AP Questions Tay/Mac #1 Name:__________________________________ 1) Use a 6th degree Maclaurin polynomial to approximate cos(.4) (1) n n x converges? 2) What are all values of x for which the series n 2 ln n (A) -e < x ≤ e (B) -1 ≤ x < 1 (C) -e ≤ x < e (D) -1 < x ≤ 1 (E) -1 ≤ x ≤ 1 3) The Taylor series for sin( x 2 ) centered at x = 0 is x2 (A) (1)k x 2 k 1 k 0 (2k 1)! (B) (1)k x 2 k k 0 (2k 1)! (D) 1 (1) k x 2 k 1 x k 1 (2k 1)! (E) (1)k x 4 k k 0 (2k 1)! (1)k x 2 k 1 (2k )! k 0 (C) 4) The figure to the right shows the graph of y = f(x) and y = T(x) where T(x) is a Taylor polynomial for f(x) centered at zero. Which of the following statements are true? I. T(0.5) is a good approximation for f(0.5) II. T(1.5) is a good approximation for f(1.5) III. T(0) = f(0) (A) I only (B) II only (C) III only (D) I and III only (E) I, II, and III 5) The first three nonzero terms in the Taylor series about x = 0 of xe-x are (A) x x 2 x3 2! (B) x x 2 x3 2! (C) x x 2 x3 2! (D) x x 2 x3 2! (E) 1 x x2 2! (1)n 1 x 2 n 1 , then f’(x) = (2n 1)! n 0 6) For all x if f(x) = (A) (1)n 1 x 2 n n 0 (2n 1)! (B) (1)n x 2 n (2n)! n 0 (D) (1)n 1 x 2 n (2n)! n 0 (E) (1)n x 2 n n 0 (2n 1)! (1)n 1 x 2 n n 0 (2n 2)! (C) x3 7) Let E be the error when the Taylor polynomial T(x) = x is used to approximate f(x) = sinx at x = 0.5. Which of the 3! following is true? (A) |E| < 0.0001 (B) 0.0001 < |E| < 0.0003 (D) 0.005 < |E| < 0.007 (E) 0.07 < |E| (C) 0.0003 < |E| < 0.005 (8) The Taylor Series of a function f(x) about x = 3 is given by f ( x) 1 3( x 3) 5( x 3)2 7( x 3)3 (2n 1)( x 3) n ... ... 1! 2! 3! n! What is the value of f’’’(3)? (A) 0 (B) 1.167 (C) 2.5 (D) 5 (E) 7 AP Questions Tay/Mac #2 Name:__________________________________ 2012 AP Test 2010 AP Test 2011 AP Test AP Questions Tay/Mac #3 Name:__________________________________ The Maclaurin series for ex is ex = 1 x x 2 x3 xn ... ... The continuous function f is defined by 2 6 n! 2009 AP Test e( x 1) 1 for x≠1 and f(1) = 1. The function f has derivatives of all orders at x = 1. f ( x) ( x 1)2 2 (a) Write the first four nonzero terms and the general term of the Taylor series for e( x 1) about x = 1. (b) Use the Taylor series found in part (a) to write the first four nonzero terms and the general term of the Taylor series for f about x = 1. (c) Use the ratio test to find the interval of convergence for the Taylor series found in part (b). (d) Use the Taylor series for f about x = 1 to determine whether the graph of f has any points of inflection. 2 2007 AP Test Let f be the function given by f ( x) e x2 . (a) Write the first four nonzero terms and the general term of the Taylor series for f about x = 0. 1 x 2 f ( x) . x 0 x4 (b) Use your answer to part (a) to find lim x (c) Write the first four nonzero terms of the Taylor series for e t 2 dt about x = 0. Use the first two terms of your 0 1/ 2 answer to estimate e t 2 dt . 0 1/ 2 (d) Explain why the estimate found in part (c) differs from the actual value of e 0 t 2 dt by less than 1 . 200 The function f is defined by the power series x 2 x 2 3x3 (1)n nx n f(x) = ... ... 2 3 4 n 1 2006 AP Test for all real numbers x for which the series converges. The function g is defined by the power series x x 2 x3 (1)n x n g(x) = 1 ... ... 2! 4! 6! (2n)! for all real numbers x for which the series converges. (a) Find the interval of convergence of the power series for f. Justify your answer. (b) The graph of y = f(x) – g(x) passes through the point (0,-1). Find y’(0) and y’’(0). Determine whether y has a relative minimum, a relative maximum, or neither at x = 0. Give a reason for you answer.