Final answer:
To approximate the average value of the function f(t) = 1/2 + sin^2(\pi t) on the interval [0, 2], we use the midpoint rule with four subintervals. The midpoints are at t = 0.25, 0.75, 1.25, and 1.75, and the approximation is the sum of the function's values at these midpoints, each multiplied by 0.5, and then divided by the interval length of 2.
Step-by-step explanation:
We are asked to approximate the average value of the function f(t) = \frac{1}{2} + \sin^2(\pi t) on the interval [0, 2] using the midpoint rule with four subintervals. To do this, we will divide the interval into four equal parts. Each subinterval will be \frac{2}{4} = 0.5 units long.
The midpoints of the subintervals will then be at t = 0.25, 0.75, 1.25, and 1.75. The midpoint rule estimates the area under the curve by evaluating the function at these midpoints and multiplying by the subinterval width. Therefore, we will evaluate the function at each midpoint and sum the results: f(0.25) + f(0.75) + f(1.25) + f(1.75), each multiplied by 0.5 (the subinterval width).
The average value of the function is computed by taking this sum and dividing it by the length of the interval, which is 2 in this case. So, the approximate average will be \(\frac{0.5 [f(0.25) + f(0.75) + f(1.25) + f(1.75)]}{2}\).