Question

CALCULUS EXPERT WANTED

Answer 1

a. Use the mean value theorem. 16 falls between 12 and 20, so

$v'(16)\approx\dfrac{v(20)-v(12)}{20-12}=\dfrac{240-200}8=5$

(Don't forget your units - 5 m/min^2)

b. $v(t)$ gives the Johanna's velocity at time $t$. The magnitude of her velocity, or speed, is $|v(t)|$. Integrating this would tell us the total distance she has traveled whilst jogging.

The Riemann sum approximates the integral as

$\displaystyle\int_0^{40}|v(t)|\,\mathrm dt=12\cdot200+8\cdot240+4\cdot220+16\cdot150=7600$

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 $|v(t)|$ is determined by the value of $v(t)$ at each sample point, 200, 240, |-220| = 220, and 150.

c. Bob's velocity is given by $B(t)$, so his acceleration is given by $B'(t)$. We have

$B'(t)=3t^2-12t$

and at $t=5$ his acceleration is $B'(5)=15$ m/min^2.

d. Bob's average velocity over [0, 10] is given by the difference quotient,

$\dfrac{B(10)-B(0)}{10-0}=\dfrac{700-300}{10}=40$ m/min