a. Use the mean value theorem. 16 falls between 12 and 20, so
[tex]v'(16)\approx\dfrac{v(20)-v(12)}{20-12}=\dfrac{240-200}8=5[/tex]
(Don't forget your units - 5 m/min^2)
b. [tex]v(t)[/tex] gives the Johanna's velocity at time [tex]t[/tex]. The magnitude of her velocity, or speed, is [tex]|v(t)|[/tex]. Integrating this would tell us the total distance she has traveled whilst jogging.
The Riemann sum approximates the integral as
[tex]\displaystyle\int_0^{40}|v(t)|\,\mathrm dt=12\cdot200+8\cdot240+4\cdot220+16\cdot150=7600[/tex]
If you're not sure how this is derived: we're given 5 sample points, so we can cut the interval [0, 40] into 4 subintervals. The lengths of each subinterval are 12, 8, 4, and 16 (the distances between each sample point), and the height of the rectangle approximating the area under the plot of [tex]|v(t)|[/tex] is determined by the value of [tex]v(t)[/tex] at each sample point, 200, 240, |-220| = 220, and 150.
c. Bob's velocity is given by [tex]B(t)[/tex], so his acceleration is given by [tex]B'(t)[/tex]. We have
[tex]B'(t)=3t^2-12t[/tex]
and at [tex]t=5[/tex] his acceleration is [tex]B'(5)=15[/tex] m/min^2.
d. Bob's average velocity over [0, 10] is given by the difference quotient,
[tex]\dfrac{B(10)-B(0)}{10-0}=\dfrac{700-300}{10}=40[/tex] m/min
