PART II
TEST AND TEST ANSWER KEYS
Test Bank . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Chapter 1P
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15
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Chapter 16
Chapter 17
Chapter 18
Chapter 19
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Preparation for Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
Limits and Their Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
Applications of Differentiation . . . . . . . . . . . . . . . . . . . . . . . 66
Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
Logarithmic, Exponential, and
Other Transcendental Functions. . . . . . . . . . . . . . . . . . . . . . . 91
Applications of Integration . . . . . . . . . . . . . . . . . . . . . . . . . . 103
Integration Techniques, L’Hôpital’s Rule,
and Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111
Infinite Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
Conics, Parametric Equations, and Polar Coordinates . . . . . . 135
Vectors and the Geometry of Space . . . . . . . . . . . . . . . . . . . . 151
Vector-Valued Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
Functions of Several Variables . . . . . . . . . . . . . . . . . . . . . . . . 174
Multiple Integration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
Vector Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
Answer Key to Chapter Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
Final Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Answer Key to Final Exams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
Gateway Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
Answer Key to Gateway Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
25
26
Chapter P
Preparation for Calculus
Test Form A
Name
__________________________________________
Date
Chapter P
Class
__________________________________________
Section _______________________
1. Find all intercepts of the graph of y (a) 2, 0
x2
.
x3
(b) 2, 0, 3, 0
(d) 2, 0, 0, 2
3
____________________________
(c)
0, 3, 3, 0
2
(e) None of these
2. Determine if the graph of y x
is symmetrical with respect to the x-axis, the y-axis, or the origin.
x2 4
(a) About the x-axis
(b) About the y-axis
(d) All of these
(e) None of these
(c) About the origin
3. Find all points of intersection of the graphs of x2 2x y 6 and x y 4.
(a) 0, 6, 0, 4
(b) 10, 14, 13, 17
(d) 5, 1, 2, 6
(e) None of these
(c) 5, 9, 2, 2
4. Which of the following is a sketch of the graph of the function y x3 1?
(a)
(b)
y
y
2
4
3
x
−2
1
2
2
−1
−2
(c)
−1
(d)
y
1
2
1
2
y
2
2
x
−3
−1
x
−2
1
−1
−1
−2
−2
(e) None of these
5. Find an equation for the line passing through the point 4, 1 and perpendicular to the
line 2x 3y 3.
(a) y 23 x 1
(b) 3x 2y 2 0
(d) 3x 2y 10
(e) None of these
(c) 2x 3y 10
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x
−2
Chapter P
6. Find the domain of f(x) 1
.
3 2x
(a)
, 32
(b)
32, (d)
, 32 32, (e) None of these
(c)
32, 7. Find f x x for f x x3 1.
(a) x3 1 x
(b) x3 3x2x 3xx2 x3 1
(c) x3 x3 1
(d) 3x 6 1
(e) None of these
1
8. If f x (a)
(d)
x
and gx 1 x2, find f g(x).
1 x2
x
1
x
(b)
1 x2
1
1 x2
(c) 1 1
x
(e) None of these
9. If the point 3, 2 lies on the graph of the equation 2x ky 11, find the value of k.
1
5
(a) 2
(b) 34
17
(c) 2
(d) 10
(e) None of these
10. Which of the following equations expresses y as a function of x?
(a) 3y 2x 9 17
(d) Neither a nor b
(b) 2x2y x 4y
(e)
3y2
x2
(c) Both a and b
5
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11. Given f x x2 3x 4, find f x 2 f 2.
(a) x2 3x 4
(b) x2 x
(d) x2 3x 4
(e) None of these
(c) x2 x 8
12. Determine which function is neither even nor odd.
(a) f x tan x
(b) f x 3x5 5x3 1
(d) f x x2 1
(e) Both a and b
(c) f x 3
x2
13. Find the point that lies on the line determined by the points (1, 2) and (3, 1).
(a) (0, 0)
(b) (5, 1)
(d) (5, 5)
(e) (2, 0)
(c) (4, 6)
14. Determine the slope of the line given by the equation 9x 5y 11.
(a)
5
9
5
(b) 9
9
(d) 5
(e) 9
(c)
9
5
Test Bank
27
28
Chapter P
Preparation for Calculus
15. Describe the transformation needed to sketch the graph of y 1
1
using the graph of f x .
x2
x
(a) Shift f x two units to the right.
(b) Shift f x two units to the left.
(c) Shift f x two units upward.
(d) Shift f x two units downward.
(e) Reflect f x about the x-axis.
16. Use the vertical line test to determine which of the following graphs represent y as a function of x.
(a)
(b)
y
y
3
3
2
2
1
1
x
−3
−1
1
2
x
−3
3
−2
−1
1
−1
−1
−2
−2
−3
−3
(c)
(d)
y
2
3
y
4
5
3
2
3
1
x
2
−4 −3
1
1
−2
x
−3
−1
−3
3
−1
−4
1
17. Let f x x
2x 1,
(a)
x < 0 . Find f 3.
x ≥ 0
1
3
(d) Undefined
(b) 1
(e)
(c) 7
22
3
18. The dollar value of a product in 1998 is $1430. The value of the product is expected to increase
$83 per year for the next 5 years. Write a linear equation that gives the dollar value V of the product
in terms of the year t. (Let t 8 represent 1998.)
(a) V 1430 83(t 8)
(b) V 83 1430t
(d) V 83 1430(t 8)
(e) V 1430 83(t 8)
(c) V 1430 83t
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(e) None of these
Chapter P
Test Bank
19. During the first and second quarters of the year, a business had sales of $150,000 and $185,000,
respectively. If the growth of sales follows a linear pattern, what will sales be during the fourth quarter?
(a) $220,000
(b) $235,000
(d) $255,000
(e) None of these
(c) $335,000
© Houghton Mifflin Company. All rights reserved.
20. In order for a company to realize a profit in the manufacture and sale of a certain item, the
revenue, R, for selling x items must be greater than the cost, C, of producing x items.
If R 79.99x and C 61x 1050, for what values of x will this product return a profit?
(a) x ≥ 55
(b) x ≥ 8
(d) x ≥ 56
(e) None of these
(c) x ≥ 18
29
30
Chapter P
Preparation for Calculus
Test Form B
Name
__________________________________________
Date
Chapter P
Class
__________________________________________
Section _______________________
1. Find all intercepts of the graph of y (a) 1, 0, 0, 1
3
(d) 3, 0, 0, 1
3
x1
.
x3
(b) 1, 0
____________________________
(c) 3, 0, 1, 0
(e) None of these
2. Determine if the graph of y x2
x2
is symmetrical with respect to the x-axis, the y-axis, or the origin.
4
(a) About the x-axis
(b) About the y-axis
(d) All of these
(e) None of these
(c) About the origin
3. Find all points of intersection of the graphs of x2 3x y 3 and x y 2.
(a) 5, 3, 1, 1
(b) 0, 3, 0, 2
(d) 5, 7, 1, 1
(e) None of these
(c) 5, 3, 1, 1
4. Which of the following is a sketch of the graph of the function y x 13?
(b)
y
y
2
2
1
1
x
−2
x
−2
2
−1
−2
(c)
2
−2
(d)
y
y
2
2
1
1
x
−2
−1
1
2
x
−2
−1
2
−2
(e) None of these
© Houghton Mifflin Company. All rights reserved.
(a)
Chapter P
Test Bank
5. Find an equation for the line passing through the point 4, 1 and parallel to the line 2x 3y 3.
(a) 2x 3y 11
(d) y 2
3x
(b) 2x 3y 5
1
(c) 3x 2y 5
(e) None of these
6. Find the domain of f x 1
3 2x
.
(a)
, 23
(b)
32, (d)
, 23 23, (e) None of these
(c)
23, 7. Find f x x for f x x2 2x 3.
(a) x2 x 3 x
(b) x2 2xx x2 2x 2x 3
(c) x2 2x 3 x
(d) 5
(e) None of these
8. If f x 1 x2 and gx (a)
(d)
1
x
1 x2
x
1
x
, find f g(x).
(b)
1 x2
1
1 x2
(c) 1 1
x
(e) None of these
© Houghton Mifflin Company. All rights reserved.
9. If the point 1, 1 lies on the graph of the equation kx2 xy y2 5, find the value of k.
(a) 7
(b) 3
(c) 5
(d) 3
(e) None of these
10. In which of the following equations is y a function of x?
(a) 2x 3y 1 0
(b) x2 3y2 7
(d) Both a and b
(e) Both a and c
(c) 2x2y 7
11. Given f x x 3 5, find f 1 f 5.
(a) 0
(b) 4
(d) 14
(e) None of these
(c) 14
12. Determine the even function.
x3
1
(a) f x sin x
(b) f x (d) f x x3 1
(e) None of these
x2
(c) f x 3x 4 5x2 1
31
32
Chapter P
Preparation for Calculus
13. Find the point that lies on the line determined by the points 1, 3 and 2, 4.
(a) 3, 2
(b) 1, 1
(d) 4, 2
(e) 4, 2
(c) 10, 0
14. Determine the slope of the line given by the equation 7x 4y 6 0.
(a)
7
4
(d) 74
(b)
4
7
(e)
3
2
(c) 7
15. Describe the transformation needed to sketch the graph of y 1
1
2 using the graph of f(x) .
x
x
(a) Shift f(x) two units to the right.
(b) Shift f(x) two units to the left.
(c) Shift f(x) two units upward.
(d) Shift f(x) two units downward.
(e) Reflect f(x) about the x-axis.
16. Use the vertical line test to determine which of the following graphs does not represent y as a
function of x.
(a)
(b)
y
y
4
2
3
2
1
1
x
x
−4
−2
1
−1
−1
1
2
−2
−3
−4
(c)
(d)
y
y
3
2
2
1
1
−2
−1
1
2
π
2
−π
2
3
−1
−1
−2
−2
−3
(e) Both a and d
17. Let f x x2 5,
3x 1,
x < 2
. Find f 1.
x ≥ 2
(a) 4
(b) 2
(d) 2
(e) 0
(c) 4
x
© Houghton Mifflin Company. All rights reserved.
x
−3
Chapter P
Test Bank
18. The dollar value of a product in 1998 is $78. The value of the product is expected to decrease
$5.75 per year for the next 5 years. Write a linear equation that gives the dollar value V of the
product in terms of the year t. (Let t 8 represent 1998.)
(a) V 78 5.75t
(b) V 78 5.75t
(d) V 78 5.75t 8
(e) V 5.75 78t 8
(c) V 78 5.75t 8
19. A business had annual retail sales of $124,000 in 1993 and $211,000 in 1996. Assuming that the
annual increase in sales follows a linear pattern, predict the retail for 2001.
(a) $356,000
(b) $435,000
(d) $298,000
(e) $327,000
(c) $646,000
© Houghton Mifflin Company. All rights reserved.
20. In order for a company to realize a profit in the manufacture and sale of a certain item, the
revenue, R, for selling x items must be greater than the cost, C, of producing x items. If
R 69.99x and C 59x 850, for what values of x will this product return a profit?
(a) x ≥ 78
(b) x ≥ 15
(d) x ≥ 13
(e) None of these
(c) x ≥ 85
33
34
Chapter P
Preparation for Calculus
Test Form C
Name
__________________________________________
Date
Chapter P
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator/utility is recommended for this test.
1. Find the x-intercepts for the graph of y 3x5 4x 4 2x3.
(a) 0, 0, 2, 0, 13 , 0
(b) 0, 0
(d)
(e) There are no x-intercepts.
2, 13 (c) 0, 0, (2, 0, 1, 0
2. Use a graphing utility to graph the equation y x 4 3x3 20.
(a)
(b)
y
y
30
6
25
4
2
x
−8 −6 −4
2
6
15
8
10
5
x
−1
(c)
(d)
y
1
2
1
2
3
4
y
30
25
25
20
15
15
5
10
x
5
−2
x
−1
1
3
(e) None of these
3. Identify the type(s) of symmetry: x 4y2 2x2y 1 0.
(a) About the x-axis
(b) About the y-axis
(d) Both a and b
(e) None of these
(c) About the origin
4. Find all points of intersection of the graphs of x2 3x y 3 and x y 2.
(a) 2, 0 and 1, 3
(b) 5, 3 and 1, 3
(d) 5, 37 and 1, 5
(e) None of these
(c) 1, 1 and 5, 7
5. A business had annual retail sales of $110,000 in 1993 and $224,000 in 1996. Assuming that the
annual increase in sales followed a linear pattern, what was the retail sales in 1995?
(a) $182,000
(b) $195,000
(d) $186,000
(e) None of these
(c) $188,000
© Houghton Mifflin Company. All rights reserved.
−3
4
−5
Chapter P
Test Bank
6. Find an equation for the horizontal line that passes through the point 3, 2.
(a) x 2
(b) y 2
(c) x 3
(d) y 3
(e) None of these
7. Write an equation satisfied by the graph obtained by shifting y 2x 5 three units to the left.
(a) y 2x 8
(b) y 2x 1
(d) y 2x 2
(e) None of these
8. Let f x 1
x
and g x) 2x 3. Find the domain of f gx.
(a) x ≥ 3
2
(b) x > 0
(d) x > 3
2
(e) None of these
9. Find the zero(s) of the function f x (a) 3 and 4
(d)
(c) y 2x 11
(c) x <
1
1
.
x3 x4
(b) 0
2
7
2
3
(c)
7
2
(e) None of these
10. Given f x 3x 1 5, find f x 1 f x.
(d) 3x 4 3x 1
© Houghton Mifflin Company. All rights reserved.
11. Find
(b) 5
(a) 3
(c) 3x 4 3x 1 10
(e) None of these
f x x f x
for f x 8x2 1.
x
1
x
(a) 8x2 1
(b) 8x (d) 16xx 8x2
(e) None of these
(c) 16x 8x
12. Determine the odd function.
x3
1
(a) f x x5 x3 x 1
(b) f x (d) f x cos x
(e) None of these
x2
(c) f x 3x2 5x 1
13. Find the point that does not lie on the line determined by the points 5, 2 and 1, 3.
(a) 0, 4
(d)
2,
12
(b) 7, 8
(e) 13, 13
(c) 11, 7
35
36
Chapter P
Preparation for Calculus
14. Find the equation of the line that passes through the point 4, 2 and is perpendicular to the line
that passes through the points 9, 7 and 11, 4.
(a) y 23 x 9 7
(b) y 32 x 9 7
(d) y 23 x 4 2
(e) None of these
15. Let f x 1x,2x,
2
(c) y 32 x 4 2
x < 1
. Find f 5.
x ≥ 1
(a) 9
(b) 25
(d) 34
(e) 1
(c) 17
16. An open box is to be made from a rectangular piece of material 9 inches by 12 inches by cutting
equal squares from each corner and turning up the sides. Let x be the length of each side of the
square cut out of each corner. Write the volume V of the box as a function of x.
(b) V 108x
(d) V x9 2x(12 2x
(e) None of these
(c) V x9 x12 x
© Houghton Mifflin Company. All rights reserved.
(a) V x3
Chapter P
37
Test Form D
Name
__________________________________________
Date
Chapter P
Class
__________________________________________
Section _______________________
1. Show that y ____________________________
x
is symmetric with respect to the origin.
x2 1
2. Find the intercepts: y 2x 1
.
3x
3. Find all points of intersection: y x2 4x and y x2.
4. Find an equation for the straight line that passes through the point 2, 3 and is parallel to
the line x 4.
5. Find an equation in general form for the straight line that passes through the point 1, 4 and is
perpendicular to the line 2x 3y 6.
6. If f x 3 x2, find:
a.
f 3
b. f 1
7. If gx x2 3x 1, find
8. If f x 1
x
c. f 2 x
gx x gx
.
x
and gx x2 5, find g f(x).
9. Find the domain: f x © Houghton Mifflin Company. All rights reserved.
Test Bank
1
.
x2 2x 2
10. Sketch a graph of y x3 1.
11. Sketch the graph of the equation 4x 2y 8 0.
12. In which of the following equations is y a function of x?
a. 3x 2y 7 0
b. 5x2y 9 2x
d. x 3y2 1
e. None of these
c. 3x2 4y2 9
13. Given f x 3x 7, find f x 1 f 2.
14. Determine whether the function is even, odd, or neither. Justify your answer.
f x x5 3x3 2x 1
38
Chapter P
Preparation for Calculus
15. Use the graph of f x x to sketch the graph of y x 1 3.
16. Find the slope and y-intercept of the line given by the equation 5x 4y 12 0.
17. Let f x a. f 0
x3 2x,4,
2
x < 2
. Evaluate:
x ≥ 2
b. f 2
c. f 3
18. A student working for a telemarket company gets paid $3 per hour plus $1.50 for each sale. Let x
represent the number of sales the student has in an 8-hour day.
a. Write a linear equation giving the day’s salary S in terms of x.
b. Use the linear equation to calculate the student’s salary on Wednesday if the student makes
14 sales that day.
© Houghton Mifflin Company. All rights reserved.
c. Use the linear equation to calculate the number of sales per day the student would have to
make in order to earn at least $100 a day.
Chapter P
Test Bank
Test Form E
Name
__________________________________________
Date
Chapter P
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator/utility is recommended for this test.
1. Given the equation: x2 y2 4x 6y 12 0.
a. Solve for y.
b. Use a graphing utility to graph the resulting equations on the same set of axes and sketch the graph.
2. Create an equation whose graph has intercepts at 5, 0, 0, 0, and 5, 0.
3. a. Use a graphing utility to graph the equation y x4 3x2 2.
b. Identify the intercepts of the graph.
c. Test for symmetry.
4. Identify the type(s) of symmetry: x2 xy y2 0.
5. Find all points of intersection of the graphs of x2 2x y 6 and x y 4.
6. Sketch the graph of the equation: 4x 2y 8 0.
7. Let f x © Houghton Mifflin Company. All rights reserved.
39
xx, 3,
x < 2
. Evaluate:
x ≥ 2
a. f 3
b. f 2
c. f 0
d. f 2
8. Let f x x x 2.
a. Use a graphing utility to graph y f x.
b. Estimate the domain and range from the graph.
c. Estimate the coordinates of any intercepts of the graph.
d. Find the intercepts analytically.
9. Use a graphing utility to estimate the zero(s) of f x x3 5x 2.
10. In which of the following is y a function of x?
a. y 3x2 9
b. x2 y2 7
d. 3x 2y 5
e. x y
11. Given f x 3x 6 , find f 0 f 3.
c. x2 y2 2
40
Chapter P
12. Find
Preparation for Calculus
gx x gx
for gx x2 3x 1.
x
13. Determine whether the function f x x 4 2x2 1 is even, odd, or neither. Justify your answer.
14. Use the graph of f shown below to sketch the graph of y f x 2.
y
2
x
−2
2
−1
−2
y f x
4
15. Find two additional points on the line that passes through the point 4, 5 and has slope m 3 .
16. Find an equation of the line that passes through the point 3, 1 and is perpendicular to the line
determined by the points 8, 9 and 10, 6.
17. For the functions f x x 2 and gx x5
, find g f(x).
3
18. A business had annual retail sales of $224,000 in 1993 and $186,500 in 1996. Assume the annual
decrease in sales follows a linear pattern.
a. Write a linear equation giving sales S in terms of the year t where t 0 corresponds to 1993.
c. Use the graph to estimate the annual retail sales for 1998 to two digits of precision (the nearest
multiple of $10,000).
d. Use the graph to estimate the first year when there will be no sales.
© Houghton Mifflin Company. All rights reserved.
b. Use a graphing utility to graph the equation.
Chapter 1
Test Bank
Test Form A
Name
__________________________________________
Date
Chapter 1
Class
__________________________________________
Section _______________________
____________________________
1. Find lim 3x2 5.
x→2
(a) 41
(b) 17
(c) 11
(d) 0
(e) None of these
2. Given lim 2x 1 3. Find such that 2x 1 3 < 0.01 whenever 0 < x 2 < .
x→2
(a) 3
(b) 0.05
(d) 0.005
(e) None of these
3. Use the graph to find lim f x if f x x→1
(a) 2
(c)
31, x,
(c) 0.03
x1
.
x1
y
(b) 1
2
3
2
(d) The limit does not exist.
1
(e) None of these
x
−1
1
−1
x2 3x 2
.
x→1
x2 1
4. Find lim
© Houghton Mifflin Company. All rights reserved.
41
(a) 0
(b)
(d) The limit does not exist.
(e) None of these
(c) 1
5. Find lim x2 4.
x→3
(a) 1
(b) 5
(d) 5
(e) None of these
(c) 1
1
2
f x
6. If lim f x and lim gx , find lim
.
x→c
x→c
x→c gx
2
3
(a) 1
3
(d) 3
(b)
1
3
(e) None of these
(c) 3
4
2
3
Chapter 1
Limits and Their Properties
x2
.
x2 4
7. Find the limit: lim
x→2
1
4
(a) 0
(b)
(d) 1
(e) None of these
8. Find the limit: lim
x 4 2
x
x→0
1
4
(b)
(d) 1
(e) None of these
x→3
(c)
x3
.
x3
(a) 0
(b) 1
(d) The limit does not exist.
(e) None of these
10. Find the limit: lim sec
x→2
(c) 3
x
.
3
(a) 2
(d)
.
(a) 0
9. Find the limit: lim
(c)
(b)
1
2
2
3
(c) x→0
x
.
tan x
4
(a) 0
(b)
(d) The limit does not exist.
(e) None of these
(c) 1
12. Find the limit: lim 2x 5.
x→3
(a) 1
(b) 0
(d) The limit does not exist.
(e) None of these
13. Find the limit: lim
x→2
(d) 2
(e) None of these
11. Find the limit: lim
(a)
3
1
4
(c) 2i
1
.
x2
(b) (e) None of these
(c) 0
© Houghton Mifflin Company. All rights reserved.
42
Chapter 1
14. Find the limit: lim
x→2
1
.
x 22
(a)
(b) (d)
1
4
(e) None of these
15. Find the limit: lim 2 x→0
(c) 0
5
.
x2
(a) 7
(b) 2
(d) 0
(e) None of these
16. At which values of x is f x (c)
x2 2x 3
discontinuous?
x2
(a) 2
(b) 1, 2, 3
3
(d) 1, , 2, 3
2
(e) None of these
17. Let f x (c) 1
1
and gx x2 5. Find all values of x for which f g(x) is discontinuous.
x1
(a) 1
(b) 1, ± 5
(d) 2, 2
(e) None of these
(c) ± 5
18. Determine the value of c so that f x is continuous on the entire real line when f x (a) 0
© Houghton Mifflin Company. All rights reserved.
(d)
Test Bank
5
6
(b)
6
5
(c) 1
(e) None of these
19. Find all vertical asymptotes of f x x3
.
x2
(a) x 2, x 3
(b) x 2
(d) x 1
(e) None of these
20. Find all vertical asymptotes of gx (c) x 3
2x 3
.
x3
2x2
3
2
3
(a) x , x 1
2
(b) x (d) y 1
(e) None of these
(c) x 1
xcx2,3,
x ≤ 5
.
x > 5
43
44
Chapter 1
Limits and Their Properties
Test Form B
Name
__________________________________________
Date
Chapter 1
Class
__________________________________________
Section _______________________
____________________________
1. Find the limit: lim 2x2 1.
x→3
(c) 17
(a) 37
(b) 19
(d) ± 2
(e) None of these
2. Given lim 2x 1 3. Find such that 2x 1 3 < 0.01 whenever 0 < x 1 < .
x→1
(a) 3
(b) 0.05
(d) 1
(e) None of these
3. Use the graph at the right to find lim f x for f x x→1
(a) 0
(c)
(c) 0.005
1
.
x1
y
(b) 1
(d) The limit does not exist.
1
x
(e) None of these
−1
1
2
3
4
x2 2x 3
.
x→1
x2 1
4. Find the limit: lim
(b) 1
(d) The limit does not exist.
(e) None of these
(c)
5. Find the limit: lim 9 x2.
x→3
(a) 0
(b) 6
(d) The limit does not exist.
(e) None of these
(c) 32
6. If lim f x 12 and lim gx 23 , find lim f xgx .
x→c
(a)
x→c
1
6
x→c
(b) 13
(d) The limit does not exist.
(c) 1
(e) None of these
x2 5x 6
.
x→1
x1
7. Find the limit: lim
(a) 0
(d)
(b) 7
(e) None of these
(c) © Houghton Mifflin Company. All rights reserved.
(a) 0
Chapter 1
8. Find the limit: lim
x→2
x2
.
x2
(a) 0
(b) 1
(d) The limit does not exist.
(e) None of these
9. Find the limit: lim
x 9 3
x
x→0
.
(a) 0
(d)
(b) 1
1
3
(c)
(e) None of these
10. Find the limit: lim csc
x→5
x
.
4
(b) 1
(a) 1
(d) (c) 2
1
(c) 2
(e) None of these
2
1 cos2 x
.
x→0
x
11. Find the limit: lim
(a) 1
(b) 0
(d) The limit does not exist.
(e) None of these
(c)
12. Find the limit: lim 2x 3.
x→2
(a) 1, 1
© Houghton Mifflin Company. All rights reserved.
(d)
(b) 1
1
2
(c) 1
(e) None of these
1
13. Find the limit: lim .
x→0 x
(a) (b) 0
(d) The limit does not exist.
(e) None of these
14. Find the limit: lim
x→1
5
.
x 12
(a) 0
(b) (d) (e) None of these
15. Find the limit: lim 2 x→1
(c) (c)
5
4
5
.
x 12
(a) (b) (d) 2
(e) None of these
(c) 3
Test Bank
45
Chapter 1
Limits and Their Properties
16. At which values of x is f x x4
discontinuous?
x2 x 2
(a) 4
(b) 1, 2, 4
(d) 1, 2, 4, 2
(e) None of these
17. Let f x (c) 1, 2
1
and gx x 1. Find all values of x for which f g(x) is discontinuous.
x
(a) 0
(b) 1
(d) 1, 1
(e) None of these
(c) 0, 1
18. Determine the value of c so that f x is continuous on the entire real line when f x (a) 4
(b) 4
(d) 1
(e) None of these
19. Find all vertical asymptotes of f x (a) x 2
(d) x 1
2
x2x3,c,
x ≤ 1
.
x > 1
(c) 0
2x 1
.
x3
1
(b) x , x 3
2
(c) x 3
(e) None of these
20. Find all vertical asymptotes of f x x2
.
x2 4
(a) x 2, x 2
(b) x 2
(d) x 2
(e) None of these
(c) x 0
© Houghton Mifflin Company. All rights reserved.
46
Chapter 1
Test Bank
Test Form C
Name
__________________________________________
Date
Chapter 1
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator/utility is recommended for this test.
1. Use a graphing utility to graph the function: f x x2 4x and then estimate lim f x (if it exists).
x→2
(a) 0
(b) 12
(c) 4
(d) 12
(e) None of these
2. Use the graph to find lim f x (if it exists).
y
x→1
(a) 1
2
(b) 2
1
y = f ( x)
x
(c) The limit does not exist.
−2
1
(d) 1
−1
(e) 3
−2
3. Find the limit: lim
x→1
(a) 3
(b) 0
(d) The limit does not exist.
(e) None of these
x→ 2
(b)
22
2
2
3
2
(c) 3
(e) None of these
6. Find the limit: lim
x→9
(d) 0
(c) x2
and then estimate lim f x.
x→0
x3
(b) 2
(d) 0
(a) 12
1
2
(e) None of these
5. Use a graphing utility to graph the function f x (a) (c) sin x
.
x
(a) 0
(d)
2
x2 x 2
.
x3
4. Find the limit: lim
© Houghton Mifflin Company. All rights reserved.
47
x2 6x 27
.
x9
(b) The limit does not exist.
(e) None of these
(c) 3
48
Chapter 1
Limits and Their Properties
1
1
x3 3
7. Find the limit: lim
.
x→0
x
(a) 1
9
(b) 0
(d) The limit does not exist.
(c)
1
9
(c)
(e) None of these
1 cos2 x
.
x→0
x
8. Find the limit: lim
(a) 1
(b) 0
(d) The limit does not exist.
(e) None of these
9. Match the graph with the correct function.
(a) f x (c) f x x3
x1
x2
x1
2x 3
y
6
(b) f x x 3
5
(d) f x x2 2x 3
x1
2
1
x
(e) None of these
−2
1
,
x3
10. Find the x-values (if any) for which f is not continuous: f x 1
,
1
2
(a) 5
(b)
2
(c) 3
(1, 4)
4
−1
1
2
3
x ≤ 5
.
x > 5
(d) 3, 5
(e) None of these
11. Use the graph to find the x-values (if any) at which f is not continuous.
(c) 2, 3
(b) 3, 3
8
6
(d) 1, 3
4
2
(e) None of these
x
−4
2
−2
−4
12. Use a graphing utility to graph f x x3 2x 5. Then use this graph to find the interval for
which the Intermediate Value Theorem guarantees the existence of at least one number c in that
interval for which f c 0.
(a) 1, 1
(b) 1, 2
(d) 3, 4
(e) None of these
(c) 2, 3
4
6
8
© Houghton Mifflin Company. All rights reserved.
(a) 3
y
Chapter 1
Test Bank
13. Use the graph to find the interval(s) for which the function f is continuous.
(a) , 2 and 2, (b) , (c) , 1 and 1, (d) 1, 2
y
6
3
(e) None of these
x
−3
3
−3
14. Find the limit:
lim
x→ 12
2x 1.
(a) 0
(b) 2
(d) The limit does not exist.
(e) None of these
15. Find the limit: lim
x→6
(c) 1
3x 18.
6x
(a) 1
(b) 1
(d) 3
(e) None of these
(c) 3
16. Which of the following statements is not true of f x x2 25?
(a) f is continuous at x 10.
(b) f is continuous on the interval , 5.
(c) f is continuous on the interval 5, .
(d) f is continuous on the interval 5, 5.
(e) f is not continuous at x 0.
17. Find the limit: lim
x→1
(a)
2
.
x1
(b) © Houghton Mifflin Company. All rights reserved.
(d) The limit does not exist.
18. Find the limit: lim
x→3
(a)
1
3
(d) 1
49
(c) 0
(e) None of these
x2 3x 2
.
x2 5x 6
(b) (c) (e) None of these
19. Find the vertical asymptote(s): f x x2
.
x2 3x 10
(a) x 2, x 5
(b) y 1
(d) x 5
(e) x 5
(c) y 0
20. f x decreases without bound as x approaches what value from the right? f x (a) 5
(b) 3
(d) 3
(e) None of these
4
x 35 x
(c) 5
6
50
Chapter 1
Limits and Their Properties
Test Form D
Name
__________________________________________
Date
Chapter 1
Class
__________________________________________
Section _______________________
____________________________
1. Calculate lim 2x2 6x 1.
x→2
2. If lim 3x 2 7, find such that 3x 2 7 < 0.003 whenever 0 < x 3 < .
x→3
3. Find the limit: lim f x if f x x→1
x2, 4,
2
x1
.
x1
x2 3x 2
.
x→1
x2 1
4. Find the limit: lim
5. Find the limit: lim 4x2 9.
x→2
1
2
6. If lim f x 2 and lim gx 3 , find lim f x gx.
x→c
x→c
x→c
x2
.
x→2 x3 8
7. Find the limit: lim
x x x
8. Find the limit: lim
x
x→0
x→1
x1
.
x1
10. Find the limit: lim cot
x→5
x
.
6
x
.
x→0 sin 3x
11. Find the limit: lim
12. Find the limit: lim 2x 1.
x→2
13. Find the limit: lim x→1
14. Find the limit: lim
x→3
1
.
x1
3
.
x2 6x 9
© Houghton Mifflin Company. All rights reserved.
9. Find the limit: lim
.
Chapter 1
15. Find the limit: lim 3x 2 x→0
1
.
x2
16. Find the value(s) of x for which f x removable or nonremovable.
17. Let f x x2
is discontinuous and label these discontinuities as
x2 4
5
and gx x4.
x1
a. Find f g(x).
b. Find all values of x for which f g(x) is discontinuous.
© Houghton Mifflin Company. All rights reserved.
x2,
18. Determine the value of c so that f x is continuous on the entire real line if f x c
,
x
19. Find all vertical asymptotes of f x if f x x2 3x 1
.
x7
20. Find all vertical asymptotes of f x if f x 2x 2
.
x 1x2 x 1
x ≤ 3
x > 3.
Test Bank
51
52
Chapter 1
Limits and Their Properties
Test Form E
Name
__________________________________________
Date
Chapter 1
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator/utility is recommended for this test.
1. Use the graph to find lim f x (if it exists).
y
x→0
4
3
y = f ( x)
1
x
−2
−1
1
2
2. Determine whether the statement is true or false. If it is false, give an example to show that it is false.
If lim f x 9, then f 3 9.
x→3
3. Calculate the limit: lim 2x2 6x 1.
x→2
4. Find the limit: lim
x2 x 2
.
x3
5. Find the limit: lim
x
.
cos x
x→1
x→ 6. Use a graphing utility to graph the function f x x3 x 5 and then estimate lim f x.
x→1
x2 4
.
x2
a. Use a graphing utility to graph the function.
b. Use the graph to estimate lim f x.
x→2
c. Find the limit by analytical methods.
8. Let f x x3 2x2
.
x2
a. Find lim f x (if it exists).
x→2
b. Identify another function that agrees with f x at all but one point.
c. Sketch the graph of f x.
9. Find the limit: lim
x→0
10. Find the limit: lim
x→0
x x 2 x 2
x
.
x x2 2x x x2 2x
.
x
© Houghton Mifflin Company. All rights reserved.
7. Let f x Chapter 1
Test Bank
11. Find the x-values (if any) for which f is not continuous.
f x 3x2x 2,3x 6,
2
x < 1
x ≥ 1
12. Use the graph to find the x-values (if any) for which f is not continuous.
y
3
y = f ( x)
2
1
x
−3
−2
−1
1
2
3
−2
−3
f x tan
13. Find the interval(s) for which the function is continuous.
x
2
y
9
6
3
x
2π
−3
−6
−9
14. Use the Intermediate Value Theorem to show that the function f x x 4 2x2 3x has a zero
in the interval 2, 1.
15. Use a graphing utility to graph f x © Houghton Mifflin Company. All rights reserved.
the function is not continuous.
16. Let f x x2x 1,3,
2
x2
. Then use the graph to determine x-values at which
x2 4
x ≤ 0
. Find each limit (if it exists).
x > 0
a. lim f x
b. lim f x
x→0
x→0
17. Use a graphing utility to find the limit: lim
x→0
18. Find all vertical asymptotes of f x 19. Find the limit: lim
x→3
20. Find the limit: lim
x→1
1
.
x3
2
.
1 x2
c. lim f x
x→0
sin 3x
. Then verify your answer analytically.
x
x2 x 2
.
x2 x 6
53
54
Chapter 2
Differentiation
Test Form A
Name
__________________________________________
Date
Chapter 2
Class
__________________________________________
Section _______________________
____________________________
1. If f x 2x2 4, which of the following will calculate the derivative of f x?
(a)
2x x2 4 2x2 4
x
2x2 4 x 2x2 4
x→0
x
(b) lim
2x x2 4 2x2 4
x→0
x
(c) lim
2x2 4 x 2x2 4
x
(e) None of these
(d)
2. Differentiate: y (a) 1
(d)
1 cos x
.
1 cos x
2 sin x
1 cos x2
(b) 2 csc x
(c) 2 csc x
(e) None of these
3. Find dydx for y x3x 1.
3x2
2x 1
(b)
(d)
7x3 x2
2x 1
(e) None of these
(c) 3x2x 1
4. Find fx for f x 2x2 57.
(a) 74x6
(b) 4x7
(d) 72x2 56
(e) None of these
5. Find
d 2y
x3
for y .
dx2
x1
(a) 0
(d)
(c) 28x2x2 56
8
x 13
(b)
8
x 13
(e) None of these
(c)
4
x 13
© Houghton Mifflin Company. All rights reserved.
x27x 6
2x 1
(a)
Chapter 2
Test Bank
6. The position equation for the movement of a particle is given by s t2 13 when s is measured
in feet and t is measured in seconds. Find the acceleration at two seconds.
(a) 342 unitssec2
(b) 18 unitssec2
(d) 90 unitssec2
(e) None of these
7. Find
(c) 288 unitssec2
dy
if y2 3xy x2 7.
dx
(a)
2x y
3x 2y
(b)
3y 2x
2y 3x
(d)
2x
y
(e) None of these
(c)
2x
3 2y
8. Find y if y sinx y.
cosx y
1 cosx y
(a) 0
(b)
(d) 1
(e) None of these
(c) cosx y
9. Differentiate: y sec2 x tan2 x.
(a) 0
(b) tan x sec4 x
(d) 4 sec2 x tan x
(e) None of these
(c) sec2 x sec2 x tan2 x
t
10. Find the derivative: s t csc .
2
(a) csc
© Houghton Mifflin Company. All rights reserved.
(d)
t
t
cot
2
2
1 2t
cot
2
2
1
t
(b) cot2
2
2
(c)
t
t
1
csc cot
2
2
2
(e) None of these
11. Find an equation for the tangent line to the graph of f x 2x2 2x 3 at the point where x 1.
(a) y 2x 2
(b) y 4x2 6x 5
(d) y 4x2 6x 2
(e) None of these
(c) y 2x 1
12. Find all points on the graph of f x x3 3x2 2 at which there is a horizontal tangent line.
(a) 0, 2, 2, 2
(b) 0, 2
(d) 2, 2
(e) None of these
(c) 1, 0, 0, 2
55
56
Chapter 2
Differentiation
13. Find the instantaneous rate of change of w with respect to z if w (a)
7
6z
(d) (b)
14
3z3
14
z
3
7
.
3z2
(c) 14
3z
(e) None of these
14. Suppose the position equation for a moving object is given by s t 3t2 2t 5 where s is measured
in meters and t is measured in seconds. Find the velocity of the object when t 2.
(a) 13 msec
(b) 14 msec
(d) 6 msec
(e) None of these
(c) 10 msec
15. A point moves along the curve y 2x2 1 in such a way that the y value is decreasing at the rate
of 2 units per second. At what rate is x changing when x 32?
(b) decreasing 13 unitsec
(d) increasing 72 unitsec
(e) None of these
(c) decreasing 72 unitsec
© Houghton Mifflin Company. All rights reserved.
(a) increasing 13 unitsec
Chapter 2
57
Test Form B
Name
__________________________________________
Date
Chapter 2
Class
__________________________________________
Section _______________________
1. If f x x2 x, which of the following will calculate the derivative of f x?
x2 x x x2 x
x→0
x
(a) lim
x x2 x x x2 x
x→0
x
(b) lim
(c)
x x2 x x x2 x
x
x2 x x x2 x
x
(e) None of these
(d)
2. Differentiate: y 3x
.
x2 1
(a)
3
1 x2
(b)
3
2x
(d)
31 x2
1 x22
(e) None of these
(c)
3x2 3
1 x23
3. Find dydx for y x 3x 1.
(a)
© Houghton Mifflin Company. All rights reserved.
Test Bank
(d)
9x 1
2x
3
2x
(b)
9
x 1
2
(c) 3x
(e) None of these
4. Find fx for f x sin3 4x.
(a) 4 cos3 4x
(b) 3 sin2 4x cos 4x
(d) 12 sin2 4x cos 4x
(e) None of these
5. Find
(c) cos3 4x
d 2y
x2
for y .
2
dx
x3
(a)
10
x 33
(b) 0
(d)
2
x 33
(e) None of these
(c)
10
x 33
____________________________
58
Chapter 2
6. Find
(a)
Differentiation
dy
if x2 y2 2xy.
dx
x
1y
(d) x
y
(b)
yx
yx
(c) 1
(e) None of these
7. Find y if x tanx y.
(a) sin2 x y
(d)
1 sec2 x
sec2 y
(b) sec2 x y
(c) tan2 x y
(e) None of these
8. Differentiate: y csc2 cot2 .
(a) cot csc4 (b) 0
(c) 4 csc2 cot (d) csc2 csc2 cot2 (e) None of these
9. Find the derivative: s t sec t.
(a) tan2t
(d) sect tant
sect tant
2t
(e) None of these
(b)
(c) sec
1
1
tan
2t
2t
10. A particle moves along the curve given by y t3 1. Find the acceleration when t 2 seconds.
2
3
(a) 3 unitssec2
(b)
unitssec2
(d) 19 unitssec2
(e) None of these
1
(c) 108
unitssec2
(a) y 4x 2
(b) 2x y 1 0
(d) 2x y 5
(e) None of these
(c) y 4x2 2x 1
12. Find point(s) on the graph of the function f x x3 2 where the slope is 3.
(a) 1, 3, 1, 3
(b) 1, 1, 1, 3
(d) 1, 3
(e) None of these
13. Find the instantaneous rate of change of w with respect to z for w (a)
3
2
(b) 2
(d)
1
z2
(e) None of these
(c)
3 2, 0
1
z
.
z
2
(c)
z2 2
2z2
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11. Find an equation for the tangent line to the graph of f x 2x2 2x 3 at the point where x 1.
Chapter 2
Test Bank
14. Suppose the position equation for a moving object is given by s t 3t2 2t 5 where s is measured
in meters and t is measured in seconds. Find the velocity of the object when t 2.
(a) 13 msec
(b) 6 msec
(d) 14 msec
(e) None of these
(c) 10 msec
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15. A point moves along the curve y 2x2 1 in such a way that the y value is decreasing at the rate of
2 units per second. At what rate is x changing when x 32 ?
(a) decreasing 72 unitsec
(b) increasing 72 unitsec
(d) decreasing 13 unitsec
(e) None of these
(c) increasing 13 unitsec
59
60
Chapter 2
Differentiation
Test Form C
Name
__________________________________________
Date
Chapter 2
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator/utility is recommended for this test.
1. Determine whether the slope at the indicated point is positive, negative, or zero.
y
3
(a) Zero
2
(b) No slope
1
x
(c) Positive
−3
−2
1
2
3
−1
(d) Negative
−2
−3
(e) None of these
2. Find the slope of the graph of f x x2 2x at the point a, f (a).
(c) f a
(a) 0
(b) 2a 2
(d) a2 2a
(e) None of these
3. Use the graph to determine all x-values at which the function is not differentiable.
y
(a) a, b, c, d, e, f, and g
(b) b and d
d
(c) a and f
a
b
c
e f g
x
(d) a, f, and g
3x2 8
and its derivative, f, on the same coordinate axes.
x2 4
Then use the graph to describe the behavior of f at that value of x where fx 0.
4. Use a graphing utility to graph f x (a) f x 0
(b) f increases without bound.
(d) f has no tangent line.
(e) None of these
5. Find the value of the derivative of the function f t (c) f has a horizontal tangent line.
t3 2
at the point 2, 3.
t
(a) 9
2
(b) 7
2
(d) 11
16
(e) None of these
(c) 12
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(e) None of these
Chapter 2
Test Bank
6. The graph at the right represents the graph of the derivative of which of the following functions?
(a) f x 2x2 1
y
(b) f x 2x 3
3
2
(c) f x 3x2 2x 1
(d) f x x3
1
x
x2
−3
−2
1
−1
(e) None of these
7. Find
−1
−2
dy
: y 4 sin x 5 cos x x.
dx
(a) 4 cos x 5 sin x 1
(b) 4 cos x 5 sin x 1
(d) 4 cos x 5 sin x
(e) None of these
(c) 4 cos x 5 sin x 1
2
8. The position function for a particular object is s 35
2 t 58t 91. Which statement is true?
(a) The initial velocity is 35.
(b) The velocity is a constant.
(c) The velocity at time t 1 is 23.
(d) The initial position is 35
2.
(e) None of these
9. Find
dy
for y csc cot .
d
(a) 0
(b) cot2 csc cot (d) csc cot csc2 (e) None of these
(c) sec tan sec2 © Houghton Mifflin Company. All rights reserved.
10. Find an equation of the tangent line to the graph of f tan at the point
(a) 4x 4y 4
(b) 42x 4y 4
(d) y x
(e) None of these
4 , 1.
(c) 4x 2y 2
1
f x
11. Let f 3 0, f3 6, g3 1 and g3 . Find h3 if hx .
3
gx
(a) 18
(b) 6
(d) 2
(e) None of these
12. Find the derivative: f x (c) 6
1
.
x3
3 3
(a)
1
33 x343
(b)
(d)
x2
3 x343
(e) None of these
x2
3 x343
(c)
x2
3 x323
2
3
61
62
Chapter 2
Differentiation
13. Find the derivative: f sin 2.
cos 2
sin 2
(d) cos (a)
(b) sec 2
(c)
cos 2
2sin 2
(e) None of these
14. Determine the slope of the graph of the relation 2x2 3xy y3 1 at the point 2, 3.
(a) 17
21
(d)
4
3
(b)
5
7
(c) 13
(e) None of these
15. A machine is rolling a metal cylinder under pressure. The radius of the cylinder is decreasing at a
constant rate of 0.05 inches per second and the volume V is 128 cubic inches. At what rate is
the length h changing when the radius r is 1.8 inches? Hint: V r2h
(b) 39.51 in.sec
(d) 43.90 in.sec
(e) None of these
(c) 2.195 in.sec
© Houghton Mifflin Company. All rights reserved.
(a) 2.195 in.sec
Chapter 2
Test Bank
Test Form D
Name
__________________________________________
Date
Chapter 2
Class
__________________________________________
Section _______________________
____________________________
1
1. Use the definition of a derivative to calculate the derivative of f x .
x
2. Differentiate: y 3. Find
2x
.
1 3x2
dy
for y 2x 1 x3.
dx
4. Find fx for f x cot3x.
5. Calculate
d 2y
1x
for y .
dx2
2x
6. The position equation for the movement of a particle is given by s t3 12 where s is measured
in feet and t is measured in seconds. Find the acceleration of this particle at 1 second.
7. Find y if y 8. Find
x
.
xy
dy
if x cos y.
dx
9. Find the derivative: f sec 2.
10. Differentiate: y sin2 x cos2 x.
© Houghton Mifflin Company. All rights reserved.
63
11. Find an equation for the tangent line to the graph of f x x 1 at the point where x 3.
12. Find the values of x for all points on the graph of f x x3 2x2 5x 16 at which the slope
of the tangent line is 4.
1
13. Find the instantaneous rate of change of R with respect to x if R 2x2 .
x
14. An object is thrown (straight down) from the top of a 220-foot building with an initial velocity of
26 feet per second.
a. Write the position equation for the movement described.
b. What is the velocity at 1 second?
15. As a balloon in the shape of a sphere is being blown up, the volume is increasing at the rate of
4 cubic inches per second. At what rate is the radius increasing when the radius is 1 inch?
64
Chapter 2
Differentiation
Test Form E
Name
__________________________________________
Date
Chapter 2
Class
__________________________________________
Section _______________________
____________________________
A graphing calculator/utility is recommended for this test.
1. At each point indicated on the graph, determine whether
the value of the derivative is positive, negative, zero, or
the function has no derivative.
y
a
b c
d
e
x
4
2. Let f x .
x
a. Use the definition of the derivative to calculate the derivative of f.
b. Find the slope of the tangent line to the graph of f at the point 2, 2.
c. Write an equation of the tangent line in part b.
d. Use a graphing utility to graph f and the tangent line on the same axes. Then sketch the graphs.
3. Find the point(s) on the graph of y 1
where the graph is parallel to the line 4x 9y 3.
x
4. A coin is dropped from a height of 750 feet. The height, s, (measured in feet), at time, t
(measured in seconds), is given by s 16t2 750.
b. Find the instantaneous velocity when t 3.
c. How long does it take for the coin to hit the ground?
d. Find the velocity of the coin when it hits the ground.
3
r . How fast is the
3
changing volume with respect to changes in r when the radius is equal to 2 feet?
5. The volume of a right circular cone of radius r and height r is given by V 6. Find the derivative: f x 5 sec x tan x.
7. Evaluate the derivative for the function f t t
2
at the point ,
.
cos t
3 3
© Houghton Mifflin Company. All rights reserved.
a. Find the average velocity on the interval 1, 3.
Chapter 2
Test Bank
8. Let f x x5 5x.
a. Calculate f x.
b. Use a graphing utility to graph f and f on the same axes. Sketch the graphs.
c. Use the graph to determine those point(s) where f has a horizontal tangent line.
d. Give the value of f at each of the points found in part c.
9. The graphs of a function f and its derivative f are given on the same
coordinate axes. Label the graphs as f or f and state the reasons for
your choice.
y
3
2
1
x
−3
2
3
10. Find f x: f x 2x2 5.
11. Find the point(s) (if any) of horizontal tangent lines: x2 xy y2 6.
© Houghton Mifflin Company. All rights reserved.
12. A balloon rises at the rate of 8 feet per second from a point on the ground 60 feet from an observer.
Find the rate of change of the angle of elevation when the balloon is 25 feet above the ground.
65