao)

Proceeding in this manner we have th e following rul e : The difieren tia l coefiicient of the pr od uc t of a finite nu mber of func tions is found by mu lti p lying the dijferen tia l coefiicien t of each fu nc tion by a ll the other fu nc tion s and adding the p rod ucts thus formed .

A EX M PLES .

f w e h ave b Ru le IV . 1 . I y

’ M f — = x ” f (x ) = m e gi x e e : (cc

= l h 2 . f x cc tan at o cc w e ave I f( ) g , d l — lo w ) t n a; og re g5 + x g x % + ce a cc 2 l n tan cc l og 5e + ce og a: sec ze + ta re .

n ” 2 3 . r e n th e f eren a efﬁ en re e x sin a a; x W it dow di f ti l co ci ts of , , log , “ 5 l ? ” t ac tan re 2 cc a: sin a: tan cc e m :e a a: sec 93 . , log , , log , log

4 nd h e efﬁ en 1 . Fi t diff erential co ci t of :e( f ” 2 F nd h d f r nt c fﬁc ent o e c os re. 5 . i t e if e e i al oe i d 21: 2 — 2x (e cos x ) e c os 51: c os as) dce dac

1 0COS as 2 2" 9— ) cos aa3 + 2e c os ce ‘ J dre dce

— 2 2 x ” 2 c os cc e —2 Sin re c os :e e

” _2 — in c os cc (c os x s as) e .

1 3 ' 4 6 x tan re + tan x r e a cc sec re. If f( ) 3, , p ov th t f ( ) FUNDAMENTAL RULES F OR DIFFERENTIA TION 27

2 9 . Quotient of two functions.

’ The di erentia l coe cient o a uotien t o two unc tions Rule V . ﬁ fﬁ f q f f

mr . o Di D r . Nu u r . o ) N m ) ( Coeff . f eu Squa re of Denomina tor or stated in s mbo s , y l ,

{new where

For Th erefore 4} (2153

lim h = 0

lim

h = 0 tall/ (513 + h) 4x(ze)

A EX M PLES . 1 h w e a e b Ru e . . If f(cc) v y l V

“ sin as—cc c os a: 2 sin as

2 sin a:

2 . Write down th e diff erential coefﬁcient s of

— ‘ x 1 ce a a zze sinh a: as 1 + 2 e tan a: , I 2 2 1: — re + a ce + a a + cc cosh ce e 1 J re a: 28 DIFFERENTIAL CALC UL US

3 n th e a e for h n a e fi n . Fi d v lu s of x which t e diff ere ti l co f cie t of

a: 10 is z ero .

4 n ﬁ r i . Fi d th e di e ent al co efﬁcient of

0 Funct n of f c i 3 . io a un t on.

u I I R le V . f where then f

For let ae -h = t r then cc r , M l ) + , f( ) .

Th erefore

= 0 7' h — lim p(t + r ) lim il (a3 + h ) lt(ze ) h = r h = 0 h

h= 0

nd s z ero 1' al o tend t z r r But as h te to , s s o e o . Th erefo e

h r s 0 r

’ ’ Th erefore f (x ) =

3 1 . Inverse functions.

h n f in Ru e VI . w e ut t e cc be n 1 w e obta n th I l p , f ( ) i g , i e resu lt o r as it ma be wr tten , y i , dx d—t x 1 . dt iix

Th is result gives a simple meth od o f d ifferen tiating inverse F r l t be th n r fun c t ons . o e e ve se func t on f h n i , i i o f (t) t e Th erefore d t _ 1 _ 1 _ “ ' _ ’ ’ dx t dab f t) f

30 DIFFERENTIAL CAL O UL US

d l og re + l o g re d re

’ z -x 5 . r e n th e d f eren a c e ﬁ en a e W it dow i f ti l o f ci ts of ( ,

a be “ 1 2 / in , tan x . A an re s We, cos n, c re, ( ) t , log cos log osh log a be

n ate h n n 6 . Differe ti t e followi g functio s

3 re re 93 " / N tan J re x — 1 , log , r Z — va -T e? ia r s (a w n

e ar ‘ “ 1 t an Sin e 93 t an re sin a sin log log , log

“ 1 -1 7 . A Art . 31 ﬁnd th e d f eren a e ﬁ en sec and er pply to i f ti l co f ci ts of re v s re. 8 . Differentiate th e following function s

n ” x 111 113 sin (U —1 sin m x t n “8 in m in ); a s s m ( y, , ( 0 ( ) , ( y 7b

o 5 9 t I Ex am h a er II I W ples on C pt .

Find f (re) i n th e following c ases

1 . 2 ' f(x ) (a: b) "

l V 2 2 “z — z a + re + a re ? ? f(cc) f (x ) (a cur 2 2 V a + re2 r/a — rez

sin e 1734 1 re re 6 . fr tan f( ) f( ) re re. 1 —Sin re 2:

8 . re Sin re Sin 2x sin f ( ) 3cc .

a + b tan re f(x ) = l 0 g a — b ta n x

- — it;232: log (a + 6 t an re) log (a b tan re)

2 2 2 “ b sec re b sec at 2a b sec re re 1 ( ) + 2 2 2 a + b tan re a -b tan re a -b tan re

2a b 2 ? — ‘2 2 a cos re b sin e

“ x 1 f( ) log tan w.

1 1 f (cc) log FUNDAMENTAL R ULES F OR DIFFEREN TIA TION 31

a + b cos re - l r _1 re = an 10 c0 t re . in f( ) t e . g S b + a c os re — “ a b = tan 1 tan E (Va b g

? “ 1 i n . 1 = re . f(re) (1 re ) S n (m ta re) 7 . f(x ) log (log )

” " re = re ere mean l o re ea e n me . f( ) log , wh log s log log g ( p t d ti s) J cc 20 f( ) . 1

= ” 22 " 1 / re . in m re. f( ) tan a . f(re) s A s

1 1

f(x ) 10 s

/ 2 - rc _10 50 4 w — 1 1 f( ) 8 ( + ) + sec w.

f(re) = (COS 3’ (c osh ce) c os x

x re i) + x .

‘l i l + x c os e

2 V 1 + re + re f(it ) 10 g / 2 4 1 + re — x

—1 mt 5” 2 3 rt a f( ) log (cosec re ) .

1 - M x = 10 f( ) 3 3 4 24 { x + a3 + 1 }

V 3 4 2 1 _1 1 e + re + x f( ) 2 e

— 2

37 f(x )

38 ° f(x ) (4r 32 DIFFEREN TIAL CALC UL US

ak n ar m T i g log ith s , _ 1 — . log f(re) z log (or 2) log (4a: 1) log (33: g 3 7 5 Therefore 3 1 6 3 2 (a: 2) 3 (4re 1) 5 (3as 5)

2 1 2re + 11 21re 1 789 30(re 2) (4a: 1 )(3re Therefore

30(4oe

m h th Th e n r e ed ab e viz . ak n t e l ar e co pou d p oc ss us ov , , of t i g og ithm of ’ n n and en d ff eren a n a led lo a rithmic d i eren tia tio n. fu ctio th i ti ti g it , is c l g ﬁ It is useful wh en th e function cont ain s variables as index or several involved r facto s.

’ a + 2ﬁnr - a 7 40 in re 1°€ x c ec 9 bre . f(re) (s ) os { m ))

2x " 1 4 1 . cc tan f ( ) f l — x

- Here f(re) = 2 tan 1 re ;

42 —1 . f(re) t an

43 fimi

4 7 . Differentiate sin re with regard to

We a e b R e h v y ul VI .

(30 8 CI}

d a:

" “ " 4 8 ff n — . Di erentiate Si (3re 4re ) with regard to see FUNDAMENTAL R ULES F OR DIFFERENTIA TION 33

49 . D f eren a e re ar an or i f ti t log with g d to t .

50 f in Sin a r e t a I S y re ( y) , p ov h t 2 8 111 (E t a) d oe sin a

i . T r re e e b Here y s a function of re he efo w h av y Rule VI . d dy i (sm y) _cos y and { re d; d x d re

dz? Therefore cos 3] Sin ( a y) 93 cos (a y) . dc; 35 Therefore

— d oe cos y - re cos (a + y) cos y sin (a + y) sin y cos (a + y)

dy 1 1 1 re t 0 r e a — If re( yﬁ y( ) , p ov th t ” 615 (1 re)

i— — 2 Fl u d If y ) ftg

t t1 t2 , ’ ’ ’ w W w ere th e t e w s and u s are nc n re i 2 s wh , , fu tio s of , u u u , 2 3

d d t t t t 5 dt, t2 3 t , 2 3 ] 12 ts d d iv Z dw w 70 w 85 ? { ﬂ s ] 2 s w w dre d rc d re du d u d a w1 2 3 , z , u a u u a d o: dx d u , 2 s , 2 s ie

k k k t t t t l s s 1 2 s l $2 ts

h cc w w w 1 l2 l, w w w erms f(re + ) f( ) , 2 3 , ““ l , s t “““ u m m I 2 s ] 2 3 1 2 us,

ns or ree th e n ne an es k l m c n a t . each of which o t i two th of i qu titi , l l e c , ere re as fac tors. Th fo w h “ x r 1 m f( ) f( ) f e”) 71: 0 h

k k k t t t t t t l s s 1 2 s ] s s lim h lim Z Z lim w w w h h 1 2 l, 1 2 3 = w h = 0 h h h h = 0 m m m h o wl w2 3 r z s u u u h h h u l u 2 u 3 l ,7 3 D 34 DIFFERENTIAL CALC UL US

etc .

’ p rove that f ( cc) 3 (a3

z - be en e b A n r e t a ( A d ot d y , p ov h t dT

’ n re nd e erm n n In th e following cases ﬁ d f ( ) a d t i e if it is conti uous for or 0 .

2 56 . re = 0 or re cos acc r n as re is z er or f eren r m z er . f( ) , 5 o di g o di f t f o o 1 ' Here re is 0 or sin + 2x cos acc r n as re is 0 or ff eren r m O. f ( ) , , o di g di t f o i 515

' non-ex s en and cc 1 S c n e ent n n f (cc) is i t t , f ( ) o s qu ly disco ti uous

1 r z r r r m r = acc n as x e o ff e en r z e . 57 . f(x ) 0 or re o di g is o di t f o o

1 “2 1 58 . rt O or e c r n as z er or f eren r m r f( ) os acco di g or is o dif t f o z e o . re

3 59 . cc = O or x c s cc r n s re is zer or f eren r m f( ) o ia o di g a o di f t f o z ero.

r= 10 — l 2 60 re : 2 — . If f( ) 0 or _ re ) according as re is one of th e num

’ or ot ﬁnd th e a es re for c re is dis i s n , v lu of whi h f ( ) H C APTER IV .

TA GENTS AND OR MAL N N S .

Tan ent : Deﬁnition and Cart esian E a i n 3 3 . g qu t o .

Let P be a ven o nt on a curve and an oth er o nt on it gi p i , Q y p i th en by th e tangent to th e curve at P is u n d erstood th e straigh t ne wh ch is th e l m t n os t on to wh c h secant P tend s as li i i i i g p i i i Q Q , n h r nd t P r n a o t e cu ve te s o . t avelli g l g , L t be th e coord n ates of P e cc, y i , re + h + h th e coordinates of . Th en , y Q X Y be n cu rrent coord nates w e h ave , i g i , for th e u at on to th e stra h t l n e P e q i ig i Q, — £ 9 (ri m y 1 . — — ’ X re (re + h) re th at is , — Y h FI G 4 . y . X —re h

nd t P Th n h nd z r d Now let Q te o . e te s to e o an th e straigh t line P Qtend s to th e straigh t line — Y y lim h ' X —cc h = 0 h

But i a fu nct on of 2 sa x . Th erefore y s i , y f( )

m k lim re h — re d li f( + ) f( ) ’ _ = f (re) = h = 0 h h = 0 dx

‘ h n ent h Th erefore th e e uation to t e t a to t e curve at P i . e. at q g , , th e oint x is p ( , y) , 36 DIFFERENTIAL CALC UL US

N TE 1 r m th e a e re a e me r ca n er re e O . It follows f o bov sult th t , g o t i lly i t p t d ,

n h n h h t . e. is th e an e t e a e e een t e e rec n t e 3g, , f t g t of gl b tw positiv di tio of m ax s and th e e rec n th e an en at th e n x on th e c r e i positiv di tio of t g t poi t ( , g) u v

th e an en e n c n ere e n c Y in r that direction of t g t b i g o sid d positiv alo g whi h c eases .

E 2 Let th e en th e arc AP mea re r me ﬁx e n NOT . l gth of su d f om so d poi t A

n th e c r e be en e b 3 . en 3 e en a nc n re th e ah o u v d ot d y Th is vid tly fu tio of , a P As n n c n erab e sciss of . this fu ctio is of o sid l importance in th e geometric al applic ations of D f eren a Ca c we r cee r e it i f ti l l ulus , p o d to p ov s

nd a en a r er viz . fu m t l p op ty,

I 2 ( S dy d x (d;

C n r c as in th e ad ne ﬁ re and en te x o st u t joi d gu , d o th e l ength of th e arc AQby s on Then we may n m FI G . 5 . a e as a a x a en is fﬁ en ssum io th t , wh Q su ci tly near P to ,

or P chord P Q R R Q.

V 2 2 chord P Q h + h ;

P R z P S sec [ RP S — P S sec L P TX

R Q==S Q-SR — SQ h tan L RP S

Therefore

But s h en s z er , a t d to o ,

Cl 2 k d H - — A/ r z et

respectively tend to

ar/ 2 618 r z r 3 ” m _r é drr A1 and + — “ ’ 5 daz dre d re d rc V ( dre

38 DIFFERENTIAL CALC UL US

Th erefore th e required equation 1s — — — Y a (1 c os t) = {X a (t + sin t)} tan

= £ — — i . e. Y X t an a 1 cos n t t an , + ( t) (t + si ) 2

= — i . Y X t n tan . e. a a t , é 5

4 . n th e e a n th e an en at th e n re on eac Fi d qu tio to t g t poi t ( , y) h following curves

’ 2 ‘ a re a . 11 e a (i) y ( ) y . (iii) y cosh

3 3 3 (iv) y re . (v) cc y 3ae g . g 2 2 2 2 2 ? 2 a re2 (vi) re rey ay . (vii) (cc y ) (

r V111 ae ( ) m a b

5 . Prove that z+ 1 touches th e curve

3. y be‘ a t h a t e point wh ere th e c urve crosses th e ax is of y .

6 n n rm n on . Fi d th e equation to th e tangent at th e poi t dete i ed by t (i) Th e ellipse re = a cos t l . y z b sin b I - s (11) Th e semi cubical p arabola re a (2 3 t ) 3 y 2a t

In th r r h h rt n of th tan 7 . e c u ve p ove t at t e p o io e n t nt r n h x f n h ge i e cepted betw ee t e a es is o c on stan t le gt .

1. 2 - 2 - dy 7 r A s _3 = h f _4 h r r th H e e T re r . T e efo e e rm + y 0 . e o e f g 3 dre dre re e uat on to th e tan ent at th e o nt ac i s q i g p i ( , y) — — g Y (X m) e

)

: % 3 i i . e. Y X x + , g; y ( y ) ’

i Y: g t . e. , a y .

Th erefore th e interc epts on th e ax es o f a: and y are respec tively ﬁ f i i n d Th r r th h t n n t a e a a y . e efo e e l ength o f th e portion of t e a ge interc epted betw een th e ax es TANGE N T S A ND NOR MAL S 39

8 r e th e r n . In th e curv e p ov that po tio of th e tangent inter c epted between th e ax es is divided at its point of contac t in to segmen ts which are in a c onstant ratio .

rmal : Deﬁnition and Cartesian E uation. 3 4 . No q

By th e norma l to a c u rve at a po int is u nderstood th e straight lin e wh ich passes th rou gh th at p oin t an d is at righ t an gles to th e h tangent at t at p oint . h n h n i S nce b Art . 33 t e e u at o to t e ta ent at a o nt 517 s i , y , q i g p i ( , y) d Y— = x —w — e ( ) . 515

u ion t o th e normal at th e oint x is th e eq at p ( , y) 1

— X —x Z y( ) , dx

X —y = ( y)g 0 .

E AM L X P ES .

n to h n rm l h 1 . F nd th e e u at o t e o a at t e o nt as on h i q i p i ( , y) t e — y ellipse - + 22

2 ‘ — z Th r r Th Here f+ § ég o. e efo e erefore th e

requ ired equ ation is

n h e e a n th e n rma at th e n x on eac th e 2 . Fi d t qu tio to o l poi t ( , y) h of

Ex . 4 Art . 33 . curves of ,

3 n th e e a n th e n rma at th e n e erm ne b t on th . Fi d qu tio to o l poi t d t i d y e

4 n rma th e c r e x 3 3 makes an an e 5 th e . If th e o l to u v y gl 4 with axis ‘ — = at a its e a n is Y cos b X sin t a. cos of , show th t qu tio q g

5 . In th ca enar a c s r e a th e en th e r n e t y y o h 2, p ov th t l gth of po tio of

e e een th e c r e and th e ax s a; ar es a ” th e norm al intercept d b tw u v i of v i s y . DIFFE RENTIAL CAL C UL US

3 3- 3- h r n th e 6 . In th e e li se 1 r e a th e en t t e l p 2 2 , p ov th t l g h of po tio of a 6 th e normal intercepted between th e curve and th e ax is of it: varies inversely as r p e pendicular from th e origin on th e tangent .

3 5 Cartes an n nd Subnorma . . i Subt ange t a l

Let P b a r Let P M b e th e ord n ate of e given p oint on a c u ve . i P and let th e tan nt nd n orm a at P meet th e ax s of ac at T , ge a l i Th n th e en th an d N resp ectively . e l g h en at P th e TM is c alled t e subtang t , length MN is c alled th e su bnorma l at P Now c onstru ct as in th e adj oining u r ﬁg e . Th en

FIG. 6.

b n — t an e . z. . s t e , u g 92 dx

Al so 4 MPN= L P TM th erefore

MN= PM tan P TX

y a subnormal = g~ Y , y . dx

N E . Th e s an en is mea re r m ar th e ri and th e OT ubt g t su d f o T tow ds ght , y - s n rma is mea re r m M th i in n r e a ub o l su d f o towards e r ght . If a y cu v tag is da: ne a e an n ca e a M es th e e and as in a g tiv qu tity, it i di t s th t li to l ft of T , th t

case . i s a s ne a e N es th e e M . ygg l o g tiv , li to l ft of

EX AM P LES .

1 . F nd th e subtan ent and subn rmal at th e o nt x on i g o p i ( , y) th e curve

n- 1 Here s y Th erefo re TANGEN T S AND NORMAL S 41

Th er fore th e subtan en t z l naz and th e su bnorma is e g , l

da: ? 6171 a y da: na

2 n rm in th e a ra 4 mi e 2 h e a a a s a 2 . . Prove that t sub o l p bol y qu l to a a: g 3 . In th e c r e : be r e a th e s an en is c n an . u v y , p ov th t ubt g t o st t m mm 4 n z r e th e n n r e as h . In th e c r e a t a a e a s t e u v a y , p ov h t subt g t v i absciss a .

m” m- n 2 n 5 . In th e c r e m u r e a th e mth er th e s an en u v y , p ov th t pow of ubt g t

ar e as th e nth n r v i s power of th e sub o m al .

6 . Fin d th e subtangent and th e subnormal at th e point determined by t on th e cycloid w = a ( t + sin t) y a (l cos t)

Pol ar r n s n nt nd Radius ector. Coo di . An e n e a V 3 6 . ate gl betwee Ta g Let 0 be a given p oint and C A a given straigh t line ; th en it is evid ent th at th e position of any point P is u niqu ely d etermined if th e distanc e OP and th e angle P OA are know n : th ese are c alled th e p olar coordina tes of P and are enera d enoted b r and 6 O is g lly y . c a led th e o e 0 A th e n t al ne l p l , i i i li , “ 7 th e mdlus v r f ecto o P , and 6 th e

’ oectoma l a n l f g e o P . Let P be a ven o nt on a c ur e gi p i v , and an oth er o nt on i L t Q y p i t . e r 6 be th e coord nates of P , i ,

6 a th e c o rd n t f + o a es o . Th en i Q , d rawi ng QM perpendicu lar to OP FIG. 7 . rodu ced w e h e p , av QM tan A QPM = MP QM 0 M OP 42 DIFFERE NTIA L OAL O UL US

l n h d n d P Th n Now et Q move alo g t e cu rve a n te to . e by d n t n n h n n L Bu t s t n eﬁ i io seca t P Qtend s to t e ta ge t P . a Q e d s to P (1 t nd z r Th r r n i LP M b ( e s to e o . e efo e d e ot n , , g [_ y p, lim tan z QP M

lim 2 = 0 [3 c os a sin a

a a

m 1 1 T li . a = 0 B d ’i‘

a d bl

_rd e That is t an ¢ , . dr

1 Th e e re r r i n t NOTE . abov sult holds t ue wh eth e 3; s positive or egative a th e point P ; it being understood that O is th e angle between OP produced and th at direction of th e t angent at P in which those points on th e cu rve l ie n ar r a er n r n wh ose vectorial a gl es e g e t th a th e vecto ial a gl e of P . ' h en th e rc A P m r d r me ﬁx e n N 2 . Let t e a ea e s OTE . l gth of su f om o d poi t ' n n A on th e curve be de oted by 3 . The

d s r 2 2 d as J

m a be ea e ce r m th e re r e in Art . 3 3 N e 2 . This result y sily d du d f o sult p ov d , ot For

2 d y d a dy /1 L — N (d o g s)

c 0 z in 0. os , y r s Th erefore

O— 9 cos r sin ,

sm 0 + r cos 0.

Hence ere a t e. d o) (d o d o) and n e en , co s qu tly , TANGE NT S AND NORMALS 43

B a r ce re i m ar a Art . 33 N te 2 th e ab e a e Of [ y p o du S il to th t of , o , ov v lu can be obtained without using th e result

d s d 2 1 —y d a: / oCla?)

n Su rm 3 7 Po ar Subta ent and bno al . . l g

Let P be a ven o n on a cu rve . Let th e stra h t ine d r n gi p i t ig l , aw th rou gh th e pole O at righ t angles to OP m eet th e t an ent and n orm a at , g l

P at T and N res ect ve . Th en th e , p i ly l ength OT is c alled th e p olar su b tan ent th e en th NO is c a ed th e g , l g ll ola r su b r l p no ma . Now constru ct as in th e adj oined ﬁ ur h n P Th r r e . T e fo e g L O s . e e

OT : P tan ( O p, d 9 = = 2 i . e. olar subt an ent r tan > r , p g q ‘ d r FI 8 G. . Also 4 ONP = A OP T th erefore

= OP c ot NO e,

- l subnormal y r c ot 5 t . e. o ar , p 9 32

h r n i r th e r r m an er er at 0 NOTE . T e pol a subta gent s measu ed to igh f o Obs v

k n r 3 — e it n a e a is a P . in an c r e 1 is ne a loo i g tow ds If y u v 33 g tiv , i dic t s th t T to th e e th e r r l ft of obse ve . m r Si ila statements hold for th e polar subnormal .

EX AM PLES .

1 . In th e logarith mic spiral prove th at th e angle

r a a between th e tangent and th e radi u s vecto is equ l to . Here w e h ave

9 °° c ot a x ae

° 7 cot a ~

Th erefore

d B 7 n ( — — — n 0 ta p r tal . d r mf d B 44 DIFFE RENTIAL GALOUL US

Hence th e angle between th e tan gent an d th e radiu s vector 1s

t a equ al o .

2 h o th at in th e cur 7 8= th o ar subtan nt i . S w ve a e p l ge s

con stant .

Here d r d d o 2d Th r n = — e efore th e p olar subta gent r a . a7

n 3 . n ) n h r 9 Find th e a gle 4 i t e cu ve cos n a .

4 . S a in th e s ra Arc me es r z a e th e ar s n rma how th t pi l of hi d , pol ub o l is

onstant .

P r n icular from Pol e on Tan ent . 3 8 . e pe d g

L t n r L et d e P be a given p oi t on a c u ve . p enote the p epp en dic l P Th n u ar f rom the p ole on the tangent a t . e

= in ( ) p T S j . Th erefore 1 1 ? 7 p 7

FIG. 9 .

Th i at s,

3 9 Ped a E uati n . l q o . By th e p eda l equa tion o f a cu rve is und erstood th e re at on betw een and r in th at r l i p c u ve . T I . o ﬁnd the p ed a l equa tion of a curve from its Cartesia n eq ua tion .

Th e tan ent at th e o nt a is g p i ( , y)

Th erefore

46 DIFFE RENTIAL OAL O UL US

Th erefore

t n H a n ,

= i . e. nB , p g .

Th erefore

p = r sin a= r c os nB

Th u s th e required equ ation is u -i-l T

n a

"c°t 3 In th e ar m c s ra r ae r e a . log ith i pi l p ov th t

= in a p r s .

h e a n th e c r = 1 4 . Find t e p edal qu tio of a dioid r a ( cos

P dal Cur es. i n e v Po ar r cal s. Inversio . l Rec p o

n d P n h r m h n L t 0 be a en o t an a ot e o t . T e f a I . e giv p i y p i o nt be ta en on OP or OP roduced su ch th at p i Q k , p ,

= 2 OP . O a c onstant sa a Q , y ,

Q is said to be th e i nverse of P w ith respec t to a c irc le of radiu s a and c entre 0 an d f P d escr be an curve d escr bes anoth er ; i i y , Q i r h f h rm r cu ve c alled t e inverse o t e fo e . r m a n n a r nd r dra n t h II . If f o give p o i t 0 p e p e ic u la be w o t e t an ent to a ven cu rve th e o cu s of th e foot of th e er end cul ar g gi , l p p i is c a d th e rs ositive ed a o f th ur th res ec t to lle fi t p p l e c ve w i p 0 . Let th ere be a ser es of curves wh c h w e ma re resent b i , i y p y O 0 0 O su c h th at each is th e first os t ve eda of . 1 , 2 , n, p i i p l th e n r d n it Th n 0 t r r s c t e cal ed o e ece . e 0 e c . a e e e v p i g 2 , 3 , , p i ly l h d i t n h d e . e seco t r tc ositive ed a ls of O O O etc . are , , , p p ; M , M , , res ect ve c a ed th e r d a f O st second etc . ne a tive e ls o . p i ly ll fi , , , g p , L 0 b III . et e a given point and OY th e p erpendicu lar from O on th e tan ent to a n ur Th n nt n in g give c ve . e if a poi Z be take OY or OY rod uc ed su ch th at , p ,

2 OY OZ = a constan t sa a , y ,

th e locu s of Z is c alled th e p olar reciproca l of th e given curve with res ect to a c rc e of rad nd n tr p i l ius a a ce e 0 . TANGE NTS AND NORMAL S 47

EXAM PLES .

1 n th e n erse th e ara a r res ec a c rc e . Fi d i v of p bol with p t to i l of

h radiu s a and th e centre at t e pole .

2 F nd th e eda a nd o ar rec roca Of the curve r" . i p l p l ip l " H a cos u .

2 Ar h Ex t . 9 H r b . 3 e e w e ave y , ,

7? t —n6 and q p " 2 a

’ Th erefore d enot n th e er en d c u ar OY b r an d th e an e , i g p p i l y gl , wh ch OY ma es w th th e n t a n e b w e h ave i k i i i i l li , y

" ’ a

+ H== (n

n + 1

Th us th e equ ation of th e p ed al is

° n + 1 2 A a a n OZ ' Th erefore th e o ar rec roca be n th g i p l ip l , i e Oy g

nverse of th e ed a is th e cu rve i p l ,

' n 1

" " 3 . n th e hth e e a h e c r e a o Fi d positiv p d l of t u v r cos n .

4 . Find th e polar recip rocal of th e conic ? 2 a x 2ha2y by 1

re ect a circ e ra s 0 and en re at th e r n with sp to l of diu c t o igi .

Exam l e on Ch a er p s pt IV. — — — 1 . In th e r e :c 2 cc 3 = x 7 a th e n cu v y( ) ( ) , show th t ta gent p aral lel to th e ax is of a: at th e points for which

a = 7 ¢ 2 ¢ 5 48 DIFFE RENTIAL OAL O UL US

2 a all th e n s th e c r e . Show th t poi t of u v

2 Q y 4a a: a sin ) ( a

i h n en i r e h e x at wh ch t e ta g t s p a all l to t a is of a: lie on a certain parabola . 3 n th e an e n ersec n th e c r . Fi d gl of i t tio of u ves ? ? “2 w y , 4 m2 y2 a ? M2

Th e angle between th e curves at any one of their points of intersection is n e et een th e an en h e c r ere th e n Of th e a gl b w t g ts to t u ves at that point . H poi ts intersection are given by 2 gu m 1)

2 2 — y g( d g—l )

l e th e an en s th e c r es at cc be Now t t g t to u v ( , y)

= Y rnX + c ,

93 a: y 21

r re th e an e e een th e c r e at a is The efo gl b tw u v s ( , y)

“ y “ 1 tan 1 t n 1 1 b e a ns 1 and tan ff f d: a y qu tio ( ) $ 1732 a , y

e Therefore th e angle requir d is 2.

n th e c n t n in r er a th e c r es 4 . Fi d o di io o d th t u v w” + y = 1 3 ’ g 6 a

n should intersect at right a gles.

n th e an e n ersec n th e ara a 5 . Fi d gl of i t tio of p bol s

6 Pr e a th e c r e . ov th t u v s “" ” ” o r a cos M , r b sin n

n intersect at right a gles.

3 : : 7 In th e e i se 1 a a sin t, r e a . ll p 5 , if p ov th t

2 a e b Art . 33 N e We h v , y , ot , ds TANGENT S AND NORMAL S 49

Therefore

tan t . $ 2

ere re a c os t . Th fo

8 . In th e cycloid sin t) = — y a (1 cos t) . prove that

d s t d s 2a cos and d t 2 d y

" " 9 . In th e c rve r a co s n9 r e a u , p ov th t

n a see it 9. d o

0 °°t 4 10 In th e e an ar ra r = a e r e a a c n an . qui gul spi l , p ov th t g;is o st t

1 1 a ere are rea r ne are b an en . Show th t th just two l st aight li s which oth t g t " “ and normal to th e curve y w and th a t th e absc issse of th e points wh ere e ith er h r r n r nd r of them touches t e cu ve and c uts it at ight a gles a e 3 a 3 esp ectively .

2 El—fl 3 cc Here Therefore th e equ ation to th e tangent is d zc 2 y

” m . T en n e we a e h , si c y , h v 50 DIFFE RENTIAL CAL O UL US

Thus th e equ ation to th e tangent is 4 3 Y : rnX 772 5 7

u n c n ac e n n W . its poi t of o t t b i g (g , 57 ) Similarly th e equation to th e normal is found to be

8 3n2 Y nX 1 27n3 ( 2 1 t h e in ere c s th e c r e at r an es e n po t wh it ut u v ight gl b i g , (972

1 n 2 be en n Now l et th e stra ight lines ( ) a d ( ) id tical . The

2 4 3 8 3n n 771; 1 m u a d 0 27 2 Therefore 3n2

2 i c ec , ” 2 or 1 0

B h — ere re ut t e value 1 must be rej ected . Th fo

n h in r n a d t e abscissae question a e 3 a d g.

2 12 S a th e n r a th e ara a 4am c e th e c r e . how th t o m l to p bol y tou h s u v

2 a 2 4 w 2 3 7 y ( a ) .

{ 13 S th e n rma th e e se 9 1 es th e r e . how that o l to llip 2+ n touch cu v a

3 g 2 (cw?) (by) (a

1 4 t th e n rma h e ra r x . Show th a o l to t t ct i a: a ( cos t log t an y = a sin t

c th e a n r = tou hes c te a y y a cosh 2.

15 r h n th e an n h n erm n . P ove th at t e equatio to t ge t at t e poi t det i ed by t on th e curve

may be writte n in th e form TANGE NT S AND NORMAL S 5 1

3 i’ z 1 r h e c r e z a w are ra n s en n r an 6. Cho ds of t u v 90 y d w ubt di g a ight gle at th e r n r e at th e an en at e r ex rem es n erse on th e c n c o igi , p ov th t g ts th i t iti i t ct o i 2 — 5x + 3wy 2aaz M r a . s [ th T ipo ,

17 . th e an en s at P R th e car If t g t , Q, to dioid r = a (1 cos 9) — — — be ara e en th e an es f OP a/O x/DR th e sum be p ll l , th of qu titi , Qj of two will

e a th e r ere O is th e e . qu l to thi d , wh pol

~2 2 1 8 . Pr e a th e n rma th e c r e 9 = a cos 20 at th e n Oz a ov th t o l to u v poi t , meets th e perp endic ular normal at a point whose distance from th e pole is

-1 } f Qu 2 3 a cos ( i z??

19 h n n t nt th e car . T e ta ge t a a poi of dioid 7‘ a (l COS 9) whose vectorial angle is 2 a meets th e curve again at a point whose vectorial

angle is 23 . Show that — cos (2B a) + 2 cos

20 . T r en e th e ance an n P on th e e n a e If l , , d ot dist s of y poi t l m isc t 2 2 - (mu y >= armh r) 9 r m th e nts _ O and th e er en c ar on th e an en at P f o poi i : ) p , , p , p p di ul s t g t

r m e e n s r e a f o th s poi t , p ov th t 1 1 3 2 - — + 2 7 2 7 1 2 2 1 . Show that th e condition of tangency of — 90 cos a + y sin a p = 0

m _‘IL m — m — — ‘ p l (a cos a )m 1 (b sin u)m 1 2 2 . Show th at each of th e curv es

nf’ ° r z ae 1 9 = a 7 sin n0 = a r n n9 = a , , , si h

h as e a e a i n th e rm B ere A and B are c n an . its p d l qu t o of fo , wh o st ts P 23 I h h . n t e equiangular spiral prove that th e loci of t e ex tremities of th e polar subtangent and subnormal are also equiangu lar r spi als . 24 . Show that th e ﬁrst positive p edal may be obtained by writing r instead r2 and in ea r in th e e a e n h e of p st d of p d l quatio of t e o riginal curv . }; 25 . Prove that th e polar reciprocal of with respect to th e c rc l — 2 2 2 e e r bed on th e ne n n Mai 5 O and x/ a 6 0 as ame er i l d sc i li joi i g ( , ) , ) di t , E CHAPT R V .

YM T OT E AS P S .

N. B. r k b th e rd a s m tote n r Th oughout this boo , y wo y p is to be u de stood in ar a s m tote n e th e n r r x re e til e a e a ed . r c y p , u l ss co t y is p ssly st t

i n f m t 1 Definit o o As ote. In en eral b an a s m e t 4 . y p g y y p tot o a i u n r d r h n n c u rve s de stoo a st aig t li e S O possessi g th e follow ing fi d n r properties : (a ) S o is at a n ite ista c e f om th e origin ; as n n f h u r m d n n a y seca t S o t e c ve is a e to te d to 8 0 (both in directio an d os t on tw o of its o nts of ntersect on w th th e cu r tend p i i ) , p i i i i ve n nfin n r m th rl m to poi ts i itely dista t f o e o g .

Th e be nner a e k at e th e m r e NOTE . gi is dvis d to loo som of asy ptotic cu v s r traced in Chapte VIII .

EX AM P LES .

1 P ro e th at th e stra h t ne = fc is an as m tote o th e . v ig li y y p t z— 2 = 2 rectangu lar h yperbola w y a .

Ta e an sec ant = mm e an d let w x d enote th e absc ssae k y y + , ” 2 i f it o n of nt r t on h h c u r Th n r th e o s p i ts i e sec i Wit t e ve . e a e roots of th e equ ation — - 2 as (mcc c) ? 1 — 2 2m . m c t e. a zc , ( )

P u t a: m th e equ ation (1) th en it becomes

— ‘2 — (l m ) 2mct

an d c cc are th rec ro ca s of th e roots of th e e u at on a , , e ip l q i

Now wh en = mx c is m ad e to t en d to = x m and c mu st y + y , tend to 1 and 0 res ec t ve and th e e u at on 2 mu st tend to th e p i ly, q i ( ) t n 2 = equ a io t 0 . Th u s wh en = mm e i mad e to tend to = zc th e rec roc a s of y + s y , ip l ac and x tend t o z ero and c on se u ent tend to become , 2 , q ly nﬁn h r = n m t T r i d n a . i ite . e efo e y az s by eﬁn itio a sy pto e

54 DIFFE RE NTIAL CAL C UL US

II . a i z r Ca se “s e o . — Th ere are in enera n l sets of values m c , g l , ( l , l ) , m _ o _ sat sf n th e tw o e u at on s : th u s th ere are o n ( 1 , n 1) i yi g q i ly n —l a s m totes of th e form = mcc c th e r e u at ons be n ( ) y p y + , i q i i g

= m w 0 z m — x ’ ' C — y 1 + h y n l l n l i

h rm z In rd r th e remaining asymptote is of t e fo rc d . o e to obtain it u t x z d in th e ex ress on on th e eft s d e o f e u ate to z ero , p p i l i q th e c oefﬁci en t of and solve for d th e simple equ ation th us

h r u red as m tote is cc z d at be n th e root of obtained ; t e eq i y p l , , i g th i s equ ation .

Th e d en is a e er th e enera r e en a e b NOTE 1 . stu t dvis d to v ify g l ul , giv bov , y 41 mean s of th e deﬁnition of Art . .

2 a r e a C ar e an e a n not th e rm NOTE . If cu v is such th t its t si qu tio is of fo n in r er ﬁnd th e a e b e n e e ar e er u se the , o d to symptot s , it will c ss y ith to

t e in Ex . 1 2 en in th e set ex a e on th e re en th e method indica d , giv of mpl s p s t

er or a th e re Art . 45 a er ran rm n th e e a n n ch apt , to pply sult of ft t sfo i g qu tio i to polar coordinates .

EXAM PLES .

F nd th e as m totes of th e cu rve 1 . i y p 3 — 2 2 2 — 2 = y yx + y + w a 0 . h P ut y = mm+ e in th e expression on t e left sid e of th e abo ve n Th en w e h ave equ atio . 3 2 2 — 2 x c —x mx c x a (m + ) ( + ) + , 3 3 — 2 2 — 2 1 m m m w 3m c 0 m etc . e. i . , ( ) ( + )

3 2 ero th coefﬁc ents of x an d 511 w e obta n th e tw o Equ ating to z e i , i equ ations m3 — m= 0 3c

wh ich are satisﬁed by th e th ree sets of values m= 0 m= m = 1

C = 1 C —1 c —1

all th e asymptotes are of th e form y z mx b eing = 1 = sc — 1 — — 1 y , y , y a: .

F nd th e a s m to tes of th e curve 2 . i y p 3 w y) 6 . AS YMP T OTE S 55

Putting y = mm+ e in w 2 — w2 —b3 y y , w e h ave 2 2 2 3 a mx c —x ma~ c —a ac mx c —b ( ) ( ) ( + ) , 3 2 — 2 — i . m m 2 tc . e. m mc c e , ( ) + x ( ) +

3 2 to z ero th e oefﬁc ents of x and 33 we h ave c i ,

m2 —m=

2mc — c = 0 whic h are satisﬁed by th e tw o sets of val u es m m = l

Th u s th ere are on tw o as m totes of th e form = mx c the r ly y p y + , i equ ation s being = = O x . , y 3 As in th e e u at on to th e cu rve th e c oefﬁc ent of is z ero th e , q i , i y , r ma n n m i o f th orm = d P tt n = d in e a s tote s e f w . u x i i g y p , i g 2 2 3 sc — x -b y y , w e h ave d 2 — 2 y d y et c .

2 u t n t z r h n f h z Th h E a o e o t e c oefﬁc e t o w e ave d 0 . us t e q i g i y , rem aining a symptote is Th e requ ired asymptotes are th erefore = = = 0 x m o. y , y ,

3 . Find th e a symptotes of th e curves z - = (i) my y 0 . ? z Q 2 2 2 z 2 (ii) some} y ) a y b x a b . 3 l l 2 3 (iii) y fin ya: 6x x y 0 .

4 n h e c r es . Fi d th e real asymptotes of t u v 3 (i) y Sassy . 2 ? 2 2 (ii a (cc y ) a (y 3x ) . 3 2 : (iii) y (x a ) ( c b) . 5 . Prove that th e a symptotes of th e curv e 3 — 2 s ax (lcc y + my ) has ky + b = 0

a r th e ri n p ss th ough o gi . 6 . Prove that th e asymptotes of th e curve

z‘ 2 ' 2 ' 2 " " acc y bzzzy a a: b y a x b y 0 ar e a a rea and ﬁnd e r e a n . lw ys l , th i qu tio s 56 DIFFE RE NTIAL CA LC UL US

4 3 Parall el As m t otes. In c onnex on w th th e enera . y p i i g l

ru e ven in Art . 42 it sh ou d be noted th a t in some c urves l gi , l , , th e tw o e u at on s obta ned b e u at n to z ero th e c oefi c ients of q i , i y q i g " ” cc and at are sat sfied b one or more va u es of m and a n , i y l y a u f h b u c h a f a rd n t th v e o 0 . T u s f e s va u e o m c co o e l i p l , i g d fin n r 41 = h ll d _an a s m e t on n i A t . x c s o u d be c a e i i give , y y + l y to h t m Bu t it is c u stom ar to ca u h stra h t p te w a ever 0 ay be . y ll s c ig line s o f th e system y = px + c asymptotes a s sati sfy th e follow ing c ondition : 0 is suc h tha t th e grea test number of the p oints of in ter sec tion of a ny secant S tend to p oints infinitely dis tant from the i w n mad to tend to = a3 or in he S is e c . g , y p + " f be absent from th e e u at on of th e c urve a statement I y q i , , s m ar to th at ven above w ou d a to th e tra h t n s f i il gi , l pply s ig li e o th e s t m = d ys e m . Th e m eth od of findi ng parallel asymptotes is made clear by th e r t w o ex ampl es wo ked below .

1 n h e th e c r e . Fi d t e asymptot s of u v 2 2 ? 3 “ 2 a: 2a zr: sca ( ( y ) a .

Putting y mac c in 2 2 " 2 z ? ‘ a: 2a re wa s a ( y ) ( y ) , we h ave 2 2 Z ‘ ' 2 2 3 i 1 m :c 2mcx - c 2a 1 a: 2mm: 6 sea — a ( ) } ) } ,

2 2 2 " z 2 2 2 ? 2 2 1 x 4mc 1 m x 4m c 2c 1 m 2a 1 m a: e tc . i . e . m , ( ) ( ) { ( ) ( )}

4 ” r th e c efﬁ ien m and ac we a e th e e a ns Equa ting to z e o o c ts of , h v two qu tio (1 m2) 2 O 4mc(1 m2) 0

re a ﬁe b th e a e m 1 m — 1 a e er 0 ma be which a s tis d y v lu s , , wh t v y . 2 e 0 so a th e e ﬁ en x z er for m : 1 en Now choos th t co f ci t of is o . Th 2 — ? 46 461 0, i . e. , C a .

th e s ra ne th e em Thus , of t ight li s of syst y = x + a

= n ao d: a are a m e . For c rre n n eac e ree o n s o ly y sy ptot s , o spo di g to h of th m , th p i t n n nﬁ h r n of intersection of S te d to poi t s i nitely distant from t e o igi . — — ar th e ra ne th e em w c n :c :l a are Simil ly, of st ight li s of syst y , o ly y asymptotes . Th e required asymptotes are therefore — z x . y i a , y w4 a

n th e as m e th e r e 2 . Fi d y ptot s of cu v

z z 2 2 2 w y Z a (x y ) . AS YMP T O TE S 57

Putting y mx c in x 2 z 2 w2 2 y a ( y ) ’ we have 2 4 3 2 2 " 2 i m w 2mm: { 0 a (1 in ) } az 2ma zor: a zo .

4 3 z r th e c e ﬁ en x a nd 513 w e a e th e e a n Equ ating to e o o f ci ts of , h v two qu tio s

z which are satisﬁed by m o,wh atev er 0 m ay be . ? r f : Now choose 0 so tha t th e co efﬁcient of st is z e o or m 0. Th en we have z - z i . : a c a o . e 0 . z , , i

: th e ra ne th e em 0 n zl: a are a e . Thus , of st ight li s of syst y , o ly y symptot s ‘ n th e r e P n a = d in Also y is absent from th e equ a tio of cu v . utti g c

‘ 2 2 2 z a x n ( y ) ’ we have 2 2 ? 2 2 y (d a ) a d .

3 r d m Th e coefﬁcien t of y is z ero wha tev e ay be . Now c hoose d so that th e 2 n e a e coefﬁci en t of y is z ero . The w h v

h m d n th e ra nes t e e at a: d: a are a e . Thus , of st ight li of syst , o ly symptot s Th e required a symptotes are th erefore a cc a y d: , i .

3 n th a es th e r e . Fi d e symptot of cu v s 2 2 3 2 2 3 8 4x 3 9x 6x 2 2 1 (i) y 5wy x y g y y x . 2 ‘2 2 3 2 2 2 3 h): b (ii) x y :23 y my a y . 4 —l . Find th e a symptotes of th e cu rves i 2 2 (i) a l/ (b WW 11 ) 2 4 “ 6 (ii) y (a: a ) a

m In i B h n r 4 4 As totes b s ect on. t e fo ow u es w h ch . y p y p y ll i g l , i are a s d d r m Ar 42 n d 43 om t m s th e e u c b e f o ts . a s e e e e u at on ily i l , i q i s of some or all of th e asymptot es o f a cu rve c an be w ritten d ow n from an inspection of th e equ ation of th e c urve ” Ru e e r m h u t n f u r l I . If y b a bsen t f o t e eq a io o a c ve of th e nth d e ree th e a s m totes a ra llel to the — ax is a re ob a ine g , y p p y t d by equa ting to zero the coeﬁicien t of the highe st p ower of y in th e equ a " t on s m lar i as be a bsent th e a s m totes ara llel i ; i i ly, f , y p p to the — x ax is a re obtained by equa ting to z ero the coeﬁicient of the hi hest ow r o g p e f x .

Ru e . L et th e s mbol E re resent a rodu c t o n di eren l II y u p p f ﬂ t linear ac tor Th n th e u t n f u r h f s. e if e q a io o a c ve of t e nth d egree be of the form wh ere P is of a d egree not h igh er th an - n 2 a ll the a s m t e a re iven b = ot s the e ua tion E 0 . y p g y q n 58 DIFFE RE NTIAL UALOUL US

1 be e ex re th e e n NOTE . If it possibl to p ss quatio of a curve in th e form 0 D y = Aat B + + + az x 2

A B 0 etc . e n c ns an s en th e ra ne , , , , b i g o t t , th st ight li y = An + B

th r is evidently an a symptote to e c u ve . This method of ﬁnding asymptotes is som etimes useful . In r er to d etermine on wh ic h side o the urve the a s m tote lies we a e o d f c y p , h v

n n er th e n C . For ex am e ear a in th e ﬁr o ly to co sid sig of pl , it is cl th t , st qua drant th e curve will approach th e a symptot e from above or b elow according ne e a s C is positive or gativ . L 2 C r l near as m t ot esn Let ere be r e c n NOTE . u vi i y p th two cu v s which o tinually approach ea ch other so th at for a common abscissa th e limit of th e f eren th e rd na e z er or for a c n rd na e th e Of th e di f ce of o i t s is o , ommo o i t limit diff erence of th e abscissee is z ero wh en that c ommon a bscissa or common ordina te tends to become inﬁnite ; then each curve is said to be a curvilinea r r a symp tote of th e othe . In ar ar one th e r e IS a ara a a d be a ara bo lic p ticul , if of cu v s p bol , it is s i to p

h er. For an ex a e a ara i m o e t e c a e see Ex . 1 a sy p t t of oth mpl of p bol symptot , 9 , 2 p . 6 . X AM L E P ES .

r te d own th e e u at on s o f th e as m tot of th r 1 . W i q i y p es e c u ve 2 2 2 2 2 = a x w y ( + y ) .

1 th re u red e at on s are By Ru le . e q i qu i Q— 2 = i = w a 0, . e. , m + a

Q— z= 0 = i . a e. + a y , . , y

r te d own th e e u at ons of th e as m totes of th e c ur 2 . W i q i y p ve 2 3 a y y) b

z z h m tot a r n — r = Th r t as es e ve b x 0 . e e By Ru le II . e y p gi y y a y fore th e required equ ation s are = = = w O 0 x . , y , y

3 r e n th e e a ns th e a es th e c r e . W it dow qu tio of symptot of u v s 2 — = 2 (i) y (a: 1 ) (x 2) oc 3 . 2 ? “’ 2 b 2 (ii) zcy (a: y ) a y % a zbz . 2 — 2 2 ? 2 2 ? ? 2 1 (iii) ccy(:c y ) (x 4y ) :cy(cr: y ) a} y .

4 r e n th e e a n th e rea a m e th e r e . W it dow qu tio s of l sy ptot s of cu v s 3 3 “ i x ( ) 5c y a . z 2 z 2 — (ii) wy 4a ( a w) . ‘z 2 = 2 2 - 2 (iii) cc y a (:c y ) .

i II. i A fuller treatment will be given n Vol . AS YMP T O TE S 59

General Rul e for ﬁnding Asymptotes from Pol ar Equation

L et the equa tion of a curve be th en if a be a root of the equa tion — 1 r 8111 O a ,

r is an a symp tote to the cu ve.

For th e vec tor a an es of th e o nts of ntersec t on Of th e , i l gl p i i i sec ant r sin (H with th e curve are th e roots of th e equ ation — sin (6

Now wh en th e sec ant is mad e to t en d to [3 and a mu st 1 tend to a and res ec t ve and bo th a an d mu st ’ p i ly, ¢( ) f (a ) Th w h n the sec an t is m ad e to t n d to t o tend to z ero . u s e e w

O f th e ro ots of th e e u at on t end to a and conse u ent q i , q ly

r r t n d 0 0 th e c orresp onding adii vecto es e to .

1 Th e ﬁr ar th e a e sen ence l ec e e ear NOTE . st p t of bov t wil b om quit cl to h r a er h e n e a a c r n a enera e rem h e t e e d if ot s th t , c o di g to g l th o which will

in C a er X . study h pt , — W) (G (O a VP

ere P is ﬁn e for a e 0 near e a a . For r 2 a wh it v lu s of ly qu l to , it follows f om ( ) th t , en < a nd en z er th e e a n < 6 en s ec me e uiva wh p( ) a t d to o , qu tio p( ) t d to b o q len t to th e equation 2 9 a P 0 ( ) ,

and c n e en its r en a . , o s qu tly , two of oots t d to

2 C rc l ar a m o e . Let th e a r e a n a c r e be NOTE . i u sy pt t s pol qu tio of u v such

en a i 0 sa en 0 en be e nﬁn e . en th e r e tha t r t ds to l mit , y , wh t ds to com i it Th ci cl r = c is sa id to be a circu la r a symp tote of th e curve ; for evidently th e c urve n ne rer th e c rc e as 9 is a e ar er and r er approaches nearer a d a to i l m d l g l a g .

EX AM PLES .

r i 26= 1 nd h a m totes of th e cu ve r s n a . . Fi t e sy p

Here and th erefore th e roots of th e equ ation

Th erefore th e requ ired

asymptotes are 60 DIFFE RE NTIAL CAL C UL US

r sin 6

Find th e asymptote of th e cissoid z a si n 6

cos 6

Find th e asymptotes of th e curve

r sin na a .

P rove that r sin 0 : a is th e only a symptote of conchoid

r = a c ec a h os + .

Exam l e on Ch a er p s pt V .

1 th e a m es th e c r e . Show that sy ptot of u v r — 2 (a rr= 2

2 a th e a m e th e r e . Show th t sy ptot s of cu v ? “ ﬂ 2 — 3 a y y ) a (w y) a 0

r a are r e an ar n th e c r e asses . fo m squ , th ough two of whos gul poi ts u v p

3 n th e a m es th e n c r es . Fi d sy ptot of followi g u v

3 3 (i) try a (a it ) . x 2 ‘ 2 3 ' 2 (ii) se y y a y a zat 0 . 2 3 a: :c 2 90 3 2a ct ” 2 2 (iii) ( y) ( y)( y) ( y ) a (cc y) 0 .

4 n th e a e th e n . Fi d symptot s of followi g c urves

r in 29 = a c s 9 (i) s o 3 . (ii) r = a (sec 9 + cos r " sin n 0 = a " (iii) .

5 . Pr e a e er c r e Of th e nth e ree h as in enera it a e ov th t v y u v d g g l symptot s, nar real or imagi y .

62 DIFFE RENTIAL CAL O UL US

e and h re re Therefore m and c are respectiv ly 4 2 0 . Thus t e qui d asymptotes

r : X a e Y j : g .

13 S at th e ra ne a: a is an a m e th e c r e . how th st ight li sy ptot to u v

if 4>(a ) and are ﬁnite.

14 n h e as m es Of th e n c r e . Fi d t y ptot followi g u v s 3 = an = e‘ x (i) y t x . (ii) y .

15 A th e eﬁn n Ex . 1 2 es a th e enera r e en in . pply d itio of to t blish g l ul giv 2 Art . 4 .

16 S a ere is an nﬁn e ser es ara e a m e th e c . how th t th i it i of p ll l sy ptot s to urv e a r + b, 0 s1n 0

e r ances r m th e are in Harm n ca Pr res n and show that th i dist f o pole o i l og sio .

x 2 a2

17 . Pr e a th e c r e : a: es a e its e a m e in ov th t u v y o li bov obliqu sy ptot sir — a ?

th e ﬁrst quadrant .

1 8 In c nn x n th e c r e . o e io with u v 2 z 2 ? c 8a 2a 3 4ar 8a 0 y( c ) ( x ) ,

ﬁnd th e e a ns th e a m e and e ermine on c s e ea qu tio of sy ptot s , d t whi h id of ch t h nﬁn sympto e t e curve lies at i ity.

a:3 —a s _ z 19 . a th e c r e h as a ara c a e a z x Show th t u v y p boli symptot y .

s 2 CD3 a 56 ere re for th e same a e x th e d ff erence e een Th fo , v lu of , i b tw a n a a: th e ordinates of cc a 3 y and ay = x 2

2 93 m nﬁn z er en en s ec e e . Hen b Art 44 c en a: ce . , whi h t ds to o wh t d to b o i it , y , a:

N e 2 th e c r es are a m c eac er and ince th e ec n ot , two u v sy ptoti to h oth ; , s s o d

c r e a ara a a ara a m e th e ﬁr . u v is p bol , it is p bolic sy ptot of st

20 . a th e c r e h e e a n th e rm Show th t u v , w os qu tio is of fo — B B — n " 1 __l A o o 0 A 0 y A x + x + o + n m h as a curvilinear asymptot e

n " -1 z A cc + A x + . + A ; y n n _l 0 ’ ’ h n r a r n 1 t e A s and th e B s being all constants and it bei g g e te th a .

21 : r . Show th at th e curve a by any 0 h as a pa abolic asymptote

2 2 bi: b _a u es i in descend in er Of w 73 y . [S gg t on z Exp and g pow s acc —a b 22 ﬂ . Find th e circul ar asymptote of th e curve r . CHAPTER VI .

U V T U C R A RE .

4 6 . Centre of Curvature : Deﬁnition and Cartesian Coordinat es.

L P n o nt on a curve and n n it et be a give p i Q a y o th er point o . Th en f M d en o te th e o n t of ntersec t on of th e n orm al s at P , i p i i i and b th e centre o curva ture o f th e curve a t P is u nd erstood Q , y f th e m t n o s t on M on th e li i i g p i i o, n ormal at P to w h ch M ten d s , i as trave n al on th e c u rve Q , lli g g , tend s t o P

Let a be th e c oord nates of , y i P as h h th e c oord n ates , . + , y + i d y of Th en re resent n — b Q . , p i g y d zb

( ac th e n orm a s at P and are Fm 1 M) , l Q , 0 , respec tively (X and — — — — b h = 0 (X tu h) ( Y y h)g(a3 ) .

Th erefore d enot n b a 3 th e c oord n ates O f M w e h ave , i g y , [ i , — (a w) (B - —

Hen c e w e h ave b subtract on , y i , 64 DIFFERE NTIAL CAL C UL US

l t nd t h n nd Now e P T nd z r n M M . t e o . e h te s to e o a d te s to Q O Th erefore d enot n b d 3 th e coord n ates of M w e h ave , i g y o, [ 0 i 0 ,

lim k_dy and

lim ec h — d d a( + ) p(x ) — __ ‘ h = 0 d az dcc

2 i g Th erefore d enotin b the s mbol w e h ave from 3 , g y y ? ( ) dd dd

— Bo Y

an d h ence w e h ave from (1 )

— = a o X

EXAM PLES .

2 — : 1 . In th e arabo a x 4a rove th at p l y, p 0 UR VA T UR E 65

3 2 y h 4 r a z 2 — — 2 . In t e ara a d x e t at a 3 x p bol y , p ov h 0 , B, 1a §

F n d th entr of ur tur at th e o n d t rm n d b 3 . i e c e c va e p i t e e i e y t on th e ellipse

3 c o sec t .

Th erefore b2 2 1 + 2 co t t — a b sm t 3 t 7 c osec a 2 2 2 2 — — a b c ot t b a b sm -t — 3 b c osec t b

= a 0 a7 a COS t 3 c osec t 2 a

2 c o t t c ot t —a c os 3 a c o sec t

3 a c o sec t 66 DIFFE RE NTIAL CAL C UL US

4 n th e cen re r a re at th e n e erm ne b t on th e . Fi d t of cu v tu poi t d t i d y astroid 3 a: a cos t

3 y : a sin t

ﬁ F r u Radius of Curvat ure : De nition and. o m ae. 4 7 . l

r n d M th n tr f u r Let P be a given point on a c u ve ; a O e ce e O c va ture of th e cu rve at P th en by th e ra dius of curvat ure of th e be w een P nd M c urve at P is u nd erstood th e dista nce t a O.

r E a (a ) Formu la f or Ca tesia n qu tion.

Let as be th e c oord nates Of P th en b Art . 46 th e , y i , y c oordinates of M0 are

Th erefore d enotin b th e rad u s of c u rvatu re at P w e h ave , g y p i ,

2= 2 p PM0

b F rm ( ) o u la f or P ed a l Equa tion .

r We h ave by A t . 39

2 l og p = 2 l og y l og 1

2 2 i r = x + y

Th erefore d fferent at n both th e s d es of 1 and , i i i g i ( )

were a re ? 2 1 dp _ d d dtc dfc d p at mdy dar C UR VA T URE 67

Th erefore

0 F ormu la or P ola r E a i n ( ) f qu t o . 8 We h ave by Art . 3 1 1 1 dr ﬂ + 2 é 4 p r r d H

Th r fore d fferent at n both th e s d es Of th s e u at on e e , i i i g i i q i , w e h ave

l dp 3 36 1 d 1 p _ “ Th erefore 3 d ? p a? d B‘

Th erefore 68 DIFFE RE NTIAL CAL C UL US

Th u s w e h ave th e formu la

1 An er m r an rm a th e n NOTE . oth i po t t fo ul is followi g E 2 E (d ) (N

” 1 i - h en t th e are ea re r a ﬁx ed where \P= tan tlZ and 3 denotes t e l g h of m su d f om

h in 93 i . e. P . Th e rm a d ma e ea point A on th e curve to t e po t ( , y) , , fo ul ( ) y b sily For deduced from th e formula (a ) . , 2 2 d s dy 1 _1 2 d s (d os) dcc 2 d v 2 61 11 2 d x d cc (it)

i1 +

th e ax a: is th e an en at A th e an e / c a e th e an le o con If is of t g t , gl 4 is ll d g f r A Man r r e r r b h e tingence of th e a c P . y w ite s d ﬁne radiu s of cu vatu e y t n d equatio ( ) .

m er rm ae f r are en Th n i a e e o e . e NOTE 2 . So oth fo ul p giv b low stude t w ll h v n n em r m th e r a no difﬁculty i deduci g th f o fo mul (a ) .

2 e e 1_ er + er 2 ‘i 9 d3 2 d s i-iiif) (it) b (at) (at)

EXAM PLES . “ = 1 . In th e catenar a c osh rove th at y y g, p

2 d} ! w _d y 1 sinh and Th erefore 2 doc a ( la: a

70 DIFFE RE NTIAL CAL C UL US

4 . n th e ra c r a re a t th e n cc on eac th e n Fi d dius of u v tu poi t ( , y) h of followi g curves 2 2 z 3 x 4 11 . (i) ay. ( ) my a (iii) a y 90 3 3 i (iv) y a log sec (v) a: y a

(Vi) y

5 In h c i : . t e yclo d w a (t + sin t) 1 y : a (1 cos t) f prove that

p 4a 0 0 8

6 . n th e ra s r a re at th e n r on ea th e n Fi d diu of cu v tu poi t (p, ) ch of followi g curves “ r sin a . (i) p (iii) u p r .

2 2 r a z (iv) p .

1 a 2 b2 r2 VI ( ) ; P a w

n th e ra s Of c r a r th n r 0 on ea th e n Fi d diu u v tu e at e poi t ( , ) ch of followi g

r : a os 0 11 r . (i) c . ( ) 1 + 20 8 9

" ” " n r a 9 r n n oz a . (iii) cos n . (iv) si h

z z 2 2 v r - a 1 cos 0 sin 0 V 0 VI ( ) ( ) z 2 a a b

2 2 r 8 . I n th e c n c x 2hx 1 r e a th e rad r a e o i a + y by , p ov th t ius of cu v tu n v a rie s inversely as th e cube O f th e central perp endicular on th e tange t .

4 8 Circ n h u a ure. . le a d C ord of C rv t

Let P be a n o nt n cu r e and M th e c entre of cu rva give p i o a v , O ture Of th e c u r e a P Th en th e c rc e wh ose c entre is M an d v t . i l O rad u M P Of th e c urve at P i s O is c alled th e circle Of curva ture ; t n and by th e chord of curva ture Of th e c u rve at P 111 a given direc io is u nd ersto od th at ch o rd o f th e c ircle of c urvature wh ich passes

th rou h P in th n r n g e give di ec tio .

EXAM PLES .

1 . In th e curve = a l o sec rove th at th e c h ord of cu rva y g 2, p

ur r n n n th t e p a allel to th e ax is of y is of c o sta t le g . C UR VA T URE 71

It is clear th at th e required ch ord

P = 2 c M P Q p os O Q.

w 2 But s nce : tan and conse u entl , i , q y, (if) d

6 a: a 4 3 4 ” E. f dufl a

- 2 a sec

= i . . a ec e , p s

Cl ‘ 1 —ii COS A M P Q= c os tan - O (

Th erefore P Q= 2a sec 2c os

2 . Show that in a parabola th e chord of c urvature through th e focus and th e c hord of c urvature p arallel to th e ax is are each four times th e focal distance h n of t e poi t .

3 . Show tha t th e chord of curvature through th e pole of th e equiangular r = in a i 2r spi al p r s s .

4 h r re r th e e th e r e . Show tha t t e chord of cu vatu th ough pol of cu v

r" a n cos

Evolu h l of a c urve is u n derstood th e 4 9 . te. By t e evo u te l u f th n r f u r oc s o e c e t e o cu rvat e .

B Art 4 h n ( 1 3 of th ntre O f ur atur at . 6 t e coord ates e ce c v e y i 0 , [ 0 th e o nt as On a curve are ven b p i ( , y) gi y 72 DIFFE RE NTIAL CAL C UL US

= 160 y +

f x be e m nated betw een th e e u at on to th ur I , y li i q i e c ve and

' e u at on s 1 an d th e resu t w l be th e e u at on to th e o u t q i ( ) l il q i e v l e .

XAM PLE E S .

1 F nd h ut of th e . i t e evol e ellip se

3 22+ Z—

B Ex 3 Art . 46 y . , ,

2 2 b a 3

Th erefo re

a: tw o 2 2 ’ a a - b

6 313 30 ' 3 2 2 b a —b 2 2 £13 y Bu t + Th erefore th e e u at on to th e evo u te is 55 q i l —b2 i ) ,

h r r n . d r f o d t e. on u s n t e or na notat on o c o ates , i g i y i i , —b2) 3

2 2 in h 4 m. . F d t e evolute of th e p arabola y a

3 P v ve r a rve is a tan ent to the evolu . ro e tha t e ry no ma l of cu g te o f tha t curve. S nc e th e norm t th ur at th o nt cc asses th rou h i al o e c ve e p i ( , y) p g th e o nt n 3 on th e evo u te in ord er t o rove th a t th e p i ( o, 1 0) l ; p tan en t to th e evo u te a t n 6 is th e same a s th s n orm a w e g l ( o, 1 0) i l , h n r r Th u w ave o ly to p ove th at th eir direc ti ons a e id enti cal . s e h ave to prove th at C UR VA T URE 73

ow on d fferent at on and reduc t on N , i i i i ,

Th erefore

4 . Prove tha t th e evolute of th e tractrix

mz a (cos t + l og tan y = a sin t

th e ca enar a c 9 is t y y osh . a

5 Conca i a d n xit P in f inﬂ xion 0 . v ty n Co ve y . o t o e .

Let P be a en o nt on a curve and S a ven stra h t line giv p i , gi ig w h ch d o s n t n n P Th n th e curve is a d to be concave i e o co tai . e s i or convex at P w th res ect to S acc ord n as ever su ﬁi cientl i p , i g y y sm all arc c ont ain i ng P li es in one Of th e tw o ac ute or tw o obtu se angles form ed by S a nd th e tangent if th e cu rve is n eith er con c ave n or c onvex at P w th res ec t to S P is sa d to be a oin t o i p , i p f in ex n h r ﬁ icn o t e cu ve . 74 DIFFE RE NTIAL CAL C UL US

" Let a be th e coord nates of P . Th en th e c urve is conc ve or , y i a c onvex at P with respec t to th e ax i s of a' ac cording a s y ygl < 0 0 r > 0 ;

in eneral P 1s i n h ur if g a point of inﬂex on o t e c ve.

N . For r s th e a e a emen a s e a s for a c n th e OTE p oof of bov st t ts , w ll dis ussio of e h N h e n c ntac t s e t e e at t e end C a er X . qu stio of o , ot of h pt

EXAM PLE S .

2 1 P rove th at th e arabo a = 4 is r h r n h . p l y acc eve yw e e co c ave wit x f regard to th e a i s o as. 2 2 d _2a 4a d H r : d n u ntl h r y e e an c o se e ' T erefo e q y 3 y 2 dd y y dx 4 2 — h an z r t r n h r w hich is less t e o a eve y poi t on t e cu ve . ? Th erefore th e curve is everywh ere conc ave with regard to th e ax is f o at .

f nfl x n n 2 . Sh ow th at th e p oint s o i e io o 2 a oc Z/ 2 2 a + x are given by and 3 . l Here — and c on se uent ilZ q ly

2 ? 2 3 d cc (a + w )

Th erefore th e poin ts of inﬁex ion are given by

2 2 — m3a —zc = 0 ( ) , — : 0 3 . i . e. , at ,

" -x 3 = n inﬁexi n ere .v a: . Show tha t th e curve y a h as poi ts of o wh

4 Find th e point of inﬁexion on th e curve 2 ’ 3bx a y 0 .

5 . Pr e a in enera th e n r 0 is a nt inﬂexion if ov th t , g l , poi t ( , ) poi of , 2 - dr d r 7 2 2 r (d o) d 02

6 In th e curve prove that there is a point of inﬂex ion when 0 UR VA T URE 75

m s n Ch a Ex a ple o pter VI .

n th e r na e th e cen re rva re th r 1 . Fi d c oo di t s of t of cu tu of e ca tena y U = a c E— and a th e ra s Of c r a re e a in en t th e y osh , show th t diu u v tu is qu l l g h to CL

r h n r a n er e e e een th e r e and th e ax f po tion of t e o m l i t c pt d b tw cu v is O x .

me 2 . In th e e an r iral r z ae r e a th e c en re r a re qui gula sp , p ov th t t of cu v tu is th e point where th e p erpendicul ar to th e radius vector through th e pole r h r inte s ects t e no m al .

3 Pr h an e e een th e e and th e n r re . ove tha t t e dist c b tw pol ce t e of curvatu correspo n ding to any point on th e curve

r ” a " cos n o

mt 2 _ { a n - (n Urn 1 M a . r s [ th T ipo ,

4 . Prove tha t 2 1 3 J ] 2 d t d t2 d i d t2 1 at th e point determined by t on th e cu rve

a: ¢(t) y : W)

' 5 . be th e rad 11 r r th e ex rem es c n a e If p, p of c u vatu e at t iti of two o jug t ame ers an e e r e a di t of llips , p ov th t

2 t p t

6 . Prove th at th e radius of curvature at any point of th e curve

2 w a bt c t

1 ' ' 2 y = a + b t -i-c t }

3 ar e as sec c ere th e n th e n en at th e n makes w th e v i s p, wh c is a gl e ta g t poi t ith ' n c li e can y 0 .

7 . If a curve be given by th e equations — 2x : « V 2 — T E H t 2t, 2 = f - 2 — y « r t 2 t V t 2t,

ﬁnd th e ra Of c r a re in er dius u v tu t ms of t.

8 = . In th e curve whose in trinsic equa tion is s a log sec prove th at t n p o c a it . [A rela tion b etween th e are 3 and th e angle of contingence it in a curve is called an intrinsic equ a tion of th e c urve ] 76 DIFFE RE NTIAL CAL C UL US

n en TP U ra n th e c r e 9 . A ta g t is d w to u v “ ere )

n P and n er ec th e ax es a: and in T and U res ectivel . at any poi t , i t s ts of y p y 1 i h a a c n c a n Pr e a th e rad s r a re at P s t of th t of o i h vi g ov th t iu of cu v tu (n 1 >

n d n at P . th e ax es of c oordinates for p rincip al ax es a touchi g TU

h c r e r e a th e c r c r a ure r th e 10 . In t e u v p ov th t ho d of u v t th ough 2 N ) e i s n er cal e a pol um i ly qu l to ’ f (r)

= 1 1 . At th e n on th e Arch imedian ra r a 0 at th e a n en poi t Spi l , which t g t k a a r an e th e rad ec r th e rd Of r a re a n ma es h lf ight gl with ius v to , cho s cu v tu lo g d ar h r re e and perpen icul to t e adius v ector a both qua l to ga .

’ 12 and be th e ra d 11 c r re t c rres n n n s th e c r e . If p p of u vatu a o po di g poi t of u v and its e e r e a volut , p ov th t

13 h a n e an ar . Prove that t e evolute of an equiangular spiral is also qui gul spiral .

Th e e a e a n th e r = i a t be th e e P an c e i n . Le p d l qu tio of u v s p r s 0 pol , y

n on th e c r e and M th e c rre nd n cen re c r a re . Let r be poi t u v , O o spo i g t of u v tu p , P M th e r na e P and r th e c rd na e Of M . en s nce coo di t s of p , , l oo i t s O Th i ,

p l b Ex . 2 e h e e e a t M x _ M . But s t see E . 3 Art . 4 sin O P touch volut o ( , 9) , O y r ] M P 1 r n th e an e O s a a e. ere re gl O ight gl Th fo

P 1 em a 7 2 (2 ) 1 hence th e evolute of th e equiangular Spiral p = r sin a is th e equiangular Spiral

i a p r s n .

14 . Show that th e evolute of th e curve

2 2 a mp h as for its equ ation 2 2 r (1 m)a 2 mp

15 Pr h . ove that t e evolute of th e c ardioid r a (1 cos 6) is th e c ardioid r $0 cos

16 . nd th e a e th e e an cc Fi symptot s of volute of y t .

CHAPTER VII .

L E NV E OPE S .

Famil of Curves — Con s d er th e e u at on 5 1 . y i q i

a in a = a: c os + y s a .

Th re resents a stra h t ne for ever va u e of a . Th us vin is p ig li y l , gi g c s e all va u es to a w e obta n an nﬁn te nu mber o f su c e siv ly l , i i i straight lines wh ich are equidi stant from th e origi n : th ese stra h t nes ma th erefore be sa d to a fa mil an d ig li y i W y, th e qu antity ( 1 wh ose different valu es serve to distingu ish th e in divid u al members of th e family may be c alled th e para meter f h am o t e f ily .

In enera d enot n b th e s mbo F zc a an ex ress on g l , i g y y l ( , y, ) p i c ont a n n cc and a th e cu rves c orres on d n to th e e u at on i i g , y , , p i g q i

F x a = 0 m a be sa d to c onst tu te a fam of wh ch th e ( , y , ) y i i ily i

r m r i a p a a ete s . Definition of E l o h f th m f nve e. B t e l 5 2 . p y e nve op e o e fa ily o curves corresponding to F x a = 0 ( , y , ) , w h ere a is th e arameter of th e fam is und erstood th e c urve p ily , wh ch tou ch es ever member of th e fam an d w h ch at each i y ily, i , i t u h d b m m m r h m o nt s o c e so e e be Of t e fa . p i , y ily For an u strat on of th s d efin t on ee th e fi u re ven in th e ill i i i i , s g gi th rd of th e fo ow n ex am es i ll i g pl .

h e n h NOTE . T stude t is advised to make ﬁgures for t e other examples .

EXAM PLES .

1 r n h . W ite dow t e equation of th e family of straight lines parallel to y 2 i h . What s t e family of curves c orresponding to z 2 2 i a: at a ara e r a ( ) ( ) y , p m te ; ii 2 2 2 a: a a ara e er a. ( ) ( ) y , p m t ? iii = 4a x —a ara e er a ( ) y ( ) , p m t ; 2 2 W r a ( ) + 1 , a a e er 25 p m t E NVE L OP E S 79

2 2 2 h at th e c rcle 33 = a is th e envelo e of th e fam 3 . Prove t i + y p ily

nd n to a: c os a sin a = a a be n th e of straigh t lines c orrespo i g + y , i g m parameter of th e fa ily . Th e straigh t line 51: cos a + 3; sin = th c rcl e th e o nt of a a touch es e i , p i

n ct b n a c os a a sin Th u s c o ta ei g ( , th e c ircle tou ch es every member of th e

fam an d is touch ed at each o nt b ily , p i y h m Th ere ore one member of t e fa ily . f by d eﬁnition th e c ircle is th e en velop e m of th e fa ily .

2 2 4 Pr e a th e c r e = a th e en . ov th t u v y is velope of th e family of circles c orresponding cc a e n th e arame er 1 1 to ( b i g p t of FIG . .

th e family.

z z 5 Pr e a th e c r e —x 0 th e en e e th e am ara a . ov th t u v y is v lop of f ily of p bol s 2 — n n : 4a ze a a e n th e arame er th e a . correspo di g to 31 ( ) , b i g p t of f mily

2 2 2 6 Ex a n wh th e a c rc e c rre n n cc —a h a s . pl i y f mily of i l s o spo di g to ( ) y m r h no en e e a e n th e ara e e t e am . v lop , b i g p t of f ily

Rul e for ﬁndin th e Enve o e of a Famil of Strai ht 5 3 . g l p y g

Lines. The equa tion of the envelop e of the fa mily of straight lines corresp onding to F a: a E —u ( , y, ) y m

a bein he arameter o the a mil is btaine b eli i i ( 1 g t p f f y , o d y m na t ng between

F cc a = 0 and 17 8 a = 0 ( , y , ) £1 6 , y, ) .

= — F cc a o i . e. u m ( , y, ) , , y

= a 0 i . e. c , y, ) , , ja

We h ave a? y = f(a ) and c on sequently th e curve wh ose equ ation is obtained by eliminating a betw een (1) and (2) is th e same as th e curve — at f (a ) y = f(a ) 80 DIFFE RE NTIAL C AL C UL US

No w th e tangent at th e point d etermined by a on th is curve is th e st ra igh t line d

ae. Y

i . e. ,

Henc e th e fam of stra h t nes corres ond n to F cc a = 0 is ily ig li p i g ( , y, ) th e same as th e family of tan gen ts to th e curve wh ose equ ation is obtain ed by eliminating a betw een a = —{ = F a: a 0 an d P a 0 . ( , y, ) Q, y, )

Zia.

M EXA PLES .

1 F nd th e envelo e o f th e fam of tra h t l nes c orres ond . i p ily s ig i p

a in d = in g to n cos + y s a . Eliminating a between — n cos a + y sin a a = 0

— i a = .n S n CO a 0 + y S ,

h h i th r u r d n l w ic s e eq i e e ve ope .

2 n h e en e e th e e r ne rr . Fi d t v lop s of famili s of st aight li s co esponding to

in a : a m r a a: s a cos ara e e . (i) y , p t

ace b — 3/ = 2 z n a b arame er a . ( ) , p t c os a sin (1

= m 91 m r cc ara ete m. iii y , P m s = mx 2 — r am am arame e m. (iv) y , p t

a: cos 30 + sin 3 0 : a co r me er 0 (v) y ( s pa a t .

cos O—zv sin 0 a — a in 9 n ( ) y s ta ara e er 0. vi log (2 g) , p m t E NVEL OP E S 8 1

2 a 2

— — 2 3 Pr e 2 n 2 n = a -n i th . ov that m + y s e envelop e of th e family of straight " " ne c rre n n 90 cos 0 + sin 9 z a 9 e n th e ara e er li s o spo di g to y , b i g p m t of th e

family .

4 . F nd th e enve o e of th e stra ht ne th e rodu c t of h i l p ig li , p w o se 2 interc epts on th e axes of c oordin ates is a Here th e family correspond s to

2 a a a

a be n th ramet r Th erefore e m nat n a betw en 1 and e a e . e i g p , li i i g ( )

a: = a + b q

2 h 4 z Th r r th e re u r d en e o i th we ave a y a . e efo e q i e v l pe s e 2 t rectan ular h erbola, m z g yp y i .

5 r ne ﬁx e en a . Find th e envelop e of a st aight li of d l gth which moves with ri n e r it s ex tremities on two straight lines at ght a gl s to each othe .

6 n h e en e e ra nes at r an e th e r i . Fi d t v lop s of st ight li ight gl s to a d i vecto res of th e following curves drawn through their ex tremities

r = 0 (i) (ii) a + b cos .

G n ral Rul e. I l h ru l n i A 5 4 . e e n enera t e e ive n rt . 53 i g , g s true = even when F x a 0 re resen ts a curve o an kind . ( , y , ) p f y

A r r r h e a e a emen e n th e c e th NOTE . igo ous p oof of t bov st t t is b yo d s op of e

re en e . S c a r as We as a sc s n th e ex ce na c a es p s t volum u h p oof , ll di u sio of ptio l s , i i n in w ll be g ve Vol . II .

EXAMPLES .

1 F nd th n e of th e f am of c rc es c orres ond n . i e e velop ily i l p i g h r m t r f h am (x a being t e p a a e e o t e f ily . — 2 z — 2 F cc a E m a a . Th erefore ( , y , ) ( ) + g d ° a —2 —a F a E .v . ( , y, ) ( ) d a

Th erefore el m n at n 0. betw een , i i i g

a = nd E x a = o F a9 o a , y , , ( , y, ) % ( )

2 2 we h ave = a wh ch is th e e u at on of th e re u red enve o e . y , i q i q i l p G 82 DIFFE RENTIAL CAL C UL US

2 f A B O are func t on s of th e c oord n ate of . s a n I , i i poi t an d a a var ab e arameter sh ow th at th e enve o e of th e fami i l p , l p ly of 2 2 es corres ondin t o Aa 2Ba C= O is B = AC curv p g + + .

3 n th n th e a e r es c rre n n . Fi d e e velopes of f mili s of cu v o spo di g to 2 = 4c x a ara e er a . (i) g ( ) , p m t z 5233 y 11 1 ara e er a . ( ) + , p m t “2 “2 “2

2 2 2 O 0 O EU aJ i 111 1 arame er a . ( ) , p t

m 4 n th e en e e x a e ses th e su e axes is . . Fi d v lop of coa i l llip of whos a

5 n h en r e escr e on th e r i i . Fi d t e velopes of ci cl s d ib d ad vectores of th e following curves as diameters 2 x 2

2 4 (u) y a (a: a ) . " " O (iii) r a cos n .

Ex Ch amples on apter VII .

1 . nd th e en e e a r ne en th e rec an e n r h Fi v lop of ight li , wh t gl u de t e perpen i n i d culars from two give points s con stan t . $ 2 a2 2 . r m an n th e e se 1 er en i ar r F o y poi t of llip , p p d cul s a e drawn to th e a 2 b2 ax es and th e ee ese er en ars are ned : , f t of th p p dicul joi show that th e straight g h r line thus formed always touches t e cu ve Go 1 . 9 f i 3 n n h e ara a r cos : a a ra . At a y poi t of t p bol 5 st ight line is drawn m aking with th e t angent an angle equal to th e angle b etween th e t angent an d th e ordinate at th e point ; prove that th e envelope of th e straight line is th e 0 b ll r e r cos a cu v 5 .

4 n h en th e c rc es c a s r th e r n nd . Fi d t e velope of i l whi h p s th ough o igi a z — 2 = 2 h ave their centres on th e equil ateral hyp erbola atz y a . “ : 5 . A r e m es en re on th e ara a 4ax and a a ci cl ov with its c t p bol y , lw ys p a sses through th e vertex of th e p arabola : show that th e envelope of th e 2 " c ircle is th e cissoid y (a: 2a ) as O.

6 . Prove th at th e ellipses s z z i bzx a y a zb and z 2 4 z 2 4 2 2 a a: sec at b y cosec a (a b )? are so related tha t th e envelop e of th e second for diff eren t values of a is th e e e th e ﬁr volut of st . E NVE L OP E S 83

7 n th e e e th e e se . Fi d volut of llip a: = a cos t y b sin b}

r n h e c r e a th e n e n consid e i g t e volute of a u v s e v lop e of its orm als .

8 n th e e a e a n th e en e e th e ami c rre . Fi d p d l qu tio of v lop of f ly of o sponding to c 9 in m9 = a co o n os m + y s s n ,

0 n h ar m m bei g t e p a eter of th e fa ily.

9 . ai / a be th e e a n a an en a c r e a If d( ) qu tio of t g t to u v , show th t th e equation to th e c orresponding n orm al is MN“)

10 Pr e a h r i c r a re th e en e e th e ne use 1 . ov th t t e ad us of u v tu of v lop of li By , en a are nc ns th e arame er t n mer cal wh , B fu tio of p t , is u i ly Z d a d B d t d t2 ”6 HAPTER C VIII .

URV E T R CING . RO ERT IE OF E C I L C R E C A P P S SP A U V S .

Introductor . Th e aim of th e resent h t r i h 5 5 . y p c ap e s to s ow h ow th e resu lts obtain ed in th e last four c h apters c an be u sed fo r trac n cu rves and to enu m erate th e m ort ant ro ert es of th e i g , i p p p i - r r h a f r r best kn ow n cu ves fo t e s ke o efe enc e . With th ese end s in ew w e ro ose t o l a d ow n ru les or tracin curves rom h i vi , p p y f g f t e r cartesia n a nd ola r e ua tions a nd to enu m erate th e r n c a p q , p i ip l ro ert es of the three conic sections the semi-cu bica l a ra bola the p p i , p , cissoid o Diocles the olium o D esca rtes th e lemnisca te o f , f f , f B ernou lli the ca rdioid the conch oid o Nicomed es the c cloid th , , f , y , e d h b -k r tena r the tra ctrix a n t e est nown s i a ls . ca y, , p It sh ou d h ow ever be u nd erst ood th at th e su b ect of mu t e l , , j l ipl o nts d oes not c ome w th n th e sco e of th e resent vo u me an d p i i i p p l , th at in c onse u enc e th e ru es w h ch w be ven be ow su fﬁc e , q , l i ill gi l , in en era for trac n on su ch c u rves a s do n o t ossess mu t l e g l , i g ly p l ip points .

i nn e i N . We c n er e ar e n s c a er r s th e OTE o s d it u c ss y to giv , thi h pt , p oof of — r er es be en era e in Ar . 5 8 62 as th e en be p op ti which will um t d ts , stud t will M n r r readily able to supply th em . a y of th ese p op e ties will be found to h ave in e r r In th r r h b een al ready proved a lie p ages . e case of eve y p op erty which t e

en ke ﬁnd ﬁ l r th e r er re erence be en . stud t is li ly to dif cu t of p oof , p op f will giv

s f r Car esian E uations 5 6 ; Rule o t q .

h s r 1 otice whet er there i an s mmet in the curve. . N y y y For ex am e th e c u rve w be s mm etr c a w th res ec t to th e pl , ill y i l i p — r — x r n r f r r w ax is o y a is ac c o di g as only even pow e s o y o 9: o c cu .

Find he re i n r r io s i whic h the curve is non 2 . t g o o eg n n

ex istent.

For ex am e f is ma nar wh en a: es betw een a and b th e pl , i y i gi y li , c urve is non-exi stent in th e region bound ed by th e lines x = a and = w b.

86 -DIFFERE NTIAL C AL C UL US

r h — x I r h (v) Th e cu rve d oes n ot c oss t e y a is . t c osses t e — = x at 2 O th e tan ent th ere be n cc 2 . x a is ( , ) , g i g vi Ev d ent th e curve h a s ( ) i ly, n o o nt of inﬁ x ion p i s e . (vii) It is no t n ecessary to ﬁnd any o th er p articul ar points h on t e curve .

Th erefore it follow s from th e foregoing statem ents th at th e form of th e curve is th at sh ow n

i n th ad n d ﬁ ur e j oi e g e .

2 . Trace th e cu rve 2 — 3 y (cc 2a) + w = 0 d f D [Cissoi o iocles]. Th e following statements h old tru e in this c ase

Fm 15 . Th ere is s mmetr th . (i) y y wi res t t th — x ec o v a . p e . i s - (11) Th e c u rve is n on ex istent on th e righ t sid e of th e line = h d f th e — x x 2a as w ell as on t e left si e o y a i s . n r l m tot iz = 2 Th ere is on o e ea as e v . w a . (iii) ly y p , , i Th e cu r e asses th rou h th e or in its tan ent th ere be n ( v) v p g ig , g i g = y O. n r an of h ax (v) Th e curve does ot c oss y t e es . (vi) Th e c urve h as no point of ﬁ x i n in e o .

vii Th e o nts a t a lie on ( ) p i ( , )

th e c urve .

Th erefore it follow s from th e foregoing sta tem ents th at th e form of th e curve is th at sh own in th e

d n ed ﬁ ur a j oi g e .

3 . Trac e th e cu rve

2 ? = 2 y(a a ) a x .

— Th e fo ow n statements ho d Fro . 1 m 6. 0 Iss0 or DIOCLEs . ll i g l tru e in th is c a se z i ( ) Th e equ ation remains u naltered by ch anging th e signs of a: and th erefore th ere is s mm tr in O u dr nt y y e y pposite q a a s . C E T A CI P OPE IE S OF P E IAL C VE S UR V R NG , R R T S C UR 87

_ bein a w a s o s t ve th e curve is non ex stent in th e (ii) z g l y p i i , i

n d urth uad r nt seco d an fo q a s . h r i n n r l m iz = T e e s o o e ea as tote v . (iii) ly y p , , y iv Th e cu rve asses th rou h th e or in it s tan ent th ere ( ) p g ig , g b n = ei g y cc . (v) Th e curve d oes not cross any of th e axes at any po int 7 r n o th er th an th e o igi . — vi Th e o nts of inﬁexion are a and a ~ 3 ( ) p i ( v / ,

ii Th n h r (v ) e p lies o t e cu ve .

Th erefore it follow s from th e foregoing statements th at th e form o f th cur 1 h h n in h d n d u r e ve s t at s ow t e a j oi e ﬁg e .

Trace th e following curves 4 2 ? 3 s . 90 g 90 a . z 2 — 2 5 . a g a:(a cc ) . — — - 6 y = (w l )(x 2)(w 3) 7 “? z . can y ) a (n 8 l 2 2 2 2 . oi y a (ac y ) . a: a 2 _ 2 9 _ ' . y :c ‘ ac — a 10 4 = 2 2 2 . x a x y ) .

FIG . 1 7 .

Rul es for Pol ar E uations. 5 7 . q

T r A r l r he u les of rt . 56 a e a lso app lica ble to the case of p o a e ua tions certain oin ts bein ke t i view q , p g p n

For ex am e th ere is s mm etr w th res ect to th e n t a ne pl , y y i p i i i l li 6 if th e equ ation rem ains u n altered by c h angi ng th e sign of . A a n r bein essen tia ll ositive th e cu r e is n on-ex stent in th e g i , g y p , v i re on bou nd ed b th e l nes 6= a f r is ne at ve or u nrea gi y i , i g i l

h en 6 n a n d 8 w lies betw ee a ) .

EXAM PLES .

1 . Tr h = 1 6 rd o d ac e t e curve r a ( + c os ) [Ca i i ].

Th e followin g statements h old true in th is c ase

Th r i h n n (i) e e s symmetry w ith respect to t e i itial li e . h - = (ii) T e cu rve is n on ex istent ou tsid e th e c ircle r 2a . Th re i n o (iii) e s asymptote . 88 DIFFERE NTIAL CAL C UL US

i h u r a ses th rou h th e or n its tan ent th ere ( v) T e c ve p s g igi , g h n n being t e i itial li e .

h th e n t al ne at 2a th e tan ent (v) T e curve crosses i i i li ( , g th ere being perp endicul ar to th e l l n initi a i e . (vi) Th e cu rve h as no poin t of in i n ﬂex o .

(vn) Th e poin ts

and lie on th e cu rve .

Th erefore it follow s fro m th e foregoing statements th at th e fo rm

Fm — A D D f h h h in th e 18 . C t i t n . R IOI . o e curve s t a s ow d adj oine ﬁgure .

= i 2 . Trace th e cu r e r a sec B b a be n ess th an b Con v , i g l [ h id f i m d c o o N co e es].

Th e follow ing statemen ts h old true in th is case

r t h n (i) Th ere is symmetry w ith espec t o t e i itial line .

‘ 1 11 Denot n c os b a th e curve is n on-ex stent in th e ( ) i g % y , i 7’ — regi on bounded by th e lines 6 a a s w ell as in th e region

3 " bou nd ed b 6 = vr a L 1 y + , . 5

Th ere is on one as m t t iz 6 = o e . r co a . (iii) ly y p , v , s

iv Th e c u rve asses th rou h th e or n its tan ents th ere ( ) p g igi , g

be n 6 = 2 7r —a i g .

v Th e cu rve crosses th e n t a ne at a + b O and b— a ( ) i i i l li ( , ) ( , h t e tangent at each of th ese points being perpendicu lar to th e . n t a n i i i l li e . i (v ) Th e cu rve h as tw o points o f inflex ion given by

3 sec ( 1 sec 6— c 6 3 se O. 2

vn Th e o nts 2b a and li n h r ( ) p i ( , ) e o t e cu ve .

90 DIFFE RE NTIAL CAL C UL US

o e th e w-i-ax s is th e n t a l ne and e th e ed al p l , i i i i l i , ; p equ ation of = 1 1s b2 2 1 1 1 r

° ? 2 ? 2 2 p 01 6 “6 2 ac y (m is th e loc u s o f a p oint wh ose distance from a - a e O is e u a to e t mes its er end cu ar d stance from _ , ) q l i p p i l i _a 0 ; th e o nts i a e O are th e 2 p i ( , ) foci of th e ellip se and th e straigh t

o a 0 o l1nes (vi are th e d i rectri ces e

(y) A ch aracteristic prop erty of 2 is th e con stancy of th e su m

F 2 — h ro . 1 f d n f . E IP t e ta r m LL . o s ces o x f o a e 0 SE i ( , y) ( , ) and — e a ,

s — 2 ( ) Th e evolute of IS (a by

S e Ex 1 Ar e . t . ( ,

. H a h III yperbola . ( ) T e simplest c artesian eq u ation is 2 m. . y— a “V 1 ; th e s m l est ol ar e u at on is r - where é a b i p p q i 0 0 8 9 :

x/ Q-- z a b O is . th e o e th e ( l , ) p l , w—ax s is th e n t al ne and i i i i li , 2 2 N/a + b th e p edal equ a

2 2 613 y

1 — 1 1 + ' 2 2 2 9 2 29 61 b a b — 2 2 H YP RB A . . E OL i (K3) 2: g= 1 is th e l ocu s

a o nt w h os d stance i a e O is e u al to e t m es its p i e i ( , ) q i C E ACI P OP E IE S OF SP E CIAL C E S 1 UR V TR NG , R R T UR V 9

er end cu ar d stance rom cc + = 0 z th e o nts ae 0 are p p i l i ‘ f g p i ( i , )

th e foci of th e hyp erbol a and th e straigh t lines n + g= o are th e

directrices . 2 ? cc y h r r t c ro er of _ 1 is th e con stanc of (y) A c a acte is i p p ty 2 2 y a b — th e d fference of th e d stanc es of w from d e O an d ae O . i i ( , y) ( , ) , ) 2 2 3 th e (a) Th e evolute of is (aw) Of .

? f at y u t1 n f h eq a o o t e evolute of 1 .

m - l iss id F lium 5 9 . Se i cubical Parabo a . C o . o .

- a m rt s an u a . i P r Th e est c a e e I Sem cu bica l a a bola . ( ) si pl i q tion is ay th e simplest p olar 2 a sin 6 wh ere O 0 IS 3 , ( , ) c os 6 — th e pole and th e cc ax is is th e initial n li e .

Th e semi -cu bic al p arabola is th e evolut e of th e c ommon parabola

? l 6a y 27 — - FIG . 23 . EM1 c UBI CAL PARAB A S OL .

. Ci a h II ssoid . ( ) T e simplest c artesian equ ation is

x O

2a — x

2a si 6 th e 8 1m 1est o ar e u atl on 18 r w h ere O O is th e p p l q ( , ) c os 6

o e and th — x i n p l e cc a is s th e initia l li e . 2 3 Th 1 id i h f h arabo a —8 (1 ) e c sso s t e p ed al o t e p l y ax . F r h in Ar o t n . r i 1 6 e t 56. e fo m of the cu r e ee F . v v , s g gi

3 3 . F h m rtes an e u a t on is w = 3a cc III olium. T e si plest c a i q i + y y 3a sin 6 c o s 6

th e ole an d th — h n t n p e w ax is is t e i i i al li e . 92 DIFFE REN TIAL CAL C UL US

Th e em -c ca ara o a is a NOTE . s i ubi l p b l lso ’ c alled Neil s p arabola a fter William Neil (1 637 who foun d th e length of any

r c r e in 1 65 . D c e th e a c of this u v 7 io l s , d erer th e d was a Greek iscov of cissoi , math ematicia n who ﬂourish ed some time b etween 250 and 100 Th e c urve

‘ x 3 + y3 = 3a xy is called Folium of Desca rte s a fter Ré né D esc artes (1596 th e French a e a an and i er was one — m th m tici ph losoph , who FIG . 24 . IUM or D CAR FOL ES TES . of th e ﬁrst to discuss its form .

mnis t ar 6 0 e ca e. C dioi C n h id . L d . o c o .

L a h n i . T m l r n t o I emnisca te. ( ) e si p est ca tesia equ a i s — z v ) 5

2 2 th e s m le t o ar e uat on is r = a cos 29 wh ere 0 O is th e i p s p l q i , ( , ) — pole and th e x axis is th e initial line ; th e ped al equ ation is 3 = 2 r a p .

(S) Th e lemni sc ate is th e pedal of th e rec tangular h yperbola 2 2 6 = Art 40 2 Ex 2 . r a See . c os . ( , , n being ( y ) A ch arac teristic property of th e l emn isc ate is th e c on stan c y o f

th roduc t of th e d stan ces of .v e p i ( , y)

from " 2”

— d a h m est A . Ca rdioi . T e FIG. 25 . L M I C or B R I E N S TE E NOULL . II ( ) Si pl c artesian equ ation is

2 2 (w + y th e s m est o ar e u at on is r = a 1 c os wh ere O 0 is th e i pl p l q i ( + ( , ) - pole an d th e x ax is is th e initi al line ; th e p ed al equ ation is

Th e c ard o d is th e ed al of th e c rc e r = 2a co s 0 an d th e (B) i i p i l ,

n er f th r = 4 b in v se o e a abo a r Ex 2 Art . n See . 0 e i p l ( , , g 1 + c os 0

Th e evo u te of th e card o d is th e c ard o d r = 9 1 -c os H ( y ) l i i i i ( ) , 3

94 DIFFERE NTIAL CAL C UL US

ﬂ ' —1- f h c oord nate l n th ese e u at o ns be n a 2a . th e origin o t e i s q i i g , ) - (1 54 5 51 3 are fou nd to be a H Sin t a nd Note th at b Art . 46 [ , y , 3 [ 0 ( ) — a (1 c os t) .

= a Th e s m lest e u at on is a c osh . Ca tenar . II . y ( ) i p q i y g

h u n orm n d r t Th e c atenary is th e cu rve in wh ic a if a p e fec ly ﬁx d n t d r o o s . ﬂex ible strin g h angs wh en su spend e f om tw e p i (See ’ ’ l 1 Minchin s Stat cs V o . I . Art . 84 or i , , ’ ’ l I A u th s Stat c s Vo . . rt . Ro i , ,

(y ) A ch aracteristic property of th e c at en ary is th e equ ality of th e ra diu s of u r a tu re an d th e ort on of th e norm a c v , p i l intercepted betw een th e curve an d th e zc — ax s th e c entre of curvatu re and th e i , — cc ax is being on opposite sid es of th e — 4 n Ex d . x 1 Art . a 5 R . r E . 2 CA NA Y u . See 7 IG 7 . c ve F . TE ( , , , Art .

(P) A sec ond c h aracteri stic property of th e c atenary is th e c onstanc y of th e length of th e p erpendi cu lar from th e foot of th e n n th e tan n ordi ate o ge t .

a h Tra ctrix . T r m r r r n t III . ( ) e simplest p a a et ic ep ese ta ion is given by th e equ ations

a: OS t l t n = in + og a y a s t.

Th e tractrix is th e curve in w h ich th e portion o f th e ta ngent — interc epted betw een th e cu rve and th e zc ax i s is o f c o nstan t

= ‘ 1 -= length a . ¢ tan gZ t an d h ence sin gt (7 ) A ch arac teristic property of th e trac trix 1s th at th e radius of cu rvatu re varies inversely as th e portion of th e no rm al intercepted betw een th e cu rve an d th e x —ax s i , th e centre of curvature and th e — x x ax i s being on opposite sid es of h r Ded u ce from 6 and — t e cu ve . [ ( ) 2 . RAC R X 8 I . FIG. T T 5 h t n r an d th e property ( ) of t e c a e a y . ]

= t n ar a co sh . See ( 5) Th e evolu te of th e tra ctrix is th e c a e y y 2 (

4 Art Ex . , . VE T ACI P OP E IE S OF SP E CIAL C VE S 9 5 C UR R NG, R R T UR

1 Th e n n a r NOTE . stude t will have o difﬁculty in deducing ( ) f om (B) if h e WW n e a in . 26 th e arc NP a t th e rc c r e e a ON nce ot s th t , Fig , of i cl is qu l to , si r n th e c ircle olls without slippi g .

2 Th e c c d a th e a r r r ek r ean n NOTE . y loi is c lled t utoch one (f om a G e wo d m i g same time) b ecaus e of th e following prOperty : If a cycloid be pl ac ed with ba e r z n a and nca rned ard a ar e a n n its s ho i o t l its co vity tu upw s , p ticl f lli g dow th e r e reac th e e n in th e a e i e a e er be ar n cu v will h low st poi t s m t m , wh t v its st ti g

n . poi t J Th e cycloid is called th e b ra chist ochrone (from a Greek word meaning shortest time) b ec aus e it is down a cycloid that a particle must fall in ord er that it may travel from one given point to anoth er in th e shortest time . Th ese prop erties were discovered resp ectively by th e Dutch m athema tici an Huygh ens (1629- 95) an d J ohann Bernoulli (1667

Fo r r e e r er e see me b k on na c e . . a and p oofs of th s p op ti s so oo dy mi s , g , T it ’ ’ ee e s D nam cs a Par c e C a er S . t l y i of ti l , h pt VI

m 6 2 Lo arith ic S iral . Archim i ir l Sine S ira . . g p ed an Sp a . p l

L ri hmi ra h o t c S i l . T m n i I . ga p e si plest equ atio s

h d t n i = in a t e pe al equ a io s p r s . (B) Th e l ogarith mic Spiral is called th e equia ngu lar sp ira l be c au se oi th e e u al t of th e an e q i y gl , betw een th e tan gent an d th e radiu s

h n t n a vector to t e c o s ant a e . , gl (y ) Th e l ogarith mic Spiral h as — LOGAEIT HM I . 2 . I P RA FIG. 9 O S L for its ed a nverse o ar rec p l , i , p l i proc al and evo lu te logarith mic E E m les i F t x . 1 3 x a ra wh ch are e u a to t . or evo u e see Spi ls i q l ( l , , p I on Chap ter V . )

a h m u t on is r = a 0 Ar i e i S ir l . T e s est e a II . ch m d an p a ( ) i pl q i

2 th e ed a e u at on is p l q i p 2 2 r + a (S) Th e Archimedian spiral is th e path d escribed by a point wh ich moves fro m th e pole w ith u n ifo rm velocity along a stra igh t line wh ile th e straigh t line rotates w ith u n iform veloc ity a bou t th e o e th e rat o of th e veloc t es p l , i i i b n e a . i g — ARCHIMEDIAN SP IRA . FIG . 30 . L (y ) A ch aracteri stic property of th e

bnorm a . See Arch imedian spiral is th e c onstan cy of its polar su l ( E 4 r x . A , t . 9 6 DIFFE RENTIAL CAL C UL US

n n a h m = S ira l . T t . Sine e s es e u at on is r a cos nB III p ( ) i pl q i , th e ped al equ ation is A c h arac teri stic property of th e sine spiral is th e equ ality

of th e an e betw een th e tan en t and th e rad u s vec tor to + n0. gl , g i , g

For th e va ues —1 1 —2 2 and of n th e (y) l , + , , + , + 5 , s ne s ra is res ec t ve a stra h t ne a c rc e a rectan u ar i pi l p i ly ig li , i l , g l l mn ar a and ard h erbo a a e scate a abo a c o d . yp l , i , p l , i i

h 9 - NOTE . T e spiral r a is calle d a fter its discoverer Archim edes (287 2 12 th e a reek e a n r Th e f mous G m a th m ticia of Sy acuse. discovery of this spiral is b r wrongly ascribed y some writ e s to Conon .

Ex m n Ch a er a pl es o pt VIII .

2 a In h - 1 . t e cissoid x

2 a y t(1 t?) r e a th e n a are c near a + and a th e n p ov th t poi ts , B , y olli if B th t poi ts a ‘ 6 are c n c c a B+ + , B , y, o cy li if 7

2 2 Pr e a th e ar rec r ca th e c re ec . ov th t pol ip o l of issoid y with sp t to

- 0 0 is a e c ca ara a . ( , ) s mi ubi l p bol 3 z i 3 Pr e a in th e c x z - i— th e n en i e h . ov th t issoid subta g t s qual to t e é z E/ portion of th e tangent intercepted between th e ax es.

4 In th e m . foliu

y “ 1 + ﬁ

r e a th e n a are c near a B 1 0 . p ov th t poi ts , B , 7 olli if y 3 3 = ' 5 . r a n P on th e r e x + 3ax an en s P P dis F om poi t cu v y y two t g t Q, Q ' n r h e n en r dr n r ti ct f om t ta g t at P a e aw to th e cu ve . Show that QQtouches ” n r 4 th e recta gular hype bola x y 9a . ’ Let th e ara e er P and be res ec e a a and a . en p m t s of , Q Q p tiv ly , , 2 Th it r m Ex 4 follows f o . th at ““ 2 u 1 s

' Now th e equation to QQis

‘ — 2 c s m ﬁe x a 3a oz 0 . whi h i pli s to y , , ’ “ ere re th e en e e is 4x z Th fo v lop of QQ y 9a .

DIFFE RENTIAL CAL C UL US

16 . In th e c a enar : a r e a t y y cosh 2, p ov th t

2 2 ? y s a up and s z a tan i ere s aken be z er a t th e n 0 a and d en e th e t, wh is t to o poi t ( , ) t ot s an e d e n h x x gl m a by th e tange t with t e a is .

1 . 7 . P rove th at th e sum of t h e curvatures a t th e points of contact of per endic ul a r n p ta gen ts to a c a tenary is con stant . x + a 1 8 = ﬂ i . Prove th a t th e ca tenary y a cosh is th e c austic of re ex on of a th e = if — Th e logarithmic curve x a log for rays perpendicular to th e x ax is . [ a

a c ustic of reﬂex ion is th e envelop e of th e reﬂ ected rays . ) 19 . In th e tra ctrix x = a ( c os t + l og tan y = a sin t CL — r e a s a 3 be n aken be z er at th e . p ov th t log , i g t to o cusp

20 . P r e a in th e c r e e n r n e a n s a e a ov th t u v , whos i t i sic qu tio is log cos c 4, th e r d th e rad r a re and th e n r a is n an th e n r a p o uct of ius of cu v tu o m l co st t , o m l n bei g terminat ed by th e asymptot e to th e curve . i Ma . r [ th T pos ,

“1 2 1 . In th e l ar ra r x 6 sin r e a th e r n th e po t ct i , p ov th t po tio of r 2

an en n er e ed be een th e c r e and th e ra n e dra n r th e t g t , i t c pt tw u v st ight li w th ough

e er end l ar th e rad e r n an en a . pol p p icu to ius v cto , is of co st t l gth

22 E . stablish th e following construction for a logarithmic spiral h avin g it s e at 0 — n M N on th e ra be n kn wn n r on ON th e pol Two poi ts , spi l i g o , co st uct triangl e ONP simil ar to th e triangl e OMN ; th en P is a third point on th e

tc . ra a rl a r n a nd en a ﬁ n e c an be d e er ned . spi l ; simil y fou th poi t th fth poi t , , t mi

23 n h r = a 0 0 " r . I t e loga ithmic spiral r a e p ove th at 2 2 2 2 2 s = ta n a 3 r + S a nd = r + S p , , p

ere s aken be z er a t th e e a nd S and S d en e re e e wh is t to o pol , n ot sp ctiv ly th e b a n en a n d th n r su t g t e sub o m al .

24 . P rove th a t th e ped al of th e involute of a circle

‘ a 6 _ + sm r a n Arch im edian r is spi al .

25 . Prove tha t th e sine spiral i ? " ( ) r cos n6 s is th e locus of a poi nt th e continued prod uct O f whos e distances from th e - er e th e re l ar n a n a n its en re a t th e e c n a n . v tic s of gu go , h vi g c t pol , is o st t CHAPTER IX .

U CC E IV E DIFFERE NT I T IO N S SS A .

‘ Definitions. Th e second di ferentia l coe icien f 6 3 . f fi t o f(x ) mea ns fi ) th e d fferen t a c oeffic ent o f an d is re resented b th e s mbo i i l i g, p y y l 2 d f(a;) ? div

In like mann er th e third difierentia l coefiicient o f f (x ) m ean s th e d (x ) difieren tial co effic ient of j L— an d i s re re sen ted b £ i ? p y 3 eta} d x

so on .

Th u s f th e su c c ess ve d fferen t a c oefﬁc ent s of r , i i i i l i y a e 2 dg/ d y d ay d ny th ese are a so d enoted b , ” 3 ’ l y g l , y 2 , 35 d oc d x

d n N . Th e a e for a; a en era e e b th e b = OTE v lu of yn is g lly ot d y sym ol (yn)x a n ‘ d ) d th a t r or br eﬂ . d be a n e e en en e b , y , g It shoul lso ot is f qu tly d ot d y i ( n) a f dg

th e symbol its va lue for a: a being d enoted by

EX AM PLES .

3 1 Th e first sec on d an d th rd d fferen t a c oeffic ent s of x are . , , i i i l i 2 res ec t ve 333 6x an d 6 all th e h h er d fferent a c oeffic en ts p i ly , , ; ig i i l i r are equ al t o z e o .

f lo c o s ﬁn d . 2 . I y g ( y g — 2 H — n Th erefore sec at} an d t a as. ere y , y 2 , 2 sin 51:

° 3 c o s 33 2 t — 37 — f = a in w b c os x rove th a O. 3 . I y s + , p 100 DIFFE RE NTIAL CAL C UL US

r = — in Th r r He e y 1 a c o s x b s x . e efo e

— in — — yz a s az b c os a: y .

4 nd th e e n ff eren a c e ﬁc en ea h . Fi s co d di ti l o f i t of ch of t e following function s

4 2 11 tan w see so. (i) x . ( ) “ 1 2 w 2x ( i n ) sm (x 1v) tan tan a: 1 —902 6 5 = x 3 as r e a If y log , p ov th t y,

‘ 1 6 If y : sin (m sin prov e th a t d z d y e ’ 1 — - x m ( y. d wz 33;

” “ ” 7 . a e be cc r e a If y , p ov th t 1 — = 1 6 0 . y , 6y2 + g , y

8 r e a . If y p ov th t

1 (23 3 ° ( )s 9 2

Standard Resu ts. 6 4 . l

f th en r -mx = m m and in enera I . I y l y 2 ( , g l , dn — 1)(m—2) (m dx "

x x x 2 f z a th en = a l o a = a lo a and in enera II . I y , y 1 g , y2 ( g ) , , g l , d” (l og a)

= n th = w= in . f s x en cos s III I y i , y 1

a nd in enera , g l , d " sm x + d x "

Proc eed n as in th e c ase of in 56 w e ﬁnd th at i g s , d ” (cos x ) _cos dx "

= ” V . f a sin bx th en I y , m “ = a e sin bx 193 c o s in bx b c o s bx y 1 + s + ) .

102 D SFFF ERE N TIAL CAL C UL US

‘ ' b ihd ‘ltlieznth diff erentia l coefﬁ ci en t of ea ch of th e following functions

6 (i) 512. (ii) log (a x b) .

3 ” 3 in . (iii) 0 0 3 x . (iv) e s bx i 2 in 3x sin 4 (v) s n x s x . ” 2 (vi) e sin x sin 2 x .

n 5 . z e sin bx r e a If y , p ov th t d M - e 2 2 2 z: O z d x

6 . sin mx ni x r e a If y cos , p ov th t n 5 = m 1 sin 2mx . yn { }

’ r m t e n 6 5 . Leibnitz s Theo e . a nd b i both fu nc tions o x f g g f , a nd b in a n ositive in te er n e g y p g ,

—l ’ H ‘ n an d” f dn f ( d f a a _ Le n, f d. C ' + C’ + + l “ 1 " n dx n dx dx " f dx ’ dx

’ For l et ) an d a ssume t h at Le bn tz s th eorem h o d s for , n q i i l an art c u ar va u e o f n sa ﬁ t so th a t y p i l l , y , m m = ' ' ’ + o — + + 0 ' — o l + ' ym fm¢ i f m l ¢l 7 fm rg r f qm

Th en d fferen t at n both s d es of th e above e u at on w e h ave , i i i g i q i ,

m m 0 0 — + + ( r 1 + 7 )fm+ 1 r¢r + f¢ m + l C — b + r fm + l r§ i + + f¢m + l y m “H ’ d s nc e . Th erefo re f Le bn tz s th eo rem h o s i Q , i i i l for n = it h d s a so f r = 1 m ol l o n 7n + . Bu t by ac tu al d ifferen ti a tion

’ H enc e Leibn i tz s th eorem h old s for n = 2 a nd th erefore it h o ld s also for n = 3 th erefore also for n = 4 ; an d so on ; an d th erefo re it is enera tru e g lly .

EXAM PLES .

Q ‘“ 1 F n h n i . i d t e nth differe t a l c oefﬁcien t o f x e 2 m Here l t b n N w = 2x e a d stan d for x and c re ec t ve . o g f sp i ly ¢1 , ( 2 2 n d l z Th re ore b a t l r . e 0 e c . are a e o A so f P2 , $ 3 , , l y ’ Leibnitz s th eorem w e h ave

2 ax n ‘” 2 " “ 2 n ‘ 2 x e a e x x 2x + O a x x 2 n ( ) g jx n - 2 “ ﬂ 2 Li 6 { a x 2anx n (n S UC CE S SI VE DIFFE RE NTIA TION 103

2 n th e h d ff r n ﬁ n . Fi d nt i e e tial coef cie t of e ach of th e following functions 3 “ 2 2 in e —2 (i) x s ax . (ii) { a x “na x n (u 1 ) a bx (iii) (iv) ( ) log (a bx ) . “ 1 1 n £ ) 3 . z x x r e a If y log , p ov th t y n 27 4 P . rove that x 17 £ 62; ) { P sin (x Qcos (x + 2) where P and Qstand for

‘ 2 x " n (n 1)x " n (u 1)(n 2)(n

n ' 1 ' 3 n x n (u 1) (n 2)x "

respectively . l n az ” —f 5 . Prove that l (c y) e a g . dril ( )

rmulae. 6 6 . Use of Partial Fractions. Recurrence Fo

U In h s f r t n I o P . . se f a rtia l Frac tions t e c a e o a f ac io al ex pression w h ose n u merator an d d en omin ator are both ration al n te ra a ebra c ex ress on s th e nth d fferent a c oeffic ent m a i g l lg i p i , i i l i y be obtained in a c onvenien t form by first resolving th e fra c tion in to p artia l fra ctions and if th e d enominato r c ont ain s som e im aginary near fac tors th e d fferen t a c oeffic ent ma be red u ced to rea l li , i i l i y ’

orm b a id o Demoivre s theorem as in Ex . 4 . f y f ,

II. R e ur F l Som et m es th e va u e of th e nth c rence ormu x . i l d ifferential co efﬁc ient for a p articu lar valu e of th e variable is most

eas obta ned b ﬁrst obta n n a recu rrence form u a a s in Ex . 6 . ily i y i i g l ,

h n e e a ar N rn. in r e A ebra th e d en a s ot b l o If , his cou s of lg , stu t com f mi i with th e resolution of a ra tional fra ction into parti al fra ction s;h e is specia lly advised to follow care fully th e v arious steps in th e solutio n s of Ex ampl es 1 n 2 n n are n en d ed ake ear th e ra a l a d give below . Th es e solutio s i t to m cl p ctic

points that are nec essary for his p resent purpos es . be n t ed at th e ar a ra n rre n d n a a r It should o th p ti l f ctio s , co spo i g to f cto r x —a in th e en n r h e ri na rac n are a a a m ed be ( ) d omi ato of t o gi l f tio , lw ys ssu to A A A A r 2 , For a u n th e b e a f ll discussio of su j ct ’ — ” - —l — a - a (x a ) (cc a y (ac ) (iv )

ar a ra n see s e d b k on A e ra or Ch . . . I . th e of p ti l f ctio s , om goo oo lg b , III , Vol , of ’ ’ author s book on Integral Calculus .

EX AM PLES .

1 nd h h d ff r nt fﬁc en t of . Fi t e nt i e e ial coe i 104 DIFFE RE NTIAL CAL C UL US

A ssume tha t 2 a x + bx + c A B G — — — — — ’ (x c )(x y) x a x ﬁ x y

w h r A B an d O are c onstants w h o se a u es are t o b d t rm n e e v e e e . , l i ed On c ear n of fra ct on s l i g i , 2 E — — a x + bx + C A(33 18)(5U E(£U 7 ) C (x a )(x

Th erefore u tt n x = c in th e above d ent t w e h a , p i g i i y , ve 2 2 a a + ba + c a + b + c ﬁ B , Sl ml larl B = an d C — — y — — — — ( a exa y) (e u xe y) (y a xy e)

Th erefore th e requ ired differenti al coefﬁci en t is 1a B (1: . — — (33 0 ) (93 (93 7 )

2 a a ba + 0 1 —1 "n ! — — — (cc exa y) (as a )

2 2 a B + bB+ c 1 a y + by + c 1 — — — — — (U a Xﬁ r) (93 13) (r a )(r (513 7 )

F nd th e nth d fferent a c oefﬁc en t o f 2 . i i i l i 2 33 — 2 (w a ) (w Assu me tha t 2 x A , A , B — 2 — — 2 (x c ) (x x a (x a ) x w h ere A A and B are c o nstan ts w h o se va u es are to be d eter 1 , , l m n d n r n oi r c t o ns e . O c ea f a i l i g i , 2Z — — — 2 x __A (x a )(x + A (x B) B(x a )

‘ Th erefore e u a tin the coe icien ts o like owers o x in the a bove , q g ﬁ f p f id en tit w e h ave y , — 2 = and A l a B A Q/S+B a o

Henc e so v n th e above s stem o f s mu tan eou s e u a t ons for A , l i g y i l q i 1 , A and B h 2 , w e ave 2 g_(a _ ( 1 A A an d B ° ] 2 — ’ — f a ﬁ ( a ﬁ)

Th erefore th e requ ired differen ti al c o efﬁcient is I B (L M A + t u + 1 (x

106 DIFFERE N TI AL CAL C UL US

—1 = “Sin x 6 . f e rove th at I y , p — 2 (1 (2n (n

and h en ce ﬁn d th e va ue of wh en x is z ero . l g ,

-1 = a Sm ”5 Here g 1 e x

i . 1 e. , ( w h en c e d fferent a t n an d d v d n b w e h ave , i i i g i i i g y (1 Th erefore

ar 2 d 2 fe _ a —o m n/2 } ew) y. M ( n

’ Bu t by Leibnitz s th eorem d n 1 —w2 — 2nw — n n n + 2 yn + 1 ( . ( )y d z

(wed

Hen c e (1) becomes — 3‘ 2 (1 x ) g (2n (n

P u tt n x = 0 in th e above e u at on w e h ave th e recurrence i g q i , formu la (9)

No = nd Th r for th f rmu a 2 es w (g l ) o a a e e e e o l ( ) gi v 2 2 {a ( m (yam

2 2 a 2772 — 3 W ( m l ( 12 (yam—3) o

2 2 2 a 2m a 2m a l a { ( { ( ( ) ,

2 — 2 2 ( 771 ) } (yam

2 4 2m 2777, 4 { ( W} W ( X} (yzm 4) o

2 2 2 2 9 { a (2m {a (2m (a 2 )a .

Th u s i (gn) o s 2 { a + (n S UC CE S SIVE DIFFE RE NTIA TION 107

2 2 2 2 2 a 2(a 2 )(a 4 ) { a (n

rd n n ac c o i g as n is odd or eve .

—1 7 . sin x r e a If y , p ov th t

(yn i 2) and ence ﬁnd h (t n) , “1 8 . : sin m sin x r e a If y ( ), p ov th t m 2 (yn +2)o ) (ynlo

and en e ﬁn d . h c (gn) 0

Ex am l es on Ch a er I X p pt .

2 1 . x = 2 r ve a th e ix d i f eren a c e ﬁ en sin If g , p o th t s th f ti l o f ci t of y with resp ect to x is l 4 z —2 4 2 — 4 5 ( y 1) cos g y( y 5) sin y .

2 x 2 = ‘ l 2 . + t an r e a If g log , p ov th t 2 ‘ 1 x 2 x 1 CU

“3 d ig 8 2513 " d x (1

3 . Th e ﬁr e nd r an d r d ff eren a e ﬁ en st , s co , thi d fou th i ti l co f ci ts of y with re e x are en e b t a b 0 re e e and e x re e sp ct to d ot d y , , , sp ctiv ly , thos of with sp ct

b 7 a . S a to y y , , B , 7 how th t 2 2 3ao 5 b 3 a ’ y 53 “ " t 7

Y = 3X Z tX and all th e e er en e n n x en If , , l tt s d ot fu ctio s of , th X Y Z 3 t X Y Z 1 l 1 1 1 3 t Y Z 2 2 X 2 2 2

Ma . r 1900 [ th T ipos , ] — If g 1 ) prove th at 2 1 2 y 1 4 9 3 d x d x ( d x ) } l{ ( d x ) } M a . r [ th T ipos , d n n m d m a x m a x n Prove th at x (6 x ) . did d x ’n

" " 1 d x x x 1 r

d n — m 1 (x + 1 1 (x + 2 1 d x n l 108 DIFFE RENTIAL CAL C UL US

J ; 9 . r e a If y 2 2 p ov th t a. + x

l n 1 nn l sm71 a d p _ ¢ cos , i l d wn a n + 33

10 : . If y p rove that

1 1 . Ii ’ r e a g 3 p ov th t x -a 3

2 ? (x a x a ) 2

a d 3 " 1 where «p tan 2 x + x

1 2 : . If y prove that

2 x l 2x n n 1 0 . ( )yn + 2 11” + 1 ( )yn

H n e P 2 e c if I ; (x show that d

d x

= 13 . a cos x b sin x a If y (log ) (log ) , show th t

2 x 2n 1 x n 1 0 . iv” + 2 ( ) yn H ( )yn

“2 14 , x 1 x r e a If y log ( ) , p ov th t (y )0 u m

1 5 «f ? ﬁn . If y (x oo d m ‘2 2 “ l : 16 . 1 + x sin m an x ﬁnd If y ( ) ( t ) , y )? ( a

1 r 7 . If y p ove th at

n _ _e n u ( —1

1 8 r . P ove th at _ d " _l n 1 1 5 2n 2 " — — 3 ( —x 5 l l " ) } d x

* r ) 1 whe e q cos x .

HAP TE C R X .

’ ’ M AND L R T O M R OL L E s T HE ORE T AY O s HE RE .

’ r m Th Th eorem f h e Mean V u Ro e s Theo e . e o t a e. 6 7 . ll l

’ L b n i r R olle s Th eorem. et x e co t nuous o e ver va lue o I . f( ) f y f ’ x in the d oma in a b a nd let u r th er let x be ( , ) f , f ( ) d m T e istent or e ve r va lu e o x w ithin the o a in a b . hen th ere x f y f ( , ) mus t be a t lea st one va lue 0 o x f , in termed ia te b w en a n et e a d b, suc h ’ tha t f In th e ca se w h e n f(x ) is repre sentabl e ra h c a th e tru th o f g p i lly , ’ R olle s th eorem bec om es Obviou s

31 . FIG . fro m th e ﬁgu re of th e c u rve y = f (x ) for th e o rd n a tes of th e o nts , i p i , w h o se ab scissee are a an d b be n e u a th ere is a t l east o ne n ter , i g q l , i m t n t at w h ic h th e t an nt i ara to th x — a x edi a e poi ge s p llel e i s .

’ r r ro o of Bo th rem N t For a o ou s f e s eo see o e B . ig p ll ,

II Th Theorem o the M a n lu L et x n inuou s e f e Va e. f ( ) be co t ’ or ever va lue o x in th e d oma in a b u rthe let x be f y f ( , ) ; f r , f ( ) - i r va l w i hi T h re ex sten t or eve u e o x t n the d oma in a b . hen t e f y f ( , ) mus t be a t lea st one va lue 0 o x in termedia te between a a nd b f , , such tha t f(b)

For c on sid er th e fu n c ti on (b — —f -- — f(x ) g w)

Now < a and ( b are O bv ou s z ero an d is ex stent its t( ) N) i ly ; i , va lu e being R OLLE ’ S THE ORE M AND TA YL OR ’ S THE ORE M 1 1 1

’ Th erefore < x sat sﬁes all th c ond t on f th r m p( ) i e i i s o Rolle s eo e . Th erefore

i . a. re) . ,

a n sh es for at east one a u e 0 of x n t rm b tw e n a nd v i l v l i e ediate e e a b . Henc e — f(5) f(a ) (b

’ N 1 R l e e re h as been cal ed b e ri er th e nd a en a OTE . o l s th o m l y som w t s fu m t l e re D ff eren a Ca s be a e th e a n r an e re th o m of i ti l lculu , c us of m y impo t t th o ms n d r -th r n n M which are ba sed on it . It is ame a fte e F e ch m athematicia ich el 1 2 i r r Rolle ( 65 who ﬁrst a ttempted a r go ous p oof of it . — h N 2 . P n h for b a w e et 1 in t e r OTE utti g , g ( ) fo m ’ f a = f a h f a 6h ( h) ( ) ( ) , 6 1 wh ere 0 < .

EX AM PLES .

1 For th e d om a n ive th e ra h s Of th e fu n c t on s . i g g p i

7 W m “ 1 ’ c os and tan an d h en c e sh ow th at Rolle s 2 tli 0 h th r n n u n h n d t eorem i s applic able to e fi st fu ctio b t o t t o t e sec o . S nc e th e sec on d fu n c t on e u a s 0 i i q l , x or -x ac c ord n as x is 0 0 o r i g , 0 th e re u red ra h s are AOB a n d D , q i g p i r h r i D OE res ec t ve . u t e t p i ly F , obv ou s from th e fi u re t h at h e i g , W il n A B h th ere IS a o t viz . O o n O w ere p i , , th e t an en t is ara e to th e x — ax s g p ll l i , 2 FI G . 3 . h n n Th th ere is n o su c poi t o D OE . u s ’ it is sh ow n th at in th e d om a n — 1 Ro e s th eorem is a li , i , ll pp t th e first fu n c t on bu t n o t to th e econd c able o i s .

In th a s wh en x is re rese ntab e ra ca ve th e 2 . e c e h f ( ) p l g p i lly , gi geo m etric a l in terpre tation of th e th eo rem o f

th e m ean valu e . Let ACB be th e graph of f(x ) in th e

d o m ain ~ a b th e o n ts w h o se a bsc sses are ( , ) , p i i

a c and b be n A O an d B res ec t ve . , i g , p i ly Th en con stru ct n a s in th e a d o ned ﬁ u re , i g j i g , h w e ave m . F . 33

an L BAR . 1 12 DIFFE RE NTIAL CAL C UL US

’ = X There or th n i Now f (c) tan L OT . f e e follo wi g s th e geo — metrical interpretation of th e th eorem of th e mean val ue If the grap h AOB of f (x ) is a continuou s curve ha ving everywhere a tan en t then there must be a t lea st one oint inter med ia te betw en g , p , e B a w i h h an ent is a ra llel to th c h r A A nd t h c t e t e o d B . a , g p

3 In e th e n a e e th e ra th e functi n n . ach of followi g c s s giv g ph of O a d decid e wh ether th e th eorem of th e m ea n va lue is applicable

(i) f (x ) t h e

(11) (iii)

— z 1 , b l . n = 0 0 (1 + sin

2 a + h . 4 . For th e n n Ax Bx C r e a c i n th e e re th e fu ctio , p ov th t th o m of

n mea value .

’ m 6 8 Ta or s Deve o ment in Finite For . L et x and its . yl l p f( ) — ’ first (n l ) difieren tia l coefiicients be continuous f or every va lue of x in the doma in a b u rther let th e n th di eren tia l ( , ) ; f , fi coe icien t o x be ex istent or ever va lu e o x within the d oma in fi f f( ) , f y f Th n here m s b a l s ne num r 0 m a b . e t u t e t ea t o be in ter edia te ( , ) , 1 h ha between 0 a nd , su c t t

’ ) b b a f a (1— f f( ) ( ) ( ) 5; (a) b ( ( n _ f f

b ( _ 1 9b a) ( )

’ ’ This is Tayl or s th eorem with th e remainder in Lagrange s form h r i d r bein , t e ema n e g

’ ’ th e equa tion ( 1) is ca lled Taylor s formul a or Taylor s development n rm in ﬁ it e fo .

Th e above th e orem m ay be proved in th e follow in g manner Consider th e fu n c tion — — — ” f(w) (b re)f (e f (w)

1 14 DIFFERE NTIAL CAL C UL US

E A X M P LES .

1 F nd th r m a nd er a ter n terms in th e x an n f . i e e i f e p sio o i in r f s n (a + h) pow e s o h . ’ Here x = sin x an d c o n se u ent all th e c ond t on s of Ta or s f( ) , q ly , i i yl

r d A so Th r or th formu la a e satisﬁe . l e ef e e

rem ain d er requ ired is

” h sin

3 5 7 “ 1 x x 2 . P rove th at tan x x + 5

m ' 1 in e am-q co s + t an 1 m 2m— 1

—1 = n Th en x at ﬁ all th e c nd t on Take f(x) t a x . f ( ) s i s es o i i s ’ mu r . 4 Ar T r f a A s Ex . t 66 a o s o o b . yl l l y , ,

—1 (n tan f (x )

— an d c onsequ en tly f 1) sin

Th erefo re 7 f(x ) = f(0) ar e) M W 3,

2 m 2 P rove th at a : 1 x log (log a ) ROL LE ’ S THE ORE M AND TA YL OR ’ S THE OR EM 1 15 “ 4 Pr e a th e re a n er a er n erm in th e ex an n e bx . ov th t m i d ft t s p sio of cos in powers of x is

n x —1 x ea fi cos ( b9x n ta n

’ ’ h r m m Macl aurin s T eo e . a or s Th e re . 6 9 . T yl o

' r T r L x be a u nc tion d e ned or a iv T l s h o em . et en I . ay o e f ( ) f ﬁ f g H Th n w h en certa in conditions a re s tis d d omain a a o x . e a e ( , + ) f , ﬁ , h ? h n s f f‘n ’ a ( ) + a (a) + ( )

t o inﬁnity H or ever va lu e o h w ithin th e d oma in 0 . f y f ( , )

’ Fo r a d isc u ss1o n o f th e c on d t o n s o f Ta o r s th eore m see i i yl , t B No e .

’ r L be nc d e ne M la rin Theo em . et x a u tion d r II . a c u s f ( ) f ﬁ f o a i Th n w he cert in nd iti ven d oma in 0 X o x . e n a co ons a re g ( , ) f , sa tis ed ﬁ , 2 ” f(x ) = f(0) f (O) g,

t o inﬁnity

r f o every va lu e of x within the d oma in .

’ ’ Mac l au rin s th eorem is a art cu ar c a se of T a or s th eorem fo r p i l yl , ’ it is o btain ed fro m T aylor s th eo rem by pu tti ng a = 0 an d w riting x a n f r d H r d X o h an espec tively .

’ e r h n r n d NOTE . T aylor s th orem is ca ll ed a fte t e E glish ju ist a ma th ema tician Brook Taylor (1 68 5 AM L EX P ES . 1 ’ . Apply Ta ylor s th eorem t o O bta in th e Binomi a l Ex pan sion

2 !

1 n . m Here f(x ) = x a n d c on sequ ently —1) (m Th erefore th e gen eral term in th e series for (a r is

and th u s w t th r u r d x n n e ge e eq i e e pa sio . 1 16 DIFFE RENTIAL CAL C UL US

’ A M l u rin h eor m to ro h 2 . pply ac a s t e p ve t at 2 4 x x

81“

os c x ,

S in f z — (c os x sin 2 — e c os x (cos x sin x )

— s in x e (2 sin x co s x + c o s x )

11 1 258 8 m 8 3 sin 5 0 , 2

0 .

_ s i“ ge cos x (3 + sin x ) sin 2x __l sin x gg {2 c os 2x (3 sin x ) + co s x sin 2x } - 3 . 2 ’ ’ ’ Th erefore = f (0) xf (O) f (O) g] 3 4 9” " + ton — o) 37 gm

2 4 x x 1 + CU+ § 8

’ ri ” 3 . Apply Macl au n s th eorem to ﬁnd th e coefﬁc ient of x —1 a m th e ex pan sion of c ”

a m Here x c nd n Art 6 a c o se u ent b Ex . 6 . 6 f( ) q ly y , , f 2 { a + (n

2 { a + (n ac cord n as n is dd r n Th r r th oe ﬁc ent re u red i g o o eve . e efo e e c f i q i 2 ja + (n n !

a ccord n a s n is o dd r n i g o eve . — 4 A ’ " 1 1 . pply Taylor s theo rem to prove tha t tan (a h ) tan a (h sin ct) sin ( 2 < sin n< + nﬁn ere p M p to i ity, wh n a = cot

1 18 DIFFE RE NTIAL CAL C UL US

b a n th e r er n for a n in flex ion fir n e a [To o t i c it io poi t of , st ot th t , if " — re re en th e an en a etc . b e n all z er h p s t t g t to W ( ) , , i g o , y , y2 as n a s " + 1 for su fii cien tl h th e same sig h y sm all v alue s of . No w by

n n Art . 5 0 th e an en r e th e r e a t a n inflex i H n d efi itio ( ) t g t c oss s cu v poi t of on . e c e = x = a a n infl ex ion 0 and a i: O. is poi t of , if ( ) Th e criteria for con c avity and convex ity m ay be obtain ed in a similar r manne . ]

h c l in cir in r e le a t a x i . I n a iven cu v t e os il a t c o t e. (II) g , g p ( , g) , , the circle which ha s a conta c t of the second ord er w ith the curve a t h irc le o cu rv ture o th e cu rv a x is the same a s t e c e t x . ( , g) , f a f ( , g) For ta e th e en era e u at on of a c rc e in th e form , k g l q i i l — 2 2 (X b) = c

Th en w h en th s is th e o sc u atin c rc e a t x W e mu st h ave , i l g i l ( , y) ,

= X x , Y= v,

_d_g_ d Y d x d X

2 di g d Y 2 2 ' d x dX

Bu t d fferen t at n w e et , i i i g g

‘ d Y X I, Y

d 2 Y 1 + + (Y —b) dX 2

Th erefore from (3) and (4) w e h ave (w

an d h enc e ROLL E ’ S THE ORE M AND TA YL OR ’ S THE ORE M 1 19

Ex n X ampl es o Ch a pt er .

1 . ein an en n er rea er an z er b a n th e re a nd er p b g y giv umb g t th o , o t i m i ’ n erm in a r d e e en b in er b — a in th e r t s , T ylo s v lopm t of f( ) pow s of ( ) , fo m _ " n p b — a 1 —0 ( ) ( ) n) a --Gb — a ﬂ ( l ) , p ot where b C n er th e nc n b x x o sid fu tio f( ) f ( ) l f ( ) b i— K J QM ,

ere K a n an ven b th e e a n c a 0 . en r ee n a s in wh is co st t gi y qu tio p( ) Th , p oc di g

r d re . Art . 68 we e t th e re e , g qui sult

‘ This form of the rema in der is d u e to S ch lbmilc h ; i t in c lud es t h e forms f L ran e and Ca ch these orms a re obta ined on b i u in n an d o ag g u y , for f su st t t g

1 res ectively for p . p “ 2 d en e th e re a n er a er n er in th e ex an n 1 x in . If R , ot m i d ft t ms p sio of ( ) a end n er x r e a en z er a s n end n ﬁn x sc i g pow s of , p ov th t R , t ds to o t s to i ity , n 1 b eing l e ss tha . ’ In a e ak n Ca r th e re a nd er we a e this c s , t i g uchy s fo m of m i , h v

(n 1 ) l 1 9 m 1 W ) m(1 0x ) (1 9x ) (n 1 ) 1 1 0 n -1 m—l m 1 9x w ( ) " , (1 6x ) ere w an for (w wh ” st ds (n 1 ) — 1 i l Now s nce x 1 O 1 — 0 1 0x and n e en l , i , , co s qu t y , must 1 +

r A r m r e a n e an a ﬁn e n er e er a e it a be . e e re m i l ss th it umb , how v l g y Th fo

m—l 6 12 33 A m(1 x ) X 70 I 74 )

B ut e kn n a a s x < 1 w end z er en n en d it is w ll ow th t , | I , n t s to o wh t s to

ﬁn . nﬁn . H en e R end z er en n en d n i ity c n t s to o wh t s to i ity

3 Pr e a a a e en ex a l ar a in Ex . 2 d r e in . ov th t st t m t , ct y simil to th t , hol s t u h h x n " 1 t e c a se of t e e p an sio of tan x .

4 . In th e a e ea th e n n n n e a e th e be a r c s of ch of followi g fu ctio s , i v stig t h viou of th e rem aind er after n terms as n tend s to inﬁnity

t in x . 1 x . (i) a . (ii) s (iii) log ( + )

5 . 1f x is ﬁn e c n n an d h a s a n n d ff eren a efﬁ en f( ) it , o ti uous , co ti uous i ti l co ci t ’ x for all a e x r O 1 r e withou t th e u se o eome ric l r f ( ) v lu s of f om to , p ov f g t a ep re n io - r 6 i r r r se ta t n tha t f(1) f(0) f wh e e s som e p ope f a ction . 120 DIFFE RE NTIAL CAL C UL US

D educe th e rel ation l f (x ) f e (ow ne w) ﬁ r ! sta ting th e further assumption s necessary .

Ma . ri [ th T pos ,

6 n er c er n n n be e ﬁed . Prove th at u d tai co ditio s to Sp ci

n —l ' " w x = 0 x 1 o 97 w f( ) f( ) f v) ( > me ) , $, (n 1 ) 0 1 where O .

Ex an x 5 1 in th e ab e r a nd r e a as x en n p d f( ) ( ov fo m , p ov th t t ds to u ity th e valu e of 0 in f tends to th e limit

M r a . [ th T ipos , 7 In th e e a n . qu tio 2 n ' w+ h = w h c - x

’ 8 Ma laurin s e rem r e a . Apply c th o to p ov th t 2 ‘ 6 x x x x x _1 B 2 + 3 et _1 1 1 a

1 z l -l : — = ﬁ d B B B Th e n u ber B B etc . are a e ere B . wh l 3 4 % [ m s 2, , c ll ’ Bernoulli s Numbers .

9 Pr e a . ov th t 2 “ " w v £ as a. 55 —3 2 2 24 1 3 6 !

r e sin 2 2 10 . P ov th at if y log (x x

(x + l ) “ r Hence or othe wise ex pand y in a sc ending powers of x a s far a s x .

Ma . r [ th T ipos ,

1 1 . A be th e rd an r ar arc a nd B a a th e arc en If cho of y ci cul , th t of h lf th 8B —A ’ th e le ngth of th e arc i s v ery nearly equ a l to [This is Huyghen s a pp ro x ima tion ]

’ 1 2 r n a t‘ . Dete mine th e para bola y a + bx cx so tha t it sha ll h ave a co t c " th e e nd rd er th e r e = x a t th e n 1 of s co o with cu v y poi t ( ,

122 DIFFE RE NTIAL CAL O UL US

2 = m n m u m f r = F r l e 1 a . f n t n x h as a o x 0 . o et s 2 . Th e u c io i i , , y Th en ? x

fo r every va lu e o f x satisfying th e c on d iti o n

— 5 O< 513 0 < .

See th e adj oin ed ﬁgure .

= = m m m f r F r l t 8 1 sa . h as a n u o x 0 . o e i i , , y Th en i x —O 0

for every valu e o f x satisfying th e c on d ition — 0 < 33 0 < 5 .

h See t e ad o n ed ﬁ u re . FIG . 36 . j i g

3 4 . S w b a ra a th e n n x h a s ne er a ax nor a ho , y g ph , th t fu ctio ith m imum n mi imum . — “ a 5 . b a ra a th e n n a x x h as a ax for x . Show, y g ph , th t fu ctio m imum

6 . G e th e ra th e n n + x for th e a n 0 4 1r and iv g ph of fu ctio g cos dom i ( , ) ,

n e ﬁnd x a and n a n d a n h e c its m a im mi im withi th at om i .

Tw o Theorems.

h r m I nc ti n d ned or d ma in a b ha T eo e . a u o x e a o s I f f( ) , ﬁ f ( , ) , eit h er a max imum or a minimum f or a va lu e 0 of x within the ’ d oma in th en c mu st be z ero i it f ( ) , f

h f r = e x as a ax x . n b Ar For o a e t . 70 ere e x , suppos f( ) m imum Th , y , th ists a quantity e 0 such th at — f (x ) f(a) < 0

for every v alue of x sa tisfying th e cond i tion

0 < x —c e I .

£ 2 92 2 5 < 0 0 r > 0

x —c > -e 0 < ( ) .

’ - in c n n ex en e . : For a a e o se ere 0 . 1 c s which f ( ) is ist t , Fig wh c MA X IMA AND MINIMA 123

lim lim 0 o o Now l et th e symbols a nd

< x end a s x e nd a b a k n a e rea er an a a n d th e to which p( ) t s t s to y t i g v lu s g t th , x n d s nd a b ak n n limit to which ¢( ) te s a x te s to y t i g value s less th an a . Th e it follows from (1) tha t

x c 0 x c if th e limit ex ists ; and that

x = c —0 x — c

’ h e ex . Bu t c ex a nd n e en b th e m exi if t limit ists f ( ) ists , , co s qu tly , oth li its st ’ ’ ’ and equ al f Therefore f (c) 1» 0 an d also f (c) «t 0 ; h en c e f (c) A r ex a a r th e a b e d in th e a e en x h a s a p oof , ctly simil to ov , hol s c s wh f ( ) min imum .

L et x a nd its rst it di eren tia l coe icien ts be Th eorem II . f ( ) fi fi fi n i uou s or ever va lue o x in a d oma in a b fu rther let co t n f y f ( , ) ,

a nd f

lu o x within th e d in b Th h s w here c is a va e oma a . en x a f ( , ) f ( ) i h r a max imum nor a mi ni mum or x = c i n is odd i n is ne t e f , f ; f even x ha s a ma x imum or a minimum or x = 0 a ccord in a s , f ( ) f g

or f 0 .

h r For u nd er t e c on d t on s b A t . 68 , i i , y ,

—c ) ,

h r 1 A b n n n w e e . so e a c o t u ou s fu n c t on th ere l , i g i i , ex ists a qu a n tity 5 0 su c h th a t h a s th e sa me sign a s fo r every valu e o f x sa ti sfyin g th e c o n di ti on — 0 < 93 0 < 8 .

Th erefo re for su c h va ues of x x — c h as th e sa me s n a s , l , f ( ) f ( ) ig x Hen c e f n is od d x — c ch an es s n w th ( , i , f ( ) f ( ) g ig i x —c an d co n se u e nt x h as ne th er a m ax m u m n or a ( ) , q ly, f( ) i i m n m u m for x = c f n is even x — c h as th e sa me s n a s i i ; i , f( ) f( ) ig and c onse uen t x h as a m ax m u m or a m n mu m fo r , q ly, f ( ) i i i x = c a c c ord in g as

< 0 or 0 . 124 DIFFE RE NTIAL CAL C UL US

E AM L X P ES .

1 P ro th a t sin x h as a m ax mu m for x and a m n mu m . ve i g i i

. 2

’ Here x s1n x an d c onse u ent x c os x a d f( ) , , q ly, f ( ) n ” — in Th er or f (x ) s x . ef e

’ ” o 0 and f f , f < f

Hence b Th eorem . sin x h a s a m ax mu m for x an d a , y II , i

minimu m for x

5 ‘ i Pr e th at x —5x 5x —1 h as a max mu m for x = 1 a 2 . ov + i m m f r = 3 n th r for = mini u o x ei e x 0 . 5 “ 3 Here x = x —5x 5x — 1 an d c on se uen t f( ) + , , q ly,

’ 4 3 2 x 5 x —4x 3x f ( ) ( ) ,

Th erefore

an d

m H ence b Th eore . w e h ave th e re u red resu t . , y II , q i l 1 r th i h m m m r = 3 . P o e at x as a ax u f c v i o x . 1 Here and c on se u ent , q ly , — los x )

x l —lo x w x -— z l o w—s f ga g ) +f( ) jx g ) . 1 ’ ” = n n 4? h h r re c 0 a d e 0 . He c e b Th r as T e efo < eo em . x f ( ) f ( ) , y II , r = a max im um fo x c .

4 r e ea th e n a e n . P ov ch of followi g st t me ts

3 x h as ne er a ax n or a n (i) ith m imum mi imum for x 0 .

x l x h as a x (ii) ( )( ma imum for x and a minimum for x 2 .

126 D IFFE RE NTIAL CAL C UL US

2 An o n t a n of a ss ned vo u m e h as a s u are ba se and . pe k ig l q f th n n er su rfa c e is th e a t o s b e h at i vert c a s d es . e e s s w s i l i I i l p i l , th e ratio of th e d epth to th e w idth

’ Let V d enote th e a ss ned vo u m e a th e w d th an d a x th e ig l , i , h nn r r a Th n d e th fu rth er let S d en ote t e e su f c e . e p , i

2 2 3 4a an d = a x S a x V .

5 h i m n m m Th erefore S w i ch s to be a i i u . [Note

this s tep ]

6 j ‘ § — — - _ Now = V fl fix i i Th erefore th e equ ation g h h a s on o ne fin te ro ot viz . x an d t s va u e o f x m a es S a ly i , , i l k m n mu m for i i , d 2 S — x 2 S d x

h an z r w h n enc e th e re u red w h ich is greater t e o e x 5. H q i 2 is 1 .

3 . In th e a se ea th e n nc n ﬁn d th e a e x fo r c of ch of followi g fu tio s , v lu s of which th e function h as a ma x imum or a min imum :

3 2 2 3 2 x 1 5x 11 x a x a x (i) 36x 2 . ( ) ( ) ( ) 2 2 3x a i (m) v sec x x . (a n y ) , x 3 cos v vi ( ) si n x cos x . ( ) 1 + 0 0 1; x

- 4 . re a n A Norm an win dow consists of a ct gle surmounted by a s emi circl e . G en th e er e er re red th e e and bread th e nd en th e iv p im t , qui h ight th of wi ow wh a n x m qu tity of light a dmitted is a m a i um . 5 . In a submarine telegraph cable th e speed of sign alling va ries as

2 x ere x th e ra th e ra th e re a the er n log 35, wh is tio of dius of co to th t of cov i g

a th e ee rea e en s ra 1 a/ show th t sp d is g t st wh thi tio is e .

6 W . ha t is th e ratio of th e height to th e radius of an open cylind ric a l can en e en th e r ace a n of giv volum , wh su f is mi imum

7 . Ex amine each of th e followi ng functions for m ax im a a nd minima

2 (i) x (x l )(x (11) (if 6 5x 1 2x 5 1 ‘ 3 1 ? (iii) 5x 40x 5 x 6ox 1 7 .

4 s (iv) co x cos 2x . MAX IMA AND MINIMA 127

2 o 2 1 o o c 0 o Pr e a x sm h as a n i nﬁni e n ber am a a nd i i 8 . ov th t t um of m m m n ma i n x

every domain containi ng 0 .

x X E ampl es on Ch apter I .

1 n a e th e ax a and n a th e n . I vestig t m im mi ima of e ch of followi g functions 2 in " in x . n in (i) s cos x (ii) s x s x . x sin . x 2 (iii) e x (iv) cos cos x cos 3x .

2 n th e ax a nd n u ere . Fi d m imum mi im m of wh

2 2 a x 2 h x y by 3 h nd t e ax ne n n . . Fi m imum co of give sla t h eight

4 a h - n . Show th t t e s emi vertic al a ngle of th e co n e of given surfa c e a d “ 1 m ax imum volum e is sin 5

2 5 . th = 9 Show tha t e shortest norm al chord of th e parabola y 4a x is 6 a «4: 6 Th . e port ion of th e tan gent to an ellipse i nt erc epted between th e ax es is a n ﬁnd n mi imum its l e gth .

r a ﬁx e n A n h r eren e a r e ra d a th e F om d poi t o t e ci cumf c of ci cl of ius , er end r A r a th e rea e p p icula Y is l et fa ll on th e tangent at P . P ove th t g t st 3 area AP Y c an have is

8 n th e n e r m r bed . Fi d h eight of th e cone of lea st volum e which c a b ci cu sc i ere about a Sph of given radius a .

9 . n th e a rea and n th e ax m r a n e a n a en Fi d positio of m imu t i gl , h vi g giv an e c an be n r e in a en r e a n d r e a th e area ann gl , which i sc ib d giv ci cl , p ov th t c ot e in h av a m imum valu e . 10 . Show th at th e tot al surfac e of a cylind er inscribed in a right circular cone c annot h ave a ma x imum valu e if th e semi -angle of th e con e ex c eed s

t an 5 .

1 1 nd th in n . Fi e area of th e l argest rectangl e that c an be in scribed a el lips e . 12 . P re n on a r e e en re C an d rad a . Pr e , Qa two poi ts ci cl whos c t is ius ov tha t th e m a x imum v alue of th e rad ius of th e circl e in scribed in th e trian gl e

13 . h e sum th e ed e a rec an ar ara e ed l and th e If t of g s of t gul p ll lopip is ,

sum th e area th e a e ; a en th e ex e th e e of s of f c s is 2 show th t wh c ss of volum

th e ara e ed er a a be oe ed e a e ed e a s of p ll lopip ov th t of cu , wh s g is its sm ll st g , is e rea t as b e th e s a e ed e be a nd ﬁnd th e en th e g possi l , m ll st g must ; l gths of 50 er ed e oth g s . Ma r s [ th . T ipo , 128 DIFFE RE NTIAL CAL C UL US

14 . Th e circl e of curva ture a t a point P of a p arabola meets th e para bola

a a n at and en re O . Pr e a th e area th e r a n e C P a g i Q, its c t is ov th t of t i gl Qis 1 m ax imum when OP makes with th e ax is an angl e whose tangent is

Ma . r [ th T ipos ,

15 Pr r h r th . ove that f(x ) is stationa y for values of x which are t e oots of e equation a nd d etermine th e condition s nec essary to distinguish th e ’ a e x rre n n th e r x = 0 as max a ni a or v lu s of i ( ) , co spo di g to oots of f ( ) , im , mi m , ne er m x ith a ima nor minim a . ' A A are th e are a th e a x e e r an e c an be If , s of two m imum isosc l s t i gl s which d escribed with their vertic es a t th e pol e a nd their ba se a ngl es o n th e c ardioid ’ ‘ “ r a 1 r e 2 5 A 2 2 ( cos p ov tha t 6 A 5 a 5 . Ma r [ th . T ipos ,

1 30 DIFFE RE N TIAL CAL C UL US

’ Cauch s Theorem Th Fu 9 7 4 . y . e ndamental Form . 0 ’ T r . Ca uc h s h e L x I y orem. et a nd t (x ) be continu ous f o ever va lu e o x in the d omain a b u rther or ever va lue o x y f ( , ) ; f , f y f ’ ' w ithin the d oma in a b let x a nd b x be ex istent a nd x x ( , ) , ( ) g ( ) l ( ) == ﬁnite and l 0 . Then

For c ons der th e fu nc t on , i i ( a F P ea i v) (a) iuse) i s». (b) _M

We no te th at 1 b ¥-1 a c ann ot be z ero for f b l( ) t( ) ; , i y ’ ' Rolle s th eorem ml (x ) mu st vani sh for some valu e o f x w ith in th e

' n h n Th u E i d oma a b w ch is c on trar to th e su os t o . s x s i ( , ) , i y pp i i ( ) ﬁn t an d n n u f r r th d oma n a e c o t u o s o eve va u e of x in e b . i i y l i ( , ) ’ A so F a a nd F b are obv ou s z ero a n d F x is ex stent its l ( ) ( ) i ly ; ( ) i , valu e bei ng

’ Th r r h r e efo e E(x ) sati sﬁes all th e c on diti on s of Rolle s t eo em . Th erefore

i . e. ,

n n van ish es for at least on e valu e 0 of x interm ed i ate betw ee a a d b. Henc e ret ai l site) — . t au Ma) We)

Th F n m n l F 1 and heir rs t . e u da e a r L x t II t o m 3. et L( ) fi — (n l ) difieren tia l coefiicients be continu ous f or every va lu e of x in

a d oma in a b and vanish or a va lue a o x w ithin this d oma in ( , ) f f , ' x x bein d i rent rom zero or a ll other l ( ) , g ﬁe f f va lu es o x u rther let a nd be ex isten t a nd f ; f , ﬁnite and Then

’ For b Cau ch s th eorem , y y , r

b n ntermed at b tw en x and a 0 bet een 0 n a 0 e e e e w a d etc . , i g i i , , , , m m Th erefo re as ( a n d r a re both z ero b su os t on , p p y pp i i ,

— e a

Now obv ou s es between x and a a n d c onse u ent it i ly li , q ly

a n a Th re ore t en d s to a s x te d s to . e f

a 0 n_1

— a mu st ex i st and be eq ual to

lim x — a

( n x = a ¢

w—a

th l tt r m t x t Bu if is a e li i e is s . t it follow s from th e re at n to an d th at th s m its l i g i li it ex i sts, n) Henc e u s n w e h a , i g ve

x = a 1L(x ) gt (a )