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Differential Calculus A TEX T -B O O K OF 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 .
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