Question

Estimate the average velocity vave of the car during the first 12 seconds. Use the Midpoint Rule with n = 3.

Answer 1

The average value of a function \(f(x)\) over an interval \([a,b]\) is given by
\[\frac{1}{b-a}\int_a^b f(x)\,\mathrm{d}x\]Here you're using \([a,b]=[0,12]\), so you're asked to approximate
\[\frac{1}{12}\int_0^{12}f(x)\,\mathrm{d}x\]With \(n=3\) partitions, you're considering as your sample points \(\{2,6,10\}\), so the integral is approximately equal to
\[\frac{1}{2}\int_0^{12}f(x)\,\mathrm{d}x\approx\frac{1}{12}\left(\frac{12}{3}f(2)+\frac{12}{3}f(6)+\frac{12}{3}f(10)\right)=\frac{f(2)+f(6)+f(10)}{3}\]