Question

CALCULUS EXPERT WANTED

if u could solve letters b and d or at least explain some of it that would be pretty awesome

Answer 1

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

`v'(16)\approx(v(20)-v(12))/(20-12)=\frac{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,

`(B(10)-B(0))/(10-0)=(700-300)/(10)=40` m/min