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# Taylor and Maclaurin Series Exercises

### Example 1

Let f(x) = sin(x). By taking derivatives, find a function g(x) of the form g(x) = a + bx + cx2 that has the same value, slope, and second derivative as f when x = 0.

Graph f and g on the same axes.

### Example 2

Find the 5th-degree Taylor polynomial centered at 0 for cos x.

### Example 3

What is the Maclaurin series for f(x) = cos x (a.k.a. the Taylor series for f(x) = cos x near x = 0) ?

### Example 4

What is the nth term, or general term, of the Taylor series for f(x) = ex near x = 0?

### Example 5

(a) Find the Taylor series for .

(b) Why does your answer make sense?

### Example 6

Which is the Taylor series for the function f(x) near x = 5 ?

(A)

(B)

(C)

(D)

### Example 7

Find the Taylor series for the function f(x) = ln x at a = 1.

### Example 8

Which of the following functions could be the second degree Taylor polynomial for the function f(x) near -π/2?

### Example 9

Find the 5th degree Taylor polynomial for the function f(x) = cos x at .

### Example 10

Find the first four nonzero terms of the Taylor series for f(x) = x1/3 centered at x = 1.