Module 22 - Power Series

Module 22 - Answers

Lesson 1

Answer 1

22.1.1

The graphs appear to coincide approximately on the interval (-1/2, 1/2).

Answer 2

22.1.2 The graph of y = 1/(1 - x) is in "thick" style and the graph of the tenth-degree partial sum
y = 1 + x + x² + ... + x¹⁰ =

(xⁿ, n, 0, 10) is in line style.

Answer 3

22.1.3

The graphs appear to converge to y = sin x around x = 0.

Answer 4

22.1.4

Answer 5

22.1.5 The graphs are shown in [-10, 10] x [-10, 10] and [0, 10] x [0, 1000] viewing windows.

As more terms are added, the interval where the partial sums are approximately the same as e^x widens. The interval of convergence for the infinite power series is (– , )

Lesson 2

Answer 1

22.2.1

The quadratic polynomial is 0.01847 greater than the actual value of y = e^x when x = -0.5 and 0.02372 less than the actual value of y = e^x when x = 0.5.

Answer 2

22.2.2

Answer 3

22.2.3 The graphs of y = ln(1 + x) (thick) and the 5th-order polynomial (line) are shown in a [-7.9, 7.9] x [-3, 3] window.

Lesson 3

Answer 1

22.3.1

Near .

Answer 2

22.3.2

Near .

Answer 3

22.3.3 The derivative of the fifth-degree Taylor polynomial centered at 1 for y = ln x is equal to the fourth-degree Taylor polynomial centered at 1 for

. This seems reasonable because

Answer 4

22.3.4 The derivative of the fifth- degree Maclaurin polynomial for y = sin x should be equal to the fourth- degree Maclaurin polynomial for y = cos x.

Self Test

Answer 1

Answer 2

The graphs are shown in a [-2, 2] x [-2, 2] window.

The interval of convergence appears to be (-1,1).

Answer 3

Answer 4

Answer 5

The 5th-order Maclaurin polynomial for