The function decreases and the slope increases as x increases. decreases as x increases. [Note: The slope is negative but becomes less negative.] 22. Both the function and the slope decrease as x increases. [Note: The slope is negative and becomes more negative.] 23. Annual World Consumption of Oil The annual world consump- tion of oil rises each year. Furthermore, the amount of the annual increase in oil consumption is also rising each year. Sketch a graph that could represent the annual world conm
The function decreases and the slope increases as x increases. decreases as x increases. [Note: The slope is negative but becomes less negative.] 22. Both the function and the slope decrease as x increases. [Note: The slope is negative and becomes more negative.] 23. Annual World Consumption of Oil The annual world consump- tion of oil rises each year. Furthermore, the amount of the annual increase in oil consumption is also rising each year. Sketch a graph that could represent the annual world conm
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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![10.
11.
12.
y=x
MA
13. Describe the way the slope changes as you move along the
graph (from left to right) in Exercise 5.
14. Describe the way the slope changes on the graph in Exercise 6.
15. Describe the way the slope changes on the graph in Exercise 8.
16. Describe the way the slope changes on the graph in Exercise 10.
Exercises 17 and 18 refer to the graph in Fig. 20.
Y
A
B
с.
F
y = f(x)
Figure 20
17. (a) At which labeled points is the function increasing?
(b) At which labeled points is the graph concave up?
(c) Which labeled point has the most positive slope?
18. (a) At which labeled points is the function decreasing?
(b) At which labeled points is the graph concave down?
2.1 Describing Graphs of Functions 139
(c) Which labeled point has the most negative slope (that is,
negative and with the greatest magnitude)?
In Exercises 19-22, draw the graph of a function y=f(x) with the
stated properties.
19. Both the function and the slope increase as x increases.
20. The function increases and the slope decreases as x increases.
21. The function decreases and the slope increases as x increases.
[Note: The slope is negative but becomes less negative.]
22. Both the function and the slope decrease as x increases. [Note:
The slope is negative and becomes more negative.]
23. Annual World Consumption of Oil The annual world consump-
tion of oil rises each year. Furthermore, the amount of the annual
increase in oil consumption is also rising each year. Sketch a
graph that could represent the annual world consumption of oil.
24. Average Annual Income In certain professions, the average
annual income has been rising at an increasing rate. Let f(T)
denote the average annual income at year T for persons in one of
these professions and sketch a graph that could represent f(T).
25. A Patient's Temperature At noon, a child's temperature is
101°F and is rising at an increasing rate. At 1 P.M. the child is
given medicine. After 2 P.M. the temperature is still increasing
but at a decreasing rate. The temperature reaches a peak of 103°
at 3 P.M. and decreases to 100° by 5 P.M. Draw a possible graph
of the function 7(1), the child's temperature at time t.
26. A Cost Function Let C(x) denote the total cost of manufactur-
ing x units of some product. Then C(x) is an increasing func-
tion for all x. For small values of x, the rate of increase of C(x)
decreases (because of the savings that are possible with "mass
production"). Eventually, however, for large values of x, the
cost C(x) increases at an increasing rate. (This happens when
production facilities are strained and become less efficient.)
Sketch a graph that could represent C(x).
27. Blood Flow through the Brain One method of determining the
level of blood flow through the brain requires the person to
inhale air containing a fixed concentration of N₂O, nitrous
oxide. During the first minute, the concentration of N₂O in
the jugular vein grows at an increasing rate to a level of .25%.
Thereafter, it grows at a decreasing rate and reaches a concen-
tration of about 4% after 10 minutes. Draw a possible graph
of the concentration of N₂O in the vein as a function of time.
28. Pollution Suppose that some organic waste products are
dumped into a lake at time r = 0 and that the oxygen content
of the lake at time t is given by the graph in Fig. 21. Describe
the graph in physical terms. Indicate the significance of the
inflection point at t = b.
Oxygen content of water
teat
b
Time (days)
Figure 21 A lake's recovery from pollution.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F638d3a15-e6c8-4dbb-af03-e006d7a575bc%2F1d9cd72d-d6d1-4421-8bf6-1f94ebfdd410%2Fsqim1rp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:10.
11.
12.
y=x
MA
13. Describe the way the slope changes as you move along the
graph (from left to right) in Exercise 5.
14. Describe the way the slope changes on the graph in Exercise 6.
15. Describe the way the slope changes on the graph in Exercise 8.
16. Describe the way the slope changes on the graph in Exercise 10.
Exercises 17 and 18 refer to the graph in Fig. 20.
Y
A
B
с.
F
y = f(x)
Figure 20
17. (a) At which labeled points is the function increasing?
(b) At which labeled points is the graph concave up?
(c) Which labeled point has the most positive slope?
18. (a) At which labeled points is the function decreasing?
(b) At which labeled points is the graph concave down?
2.1 Describing Graphs of Functions 139
(c) Which labeled point has the most negative slope (that is,
negative and with the greatest magnitude)?
In Exercises 19-22, draw the graph of a function y=f(x) with the
stated properties.
19. Both the function and the slope increase as x increases.
20. The function increases and the slope decreases as x increases.
21. The function decreases and the slope increases as x increases.
[Note: The slope is negative but becomes less negative.]
22. Both the function and the slope decrease as x increases. [Note:
The slope is negative and becomes more negative.]
23. Annual World Consumption of Oil The annual world consump-
tion of oil rises each year. Furthermore, the amount of the annual
increase in oil consumption is also rising each year. Sketch a
graph that could represent the annual world consumption of oil.
24. Average Annual Income In certain professions, the average
annual income has been rising at an increasing rate. Let f(T)
denote the average annual income at year T for persons in one of
these professions and sketch a graph that could represent f(T).
25. A Patient's Temperature At noon, a child's temperature is
101°F and is rising at an increasing rate. At 1 P.M. the child is
given medicine. After 2 P.M. the temperature is still increasing
but at a decreasing rate. The temperature reaches a peak of 103°
at 3 P.M. and decreases to 100° by 5 P.M. Draw a possible graph
of the function 7(1), the child's temperature at time t.
26. A Cost Function Let C(x) denote the total cost of manufactur-
ing x units of some product. Then C(x) is an increasing func-
tion for all x. For small values of x, the rate of increase of C(x)
decreases (because of the savings that are possible with "mass
production"). Eventually, however, for large values of x, the
cost C(x) increases at an increasing rate. (This happens when
production facilities are strained and become less efficient.)
Sketch a graph that could represent C(x).
27. Blood Flow through the Brain One method of determining the
level of blood flow through the brain requires the person to
inhale air containing a fixed concentration of N₂O, nitrous
oxide. During the first minute, the concentration of N₂O in
the jugular vein grows at an increasing rate to a level of .25%.
Thereafter, it grows at a decreasing rate and reaches a concen-
tration of about 4% after 10 minutes. Draw a possible graph
of the concentration of N₂O in the vein as a function of time.
28. Pollution Suppose that some organic waste products are
dumped into a lake at time r = 0 and that the oxygen content
of the lake at time t is given by the graph in Fig. 21. Describe
the graph in physical terms. Indicate the significance of the
inflection point at t = b.
Oxygen content of water
teat
b
Time (days)
Figure 21 A lake's recovery from pollution.
![140 CHAPTER 2 Applications of the Derivative
29. Number of U.S. Farms Figure 22 gives the number of U.S.
farms in millions from 1920 (r= 20) to 2000 (t = 100). In
what year was the number of farms decreasing most rapidly?
Y
U.S. Farms (millions)
0 10
(1900)
Consumer price index
(dollars)
175
160
Time (years)
og Figure 22 Number of U.S. farms.
30. Consumer Price Index Figure 23 shows the graph of the con-
sumer price index for the years 1983 (r= 0) through 2002
( 19). This index measures how much a basket of com-
modities that costs $100 in the beginning of 1983 would cost
at any given time. In what year was the rate of increase of the
index greatest? The least?
190%
145
130
115
100
20 30
Tod
30 40 50 60 70 80 90 100
(2000)
4
070
8
11
O To
0
et (1983)
basse gins Time (years)
Figure 23 Consumer price index.
31. Velocity of a Parachutist Let s(t) be the distance (in feet) trav-
eled by a parachutist after t seconds from the time of opening
the chute, and suppose that s() has the line y = -15t + 10
pibal
Solutions to Check Your Understanding 2.1
12
16
t
19
(2002)
1. The curve is concave up, so the slope increases. Even though
the curve itself is decreasing, the slope becomes less negative
as we move from left to right.
2. At x = 3. We have drawn in tangent lines at three points in
Fig. 24. Note that as we move from left to right, the slopes
decrease steadily until the point (3, 2), at which time they start
to increase. This is consistent with the fact that the graph is
concave down (hence slopes are decreasing) to the left of (3, 2)
and concave up (hence, slopes are increasing) to the right of
(3, 2). Extreme values of slopes always occur at inflection
points.
as an asymptote. What does this imply about the velocity of
the parachutist? [Note: Distance traveled downward is given a
negative value.]
32. Let P(1) be the population of a bacteria culture after / days and
suppose that P() has the line y = 25,000,000 as an asymptote.
What does this imply about the size of the population?
In Exercises 33-36, sketch the graph of a function having the given
properties.
33. Defined for 0≤x≤ 10; relative maximum point at x = 3;
absolute maximum value at x = 10
34. Relative maximum points at x = 1 and x = 5; relative mini-
mum point at x = 3; inflection points at x = 2 and x = 4
35. Defined and increasing for all x ≥ 0; inflection point at x = 5;
asymptotic to the line y = (2)x+ 5
36. Defined for x ≥ 0; absolute minimum value at x = 0; relative
maximum point at x = 4; asymptotic to the line y = () +1
37. Consider a smooth curve with no undefined points.
(a) If it has two relative maximum points, must it have a rela-
tive minimum point?
(b) If it has two relative extreme points, must it have an inflec-
tion point?
38. If the function f(x) has a relative minimum at x = a and a
relative maximum at x = b, must f(a) be less than f(b)?
TECHNOLOGY EXERCISES
39. Graph the function.
f(x) =
1
x³2x²+x-2
in the window [0, 4] by [-15, 15]. For what value of x does
f(x) have a vertical asymptote?
40. The graph of the function
f(x) =
2x²-1
5x+6
has a horizontal asymptote of the form y= c. Estimate the
value of e by graphing f(x) in the window [0, 50] by [-1, 6].
41. Simultaneously graph the functions
Figure 24
1
y=+x and y=x
X
in the window [-6, 6] by [-6, 6]. Describe the asymptote of
the first function.
(1, 1)
(3, 2)
(5, 4)
A](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F638d3a15-e6c8-4dbb-af03-e006d7a575bc%2F1d9cd72d-d6d1-4421-8bf6-1f94ebfdd410%2Fo0ntycs_processed.jpeg&w=3840&q=75)
Transcribed Image Text:140 CHAPTER 2 Applications of the Derivative
29. Number of U.S. Farms Figure 22 gives the number of U.S.
farms in millions from 1920 (r= 20) to 2000 (t = 100). In
what year was the number of farms decreasing most rapidly?
Y
U.S. Farms (millions)
0 10
(1900)
Consumer price index
(dollars)
175
160
Time (years)
og Figure 22 Number of U.S. farms.
30. Consumer Price Index Figure 23 shows the graph of the con-
sumer price index for the years 1983 (r= 0) through 2002
( 19). This index measures how much a basket of com-
modities that costs $100 in the beginning of 1983 would cost
at any given time. In what year was the rate of increase of the
index greatest? The least?
190%
145
130
115
100
20 30
Tod
30 40 50 60 70 80 90 100
(2000)
4
070
8
11
O To
0
et (1983)
basse gins Time (years)
Figure 23 Consumer price index.
31. Velocity of a Parachutist Let s(t) be the distance (in feet) trav-
eled by a parachutist after t seconds from the time of opening
the chute, and suppose that s() has the line y = -15t + 10
pibal
Solutions to Check Your Understanding 2.1
12
16
t
19
(2002)
1. The curve is concave up, so the slope increases. Even though
the curve itself is decreasing, the slope becomes less negative
as we move from left to right.
2. At x = 3. We have drawn in tangent lines at three points in
Fig. 24. Note that as we move from left to right, the slopes
decrease steadily until the point (3, 2), at which time they start
to increase. This is consistent with the fact that the graph is
concave down (hence slopes are decreasing) to the left of (3, 2)
and concave up (hence, slopes are increasing) to the right of
(3, 2). Extreme values of slopes always occur at inflection
points.
as an asymptote. What does this imply about the velocity of
the parachutist? [Note: Distance traveled downward is given a
negative value.]
32. Let P(1) be the population of a bacteria culture after / days and
suppose that P() has the line y = 25,000,000 as an asymptote.
What does this imply about the size of the population?
In Exercises 33-36, sketch the graph of a function having the given
properties.
33. Defined for 0≤x≤ 10; relative maximum point at x = 3;
absolute maximum value at x = 10
34. Relative maximum points at x = 1 and x = 5; relative mini-
mum point at x = 3; inflection points at x = 2 and x = 4
35. Defined and increasing for all x ≥ 0; inflection point at x = 5;
asymptotic to the line y = (2)x+ 5
36. Defined for x ≥ 0; absolute minimum value at x = 0; relative
maximum point at x = 4; asymptotic to the line y = () +1
37. Consider a smooth curve with no undefined points.
(a) If it has two relative maximum points, must it have a rela-
tive minimum point?
(b) If it has two relative extreme points, must it have an inflec-
tion point?
38. If the function f(x) has a relative minimum at x = a and a
relative maximum at x = b, must f(a) be less than f(b)?
TECHNOLOGY EXERCISES
39. Graph the function.
f(x) =
1
x³2x²+x-2
in the window [0, 4] by [-15, 15]. For what value of x does
f(x) have a vertical asymptote?
40. The graph of the function
f(x) =
2x²-1
5x+6
has a horizontal asymptote of the form y= c. Estimate the
value of e by graphing f(x) in the window [0, 50] by [-1, 6].
41. Simultaneously graph the functions
Figure 24
1
y=+x and y=x
X
in the window [-6, 6] by [-6, 6]. Describe the asymptote of
the first function.
(1, 1)
(3, 2)
(5, 4)
A
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