AP Calculus BC
So You Want More Practice, eh?
Name_________________________
1. Show that the function f satisfies the hypotheses of the Mean Value Theorem on the
given interval. Find each value of c that satisfies the Mean Value Theorem.
f(x)  3x2  x  5 , [-1, 4]
If numbers
are “bad”
2. You are designing a rectangular poster to contain 256 in 2 of printing with a 3-in margin
at the top and bottom and a 2-in margin at each side. What overall dimensions will
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minimize the amount of paper used? Round your answer to the nearest inch.
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3. A 15-foot ladder is leaning against a building as shown, so that the top of the ladder is
at (0, y) and the bottom is at (x, 0). The ground is slippery so the ladder is falling down
at a rate of 12 feet per second at the instant when x = 9 feet. Find the rate at which
the ladder is moving away from the building at this instant.

4. The side of a cube measures 10  0.5 cm. Using differentials, estimate the percentage
error in the volume calculation if this measurement for the side is used.
5. A receptacle is in the shape of an inverted square pyramid. It has a 6” x 6” square base
and a height of 10”. Suppose that the receptacle is being filled with water at the rate
of 0.2 cubic inches per second. How fast is surface area of the water increasing when
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the water is 2 inches deep? ( V   h )
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6. For f(x) =
a. f
2x  7 , x0 = 9, and dx = 0.1, find the following:
b. df
c. the error
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7. If f  x    x  1  x  2  , then find the value of x at each point where f has a:
a. local maximum
b. local minimum
c. point of inflection
8. If the point (1, 6) is a point of inflection of the curve y  x3  ax2  bx  1, find the value
of a and the value of b.
9. If g is a differentiable function such that g(x) < 0 for all real numbers x and if
f  x    x2  4  g  x  , which of the following is true.
(a)
(b)
(c)
(d)
(e)
f has a relative maximum at x = -2 and a relative minimum at x = 2.
f has a relative minimum at x = -2 and a relative maximum at x = 2.
f has relative minima at x = -2 and x = 2.
f has relative maxima at x = -2 and x = 2.
It cannot be determined if f has any relative extrema.
10. If the base of a triangle is increasing at a rate of 3 in/min while its height h is
decreasing at a rate of 3 in/min, which of the following must be true about the area A
of the triangle?
(a)
(b)
(c)
(d)
(e)
A is always increasing.
A is always decreasing.
A is decreasing only when b < h.
A is decreasing only when b > h.
A remains constant.
11. Let f  x   x3  3a2x  2a3 where a is a positive constant.
(a) Find the intervals where f(x) is decreasing.
(b) Find the relative maximum value of the function.
(c) Find the point of inflection.
For #12 – 14, suppose f   x   2x  1  x  5  .
12. Use an interval test to show the signs off   x  .
True or False. Justify your answer with a graph.
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_____ 13. The slope of f   x  is increasing to the left of  and decreasing to the
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right of  .
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_____ 14. f   x  is decreasing then increasing at x = 5.
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_____ 15. If f   x    x  3  x  7  , then f  x  has points of inflection at x = -3 and 7.