### Home > APCALC > Chapter 12 > Lesson 12.2.1 > Problem 12-67

12-67.

A function has derivatives of all orders for all

*. Some values of*

*are given in the table below. It is known that*

*.*

Use a trapezoidal sum with four subintervals to approximate

. Use your approximation from part (a) to estimate the value of

. Justify your estimate with your work.Use Euler’s Method, starting with

, with four steps of equal size, to approximate. Show your work.At

, the slope is . Since the step size is , then Starting at

, the next point is: If

and, write a third-degree Taylor polynomial aboutand use it to approximate.How close were your estimates of

in parts (b) through (d)? Explain why this happened.