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# Series

# 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* + *cx*^{2} 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 *n*th term, or general term, of the Taylor series for *f*(*x*) = *e*^{x} 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*) = *x*^{1/3} centered at *x* = 1.