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mamaluj [8]
3 years ago
10

CALCULUS EXPERT WANTED

Mathematics
1 answer:
Vlad [161]3 years ago
6 0

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

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