Section 11.6 – Taylor’s Formula with Remainder

But there is some stuff we left out. The remainder. We have created Taylor (Maclaurin) Polynomials to approximate other functions. For example, But there is some stuff we left out. The remainder. So how big is the remainder? Which gives us the following result (after some work):

The Lagrange Remainder of a Taylor Polynomial where z is some number between x and c This is not the most useful thing in the world. However, it does lead to something good. 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. This is good enough.

Use this range for 2 decimal places. Use a Taylor Polynomial to estimate cos(0.2) to 2 decimal places If x = 0.2, Alternating Series Test works for convergence Use this range for 2 decimal places. Use P2

Use a Taylor Polynomial to estimate with two 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.

P5