1.1 SOLUTIONS
cn 1
,2 Chapter One /SOLUTIONScn cn
CHAPTER ONE cn
c n
1. (a) The story in (a) matches Graph (IV), in which the person forgot her books and had to return home.
c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
(b) The story in (b) matches Graph (II), the flat tire story. Note the long period of time during which the distance f
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
rom home did not change (the horizontal part).
cn cn cn cn cn cn cn
(c) The story in (c) matches Graph (III), in which the person started calmly but sped up later.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
The first graph (I) does not match any of the given stories. In this picture, the person keeps going away from ho
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
me, but his speed decreases as time passes. So a story for this might be: I started walking to school at a good pace, but
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
since I stayed up all night studying calculus, I got more and more tired the farther I walked.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
2. The height is going down as time goes on. A possible graph is shown in Figure 1.1. The graph is decreasing.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
height
time
Figure 1.1 cn
3. The amount of carbon dioxide is going up as time goes on. A possible graph is shown in Figure 1.2. The graph is increasing.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
CO2
time
Figure 1.2 cn
, 1.1 SOLUTIONS
cn 3
4. The number of air conditioning units sold is going up as temperature goes up. A possible graph is shown in Figure
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
1.3. The graph is increasing.
cn cn cn cn
AC units cn
temperature
Figure 1.3 cn
5. The noise level is going down as distance goes up. A possible graph is shown in Figure 1.4. The graph is decreasing.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
noise level cn
distance
Figure 1.4 cn
6. If we let t represent the number of years since 1900, then the population increased between t = 0 and t = 40, sta
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
yed approximately constant between t = 40 and t = 50, and decreased for t ≥ 50. Figure 1.5 shows one possible graph
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. Many other answers are also possible.
cn cn cn cn cn cn
population
years
20 c n c n 40 c n c n 60 c n c n 80 c n 100 120 since 1900
c n cn
cn
Figure 1.5 cn
, 4 Chapter One /SOLUTIONS cn cn
7. Amount of grass G = f (r) increases as the amount of rainfall r increases, so f (r) is an increasing function.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
8. We are given information about how atmospheric pressure P = f (ℎ) behaves when the altitude ℎ decreases: as altitud
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
e ℎ decreases the atmospheric pressure P increases. This means that as altitude ℎ increases the atmospheric pressur
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
e P decreases. Therefore, P = f (ℎ) is a decreasing function.
cn c n cn cn c n cn n
c cn cn cn cn
9. We are given information about how battery capacity C = f (T ) behaves when air temperature T decreases: as T decreas
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
es, battery capacity C also decreases. This means that increasing temperature T increases battery capacity C. Ther
cn cn cn cn cn cn cn cn cn cn cn c n cn cn cn cn
efore, C = f (T ) is an increasing function.
c n c n cn c
n cn cn cn cn cn
10. Time T = f (m) increases as m increases, so f (m) is increasing.
cn c n c n cn cn cn cn cn cn cn cn cn cn
11. The attendance A = f (P ) decreases as the price P increases, so f (P ) is a decreasing function.
cn cn cn cn n
c cn cn cn cn cn cn cn cn cn n
c cn cn cn cn cn
12. The cost of manufacturing C = f (v) increases as the number of vehicles manufactured v increases, so f (v) is an increasi
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
ng function.
cn
13. We are given information about how commuting time, T = f (c) behaves as the number of cars on the road c decreas
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
es: as c decreases, commuting time T also decreases. This means that increasing the number of cars on the road c incr
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
eases commuting time T . Therefore, T = f (c) is an increasing function.
cn cn cn cn cn cn c n cn n
c cn cn cn cn
14. The statement f (4) = 20 tells us that W = 20 when t = 4. In other words, in 2019, Argentina produced 20 million me
cn cn cn cn cn cn cn cn cn c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
tric tons of wheat.
cn cn cn
15. (a) If we consider the equation
c n cn cn cn cn
C = 4T − 160
c n cn c n cn
simply as a mathematical relationship between two variables C and T , any T value is possible. However, if we thi
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
nk of it as a relationship between cricket chirps and temperature, then C cannot be less than 0. Since C = 0
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn c n cn cn
leads to 0 = 4T − 160, and so T = 40 F, we see that T cannot be less than 40 F. In addition, we are told tha
cn c n cn cn cn cn cn cn cn c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
t the function is not defined for temperatures above 134 . Thus, for the function C = f (T ) we have
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn n
c cn cn cn
Domain = All T values between 40 F and 134 F
cn cn cn c n cn cn cn cn cn c n
= All T values with 40 ≤ T ≤ 134
cn cn c n cn cn cn cn c n cn
= [40, 134].
cn cn
(b) Again, if we consider C = 4T − 160 simply as a mathematical relationship, its range is all real C values. Howev
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
er, when thinking of the meaning of C = f (T ) for crickets, we see that the function predicts cricket chirps per
cn cn cn cn cn cn cn c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn c
minute between 0 (at T = 40 F) and 376 (at T = 134 F). Hence,
n cn cn cn cn c n cn cn cn cn cn cn c n cn cn cn
Range = All C values from 0 to 376cn cn cn cn cn cn cn cn
= All C values with 0 ≤ C ≤ 376
cn cn cn cn cn cn cn cn cn
= [0, 376].
cn cn
16. (a) The statement f (19) = 415 means that C = 415 when t = 19. In other words, in the year 2019, the concentratio
cn cn cn n
c cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
n of carbon dioxide in the atmosphere was 415 ppm.
cn cn cn cn cn cn cn cn cn
(b) The expression f (22) represents the concentration of carbon dioxide in the year 2022.
cn cn cn cn cn cn cn cn cn cn cn cn cn
17. (a) At p = 0, we see r = 8. At p = 3, we see r = 7.
c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
(b) When p = 2, we see r = 10. Thus, f (2) = 10. cn cn cn cn cn cn cn cn cn cn n
c cn cn
18. Substituting x = 5 into f (x) = 2x + 3 gives cn cn cn cn cn cn cn cn cn cn cn
f (5) = 2(5) + 3 = 10 + 3 = 13.
n
c cn cn cn cn cn cn cn cn cn cn
19. Substituting x = 5 into f (x) = 10x − x2 gives cn cn cn cn cn cn cn cn cn cn cn
f (5) = 10(5) − (5)2 = 50 − 25 = 25.
n
c cn cn cn cn cn cn cn cn cn cn
20. We want the y-coordinate of the graph at the point where its x-coordinate is 5. Looking at the graph, we see that the
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
y-coordinate of this point is 3. Thus cn cn cn cn cn cn
f (5) = 3. n
c cn cn
cn 1
,2 Chapter One /SOLUTIONScn cn
CHAPTER ONE cn
c n
1. (a) The story in (a) matches Graph (IV), in which the person forgot her books and had to return home.
c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
(b) The story in (b) matches Graph (II), the flat tire story. Note the long period of time during which the distance f
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
rom home did not change (the horizontal part).
cn cn cn cn cn cn cn
(c) The story in (c) matches Graph (III), in which the person started calmly but sped up later.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
The first graph (I) does not match any of the given stories. In this picture, the person keeps going away from ho
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
me, but his speed decreases as time passes. So a story for this might be: I started walking to school at a good pace, but
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
since I stayed up all night studying calculus, I got more and more tired the farther I walked.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
2. The height is going down as time goes on. A possible graph is shown in Figure 1.1. The graph is decreasing.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
height
time
Figure 1.1 cn
3. The amount of carbon dioxide is going up as time goes on. A possible graph is shown in Figure 1.2. The graph is increasing.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
CO2
time
Figure 1.2 cn
, 1.1 SOLUTIONS
cn 3
4. The number of air conditioning units sold is going up as temperature goes up. A possible graph is shown in Figure
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
1.3. The graph is increasing.
cn cn cn cn
AC units cn
temperature
Figure 1.3 cn
5. The noise level is going down as distance goes up. A possible graph is shown in Figure 1.4. The graph is decreasing.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
noise level cn
distance
Figure 1.4 cn
6. If we let t represent the number of years since 1900, then the population increased between t = 0 and t = 40, sta
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
yed approximately constant between t = 40 and t = 50, and decreased for t ≥ 50. Figure 1.5 shows one possible graph
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
. Many other answers are also possible.
cn cn cn cn cn cn
population
years
20 c n c n 40 c n c n 60 c n c n 80 c n 100 120 since 1900
c n cn
cn
Figure 1.5 cn
, 4 Chapter One /SOLUTIONS cn cn
7. Amount of grass G = f (r) increases as the amount of rainfall r increases, so f (r) is an increasing function.
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
8. We are given information about how atmospheric pressure P = f (ℎ) behaves when the altitude ℎ decreases: as altitud
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
e ℎ decreases the atmospheric pressure P increases. This means that as altitude ℎ increases the atmospheric pressur
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
e P decreases. Therefore, P = f (ℎ) is a decreasing function.
cn c n cn cn c n cn n
c cn cn cn cn
9. We are given information about how battery capacity C = f (T ) behaves when air temperature T decreases: as T decreas
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
es, battery capacity C also decreases. This means that increasing temperature T increases battery capacity C. Ther
cn cn cn cn cn cn cn cn cn cn cn c n cn cn cn cn
efore, C = f (T ) is an increasing function.
c n c n cn c
n cn cn cn cn cn
10. Time T = f (m) increases as m increases, so f (m) is increasing.
cn c n c n cn cn cn cn cn cn cn cn cn cn
11. The attendance A = f (P ) decreases as the price P increases, so f (P ) is a decreasing function.
cn cn cn cn n
c cn cn cn cn cn cn cn cn cn n
c cn cn cn cn cn
12. The cost of manufacturing C = f (v) increases as the number of vehicles manufactured v increases, so f (v) is an increasi
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
ng function.
cn
13. We are given information about how commuting time, T = f (c) behaves as the number of cars on the road c decreas
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
es: as c decreases, commuting time T also decreases. This means that increasing the number of cars on the road c incr
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
eases commuting time T . Therefore, T = f (c) is an increasing function.
cn cn cn cn cn cn c n cn n
c cn cn cn cn
14. The statement f (4) = 20 tells us that W = 20 when t = 4. In other words, in 2019, Argentina produced 20 million me
cn cn cn cn cn cn cn cn cn c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
tric tons of wheat.
cn cn cn
15. (a) If we consider the equation
c n cn cn cn cn
C = 4T − 160
c n cn c n cn
simply as a mathematical relationship between two variables C and T , any T value is possible. However, if we thi
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
nk of it as a relationship between cricket chirps and temperature, then C cannot be less than 0. Since C = 0
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn c n cn cn
leads to 0 = 4T − 160, and so T = 40 F, we see that T cannot be less than 40 F. In addition, we are told tha
cn c n cn cn cn cn cn cn cn c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
t the function is not defined for temperatures above 134 . Thus, for the function C = f (T ) we have
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn n
c cn cn cn
Domain = All T values between 40 F and 134 F
cn cn cn c n cn cn cn cn cn c n
= All T values with 40 ≤ T ≤ 134
cn cn c n cn cn cn cn c n cn
= [40, 134].
cn cn
(b) Again, if we consider C = 4T − 160 simply as a mathematical relationship, its range is all real C values. Howev
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
er, when thinking of the meaning of C = f (T ) for crickets, we see that the function predicts cricket chirps per
cn cn cn cn cn cn cn c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn c
minute between 0 (at T = 40 F) and 376 (at T = 134 F). Hence,
n cn cn cn cn c n cn cn cn cn cn cn c n cn cn cn
Range = All C values from 0 to 376cn cn cn cn cn cn cn cn
= All C values with 0 ≤ C ≤ 376
cn cn cn cn cn cn cn cn cn
= [0, 376].
cn cn
16. (a) The statement f (19) = 415 means that C = 415 when t = 19. In other words, in the year 2019, the concentratio
cn cn cn n
c cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
n of carbon dioxide in the atmosphere was 415 ppm.
cn cn cn cn cn cn cn cn cn
(b) The expression f (22) represents the concentration of carbon dioxide in the year 2022.
cn cn cn cn cn cn cn cn cn cn cn cn cn
17. (a) At p = 0, we see r = 8. At p = 3, we see r = 7.
c n cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
(b) When p = 2, we see r = 10. Thus, f (2) = 10. cn cn cn cn cn cn cn cn cn cn n
c cn cn
18. Substituting x = 5 into f (x) = 2x + 3 gives cn cn cn cn cn cn cn cn cn cn cn
f (5) = 2(5) + 3 = 10 + 3 = 13.
n
c cn cn cn cn cn cn cn cn cn cn
19. Substituting x = 5 into f (x) = 10x − x2 gives cn cn cn cn cn cn cn cn cn cn cn
f (5) = 10(5) − (5)2 = 50 − 25 = 25.
n
c cn cn cn cn cn cn cn cn cn cn
20. We want the y-coordinate of the graph at the point where its x-coordinate is 5. Looking at the graph, we see that the
cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn cn
y-coordinate of this point is 3. Thus cn cn cn cn cn cn
f (5) = 3. n
c cn cn
