Section 11.6 – Taylor’s Formula with Remainder

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Presentation transcript:

Section 11.6 – Taylor’s Formula with Remainder

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]

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)

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?

Estimate the error that results when arctan x is replaced by

Estimate the error that results when ln(x + 1) is replaced by F ‘’’ (x) has a maximum value at x = -0.1

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 0.000457.

Use a Taylor Polynomial to estimate cos(0.2) to 3 decimal places If x = 0.2, Alternating Series Test works for convergence

Use a Taylor Polynomial to estimate with three decimal place accuracy. Satisfies Alternating Series Test

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)

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

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.

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)

The first four derivatives of Estimate the error involved in the approximation in (b). Show your reasoning.