Name:_____________________________
Date:____________Pd:_____
Calculus AB: First Semester Exam Review
This review is intended to be representative of the types of problems that we have done this semester, and of the types of problems that you
will be doing for the exam. It is not a duplicate of the exam. You will still come across some problems on the exam that while you’ve never done
the specific problem, we’ve done the content many times. The exam is all multiple choice, AP-style problems. You will also want to review any
test or quiz you’ve done.
Only use a calculator where necessary – the exam will be split into calculator and noncalculator sections.
1. Write the equation of the line tangent to the graph of fx  x 4  2x2 at the point x  1 .
2. Where does the graph of y  5x 4  x 5 have a point of inflection?
y  f'x
3. The graph of f' , the derivative of f, is shown in the figure to the
right. Describe all relative extrema of f on the open interval (a, b).
a
b
4. If a function f is continuous for all x and if f has a relative
maximum at (-1, 4) and a relative minimum at (3, -2), which of the following statements must be
true?
A) The graph of f has a point on inflection somewhere between x  1 and x  3
B) f' 1  0
C) The graph of f has a horizontal asymptote
D) The graph of f has a horizontal tangent line at x  3
E) The graph of f intersects both axes
 x2
if x  3
5. Determine if the function given by fx   
is continuous and/or
6
x

9
if
x

3

differentiable at x  3 .
6. The Mean Value Theorem guarantees the existence of a special point on the graph of y  x
between (0, 0) and (4, 2). What are the coordinates of this point?


7. What are all values of x for which the function f defined by fx  x2  3 e x is increasing?
8. The function given by fx  x3  3x2 for all real numbers x has a relative maximum where?
9. A particle moves along the x-axis so that its position at time t is given by xt  t2  6t  5 .
For what value of t is the velocity of the particle zero?
10. The graph of fx  
5
is concave downward which values of x?
x 2
11. The graph of a function f is shown below.
y
x
Which of the graphs A through E could be the graph of the derivative of f ?
A)
y
B)
x
y
C)
y
x
D)
x
y
E)
x
12. The graph of the function y  x3  5x2  4x  3 sin x changes concavity where?
13. The graph of the function f shown in the figure to
the right has a vertical tangent at the point (2, 0) and
horizontal tangents at the points (1, -1) and (3, 1). For
what values of x on  2  x  4 is f not differentiable?
y
x
14. If y  2x  8 , what is the minimum value of the product xy?
15. Write the equation of the line tangent to the graph of y  x  cos x at the point (0, 1).
16. The function f given by fx  3x 5  4x3  3x has a relative maximum at where?
17. Find the intervals where f is increasing and where it is decreasing for the function given by
fx  x3  12x  24 .
18. If fx  sine x  , then f'x 
19. A railroad track and a road cross at right angles. An observer stands on the road 90
meters north of the crossing and watches a westbound train traveling at 80 meters per second.
At how many meters per second is the train moving away from the observer 8 seconds after it
passes through the intersection?
For the next two questions, refer to the graph below.
y  f'x
20. How many points of inflection does the graph of f have?
21. At what value of x does the absolute minimum of f occur?
22. If x2  xy  10 then when x  2 , find
dy
.
dx
23. The function f is continuous on the closed interval [0, 2] and has values that are given in
the table below. The equation fx  21 must have at least two solutions in the interval [0, 2] for
what value of k?
x
f(x)
a) 0
b)
1
2
c) 1
0
1
d) 2
1
k
2
2
e) 3
24. The radius of a circle is decreasing at a constant rate of 0.1 centimeter per second. In
terms of the circumference C, what is the rate of change of the area of the circle, in square
centimeters per second?
25. The figure at right shows the graph of the
velocity of a moving object as a function of time.
At which of the marked points is the speed the
greatest?
A
B
C
E
D
26. The equation of the tangent line to the curve 3x2  xy 2  12 at the point (4, 3) is:
27. If the graph of fx  2x2 
k
x
has a point of inflection at x  1 , then find the value of k.
28. What is the x-coordinate of the point of inflection on the graph of y  31 x3  5x2  24 ?
29. The graph of a twice-differentiable function f is shown to the
right. Which statement is true?
fx 
1
a) f1  f'1  f" 1
b) f1  f" 1  f'1
c) f'1  f1  f" 1
d) f" 1  f1  f'1
e) f" 1  f'1  f1
30. If the base b of a triangle is increasing at a rate of 3 inches per minute while its height h
is decreasing at a rate of 3 inches per minute, 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
31. Let f be the function given by fx  3e 2x and let g be the function given by gx  6x 3 . At
what value of x do the graphs of f and g have parallel tangents?
32. If fx  x 2x  3 , then f'x 
33. The graph of a function is shown. In the given table, indicate whether f, f’, and f” at each
marked point is positive, negative, or zero.
D
C
B
A
E
Point
A
B
C
D
E
f
f’
f”
34. The graph of the function f is shown in the figure to the right. Which of the following
statements about f is true?
a) limfx  limfx 
x2
x3
b) limfx  limfx
x2
x 4
c) lim fx   limfx
x3
x2
d) limfx   3
x 2
e) limfx   4
x 2
35. If fx  x3  x  x1 , then f' 1 
x 2  5x  8 if x  3
36. For fx   
, what value of k would make fx  continuous?
if x  3
 kx  2
d2 y
37. If x  y  25 , what is the value of
at the point (4, 3)?
dx 2
2
2
38. Let f be the function given by fx  2e 6x , and let g be the function given by gx  3x 4 .
At what value of x do the graphs of f and g have parallel tangent lines?
39. If fx  4 sin x  2 , then f' 0 
40. The equation of the line tangent to the graph of y  cos2x at x 



4
is:
41. What are all the values of x for which the function fx  x2  3x e x is decreasing?
42.


d 2 ln x 4
x e

dx
 x2  9

x  3 if x  3
43. For fx   
, what value of k would make fx  continuous at x  3 ?

if x  3
 k
Determine the limits of the following functions. If a limit does not exist, give the reason:
44. lim f(x) 
45.
46.
47. lim f(x) 
x6
lim f(x) 
x 2
lim f(x) 
x 2
x2
48. At what point on the graph of y  21 x 2 is the tangent line parallel to the line 2x  4y  3 ?
49. The table below gives the values of three functions, r, q, and s near x  0 .
x
rx 
qx 
sx 
-0.3
4.798
6.981
14.5
-0.2
4.825
6.983
17.4
-0.1
4.835
6.989
27.32
0
4.840
Und.
Und.
0.1
5.722
6.989
-15.75
0.2
5.725
6.983
-11.38
0.3
5.730
6.981
-8.42
Based on the values given, for which of the functions does it appear that the limit as x
approaches zero exists?
a) r only
b) q only
c) s only
d) r and s only
e) r, q, and s
50. Where does the graph of y  3x2  x3 have a relative maximum?
51. If, for all values of x, f'x  0 and f" x  0 , which of the following curves could be part
of the graph of f ?
a)
b)
d)
e)
c)
52. A particle moves along the x-axis so that its position function at any time t  0 is given by
xt  t 4  10t3  29t2  36t  2 . For which value of t (1, 2, 3, 4, or 5) is the speed the greatest?
53. The graphs of the derivatives of the functions f, g, and h are shown below.
y
y
y
y  h'x
y  f'x
a
y  g'x
b
x
a
b
x
a
b
Which of the functions f, g, or h have a relative maximum on the open interval a  x  b ?
54. Let f be a function defined for all real numbers x. If f'x  
decreasing?
4  x2
x 2
, then where is f
x
55. Let f be a function such that lim
h 0
f2  h  f2
 5 . Which of the following must be true?
h
I. f is continuous at x  2
II. f is differentiable at x  2
III. The derivative of f is continuous at x  2
a) I only
b) II only
c) I and II only
d) I and III only
e) II and III only
56. If a person holding a radar gun is watching an airplane through binoculars, and the radar
shows that the plane is flying at an altitude of 5 miles, and at a speed of 180 miles per hour.
Then 10 minutes after the plane flew over the person holding the radar, at what rate is the
angle of elevation between the ground and the binoculars changing?
57. If the radius of a sphere is increasing at the rate of 2 inches per second, how fast, in cubic
inches per second, is the volume increasing when the radius is 10 inches? V  34 r3 
58. A mysterious sphere is dropped by aliens in a pond, forming ripples of concentric circles on
the surface of the water. The circumference of the outermost ripple is increasing at a rate of
24 feet per minute. At the moment when the circumference is 16 feet, at what rate is the
area inside the ripple changing?
59. The graph of the function f is shown in the figure to the right. Which of the following
statements about f is true?
a) limfx  limfx
x a
x b
b) limfx   2
3
x a
c) limfx   2
2
d) limfx   1
1
x b
x b
e) limfx  does not exist
xa
a
b
60. A body’s position at time t sec is st  3t3  18t2  18t  6 meters. Find the body’s velocity
each time the acceleration is zero.
61. Write the equation of the tangent line to the curve x 2  y 2  169 at the point (5, -12).
62. Find the equation of the tangent line to the curve y  x3  6x2 at its point of inflection.
63.
d
csc3 2x 4 
dx


64. A particle moves along the x-axis in such a way that its position at time t is given by
1t
. What is the acceleration of the particle at time t  0 ?
xt  
1t
65. The graph of y  3x 4  16x3  24x2  48 is concave down for what x-values?
66. If xy  y 2  28 , then when y  4 ,
dy

dx
67. Where does the graph of the function y  x3  6x2  7x  2 cos x change concavity?


68.
2
d
xe ln x 
dx
69.
d
cos 2 x 3  
dx
70. A bug begins to crawl up a vertical wire at time
t  0 . The velocity, v, of the bug at time t on
0  t  8 is given by the function whose graph is
shown to the right. At what value of t does the bug
change direction?
71. A railroad track and a road cross at right angles. An observer stands on the road 70 meters
south of the crossing and watches an eastbound train traveling at 60 meters per second. At
how many meters per second is the train moving away from the observer 4 seconds after it
passes through the intersection?
72. Once upon a time, someone tried to see if cows would really get swept away by tornados like
in the movie “Twister” so they tied a poor, unsuspecting cow to the end of a 20 foot long rope
(the other end was staked into the ground). Along came a tornado, and sure enough, the cow
was pulled up into the air at a rate of 3.5 feet a second. At what rate is the angle between the
rope and the ground changing when the cow is 12 feet in the air?
73. A fishing line is reeled in at a rate of 1 foot per second from a bridge 15 feet above the
water. At what rate is the angle between the line and the water changing when 25 feet of line
is out?
74. Given the position function st  t 5  4t3  3t  5 , find the following:
a) Where is the particle moving right?
b) Moving left?
c) Where is the particle speeding up?
d) Slowing down?
75. Given the velocity function vt  cos 2 t  2 sint , find the following on the interval (0, 2):
a) Where is the particle moving right?
b) Moving left?
c) Where is the particle speeding up?
d) Slowing down?
76. Given a function fx  2 x  2  2 , determine a left-bound approximation of the area
between the curve and the x-axis on the interval [2, 4]. Use 4 sub-intervals.
77. The graph of the derivative of f is shown in the figure below.
y
y  f'x
2 x
2
Which of the following could be the graph of f?
y
a)
2
2
x
y
d)
x
y
c)
2
2
x
y
e)
2
2
y
b)
2
2
x
2
2
x
Use the following table of f(x), a twice differentiable function for all values of x, for problems
78 through 82:
x
f(x)
3
11
6
16
7
32
10
8
78. Is f(x) guaranteed to have at least one relative extrema? Why or why not?
79. Demonstrate the mean value theorem using a complete sentence with correct notation
(there are many correct answers).
80. Demonstrate the intermediate value theorem using a complete sentence with correct
notation (there are many correct answers).
81. Is it possible to determine any relative maxima or minima on [3, 10]? If so, state where
they occur. If not, state why not.
82. Is there a number c in the open interval (3, 10) such that fc  fx ?
83. Given that fx  34 x 3  2x 2  24x  10 , find the minimum value of fx  on [-3, 4].
84. Find the average rate of change of the following functions over the interval 2  x  6
b) y  sin3 x
c) y  lnx  1
2  x1  x
x  3x  1x  2
x 4  81
b) lim 2
x 3 x  x  6
ln 23  h  ln 23
c) lim
h 0
h
cos3 1  h  21
h 0
h
e) lim
3x 3  3x 2
x 0 5x 3  10 x 2
h) lim
a) y  x 4  3x
85. Find the following limits:
a) lim
d) lim
g) lim

x3  x  1
x   2x 2  4
f) lim
x 2
x  4  2x  x 4
i) lim
x 2
2
 
tanx   tan2
x 2
x 2  b  a x  ab
x a
abx 2  a 2bx
2
