CONTENTS
Chapter 0 Functions .............................................................................................1
Chapter 1 The Derivative ..................................................................................26
Chapter 2 Applications of the Derivative ..........................................................73
Chapter 3 Techniques of Differentiation .........................................................121
Chapter 4 The Exponential and Natural Logarithmic Functions .....................147
Chapter 5 Applications of the Exponential and Natural Logarithm
Functions .........................................................................................177
M
Chapter 6 The Definite Integral .......................................................................196
Chapter 7 Functions of Several Variables .......................................................230
EL
Chapter 8 The Trigonometric Functions .........................................................267
Chapter 9 Techniques of Integration ...............................................................286
AN
Chapter 10 Differential Equations .....................................................................327
Chapter 11 Taylor Polynomials and Infinite Series...........................................360
Chapter 12 Probability and Calculus .................................................................382
IE
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IE
AN
EL
M
, Chapter 0 Functions
0.1 Functions and Their Graphs 16. h( s ) =
s
1. (1 + s )
1 1
⎛1⎞ 1
h⎜ ⎟ = 2 = 2
=
2. ⎝ 2 ⎠ 1+ 1
2 ( ) 3
2
3
3. ⎛ 3⎞ − 32 −3
h ⎜− ⎟ = = 12 = 3
4.
⎝ 2 ⎠ 1+ − 3 −2 ( 2)
a +1 a +1
h(a + 1) = =
5. 1 + (a + 1) a + 2
6. 17. f ( x) = 3 x + 2, h ≠ 0
f (3 + h ) = 3 (3 + h ) + 2 = 9 + 3h + 2 = 3h + 11
M
⎛ 3⎞ f (3) = 3 (3) + 2 = 11
7. [2, 3) 8. ⎜ −1, ⎟
⎝ 2⎠
f (3 + h ) − f (3) (3h + 11) − 11 3h
9. [–1, 0) 10. [–1, 8) = = =3
h h h
EL
11. (−∞, 3) 12. ⎡⎣ 2, ∞ ) 18. f ( x) = x 2 , h ≠ 0
f (1 + h ) = (1 + h ) = 1 + 2h + h 2
2
13. f ( x) = x 2 − 3 x
f (1) = 12 = 1
AN
f (0) = 0 2 − 3(0) = 0
f (5) = 5 2 − 3(5) = 25 − 15 = 10 ( 2
f (1 + h ) − f (1) 1 + 2h + h − 1
=
)
h h
f (3) = 3 2 − 3(3) = 9 − 9 = 0
2h + h 2
= = 2+h
f (−7) = (−7) 2 − 3(−7) = 49 + 21 = 70 h
IE
14. f ( x) = x 3 + x 2 − x − 1 19. a. k ( x ) = x + 273
3 2 5933 = x + 273 ⇒ x = 5660
f (1) = 1 + 1 − 1 − 1 = 0
The boiling point of tungsten is 5660°C.
??
f (−1) = (−1) 3 + (−1) 2 − (−1) − 1 = 0
9
3 2 b. f ( x) = x + 32
⎛1⎞ ⎛1⎞ ⎛1⎞ ⎛1⎞ 9 5
f ⎜ ⎟ = ⎜ ⎟ + ⎜ ⎟ − ⎜ ⎟ −1 = −
⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ 8 9
f ( x ) = (5660) + 32 = 10220
??
5
f (a) = a 3 + a 2 − a − 1
The boiling point of tungsten is 10220°F.
15. f ( x) = x 2 − 2 x 20. a. f (0) represents the number of laptops sold
2 in 2015.
f (a + 1) = (a + 1) − 2(a + 1)
= (a 2 + 2a + 1) − 2a − 2 = a 2 − 1 b. f (5) = 150 + 2(5) + 5 2
f (a + 2) = (a + 2) 2 − 2(a + 2) = 150 + 10 + 25 = 185
In 2020, the company will sell 185
= (a 2 + 4a + 4) − 2a − 4 = a 2 + 2a laptops.
8x
21. f ( x) =
( x − 1)( x − 2)
all real numbers such that x ≠ 1, 2 or
(−∞, −1) (−1, 2) (2, ∞ )
Copyright © 2023 Pearson Education Inc. 1
, 2 Chapter 0 Functions
1 37. positive 38. negative
22. f (t ) =
t 39. [−1, 3] 40. −1, 5, 9
all real numbers such that t > 0 or (0, ∞ )
41. (−∞, − 1] [5, 9] 42. [ −1, 5] [9, ∞ ]
1
23. g ( x ) = 43. f (1) ≈ .03; f (5) ≈ .037
3− x
all real numbers such that x < 3 or ( −∞, −3) 44. f (6) ≈ .03
24. g ( x) =
4 45. [0, .05] 46. t ≈ 3
x ( x + 2)
⎛ 1⎞
all real numbers such that x ≠ 0, –2 or 47. f ( x) = ⎜ x − ⎟ ( x + 2)
⎝ 2⎠
(−∞, −2) (−2, 0) (0, ∞ )
⎛ 1⎞ 25
25. f (3) = ⎜ 3 − ⎟ (3 + 2) =
⎝ 2 ⎠ 2
M
No, (3, 12) is not on the graph.
48. f(x) = x(5 + x)(4 – x)
f(–2) = –2(5 + (–2))(4 – (–2)) = –36
No, (–2, 12) is not on the graph.
EL
3x − 1
26. 49. g ( x) =
x2 + 1
3 (1) − 1 2
g (1) = = =1
AN
(1) 2
+1 2
Yes, (1, 1) is on the graph.
x2 + 4
50. g ( x) =
x+2
27.
IE
( 4) 2 + 4 20 10
g ( 4) = = =
4+2 6 3
⎛ 1⎞
No, ⎜ 4, ⎟ is not on the graph.
??
⎝ 4⎠
51. f ( x) = x 3
28. f (a + 1) = (a + 1) 3
??
⎛5⎞
52. f ( x) = ⎜ ⎟ − x
⎝x⎠
5
f (2 + h) = − (2 + h)
(2 + h)
5 − (2 + h) 2 1 − 4h − h 2
= =
29. function 30. not a function (2 + h) 2+h
31. not a function 32. not a function ⎪⎧ x for 0 ≤ x < 2
53. f ( x) = ⎨
33. not a function 34. function ⎪⎩1 + x for 2 ≤ x ≤ 5
35. f (0) = 1; f (7 ) = −1 f (1) = 1 = 1
f (2) = 1 + 2 = 3
36. f ( 2) = 3; f ( −1) = 0 f (3) = 1 + 3 = 4
Copyright © 2023 Pearson Education Inc.
Chapter 0 Functions .............................................................................................1
Chapter 1 The Derivative ..................................................................................26
Chapter 2 Applications of the Derivative ..........................................................73
Chapter 3 Techniques of Differentiation .........................................................121
Chapter 4 The Exponential and Natural Logarithmic Functions .....................147
Chapter 5 Applications of the Exponential and Natural Logarithm
Functions .........................................................................................177
M
Chapter 6 The Definite Integral .......................................................................196
Chapter 7 Functions of Several Variables .......................................................230
EL
Chapter 8 The Trigonometric Functions .........................................................267
Chapter 9 Techniques of Integration ...............................................................286
AN
Chapter 10 Differential Equations .....................................................................327
Chapter 11 Taylor Polynomials and Infinite Series...........................................360
Chapter 12 Probability and Calculus .................................................................382
IE
??
??
, ??
??
IE
AN
EL
M
, Chapter 0 Functions
0.1 Functions and Their Graphs 16. h( s ) =
s
1. (1 + s )
1 1
⎛1⎞ 1
h⎜ ⎟ = 2 = 2
=
2. ⎝ 2 ⎠ 1+ 1
2 ( ) 3
2
3
3. ⎛ 3⎞ − 32 −3
h ⎜− ⎟ = = 12 = 3
4.
⎝ 2 ⎠ 1+ − 3 −2 ( 2)
a +1 a +1
h(a + 1) = =
5. 1 + (a + 1) a + 2
6. 17. f ( x) = 3 x + 2, h ≠ 0
f (3 + h ) = 3 (3 + h ) + 2 = 9 + 3h + 2 = 3h + 11
M
⎛ 3⎞ f (3) = 3 (3) + 2 = 11
7. [2, 3) 8. ⎜ −1, ⎟
⎝ 2⎠
f (3 + h ) − f (3) (3h + 11) − 11 3h
9. [–1, 0) 10. [–1, 8) = = =3
h h h
EL
11. (−∞, 3) 12. ⎡⎣ 2, ∞ ) 18. f ( x) = x 2 , h ≠ 0
f (1 + h ) = (1 + h ) = 1 + 2h + h 2
2
13. f ( x) = x 2 − 3 x
f (1) = 12 = 1
AN
f (0) = 0 2 − 3(0) = 0
f (5) = 5 2 − 3(5) = 25 − 15 = 10 ( 2
f (1 + h ) − f (1) 1 + 2h + h − 1
=
)
h h
f (3) = 3 2 − 3(3) = 9 − 9 = 0
2h + h 2
= = 2+h
f (−7) = (−7) 2 − 3(−7) = 49 + 21 = 70 h
IE
14. f ( x) = x 3 + x 2 − x − 1 19. a. k ( x ) = x + 273
3 2 5933 = x + 273 ⇒ x = 5660
f (1) = 1 + 1 − 1 − 1 = 0
The boiling point of tungsten is 5660°C.
??
f (−1) = (−1) 3 + (−1) 2 − (−1) − 1 = 0
9
3 2 b. f ( x) = x + 32
⎛1⎞ ⎛1⎞ ⎛1⎞ ⎛1⎞ 9 5
f ⎜ ⎟ = ⎜ ⎟ + ⎜ ⎟ − ⎜ ⎟ −1 = −
⎝2⎠ ⎝2⎠ ⎝2⎠ ⎝2⎠ 8 9
f ( x ) = (5660) + 32 = 10220
??
5
f (a) = a 3 + a 2 − a − 1
The boiling point of tungsten is 10220°F.
15. f ( x) = x 2 − 2 x 20. a. f (0) represents the number of laptops sold
2 in 2015.
f (a + 1) = (a + 1) − 2(a + 1)
= (a 2 + 2a + 1) − 2a − 2 = a 2 − 1 b. f (5) = 150 + 2(5) + 5 2
f (a + 2) = (a + 2) 2 − 2(a + 2) = 150 + 10 + 25 = 185
In 2020, the company will sell 185
= (a 2 + 4a + 4) − 2a − 4 = a 2 + 2a laptops.
8x
21. f ( x) =
( x − 1)( x − 2)
all real numbers such that x ≠ 1, 2 or
(−∞, −1) (−1, 2) (2, ∞ )
Copyright © 2023 Pearson Education Inc. 1
, 2 Chapter 0 Functions
1 37. positive 38. negative
22. f (t ) =
t 39. [−1, 3] 40. −1, 5, 9
all real numbers such that t > 0 or (0, ∞ )
41. (−∞, − 1] [5, 9] 42. [ −1, 5] [9, ∞ ]
1
23. g ( x ) = 43. f (1) ≈ .03; f (5) ≈ .037
3− x
all real numbers such that x < 3 or ( −∞, −3) 44. f (6) ≈ .03
24. g ( x) =
4 45. [0, .05] 46. t ≈ 3
x ( x + 2)
⎛ 1⎞
all real numbers such that x ≠ 0, –2 or 47. f ( x) = ⎜ x − ⎟ ( x + 2)
⎝ 2⎠
(−∞, −2) (−2, 0) (0, ∞ )
⎛ 1⎞ 25
25. f (3) = ⎜ 3 − ⎟ (3 + 2) =
⎝ 2 ⎠ 2
M
No, (3, 12) is not on the graph.
48. f(x) = x(5 + x)(4 – x)
f(–2) = –2(5 + (–2))(4 – (–2)) = –36
No, (–2, 12) is not on the graph.
EL
3x − 1
26. 49. g ( x) =
x2 + 1
3 (1) − 1 2
g (1) = = =1
AN
(1) 2
+1 2
Yes, (1, 1) is on the graph.
x2 + 4
50. g ( x) =
x+2
27.
IE
( 4) 2 + 4 20 10
g ( 4) = = =
4+2 6 3
⎛ 1⎞
No, ⎜ 4, ⎟ is not on the graph.
??
⎝ 4⎠
51. f ( x) = x 3
28. f (a + 1) = (a + 1) 3
??
⎛5⎞
52. f ( x) = ⎜ ⎟ − x
⎝x⎠
5
f (2 + h) = − (2 + h)
(2 + h)
5 − (2 + h) 2 1 − 4h − h 2
= =
29. function 30. not a function (2 + h) 2+h
31. not a function 32. not a function ⎪⎧ x for 0 ≤ x < 2
53. f ( x) = ⎨
33. not a function 34. function ⎪⎩1 + x for 2 ≤ x ≤ 5
35. f (0) = 1; f (7 ) = −1 f (1) = 1 = 1
f (2) = 1 + 2 = 3
36. f ( 2) = 3; f ( −1) = 0 f (3) = 1 + 3 = 4
Copyright © 2023 Pearson Education Inc.
