The response shown in Figure 2 is obtained from the following state equation

By using a partition size of 0.5, calculate the values for state x by integrating the curve shown in Figure 2 at each time interval using the Midpoint Rule (from 0-2.5s)

Therefore:

So we deduce that

The response shown in Figure 2 is obtained from the following state equation

By using a partition size of 0.5, calculate the values for state x by integrating the curve shown in Figure 2 at each time interval using Simpson's Rule (from 0-2.5s)

Therefore:

So we deduce that

The response shown in Figure 2 is obtained from the following state equation

By using a partition size of 0.5, calculate the values for state x by integrating the curve shown in Figure 2 at each time interval using the Trapezoid Rule (from 0-2.5s)

Therefore:

So we deduce that:

Find the absolute error of the value of the integral of:

From 0 to 2 seconds
Comparing the value estimated by the Trapezoid Rule and Simpson's Rule, against a calculated value for the integral

From the previous question calculate the relative error for both the Trapezoid and Simpson's Rule

From the following set of data calculate the interpolated

So our final answer is