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1 CALCULUS BC WORKSHEET ON POWER SERIES. Derive the Taylor series formula by fillig i the blaks below. 4 5 Let f a a c a c a c a4 c a5 c a c What happes to this series if we let = c? f c so a Now differetiate f to fid f ad f c. f f c so a Differetiate agai, ad fid f ad f c. f f c so a Now fid f ad f c. f f c = so a Do you see a patter? What is f c? f c so a Now substitute your results ito 4 5 f a a c a c a c a4 c a5 c a c f c c c c O problem, fid a Taylor series for f cetered at the give value of c. Give the first four ozero terms ad the geeral term for the series.. f e, c TURN->>>
2 O problem - 4, fid a Taylor series for f cetered at the give value of c. Give the first four ozero terms. (You do ot eed to give the geeral term.). f si, c 4. f cos, c O problems 5 8, fid a Maclauri series for f. Give the first four ozero terms ad the geeral term for each series. 5. f si 6. f cos 7. f e 8. cos f si Hit: Use the fact that si.
3 CALCULUS BC WORKSHEET ON POWER SERIES Work the followig o otebook paper. Do ot use your calculator. Show all work. O problems -, fid a series for the give fuctio. Give the first four ozero terms ad the geeral term for the series.. f e cetered at. g e cetered at. Use your aswer to problem to fid a series for h e cetered at O problem 4-5, fid a series for the give fuctio. Give the first four ozero terms. (You do ot eed to give the geeral term.) 4. f cos cetered at 5. g cos cetered at 6. Let f be the fuctio defied by f. (a) Write the first four terms ad the geeral term of the Taylor series epasio of f about =. (b) Use the result from part (a) to fid the first four terms ad the geeral term of the series epasio about = for l. (c) Use the series i part (b) to fid a approimatio for l so that the error i your approimatio is less tha. How may terms were eeded? Justify your aswer. 7. Fid the Taylor series for f cetered at = 4 if first four ozero terms ad the geeral term. f! 4 ad f 4. Give the
4 Aswers e e e. e e!!!.!!! We ca get this aswer by substitutig i place of ito the Maclauri series for because the derivatives of of e evaluated at =. e evaluated at have the same values as the derivatvives e.!!! ! 4! 6! 5.!! 6. (a) 4 (b) 4 (c) l. Two terms are eeded. Sice the terms of the series are alteratig, decreasig i 8 magitude, ad havig a limit of, Error
5 CALCULUS BC WORKSHEET ON POWER SERIES Work the followig o otebook paper. No calculator ecept o 6(c). O problems, fid a power series for the give fuctio, cetered at the give value of c, ad fid its iterval of covergece. Give the first four ozero terms ad the geeral term of the power series f, c. f, c. f, c 4 4. (a) Fid the first four ozero terms of the power series for f si, cetered at. 6 (You do ot eed to write the geeral term.) (b) Fid the first four ozero terms of the power series for g si, 6 cetered at. (You do ot eed to write the geeral term.) 6 (c) Could the aswers to (a) ad (b) be foud by substitutio, or is it ecessary to fid derivatives ad use the Taylor formula? Eplai. 5. (a) Fid the first four ozero terms of the Taylor series about = for f. (b) Use the results foud i part (a) to fid the first four ozero terms i the Taylor series about = for g. (c) Fid the first four ozero terms i the Taylor series epasio about = for the fuctio h such that h ad h Let f be the fuctio give by f e. (a) Fid the first four ozero terms ad the geeral term of the power series for f about =. (b) Fid the iterval of covergece of the power series for f about =. Show the aalysis that leads to your coclusio. (c) Let g be the fuctio give by the sum of the first four ozero terms of the power series for f about =. Show that f g. for The Maclauri series for f is give by!! 4!! (a) Fid 7 f, f, f, ad f. (b) For what values of does the give series coverge? Show your reasoig. (c) Let g f. Write the Maclauri series for g, showig the first three ozero terms ad the geeral term. (d) Write g i terms of a familiar fuctio without usig series. The, write f i terms of the same familiar fuctio.
6 Aswers to Worksheet o Power Series (a) 4 or or 9 7 (b) 6 6 6!! ! 5! 7! (c) The aswer to (a) must be foud by takig derivatives ad usig the Taylor formula. Sice oe of the derivatives of the fuctio o (a) give a value of whe evaluated at 6, substitutig ito the Maclauri series for si will ot give the correct series. The aswer to (b) ca be foud by substitutig ito the Maclauri series for si because all of the derivatives of g whe evaluated at 6 will give the same values that the derivatives of si gives whe evaluated at, BC (a) (b) (c) BC 4 (a)!!! (b) Coverges for all (c) Sice f is a alteratig series whose terms decrease i magitude ad have a limit of, f g.. by the Alteratig Series Remaider. 7. Modificatio of 996 BC!!! 7 7! (a) f or ; f or ; f or ; f or!! 4! 4 8! 8 (b) Coverges for all (c)!!! (d) g e f e if if \
7 CALCULUS BC WORKSHEET 4 ON POWER SERIES Work the followig o otebook paper. Fid the sum of each of the followig coverget series !!!! ,. 4.! 5! 7!!! 4!! 4 5. Let f. 4 (a) For what values of does the series for f coverge? (b) Fid the first four ozero terms ad the geeral term of the power series for g f. (c) Use the series foud i part (b) to fid the value of g 5. Show the steps that lead to your aswer. 6. The fuctio f is defied by f. (a) Write the Maclauri series for f. Give the first four ozero terms ad the geeral term. For what values of does it coverge? (b) Fid the first three ozero terms ad the geeral term for the Maclauri series for f. (c) Use your results from part (b) to fid the sum of the ifiite series Show the steps that lead to your aswer. 7. Let f be the fuctio defied by f. (a) Write the Maclauri series for f. Give the first four ozero terms ad the geeral term. For what values of does it coverge? (b) Use your aswer to (a) to fid the first four ozero terms ad the geeral term of the Maclauri series for f t dt. For what values of does this series coverge? (c) Use your aswer to (b) to fid the value of 5 7. Show the steps that lead to your aswer The Maclauri series for f is give by 7!!! (a) For what values of does the series for f coverge? (b) Fid the first three ozero terms ad the geeral term for the Maclauri series for f. (c) Use your aswer to (b) to fid the value of f. Show the steps that lead to your aswer.
8 Aswers. e. si cos 5. (a) Coverges for. (b) g (c) g, the sum is Modificatio of 6 Form B BC (a). Coverges for. (b) Coverges for. 7 (c) The series is f. f or (a). Coverges for. (b) 5 7 (c) This is g., Coverges for. 5 7 g dt arcta t arcta so g arcta. t 4 8. Modificatio of 98 BC 5 (a) Coverges for all real umbers (b) !! 7 7 c f 7 7!! !! 7e
9 CALCULUS BC *** - USE SOME OF THESE FOR TEST???? ******** WORKSHEET 5 ON POWER SERIES Work the followig o otebook paper. Fid the radius ad iterval of covergece for each of the followig series. All work must be show.. 4.! Fid the Taylor series for f si cetered at (You do ot eed to give the geeral term.) 6. (a) Fid a power series for. c 5 5!. Give the first four ozero terms. f e cetered at. Give the first four ozero terms ad the geeral term. For what values of does the series coverge? (b) Use your aswer to (a) to fid a power series for g e cetered at. Give the first four ozero terms ad the geeral term. 7. (a) Fid a Maclauri series for f. Give the first four ozero terms ad the geeral term. For what values of does the series coverge? (b) Use your aswer to (a) to fid a Maclauri series for f. Give the first four ozero terms ad the geeral term. (c) Fid the sum of 8. If f is a fuctio such that for f about =. 9. The Maclauri series for f is give by f cos, fid the coefficiet of 9 i the Taylor series 5 7 9!! 5! 7!!. (a) Fid f 7. (b) Fid lim 5 f.
10 Aswers to Worksheet 5 o Power Series. 7. Cov. oly if = !! 6. (a), coverges for all!!! (b)!!! (a), coverges for 5 7 (b) (c) This is f. Sice f, f ! 9. (a) 7! or 4 (b) 5! or! 6
e to approximate (using 4
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