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AP Calculus AB
Applications of Derivatives with Optimization and Differentials
Study Guide (also use last couple tests and study guides)
When necessary, round your answers to 3 decimal places.
1. An open box is made from cutting equal squares from the corners and turning up the
sides. What size square should be cut from a 12” by 36” piece of cardboard to maximize
the volume?
2. If the product of two positive numbers is 176, what do the two numbers need to be to
make the sum of the first one plus 3 times the second one a minimum?
3. What point on the graph of f ( x)  x 2 is closest to the point (4, 2)?
v
where v is
84  .02v 2
the speed of the traffic in miles per hour. What speed would maximize the flow rate?
4. The flow rate (cars per hour) on a busy highway is given by F 
5. A rancher has 300 feet of fencing with which to build 3 equal
adjacent rectangular corrals. What should the dimension of each
corral be to maximize the enclosed area?
AP Calculus AB
Applications of Derivatives with Optimization and Differentials
Study Guide (also use last couple tests and study guides)
Solve using differentials.
9. The radius of a sphere is measured to be 8 inches, with a possible error of .03 inch. Use
differentials to approximate the following:
a. the maximum possible error in the volume of the sphere,
b. the relative error in finding the volume,
c. the maximum possible error in the surface area of the sphere
d. the relative error in finding the surface area.
Use the Table to answer number 10.
x
f(x)
6.3
2.05
6.4
2.38
6.5
2.79
6.6
3.34
10. Let f be a function such that f ' ' ( x)  0 for all x in [6, 7]. Given the selected values of f
in the table above, which much be true about f ' (6.5) ?
(A)
(B)
(C)
(D)
(E)
f ' (6.5)  0
0  f ' (6.5)  3.3
3.3  f ' (6.5)  4.1
4.1  f ' (6.5)  5.5
f ' (6.5)  5.5