Module 24  Answers 
Lesson 1 
Answer 1 
24.1.1
1 + x 1 + x + x^{2} 1 + x + x^{2} + x^{3} 
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 e^{x} 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 fifthorder Maclaurin polynomial.
The fifthorder 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 secondorder 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 secondorder Taylor polynomial for 1/x centered at 1.
[0, 3, 1] x [1, 3, 1] 
Self Test 
Answer 1 
The fifthorder 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 secondorder Taylor polynomial centered at 2 is for e^{x} is . 
Answer 4 
The fourthorder Maclaurin polynomial for e^{x2} is . 
Answer 5 
[2, 2, 1] x [1, 5, 1] 
©Copyright 2007 All rights reserved.  Trademarks  Privacy Policy  Link Policy 