e to approximate (using 4

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1 Review: Taylor Polyomials ad Power Series Fid the iterval of covergece for the series Fid a series for f ( ) d ad fid its iterval of covergece Let f( ) Let f arcta a) Fid the rd degree Maclauri polyomial b) Approimate arcta() usig the polyomial of part a Fid the th ozero term of the Taylor series for f () l cetered at = 5 Use the Maclauri series for terms) the value of e d 6 Fid a series cetered aroud = for e to approimate (usig f( ) 7 Let f( ) ad g ( ) a) Fid a power series aroud, for f b) For what values does the above series coverge? c) Use the series i part a) to determie a power series for g d) For what values of does this series coverge? Use a appropriate idetity to fid a series for f () cos 9 Fid the third o-zero term of the Taylor polyomial cetered at for f cos Fid the radius of covergece of the power series Determie the iterval of covergece of the series Fid a power series epasio for Use the third Maclauri polyomial to approimate the value of e Roud your aswer to four decimal places Let the fuctio f( ) of covergece of f ( ) d Fid the iterval 5 Fid a power series, cetered at, for the fuctio f 6 Fid the power series for cos cos 7 Usig the trigoometric idetity si ad the power series for cos, fid a power series for the fuctio f si Fid the Taylor series, cetered at, for the fuctio f l 9 Fid a Taylor series cetered at zero for the fuctio f l Let f be the fuctio defied by f( ) for all! values of for which the series coverge (a) Fid the radius of covergece of this series (b) Use the first three terms of this series to fid a approimatio of f ( ) (a) Fid the first four ozero terms i the Taylor series epasio about = for f ( ) (b) Use the results foud i part (a) to fid the first four ozero terms i the Taylor series epasio about = for g( ) (c) Fid the first four ozero terms i the Taylor series epasio about for the fuctio h such that h ( ) (a) Fid the first five terms i the Taylor series about = for f( ) (b) Fid the iterval of covergece for the series i part a) (c) Use partial fractios ad the result from part a) to fid the first five terms i the Taylor series about for g ( )

2 Let f be the fuctio give by f() t t ad G be the fuctio give by G() f () t dt (a) Fid the first four ozero terms ad the geeral term for the power series epasio of f (t) about t = (b) Fid the first four ozero terms ad the geeral term for the power series epasio of G() about = (c) Fid the iterval of covergece of the power series i part (b) (Your solutio must iclude a aalysis that justifies your aswer) Let f be a fuctio that has derivatives of all orders for all real umbers Assume f (), f (), f (), ad f () (a) Write the secod-degree Taylor polyomial for f about = ad use it to approimate f (7) (b) Write the third-degree Taylor polyomial for f about = ad use it to approimate f () (c) Write the secod-degree Taylor polyomial for f, the derivative of f about = ad use it to approimate f () / 5 Let f be the fuctio give by f e (a) Write the first four ozero terms ad the geeral term for the Taylor series epasio of f about = (b) Use the result from part (a) to write the first three ozero terms ad the geeral term of the series / e epasio about = for g() = 6 Let P75 6 be the fourth-degree Taylor polyomial for the fuctio f about Assume f has derivatives of all orders for all real umbers (a) Fid f () ad f (b) Write the secod degree Taylor polyomial for f about ad use it to approimate f (c) Write the fourth degree Taylor polyomial for g() f(t)dt about (d) Ca f () be determied from the iformatio give? Justify your aswer 7 Let f be a fuctio that has derivatives of all orders for all real umbers Assume f () 5, f, f, ad f (a) Write the third-degree Taylor polyomial for f about = ad use it to approimate f () (b) Write the fourth-degree Taylor polyomial for g, where g f, about = (c) Write the third-degree Taylor polyomial for h, where h( ) f ( t )dt, about = The Taylor series about = 5 for a certai fuctio f coverges to f() for all i the iterval of covergece The th derivative of f at = 5 is give by ( ) ( )! f (5), ad f ( 5 ) ( ) (a) Write the third-degree Taylor polyomial for f about = 5 (b) Fid the radius of covergece of the Taylor series for f about = 5 9 The power series coverges for which values of a = b - < < c - < d - e is ay real umber Which of the followig series is the power series epasio for f cos a b c d 7 7 e 5 Which are the values of c for which the series 9 c coverges a c < b < c < c c < 9 d c < 9 or c > 9 e c < or c > g ta? The Maclauri series for is What is the Maclauri series for a b f c d e

3 The Maclauri series! fuctio? 9 represets which a cos b si c cos9 d ta e e If the first five terms of the Taylor epasio for f 5 about = are 7 6 a b c 9 d 6 e, the 5 Which of the followig series diverge? I 5 7 II III l f = a I oly b II oly c I ad II oly d I ad III oly e I, II, ad III 6 The sith degree term of the Taylor series for about = has coefficiet a -/ b -/6 c /7 d /6 e -/6 f e 7 Which of the followig is a term i the Taylor series about = for the fuctio f cos? (A) (B) (C) (D) 5 6 (E) 6 5 Fid the values of for which the series coverges (A) = (B) 5 (C) 5 (D) 5 (E) All real umbers The coefficiet of (A) 6 (B) i the Taylor series for (C) (D) (E) The Taylor polyomial of order at = for f is (A) (B) 6 (C) (D) 6 (E) e at = is The fuctio f has a Taylor series about = that coverges to f () for all i the iterval of covergece The th derivative of f at = is give by! f for, ad f (a) Write the first four terms ad the geeral term of the Taylor series for f about = (b) Fid the radius of covergece for the Taylor series for f about = Show the wor that leads to your aswer (c) Let g be a fuctio satisfyig g() = ad g'() = f () for all Write the first four terms ad the geeral term of the Taylor series for g about = (d) Does the Taylor series for g as defied i part (c) coverge at =? Give a reaso for your aswer 5 The fuctio / is defied by the power series 6 f for all real!! 5! 7! umbers Fid f ad f Determie whether f has a local maimum, a local miimum, or either at = Give a reaso for your aswer f cos 9 Let (A) (B) (C) (E) The series diverges Evaluate f (D) 9 Fid the sum of the geometric series 9 (A) 5 The series series for (B) 5 (C) (D) (E) 7 is the Maclauri!!! (A) l (B) l (D) e (E) e (C) e

4 Aswers: 7 9, f( ) d C ; - < a b a b - < < c d 5 cos cos 9! 5! 6 [-, 5) () () e e ()!!!! (, ] 5 6! cos 7 si cos cos ; cos!! cos cos si cos!!!!!! si! 9 5 a b a b c a 6 b c 7 5 l( ) d 6 a 5 7 b 5 7 a T ( ) f (7) b T ( ) f() () 65 c T ( ) f () 5 5 a b e!!! e!!! e!!!!!!!!! c - < <

5 f 6a f() = P() = 7 f! b c P() 75 P g, P(t)dt P() 6 f( ) t 5t t dt 5 7t t t t ( ) d No The iformatio give provides values for f ( ), f ( ), f ( ), f ( ), ad f ( ) oly 7 a b c P P ( f)( ) 5 ( g)( ) P( f)( ) P ( h)( ) 5t t dt f() P ( f)() 5 5t t t 5 5 () () () a b!!! ( ) f(5), f(5), f(5) f ( 5 ) ( ) () () (5) a! ( ) P ( f,5)( ) ( 5) ( 5) ( 5) 6 6 ( ) ( 5 ) The radius of covergece is ( ) lim lim 5 ( ) ( 5 ) ( ) 5 9 C D E B A C 5 D 6 A 7 C C 9 B E D D B (a) (b) Radius = (c) 9 7 (d) No, = - is outside the iterval of covergece, which is - < < 5 f ad f Sice f ad f, f has a local maimum at = by the Secod Derivative Test 5