Calculus Mock Exam Practice

Question

For 0 ≤  t ≤  6 seconds, a screen saver on a computer screen shows two circles that start as dots and expand outwards.

1) At the instant that the first circle has a radius of 9 centimeters, the radius is increasing at a rate of 3/2 cm/sec. Find the rate at which the area of the circle is changing at that instant.

2)The radius of the first circle is modeled by g(t) = 12 - 12e^-0.5t for 0 ≤  t ≤  where g(t) is measured in centimeters and t is measured in seconds. At what time t is the radius of the circle increasing at a rate of 3 cm/sec.

3) A model for the radius of the second circle given b the function f for 0 ≤  t ≤  6, where f(t) is measured in cm and t is measured in seconds. The rate of change of the radius of the second circle is given by f¹(t) = t²-4t+4. Based on this model, by how many cm is the radius of the second circle increasing from time t =0 to t = 3?



Answer 1

1) A = πr²

A' = 2πrr'

when r = 9 and r' = 3/2

A' = 2π(9)(3/2) = 27π cm²/sec

2) g(t) = 12-12e^-t/2

g'(t) = -12(-1/2)e^-t/2 = 6e^-t/2

When g'(t) = 3

6e^-t/2=3

-t/2 = ln(1/2)

t = -2ln(1/2) = 1.386 seconds

3) f'(t) = t²-4t+4

∫₀³ t²-4t+4 dt = t³/3-4t²/2+4t |₀³ = 9-18+12 = 3 cm