Question

Q. Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

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

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

(c) A function $f$on an interval $[a,b]$for which Simpson’s Rule has no error at all.

Short answer

Part (a): An example for a function $f$on an interval $[a,b]$for which the left sum and the trapezoid sum are over-approximations is a function which is concave upwards and decreasing in the interval $\left[a,b\right]$.

Part (b): An example for a function $f$on an interval $[a,b]$for which the right sum and the trapezoid sum are over-approximations is a function which is concave upwards and decreasing in the interval $\left[a,b\right]$.

Part (c): An example for a function $f$on an interval $[a,b]$for which Simpson’s Rule has no error at all is a quadratic function.

Step-by-step solution

Part (a) Step 1. Given information

We need to write an example for a function $f$on an interval$[a,b]$for which the left sum and the trapezoid sum are over-approximations for every $n$.

Part (a) Step 2. Example for the given function

The left sum is known to be over - approximation for a function which is decreasing monotonically in the interval $[a,b]$.The trapezoid sum is known to be over - approximation for a function which is concave upwards in 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 in the interval $\left[a,b\right]$.

Part (a) Step 3. An example graph for the given function is fx=sin x;π,3π2 which is shown below,

Part (b) Step 1. An example for the given function:

The right sum is known to be over - approximation for a function which is increasing monotonically in the interval $\left[a,b\right]$.The trapezoid sum is known to be over - approximation for a function which is concave upwards in the interval $\left[a,b\right]$.

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

Part (b) Step 2. An example graph for the given function is, fx=sin x;3π2,2π which is shown below,

Part (c) Step 1. An example for the given function:

The Simpson's rule approximates the area under the curve by comparing a parabolic shape for the function. Thus, the function itself is a parabolic function, the Simpson's rule would not require any approximation. Hence the error would be $0$.

Thus to have a function, which would give no error with Simpson's rule create a quadratic function.

Part (c) Step 2. An example graph for the given function: