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Taylor and Maclaurin Series Examples

Example 1

Graph the function f(x) = ex. Then add, on the same set of axes:

(a) the first-degree Taylor polynomial for f(x) at 0.

(b) the second-degree Taylor polynomial for f(x), centered at 0.

(c) the 3rd-degree Taylor polynomial for f(x) at 0.

(d) the 4th-degree Taylor polynomial for f(x) at 0.

(e) the 5th-degree Taylor polynomial for f(x) at 0.

Example 2

What is the Maclaurin series for f(x) = ex?

Example 3

Find the MacLaurin series for f(x) = sin x.

Example 4

Find the second-degree Taylor polynomial for  near x = 4.

Example 5

Find the Taylor series for the function sin x near .