$[10$ points$]$ The function $v(t),$ is a continuous and differentiable function giving the velocity, of a car $,$ in $f\frac{t}{sec},$ traveling on a straight road, for $0t50.$ Values of the function are given for several values of $t$ at $5-$second intervals as shown in the table below.

$\backslash$table$[[t(sec),0,5,10,15,20,25,30,35,40,45,50],[v(t)(f\frac{t}{sec}),0,10,21,28,54,66,78,82,74,60,72]]$

a$)$ Give reasonable intervals of $t$ where the acceleration of the car is positive. Justify your answers.

b$)$ Find the average acceleration of the car $,$ in $f\frac{t}{se{c}^2},$ over the interval $0t50.$

c$)$ Find an approximation for the acceleration of the car, in $f\frac{t}{se{c}^2}$ at $t=40.$ Show the computation leading to your answer.

d$)$ Approximate the value of ${\displaystyle \underset{0}{\overset{50}{}}}v(t)dt$ with a Riemann sum, using the midpoints of five subintervals of equal length. Using correct units, explain the meaning of this integral.