STUDY NOTES & REFERENCE GUIDE
Covers All Chapters
SOLUTIONS MANUAL
officialruss23 - Stuvia | Page 1 of 488
, STUDY NOTES & REFERENCE GUIDE
CONTENTS
Chapter 0 Functions ............................................................................................. 1
··
Chapter 1 The Derivative ................................................................................... 26 ···
Chapter 2 Applications of the Derivative........................................................... 77
··
Chapter 3 Techniques of Differentiation.......................................................... 115
···
Chapter 4 The Exponential and Natural Logarithmic Functions ..................... 141
···
Chapter 5 Applications of the Exponential and Natural Logarithm ··
Functions ......................................................................................... 168
·
Chapter 6 The Definite Integral ....................................................................... 182
·
Chapter 7 Functions of Several Variables ........................................................ 215
···
Chapter 8 The Trigonometric Functions .......................................................... 245 ··
Chapter 9 Techniques of Integration. ............................................................... 263
Chapter 10 Differential Equations ..................................................................... 302
Chapter 11 Taylor Polynomials and Infinite Series ........................................... 334
Chapter 12 Probability and Calculus.................................................................. 356
officialruss23 - Stuvia | Page 2 of 488
, STUDY NOTES & REFERENCE GUIDE
Chapter 0 Functions
···
0.1 Functions and Their Graphs t
·
16. h(t)
1. (1 t)
1 1
1 2 2 1
2. h 3
2 1 1 3
2 2
3 3
3. 3
h 2 2 3
2 3 1
1 2
4. 2
a 1 a 1
h(a 1)
5. 1 (a 1) a 2
6. 17. ƒ (x) x 2 2x
3 ƒ (a 1) (a 1) 2 2(a 1)
7. [2, 3) 8. 1,
2 (a 2 2a 1) 2a 2 a2 1
ƒ (a 2) (a 2) 2 2(a 2)
9. [–1, 0) 10. [–1, 8)
(a 2 4a 4) 2a 4 a2 2a
11. ,3 12. 2,
18. ƒ (x) x 2 4x 3
13. ƒ (x) x 2 3x ƒ (a 1) (a 1)2 4(a 1) 3
ƒ (0) 02 3(0) 0 (a 2 2a 1) (4a 4) 3
2 a 2 2a
ƒ (5) 5 3(5) 25 15 10
ƒ (a 2) (a 2) 2 4(a 2) 3
ƒ (3) 3 2
3(3) 9 9 0 2
(a 4a 4) (4a 8) 3
ƒ ( 7) ( 7)2 3( 7) 49 21 70 a 2
1
19. a. ƒ (0) represents the number of laptops sold
14. ƒ (x) x3 x2 x 1
in 2010.
ƒ (1) 13 12 1 1 0
b. ƒ (6) 150 2(6) 62
ƒ ( 1) ( 1) 3
( 1) 2
( 1) 1 0
150 12 36 198
3 2
1 1 1 1 9
ƒ 1 100x
2 2 2 2 8 20. R(x) , x 0
b x
ƒ (a) a3 a 2 a 1 a. b = 20, x 60
100(60)
15. g(t) t3 3t 2 t R(60) 75
20 60
g(2) 23 3(2)2 2 8 12 2 2 The solution produces a 75% response.
3 2
1 1 1 1 b. If R(50) = 60, then
g 3
2 2 2 2 100(50)
60
1 3 1 11 b 50
8 4 2 8 60b 3000 5000
2 3
2 2 2 100
2 b
g 3 3
3 3 3 3 This particular frog has a positive constant
8 12 2 10
.37037 of 33.3.
27 9 3 27
g(a) a3 3a 2 a
officialruss23 - Stuvia | Page 3 of 488
, STUDY NOTES & REFERENCE GUIDE
2
8x 39. 40.
21. ƒ (x)
(x 1)(x 2) 41. , 1 5, 9 42. 1, 5 9,
all real numbers such that x 1, 2 or
, 1 1, 2 2, 43. ƒ 1 .03; ƒ 5 .037
22. ƒ (t)
1 44. ƒ 6 .03
45. 0, .05 46. t 3
all real numbers such that t > 0 or 0,
1
1 47. ƒ (x) x x 2
23. g(x) ____ 2
3 x
1 25
all real numbers such that x < 3 or , 3 ƒ (3) 3 (3 2)
2 2
4 Thus, (3, 12) is not on the graph.
24. g(x)
x(x 2) 48. ƒ(x) = x(5 + x)(4 – x)
all real numbers such that x 0, –2 or ƒ(–2) = –2(5 + (–2))(4 – (–2)) = –36
So (–2, 12) is not on the graph.
, 2 2, 0 0,
3x 5 49. g(x) 3x 1
25. ƒ x x2 1
x2 x 6
1 1
The denominator is zero for x x = 2, 1 3 1 2
g 2 2
so the domain consists of all real numbers 3 1 2 5 5
such that x 2
1 4
, 3 3, 2 2, . 1 2
So , is on the graph.
2 5
1
26. ƒ x
3x 2 1 (x 2 4)
The domain consists of all real numbers, or 50. g(x)
(x 2)
, . 2
40
2 4
g 2 3
2
9 5
27. ƒ x 2x 7 x 3 2 8 3
3 3
The domain consists of all real numbers 2 5
So , is on the graph.
greater than or equal to 0, or 0, .
3 3
2x 1 51. ƒ (x) x3
28. ƒ x
1 x
ƒ (a 1) (a 1)3
The denominator is greater than 0 for all x < 1,
1 5
and the numerator is defined for all x .
2 52. ƒ (x) x
x
1 1,1 .
Thus, the domain is x 1, or 5
2 2 ƒ (2 h) (2 h)
(2 h)
29. function 30. not a function 5 (2 h)2 1 4h h 2
31. not a function 32. not a function (2 h) 2 h
33. not a function 34. function x for 0 x 2
53. ƒ (x)
35. ƒ 0 1; ƒ 7 1 1 x for 2 x 5
ƒ (1) 1 1; ƒ (2) 1 2 3
36. ƒ 2 3; ƒ 1 0
ƒ (3) 1 3 4
37. positive 38. negative
officialruss23 - Stuvia | Page 4 of 488
Covers All Chapters
SOLUTIONS MANUAL
officialruss23 - Stuvia | Page 1 of 488
, STUDY NOTES & REFERENCE GUIDE
CONTENTS
Chapter 0 Functions ............................................................................................. 1
··
Chapter 1 The Derivative ................................................................................... 26 ···
Chapter 2 Applications of the Derivative........................................................... 77
··
Chapter 3 Techniques of Differentiation.......................................................... 115
···
Chapter 4 The Exponential and Natural Logarithmic Functions ..................... 141
···
Chapter 5 Applications of the Exponential and Natural Logarithm ··
Functions ......................................................................................... 168
·
Chapter 6 The Definite Integral ....................................................................... 182
·
Chapter 7 Functions of Several Variables ........................................................ 215
···
Chapter 8 The Trigonometric Functions .......................................................... 245 ··
Chapter 9 Techniques of Integration. ............................................................... 263
Chapter 10 Differential Equations ..................................................................... 302
Chapter 11 Taylor Polynomials and Infinite Series ........................................... 334
Chapter 12 Probability and Calculus.................................................................. 356
officialruss23 - Stuvia | Page 2 of 488
, STUDY NOTES & REFERENCE GUIDE
Chapter 0 Functions
···
0.1 Functions and Their Graphs t
·
16. h(t)
1. (1 t)
1 1
1 2 2 1
2. h 3
2 1 1 3
2 2
3 3
3. 3
h 2 2 3
2 3 1
1 2
4. 2
a 1 a 1
h(a 1)
5. 1 (a 1) a 2
6. 17. ƒ (x) x 2 2x
3 ƒ (a 1) (a 1) 2 2(a 1)
7. [2, 3) 8. 1,
2 (a 2 2a 1) 2a 2 a2 1
ƒ (a 2) (a 2) 2 2(a 2)
9. [–1, 0) 10. [–1, 8)
(a 2 4a 4) 2a 4 a2 2a
11. ,3 12. 2,
18. ƒ (x) x 2 4x 3
13. ƒ (x) x 2 3x ƒ (a 1) (a 1)2 4(a 1) 3
ƒ (0) 02 3(0) 0 (a 2 2a 1) (4a 4) 3
2 a 2 2a
ƒ (5) 5 3(5) 25 15 10
ƒ (a 2) (a 2) 2 4(a 2) 3
ƒ (3) 3 2
3(3) 9 9 0 2
(a 4a 4) (4a 8) 3
ƒ ( 7) ( 7)2 3( 7) 49 21 70 a 2
1
19. a. ƒ (0) represents the number of laptops sold
14. ƒ (x) x3 x2 x 1
in 2010.
ƒ (1) 13 12 1 1 0
b. ƒ (6) 150 2(6) 62
ƒ ( 1) ( 1) 3
( 1) 2
( 1) 1 0
150 12 36 198
3 2
1 1 1 1 9
ƒ 1 100x
2 2 2 2 8 20. R(x) , x 0
b x
ƒ (a) a3 a 2 a 1 a. b = 20, x 60
100(60)
15. g(t) t3 3t 2 t R(60) 75
20 60
g(2) 23 3(2)2 2 8 12 2 2 The solution produces a 75% response.
3 2
1 1 1 1 b. If R(50) = 60, then
g 3
2 2 2 2 100(50)
60
1 3 1 11 b 50
8 4 2 8 60b 3000 5000
2 3
2 2 2 100
2 b
g 3 3
3 3 3 3 This particular frog has a positive constant
8 12 2 10
.37037 of 33.3.
27 9 3 27
g(a) a3 3a 2 a
officialruss23 - Stuvia | Page 3 of 488
, STUDY NOTES & REFERENCE GUIDE
2
8x 39. 40.
21. ƒ (x)
(x 1)(x 2) 41. , 1 5, 9 42. 1, 5 9,
all real numbers such that x 1, 2 or
, 1 1, 2 2, 43. ƒ 1 .03; ƒ 5 .037
22. ƒ (t)
1 44. ƒ 6 .03
45. 0, .05 46. t 3
all real numbers such that t > 0 or 0,
1
1 47. ƒ (x) x x 2
23. g(x) ____ 2
3 x
1 25
all real numbers such that x < 3 or , 3 ƒ (3) 3 (3 2)
2 2
4 Thus, (3, 12) is not on the graph.
24. g(x)
x(x 2) 48. ƒ(x) = x(5 + x)(4 – x)
all real numbers such that x 0, –2 or ƒ(–2) = –2(5 + (–2))(4 – (–2)) = –36
So (–2, 12) is not on the graph.
, 2 2, 0 0,
3x 5 49. g(x) 3x 1
25. ƒ x x2 1
x2 x 6
1 1
The denominator is zero for x x = 2, 1 3 1 2
g 2 2
so the domain consists of all real numbers 3 1 2 5 5
such that x 2
1 4
, 3 3, 2 2, . 1 2
So , is on the graph.
2 5
1
26. ƒ x
3x 2 1 (x 2 4)
The domain consists of all real numbers, or 50. g(x)
(x 2)
, . 2
40
2 4
g 2 3
2
9 5
27. ƒ x 2x 7 x 3 2 8 3
3 3
The domain consists of all real numbers 2 5
So , is on the graph.
greater than or equal to 0, or 0, .
3 3
2x 1 51. ƒ (x) x3
28. ƒ x
1 x
ƒ (a 1) (a 1)3
The denominator is greater than 0 for all x < 1,
1 5
and the numerator is defined for all x .
2 52. ƒ (x) x
x
1 1,1 .
Thus, the domain is x 1, or 5
2 2 ƒ (2 h) (2 h)
(2 h)
29. function 30. not a function 5 (2 h)2 1 4h h 2
31. not a function 32. not a function (2 h) 2 h
33. not a function 34. function x for 0 x 2
53. ƒ (x)
35. ƒ 0 1; ƒ 7 1 1 x for 2 x 5
ƒ (1) 1 1; ƒ (2) 1 2 3
36. ƒ 2 3; ƒ 1 0
ƒ (3) 1 3 4
37. positive 38. negative
officialruss23 - Stuvia | Page 4 of 488
