Name: _______________________________________________
Date: __________ Per: ____
Calculus AB Midterm Review
Show your work on your own paper. DUE 4 minutes BEFORE your final. 3 homework grades and a small final exam bonus.
Solve each trig equation such that 0  x  2 .
2.
3
2
cot x  0
3.
1.
sin x 
4.
lim
5.
How does the answer to number 12 relate to the graph of f  x  
6.
If f  x  
7.
8.
4 x 2  3x  2
x 
2  3x 2
4 x 2  3x  2
?
2  3x 2
ax  b
has a horizontal asymptote at y = 3 and a vertical asymptote of x = -6 then a + c is what?
xc
sinx
Graph f  x  
.
x
x 2
. Find
x 2
lim g  x 
Graph g  x  
a.
b.
x2
lim g  x 
c.
x2
Find each limit.
3sin x
x2  4 x
2x  3  3
10. lim
x0
x
2x 1

11. lim x  3 2
x 1
x 1
9.
lim
x3  8
x 2 x  2
1
1
14. lim 1  x
x 0
x
x 0
13.
lim
15. lim
x1
12.
csc x  2
lim 2tan x
2
x
3
2ax  5; x  2
16. Find the value of “a” such that g  x   
2
3x  a; x  2
17. Use the figure to find each.
a.
b.
c.
f  2
lim f  x 
x 2
lim f  x 
x2
d.
lim f  x 
e.
lim f  x 
f.
Name the intervals for which f(x) is continuous.
x 3
x 2
is continuous.
x  6x  5
x2  1
lim g  x 
x 2
18. Use the graphs of f and g to answer each.
a.
b.
c.
f x
g x
d.
lim f  x   g  x 
e.
lim
x2
x1
lim g  f  x  
x 2
lim
x3
g x
f x
lim f  x   g  x 
x2
19. If f  x   3x  4 then lim
2
x 0
f  x   f 0
=?
x2
2 x  2; x  1
 2
20. g  x    x  5; 1  x  2
 2; x  2

a.
b.
c.
d.
Graph g  x  .
lim f  x  
x2
lim f  x  
x2
Name the location of discontinuities of f(x). Label them as removable or non-removable.
21. Define the Intermediate Value Theorem.
22. Use the Intermediate Value Theorem to decide if a root exists on the interval  3,1 for g  x   x  2x  x  3 .
3
2 x  3; x  3
.
2
2 x  9; x  3
23. Use the 3 step test to decide if g(x) is continuous at x=3 if g  x   
24. Draw a sketch such that: lim f  x   2 , lim f  x   2 , lim f  x    , lim f  x   
x 
x 
x3
x3
ax  b; x  3

25. h  x   2; x  3
2bx  a; x  3

What values of a and b will make h(x) continuous at x=3?
Graph f  for each function f .
26. –
28. –
27. –
29. –
2
Sketch a possible f for each f’.
30. –
31. -
32. –
33. –
34. Place in order from least to greatest based on the figure.
a.
f a 
b.
f   b
c.
f  c 
d.
f d 
35. Express the average rate of change from t to g on the function h(x).
mx 2  2 if x  1
. If f is differentiable at x = 1, what are the values of k and m?
k x if x  1
36. Let f be the function defined by f  x   
Find the derivative of each. Simplify. No negative exponents in your answer.
37.
f  x   5x 3  2x2  6x  3
38.
f x 
39.
x 2  5x
x3
42.
f  x   x sin x
f  x   tan2x
43.
40.
f  x   sin x cos x
f  x   3x 2  3x  2 
44.
f  x   x sec x
41.
f x 
45.
f x 
3x
x2  1
46. If f  x  
5
2x 3
 5x  1
5
1 3
x  24 x , find f  9 .
6
47. The graph of the rate the water level is changing over 8 hours is shown.
a. What are the units of the slope of the graph?
b. Describe what happens to the water level over the 8 hour period.
c. When is the water level the lowest?
d. What is the water level doing on day 3?
e. Is it possible to know if the aquifer is ever full or empty over the 8 hour period?
f. If on day 0 the water level was 5 gal , graph the water level during the 8 hours.
2
ax  bx ; x  2
. Find a and b so that f  x  is both continuous and differentiable.
2
ax  2x ; x  2
48. Consider f  x   
49. MULTIPLE CHOICE: If g 2  3 and g 2  1 , what is the value of
a.
b.
-3
-1
d  g x 

 at x  2 ?
dx  x 2 
c.
d.
0
2
x
f(x)
50. According to the table for a differentiable function, what is the best estimate of f 1.70  .
Find the slope of the tangent line at the indicated point.
51.
f  x   sin2x;  ,0
1.7
3.35
1.8
3.51
1.9
3.60
2.0
3.34
52. g  x   2x  4 x  3x  5;  1,10 
3
2
Find the equation of the tangent line at the given point.
53.
f  x   x sin x;  0,0
54.
2
f  x    x; 1,3
x
55. MULTIPLE CHOICE: If y  x  f  x  , then y  
2
a.
x2 f  x   x f  x   2 f  x 
c.
x2 f  x   2x f  x   f  x 
b.
x f  x   x f  x   f  x 
d.
x2 f  x   4x f  x   2 f  x 
61-62 Use the definition of the derivative to find g’(x) for each g(x).
56. g  x   x  2
2
1
x 3
2
58. If f  x   3x  5x  4 , find the average rate of change for f(x) from x=4 to x=6.
57. g  x  
59. What is another name for the average rate of change?
60. Find the instantaneous rate of change at x=3 for f  x   3x  5x  4 .
2
61. Find the location of any horizontal tangent line(s) for f  x   2x  4 x  5 .
4
2
62. Use implicit differentiation to find
dy
2
when x=2 for 3x  2xy  3y  1 .
dx
63. Use implicit differentiation to find
dy
2
for 2x  xy  2y  1 .
dx
64. If y  sin2 x , find y  .
65. Locate the absolute minimum and maximum on 1,6 for f  x   2x  3x  12x  2 .
3
2
66. Locate the interval(s) that f  x   2x  3x  12x  2 is increasing or decreasing.
3
2
67. Locate the interval(s) that f  x   2x  3x  12x  2 is concave up or concave down.
3
2
Related Rates
68. A person is standing at the end of a pier 16 feet above the water and is pulling a rope attached to a rowboat at the waterline at a rate of 6
feet of rope per minute. How fast is the boat moving in the water when it is 16 feet from the pier?
69. Consider a piece of ice in the shape of a sphere that is melting at the rate of 2
a.
b.
cm3
min
.
How fast is the radius changing at a moment when the radius is 8 cm?
How fast is the surface area of the ice changing at the same instant?
70. Air is being pumped into a spherical balloon at the rate of 4.25 cubic inches per minute. Find the rate of change in the radius when the
radius is 35 in.
71. MULTIPLE CHOICE: In the figure, a hot air balloon rising straight up from the ground is tracked by a
television camera 300 ft from the liftoff point. At the moment that the camera’s elevation angle is
 / 6 , the balloon is rising at the rate of 80 ft/min. At what rate is the angle of elevation changing at
that moment?
a. 0.12 radians per minute
d. 0.40 radians
b. 0.16 radians per minute
per minute
c. 0.20 radians per minute
72. All of the edges of a cube are expanding at a rate of 3 cm/s.
a. How fast is the volume changing, when the edge is 1 cm?
b. How fast is the surface area changing, when the edge is 1 cm?
73. At a sand and gravel plant, sand is falling off a conveyor and onto a conical pile at the rate of 10 cubic ft per minute. The diameter of the
base of the cone is 3 times the height of the pile. At what rate is the height of the pile changing when the pile is 15 ft high?
74. At noon, ship A is 160 km west of ship B. Ship A is sailing east at 35 km/h and ship B is sailing north at 25 km/h. How fast is the distance
between the ships changing a 3 PM?
75. MULTIPLE CHOICE: The radius of a circle is changing at the rate of 1/  inches per second. At what rate, in square inches per second, is
the circle’s area changing when r  5 inches.
5
a.

c.
b.
10
d.
10

15
76. Define Rolles Theorem.
77. Define the Mean Value Theorem.
78. Let f be the function given by f  x   sin x  . What are the values of c that satisfy Rolle’s Theorem on the closed interval  0,2 ?
79. Use the Mean Value Theorem to find a “c” such that f   c  
80. MULTIPLE CHOICE: Let f be the function given by f  x  
interval  1,2 ?
a.
b.
f  b  f a 
3
2
for f  x   x  x  2x on  1,1 .
ba
x
. What are the values of c that satisfy the Mean Value Theorem on the
x 2
-4 only
0 only
81. Find the first and second derivative of f  x  
c.
d.
0 and 3/2
-4 and 0
c.
x sec2  xy 
d.

1
.
x 3
1
decreasing?
x 3
1
83. Over what intervals is f  x  
concave up?
x 3
82. Over what intervals is f  x  
84. MULTIPLE CHOICE:If xy  tan xy    , then
a.
y sec2  xy 
b.
y cos2  xy 
dy

dx
Use the diagram to answer 85-95.
Consider the graph of f  defined on the closed interval [-5,4].
85.
86.
87.
88.
Name the interval(s) that f is increasing.
Does f have a local minimum? If so where? Justify your answer.
Where does f have point(s) of inflection?
Where does f  have a local maximum?
89.
90.
91.
92.
Where is f  decreasing?
On what intervals is f concave up?
Where does f achieve its minimum value on the interval [-2, 6]?
Which is larger f(0) or f(2). Explain your reasoning.
93. Which is larger f  0  , or f  2 ? Explain your reasoning.
94. If f  0  2 , graph f. Clearly show changes in concavity.
95. The graph of the second derivative f  is shown.
At what values does f have a point of inflection?
y
x
Use the graph of the second derivative to answer each.
96. Where is f concave down?
97. Where are the point(s) of inflection on f?
98. Where is the graph of f’ increasing?
99. On what interval(s) is f’ decreasing and concave down?
100. MULTIPLE CHOICE: At what values of x does f  x   x  2x
a.
b.
2/3
have a relative minimum?
64
27
16
9
c.
d.
4
3
2
For questions 106-110 consider the equation of the derivative of t on the closed interval [1,4]. Round all answers to the nearest hundredth.
t   x     e x  1  sin  2 x   3.3
101. Name the intervals that t(x) is increasing.
102. Name the intervals that t(x) is concave down.
103. Name the points of inflection for t.
104. Where is t concave up AND Increasing?
105. Find t    .
106. MULTIPLE CHOICE: Let f be a twice differentiable function f with f   0 and f   0 in the closed interval 2,8 . Which of the following
could be a table of values for f?
a.
b.
c.
For questions 112 -116 consider the equation of r’’’ on the closed interval [3,6]. Round all answers to the nearest hundredth.
r   x   ln x 3  cos  2 x   1
107. Find the average rate of change of r   x  from x = 2 to x =
4.
108. Find the average value of r   x  from x = 2 to x = 4.
112. At which of the five points on the graph is
109. Find the interval(s) where R’ is concave up.
110. r  3.721
111. r  3.721
dy
d 2y
 0 and 2  0 ?
dx
dx
113. What is the Fundamental Theorem of Calculus?
114. Make a chart of f, f’ and f’’ for f  x  
a.
b.
c.
d.
1 3
x  4 x  3 . Name the following:
3
the intervals that f is increasing and decreasing.
the intervals the f is concave up and concave down.
the point(s) of inflection of f.
the local maximum and minimum values for f.
115. Repeat 119 a-d for f  x  
x
and name the horizontal and vertical asymptotes.
x 3
d.
OPTIMIZATION
116. A farmer has 1500 feet of fencing to construct a rectangular pig pen. If he uses the side of his barn as on of the sides of the pen (as any
good pig farmer would), what are the dimensions of the pen that would maximize the area of the pen?
117. Determine the dimensions of a rectangular solid with a square base that has maximum volume if the surface area is 337.5 square
centimeters.
118. Your company designs cylindrical containers that have a volume of 16 cubic cm. The top and the bottom are made of a sturdy material
that cost $1.50 per sq. cm while the sides are made of a thinner material costing $1 per sp. cm. What is the minimum cost per container
and what is the height of that container?
119. The graph of y  f (x) is shown. If A and B are positive numbers that
represent the areas of the shaded regions, what is the value of
3

3
3
f  x  dx  2  f  x  dx in terms of A and B?
1
a.
b.
c.
d.
A  B
AB
A  2B
AB
Use the table at the right to find f   3 for each.
120. f  x  
g x
h x 
121. f  x   h  x   g  x 

122. f  x   h g  x 
123. f  x   h  x  

4
x
g x
g x 
h x 
h x 
-2
3
5
4
5
1
3
-4
-1
2
-2
3
8
7
6
124. f  x   g  x   h  x 

125. f  x   g h  x 

3
2
126. MULTIPLE CHOICE : If the graph of y  ax  6 x  bx  4 has a point of inflection at 2, 2 , what is the value of a  b ?
a.
b.
-2
3
The rate that water flows into a reservoir is recorded below. Include units in your
answer.
c.
d.
6
10
Time (hour)
Rate (thousands of gallons
per hour)
0
8
2
6
127. Use a trapezoidal approximation with 3 subintervals to estimate that amount of water that has flowed into the reservoir.
128. Use a right hand approximation with 3 subintervals to estimate that amount of water that has flowed into the reservoir.
4
10
6
8
21. GOOGLE IT!
g  3  3
22.
g 1  5
ANSWERS
1.
2.
3.
 2
,
3 3
 3
,
2 2
7 11
,
6 6
Yes! There is a root, since 0 is between -3
and 5.
23. STEP1: g 3  9 and therefore exists
STEP2: lim g  x   9 , lim g  x   9 and
4. -4/3
5. horizontal asymptote
6. a  c  9
x3
therefore
x3
lim g  x  exists.
x 3
STEP3: g 3  lim g  x   9
x3
CONCLUSION: g  x  is continuous at x = 3
7.
8. -1, 1, dne
9. ¾
10.
3
3
11. 3/8
12. 2 3
13. 12
14. -1/2
15. -1
16. -17/5
17. –
a.
b.
c.
d.
e.
f.
24.
25. a  10/19, b  8/19
26.
27.
1
3
3
1
DNE
 , 3 3, 2 2,22,
18. –
a.
b.
c.
d.
e.
28.
29.

-1
dne
0
30.

19. 3
20. –
a.
31.
b. -2
c. -1
d. -1 ; REMOVEABLE
2; NON-REMOVEABLE
51. 2
52. -5
53. y  0
54. y   x  4
55. d
56. 2x
32.
57.
33.
34. f d  , f  c  , f a  , f  b
35.
h  g   h t 
g t
2
3
36. m   , k  
37. 15x2  4 x  6
38. 
1 10

x2 x3
39. x cos x  sin x
40. cos2 x  sin2 x  cos2x
41.
3x 2  3
x
2
 1
2
42. 2sec2 2x
43. 3x  3x  24 21x  4 
44. sec x 1  x tan x 
45.
2 x 2 10 x  3 
 5 x  1 6
46. 28/27
47. –
a. Rate that H2O depth changes
 x  3 2
58. 25
59. slope of the secant line
60. 13
61. x  0, 1
62. -10
63.
8
3
1
dy 4 x  y

dx
x 2
64. 4sin2x
65. Abs Min -22 at x = 2, Abs Max 250 at x=6
66. DEC:  1,2 INC:  , 12,  
67. cc : 1/2, cc :  ,1/2
68. -8.485 ft/min
69. -0.00249 cm/min, -0.5 cm2/min
70. 0.00028 in/min
71. c
72. –
a. 9 cm3/s
b. 36 cm2/min
73. 0. 0063 ft/min
74. -0.54 kph
75. b
76. Google!
77. Google too!
78.
1
3
and
2
2
measured in m/day.
b. DEC:  0,0.5  3,7 INC:  0.5,3  7,8 
79. -1/3
80. b
c. 7
d. max; H2O level starts to go down
81. f  
e. no
82. Nowhere!
83.  , 3
f.
48. a  2, b  2
49. b
50. 1.6
1
 x  32
, f  
2
 x  3 3
84. d
85.  5, 23,6
86. 3 is a relative minimum since f  switches
from negative to positive
87. x  4,1.5,0,2
88. x  1
89.  4, 1.5 0,2
115.
90.  5, 4  1.5,02,6 
b. cc :  , 3 cc :  3,  
91. 3
92. f  0  is larger because f  (the slope) is
negative on interval 0,2 .
116.
117.
118.
119.
120.
121.
122.
123.
124.
125.
126.
127.
128.
93. f 2 since at x = 2 f  is cc  , while at x = 0
f
–
a. INC:  , 3  3,   DEC: none
is cc 
94.
95. a and O
96.  , 3 0,2
97. -3, 0, 2
98.  3,02, 
c. none
d. REL MAX: none REL MIN:none
375 by 750
7.5 by 7.5 by 7.5
R=1.193, h = 3.579, Cost $40.25
B
-27/4
11
-24
-224
43
21
C
48000 gal
48000 gal
99.  0,1
100.
101.
a
1.81,3.07
2.57,4 
102.
103.
2.57
104. 1.81,2.57
105.
106.
107.
108.
109.
110.
111.
112.
113.
114.
48.281
g x
x
-2
2
g  2   f t  dt
2
2
g  2    f  t  dt
Absolute Max
is  at x=-2

Absolute Min
is  at x=2
0
5
5
g  5   f  t  dt
0
c
0.377
2.56
3,4.047 5.398,6
2.578
-6.901
b
Google it!
–
a. INC:  , 2 2,  DEC:  2,2
b. cc :  0,  cc :  ,0
c.  0,3
d. REL MAX:  2,25/3 REL MIN:
2, 7/3

0
5