Differentiation 140 Chapter 2 3rt3 See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. 69. y= 26 sec 4x, (0, 25) 70, v 2.4 Exercises Finding an Exercises 71-78, (a) find an line to the graph of the fun (b) use a graphing utility to its tangent line at the point feature of a graphing utility Equation of Finding a Derivative of a Trigonometric Function In Exercises 35-54, find the derivative of the trigonometric function. 36. y = sin X CONCEPT CHЕСК 1. Chain Rule Describe the Chain Rule for the composition of two differentiable functions in your own words. 35. y=cos 4x 38. h(x) sec 6x 40. y csc(1 - 2x)2 42. g(e) = sec(0) tan(0) 71. f(x) = /2x2 7, (4, 5) 73. y= (4x3 + 3)2, (-1, 1) 72. 37. g(x) = 5 tan 3x 2. General Power Rule What is the difference between 39. y= sin(Tx)2 41. h(x) = sin 2x cos 2x 74. the (Simple) Power Rule and the General Power Rule? 76. COS V 75. f(x) = sin 8x, (T, 0) 44. g(v) = cot x Decomposition of a Composite Function In Exercises 3-8, complete the table. 43. f(x)sin x CSC V 78. 46. g(t) = 5 cos2 t 45. y=4 sec2x 47. f(e) sin 20 49. f(t) 3 sec (nt - 1)2 77. f(x) = tan2 x, 48. h(t) = 2 cot2(t + 2) y = f(g(x)) u g(x) y = f(u) AFamous Curves In Exercises 79 50. y 5 cos(x)2 of the tangent line to the graph at t graphing utility to graph the functio point in the same viewing window. 3. y=(6x 5)4 ес A 52. y = cos(5x + csc A 4. y = 3/4x+ 3 51. y sin(3x +cos A 54. y cossin(tan mx) 1 5. y 3x+ 53. y sincot 3x A 80 В 79. Semicircle AFinding a Derivative Using Technology In Exercises 55-60, use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the 2 6. y t10 f(x) = 25 - x y 7. y csc3 x 5x 8. y sin 2 graph of the derivative. (3, 4) 4 x1 x2 1 Finding 2x a Derivative In Exercises 9-34, find 2 55. y 56. у %3 the derivative of the function. x+1 X 2 -6-4 -2 xt 1 58. g(x) = x - 1 + +1 57. y = -4 9. y= (2x- 7)3 10. y 5(2 x)4 12. f(t) (9t + 2)2/3 14. g(x) = 4-3x2 16. y 29 - x X 11. g(x) = 3(4 - 9x)5/6 COS TX +1 1 81. Horizontal Tangent Line interval (0, 2T) at which the g has a horizontal tangent. 59. у 3 60. y x2 tan 13. h(s) = 25s2 + 3 15. y 3/6x 1 Slope of a Tangent Line itExercises 61 and 62, find the slope of the tangent line to the sine function at the origin. Compare this value with the mmber of complete cycles in the interval [0, 27]. 82. Horizontal Tangent Li which the graph of 17. у 18. s(t)45t 2 6 19. g(s)= 2) f(x)= 1 20. y= (t 2)4 61. I 1 y sin 3x 21. y = 1 2 22. g(t) y=sin has a horizoal tangent. /3x +5. 23. f(x) = x(x - 2)7 25. y x1- 2-2 24. f(x)= x(2x - 5)3 26. y 16- 1 Finding a Secend Derivati second derivative of the functi 1 2т 37T 2T 2 -1 27. y 83. f(x) = 5(2 -2t 28. y 11 -2+ 1 4 - (3) 85. f(x) = 11x t 512 29. g(x) = Finding the Slope of a Graph In Exercises 63-70, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. 30. h(t) 87. f(x) = sin 1 + t 4 31. s(t) = t + 3 Evaluating a Second D evaluate the second deriva 63. y x2+8x, (1,3) 64. y 65. f(x) = 5(x3 - 2)-1, (-2,-3) 3x2 2 32. gx) =2x +3. /3x3 +4x, (2, 2) point. Use a computer algeb 33. f(x) = ((x2 + 3)5+ x2 1 66. f(x) 3x) 89. hx) 4. 16/ 34. g(x) = (2 + (x +1)4)3 (3.x + 1)3,1. 4 67. y x2)2' (0, 1) 4 68. y x22r)3 (1,-4) 91. f(x) = cos x2, (0, 1) XI2 S A AAAA AAA AA

Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Differentiation
140
Chapter 2
3rt3
See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises.
69. y= 26 sec 4x, (0, 25) 70, v
2.4 Exercises
Finding an
Exercises 71-78, (a) find an
line to the graph of the fun
(b) use a graphing utility to
its tangent line at the point
feature of a graphing utility
Equation of
Finding a Derivative of a Trigonometric
Function In Exercises 35-54, find the derivative
of the trigonometric function.
36. y = sin X
CONCEPT CHЕСК
1. Chain Rule Describe the Chain Rule for the
composition of two differentiable functions in your own
words.
35. y=cos 4x
38. h(x) sec 6x
40. y csc(1 - 2x)2
42. g(e) = sec(0) tan(0)
71. f(x) = /2x2 7, (4, 5)
73. y= (4x3 + 3)2, (-1, 1)
72.
37. g(x) = 5 tan 3x
2. General Power Rule
What is the difference between
39. y= sin(Tx)2
41. h(x) = sin 2x cos 2x
74.
the (Simple) Power Rule and the General Power Rule?
76.
COS V
75. f(x) = sin 8x, (T, 0)
44. g(v) =
cot x
Decomposition of a Composite Function
In Exercises 3-8, complete the table.
43. f(x)sin x
CSC V
78.
46. g(t) = 5 cos2 t
45. y=4 sec2x
47. f(e) sin 20
49. f(t) 3 sec (nt - 1)2
77. f(x) = tan2 x,
48. h(t) = 2 cot2(t + 2)
y = f(g(x))
u g(x)
y = f(u)
AFamous Curves In Exercises 79
50. y 5 cos(x)2
of the tangent line to the graph at t
graphing utility to graph the functio
point in the same viewing window.
3. y=(6x 5)4
ес
A
52. y = cos(5x + csc
A
4. y =
3/4x+ 3
51. y sin(3x +cos
A
54. y cossin(tan mx)
1
5. y 3x+
53. y sincot 3x
A
80
В
79. Semicircle
AFinding a Derivative Using Technology In Exercises
55-60, use a computer algebra system to find the derivative
of the function. Then use the utility to graph the function and
its derivative on the same set of coordinate axes. Describe the
behavior of the function that corresponds to any zeros of the
2
6. y t10
f(x) = 25 - x
y
7. y csc3 x
5x
8. y sin
2
graph of the derivative.
(3, 4)
4
x1
x2 1
Finding
2x
a Derivative In Exercises 9-34, find
2
55. y
56. у %3
the derivative of the function.
x+1
X
2
-6-4 -2
xt 1
58. g(x) = x - 1 + +1
57. y =
-4
9. y= (2x- 7)3
10. y 5(2 x)4
12. f(t) (9t + 2)2/3
14. g(x) = 4-3x2
16. y 29 - x
X
11. g(x) = 3(4 - 9x)5/6
COS TX +1
1
81. Horizontal Tangent Line
interval (0, 2T) at which the g
has a horizontal tangent.
59. у 3
60. y x2 tan
13. h(s) = 25s2 + 3
15. y 3/6x 1
Slope of a Tangent Line itExercises 61 and 62, find
the slope of the tangent line to the sine function at the origin.
Compare this value with the mmber of complete cycles in the
interval [0, 27].
82. Horizontal Tangent Li
which the graph of
17. у
18. s(t)45t
2
6
19. g(s)= 2)
f(x)= 1
20. y=
(t 2)4
61.
I
1
y sin 3x
21. y =
1
2
22. g(t)
y=sin
has a horizoal tangent.
/3x +5.
23. f(x) = x(x - 2)7
25. y x1-
2-2
24. f(x)= x(2x - 5)3
26. y 16-
1
Finding a Secend Derivati
second derivative of the functi
1
2т
37T 2T
2
-1
27. y
83. f(x) = 5(2
-2t
28. y
11
-2+
1
4
- (3)
85. f(x) =
11x
t 512
29. g(x) =
Finding the Slope of a Graph In Exercises 63-70, find
the slope of the graph of the function at the given point. Use the
derivative feature of a graphing utility to confirm your results.
30. h(t)
87. f(x) = sin
1 + t
4
31. s(t) =
t + 3
Evaluating a Second D
evaluate the second deriva
63. y x2+8x, (1,3)
64. y
65. f(x) = 5(x3 - 2)-1, (-2,-3)
3x2 2
32. gx) =2x +3.
/3x3 +4x,
(2, 2)
point. Use a computer algeb
33. f(x) = ((x2 + 3)5+ x2
1
66. f(x) 3x)
89. hx)
4.
16/
34. g(x) = (2 + (x +1)4)3
(3.x + 1)3,1.
4
67. y x2)2' (0, 1)
4
68. y x22r)3 (1,-4)
91. f(x) = cos x2, (0, 1)
XI2
S A
AAAA AAA AA
Transcribed Image Text:Differentiation 140 Chapter 2 3rt3 See CalcChat.com for tutorial help and worked-out solutions to odd-numbered exercises. 69. y= 26 sec 4x, (0, 25) 70, v 2.4 Exercises Finding an Exercises 71-78, (a) find an line to the graph of the fun (b) use a graphing utility to its tangent line at the point feature of a graphing utility Equation of Finding a Derivative of a Trigonometric Function In Exercises 35-54, find the derivative of the trigonometric function. 36. y = sin X CONCEPT CHЕСК 1. Chain Rule Describe the Chain Rule for the composition of two differentiable functions in your own words. 35. y=cos 4x 38. h(x) sec 6x 40. y csc(1 - 2x)2 42. g(e) = sec(0) tan(0) 71. f(x) = /2x2 7, (4, 5) 73. y= (4x3 + 3)2, (-1, 1) 72. 37. g(x) = 5 tan 3x 2. General Power Rule What is the difference between 39. y= sin(Tx)2 41. h(x) = sin 2x cos 2x 74. the (Simple) Power Rule and the General Power Rule? 76. COS V 75. f(x) = sin 8x, (T, 0) 44. g(v) = cot x Decomposition of a Composite Function In Exercises 3-8, complete the table. 43. f(x)sin x CSC V 78. 46. g(t) = 5 cos2 t 45. y=4 sec2x 47. f(e) sin 20 49. f(t) 3 sec (nt - 1)2 77. f(x) = tan2 x, 48. h(t) = 2 cot2(t + 2) y = f(g(x)) u g(x) y = f(u) AFamous Curves In Exercises 79 50. y 5 cos(x)2 of the tangent line to the graph at t graphing utility to graph the functio point in the same viewing window. 3. y=(6x 5)4 ес A 52. y = cos(5x + csc A 4. y = 3/4x+ 3 51. y sin(3x +cos A 54. y cossin(tan mx) 1 5. y 3x+ 53. y sincot 3x A 80 В 79. Semicircle AFinding a Derivative Using Technology In Exercises 55-60, use a computer algebra system to find the derivative of the function. Then use the utility to graph the function and its derivative on the same set of coordinate axes. Describe the behavior of the function that corresponds to any zeros of the 2 6. y t10 f(x) = 25 - x y 7. y csc3 x 5x 8. y sin 2 graph of the derivative. (3, 4) 4 x1 x2 1 Finding 2x a Derivative In Exercises 9-34, find 2 55. y 56. у %3 the derivative of the function. x+1 X 2 -6-4 -2 xt 1 58. g(x) = x - 1 + +1 57. y = -4 9. y= (2x- 7)3 10. y 5(2 x)4 12. f(t) (9t + 2)2/3 14. g(x) = 4-3x2 16. y 29 - x X 11. g(x) = 3(4 - 9x)5/6 COS TX +1 1 81. Horizontal Tangent Line interval (0, 2T) at which the g has a horizontal tangent. 59. у 3 60. y x2 tan 13. h(s) = 25s2 + 3 15. y 3/6x 1 Slope of a Tangent Line itExercises 61 and 62, find the slope of the tangent line to the sine function at the origin. Compare this value with the mmber of complete cycles in the interval [0, 27]. 82. Horizontal Tangent Li which the graph of 17. у 18. s(t)45t 2 6 19. g(s)= 2) f(x)= 1 20. y= (t 2)4 61. I 1 y sin 3x 21. y = 1 2 22. g(t) y=sin has a horizoal tangent. /3x +5. 23. f(x) = x(x - 2)7 25. y x1- 2-2 24. f(x)= x(2x - 5)3 26. y 16- 1 Finding a Secend Derivati second derivative of the functi 1 2т 37T 2T 2 -1 27. y 83. f(x) = 5(2 -2t 28. y 11 -2+ 1 4 - (3) 85. f(x) = 11x t 512 29. g(x) = Finding the Slope of a Graph In Exercises 63-70, find the slope of the graph of the function at the given point. Use the derivative feature of a graphing utility to confirm your results. 30. h(t) 87. f(x) = sin 1 + t 4 31. s(t) = t + 3 Evaluating a Second D evaluate the second deriva 63. y x2+8x, (1,3) 64. y 65. f(x) = 5(x3 - 2)-1, (-2,-3) 3x2 2 32. gx) =2x +3. /3x3 +4x, (2, 2) point. Use a computer algeb 33. f(x) = ((x2 + 3)5+ x2 1 66. f(x) 3x) 89. hx) 4. 16/ 34. g(x) = (2 + (x +1)4)3 (3.x + 1)3,1. 4 67. y x2)2' (0, 1) 4 68. y x22r)3 (1,-4) 91. f(x) = cos x2, (0, 1) XI2 S A AAAA AAA AA
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