Calculus A New Horizon

Howard Anton

Chapter 1

Functions - all with Video Answers

Educators

Section 1

Functions and the Analysis of Graphical Information

Problem 1

Use the cigarette consumption graph in Figure $1.1 .2 b$ to answer the following questions, making reasonable approximations where needed.
(a) When did the annual cigarette consumption reach 3000 per adult for the first time?
(b) When did the annual cigarette consumption per adult reach its peak, and what was the peak value?
(c) Can you tell from the graph how many cigarettes were consumed in a given year? If not, what additional information would you need to make that determination?
(d) What factors are likely to cause a sharp increase in annual cigarette consumption per adult?
(e) What factors are likely to cause a sharp decline in annual cigarette consumption per adult?

Carson Merrill

Carson Merrill

Numerade Educator

Problem 2

The accompanying graph shows the median income in U.S. households (adjusted for inflation) between 1975 and 1995 Use the graph to answer the following questions, making reasonable approximations where needed.
(a) When did the median income reach its maximum value. and what was the median income when that occurred?
(b) When did the median income reach its minimum value, and what was the median income when that occurred?
(c) The median income was declining during the 4 -year period between 1989 and $1993,$ Was it declining more
rapidly during the first 2 years or the second 2 years of that period? Explain your reasoning.
(FIGURE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 3

Use the accompanying graph to answer the following questions, making reasonable approximations were needed.
(a) For what values of $x$ is $y=1 ?$
(b) For what values of $x$ is $y=3 ?$
(c) For what values of $y$ is $x=3 ?$
(d) For what values of $x$ is $y \leq 0 ?$
(e) What are the maximum and minimum values of $y$ and for what values of $x$ do they occur?
(FIGURE CAN'T COPY)

Melissa Munoz

Numerade Educator

Problem 4

Use the table in the accompanying figure to answer the questions posed in Exercise $3 .$
(FIGURE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 5

Use the equation $y=x^{2}-6 x+8$ to answer the following questions.
(a) For what values of $x$ is $y=0 ?$
(b) For what values of $x$ is $y=-10 ?$
(c) For what values of $x$ is $y \geq 0 ?$
(d) Does $y$ have a minimum value? A maximum value? If
so, find them.

Carson Merrill

Numerade Educator

Problem 6

Use the equation $y=1+\sqrt{x}$ to answer the following questions.
(a) For what values of $x$ is $y=4 ?$
(b) For what values of $x$ is $y=0 ?$
(c) For what values of $x$ is $y \geq 6 ?$
(d) Does $y$ have a minimum value? A maximum value? If
so, find them.

Carson Merrill

Numerade Educator

Problem 7

(a) If you had a device that could record the Earth's population continuously. would you expect the graph of population versus time to be a continuous (unbroken) curve? Explain what might cause breaks in the curve.
(b) Suppose that a hospital patient receives an injection of an antibiotic every 8 hours and that between injections the concentration $C$ of the antibiotic in the bloodstream decreases as the antibiotic is absorbed by the tissues. What might the graph of $C$ versus the elapsed time $l$ look like?

Carson Merrill

Numerade Educator

Problem 8

(a) If you had a device that could record the temperature of a room continuously over a 24 -hour period. would you expect the graph of temperature versus time to be a continuous (unbroken) curve? Explain your reasoning.
(b) If you had a computer that could track the number of boxes of cereal on the shelf of a market continuously
over a 1 -week period. would you expect the graph of the number of boxes on the shelf versus time to be a continuous (unbroken) curve? Explain your reasoning.

Carson Merrill

Numerade Educator

Problem 9

A construction company wants to build a rectangular enclosure with an area of 1000 square feet by fencing in three sides and using its office building as the fourth side. Your objective as supervising enginect is to design the enclosure so that it uses the least amount of fencing. Proceed as follows.
(a) Let $x$ and $y$ be the dimensions of the enclosure, and let L be the length of fencing required for those dimensions. since the area must be 1000 square feet. we must have $x y=1000 .$ Find a formula for $L$ in terms of $x$ and
$y,$ and then express $L$ in terms of $x$ alone by using the area equation.
(b) Are there any restrictions on the value of $x ?$ Explain.
(c) Make a graph of $L$ versus $x$ over a reasonable interval, and use the graph to estimate the value of $x$ that results in the smallest value of $L$
(d) Estimate the smallest value of $L$

Carson Merrill

Numerade Educator

Problem 10

A manufacturer constructs open boxes from sheets of cardboard that are 6 inches square by cutting small squares from the corners and folding up the sides (as shown in the accompanying figure). The Research and Development Department asks you to determine the size of the square that produces a box of greatest volume. Proceed as follows.
(a) Let $x$ be the length of a side of the square to be cut, and let $V$ be the volume of the resulting box. Show that
$$
V=x(6-2 x)^{2}
$$
(b) Are there any restrictions on the value of $x ?$ Explain.
(c) Make a graph of $V$ versus $x$ over an appropriate interval, and use the graph to estimate the value of $x$ that results in the largest volume.
(d) Estimate the largest volume.
(FIGURE CAN'T COPY)

Carson Merrill

Numerade Educator

Problem 11

A soup company wants to manufacture a can in the shape of a right circular cylinder that will hold $500 \mathrm{cm}^{3}$ of liquid. The material for the top and bottom costs 0.02 cent/cm". and the material for the sides costs $0.01 \mathrm{cent} / \mathrm{cm}^{2}$
(a) Use the method of Exercises 9 and 10 to estimate the radius $r$ and height $h$ of the can that costs the least to manufacture. [Suggestion: Express the cost $C \text { in terms of } r .]$
(b) Suppose that the tops and bottoms of radius $r$ are punched out from square sheets with sides of length
$2 r$ and the scraps are waste. If you allow for the cost of the waste, would you expect the can of least cost to be taller or shorter than the one in part (a)? Explain.
(c) Estimate the radius, height, and cost of the can in part
(b), and determine whether your conjecture was correct.

Carson Merrill

Numerade Educator

Problem 12

The designer of a sports facility wants to put a quarter-mile
$(1320 \mathrm{ft})$ running track around a football field, oriented as in the accompanying figure. The football field is 360 ft long (including the end zones) and 160 ft wide. The track consists of two straightaways and two semicireles.
(a) Show that it is possible to construct a quarter-mile track around the football field. ISnggestion: Find the shortest track that can be constructed around the ficld. $]$
(b) Let $L$ be the length of a straightaway (in fect), and let $x$ be the distance (in feet) between a sideline of the football field and a straightaway. Make a graph of $L$ versus $x$
(c) Use the graph to estimate the value of $x$ that produces the shortest straightaways, and then find this value of $x$ exactly.
(d) Use the graph to estimate the length of the longest possible straightaways, and then find that length exactly.

Carson Merrill

Numerade Educator