AP Calculus AB: Chapter 4 Review
NC1. Let f be a function that has domain; the closed interval [–1, 6] and range the closed interval [-10,2].
Let f have the derivative f ¢ that is continuous and have the graph shown in the figure above.
a. Find all values of x for which f has a relative minimum. Justify your answer.
b. Find the intervals on which f is increasing. Justify your answer.
c. Find the intervals on which f is concave up. Justify your answer.
d. Find all values of x for which f has a point of inflection. Justify your answer.
C2. A gravel plant crushes gravel into sand. The sand falls from a spout and forms a right circular cone
whose radius is always twice its height. The radius of the base is increasing at the rate of 6 inches per
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minutes. The volume of a cone with radius r and height h is given by V = p r 2 h . Find the rate at which
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the volume of the sand cone is changing when its height is 5ft. Specify units.
NC3.
The graph of a function f is shown above. Which of the following could be the graph of f ¢(x) ?
C4.
The graph of f ¢ , the derivative of f, is shown above for -2 £ x £ 5. On what intervals is f increasing?
(A) [ –2, 1] only
(B) [ –2, 3] only
(C) [ 3, 5] only
(D) [0, 1.5] and [3, 5]
(E) [ –2, –1], [ 1, 2], and [ 4, 5].
NC5. Consider the function f (x) = x k - x , where k is a real number.
a. Find f ¢(x) and f ¢¢(x) .
b. For what constant k does f(x) have a relative maximum at x = 2?
C6. Find the dimensions of a cylindrical can with a bottom but no top of volume 12 m3 that uses the least
amount of metal. V = p r 2 h and SA = p r 2 + 2p rh The radius must be between 1 and 4m.
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C7. Given f ¢(x) = xcos x 2 - x , state the intervals on which the graph of f(x) is concave down on the
interval [0,4]. Justify your answer.