Lagrange Error Bound Worksheet

Lagrange Error Bound Worksheet

Read each question carefully and show all your work. Use the Lagrange Error Bound to determine the maximum possible error in Taylor series approximations.

1. State the formula for the Lagrange Error Bound (also known as the Remainder Estimation Theorem) for a Taylor series centered at c.

2. For a Taylor series approximation of a function f(x) about x=a, if the (n+1)th derivative of f(x) is bounded by M on the interval [a, x], which of the following represents the Lagrange Error Bound?

3. Find the Taylor polynomial of degree 2 for f(x) = e^x centered at x = 0.

4. Use the Taylor polynomial from Question 3 to approximate e^(0.5).

5. Use the Lagrange Error Bound to find an upper bound for the error in the approximation of e^(0.5) using the Taylor polynomial of degree 2 centered at x = 0. (Hint: Consider the interval [0, 0.5] and the third derivative of e^x).

6. The Lagrange Error Bound provides the exact error of a Taylor series approximation.

7. The Lagrange Error Bound is useful for determining the   number of terms needed to achieve a desired level of accuracy for a Taylor series approximation.

8. Consider the Taylor series for sin(x) centered at x=0. If you want to approximate sin(0.1) with an error less than 0.0001, what is the minimum degree of the Taylor polynomial you should use?