The Fundamental Theorem of Calculus Without a doubt, the birth of calculus is a glorious yet traumatic time for mathematics. It

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1 The Fundmentl Theorem of Clculus Wthout dout, the rth of clculus s glorous yet trumtc tme for mthemtcs It s pproprte we strt the course y vstng there snce t s oth the power of the suject s well s the open wounds t left ehnd tht drove mthemtcl ctvty for severl consequent decdes f not centures Its two cretors-dscoverers Isc Newton (4-77) nd Gottfred Lenz (4-7) would proly not recognze wht we tody refer to s the fundmentl theorem of clculus s such If one does GOOGLE serch of the FTC, one usully encounters some knd of sttement connectng the notons of ntegrton nd dfferentton s nverse opertons Or equvlently, the connecton etween the tngents of the grph of one functon nd the re under the curve of nother Yet, y the tme Newton nd Lenz come round, Fermt, Descrtes nd Huygens mong mny others hd lredy dscussed dervtves nd ther connecton wth tngents nd etreml vlues of functons, nd the res under curves of vred complety hd een computed y sclly dong Remnn sums ntegrton Fnlly, the connecton etween the two processes of ntegrton nd dfferentton hd een foreseen y Gregory nd Brrow, nd Newton hd een eposed to these des n Brrow's lectures As we wll see elow, they (Lenz nd Newton) hd ther own des on wht clculus ws ll out, nd they re dfferent from wht student receves n ther freshmn course Lenz & the FTC Lke Fermt, Gottfred Lenz, ws not mthemtcn y trde He ws dplomt who trveled wdely, nd s such cme to meet nd dscuss mthemtcs wth ll the est-known mthemtcns nd scentsts of hs tme, ncludng Huygens nd Newton It s, n sense, unfortunte tht Lenz met nd corresponded wth Newton, snce he wll, mny yers fter the meetng, e ccused of plgrzng hs des on clculus from Newton A long, scndlous dspute followed, nd lthough hs nme ws eventully clered, the dspute left tter tste n the soul of mthemtcns on oth sdes of the Englsh Chnnel Ths led to prtl solton of Englsh mthemtcns from those n the Contnent, where clculus, nd ts consequent dscplnes such s dfferentl equtons, wll eplode nto mssve nd powerful dscplne Lenz would de uneknownst to the world outsde of mthemtcs nd n reltve poverty It s fortunte, however, tht Lenz met Huygens, snce t s queston posed to hm y Huygens tht possly stmulted Lenz's dscovery of the connecton etween ntegrton nd dfferentton Huygens sked wht the sum of the recprocls of the trngulr numers dded to:

2 ? 5 Severl centures pror, Oresme hd proven the dvergence of the hrmonc seres We puse to gve dfferent proof of ths dvergence tht ws gven shortly fter Lenz: A Dfferent Argument for the Dvergence of the Hrmonc Seres Frst we need smple fct Let n e postve nteger Then snce n n n n n the LFS equls Thus the sum of three consecutve recprocls s greter n n thn tmes the mddle recprocl: For emple, f n 7, then n n n n 7 8 7, snce the left-hnd sde s Assume then, y wy of contrdcton, tht the hrmonc seres converges Let S denote the totl sum We know then tht S whch s certnly nonsenscl Very elegnt ndeed! S 4 5 As t turned out, the queston Huygens sked hd een nswered lredy,, ut unknown to Lenz, he plunged hed nto the prolem He understood tht from gven sequence: :,,, 4,, one could otn two other ones, the sum nd the dfference The sum (or, s we would sy tody, the seres), ( ), s defned s follows ( ):,,,, 4 For emple, f we strt wth the smple sequence,,, nd tke consecutve sums of t, we otn the followng confgurton: : ( ) : ( ) : ( ) : ( ) :

3 If we thnk of ths s mtr wth the rows nd columns leled y,,,, then we see tht we re delng wth Pscl's trngle snce the, j entry s nothng ut j j j It s n ths form tht Pscl orgnlly wrote hs trngle, nd we cn see the dgonls of our tle re ndeed the rows of the trdtonl Pscl trngle The second one: the dfference, ( ), s defned s follows, :,,, where:,, 4, etceter Thus, for emple, f s the sequence of trngulr numers, :,,,, 5, then ( ) s,,4,5, In the contet of sequences, the relton etween the dfference nd the sum, nsd ther conctenton, s esly understood If :,,, 4, Then ( ( )),, 4,, whch s lmost (ll we would need to recover s ttch t the egnnng Also ( ( )),, 4,, nd gn s esly recoverle from ths sequence, once s gven Lenz set out to prllel constructon to the tle ove, ut nsted of consecutve sums he took consecutve dfferences Snce he ws strtng wth decresng sequence, he needed to modfy the dfference to men:, 4, etceter, Hs frst (row) sequence ws the sequence of hrmonc numers:,,,, 4 : / / /4 /5 / /7 /8 /9 / / / ( ) : / / / / / /4 /5 /7 /9 / / /5 ( ) : / / / / /5 /8 /5 / /495 / /858 /9 ( ) : /4 / / /4 /8 /54 /84 / /98 /8 /44 /54 4 ( ) : /5 / /5 /8 / / / /9 /45 / /55 /84 Then he redly oserved tht the second row conssted of the hlves of the recprocls of the trngulr numers Hence f we let :,,, 4, nd ( ) :,,,, then we know,,, So, n smlr fshon to the dscusson ove, ( ( )) :,, 4, So the sum of ll the recprocls of the trngulr numers, whch s lm ( ) lm ( ( )) snce n s n n n Ths ws very ectng to hm, snce he relzed he could dequtely dd one sequence y

4 4 smply tkng dfferences of nother Although certnly tht remnds one of the sc des ehnd the fundmentl theorem of clculus, t dd not ecome rel clculus untl he pushed t further The noton of dfferentl showed up ectly s n pplcton of the dfference of sequence ppled to vrles stll n sequentl mode of thnkng Functon were nonestent only vrles were consdered Ths of course hd postve nd negtve effect on the postve sde one could consder rtrry equtons such s the folum, wth equton y y, whch ws orgnlly chllenge to the prevous generton (tht of Fermt & Descrtes)for fndng ts tngents Thus, the vrle would e consdered s tkng sequence of vlues, nd ts dfference would e consdered new vrle clled d As the sequence for would get more crowded, the vrle d would ecome nfntesml n nture Lenz developed much of our modern notton for the clculus such s d, dy d nd nd t s ths notton (s opposed to Newton's fluon notton) tht wll e dopted n the rest of Europe Smlrly, ntegrls would occur s the sum of the sequence of, nd would e referred to s Of course these would ecome nfntely lrge, ut snce most of the tme the ntegrl would e ppled to nfntesml qunttes, they would ecome just rel qunttes Hs sc elef ws tht there were levels of mgntude: y, were vrles whch were t the rel level; level d, dy were dfferentls whch were nfntesmls; level, y were ntegrls whch were very lrge; level d d( d), d y were second dfferentls (lower level) level d, dy were rel qunttes gn up from to Thus the dfferentl lowered the level whle the ntegrl rsed the level nd hence even tody we use d to denote rel quntty, nd every ntegrl hs to hve dfferentl nsde to mke sure the levels re correct Hs key elef ws tht qunttes t dfferent levels were not comprle For emple dy snce s t hgher level thn dy, nd d d y d for the sme reson Also multplcton would hve n effect on the level: two rel qunttes multpled would

5 5 remn rel thnk of, ut rel quntty multpled y dfferentl would e dfferentl: whle the product of two dfferentls would e second dfferentl: To compute the dfferentl of vrle ws smple: d ws the dfferentl of, dy ws the dfferentl of y, etceter Now f z y then t ws essentl tht dz d dy Nturlly ths s very true for sequences The followng s stll known tody s Lenz s Rule: d( y) dy yd Hs technque for computng dfferentls conssted of susttutng every vrle y tself dded to ts dfferentl, nd then sutrctng the orgnl epresson nd soltng the level quntty: d( y) ( d)( y dy) y y dy yd d( dy) y dy yd d( dy) nd the end result would e (Lenz Rule) d( y) dy yd snce d( dy ) s t level nd hence of no consequence to dfferentl The noton of lmt would hve to wt nother century efore t ws developed s n essentl ngredent to dfferentl clculus, nd the word dervtve ws nowhere to e found Alger ws the nstrument of choce For emple, suppose we wnted the quotent rule: dz where z Then wy to proceed ws to consder zy y nd tke dfferentls otnng zdy ydz d Isoltng dz we get d dy d zdy y yd dy dz y y y The use of nfntesmls s not consdered cceptle tody (lthough s eng reevluted presently), yet t s qute effectve nd Euler used t effectvely nd unworredly Consdered even wth less regrd s method used y Euler nd others nvolvng nfnte qunttes For emple n 74, Euler would rgue s follows Gven tht N ln N CN, n n nd tht for ll nfntely lrge N, C N C 577, where C s the so clled Euler constnt, (the modern dy usge s to denote ths numer) then one cn rgue tht

6 n N n n 4 N N 4 N N 4 N N N N ln N CN ln N C N ln N ln N ln ln n n n n N By smlr rgument, Euler would otn, 4 5 ln nd the two tles elow show he ws perhps justfed n hs clcultons: n n Coeffcent n Ln Coeffcent n Ln Sum Sum

7 7 Ael (n the erly prt of the 9 th century) would e one of the frst to gve n cceptle rgument for ths fct Lenz ccomplshed lot usng nfntesmls one of hs key devces ws the nfntesml trngle where the hypotenuse ws leled ds where s stood for length Untl tody we use the crucl fct tht ( ds) ( d) ( dy ) nd so ds s n order to compute the lengths of curves ds ( d) ( dy ) nd nturlly d dy From the sme nfntesml trngle, Lenz relzed tht the slope of tngent lnes ws nothng ut dy nd nturlly tht the re under the curve ws yd d He clerly stted tht d nd d Thus, t s possle tht Lenz would not hve een tht mpressed wth the FTC s we know t Insted we dscuss dfferent de tht Lenz found much more ntrgung thn d But frst we do n emple from the frst course n clculus Emple Tody we use very useful technque clled the susttuton method wth lttle geometrcl thought of ts mplctons On the pcture on the left, the green lne represents the grph of the functon y e, whle the red lne s the grph of y e If we were to sk for the re under the red grph (nd ove the s), we would e skng for e d, nd we would fnd ths defnte ntegrl y usng the susttuton whch would mmedtely leds us to the ntegrl u, e d nd the re under the green lne, nd lthough we cn ccept tht the res re the sme, t s not redly dscernle why t s so It ws Lenz who, n the 7's, dscovered ths generl prncple or technque to evlute res He referred to t y trnsmutton of res A technque sclly equvlent to the Fundmentl Theorem of Clculus, whch we now gve n some detl, nd n, more or less, modern notton nd deology It s mportnt to oserve tht Lenz s stll rooted n Greek geometrcl methods such s smlrty of trngles, even f t s t n nfntesml level

8 8 Lenz & Trnsmutton of Ares Suppose we hve n ntervl [, ] nd we hve functon y f ( ) defned on ths ntervl We re nterested n the re under the curve of ths functon Strt y consderng two neghorng (very close) ponts P nd Q on the grph of ths functon, where P hs coordntes nd y f ( ), nd Q 's coordntes re d nd y dy, where d s smll chnge n, nd dy s the correspondng smll chnge n y We wll let O denote the orgn Contnung n the lnguge of ndvsles, we let the length of the curve from pont P to pont Q e denoted y ds (recll tht s usully denotes length of curve n Clculus) Consder the tngent lne to the curve t the pont P nd suppose t ntersects the y t T (, z ) s Snce the rght trngle TUP s smlr to the rght trngle PRQ, we hve tht y z dy d, nd solvng for z we get, dy z y d We cn use ths epresson to defne new functon z of, whose grph s gven y: At the orgn O, drw the perpendculr to the tngent lne TP nd let t ntersect ths lne t pont S, whch hs hypotenuse z, nd let k e the dstnce from S to the orgn Snce

9 9 STO PTU 9, nd PTU QPR, we hve STO PQR Hence rght trngle OST s smlr to rght trngle PRQ, nd we hve d ds k z Consder now the nfntesml trngle OPQ : Its se s ds nd ts heght s k, hence ts re s whch y the smlrty ove equls zd kds, But s the pcture on the zd left llustrtes, equls hlf of the re n the ndcted nfntesml rectngle under the functon z zd re OPQ Hence we hve tht the re under the grph of z, whch s gven y zd s twce the re of the shpe mde from ll the nfntesml trngles Hence, n the pcture on the rght, the re on the left s hlf of the re on the rght

10 But y cuttng nd pstng n the pcture elow, we get tht the re under f ( ) equls the re OCD OBC OAD yd zd f ( ) f ( ) But y the result ove, OCD equls one hlf the re under z Esly, OBC equls f ( ) nd OAD s f ( ), so symolclly we hve: nd we hve echnged one computton of res for nother tht my turn out to e smpler thn the orgnl s we wll eemplfy elow But efore we do tht let's mke couple of oservtons Frst, f ( ) f ( ) cn e smplfed usng stndrd evluton notton to [ y dy z y d to susttute n the equton ove to otn ] Second, we cn use the fct tht yd zd f ( ) f ( )

11 of hs des dy y d [ y ] d [ yd dy y ] whch fter clerng nd multplyng y yelds the ntegrton y prts formul due to Lenz: f yd [ y] dy, whch s tntmountn the pctureto the three shded res fllng n the rectngleconfrmng the geometrc resonng he hd employed n the more sophstcted equton ove Lenz hmself ws plesed wth the followng pplcton Consder crcle of rdus centered t the pont (,) so tht t s tngent to the y s f nd so hence dy d Then ts upper semcrcle hs the equton y Snce, y z y y y, dfferenttng, we get ydy d d Solvng for, we hve z z But, then we hve 4 yd clsscl fct zd y ove

12 dz y pcture: z dz z y susttuton 4 z z z z dz y geometrc seres z z z z y term-wse ntegrton y evluton However, lthough one cn fnd the epresson qute eutful, t s not very prctcl for dong computtont converges too slowly For emple, f we use the fct tht ny lterntng seres toggles etween eceedng nd eng less thn the true sum, we hve the not very good estmte even fter terms Although f we verged the two terms, we would otn the superor 4555 After terms we get the mproved And fter 5, we hve Fnlly t, terms we get two-dgt ccurcy: , wth the verge eng 4595, whch s qute stsfyng Newton & the FTC One could sy tht Newton nd Lenz dd understnd thoroughly the fundmentl theorem of clculus (s we cll t tody), nd lso oth pprected the power nd rnge of the suject Certnly, Newton used Clculus-type thnkng to push the fronters of mechncs nd physcs Born on Chrstms Dy, 4, to reltvely poor wdow, Isc Newton showed promse s student, nd thus rother of hs mother greed to support hm n college He ttended Cmrdge Unversty In 5, durng n outrek of the plgue, he ws sent home, nd t ws durng tht perod tht he developed some of hs est des Soon fter tht, hs techer, Isc Brrow resgned hs poston so tht Newton could e pponted to follow

13 hm For the net yers Newton ws professor t Cmrdgels, terrle lecturer, hrdly ny one would ttend hs lectures, ut wdely known scholr In 9, he suffered nervous rekdown, prtlly cused y the stress suffered durng the dspute wth Lenz After he recovered, he ws pponted n chrge of the Royl Mnt where he spent the remnder of hs lfe When he ded, he ws the most fmous scentst n the world, nd ws ured wth ll the glory nd ceremony t Westmnster Aey Sr Isc Newton s one of the most dstngushed nmes n the hstory of mthemtcs nd scence He cn e consdered one of the founders of modern scence, nd hs ook Phlosophe Nturls Prncp Mthemtc (87) (often referred smply s the Prncp) s mjor ook n Western cvlzton He s stll consdered one of the most nfluentl thnkers of ll tme, nd Newton would e n the top three lst of nyone s choce of mthemtcns (or physcsts for tht mtter) Beng physcst, he ws nterested n the noton of nstntneous velocty (whch s closer to the lmtng de for dervtve), nd t s n those eplortons tht he cme wth the de of fluon (hs word for the dervtve wth respect to tme) In fct, one of hs fmous lw of moton, the second one, sttes tht Force = mss velocty where the dot stood for fluon In the more fmlr Lenz's notton the clm ecomes: F d ( m v ), so f mss s constnt, we get the more common verson of the second dt lw: F m where stnds for ccelerton However, f mss s not constnt, we get dfferent lwone tht s vld t the very hgh speeds of tomc prtcles of modern physcs A very esy pplcton, when ths lw s lnked wth Glleo's concluson tht grvty s constnt, s the clculton of the pth followed y projectle such s cnnon ll: Suppose projectle s shot from the orgn t ngle wth speed v Wht s the pth of the projectle? If we seprte the force nto ts two components, one n the - drecton nd one n the y-drecton (ths de s older thn Newton), we get tht F g whle Fy g, constnt But then whle y, constnt But, y ntegrtng m ccelertons, we get veloctes, v v cos( ) nd g v sn( ) y t v, f we ssume tme s mesured so m t s when we shot the projectle Hence, s v cos( ) t g nd sy t v sn( ) t, nd f we grph ths pth, we m get prol Ths knd of thnkng llowed Newton to use mthemtcs to conclude physcl fcts sed on few premses In smlr fshon

14 4 It ws durng tht htus from school when he ws n hs erly twentes tht Newton my hve ndependently rrved to hs lw of grvtton: two ojects ttrct ech other wth force proportonl to ther msses nd nversely proportonl to the squre of ther dstnce Others hd proposed t (Hooke for one) But he ws defntely the frst one to hve used t mthemtclly to prove physcl clms He redly used mthemtcs (clculus des) to nfer Kepler's frst two lws of plnetry moton from the lw of grvtton, nd nturlly ths served s mjor pece of evdence of support for ths lw But oth of these results were pulshed much lter thn they were relzed, nd n fct closer to Newton s pprecton for clculus, nd the power to clculte s hs noml theorem whch we look t net It ws one of hs frst successes, nd defnte step towrd clculus It s the etenson of the trdtonl (or fnte) noml theorem to the cse for other eponents esdes postve ntegers Newton Etends the Bnoml Theorem Wht proly strted s technque to mprove the computton of squre roots, nd other roots, ecme roder wepon, nd mde hm super mnpultor of seres, whch ws crtcl to hs whole vew of clculus We recll tht where n s n rtrry postve nteger Recll tht n n!!( n )! n ( ), n n nd f we cncel the ( n )! n the denomntor, we rrve t n n n n Although the generlzton ws not so ovous n Newton s orgnl nd more cumersome notton, t s very cler n our present notton Let e n rtrry numer now: where of course ( ), s defned y ( ) ( ) ( ( )) ( ) ( ) Mke the followng oservtons: there re fctors n the numertor nd fctors n the denomntor n f n s postve nteger, then for n,, nd thus Newton s verson

15 5 s true etenson of the fnte cse Note the recurson, s polynoml n of degree wth roots,,, We compute few of these polynomls (usng mnly the recurson gven ove, y defnton or greement; ; ( ) ( ) ( ) ; ; ; ; ; So, for emple, f we re nterested n tkng squre roots, we let followng coeffcents, nd we get the So f we put them together wth the noml theorem, we get tht hgher order terms, so f s smll, we should hve resonle ppromton For emple, f we re nterested n 7, then we cn hndle t ths wy, 7 9, so we let 9, nd we get for, whch 9

16 ppromtes to , whch when multpled y, gves , good estmte for 7 Sometmes, closed epresson for the coeffcents s desred (nd cn e found), lthough t my e dffcult to fnd the pttern t frst Let us revst the coeffcents we just hve computed:,, 8,, 5 8, 7 5, 4, nd t frst the pttern does elude us 48 But let us go ck to the defnton: ( ) ( ) ( ) 5 ( ) ( ) 5 ( )!! Cn we smplfy ths further? Perhps smplfy s the wrong word, nd wht we re tryng to do s reduce the epresson to more fmlr functons If we could reduce everythng to the fctorl, we would e t pece We need to understnd then the product of consecutve odds: ( )! 5 ( ) 4 ( 4) ( )! ( ) ( ) ( ) ( ( )) nd so we conclude tht ( ) ( )! ( )!! ( )! ( )!, for ny Now whether ths epresson s computtonlly cceptle depends very much on our control of the fctorl But we do hve wht s clled closed epresson Newton Solves Equtons Another erly mthemtcl contruton usng clculus-type resonng s hs useful wy to fndng roots of functons The pursut of fndng solutons to equtons s n ncent prt of our suject, nd methods heve een developed through the ges To wt, the Mesopotmns hd process for ppromtng squre roots tht we now revst Let n e postve numer The de ehnd the computton of ts squre root s smple Tke ny guess, let us cll t Then compnon guess s n snce n n, nd tht s the property we re lookng for So ny tme we hve guess, we relly hve two guesses When wll we hve succeeded? When our two guesses re close to one nother, for then we re ndeed close to squre root Wht should we do wth the two guesses then? Wht s more resonle thn to verge them? Indeed, tht s wht we do Ths s n terton, never-endng process Ths de dd not prtculrly seem troulesome to the Bylonns,

17 7 ut to ther successors, the Greeks, wth ther more rgorous requrements, nfnte processes ndeed seemed untdy Returnng to the lgorthm, more precsely, gven n tke to e ny (postve) ntl guess for ts squre root As mentoned ove, we re gong to keep refnng ths guess y defnng sequence tht ctully does converge to Defne k k n k For emple, let n 57,94,, An ntl guess tht s very d wll just prolong the lgorthm, ut not deter t The followng tle llustrtes how quckly we wll hve qute few decml plces correctly computed Rememer, wht we wnt s the two guesses to e close to ech other Wht we see n these numers of course depends on who we re A Mesopotmn would perhps ccept ths nswer s fnshed, whle some Greeks would perhps not ccept t s ever fnshed By Newton s tme, polynomls of rtrry degree nd nfnte seres hve ecome cceptle Two technques for solvng such equtons were lredy wdely used These stem from t lest the tme of Islm f not erler They re the secton method nd the method of flse poston from whch nother vrnt cme out lter clled the secnt method To fnd soluton to p ( ) oth methods strt wth n nd gven where p( ) nd p( ) dsgree n sgn, one s postve, one s negtve Hence one epects root etween nd Both procedures shrnk the ntervl where the root s locted The secton method smply goes to the mdpont etween nd,, nd decdes y checkng the sgn t the mdpont n whch of the two hlves the root s locted n We wll llustrte ths technque frst (the emple s due to Newton) n We re solvng 5 We strt wth nd s our orgnl guess, nd then go from there The tle s frly self-eplntory Bsecton Method - + New New Vlue

18 8 The method of flse poston s more complcted n tht t uses the secnt to the grph gven y the two pont nd In more detl, t tkes s ts new guess the ntersecton of the -s wth the lne gong through the ponts (, p( )) nd (, p( )) Thus, for the method of flse poston method, the new pont s gven y the epresson: Method of Flse Poston - + New New Vlue followng the tngent lne to the functon t guess p( ) p( ) c p( ) p( ) Ovously, the computton of the new guess s hrder, ut s we cn see, t lest n ths cse, the method of flse poston works consderly fster thn the secton method Tcklng the sme cuc, we get Newton took ths one step further ecuse he could compute tngents effcently nd one could refer to Newton s Method s the Tngent Method The reson for ths nme s ecuse t fnds ts new guess y Strt wth one guess (s opposed to two guesses necessry n the other methods), nd then follow one's nose y usng the tngent lne t the pont of the orgnl guess How do we smply fnd the new guess,? We follow the tngent lne t the pont of the grph correspondng to our ntl guess: Geometrclly then, we hve n de of where to locte our new ppromton But how do we fnd the pont effcently? Eslyuse the slope: = rse run, f( ) f '( ), nd solvng for, we get slope of the tngent lne

19 9 f( ) f '( ) Once we get, we cn use the sme epresson to get, nd then, etceter, contnung the terton When do we stop? If there s not much chnge from one to the net, most proly we re close to root, nd we cn stop When we pply t to the cuc we looked t efore we get the followng very short tle But we lso should remrk tht dependng on the nture of the equton, the method cn e unstle, nd not led to root t llthe curous reder my ttempt to fnd root of y Newton's method Nevertheless, the method s so useful t s progrmmed n most hnd-held clcultors Newton s f ( ) Needless to sy, the method just descred hs een polshed through tme, nd we now spend some tme descrng wht Newton orgnlly dd He used one of the orgnl des ehnd clculus Tht de s gnorng terms of hgher order thn, n other words, gnorng everythng ecept lner termsctully, tht s wht ppromtng curve y the tngent lne s ll out Newton ws n epert t tht technque We llustrte, nlytclly, hs orgnl thnkng ehnd hs method wth the sme emple, the cuc 5 We strt wth one ppromton,, the sme s ove, nd so we let p, nd otn p p p rememer s supposed to e root Ignorng ll ut the lner term, we get p, p nd we hve etter ppromton Agn, we let p, nd susttutng, we get p p p, nd thus, y gn consderng only the lner term, we hve tht p 5487, whch quckly gves us n estmte for just s efore Newton would use ths technque n ndeftgle nd unrelentng wys Newton fnds the Seres for the Sne & the Cosne

20 Most of us tke Tylor's seres epnsons very much for grnted Yet ths most wonderful nd powerful theorem could not hve even een thought of wthout enough emples to hnt t ts estence Before he could fnd the seres for the sne functon, Newton computed the seres for the rcsne Consder the crcle y on the frst qudrnt And tke n rtrry pont P on the s, t dstnce from the orgn O Then we know the heght of the crcle t tht pont s ( ), whch y hs noml theorem Newton could epnd nto He could then ntegrte ths seres to otn the totl shded re OPQR, consstng of the trngle OPQ nd the secton of the crcle OQR : But the re of the trngle OPQ s known, = And thus the secton of the crcle OQR s gven y the dfference of the two seres: But, nd cos sn( ), nd so rcsn( ), nd we hve tht rcsn( ) One of the est erly emples s provded y the seres epnson of the sne (nd lso the cosne)whch Newton developed Strt wth the seres of the Arcsne tht we hve ove: rcsn( ) The de s to solve for n terms of, sn( ) At ll tmes, we wll elmnte ll nonlner terms, contnung wth the sc de ehnd hs method

21 Droppng ll nonlner terms, we hve s frst ppromton,, whch s the pcture ndctes fts the sne functon for smll s We re gong to llustrte the method y usng the frst four terms of the seres for the Arcsne: 5 4 We let now Thus, 5 7 p, nd we get: ( p) ( p) ( p) 5( p) Ignorng the hgher order terms nd solvng for p, we get tht: p nd so hgher order terms,, p hgher order terms = Now we let p, nd susttute n, nd we otn: p p 4 5 p 5 p 7 And so when we epnd,,,8 5 hgher order terms n 75,49 hgher order terms n 5, 74, 74,49 + hgher order terms n p = p Smplfyng, we hve In fct, the hgher order terms n re,8 7 +5,9,84 9,58,8 +,88,4 87, ,9 7, Smlrly, for the other frcton

22 5 hgher order terms n ( hgher order terms n ) p When solvng for p nd gnorng nythng ut the lowest term order we get p 5 5, nd so we hve Contnung n ths wy, he eventully predcted the correct pttern: sn( )! 5! 7! 9!! From ths seres he used the fct tht cos( ) sn ( ), nd susttuted the seres for the sne nto the seres for clculton n solutely formdle Newton dd ths n order to estlsh 4 8 cos( )! 4!! 8!! Dd he know tht the dervtve of the sne functon s the cosne? Perhps not, snce he would most proly hve used the dervtve to fnd the seres for the cosne nsted To e strctly ccurte hstorclly, we hve gven modern representton of Newton s ctul clcultons At tht tme t ws more customry to vew the sne s correspondng to n rc more thn n ngle so n fct rdns were not needed

23 Net we gve the pctures for the net eght ppromtons of the sne seres And we see stedy mprovement n our ppromtons As we end ths chpter, we need to menton tht the foundtons of clculus were prolemtc n oth sdes of the Englsh Chnnel But, ronclly, t ws the rrognce of Hlley (of Hlley s Comet nd frend of Newton s) tht would prompt Bshop Berkeley to utter these words to dscredt clculus: He who cn dgest second or thrd fluon, second or thrd dfference, need not, methnks, e squemsh out ny pont n Dvnty And wht re these fluons? The veloctes of evnescent ncrements And wht re these evnescent ncrements? They re nether fnte qunttes, nor qunttes nfntely smll, nor yet nothng My we not cll them ghosts of deprted qunttes? These crtcsms would resonte wth the mthemtcl communty for long tme yet they proceeded to use clculus n powerful nd surprsng wys