69. Suppose that f(1) = 2 f'(1) = 3 f(2) = 1 f'(2) = 2 13D %3D g(1) 3 g'(1) = 1 g(2) = 1 g'(2) = 4 %3D (a) If S(x) = f(x) + g(x), find S'(1). (b) If P(x) = f(x)g(x), find P'(2). (c) If Q(x) = f(x)/g(x), find Q'(1). (d) If C(x) = f(g(x)), find C'(2). %3D
69. Suppose that f(1) = 2 f'(1) = 3 f(2) = 1 f'(2) = 2 13D %3D g(1) 3 g'(1) = 1 g(2) = 1 g'(2) = 4 %3D (a) If S(x) = f(x) + g(x), find S'(1). (b) If P(x) = f(x)g(x), find P'(2). (c) If Q(x) = f(x)/g(x), find Q'(1). (d) If C(x) = f(g(x)), find C'(2). %3D
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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Question
Could you help me with #69 d please?
![CHAPTER 3 Differentiation Rules
82. (a) Graph the function f(x) 2 sin x in the viewing
rectangle [0, 8] by [-2, 8].
(b) On which interval is the average rate of change larger:
[1.2] or [2, 3]?
(c) At which value of x is the instantaneous rate of change
larger: x 2 or x 5?
(d) Check your visual estimates in part (c) by computing
f'(x) and comparing the numerical values of f'(2)
and f'(5).
sin x + coS x,
65. At what points on the curve y
0<x<2m, is the tangent line horizontal?
111
66. Find the points on the ellipse x + 2y 1 where the
tangent line has slope I.
%3D
67. If f(x) (x- a)(x-b)(x- c), show that
D-X
9-8
[In(x+4)] is the tangent
68. (a) By differentiating the double-angle formula
cos 2x cos'x - sin'r
83. At what point on the curve y
horizontal?
obtain the double-angle formula for the sine function.
(b) By differentiating the addition formula
84. (a) Find an equation of the tangent to the curve y = e' that
4y 1.
is parallel to the line x-
(b) Find an equation of the tangent to the curve y= e' that
passes through the origin.
sin(x + a) = sin x cos a + cos x sin a
obtain the addition formula for the cosine function.
= ax + bx + c that passes through the
85. Find a parabola y
point (1, 4) and whose tangent lines at x=-1 and x= 5
have slopes 6 and -2, respectively.
69. Suppose that
f(1) 2 f'(1) = 3 f(2) = 1 f'(2) = 2
g(1) = 3 g'(1)= 1 g(2) = 1 g'(2) = 4
%3D
86. The function C(t) =
K(e at-e bi), where a, b, and K are
10
(a) If S(x) f(x) + g(x), findS'(1).
(b) If P(x) = f(x)g(x), find P'(2).
(c) If Q(x) = f(x)/g(x), find Q'(1).
(d) If C(x) = f(g(x)), find C'(2).
positive constants and b > a, is used to model the concen-
tration at time t of a drug injected into the bloodstream.
(a) Show that lim,. C(t)30.
(b) Find C'(t), the rate of change of drug concentration in
%3D
70. If f and g are the functions whose graphs are shown, let
P(x) = f(x)g(x). Q(x) = f(x)/g(x), and C(x) = f(g(x)).
Find (a) P'(2), (b) Q'(2), and (c) C'(2).
the blood.
(c) When is this rate equal to 0?
%3D
%3|
%3D
87. An equation of motion of the form s= Ae cos(wt + 8)
represents damped oscillation of an object. Find the velocity
and acceleration of the object.
88. A particle moves along a horizontal line so that its coor-
dinate at time t is x =
Vb? + c?t?,t 0, where b and c
are positive constants.
(a) Find the velocity and acceleration functions.
(b) Show that the particle always moves in the positive
direction.
1.
89. A particle moves on a vertical line so that its coordinate at
time t is y =t
(a) Find the velocity and acceleration functions.
(b) When is the particle moving upward and when is it
moving downward?
(c) Find the distance that the particle travels in the time
interval 0 <I<3.
71-78 Find f' in terms of g
2-12t +3, t 0.
%3D
73. f(x) = [g(x)]
77. f(x) = In g(x)|
78. f(x) = g(ln x)
(d) Graph the position, velocity, and acceleration functions
79-81 Find h' in terms of f' and g'.
(e) When is the particle speeding up? When is it slowing
79. h(x)=
(X)6(x)
80. h(x) =
down?
90. The volume of a right circular cone is V=rr'h, where
r is the radius of the base and h is the height.
(a) Find the rate of change of the volume with respect to
81. h(x) = f(g(sin 4.x))
the height if the radius is constant.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2f278d17-2467-43b3-8a5e-51fb69231928%2F6ff8ab46-a389-4de0-8969-8256cf82ef7f%2Fpbb3vt4.jpeg&w=3840&q=75)
Transcribed Image Text:CHAPTER 3 Differentiation Rules
82. (a) Graph the function f(x) 2 sin x in the viewing
rectangle [0, 8] by [-2, 8].
(b) On which interval is the average rate of change larger:
[1.2] or [2, 3]?
(c) At which value of x is the instantaneous rate of change
larger: x 2 or x 5?
(d) Check your visual estimates in part (c) by computing
f'(x) and comparing the numerical values of f'(2)
and f'(5).
sin x + coS x,
65. At what points on the curve y
0<x<2m, is the tangent line horizontal?
111
66. Find the points on the ellipse x + 2y 1 where the
tangent line has slope I.
%3D
67. If f(x) (x- a)(x-b)(x- c), show that
D-X
9-8
[In(x+4)] is the tangent
68. (a) By differentiating the double-angle formula
cos 2x cos'x - sin'r
83. At what point on the curve y
horizontal?
obtain the double-angle formula for the sine function.
(b) By differentiating the addition formula
84. (a) Find an equation of the tangent to the curve y = e' that
4y 1.
is parallel to the line x-
(b) Find an equation of the tangent to the curve y= e' that
passes through the origin.
sin(x + a) = sin x cos a + cos x sin a
obtain the addition formula for the cosine function.
= ax + bx + c that passes through the
85. Find a parabola y
point (1, 4) and whose tangent lines at x=-1 and x= 5
have slopes 6 and -2, respectively.
69. Suppose that
f(1) 2 f'(1) = 3 f(2) = 1 f'(2) = 2
g(1) = 3 g'(1)= 1 g(2) = 1 g'(2) = 4
%3D
86. The function C(t) =
K(e at-e bi), where a, b, and K are
10
(a) If S(x) f(x) + g(x), findS'(1).
(b) If P(x) = f(x)g(x), find P'(2).
(c) If Q(x) = f(x)/g(x), find Q'(1).
(d) If C(x) = f(g(x)), find C'(2).
positive constants and b > a, is used to model the concen-
tration at time t of a drug injected into the bloodstream.
(a) Show that lim,. C(t)30.
(b) Find C'(t), the rate of change of drug concentration in
%3D
70. If f and g are the functions whose graphs are shown, let
P(x) = f(x)g(x). Q(x) = f(x)/g(x), and C(x) = f(g(x)).
Find (a) P'(2), (b) Q'(2), and (c) C'(2).
the blood.
(c) When is this rate equal to 0?
%3D
%3|
%3D
87. An equation of motion of the form s= Ae cos(wt + 8)
represents damped oscillation of an object. Find the velocity
and acceleration of the object.
88. A particle moves along a horizontal line so that its coor-
dinate at time t is x =
Vb? + c?t?,t 0, where b and c
are positive constants.
(a) Find the velocity and acceleration functions.
(b) Show that the particle always moves in the positive
direction.
1.
89. A particle moves on a vertical line so that its coordinate at
time t is y =t
(a) Find the velocity and acceleration functions.
(b) When is the particle moving upward and when is it
moving downward?
(c) Find the distance that the particle travels in the time
interval 0 <I<3.
71-78 Find f' in terms of g
2-12t +3, t 0.
%3D
73. f(x) = [g(x)]
77. f(x) = In g(x)|
78. f(x) = g(ln x)
(d) Graph the position, velocity, and acceleration functions
79-81 Find h' in terms of f' and g'.
(e) When is the particle speeding up? When is it slowing
79. h(x)=
(X)6(x)
80. h(x) =
down?
90. The volume of a right circular cone is V=rr'h, where
r is the radius of the base and h is the height.
(a) Find the rate of change of the volume with respect to
81. h(x) = f(g(sin 4.x))
the height if the radius is constant.
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