Test Bank . ... Chapter 14. Integration. .... Use the vertical line test to determine
which of the following graphs represent y as a function of x. (a). (b). (c). (d).
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
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(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
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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
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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
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)
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(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
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(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
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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)
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(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
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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
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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
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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
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(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
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(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
© Houghton Mifflin Company. All rights reserved.
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
© Houghton Mifflin Company. All rights reserved.
(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
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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