9. (Calculator Permitted) Use your answer from problem 1 to approximate f to four decimal places.

Transcription

1 Calculus Maimus WS.: Taylor Polyomials Taylor Polyomials O problems -5, fid a Maclauri polyomial of degree for each of the followig.. f( ) = e, =. f( ) = e, = 4. f( ) = cos, = 8 4. f( ) = e, = 4 5. f( ) = +, = 5 O problems 6-8, fid a Taylor polyomial of degree cetered at = 6. f( ) c for each of the followig. =, = 5, c = 7. f( ) = l, = 5, c = 8. f( ) = si, = 6, c = π 4 9. (Calculator Permitted) Use your aswer from problem to approimate f to four decimal places. 0. (Calculator Permitted) Use your aswer from problem 7 to approimate f (.) to four decimal places.. Suppose that fuctio f( ) is approimated ear = 0 by a sith-degree Taylor polyomial P 6 () = Give the value of each of the followig: (a) f (0) (b) f (0) (c) f (0) (d) f (5) (0) (e) f (6) (0). (Calculator Permitted) Suppose that g is a fuctio which has cotiuous derivatives, ad that g(5) =, g (5) =, g (5) =, g (5) = (a) What is the Taylor polyomial of degree for g ear 5? What is the Taylor polyomial of degree ear 5? (b) Use the two polyomials that you foud i part (a) to approimate g(4.9). For problems -6, suppose that P () = a+b+c is the secod degree Taylor polyomial for the fuctio f about = 0. What ca you say about the sigs of a, b, ad c, if f has the graphs give below?

2 7. Show how you ca use the Taylor approimatio si for ear 0 to fid! lim si Use the fourth-degree Taylor approimatio of 4 cos + for ear 0 to fid! 4! lim cos Estimate the itegral si 0 t t dt usig a Taylor polyomial for si t about t = 0 of degree 5. Multiple Choice 0. If f ( 0) = 0, f ( 0) =, f ( 0) = 0, ad ( ) Taylor polyomial geerated by f ( ) at = 0? (A) + (B). Which of the followig is the coefficiet of 4 (A) 7 8 f 0 =, the which of the followig is the third-order + (C) + (D) (B) 9 (C) 4 + cos? i the Maclauri polyomial geerated by ( ) (D) 0. Which of the followig is the Taylor polyomial geerated by f ( ) cos 4 π (A) +! 4! (D) = at π =? 4 (B) + + (C) +! 4!! 4! 4 π π + 6. (Calculator Permitted) Which of the followig gives the Maclauri polyomial of order 5 approimatio to si(.5)? (A) (B) (C) (D) Which of the followig is the quadratic approimatio for f ( ) = e at = 0? (A) + (B) (C) + + (D) +

3 Lagrage Error Boud. (a) Fid the fourth-degree Taylor polyomial for cos about = 0. The use your polyomial to approimate the value of cos0.8, ad use Taylor s Theorem to determie the accuracy of the approimatio. Give three decimal places. (b) Fid the iterval [a,b] such that (c) Could cos0.8 equal 0.695? Show why or why ot.. (a) Write a fourth-degree Maclauri polyomial for f ( ) = e. The use your polyomial to approimate e, ad fid a Lagrage error boud for the maimum error whe. Give three decimal places. (b) Fid a iterval [ ab], such that a e b.. f ( 5) = 8, f ( 5) = 0, f ( 5) = 48, ad (4) f ( ) 75 for all i the iterval [5,5.]. (a) Fid the third-degree Taylor polyomial about = 5 for f ( ). Let f be a fuctio that has derivatives of all orders for all real umbers. Assume that f ( 5) = 6, (b) Use your aswer to part (a) to estimate the value of f (5.). What is the maimum possible error i makig this estimate? Give three decimal places. (c) Fid a iterval [ ab], such that a f (5. ) b. Give three decimal places. (d) Could f (5.) equal 8.54? Show why or why ot. Review (Problems 4-7): 4. Fid the first four ozero terms of the power series fo f ( ) = si cetered at = 4 π. 5. Fid the first four ozero terms ad the geeral term for the Maclauri series for (a) f ( ) = cos( ) (b) g( ) = +

4 6. Fid the radius ad iterval of covergece for (a) = 0 ( ) ( ) (b) ( )!( 5) = 0 cos 7. Use the Maclauri series for cos to fid lim The Taylor series about = for a certai fuctio f coverges to ( ) covergece. The th derivative of f at = is give by ( )! ( + ) ( ) f ( ) = ad f ( ) = 5 (a) Write the fourth-degree Taylor polyomial for f about =. (b) Fid the radius of covergece of the Taylor series for f about =. f for all i the iterval of (c) Show that the third-degree Taylor polyomial approimates f ( 4) with a error less tha 9. Le f be a fuctio that has derivatives of all orders o the iterval (,). Assume f ( 0) =, f (0) =, f (0) = 4, f ( 0 ) =, ad f (4) ( ) 6 for all i the iterval (,). 8 (a) Fid the third-degree Taylor polyomial about = 0 for the fuctio f (b) Use your aswer to part (a) to estimate the value of f (0.5). (d) What is the maimum possible error for the approimatio made i part (b)? 0. Let f be the fuctio defied by f ( ) =. (a) Fid the secod-degree Taylor polyomial about = 4 for the fuctio f. (b) Use your aswer to part (a) to estimate the value of f (4.). (c) Fid a boud o the error for the approimatio i part (b).. Let f ( ) =0 = for all for which the series coverges. (a) Fid the iterval of covergece of this series. (b) Use the first three terms of this series to approimate f. (c) Estimate the error ivolved i the approimatio i part (b). Show your reasoig.

5 . Let f be the fuctio give by f ( ) =cos π + ad let P( ) be the fourth-degree Taylor 6 polyomial for f about = 0. (a) Fid P ( ). (b) Use the Lagrage error boud to show that f P 6 < (Review) Use series to fid a estimate for Justify. I 0 = e d that is withi 0.00 of the actual value. 4. 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 =. Show that the sith-degree Taylor polyomial for f about = 5 approimates f (6) with a error less tha Suppose a fuctio f is approimated with a fourth-degree Taylor polyomial about =. If the maimum value of the fifth derivative betwee = ad = is 0.0, that is, f (5) ( ) < 0.0, the the maimum error icurred usig this approimatio to compute f () is (A) (B) (C) (D) What are all the values of for which the series =! coverges? (A) (B) < < (C) < (D) < All real 7. The coefficiet of 6 i the Taylor series epasio about = 0 for f ( (A) 6 (B) 0 (C) 0 (D) 6 ) = si ( ) is 4 8. The maimum error icurred by approimatig the sum of the series + + L by the!! 4! 5! sum of the first si terms is (A) (B) (C) 0. (D) Noe of these

6 9. If f is a fuctio such that f ( ) si ( ) about = 0 is (A) 7! =, the the coefficiet of 7 (B) 7 (C) 0 (D) 4 i the Taylor series for f ( ) 7! 0. Now that you have fiished the last questio of the last ew cocept worksheet of your high school career, how do you feel? (Show your work) (A) Relieved (B) Very Sad (C) Euphoric (D) Tired All of these Page of