12 3 5 Figure 23 -2 + 25. Explain why f(x) must be increasing at x = 6. 26. Explain why f(x) must be decreasing at x = 4. 27. Explain why f(x) has a relative maximum at x = 3. 28. Explain why f(x) has a relative minimum at x = 5. 29. Explain why f(x) must be concave up at x = 0. 30. Explain why f(x) must be concave down at x = 2. 31. Explain why f(x) has an inflection point at x = 1. 32. Explain why f(x) has an inflection point at x = 4. 33. If f(6)=3, what is the equation of the tangent line to the graph of y=f(x) at x = 6? 34. If f(6)= 8, what is an approximate value of f(6.5)? 35. If f(0) = 3, what is an approximate value of f(.25)? 36. If f(0) = 3, what is the equation of the tangent line to the graph of y=f(x) at x = 0? 37. Level of Wa 6 43.
12 3 5 Figure 23 -2 + 25. Explain why f(x) must be increasing at x = 6. 26. Explain why f(x) must be decreasing at x = 4. 27. Explain why f(x) has a relative maximum at x = 3. 28. Explain why f(x) has a relative minimum at x = 5. 29. Explain why f(x) must be concave up at x = 0. 30. Explain why f(x) must be concave down at x = 2. 31. Explain why f(x) has an inflection point at x = 1. 32. Explain why f(x) has an inflection point at x = 4. 33. If f(6)=3, what is the equation of the tangent line to the graph of y=f(x) at x = 6? 34. If f(6)= 8, what is an approximate value of f(6.5)? 35. If f(0) = 3, what is an approximate value of f(.25)? 36. If f(0) = 3, what is the equation of the tangent line to the graph of y=f(x) at x = 0? 37. Level of Wa 6 43.
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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![2.2 The First- and Second-Derivative Rules 147
Exercises 25-36 refer to Fig. 23, which contains the graph of '(x). 41. By looking at the second derivative, decide which of the
the derivative of the function f(x).
curves in Fig. 25 could be the graph of f(x)=x5/2
M
3+
2
14
0/1
-1+
-2
1
v=J'(x)
Figure 23
25. Explain why f(x) must be increasing at x = 6.
26. Explain why f(x) must be decreasing at x = 4.
27. Explain why f(x) has a relative maximum at x = 3.
28. Explain why f(x) has a relative minimum at x = 5.
29. Explain why f(x) must be concave up at x = 0.
30. Explain why f(x) must be concave down at x = 2.
31. Explain why f(x) has an inflection point at x = 1.
32. Explain why f(x) has an inflection point at x = 4.
ut
od lo sgada luneus
Figure 24
2 3
33. If f(6)=3, what is the equation of the tangent line to the
graph of y=f(x) at x = 6?
34. If f(6)= 8, what is an approximate value of f(6.5)?
35. If f(0) = 3, what is an approximate value of f(.25)?
Y
4/5
36. If f(0) = 3, what is the equation of the tangent line to the
graph of y=f(x) at x = 0?
I
37. Level of Water from Melting Snow Melting snow causes a river
to overflow its banks. Let h(t) denote the number of inches of
water on Main Street r hours after the melting begins.
(a) If h'(100)=, by approximately how much will the water
level change during the next half hour?
(b) Which of the following two conditions is the better news?
(i) h(100) = 3, h'(100) = 2, h"(100) = -5
(ii) h(100) = 3, h'(100) = -2, h"(100) = 5
38. Changes in Temperature T(t) is the temperature on a hot
summer day at time / hours.
og muniat
(a) If T'(10) = 4, by approximately how much will the tem-
perature rise from 10:00 to 10:45?
(6,2)
+
6
(b) Which of the following two conditions is the better news
if you do not like hot weather? Explain your answer.
(i) T(10) = 95, T'(10) = 4, T"(10) = -3
(ii) T(10) = 95, T'(10) = -4, 7"(10) = 3
39. Decide which of the curves in Fig. 24 could not be the graph
of f(x) = (3x² + 1) for x ≥ 0. Decide which of the curves in
Fig. 24 could not be the graph of f(x) = (3x² + 1)4 for x ≥ 0
by considering the derivative of f(x). Explain your answer.
Y
to play
II
x
de-
Figure 25
42. Match each observation (a)-(e) with a conclusion (A)-(E).
Observations
(a) The point (3, 4) is on the graph of f'(x).
(b) The point (3, 4) is on the graph of f(x).
(c) The point (3, 4) is on the graph of f(x).
(d) The point (3, 0) is on the graph of f'(x), and the point
(3, 4) is on the graph of f(x).
(e) The point (3, 0) is on the graph of f'(x), and the point
(3,-4) is on the graph of f"(x).
Conclusions
(A) f(x) has a relative minimum point at x = 3.
(B) When x = 3, the graph of f(x) is concave up.
(C) When x = 3, the tangent line to the graph of y=f(x)
has slope 4.
(D) When x = 3, the value of f(x) is 4.
(E) f(x) has a relative maximum point at x = 3.
00392
prirole
abivos
be noire
rond -.05
43. Number of Farms in the U.S. The number of farms in the
United States / years after 1925 is f(t) million, where is the
function graphed in Fig. 26(a). [The graphs of f'(t) and f"(t)
are shown in Fig. 26(b).]
6
5
4
3
2
1
0
(1925)
201
y
لال
10 20 30
40. By looking at the first derivative, decide which of the curves in
Fig. 24 could not be the graph of f(x)= x³9x² + 24x + 1 om to and domibus
for x ≥ 0. [Hint: Factor the formula for f'(x).]
Figure 26
x
11
y=f(t)
t
30 40 50 60 70
(a)
(b)
10 20 30 40 50 60 70
+1
y=J")
y=f'(0)
ne
when
E.S](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F638d3a15-e6c8-4dbb-af03-e006d7a575bc%2F328e5026-8da0-4f9e-a2e8-c7abc5a0c2d9%2F8kaud78_processed.jpeg&w=3840&q=75)
Transcribed Image Text:2.2 The First- and Second-Derivative Rules 147
Exercises 25-36 refer to Fig. 23, which contains the graph of '(x). 41. By looking at the second derivative, decide which of the
the derivative of the function f(x).
curves in Fig. 25 could be the graph of f(x)=x5/2
M
3+
2
14
0/1
-1+
-2
1
v=J'(x)
Figure 23
25. Explain why f(x) must be increasing at x = 6.
26. Explain why f(x) must be decreasing at x = 4.
27. Explain why f(x) has a relative maximum at x = 3.
28. Explain why f(x) has a relative minimum at x = 5.
29. Explain why f(x) must be concave up at x = 0.
30. Explain why f(x) must be concave down at x = 2.
31. Explain why f(x) has an inflection point at x = 1.
32. Explain why f(x) has an inflection point at x = 4.
ut
od lo sgada luneus
Figure 24
2 3
33. If f(6)=3, what is the equation of the tangent line to the
graph of y=f(x) at x = 6?
34. If f(6)= 8, what is an approximate value of f(6.5)?
35. If f(0) = 3, what is an approximate value of f(.25)?
Y
4/5
36. If f(0) = 3, what is the equation of the tangent line to the
graph of y=f(x) at x = 0?
I
37. Level of Water from Melting Snow Melting snow causes a river
to overflow its banks. Let h(t) denote the number of inches of
water on Main Street r hours after the melting begins.
(a) If h'(100)=, by approximately how much will the water
level change during the next half hour?
(b) Which of the following two conditions is the better news?
(i) h(100) = 3, h'(100) = 2, h"(100) = -5
(ii) h(100) = 3, h'(100) = -2, h"(100) = 5
38. Changes in Temperature T(t) is the temperature on a hot
summer day at time / hours.
og muniat
(a) If T'(10) = 4, by approximately how much will the tem-
perature rise from 10:00 to 10:45?
(6,2)
+
6
(b) Which of the following two conditions is the better news
if you do not like hot weather? Explain your answer.
(i) T(10) = 95, T'(10) = 4, T"(10) = -3
(ii) T(10) = 95, T'(10) = -4, 7"(10) = 3
39. Decide which of the curves in Fig. 24 could not be the graph
of f(x) = (3x² + 1) for x ≥ 0. Decide which of the curves in
Fig. 24 could not be the graph of f(x) = (3x² + 1)4 for x ≥ 0
by considering the derivative of f(x). Explain your answer.
Y
to play
II
x
de-
Figure 25
42. Match each observation (a)-(e) with a conclusion (A)-(E).
Observations
(a) The point (3, 4) is on the graph of f'(x).
(b) The point (3, 4) is on the graph of f(x).
(c) The point (3, 4) is on the graph of f(x).
(d) The point (3, 0) is on the graph of f'(x), and the point
(3, 4) is on the graph of f(x).
(e) The point (3, 0) is on the graph of f'(x), and the point
(3,-4) is on the graph of f"(x).
Conclusions
(A) f(x) has a relative minimum point at x = 3.
(B) When x = 3, the graph of f(x) is concave up.
(C) When x = 3, the tangent line to the graph of y=f(x)
has slope 4.
(D) When x = 3, the value of f(x) is 4.
(E) f(x) has a relative maximum point at x = 3.
00392
prirole
abivos
be noire
rond -.05
43. Number of Farms in the U.S. The number of farms in the
United States / years after 1925 is f(t) million, where is the
function graphed in Fig. 26(a). [The graphs of f'(t) and f"(t)
are shown in Fig. 26(b).]
6
5
4
3
2
1
0
(1925)
201
y
لال
10 20 30
40. By looking at the first derivative, decide which of the curves in
Fig. 24 could not be the graph of f(x)= x³9x² + 24x + 1 om to and domibus
for x ≥ 0. [Hint: Factor the formula for f'(x).]
Figure 26
x
11
y=f(t)
t
30 40 50 60 70
(a)
(b)
10 20 30 40 50 60 70
+1
y=J")
y=f'(0)
ne
when
E.S
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