AP Calculus – Optimization Problems
Show your work in order to receive credit!
1.
Name: ___________________________________
A point moves on the x-axis in such a way that its velocity at time t, (t > 0) is given by v 
ln t
. At
t
what value of t does v attain its maximum? (No calculator)
1
A. 1
B. e 2
2. The derivative of f ( x) 
C. e
D. e
3
2
E. There is no maximum value for v.
x 4 x5
 attains its maximum value at x =
3
5
(No calculator)
A. -1
3.
B. 0
C. 1
D. 4/3
E. 5/3
The absolute maximum value of f(x) = x3 – 3x2 + 12 on the closed interval [ -2, 4] occurs at x =
(No calculator)
A. 4
B. 2
C. 1
D. 0
E. -2
4.
The volume of a cylindrical tin can with a top and bottom is to be 16π cubic inches. If a minimum
amount of tin is to be used to construct the can, what must be the height, in inches, of the can?
(No Calculator)
A. 23 2
5.
C. 23 4
D. 4
E. 8
If y = 2x – 8, what is the minimum value of the product xy?
(Calculator permitted)
A. -16
6.
B. 2 2
B. -8
C. -4
D. 0
E. 2
The maximum acceleration attained on the interval, 0 < t < 3 by the particle whose velocity is given by
v(t) = t3 – 3t2 + 12t + 4.
(No calculator)
A. 9
B. 12
C. 14
D. 21
E. 40
Free-response questions:
1.
a. For -4 < x < 4, find all values of x at which the graph of f has a point of inflection. Justify your answer.
b. For -4 < x < 4, find the value of x at which f has an absolute maximum. Justify your answer.
2. The rate at which people enter an auditorium for a rock concert is modeled by the function R given by R(t) =
1380t2 – 675t3 for 0 < t < 2; R(t) is measured in people per hour. No one is in the auditorium at time t = 0,
when the doors open. The doors close and the concert begins at time t = 2.
Find the time when the rate at which people enter the auditorium is a maximum. Justify your answer.