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5.7. Numerical Integration

Short Answer

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Step by step solution

01

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 .

02

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=x2(x+2)on the interval[-4,3].

The graph is shown below:

03

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].

04

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=x2(x+2)on the interval [-4,3].

The graph is shown below:

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