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 tenthdegree partial sum y = 1 + x + x^{2} + ... + x^{10} = (x^{n}, 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 5thorder 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 fifthdegree Taylor polynomial centered at 1 for y = ln x is equal to the fourthdegree 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 5thorder Maclaurin polynomial for 
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