AP Calculus Taylor Series Student Handout 2016-2017 EDITION Use the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/S_SSS Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 2 Taylor Series Students should be able to: Construct and use Taylor polynomials. Write a power series representing a given function. Derive a power series for a given function by various methods (e.g. algebraic processes, substitutions, using properties of geometric series, and operations on known series such as term by term integration or term by term differentiation). Determine the radius and interval of convergence of a power series. Use Lagrange error bound and in some cases where the signs of a Taylor polynomial are alternating, use the alternating series error bound to bound the error of a Taylor polynomial approximation to a function. Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 3 Multiple Choice: x f (x) f (x) f (x) f (x) 0 3 -2 1 4 1 2 -3 3 -2 2 -1 1 4 5 1. (calculator not allowed) Selected values of a function f and its first three derivatives are indicated in the table above. What is the third degree Taylor polynomial for f about x 1 ? 3 1 (A) 2 3x x2 x3 2 3 3 2 1 3 (B) 2 3x 1 x 1 x 1 2 3 3 2 2 3 (C) 2 3x 1 x 1 x 1 2 3 (D) 2 3x 1 3x 1 2 2x 1 3 2. (calculator not allowed) The third-degree Taylor polynomial for the function f about x 0 is 2 3 T (x) 3 4x 2x 3x Which of the following tables gives the values of f and its first three derivatives at x 0 ? (A) x f (x) f (x) f (x) f (x) 0 3 -8 6 -12 (B) x f (x) f (x) f (x) f (x) 0 3 -4 2 -3 (C) x f (x) f (x) f (x) f (x) 0 3 -4 4 -18 (D) x f (x) f (x) f (x) f (x) 0 3 -4 4 -9 Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 4 3. (calculator not allowed) x 42n What is the radius of convergence of the series ? n n0 3 (A) 23 (B) 3 (C) 3 3 (D) 2 (E) 0 4. (calculator not allowed) 2 4 6 2n x x x n x For x 0 , the power series 1 1 3! 5! 7! (2n 1)! (A) cos x (B) sin x sin x (C) x 2 (D) ex ex x x2 (E) 1 e e 5. (calculator not allowed) What is the coefficient of x4 in the Taylor series for cos2 x about x 0 ? 1 (A) 2 1 (B) 8 1 (C) 3 3 (D) 4 Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 5 6. (calculator allowed) (5) sin x The function f has derivatives of all orders for all real numbers, and f (x) e . If the fourth-degree Taylor polynomial for f about x 0 is used to approximate f on the interval [0,1], what is the Lagrange error bound for the maximum error on the interval [0,1]? (A) 0.008 (B) 0.019 (C) 0.023 (D) 0.025 7. (calculator not allowed) Let f be the function given by f ()xx ln(3 ). The third-degree Taylor polynomial for f about x 2 is (2)(2)xx23 (A) (2)x 23 (2)(2)xx23 (B) (2)x 23 (C) (2)(2)(2)xx23 x (2)(2)xx23 (D) (2)x 23 (2)(2)xx23 (E) (2)x 23 8. (calculator not allowed) What is the polynomial approximation for the value of sin 1 obtained by using the fifth- degree Taylor polynomial about x 0 for sin x ? 11 (A) 1 224 11 (B) 1 24 11 (C) 1 35 11 (D) 1 48 11 (E) 1 6 120 Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 6 9. (calculator not allowed) 1 What is the coefficient of x2 in the Taylor series for about x 0? 1 x 2 1 (A) 6 1 (B) 3 (C) 1 (D) 3 (E) 6 10. (calculator not allowed) If f ()xx sin(2), x which of the following is the Taylor series for f about x 0? xxx357 (A) x ... 2! 4! 6! 41664xxx35 7 (B) x ... 2! 4! 6! 832128xx35 x 7 (C) 2...x 3! 5! 7! 222xxx468 (D) 2x2 ... 3! 5! 7! 8xx46 32 128 x 8 (E) 2x2 ... 3! 5! 7! 11. (calculator not allowed) xxx456 xn 3 A function f has Maclaurin series given by ... ... Which of the 2! 3! 4! (n 1)! following is an expression for f ()?x (A) 3sinx xx 32 (B) cos(x2 ) 1 (C) x22cos xx (D) x232exxx 2 (E) exx 2 1 Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 7 12. (calculator allowed) Let P()xxxxx 32345 5 7 3 be the fifth-degree Taylor polynomial for the function f about x. What is the value of f (0)? (A) 30 (B) 15 (C) 5 5 (D) 6 1 (E) 6 13. (calculator allowed) Let f be a function with ff(3) 2, (3) 1, f (3) 6, and f (3) 12. Which of the following is the third-degree Taylor polynomial for f about x 3? (A) 2 (xx 3) 3( 3)23 2( x 3) (B) 2 (xx 3) 3( 3)23 4( x 3) (C) 2 (xx 3) 6( 3)23 12( x 3) (D) 232x xx23 (E) 2612x xx23 Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 8 Free Response : 14. (calculator allowed) (4) x hx() hx() hx() hx() hx() 1 11 30 42 99 18 488 448 584 2 80 128 3 3 9 753 1383 3483 1125 3 317 2 4 16 16 Let h be a function having derivatives of all orders for x 0. Selected values for h and its first four derivatives are indicated in the table above. The function h and these four derivatives are increasing on the interval 13. x (a) Write the first degree Taylor polynomial for h about x 2 and use it to approximate h(1.9). Is this approximation greater or less than h(1.9) ? Explain your answer. (b) Write the third-degree Taylor polynomial for h about x = 2 and use it to approximate h(1.9). (c) Use the Lagrange error bound to show that the third-degree Taylor polynomial for h about x 2 approximates h(1.9) with an error less than 310. 4 Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 9 15. (calculator not allowed) 2 Let f be the function given by f ()xe x . (a) Write the first four nonzero terms and the general term of the Taylor series for f about x 0 . 1()x2 fx (b) Use your answer from part (a) to find lim . x0 x4 x 2 (c) Write the first four nonzero terms of the Taylor Series for et dt. Use the first two 0 1/2 2 terms of your answer to estimate et dt . 0 1 2 (d) Explain why the estimate found in part (c) differs from the actual value of 2 edtt by 0 1 less than . 200 Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 10 16. (calculator not allowed) Let f be a function with derivatives of all orders and for which f(2) = 7. When n is odd, the nth derivative of f at x = 2 is 0. When n is even and n > 2, the nth derivative at x = 2 is (1)!n given by f ()n (2) . 3n (a) Write the sixth-degree Taylor polynomial for f about x = 2. (b) In the Taylor series for f about x = 2, what is the coefficient of (2)x 2n for n 1? (c) Find the interval of convergence of the Taylor series for f about x = 2. Show the work that leads to your answer. Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 11 17. (calculator not allowed) The Maclaurin series for a function f if given by (3)nn11 3 (3) xxnn x23 3 x ... x ... and converges to f ()x for x R , where R n1 nn2 is the radius of convergence of the Maclaurin series. (a) Use the ratio test to find R. (b) Write the first four nonzero terms of the Maclaurin series for f ', the derivative of f. Express f ' as a rational function for x R . (c) Write the first four nonzero terms of the Maclaurin series for ex . Use the Maclaurin series for ex to write the third-degree Taylor polynomial for gx() efx () x about x 0 . Copyright © 2016 National Math + Science Initiative, Dallas, Texas. All rights reserved. Visit us online at www.nms.org 12 Taylor Series Reference Taylor series provide a way to find a polynomial “look-alike” to a non-polynomial function.