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Taylor Series Section 9.2b

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**Do Now Find the fourth order Taylor polynomial that approximates near**

Before finding a bunch of derivatives, remember that we can use a known power series to generate another… From the Table of Maclaurin Series on p.477: Therefore,

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**Do Now Find the fourth order Taylor polynomial that approximates near**

The Taylor polynomial: Support by graphing both functions in [–3, 3] by [–2, 2]

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**Practice Problems: #16 on p.478**

Let f be a function that has derivatives of all orders for all real numbers. Assume Write the third order Taylor polynomial for f at x = 0 and use it to approximate f(0.2).

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**Practice Problems: #16 on p.478**

Let f be a function that has derivatives of all orders for all real numbers. Assume (b) Write the second order Taylor polynomial for , the derivative of f, at x = 0 and use it to approximate Second order Taylor polynomial for :

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**Practice Problems: #18 on p.478**

The Maclaurin series for f(x) is (a) Find and In this case, Similarly,

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**Practice Problems: #18 on p.478**

The Maclaurin series for f(x) is (b) Let g(x) = x f(x). Write the Maclaurin series for g(x), showing the first three nonzero terms and the general term. Multiply each term of f(x) by x: (c) Write g(x) in terms of a familiar function without using series.

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**Practice Problems: #20 on p.478**

Let and Find the first four terms and the general term for the Maclaurin series generated by f. From the Table of Maclaurin Series on p.477: So factor out 2 and substitute for

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**Practice Problems: #20 on p.478**

Let and Find the first four terms and the general term for the Maclaurin series generated by f.

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**Practice Problems: #20 on p.478**

Let and (b) Find the first four nonzero terms and the Maclaurin series for G. The first term: To find the other terms, integrate the terms of the series for f:

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**Practice Problems: #20 on p.478**

Let and (b) Find the first four nonzero terms and the Maclaurin series for G.

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In this section we develop general methods for finding power series representations. Suppose that f (x) is represented by a power series centered at.

In this section we develop general methods for finding power series representations. Suppose that f (x) is represented by a power series centered at.

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