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**Section 11.6 – Taylor’s Formula with Remainder**

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**The Lagrange Remainder of a Taylor Polynomial**

where z is some number between x and c The Error of a Taylor Polynomial where M is the maximum value of on the interval [b, c] or [c, b]

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**Let f be a function that has derivatives of all orders on the**

Interval (-1, 1). Assume f(0) = 1, f ‘ (0) = ½, f ”(0) = -1/4, f ’’’(0) = 3/8 and for all x in the interval (0, 1). a. Find the third-degree Taylor polynomial about x = 0 for f. b. Use your answer to part a to estimate the value of f(0.5)

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**Let f be a function that has derivatives of all orders on the**

Interval (-1, 1). Assume f(0) = 1, f ‘ (0) = ½, f ”(0) = -1/4, f ’’’(0) = 3/8 and for all x in the interval (0, 1). What is the maximum possible error for the approximation made in part b?

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**Estimate the error that results when arctan x is replaced by**

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**Estimate the error that results when ln(x + 1) is replaced by**

F ‘’’ (x) has a maximum value at x = -0.1

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**Find an approximation of ln 1.1 that is accurate to three decimal**

places. We just determined that the error using the second degree expansion is

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**Use a Taylor Polynomial to estimate cos(0.2) to 3 decimal places**

If x = 0.2, Alternating Series Test works for convergence

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**Use a Taylor Polynomial to estimate**

with three decimal place accuracy. Satisfies Alternating Series Test

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**Suppose the function f is defined so that**

a. Write a second degree Taylor polynomial for f about x = 1 b. Use the result from (a) to approximate f(1.5)

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**Suppose the function f is defined so that**

for all x in [1, 1.5], find an upper bound for the approximation error in part b if

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**The first four derivatives of**

Find the third-degree Taylor approximation to f at x = 0 Use your answer in (a) to find an approximation of f(0.5) Estimate the error involved in the approximation in (b). Show your reasoning.

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**The first four derivatives of**

Find the third-degree Taylor approximation to f at x = 0 Use your answer in (a) to find an approximation of f(0.5)

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**The first four derivatives of**

Estimate the error involved in the approximation in (b). Show your reasoning.

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Remainder Theorem. The n-th Talor polynomial The polynomial is called the n-th Taylor polynomial for f about c.

Remainder Theorem. The n-th Talor polynomial The polynomial is called the n-th Taylor polynomial for f about c.

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