SERIES AND ERROR n 1. for all n.

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1 SERIES AND ERROR The AP Calculus BC course descriptio icludes two kids of error bouds: Alteratig series with error boud Lagrage error boud for Taylor polyomials Both types of error have bee tested o the AP test. We will look at alteratig series first. A alteratig series is a series whose terms are alterately positive ad egative. + Examples: = 4 = = ( )!! 4!! = I geeral, just kowig that lim a = 0 tells us very little about the covergece of the series a ; = however, it turs out that a alteratig series must coverge if its terms cosistetly shrik i size ad approach zero. The Alteratig Series Test tells us how to show that a alteratig series coverges. Alteratig Series Test If a > 0, the a alteratig series + a or ( ) are satisfied: ) lim a = 0 ) {a } is a decreasig sequece; that is, a+ < a for all. = a coverges if both of the followig coditios = Note: This does ot say that if lim a 0, the series diverges by the Alteratig Series Test. The Alteratig Series Test ca oly be used to prove covergece. If lim a 0, the the series diverges by the th Term Test for Divergece, ot by the Alteratig Series Test. The Alteratig Series Remaider tells us how to estimate the error foud whe we fid the sum of the first terms of a coverget alteratig series. Alteratig Series Remaider Suppose a alteratig series satisfies the coditios of the Alteratig Series Test: amely, that lim a = 0 ad { a } is a decreasig sequece ( a+ < a). If the series has a sum S, the Remaider = R = S S a +, where S is the th partial sum of the series. I other words, if a alteratig series satisfies the coditios of the Alteratig Series Test, you ca approximate the sum of the series by usig the th partial sum, S, ad your error will have a absolute value o greater tha the first term left off, a + ( ) Ex. Approximate the sum of by usig its first six terms, ad fid the error. =! Use your results to fid a iterval i which S must lie.

2 + Ex. Approximate the sum of with a error less tha = Lagrage Form of the Remaider (also called Lagrage Error Boud or Taylor's Theorem Remaider) Whe a Taylor polyomial is used to approximate a fuctio, we eed a way to see how accurately the polyomial approximates the fuctio. f x P x R x R x = f x P x = + so Writte i words: Fuctio = Polyomial + Remaider so Remaider = Fuctio - Polyomial Taylor's Theorem: If a fuctio f is differetiable through order + i a iterval cotaiig c, the for each x i the iterval, there exists a umber z betwee x ad c such that ( f ( c) ) f f = f ( c) + f ( c)( x c) + ( x c ) ( x c ) + R ( x )!! ( + f ) ( z) + where the remaider R ( x ) (or error) is give by R = ( x c) (The Lagrage Remaider) +! Historically the remaider was ot due to Taylor but to a Frech mathematicia, Joseph Louis Lagrage R x is called the Lagrage form of the remaider. (76-8). For this reaso, Whe applyig Taylor's Formula, we would ot expect to be able to fid the exact value of z. Rather, we + would attempt to fid bouds for the derivative f z from which we will be able to tell how large the remaider R x is. Thus, for the purpose of approximatig values of a fuctio, we restate Taylor's Formula i the followig way: Taylor's Iequality P x is the th-degree polyomial approximatio for the fuctio f about x = c ad M is Suppose that the maximum value of f ( + ) o the iterval [c, b] (or [b, c] if b < c). The the error i usig the M b c +! + polyomial value P ( b ) to estimate f(b) is bouded by Taylor's Formula satisfies the iequality M R x b c + +!. That is, the remaider R Ex. Let f be a fuctio with 5 derivatives o the iterval [, ] ad assume that ( 5 ) x i f x < 0. for all x i the iterval [, ]. If a fourth-degree Taylor polyomial for f at c = is used to estimate f (), how accurate is this approximatio? Give three decimal places.

3 Ex. (a) Fid the fifth-degree Maclauri polyomial for si x. The use your polyomial to approximate si, ad use Taylor's Theorem to fid the maximum error for your approximatio. Give three decimal places. (b) Fid a iterval [a, b] such that a si b. Give three decimal places. (c) Could si equal 0.9? x Ex. (a) Write the fourth-degree Maclauri polyomial for f = e. The use your polyomial to approximate e, ad fid a Lagrage error boud for the maximum error whe x. Give three decimal places, (b) Fid a iterval [a, b] such that a<e<b. Give three decimal places. Ex. 4 The fuctio f has derivatives of all orders for all real umbers x. Assume that f = 6, f = 4, f = 7, f = 8 (a) Write the third-degree Taylor polyomial for f about x =, ad use it to approximate f(.). Give three decimal places. (b) The fourth derivative of f satisfies the iequality ( 4 ) f x 9 for all x i the closed iterval [,.]. Use the Lagrage error boud o the approximatio of f(.) foud i part (a) to fid a a f. b. Give three decimal places. iterval [a, b] such that (c) Could f(.) equal 6.9? Show why or why ot.

4 CALCULUS BC POWER SERIES & ERRORS WORKSHEET. (a) Fid the fourth-degree Taylor polyomial for cos x about x = 0. The use your polyomial to approximate the value of cos 0.8, ad use Taylor's Theorem to determie the accuracy of the approximatio. Give three decimal places. a cos 08. b. (b) Fid a iterval [a, b] such that (c) Could cos ( 08. ) equal 0.695? Show why or why ot. x. (a) Write the fourth-degree Maclauri polyomial for f = e. The use your polyomial to approximate e, ad fid a Lagrage error boud for the maximum error whe x. Give three decimal places. (b) Fid a iterval [a, b] such that a e b.. Let f be a fuctio that has derivatives of all orders for all real umbers x Assume that f 5 = 6, f 5 = 8, f 5 = 0, f 5 = 48, ad for all x i the iterval [5, 5.]. ( 4 ) f x 75 (a) Fid the third-degree Taylor polyomial about x = 5 for f ( x ) (b) Use your aswer to part (a) to estimate the value of f(5.). What is the maximum possible error i makig this estimate? Give three decimal places. a f 5. b. Give three decimal places. (c) Fid a iterval [a, b] such that (d) Could f ( 5. ) equal 8.54? Show why or why ot. (e) Let g x f ( x ) =. Fid the Maclauri series for g(x). (Write as may ozero terms as possible.) =. Fid the Maclauri series for (f) Let h(x) be a fuctio that has the properties h ( 0) = 5 ad h f h( x ). (Write as may terms as possible.) 4. The Taylor series about x = for a certai fuctio f coverges to f ( x ) for all x i the iterval of covergece. The th derivative of f at x = is give by f ( ) ( ) (a) Write the fourth-degree Taylor polyomial for f about x =. ( )! ( + ) = 5 (b) Fid the radius of covergece of the Taylor series for f about x =. f = ad ( ) (c) Show that the third-degree Taylor polyomial approximates f (4) with a error less tha The Taylor series about x = 5 for a certai fuctio f coverges to f(x) for all x i the iterval of ( covergece. The th derivative of f at x = 5 is give by ) ( )! f ( 5) = ad f ( 5) =. Show that the + sixth-degree Taylor polyomial for f about x = 5 approximates f(6) with a error less tha 6. Let f be a fuctio that has derivatives of all orders o the iterval (-, ). Assume ( 4 f ( 0) =, f ( 0) =, f ( 0) =, f ( 0) =, ad f ) 6 for all x i the iterval (0,). 4 8 (a) Fid the third-degree Taylor polyomial about x = 0 for the fuctio f (b) Use your aswer to part (a) to estimate the value of f(0.5). (c) What is the maximum possible error for the approximatio made i part (b)? 7. Let f be the fuctio defied by f = x. (a) Fid the secod-degree Taylor polyomial about x = 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 approximatio i part (b). 000.

5 8. Let f x = for all x for which the series coverges. = 0 (a) Fid the iterval of covergece of this series. (b) Use the first three terms of this series to approximate f. (c) Estimate the error ivolved i the approximatio i part (b). Show your reasoig. π f x = cos x+ ad let P(x) be the fourth-degree Taylor polyomial for 6 f about x = 0. (a) Fid P(x). (b) Use the Lagrage error boud to show that f P < Let f be the fuctio give by 0. Use series to fid a estimate for x e dx that is accurate to three decimal places. Justify. 0. Suppose a fuctio f is approximated with a fourth-degree Taylor polyomial about x =. If the maximum value of the fifth derivative betwee x = ad x = is 0.0, that is, ( 5 ) f x 0.0, the the maximum error icurred usig this approximatio to compute f() is (A) (B) (C) (D) (E) The maximum error icurred by approximatig the sum of the series by the sum!! 4! 5! of the first six terms is (A) (B) (C) 0. (D) (E) Noe of these

6 Aswers to Power Series Worksheet 4 x x. (a) + ; 0.00! 4! cos (b) (c) Yes, cos(0.8) ca equal sice it lies i the iterval foud i (b). 4 x x x. (a) + x ; 0.75; (b) 05. e (a) 6+ 9( x 5) + 5( x 5) + 8( x 5) (b) f (c) (d) No, f(5.) ca't equal 8.54 because it is't i the iterval foud i (c). (e) ( x ) ( x ) ( x ) ( x ) 4 4. (a) (b) 5 (c) Error < < Sice this is a coverget alteratig series with decreasig terms, we ca use the Alteratig Series Remaider, so that the error is less tha the first trucated term. 5. f ( 6) = This is a alteratig series whose terms are decreasig i size so the error ivolved i approximatig f(6) with the sixth-degree Taylor polyomial is less i magitude tha the seveth-degree term. Error < < by the Alteratig Series Remaider 6. (a) + x x + x 8 6 (c) By the Alteratig Series Remaider, the error is at most the first omitted term, 64 4 x 9 x 7x 8 x P x = + +!! 4! ( 5 ) 5 5 f z x 0 4x (b) R4 = so 5! 5! 9. (a) 5 4 R4 = = < 5 6 5! 6 5! Because this is a alteratig series whose terms decrease i value, we ca trucate after 6 terms ad have a error correct to three decimal places. x e dx + + = 0 5 (!) 7 (!) 9 ( 4!) ( 5!) E. A (b) 57 = (c) The error is at most 64 = ( x 4) x 4 7. (a) (b) (c) The maximum value of the third derivative f = o [4, 4.] occurs at x = 4 ad is 5 8x 56. The R x. =.! 8. (a) (-, ) (b) + = 4 6 6