AP Calculus BC
Review Unit 5 Applications of Derivatives
Name: ________________________
NO calculator allowed unless otherwise indicated. On the test, you will be required to show all of your work. The test will be divided into
calculator and no calculator parts. This review is not comprehensive. Please look back over your notes, your homework, and your quizzes to
help you study for the test. This review will count as part of your test grade!
Topics to review for test:

Absolute extrema on an interval

Mean Value Theorem

Relative extrema

Intervals of increasing and decreasing




Concavity and points of inflection
Second derivative test
Graphing derivatives
Graphing functions from derivatives
1. The figure shows the graph of f’(x), the derivative of a function, f.
The derivative is continuous and f’(2) < 0. Which of the following is true?
I.
II.
III.
The function has at least one relative minimum point.
The function has at least one relative maximum point.
The function has no relative maximum point.
(A) I only
(B) II only
(C) III only
(D) I and II
(E) I and III
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2. The figure shows the graph of the derivative of a function.
The derivative is never negative. Which statement is false?
(A)
(B)
(C)
(D)
(E)
The function has no relative maximum value.
The function has no relative minimum value.
The function is always concave up.
The function has exactly three points of inflection.
The function is increasing for all x.
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3. The figure shows the graph of f’, the derivative of a function f.
Which of the following statements must be false about f?
(A) f has a relative maximum at x = 2.
(B) f has a point of inflection at x = 2.
(C) f has a critical point at x = 4.
(D) f has a relative minimum at x = 0.
(E) f is concave up for all x < 2.
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f ( x)  3x  x3 for all values of x on the closed interval 0  x  3 , for what value of x is the instantaneous
rate of change of y with respect to x the same as the average rate of change in the interval 0  x  3 ? Calculator
4. If
allowed.
(A) 0.984 only
(B) 1.244 only
(C) 2.727 only
(D) 0.984 and 2.804
(E) 1.244 and 2.727
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5. The graph of a function f is shown. Which of the following is true?
(A) An absolute maximum does not exist
(B) The function is continuous for all values in the domain
(C) The function is differentiable for all values in the domain
(D)
lim f ( x)  lim f ( x)
x a 
x a
(E) The function does not have an absolute minimum value
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6. If
  for all x, then determine the smallest value of x at which f has a relative minimum. Calculator
f ' ( x)  sin 2 x
allowed.
(A) 0
(B) 0.651
(C) 1.652
(D) 2.236
(E) 2.651
7. The graph of the derivative of the function
f
,
f ' , is shown in the figure.
At which value(s) of x does the function f have a relative maximum in the interval
0  x  7?
(A) 2, 4, and 6
(B) 2 and 4
(C) 2 and 6
(D) 4 only
(E) 3 only
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8. At which point on the graph of f shown is f ' ( x )  0 and f ' ' ( x )  0 ?
(A) A
(B) B
(C) C
(D) D
(E) E
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9. The graph of the derivative of f , f ' ( x ) , is shown in the figure.
Which of the following could be the graph of f (x ) ?
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10. If f ( x)  2 sin( x)  sin( 4 x) for all values of x on the closed interval [0, 2], for how many value(s) of x is the
instantaneous rate of change of y with respect to x the same as the average rate of change in the interval
[0, 2]? Calculator allowed.
(A) none
(B) One
(C) Two
(D) Three
(E) Four
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12. The first derivative of the function
f is defined by f ' ( x)  sin( x 2  1) for 2  x  4 . On what interval(s)
2  x  4 ? Calculator allowed.
(A) 2  x  2.299 only
(B) 2  x  2.618 only
(C) 2.299  x  2.903 only
(D) 2  x  2.618 and 3.162  x  3.625
(E) 2.299  x  2.903 and 3.401  x  3.835
is f increasing for
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13. The first derivative of the function
f is defined by f ' ( x)  sin( x 2  1) for 2  x  4 . On what interval(s)
2  x  4 ? Calculator allowed.
(A) 2  x  2.299 only
(B) 2.618  x  3.162 and 3.625  x  4
(C) 2.299  x  2.903 only
(D) 2  x  2.618 and 3.162  x  3.625
(E) 2.299  x  2.903 and 3.401  x  3.835
is f concave down for
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14. The first derivative of the function
f is defined by f ' ( x)  sin( x 2  1) for 2  x  4 . How many points
of inflection does the graph of f have? Calculator allowed.
(A) One
(B) Two
(C) Three
(D) Four
(E) Five
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15. Let f(x) be a twice differentiable function defined for all real numbers. f(x) is not constant. The table gives values of the
first and second derivative at several points. The function has no critical points other than those mentioned in the table.
x
-2
-1
0
1
2
f(x)
1
5
2
0
-10
f'(x)
0
-10
0
5
0
(A) List all relative minima. Justify your answer.
f"(x)
-5
33
28
0
-10
(B) List all relative maxima. Justify your answer.
16.
f ( x)  x 2 ( x 2  4)
on
1  x  2
(A) Find all critical numbers on the interval
1  x  2 .
(B) Locate all absolute extreme values on the interval
1  x  2 . Label each and justify your answer.
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17.
f ( x )  cos x
on
 1
0, 6 
 1
 6  .
(A) Find all critical numbers on the interval 0,
 1
 6  . Label each and justify your answer.
(B) Locate all absolute extreme values on the interval 0,
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18. Consider the following graph.
Which of the points are absolute maxima?
Which of the points are absolute minima?
Which of the points are relative maxima?
Which of the points are relative minima?
Which of the points are neither a maximum or minimum?
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19. A test plane flies in a straight line with positive velocity v(t), in miles per minute, where v is a differentiable function of t.
Selected values of v(t) for [0,40] are shown in the table.
t(minutes)
0
5
10
15
20
25
30
35
40
v(t) (mpm)
7.0
9.2
9.5
7.0
4.5
2.4
2.4
4.3
7.3
Based on the values in the table, what is the smallest number of instances at which the acceleration of the plane could equal
zero on the open interval (0,40)? Justify your answer.
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20.
f ( x)  3 x 5  5 x 3  1
(A) Find all critical points for the function.
(B) Find the intervals on which the function is increasing and decreasing. Justify your answer.
(C) Find all x-coordinate(s) where the function has relative extrema. Justify your answer.
21. Let
g ( x)  x  2 sin x
on the interval
[  ,  ]
(A) For what values of x does the graph of g have a horizontal tangent?
(B) On what intervals is the graph of g concave down? Justify your answer.
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22. A function f is continuous on the interval [-5, 5]. The first and second derivatives have the values given in the table.
X
(-5, -1)
-1
(-1, 2)
2
(2, 4)
4
(4, 5)
f’(x)
Positive
0
Negative
Negative
Negative
0
Positive
f”(x)
Negative
Negative
Negative
0
Positive
Positive
Positive
y
(A) What are the x-coordinates of all of the relative maxima and minima of f on [-5, 5]?
Justify your answer
(B) What are the x-coordinates of all points of inflection of f on the interval [-5, 5]?
Justify your answer.
x










(C) Sketch a possible graph for f which satisfies all of the given properties.
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23. The figure below shows the graph of f’, the derivative of a function f.
The domain of f is the set of all real numbers x such that  7 
x  7.
(A) For what value(s) of x does f have a relative maximum? Why?
(B) For what value(s) of x does f have a relative minimum? Why?
y
(C) On what intervals is the graph of f concave upward? Use f’ to justify your answer.
x












(D) Suppose that f(1) = 0. In the xy-plane provided, draw a
sketch that shows the general shape of the graph of the function f on the
open interval (-3, 5).
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24. Given the graph of f, graph f’.
f(x)
f’(x)
y
x

