SOLUTION MANUAL
SOLUTION MANUAL
Solutions to Tech Tutors
757
Solutions to
T E C H T U T O R S Chapter 1 Section 1.3 (page 29) The lines appear perpendicular in the setting − 9 ≤ x ≤ 9 and − 6 ≤ y ≤ 6.
Section 1.6 (page 61) Most calculators set in connected mode will join the two branches of the graph with a nearly vertical line near x = 2. This line is not part of the graph.
Chapter 4 Section 4.5 (page 287) Answers will vary.
Chapter 6 Section 6.3 (page 395) 1.46265
Chapter 8 Section 8.4 (page 537) Answers will vary.
Section 8.5 (page 546) Answers will vary.
Chapter 11 Section 11.3 (page 676) y = x 2 (ln x + 1)
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
CONTENTS Chapter 1
Functions, Graphs, and Limits ............................................................... 1
Chapter 2
Differentiation.......................................................................................66
Chapter 3
Applications of the Derivative ...........................................................145
Chapter 4
Exponential and Logarithmic Functions ...........................................247
Chapter 5
Integration and Its Applications.........................................................309
Chapter 6
Techniques of Integration ..................................................................367
Chapter 7
Functions of Several Variables ..........................................................416
Chapter 8
Trigonometric Functions....................................................................497
Chapter 9
Probability and Calculus ....................................................................548
Chapter 10
Series and Taylor Polynomials ..........................................................580
Chapter 11
Differential Equations ........................................................................646
Appendix A
Precalculus Review ............................................................................683
Appendix B
Alternate Introduction to the Fundamental Theorem of Calculus ..........................................................................................700
Checkpoints .............................................................................................................704 Tech Tutors .............................................................................................................757
APPENDICES Appendix A A Precalculus Review ........................................................................684 A.1
The Real Number Line and Order .....................................................684
A.2
Absolute Value and Distance on the Real Number Line..................686
A.3
Exponents and Radicals .....................................................................689
A.4
Factoring Polynomials........................................................................691
A.5
Fractions and Rationalization.............................................................696
Appendix B
Alternative Introduction to the Fundamental Theorem of Calculus ..........................................................................................700
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
A P P E N D I X A A Precalculus Review Section A.1 The Real Number Line and Order 1. Because 0.25 = 14 , it is rational.
x−2 < 2 4 0 < x−2 < 8
13. 0 <
2. Because −3678 = − 3678 , it is rational. 1
2 < x < 10
3π 3. Because π is irrational, is irrational. 2
4. Because
(b) No, if x = 10, then x is not less than 10. (c) No, if x = 0, then x is not greater than 2.
2 is irrational, 3 2 − 1 is irrational.
5. Because 4.3451 has a repeating decimal expansion, it is rational. 6.
(a) Yes, if x = 4, then 2 < x < 10.
3− x ≤1 2 −2 < 3 − x ≤ 2
14. −1 <
22 is rational. 7
−5 < − x ≤ −1 5 > x ≥ 1 or 1 ≤ x < 5
7. Because 3 64 = 4, it is rational.
(a) No, if x = 0, then x is not greater than or equal to 1.
8. Because 0.8177 has a repeating decimal expansion, it is rational. 9. Because 60 is not the cube of a rational number, 3 60 is irrational.
(b) Yes, if x = 1, then 1 ≤ x < 5. (c) No, if x = 5, then x is not less than 5. 15.
10. Because e is irrational, 2e is irrational.
x−5 ≥ 7
16.
1 2x 2
(a) Yes, if x = 3, then x = 3 = 15 is greater than 12 . 5 5
x −2
−1
0
1
2
4x + 1 < 2x
17.
4x + 1 − 2x − 1 < 2x − 2x − 1 2 x < −1
( ) < 12 (−1)
1 2x 2
24 . = 10
x < − 12
x 12. x + 1 < 3 3x + 3 < x
− 12 x −2
2x + 3 < 0 3 2
18.
0
1
2
2x + 7 < 3 2 x < −4
( ) < 12 (−4)
1 2x 2
3 (b) No, if x = 4, then x is not less than − . 2 3 (c) Yes, if x = −4, then x is less than − . 2
−1
2x + 7 − 7 < 3 − 7
3 (a) No, if x = 0, then x is not less than − . 2
684
16
3 2
1 3 2
25 is greater than (c) Yes, if x = 52 , then x = 52 = 10
x < −
14
x > 32
than 12 . 5
12 5
12
2x > 3
( )> ()
x > 12 5
(b) No, if x = −3, then x = −3 = − 15 is not greater 5
10
x ≥ 12
11. 5 x − 12 > 0 5 x > 12
x
x −5+5 ≥ 7+5
x < −2 x −6
−4
−2
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section A.1 4 − 2 x < 3x − 1
19.
26.
4 − 2 x − 4 − 3x < 3x − 1 − 4 − 3x −5 x < −5 − 15 ( −5 x) > − 15 ( −5)
The Real Number Line and Order
x x − > 5 2 3 x⎞ ⎛x 6⎜ − ⎟ > 6(5) 3⎠ ⎝2
685
x −10
0
10
20
30
40
0
1
50
3 x − 2 x > 30
x >1
x > 30 x
−2
0
2
x − 4 ≤ 2x + 1
20.
2x2 − x − 6 < 0
(2 x + 3)( x − 2) < 0
x − 4 − x − 1 ≤ 2x + 1 − x − 1 −5 ≤ x
21.
−2
−4 < 2 x − 3 < 4
− 12
7 2
−2
0
2
4
x
−3
−2
−1
−
0
1
2
3
3 4
−
1 4 x
x + 1 − 1 > 14 − 1
−1
2
3
4
5
(
) ( 12 , 4), and
x = 4. By testing the intervals −∞, 12 ,
29. Let E represent the earnings per share, in dollars. Then 4.1 ≤ E ≤ 4.25.
31. Let p represent the percent of Americans who conduct banking transactions online. Then p ≤ 40. x
−3
−2
−1
0
1
2
32. Let I represent the net income, in millions of dollars. Then I ≥ 239.
3
33. A = 20 and r = 220 − A = 200. Let T be the target heart rate. Then
−3 < − x < 3 3 > x > −3
(0.60)( 200) ≤ T ≤ (0.90)(200) x 4
6
8
120 ≤ T ≤ 180 So, the target heart rate for a 20-year-old is between 120 beats per minute and 180 beats per minute.
3 x + 2 x > 30
34. C = 0.35m + 2500 < 13,000
5 x > 30
0.35m < 10,500
1 1 (5 x) > (30) 5 5 x > 6
1
30. Let p represent the daily oil production, in millions of barrels. Then 2 < p < 2.4.
− 34 < x < − 14
x x 25. + > 5 2 3 ⎛ x⎞ ⎛ x⎞ 6⎜ ⎟ + 6⎜ ⎟ > 6(5) ⎝ 2⎠ ⎝ 3⎠
0
0
1 2
− 14 > x > − 43
x <1 3 ⎛ x⎞ 3( −1) < 3⎜ − ⎟ < 3(1) ⎝ 3⎠
x −1
(4, ∞), the solution set is 12 < x < 4.
> x + 1 > 14
−1 < −
2x − 9x + 4 < 0
Zeros of the polynomial ( 2 x − 1)( x − 4) are x = 12 and
−3 ≤ x < 2
24.
1 2
2
(2 x − 1)( x − 4) < 0
0−3 ≤ x +3−3 < 5−3
3 −1 > 4
2x2 + 1 < 9 x − 3
28. x
0 ≤ x +3 < 5
3 4
2
− 32 < x < 2.
−1 < 2 x < 7 −1 2x 7 < < 2 2 2 1 7 − < x < 2 2
23.
−1
(−∞, − 32 ), (− 32 , 2), and (2, ∞), the solution set is
−4 + 3 < 2 x − 3 + 3 < 4 + 3
22.
x −2
x = − 32 and x = 2. By testing the intervals
x −4
3
−2
Zeros of the polynomial ( 2 x + 3)( x − 2) are
−5 −6
2 x2 − x < 6
27.
m < 30,000 So, the number of miles driven must be less than 30,000.
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
686
Appendix A
A Precalculus Review
35. R = 115.95 x and C = 95 x + 750 and because R > C , you can write
37. (a) False. Because a < b, −2a > −2b.
(b) True. Because a < b, a + 2 < b + 2.
115.95 x > 95 x + 750
(c) True. Because a < b, 6a < 6b.
20.95 x > 750 750 x > = 35.7995… 20.95 x ≥ 36.
(d) False, if ab > 0, then
1 1 > . a b
38. (a) True. Because a < b, a − 4 < b − 4.
(b) False. Because a < b, − a > −b and 4 − a > 4 − b.
So, this product will return a profit if x ≥ 36 units. 36. Revenue: R = 4.50 x
(c) True. Because a < b, − 3a > −3b.
Cost: C = 2.75 x + 220
(d) True. Because a < b, 14 a < 14 b.
Profit: P = R − C = 4.50 x − ( 2.75 x + 220) = 1.75 x − 220 So, 60 ≤ 1.75 x − 220 ≤ 270 280 ≤ 1.75 x ≤ 490 160 ≤ x ≤ 280 So, the daily donut sales vary between 160 dozen donuts per day and 280 dozen donuts per day.
Section A.2 Absolute Value and Distance on the Real Number Line 1. (a) d = 126 − 75 = 51
61 6. (a) d = − 18 − 15 = 23 5 3
(b) d = 75 − 126 = − 51
( )
61 (b) d = 15 − − 18 = 23 5 3
(c) d = 126 − 75 = 51
61 (c) d = − 18 − 15 = − 23 5 3
2. (a) d = −126 − ( − 75) = 51
(b) d = − 75 − ( −126) = 51
7. x ≤ 2
(c) d = −126 − ( − 75) = − 51
8. x < 3
3. (a) d = 9.34 − ( − 5.65) = 14.99
9. x > 2
(b) d = − 5.65 − 9.34 = −14.99
10. x ≥ 3
(c) d = 9.34 − ( − 5.65) = 14.99 4. (a) d = − 2.05 − 4.25 = 6.3
(b) d = 4.25 − ( − 2.05) = 6.3 (c) d = − 2.05 − 4.25 = − 6.3 5. (a) d = 16 − 112 = 128 5 75 75
11. x − 5 ≤ 3 12. x + 4 < 3 13. x − 2 > 2 14. x − 22 > 2
(b) d = 112 − 16 = − 128 75 5 75
15. x − 5 < 3
(c) d = 16 − 112 = 128 5 75 75
16. x − 2 > 5 17.
y − a ≤ 2
18.
y −c < h
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
Section A.2 19. −4 < x < 4 20.
Absolute Value and Distance on the Real Number Line
−4
−2
2
0
4
10 − x < −4
27.
x −6
6
x −3
( ) < 12 (2 x) < 12 (6)
1 −6 2
−2
−1
0
1
2
− x > −6
x > 14
x < 6
3
x 6
x or < −3 2 ⎛ x⎞ 2⎜ ⎟ < 2( −3) ⎝ 2⎠
x > 3 2 ⎛ x⎞ 2⎜ ⎟ > 2(3) ⎝ 2⎠
x < −6
x > 6
0
3x > 12 −4
−2
2
0
4
10
0
1
2
3
4
5
6
7
− 12 ( −10) > − 12 (−2) x > − 12 (−8)
8
5 > x > 4
3x + 1 ≥ 4
or
x 2
3x + 1 − 1 ≥ 4 − 1 3x ≥ 3
−5 3x ≤ 3 3 5 x ≤ − 3
3x 3 ≥ 3 3 x ≥1
30.
x 2
3
x −3 ≤ −5 2 x −3 (2) ≤ −5(2) 2 x − 3 ≤ −10
6
2x <1 3 2x −1 − 1 < 1 − −1<1−1 3 2x −2 < − < 0 3 ⎛ 2x ⎞ 3( −2) < 3 ⎜ − ⎟ < 3(0) ⎝ 3⎠ −1 < 1 −
−
1 1 1 ( −6 ) > − ( − 2 x ) > − ( 0 ) 2 2 2 3 > x > 0 0 < x < 3
x − 3 + 3 ≥ 10 + 3
x ≤ −7
x −1
x ≥ 13
−7
4
−6 < −2 x < 0
x −3 ≥ 5 2 x −3 ( 2) ≥ 5( 2) 2 x − 3 ≥ 10
or
x − 3 + 3 ≤ −10 + 3
31.
13 x 0
50
−10 < −2 x < −8
− 35
−10
40
−1 − 9 < 9 − 2 x − 9 < 1 − 9
3 x ≤ −5
1
30
4 < x < 5
3x + 1 ≤ −4
0
20
−1 < 9 − 2 x < 1
29.
6
x
3x + 1 − 1 ≤ −4 − 1
25.
x ≤ 5
45
3 < x < 7
−1
− x ≥ −5
x ≥ 45
x −6
3x 12 > 3 3 x > 4
−2 + 5 < x − 5 + 5 < 2 + 5
−2
− x ≤ −45 5
−2 < x − 5 < 2
−3
25 − x − 25 ≥ 20 − 25
x
or
−12 3x < 3 3 x < −4
25 − x ≥ 20
or
6
22. 3x < −12
24.
14
25 − x − 25 ≤ −20 − 25
0
23.
10
25 − x ≤ −20
28.
x −6
10 − x − 10 > 4 − 10
− x < −14
−3 < x < 3
21.
10 − x > 4
or
10 − x − 10 < −4 − 10
−6 < 2 x < 6
687
0
1
2
3
−b ≤ x − a ≤ b −b + a ≤ x − a + a ≤ b + a
10
a −b ≤ x ≤ a + b 26.
−5 < 2 x + 1 < 5 x
−5 − 1 < 2 x + 1 − 1 < 5 − 1
a−b
a
a+b
−6 < 2 x < 4 6 2x 4 < < 2 2 2 −3 < x < 2
−
x −3
−2
−1
0
1
2
3
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.
688
Appendix A
A Precalculus Review
2 x − a ≤ −b
32.
2x − a ≥ b
or
2 x − a + a ≤ −b + a
2x − a + a ≥ b + a
2x ≤ a − b
2x ≥ a + b
a −b 2x ≤ 2 2 a −b x ≤ 2
a +b 2x ≥ 2 2 a +b x ≥ 2
5 + 5 2
40. Midpoint = 6 41.
33.
a 2
−25,000 ≤ x − 200,000 ≤ 25,000 175,000 ≤ x ≤ 225,000
The low production level is 175,000 barrels of oil, and the high production level is 275,000 barrels of oil.
a+b 2
3x − a < 2b 4 ⎛ 3x − a ⎞ 4( −2b) < 4⎜ ⎟ < 4( 2b) ⎝ 4 ⎠
43. x − 20 ≤ 0.75
−0.75 ≤ x − 20 ≤ 0.75 19.25 ≤ x ≤ 20.75 The lowest and highest acceptable weights for a 20-ounce cereal box are 19.25 ounces and 20.75 ounces.
−8b < 3 x − a < 8b −8b + a < 3 x − a + a < 8b + a 1 1 (a − 8b) < x < (a + 8b) 3 3
44.
w − 57.5 ≤1 7.5 w − 57.5 ≤1 7.5 −7.5 ≤ w − 57.5 ≤ 7.5 −1 ≤
x
a −
34.
a 3
a + 8b 3
5x < −b 2
50 ≤ w ≤ 65 The guidelines specify that the weights for male collies lie between 50 pounds and 65 pounds
5x > b 2 −5 x > b − a 2 5x < a −b 2 2 x < ( a − b) 5
a −
or
5x − a < −b − a 2 −5 x < −a − b 2 ⎛ −5 x ⎞ −1⎜ ⎟ < −1( − a − b) ⎝ 2 ⎠
a −
45. (a)
E − 4750 ≤ 237.50 (b) $5116.37 is not within 5% of the specified budgeted amount; at variance. 46. (a)
2 ( a + b) 5
2 2 a− b 5 5
8 + 24 = 16 2
36. Midpoint =
7.3 + 12.7 = 10 2
37. Midpoint =
−6.85 + 9.35 = 1.25 2
39. Midpoint =
−4.6 + ( −1.3) 2
− 12 + 43 2
1
I − 15,000 ≤ 500 0.05(15,000) = 750
x
35. Midpoint =
38. Midpoint =
E − 4750 ≤ 500 0.05( 4750) = 237.50
5x > a +b 2 2 ⎛ 5x ⎞ 2 ⎜ ⎟ > ( a + b) 5⎝ 2 ⎠ 5 x >
5 3
42. x − 200,000 ≤ 25,000
−2b <
a − 8b 3
=
p − 33.15 ≤ 2
x a−b 2
2
2a 5
2 2 a+ b 5 5
I − 15,000 ≤ 750 (b) $14,695.00 is within $500 of the specified budgeted amount; not at variance. 47. (a)
= −2.95
1 = 4 = 2 8
E − 20,000 ≤ 500 0.05( 20,000) = 1000 E − 20,000 ≤ 1000
(b) $22,718.35 is not within $500 of the specified budgeted amount; at variance. 48. (a)
T − 7500 ≤ 500 0.05(7500) = 375 T − 7500 ≤ 375
(b) $8691.00 is not within 5% of the specified budgeted amount; at variance.
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Section A.3
Exponents and Radicals
689
49. Let r be the percent of defective units. 0.0005 − 0.0001 ≤ r ≤ 0.0005 + 0.0001 0.0004 ≤ r ≤ 0.0006
Number of defective units = x = 150,000r 0.0004(150,000) ≤ 150,000r ≤ 0.0006(150,000) 60 ≤
≤ 90
x
Cost of refunds = C = 195.99 x 60(195.99) ≤ 195.99 x ≤ 90(195.99) 11,759.40 ≤
≤ 17,639.10
C
The total cost of refunds should be between $11,759.40 and $17,639.10.
Section A.3 Exponents and Radicals 1. −2(3) = −2( 27) = −54 3
15. ( −32)
62 36 2. = = 12 3 3
3. 4( 2)
17. 500(1.01)
60
1 + 3−1 1+13 43 = = = 4 3−1 13 13
19.
3
− 54 ≈ − 3.7798
20.
6
325 ≈ 2.6221
=
−2
= 3−
4
(3)
2
= 3−
4 23 = 9 9
3
10,000
(1.1)120
0
0
3
=
1 1 ( −4)
3
0
=
1 = −64 1 ( −64)
1
(−2)
2
=
1 4
2
3
1 1 = 2 4
14. 16−3 4 =
1 = 163 4
3
1
( 16 ) 4
= 6 y −2 ( 2−3 y −12 )
23. 10( x 2 ) = 10 x 4
( 19 ) = ( 19 ) = ( 13 ) = 271
13. 4−1 2 =
−3
22. z −3 (3z 4 ) = 3 z −3 + 4 = 3 z
27 2 = 3 729 = 9 3
=
⎛1⎞ = 6⎜ ⎟ y −14 ⎝8⎠ 3 = 4 y14
9. 6(10) − ⎡⎣6(10)⎤⎦ = 6(1) − (60) = 6 − 1 = 5 −3
( −32 )
2
⎛1⎞ = 6⎜ 3 ⎟ y −2 −12 ⎝2 ⎠
3
1
1
5
≈ 0.1079
21. 6 y −2 ( 2 y 4 )
8. 5( −3) = 5( −27) = −135
( −4 )
=
≈ 908.3483
18.
−2
7. 3( −2) − 4( −2) = 3( 4) − 4( −8) = 12 + 32 = 44
12.
(−32)
25
3
()
2
11.
1
16. (102 3 ) = 102 = 100
= 4 18 = 12
6. 3 − 4(3)
10.
=
7 7 = 2 5 25
4. 7(5)
5.
−3
−2 5
3
=
24. ( 4 x 3 ) = 16 x 6 2
1
(2)
3
=
1 8
25.
7 x2 = 7 x 2 + 3 = 7 x5 x −3
26.
x −3 1 = x −3 − (1 2) = x −7 2 = 7 2 x x
© 2013 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part.