Module 24 - Answers Lesson 1 Answer 1 24.1.1   1 + x 1 + x + x2 1 + x + x2 + x3  Answer 2 24.1.2 The graphs appear to coincide when x is between -0.5 and 0.5.  Answer 3 24.1.3   Answer 4 24.1.4 The partial sums appear to converge to y = sin x.  Answer 5 24.1.5 [-10,10,1] x [-50,250,50] As more terms are added, the interval where each partial sum matches ex widens.  Lesson 2 Answer 1 24.2.1 f(-0.5) 0.60653 p(-0.5) = 0.625 f(0.5) 1.6487 p(0.5) = 1.625 In each case, the values of the function and the approximating polynomials are close.  Answer 2 24.2.2   Letting f(x) = ln(1 + x), the table below lists the coefficients of the fifth-order Maclaurin polynomial. The fifth-order Maclaurin polynomial approximates f(x) = ln(1 + x) for values of x close to 0.  Answer 3 24.2.3 [-2, 2, 1] x [-3, 3, 1]  Lesson 3 Answer 1 24.3.1  is the second-order Taylor polynomial for ln x centered at 1. [0, 3, 1] x [-3, 1, 1]  Answer 2 24.3.2 1 - (x - 1) + (x - 1)2 is the second-order Taylor polynomial for 1/x centered at 1. [0, 3, 1] x [-1, 3, 1]  Self Test Answer 1 The fifth-order Maclaurin polynomial for tan-1(x) is . Answer 2 The interval of convergence appears to be (-1,1). [-2, 2, 1] x [-2, 2, 1] Answer 3 The second-order Taylor polynomial centered at 2 is for ex is . Answer 4 The fourth-order Maclaurin polynomial for ex2 is . Answer 5 [-2, 2, 1] x [-1, 5, 1] ©Copyright 2007 All rights reserved. | Trademarks | Privacy Policy | Link Policy