AP Calculus
Midterm Review
Calculator Portion of Exam:
1. A particle moves along a vertical line so that its position at any time t  0 is given by y(t )  t 2  2t  3 where t
is measured in seconds and y in meters.
a. What is the initial position of the particle?
b. When is the particle moving up?
c. Find the velocity and acceleration of the particle when y(t )  0 .
d. Find the speed of the particle when y(t )  0 . Is the speed of the particle increasing or decreasing at this time?
2. A particle moves along the x-axis so that its position at any time 0  t  3 is given by
x(t )  4cos  t   1
where t is measured in seconds and x in meters.
a. Find the interval(s) on which the particle is moving to the right and the interval(s) on which the particle is
moving to the left.
b. Find the interval(s) on which the speed of the particle is increasing and the interval(s) on which the speed
of the particle is decreasing.
c. Find the maximum speed of the particle on 0  t  3 and the time(s) at which it occurs.
3x  2 x 2  x3
3. For the function: f  x  
, do the following:
x2  9
a. Identify all zeros, coordinates  x, y  of holes and equations of vertical asymptotes.
b. Find an end-behavior model for the function.
c. Identify any discontinuities as infinite or removable.
4. Find the extreme values of f  x   4cos  x  1 on 0.63  t  1.7 , and where they occur.
5. Given that f '  x  
3
on 1.6  t  4.3 , find the x-coordinate of the maximum value for f  x  on the
x 1
interval. Explain your reasoning.
6. A fire has started in a dry, open field and spreads in the form of a circle.
a. If the radius of the circle increases at a constant rate of 6 ft/min, find the rate at which the fire area is
increasing when the radius is 150 ft.
b. If the area of the circle is increasing at a constant rate of 1200 ft 2 / min , find the rate at which the
radius is increasing when the radius is 95 ft.
7. A highway patrol airplane flies 3 miles above a level, straight road at a constant rate of 120 mph. The pilot sees
an oncoming car and with radar determines that at the instant the line-of-sight distance from the plane to the car is
5 miles, the line-of-sight distance is decreasing at a rate of 160 mph. Find the car’s speed along the highway.
8. A piece of cardboard measures 10 by 15 inches. Two equal squares are removed from the corners of a 10-in. side
and two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular
box with a lid (as shown).
a. Find a formula for the volume of the box as a
function of x and state its domain.
b. Find the maximum volume that the box could have.
Answers:
1. a. y  3 meters
b. 0  t  1 sec
c. v  3  4 m/sec, a  3  2 m/sec2
d. speed = 4 m/sec. Speed is increasing since v  3 and a  3 are both negative.
2. a. Moving right on 1, 2  and left on  0, 1
b. Speed increasing on  0, 0.5  1, 1.5 and decreasing on  0.5, 1  1.5, 2 
c. Maximum speed is 12.566 and this occurs at both t  0.5 and t  1.5
3. a. Zeros: x  1 and x  0 , hole:  3,  2 , vertical asymptote: x  3
b. y   x
c. x  3 : infinite discontinuity; x  3 : removable discontinuity
4. Minimum of 5 at x  1 and maximum of 1.351 at x  1.7 .
5. The maximum of f  x  on 1.6  t  4.3 occurs at x  4.3 because f '  x   0 on the entire interval, so f  x 
will always be increasing.
6. a.
dA
dr
 1800 ft 2 / min b.
 6.316 ft / min
dt
dt
7. 80 mph
8. a. V  x   x 10  2 x  7.5  x  , Domain:  0, 5
b. Max occurs when x  1.962 and the volume is V  66.019 in 3