Question

5.7. Numerical Integration

Short answer

an

Step-by-step solution

Part (a) Step 1. A function f on an interval [a, b] for which the left sum and the trapezoid sum are over-pproximations for every n.

The left sum is over-approximation when the function is decreasing monotonically on the interval$[a,b]$.

The trapezoid sum is over-approximation when the function is concave upwards on the interval$[a,b]$.

Thus, to have a function whose left sum and trapezoid sum is an over approximation, create a function which is concave upwards and decreasing on the interval .

Part (a) Step 2. Draw the graph of the function which which is concave upwards and decreasing on the interval [a, b].

Assume the graph of the function $y=x^2(x+2)$on the interval$[-4,3]$.

The graph is shown below:

Part (b) Step 2. A function f on an interval [a, b] for which the right sum and the trapezoid sum are over-approximations for every n.

The right sum is over-approximation when the function is increasing monotonically on the interval $[a,b]$.

The trapezoid sum is over-approximation when the function is concave upwards on the interval $[a,b]$.

Thus, to have a function whose right sum and trapezoid sum is an over approximation, create a function which is concave upwards and increasing on the interval $[a,b]$.

Part (b) Step 2. Draw the graph of the function which which is concave upwards and increasing on the interval [a, b].

Assume the graph of the function $y=x^2(x+2)$on the interval $[-4,3]$.

The graph is shown below: