Math Activity 9( Due with Fial Eam) Usig first ad secod Taylor polyomials with remaider, show that for, 8 Usig a secod Taylor polyomial with remaider, fid the best costat C so that for, C 9 The th Derivative Test : Suppose that f has cotiuous derivatives ad f c f c f c, but f c Case I: If is eve ad z f c, the from Taylor s Theorem, we kow that f f f c c for some z betwee ad c Or equivaletly,! f z f f c c Sice f is cotiuous ad f c, for close! f z to c, c This meas that f f c, or f f c for! close to c So f has a local miimum at c a) Ivestigate Case II: If is eve ad f c b) Ivestigate Case III: If is odd ad f c Test as a local etremum for the followig fuctios: a) b) si c) si d) cos 6 f 5 Let a) Fid the secod Maclauri polyomial for f b) Fid the fourth Maclauri polyomial for f c) Fid the fourth Taylor polyomial cetered at for f 6 Eplai why the polyomial the fuctio graphed: caot be the fourth Maclauri polyomial for 8
7 Let M a a a be the secod Maclauri polyomial geerated by the fuctio f graphed below Determie the sigs of a, a, ad a 8 Give the graph of the differetiable fuctio f, which has a miimum at ad iflectio poit at, determie the sigs of the coefficiets i the followig Taylor polyomials for f: P a a a a) b) P a a a c) P a a a d) P a a a 9 How accurate are the followig Maclauri polyomial approimatios if? a) e b) si 6 5 6 actually M 6 c) l d) 8
For f si Ad the remaider is fied value of,, the th Maclauri polyomial is f!!! P f z! R, where f z f z!! f z R, but si z ; k cos z ; k So for a si z ; k cos z ; k f z, so R! a) Perform the ratio test o the series! b) What does the result of part a) tell you about lim! for ay value of? c) What does the result of part b) tell you about R lim Use the first Maclauri polyomial with remaider for f to get a iequality betwee f ad?, with ad Fid the fourth Taylor polyomial cetered at for the fuctio f, ad show that it represets f eactly Fid the third Taylor polyomial cetered at for the fuctio show that it represets f eactly The fourth Maclauri polyomial for si, P the coefficiet of a) Show that if 5 is zero So, the si R 6 R 65 f 5, ad, is really a third degree polyomial sice b) Approimate 5 si d with P d, ad give a upper boud o the error 5
5 Fid Maclauri series for the followig fuctios: a) cos b) e c) ta d) si e) e f) cos 6 Epress the followig atiderivatives as ifiite series: a) si cos d b) d c) e d 7 Use series to approimate the followig defiite itegrals to withi of their eact values: d b) a) cos e d 8 Use series to evaluate the followig limits: ta a) lim b) lim cos e c) si lim 6 5 ta d) lim si e) lim cos si ta lim si ta f) 9 Use multiplicatio, divisio, or a trigoometric idetity to fid at least the first three ozero terms i the Maclauri series of the followig fuctios: a) e cos b) sec c) si ta d) ta Use Maclauri series to fid the sum of the followig series: b)! a) c) 6!! d) 5! e) 9 7 8!!! 5!
l l g) f f) l!! si si si!!! 6 8! 5! 7! 9! h) f si i) f Fid the followig derivatives of the give fuctio at :! a) 5 f, f si b) 5 f, si f c) e) 6 f, f cos 9 f, f e d) f) 7 f, t f e dt f, f l Give the two Maclauri series: a b ad a ad b for,,,, fid a equatio relatig {Hit: Partial fractios} What is the coefficiet of i the Maclauri series for e? Cosider the improper itegral e e 6 d For large, e e e 6 e lim lim So the improper itegral coverges e, ad a) Make the substitutio u e to covert the itegral ito a differet improper itegral b) Use the series l the improper itegral ad the fact that 6 to fid the value of
5 Cosider the power series f a, where a ad a a a) Fid the first four terms ad the geeral term of the series for b) What fuctio is represeted by this power series? c) Fid the eact value of f 6 Suppose that f has derivatives of all orders for all umbers ad that f, f 7, ad f 5 a) Fid the third Maclauri Polyomial for f, ad use it to approimate f b) Fid the fourth Maclauri Polyomial for the fuctio g if g f f, 7 Show that the series coverges to if, ad coverges to if This meas that the fuctio defied by the series is discotiuous at {Hit: Use Mathematical Iductio to show that s for } Everyoe is familiar with the Quadratic Formula(Babylo, circa 6 B C) with a, the solutio(s) is give by a b c b b ac a For Most people are ot familiar with the Cubic Formula(Italy, circa 5 AD) c d, a solutio is give by with a, divide through by a to get ay by cy d b substitute y For d c d d c d For a geeral cubic, 7 7 y by cy d b to get b b b b bc 7 Now b c d, which epads ito c The Cubic Formula ca be applied to the last equatio, ad subtractig b from its solutios gets you back to solutios of the origial equatio There is also a Quartic Formula, but it was prove that there is o Quitic Formula or ay higher power formula like the previous oes that epress the solutios usig roots of the coefficiets Usig substitutios, it was show that the geeral quitic could be trasformed 5 ito a The Germa mathematicia Gotthold Eisestei(8-85) at the age of 5 foud a power series solutio of the reduced quitic
8 Determie the iterval of covergece of Eisestei s Quitic Power Series Solutio: 5! a!! 9 Fid all fuctios which ca be represeted by a power series cetered at which solve the f f, ad determie their iterval of covergece fuctioal equatio {Hit: If f a a a a a, the the fuctioal equatio leads to 6 8 a a a a a a a a a a Match the correspodig coefficiets, ad solve for them} e 6 ; ad! 6 ; Use the previous series to fid e! e e e e Maclauri series for cosh ad sih Use these series to fid the eact value of!! 6!! 5! 7! f f For fuctios defied usig series of the form where each f cotiuous, it was show by Weirstrass that if there is a coverget series f M, the f is a cotiuous fuctio Further, if the series f same property, the a) Show that the fuctio f b) Show that the fuctio f formula for its derivative c) Show that the fuctio f f is differetiable, ad f f si is cotiuous everywhere is M with has the si is differetiable everywhere, ad give a si is cotiuous everywhere
d) Show that the curret method is ot eough to show that f differetiable si is Euler calculated the value of the itegral used the sie series to get si l d i the followig maer First he l l l l l 5 7 6! 5! 7! si l l l l l l! 5! 7! The he itegrated term by term to get 6 l l l d d d sil d l! 5! 7! Use the fact that l d!, alog with the series for the iverse taget to get the eact value of the itegral a) Fid the smallest iteger,, so that umber lim cos si e e is a o-zero b) For the value of from part a), what s the value of the limit? {Hit: cos si e e 6 5 7! 6!! 5! 7!!!!! }