Chapter 5.5, Problem 22E

To determine

To Sketch : The graph of a function on a closed interval for that the Trapezoidal Rule obvious gives a better approximation than Simpson’s rule for $n=4$ .

Answer to Problem 22E

The graph is:

  

Explanation of Solution

Given information:

For a given value of $n$ , Simpson's rule provides a more accurate approximation to an integral than the trapezoidal method.

Graph:

Because parabolic arches, which fit function curves better than trapezoids, are used to approximate the curve in Simpson's rule, it typically provides a better approximation. The Trapezoidal Rule, however, will provide a more accurate approximation if the function consists of line segments rather than curves. Below is an example that might be used. The Trapezoidal Rule would provide an exact number for the area while Simpson's Rule would provide an estimate if the graph were divided into four subintervals, which would result in four trapezoids.

  

Therefore, the required graph of a function on a closed interval for that the Trapezoidal Rule obvious gives a better approximation than Simpson’s rule for $n=4$ is shown.