Applied Calculus

Deborah Hughes-Hallett, Patti Frazer Lock, Andrew M. Gleason

Chapter 1

Functions And Change - all with Video Answers

Educators

Section 1

What Is A Function?

Problem 1

Which graph in Figure $1.5$ best matches each of the following stories? $^{3}$ Write a story for the remaining graph.
(a) I had just left home when I realized I had forgotten my books, and so I went back to pick them up.
(b) Things went fine until I had a flat tire.
(c) I started out calmly but sped up when I realized I was going to be late.

Anand Jangid

Anand Jangid

Numerade Educator

Problem 2

The population of a city, $P$, in millions, is a function of $t$, the number of years since 1970 , so $P=f(t)$. Explain the meaning of the statement $f(35)=12$ in terms of the population of this city.

Anand Jangid

Numerade Educator

Problem 3

Let $W=f(t)$ represent wheat production in Argentina, ${ }^{4}$ in millions of metric tons, where $t$ is years since 1990 . Interpret the statement $f(12)=9$ in terms of wheat production.

AN

Andrew Noble

Numerade Educator

Problem 4

The concentration of carbon dioxide, $C=f(t)$, in the atmosphere, in parts per million (ppm), is a function of years, $t$, since 1960 .
(a) Interpret $f(40)=370$ in terms of carbon dioxide. ${ }^{5}$
(b) What is the meaning of $f(50)$ ?

Jasper Baltz

Numerade Educator

Problem 5

The population of Washington DC grew from 1900 to 1950, stayed approximately constant during the $1950 \mathrm{~s}$. and decreased from about 1960 to 2005 . Graph the population as a function of years since 1900 .

Jasper Baltz

Numerade Educator

Problem 6

(a) The graph of $r=f(p)$ is in Figure $1.6 .$ What is the value of $r$ when $p$ is $0 ?$ When $p$ is 3 ?
(b) What is $f(2)$ ?

Jasper Baltz

Numerade Educator

Problem 7

For the functions, find $f(5)$.
$$
f(x)=2 x+3
$$

M S

Numerade Educator

Problem 8

For the functions, find $f(5)$.
$$
f(x)=10 x-x^{2}
$$

Jasper Baltz

Numerade Educator

Problem 9

For the functions, find $f(5)$.

M S

Numerade Educator

Problem 10

For the functions, find $f(5)$.

M S

Numerade Educator

Problem 11

For the functions, find $f(5)$.
$$
\begin{array}{c|c|c|c|c|c|c|c|c}
\hline x & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\
\hline f(x) & 2.3 & 2.8 & 3.2 & 3.7 & 4.1 & 5.0 & 5.6 & 6.2 \\
\hline
\end{array}
$$

M S

Numerade Educator

Problem 12

Let $y=f(x)=x^{2}+2$.
(a) Find the value of $y$ when $x$ is zero.
(b) What is $f(3)$ ?
(c) What values of $x$ give $y$ a value of 11 ?
(d) Are there any values of $x$ that give $y$ a value of 1 ?

Jasper Baltz

Numerade Educator

Problem 13

The use of CFCs (chlorofluorocarbons) has declined since the 1987 Montreal Protocol came into force to reduce the use of substances that deplete the ozone layer. World annual CFC consumption, $C=f(t)$, in million tons, is a function of time, $t$, in years since 1987 . (CFCs are measured by the weight of ozone that they could destroy.)
(a) Interpret $f(10)=0.2$ in terms of $\mathrm{CFCs} .{ }^{6}$
(b) Interpret the vertical intercept of the graph of this function in terms of CFCs.
(c) Interpret the horizontal intercept of the graph of this function in terms of CFCs.

Jasper Baltz

Numerade Educator

Problem 14

Figure $1.7$ shows the amount of nicotine, $N=f(t)$, in $\mathrm{mg}$, in a person's bloodstream as a function of the time, $t$, in hours, since the person finished smoking a cigarette.
(a) Estimate $f(3)$ and interpret it in terms of nicotine.
(b) About how many hours have passed before the nicotine level is down to $0.1 \mathrm{mg}$ ?
(c) What is the vertical intercept? What does it represent in terms of nicotine?
(d) If this function had a horizontal intercept, what would it represent?

Jasper Baltz

Numerade Educator

Problem 15

The number of sales per month, $S$, is a function of the amount, $a$ (in dollars), spent on advertising that month, so $S=f(a)$.
(a) Interpret the statement $f(1000)=3500$.
(b) Which of the graphs in Figure $1.8$ is more likely to represent this function?
(c) What does the vertical intercept of the graph of this function represent, in terms of sales and advertising?

Jasper Baltz

Numerade Educator

Problem 16

A deposit is made into an interest-bearing account. Figure $1.9$ shows the balance, $B$, in the account $t$ years later.
(a) What was the original deposit?
(b) Estimate $f(10)$ and interpret it.
(c) When does the balance reach $\$ 5000$ ?

Jasper Baltz

Numerade Educator

Problem 17

When a patient with a rapid heart rate takes a drug, the heart rate plunges dramatically and then slowly rises again as the drug wears off. Sketch the heart rate against time from the moment the drug is administered.

Jasper Baltz

Numerade Educator

Problem 18

In the Andes mountains in Peru, the number, $N$, of species of bats is a function of the elevation, $h$, in feet above sea level, so $N=f(h)$.
(a) Interpret the statement $f(500)=100$ in terms of bat species.
(b) What are the meanings of the vertical intercept, $k$, and horizontal intercept, $c$, in Figure $1.10 ?$

Jasper Baltz

Numerade Educator

Problem 19

After an injection, the concentration of a drug in a patient's body increases rapidly to a peak and then slowly decreases. Graph the concentration of the drug in the body as a function of the time since the injection was given. Assume that the patient has none of the drug in the body before the injection. Label the peak concentration and the time it takes to reach that concentration.

Jasper Baltz

Numerade Educator

Problem 20

An object is put outside on a cold day at time $t=0$. Its temperature, $H=f(t)$, in ${ }^{\circ} \mathrm{C}$, is graphed in Figure $1.11$.
(a) What does the statement $f(30)=10$ mean in terms of temperature? Include units for 30 and for 10 in
your answer.
(b) Explain what the vertical intercept, $a$, and the horizontal intercept, $b$, represent in terms of temperature of the object and time outside.

Gregory Higby

Numerade Educator

Problem 21

Financial investors know that, in general, the higher the expected rate of return on an investment, the higher the corresponding risk.
(a) Graph this relationship, showing expected return as a function of risk.
(b) On the figure from part (a), mark a point with high expected return and low risk. (Investors hope to find such opportunities.)

Jasper Baltz

Numerade Educator

Problem 22

In tide pools on the New England coast, snails eat algae. Describe what Figure $1.12$ tells you about the effect of snails on the diversity of algae. ${ }^{7}$ Does the graph support the statement that diversity peaks at intermediate predation levels?

Jasper Baltz

Numerade Educator

Problem 23

(a) A potato is put in an oven to bake at time $t=0$. Which of the graphs in Figure $1.13$ could represent the potato's temperature as a function of time?
(b) What does the vertical intercept represent in terms of the potato's temperature?

Jasper Baltz

Numerade Educator

Problem 24

Figure $1.14$ shows fifty years of fertilizer use in the US, India, and the former Soviet Union. ${ }^{8}$
(a) Estimate fertilizer use in 1970 in the US, India, and the former Soviet Union.
(b) Write a sentence for each of the three graphs describing how fertilizer use has changed in each region over this 50 -year period.

Jasper Baltz

Numerade Educator

Problem 25

The gas mileage of a car (in miles per gallon) is highest when the car is going about 45 miles per hour and is lower when the car is going faster or slower than $45 \mathrm{mph}$. Graph gas mileage as a function of speed of the car.

Jasper Baltz

Numerade Educator

Problem 26

The six graphs in Figure $1.15$ show frequently observed patterns of age-specific cancer incidence rates, in number of cases per 1000 people, as a function of age. $^{9}$ The scales on the vertical axes are equal.
(a) For each of the six graphs, write a sentence explaining the effect of age on the cancer rate.
(b) Which graph shows a relatively high incidence rate for children? Suggest a type of cancer that behaves this way.
(c) Which graph shows a brief decrease in the incidence rate at around age 50 ? Suggest a type of cancer that might behave this way.
(d) Which graph or graphs might represcnt a cancer that is caused by toxins which build up in the body over time? (For example, lung cancer.) Explain.

Jasper Baltz

Numerade Educator