Taylor Series (BC Only)

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1 Studet Study Sessio Taylor Series (BC Oly) Taylor series provide a way to fid a polyomial look-alike to a o-polyomial fuctio. This is doe by a specific formula show below (which should be memorized): Taylor Series cetered at = (Maclauri Series). Let f be a fuctio with derivatives of all orders o a iterval cotaiig =. The f,cetered at =, ca be represeted by ( ) f() f() f() f() 3 f () f( )......!!! 3!! A Taylor series ca be cetered at ay other locatio as well by the formula below: Taylor Series cetered at = a Let f be a fuctio with derivatives of all orders o a iterval cotaiig = a. The f, cetered at = a, ca be represeted by ( ) f( a) f( a) f( a) f( a) 3 f ( a) f( ) ( a) ( a) ( a) ( a)... ( a)...!!! 3!! Geerally, it is ot ecessary to simplify results o the Free Respose sectio. Aswers will be simplified o the Multiple Choice sectio. The seve formulas (recommeded to be memorized) are as follows: ;... ( )... ( ) ; e......,!! 3!!! si... ( )... ( ) ; 3! 5! 7! ()! ()! 4 6 cos... ( )... ( ) ;! 4! 6! ( )! ( )! 4 l( )... ( )... ( ) ; arcta... ( )... ( ) ; () ()

2 There are three mai types of questios asked o the eam: Write a fuctio i terms of a series Fid a error boud o a th degree Taylor Polyomial Fid a iterval of covergece Taylor Series (BC oly) Studet Study Sessio Error Bouds To determie a error boud for a Taylor polyomial, first classify the polyomial as either a alteratig or o-alteratig series. Their error bouds are foud as follows: Alteratig Series Whe a series is alteratig, the error is maimized i the et uused term evaluated at the differece betwee the ceter of the covergece ad the -coordiate beig evaluated. No-Alteratig Series If a series is o-alteratig, the error is still tied up i the et term by the formula f ( z) ( Error a) where f ) () z is the maimum value that the (+) derivative ca ( )! take o the iterval. Iterval of Covergece for Taylor Series Whe lookig for the iterval of covergece for a Taylor Series, refer back to the iterval of covergece for each of the basic Taylor Series formulas. Fit your fuctio to the fuctio beig tested. Sometimes, the eam will maipulate a Taylor series to a power series before askig for the iterval of covergece. The most commo test to fid the iterval of covergece for a power series is the Ratio a Test, which says that lim L. If L <, the series coverges. If L >, the series diverges. If L =, a the test fails ad aother test should be used. Whe usig the Ratio Test, it is importat to remember that the Ratio Test oly checks the ope iterval. The edpoits of the iterval must be checked separately to determie if the iterval is ope or closed. If a series is kow to be geometric, the edpoits do ot eed to be checked sice covergece requires r - therefore the edpoits caot be icluded.

3 Multiple Choice Taylor Series (BC oly) Studet Study Sessio. (calculator ot allowed) Let f be the fuctio give by f ( ) l(3 ). The third-degree Taylor polyomial for f about is (C) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ). (calculator ot allowed) What is the polyomial approimatio for the value of si obtaied by usig the fifth-degree Taylor polyomial about for si? (C)

4 Taylor Series (BC oly) Studet Study Sessio 3. (calculator ot allowed) What is the coefficiet of 6 3 (C) 3 6 i the Taylor series for about? 4. (calculator ot allowed) If f ( ) si( ), which of the followig is the Taylor series for f about? ! 4! 6! ! 4! 6! (C) ! 5! 7! ! 5! 7! ! 5! 7!

5 5. (calculator ot allowed) A fuctio f has Maclauri series give by is a epressio for f ( )? ! 3! 4! ( )! Taylor Series (BC oly) Studet Study Sessio Which of the followig (C) 3si 3 cos( ) cos e e 6. (calculator allowed) 4 5 Let P( ) be the fifth-degree Taylor polyomial for the fuctio f about. What is the value of f ()? 3 5 (C) (calculator allowed) Let f be a fuctio with f(3), f(3), f(3) 6, ad f (3). Which of the followig is the third-degree Taylor polyomial for f about 3? (C) ( 3) 3( 3) ( 3) ( 3) 3( 3) 4( 3) ( 3) 6( 3) ( 3) 3 6

6 Free Respose Taylor Series (BC oly) Studet Study Sessio 8. (calculator allowed) h () h ( ) h ( ) h ( ) h (4) ( ) Let h be a fuctio havig derivatives of all orders for. Selected values for h ad its first four derivatives are idicated i the table above. The fuctio h ad these four derivatives are icreasig o the iterval 3. (a) Write the first degree Taylor polyomial for h about = ad use it to approimate h (.9). Is this approimatio greater or less tha h (.9)? Eplai your aswer (b) Write the third-degree Taylor polyomial for h about = ad use it to approimate h (.9). (c) Use the Lagrage error boud to show that the third-degree Taylor polyomial for h about 4 approimates h (.9) with a error less tha 3.

7 9. (calculator ot allowed) Taylor Series (BC oly) Studet Study Sessio Let f be the fuctio give by f ( ) e. (a) Write the first four ozero terms ad the geeral term of the Taylor series for f about =. (b) Use your aswer from part (a) to fid f( ) lim. 4

8 Taylor Series (BC oly) Studet Study Sessio (c) Write the first four ozero terms of the Taylor Series for e t dt. Use the first two terms of your aswer to estimate / e t dt. (d) Eplai why the estimate foud i part (c) differs from the actual value of tha. t e dt by less

9 . (calculator ot allowed) Taylor Series (BC oly) Studet Study Sessio Let f be a fuctio with derivatives of all orders ad for which f() = 7. Whe is odd, the th derivative of f at = is. Whe is eve ad >, the th derivative at = is give by ( ) ( )! f (). 3 (a) Write the sith-degree Taylor polyomial for f about =. (b) I the Taylor series for f about =, what is the coefficiet of ( ) for? (c) Fid the iterval of covergece of the Taylor series for f about =. Show the work that leads to your aswer.

10 Taylor Series (BC oly) Studet Study Sessio. (calculator ot allowed) The fuctio f is defied by f( ) 3. The Maclauri series for f is give by , which coverges to f ( ) for. (a) Fid the first three ozero terms ad the geeral term for the Maclauri series for f (). (b) Use your results from part (a) to fid the sum of the ifiite series (c) Fid the first four ozero terms ad the geeral term for the Maclauri series represetig f () t dt. (d) Use the first three ozero terms of the ifiite series foud i part (c) to approimate / / f () t dt. What are the properties of the terms of the series represetig f () t dt that guaratee that this approimatio is withi, of the eact value of the itegral?

11 . (calculator ot allowed) Let f be the fuctio give by f( ) 6e /3 for all. Taylor Series (BC oly) Studet Study Sessio (a) Fid the first four ozero terms ad the geeral term for the Taylor series for f about =. (b) Let g be the fuctio give by g ( ) ftdt ( ). Fid the first four ozero terms ad the geeral term for the Taylor series for g about = (c) The fuctio h satisfies h ( ) kf( a) for all, where a ad k are costats. The Taylor series for h about = is give by h ( ).! 3!! Fid the values of a ad k.

e to approximate (using 4

e to approximate (using 4 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

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