zotero-db/storage/SND5JZSJ/.zotero-ft-cache

A graphical representation of a function––here the number of hours of daylight as a function of the time of year at various latitudes–– is often the most natural and convenient way to represent the function.
Functions and Models

The fundamental objects that we deal with in calculus are functions. This chapter prepares the way for calculus by discussing the basic ideas concerning functions, their graphs, and ways of transforming and combining them. We stress that a function can be represented in different ways: by an equation, in a table, by a graph, or in words. We look at the main types of functions that occur in calculus and describe the process of using these functions as mathematical models of real-world phenomena. We also discuss the use of graphing calculators and graphing software for computers.

|||| 1.1 Four Ways to Represent a Function

Year
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Population (millions)
1650 1750 1860 2070 2300 2560 3040 3710 4450 5280 6080

Functions arise whenever one quantity depends on another. Consider the following four situations. A. The area A of a circle depends on the radius r of the circle. The rule that connects r
and A is given by the equation A ෇ ␲r 2. With each positive number r there is associated one value of A, and we say that A is a function of r. B. The human population of the world P depends on the time t. The table gives estimates of the world population P͑t͒ at time t, for certain years. For instance,
P͑1950͒ Ϸ 2,560,000,000
But for each value of the time t there is a corresponding value of P, and we say that P is a function of t. C. The cost C of mailing a ﬁrst-class letter depends on the weight w of the letter. Although there is no simple formula that connects w and C, the post ofﬁce has a rule for determining C when w is known. D. The vertical acceleration a of the ground as measured by a seismograph during an earthquake is a function of the elapsed time t. Figure 1 shows a graph generated by seismic activity during the Northridge earthquake that shook Los Angeles in 1994. For a given value of t, the graph provides a corresponding value of a.
a {cm/s@}
100

50

FIGURE 1 Vertical ground acceleration during
the Northridge earthquake

5

10

15

20

25

30 t (seconds)

_50

Calif. Dept. of Mines and Geology

12 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS

Each of these examples describes a rule whereby, given a number (r, t, w, or t), another number (A, P, C, or a) is assigned. In each case we say that the second number is a function of the ﬁrst number.

A function f is a rule that assigns to each element x in a set A exactly one element, called f ͑x͒, in a set B.

x

f

(input)

ƒ (output)

FIGURE 2 Machine diagram for a function ƒ

x

ƒ

a

f(a)

f

A

B

FIGURE 3 Arrow diagram for ƒ

We usually consider functions for which the sets A and B are sets of real numbers. The set A is called the domain of the function. The number f ͑x͒ is the value of f at x and is read “ f of x.” The range of f is the set of all possible values of f ͑x͒ as x varies throughout the domain. A symbol that represents an arbitrary number in the domain of a function f is called an independent variable. A symbol that represents a number in the range of f is called a dependent variable. In Example A, for instance, r is the independent variable and A is the dependent variable.
It’s helpful to think of a function as a machine (see Figure 2). If x is in the domain of the function f, then when x enters the machine, it’s accepted as an input and the machine produces an output f ͑x͒ according to the rule of the function. Thus, we can think of the domain as the set of all possible inputs and the range as the set of all possible outputs.
The preprogrammed functions in a calculator are good examples of a function as a machine. For example, the square root key on your calculator computes such a function.
You press the key labeled s (or sx ) and enter the input x. If x Ͻ 0, then x is not in the
domain of this function; that is, x is not an acceptable input, and the calculator will indicate an error. If x ജ 0, then an approximation to sx will appear in the display. Thus, the sx key on your calculator is not quite the same as the exact mathematical function f deﬁned by f ͑x͒ ෇ sx.
Another way to picture a function is by an arrow diagram as in Figure 3. Each arrow connects an element of A to an element of B. The arrow indicates that f ͑x͒ is associated with x, f ͑a͒ is associated with a, and so on.
The most common method for visualizing a function is its graph. If f is a function with domain A, then its graph is the set of ordered pairs

Խ ͕͑x, f ͑x͒͒ x ʦ A͖

(Notice that these are input-output pairs.) In other words, the graph of f consists of all points ͑x, y͒ in the coordinate plane such that y ෇ f ͑x͒ and x is in the domain of f .
The graph of a function f gives us a useful picture of the behavior or “life history” of a function. Since the y-coordinate of any point ͑x, y͒ on the graph is y ෇ f ͑x͒, we can read the value of f ͑x͒ from the graph as being the height of the graph above the point x (see Figure 4). The graph of f also allows us to picture the domain of f on the x-axis and its range on the y-axis as in Figure 5.

y

{ x, ƒ}

y

ƒ
f (2) f (1)

0

1

2

x

x

FIGURE 4

range

y ϭ ƒ(x)

0 FIGURE 5

x domain

SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ❙❙❙❙ 13
EXAMPLE 1 The graph of a function f is shown in Figure 6. (a) Find the values of f ͑1͒ and f ͑5͒. (b) What are the domain and range of f ?
y

1

01

x

FIGURE 6

|||| The notation for intervals is given in Appendix A.

SOLUTION
(a) We see from Figure 6 that the point ͑1, 3͒ lies on the graph of f , so the value of f at 1 is f ͑1͒ ෇ 3. (In other words, the point on the graph that lies above x ෇ 1 is 3 units above the x-axis.)
When x ෇ 5, the graph lies about 0.7 unit below the x-axis, so we estimate that f ͑5͒ Ϸ Ϫ0.7.
(b) We see that f ͑x͒ is deﬁned when 0 ഛ x ഛ 7, so the domain of f is the closed interval ͓0, 7͔. Notice that f takes on all values from Ϫ2 to 4, so the range of f is
Խ͕y Ϫ2 ഛ y ഛ 4͖ ෇ ͓Ϫ2, 4͔

y

y=2 x-1

01

x

-1 2

FIGURE 7

EXAMPLE 2 Sketch the graph and ﬁnd the domain and range of each function.

(a) f͑x͒ ෇ 2x Ϫ 1

(b) t͑x͒ ෇ x 2

SOLUTION
(a) The equation of the graph is y ෇ 2x Ϫ 1, and we recognize this as being the equation of a line with slope 2 and y-intercept Ϫ1. (Recall the slope-intercept form of the equation of a line: y ෇ mx ϩ b. See Appendix B.) This enables us to sketch the graph of f in Figure 7. The expression 2x Ϫ 1 is deﬁned for all real numbers, so the domain of f is the set of all real numbers, which we denote by ‫ޒ‬. The graph shows that the range is also ‫ޒ‬.
(b) Since t͑2͒ ෇ 22 ෇ 4 and t͑Ϫ1͒ ෇ ͑Ϫ1͒2 ෇ 1, we could plot the points ͑2, 4͒ and ͑Ϫ1, 1͒, together with a few other points on the graph, and join them to produce the graph (Figure 8). The equation of the graph is y ෇ x 2, which represents a parabola (see Appendix C). The domain of t is ‫ޒ‬. The range of t consists of all values of t͑x͒, that is, all numbers of the form x 2. But x 2 ജ 0 for all numbers x and any positive number y is a
Խ square. So the range of t is ͕y y ജ 0͖ ෇ ͓0, ϱ͒. This can also be seen from Figure 8.

y (2, 4)

FIGURE 8

y=≈

(_1, 1) 1

01

x

14 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS
P 6x10 '

Representations of Functions

There are four possible ways to represent a function:

s verbally s numerically s visually s algebraically

(by a description in words) (by a table of values) (by a graph) (by an explicit formula)

If a single function can be represented in all four ways, it is often useful to go from one representation to another to gain additional insight into the function. (In Example 2, for instance, we started with algebraic formulas and then obtained the graphs.) But certain functions are described more naturally by one method than by another. With this in mind, let’s reexamine the four situations that we considered at the beginning of this section.
A. The most useful representation of the area of a circle as a function of its radius is probably the algebraic formula A͑r͒ ෇ ␲r 2, though it is possible to compile a table of values or to sketch a graph (half a parabola). Because a circle has to have a positive
Խ radius, the domain is ͕r r Ͼ 0͖ ෇ ͑0, ϱ͒, and the range is also ͑0, ϱ͒.
B. We are given a description of the function in words: P͑t͒ is the human population of the world at time t. The table of values of world population on page 11 provides a convenient representation of this function. If we plot these values, we get the graph (called a scatter plot) in Figure 9. It too is a useful representation; the graph allows us to absorb all the data at once. What about a formula? Of course, it’s impossible to devise an explicit formula that gives the exact human population P͑t͒ at any time t. But it is possible to ﬁnd an expression for a function that approximates P͑t͒. In fact, using methods explained in Section 1.5, we obtain the approximation

P͑t͒ Ϸ f ͑t͒ ෇ ͑0.008079266͒ и ͑1.013731͒t

and Figure 10 shows that it is a reasonably good “ﬁt.” The function f is called a mathematical model for population growth. In other words, it is a function with an explicit formula that approximates the behavior of our given function. We will see, however, that the ideas of calculus can be applied to a table of values; an explicit formula is not necessary.

P 6x10 '

1900 1920 1940 1960 1980 2000 t FIGURE 9

1900 1920 1940 1960 1980 2000 t FIGURE 10

a {cm/s@}
400
200

SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION ❙❙❙❙ 15
The function is typical of the functions that arise whenever we attempt to apply calculus to the real world. We start with a verbal description of a function. Then we may be able to construct a table of values of the function, perhaps from instrument readings in a scientiﬁc experiment. Even though we don’t have complete knowledge of the values of the function, we will see throughout the book that it is still possible to perform the operations of calculus on such a function.
C. Again the function is described in words: is the cost of mailing a ﬁrst-class letter with weight . The rule that the U.S. Postal Service used as of 2002 is as follows: The cost is 37 cents for up to one ounce, plus 23 cents for each successive ounce up to 11 ounces. The table of values shown in the margin is the most convenient representation for this function, though it is possible to sketch a graph (see Example 10).
D. The graph shown in Figure 1 is the most natural representation of the vertical acceleration function . It’s true that a table of values could be compiled, and it is even possible to devise an approximate formula. But everything a geologist needs to know—amplitudes and patterns—can be seen easily from the graph. (The same is true for the patterns seen in electrocardiograms of heart patients and polygraphs for liedetection.) Figures 11 and 12 show the graphs of the north-south and east-west accelerations for the Northridge earthquake; when used in conjunction with Figure 1, they provide a great deal of information about the earthquake.

_200

5

10

15

20

25

30 t

(seconds)

_400

Calif. Dept. of Mines and Geology

FIGURE 11 North-south acceleration for the Northridge earthquake

In the next example we sketch the graph of a function that is deﬁned verbally.
EXAMPLE 3 When you turn on a hot-water faucet, the temperature T of the water depends on how long the water has been running. Draw a rough graph of T as a function of the time t that has elapsed since the faucet was turned on.
SOLUTION The initial temperature of the running water is close to room temperature because of the water that has been sitting in the pipes. When the water from the hotwater tank starts coming out, T increases quickly. In the next phase, T is constant at the temperature of the heated water in the tank. When the tank is drained, T decreases to the temperature of the water supply. This enables us to make the rough sketch of T as a function of t in Figure 13.

46 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS

2. Explain how the following graphs are obtained from the graph

of y ෇ f ͑x͒.

(a) y ෇ 5 f ͑x͒

(b) y ෇ f ͑x Ϫ 5͒

(c) y ෇ Ϫf ͑x͒

(d) y ෇ Ϫ5 f ͑x͒

(e) y ෇ f ͑5x͒

(f) y ෇ 5 f ͑x͒ Ϫ 3

3. The graph of y ෇ f ͑x͒ is given. Match each equation with its

graph and give reasons for your choices.

(a) y ෇ f ͑x Ϫ 4͒

(b) y ෇ f ͑x͒ ϩ 3

(c)

y

෇

1 3

f ͑x͒

(d) y ෇ Ϫf ͑x ϩ 4͒

(e) y ෇ 2 f ͑x ϩ 6͒

y

@

6!

f 3

#

$

_6

_3

0

3

6

x

_3 %

4. The graph of f is given. Draw the graphs of the following

functions.

(a) y ෇ f ͑x ϩ 4͒

(b) y ෇ f ͑x͒ ϩ 4

(c) y ෇ 2 f ͑x͒

(d) y ෇ Ϫ12 f ͑x͒ ϩ 3

y

1

01

x

5. The graph of f is given. Use it to graph the following

functions. (a) y ෇ f ͑2x͒ (c) y ෇ f ͑Ϫx͒

(b)

y

෇

f

(

1 2

x)

(d) y ෇ Ϫf ͑Ϫx͒

y

1

01

x

6–7 |||| The graph of y ෇ s3x Ϫ x 2 is given. Use transformations to create a function whose graph is as shown.
y

1.5

y=œ„3„x„-„„≈„

0

3

x

6. y 3

0

2

5x

7. _4

y

_1 0

x

_1

_2.5

s

s

s

s

s

s

s

s

s

s

s

s

8. (a) How is the graph of y ෇ 2 sin x related to the graph of y ෇ sin x? Use your answer and Figure 6 to sketch the graph of y ෇ 2 sin x.
(b) How is the graph of y ෇ 1 ϩ sx related to the graph of y ෇ sx? Use your answer and Figure 4(a) to sketch the graph of y ෇ 1 ϩ sx.

9–24 |||| Graph the function, not by plotting points, but by starting with the graph of one of the standard functions given in Section 1.2, and then applying the appropriate transformations.

9. y ෇ Ϫx 3

10. y ෇ 1 Ϫ x 2

11. y ෇ ͑ x ϩ 1͒2

12. y ෇ x 2 Ϫ 4x ϩ 3

13. y ෇ 1 ϩ 2 cos x

14. y ෇ 4 sin 3x

15. y ෇ sin͑ x͞2͒

16.

y෇

1 xϪ4

17. y ෇ sx ϩ 3

18. y ෇ ͑ x ϩ 2͒4 ϩ 3

19.

y

෇

1 2

͑x2

ϩ

8x͒

21.

y෇

2 xϩ1

20. y ෇ 1 ϩ s3 x Ϫ 1

ͩ ͪ 22.

y෇

1 tan

xϪ

␲

4

4

23. y ෇ Խsin x Խ

Խ Խ 24. y ෇ x 2 Ϫ 2 x

s

s

s

s

s

s

s

s

s

s

s

s

25. The city of New Orleans is located at latitude 30ЊN. Use Figure 9 to ﬁnd a function that models the number of hours of daylight at New Orleans as a function of the time of year. Use the fact that on March 31 the Sun rises at 5:51 A.M. and sets at 6:18 P.M. in New Orleans to check the accuracy of your model.
26. A variable star is one whose brightness alternately increases and decreases. For the most visible variable star, Delta Cephei, the time between periods of maximum brightness is 5.4 days, the average brightness (or magnitude) of the star is 4.0, and its brightness varies by Ϯ0.35 magnitude. Find a function that models the brightness of Delta Cephei as a function of time.

Խ Խ 27. (a) How is the graph of y ෇ f ( x ) related to the graph of f ? Խ Խ (b) Sketch the graph of y ෇ sin x . Խ Խ (c) Sketch the graph of y ෇ s x .
28. Use the given graph of f to sketch the graph of y ෇ 1͞f ͑x͒. Which features of f are the most important in sketching y ෇ 1͞f ͑x͒? Explain how they are used.
y

1

01

x

29–30 |||| Use graphical addition to sketch the graph of f ϩ t. 29. y

g

f

0

x

30.

y

f

0

g

x

s

s

s

s

s

s

s

s

s

s

s

s

31–32 |||| Find f ϩ t, f Ϫ t, f t, and f͞t and state their domains. 31. f ͑x͒ ෇ x 3 ϩ 2x 2, t͑x͒ ෇ 3x 2 Ϫ 1 32. f ͑x͒ ෇ s1 ϩ x, t͑x͒ ෇ s1 Ϫ x

s

s

s

s

s

s

s

s

s

s

s

s

33–34 |||| Use the graphs of f and t and the method of graphical addition to sketch the graph of f ϩ t.

33. f ͑x͒ ෇ x, t͑x͒ ෇ 1͞x

34. f ͑x͒ ෇ x 3, t͑x͒ ෇ Ϫx 2

s

s

s

s

s

s

s

s

s

s

s

s

35–40 |||| Find the functions f ‫ ؠ‬t, t ‫ ؠ‬f , f ‫ ؠ‬f , and t ‫ ؠ‬t and their domains. 35. f ͑x͒ ෇ 2x 2 Ϫ x, t͑x͒ ෇ 3x ϩ 2 36. f ͑x͒ ෇ 1 Ϫ x 3, t͑x͒ ෇ 1͞x 37. f ͑x͒ ෇ sin x, t͑x͒ ෇ 1 Ϫ sx

SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS ❙❙❙❙ 47

38. f ͑x͒ ෇ 1 Ϫ 3x, t͑x͒ ෇ 5x 2 ϩ 3x ϩ 2

39.

f ͑x͒

෇

x

ϩ

1 ,

x

t͑x͒

෇

x x

ϩ ϩ

1 2

40. f ͑x͒ ෇ s2x ϩ 3, t͑x͒ ෇ x 2 ϩ 1

s

s

s

s

s

s

s

s

s

s

s

s

41–44 |||| Find f ‫ ؠ‬t ‫ ؠ‬h.

41. f ͑x͒ ෇ x ϩ 1, t͑x͒ ෇ 2 x , h͑x͒ ෇ x Ϫ 1

42. f ͑x͒ ෇ 2x Ϫ 1, t͑x͒ ෇ x 2, h͑x͒ ෇ 1 Ϫ x

43. f ͑x͒ ෇ sx Ϫ 1, t͑x͒ ෇ x2 ϩ 2, h͑x͒ ෇ x ϩ 3

44.

f ͑x͒ ෇

2 xϩ1,

t͑x͒ ෇ cos x,

h͑x͒ ෇ sx ϩ 3

s

s

s

s

s

s

s

s

s

s

s

s

45–50 |||| Express the function in the form f ‫ ؠ‬t.

45. F͑x͒ ෇ ͑x 2 ϩ 1͒10

46. F͑x͒ ෇ sin(sx )

47.

G͑x͒

෇

x2 x2 ϩ

4

48.

G͑x͒ ෇

1 xϩ3

49. u͑t͒ ෇ scos t

50.

u͑t͒ ෇

tan t 1 ϩ tan t

s

s

s

s

s

s

s

s

s

s

s

s

51–53 |||| Express the function in the form f ‫ ؠ‬t ‫ ؠ‬h.

51. H͑x͒ ෇ 1 Ϫ 3x2

52. H͑x͒ ෇ s3 sx Ϫ 1

53. H͑x͒ ෇ sec4(sx )

s

s

s

s

s

s

s

s

s

s

s

s

54. Use the table to evaluate each expression.

(a) f ͑ t͑1͒͒

(b) t͑ f ͑1͒͒

(d) t͑ t͑1͒͒

(e) ͑ t ‫ ؠ‬f ͒͑3͒

(c) f ͑ f ͑1͒͒ (f) ͑ f ‫ ؠ‬t͒͑6͒

x

123456

f ͑x͒ 3 1 4 2 2 5

t͑x͒ 6 3 2 1 2 3

55. Use the given graphs of f and t to evaluate each expression,

or explain why it is undeﬁned.

(a) f ͑ t͑2͒͒

(b) t͑ f ͑0͒͒

(c) ͑ f ‫ ؠ‬t͒͑0͒

(d) ͑ t ‫ ؠ‬f ͒͑6͒

(e) ͑ t ‫ ؠ‬t͒͑Ϫ2͒

(f) ͑ f ‫ ؠ‬f ͒͑4͒

y

g

f

2

0

2

x

48 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS

56. Use the given graphs of f and t to estimate the value of f ͑ t͑x͒͒ for x ෇ Ϫ5, Ϫ4, Ϫ3, . . . , 5. Use these estimates to sketch a rough graph of f ‫ ؠ‬t.
y
g

1

01

x

f

57. A stone is dropped into a lake, creating a circular ripple that travels outward at a speed of 60 cm͞s. (a) Express the radius r of this circle as a function of the time t (in seconds). (b) If A is the area of this circle as a function of the radius, ﬁnd A ‫ ؠ‬r and interpret it.

58. An airplane is ﬂying at a speed of 350 mi͞h at an altitude of one mile and passes directly over a radar station at time t ෇ 0. (a) Express the horizontal distance d (in miles) that the plane has ﬂown as a function of t. (b) Express the distance s between the plane and the radar station as a function of d. (c) Use composition to express s as a function of t.

59. The Heaviside function H is deﬁned by

ͭ H͑t͒ ෇

0 1

if t Ͻ 0 if t ജ 0

It is used in the study of electric circuits to represent the sudden surge of electric current, or voltage, when a switch is instantaneously turned on. (a) Sketch the graph of the Heaviside function.

(b) Sketch the graph of the voltage V͑t͒ in a circuit if the switch is turned on at time t ෇ 0 and 120 volts are applied instantaneously to the circuit. Write a formula for V͑t͒ in terms of H͑t͒.
(c) Sketch the graph of the voltage V͑t͒ in a circuit if the switch is turned on at time t ෇ 5 seconds and 240 volts are applied instantaneously to the circuit. Write a formula for V͑t͒ in terms of H͑t͒. (Note that starting at t ෇ 5 corresponds to a translation.)
60. The Heaviside function deﬁned in Exercise 59 can also be used to deﬁne the ramp function y ෇ ctH͑t͒, which represents a gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y ෇ tH͑t͒. (b) Sketch the graph of the voltage V͑t͒ in a circuit if the switch is turned on at time t ෇ 0 and the voltage is gradually increased to 120 volts over a 60-second time interval. Write a formula for V͑t͒ in terms of H͑t͒ for t ഛ 60. (c) Sketch the graph of the voltage V͑t͒ in a circuit if the switch is turned on at time t ෇ 7 seconds and the voltage is gradually increased to 100 volts over a period of 25 seconds. Write a formula for V͑t͒ in terms of H͑t͒ for t ഛ 32.
61. (a) If t͑x͒ ෇ 2x ϩ 1 and h͑x͒ ෇ 4x 2 ϩ 4x ϩ 7, ﬁnd a function f such that f ‫ ؠ‬t ෇ h. (Think about what operations you would have to perform on the formula for t to end up with the formula for h.)
(b) If f ͑x͒ ෇ 3x ϩ 5 and h͑x͒ ෇ 3x 2 ϩ 3x ϩ 2, ﬁnd a function t such that f ‫ ؠ‬t ෇ h.
62. If f ͑x͒ ෇ x ϩ 4 and h͑x͒ ෇ 4x Ϫ 1, ﬁnd a function t such that t ‫ ؠ‬f ෇ h.
63. Suppose t is an even function and let h ෇ f ‫ ؠ‬t. Is h always an even function?
64. Suppose t is an odd function and let h ෇ f ‫ ؠ‬t. Is h always an odd function? What if f is odd? What if f is even?

|||| 1.4 Graphing Calculators and Computers
In this section we assume that you have access to a graphing calculator or a computer with graphing software. We will see that the use of such a device enables us to graph more complicated functions and to solve more complex problems than would otherwise be possible. We also point out some of the pitfalls that can occur with these machines.
Graphing calculators and computers can give very accurate graphs of functions. But we will see in Chapter 4 that only through the use of calculus can we be sure that we have uncovered all the interesting aspects of a graph.
A graphing calculator or computer displays a rectangular portion of the graph of a function in a display window or viewing screen, which we refer to as a viewing rectangle. The default screen often gives an incomplete or misleading picture, so it is important to choose the viewing rectangle with care. If we choose the x-values to range from a minimum value of Xmin ෇ a to a maximum value of Xmax ෇ b and the y-values to range from

SECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS ❙❙❙❙ 49

(a, d )

y=d

(b, d ) a minimum of Ymin ෇ c to a maximum of Ymax ෇ d, then the visible portion of the graph

lies in the rectangle

x=a

x=b

Խ ͓a, b͔ ϫ ͓c, d͔ ෇ ͕͑x, y͒ a ഛ x ഛ b, c ഛ y ഛ d͖

(a, c)

y=c

( b, c )

FIGURE 1 The viewing rectangle ͓a, b͔ by ͓c, d ͔

shown in Figure 1. We refer to this rectangle as the ͓a, b͔ by ͓c, d͔ viewing rectangle. The machine draws the graph of a function f much as you would. It plots points of the
form ͑x, f ͑x͒͒ for a certain number of equally spaced values of x between a and b. If an x-value is not in the domain of f , or if f ͑x͒ lies outside the viewing rectangle, it moves on to the next x-value. The machine connects each point to the preceding plotted point to form a representation of the graph of f .

2

_2

2

_2 (a) ͓_2, 2͔ by ͓_2, 2͔
4

EXAMPLE 1 Draw the graph of the function f ͑x͒ ෇ x 2 ϩ 3 in each of the following view-

ing rectangles.

(a) ͓Ϫ2, 2͔ by ͓Ϫ2, 2͔ (c) ͓Ϫ10, 10͔ by ͓Ϫ5, 30͔

(b) ͓Ϫ4, 4͔ by ͓Ϫ4, 4͔ (d) ͓Ϫ50, 50͔ by ͓Ϫ100, 1000͔

SOLUTION For part (a) we select the range by setting Xmin ෇ Ϫ2, Xmax ෇ 2, Ymin ෇ Ϫ2, and Ymax ෇ 2. The resulting graph is shown in Figure 2(a). The display window is blank! A moment’s thought provides the explanation: Notice that x 2 ജ 0 for all x, so x 2 ϩ 3 ജ 3 for all x. Thus, the range of the function f ͑x͒ ෇ x2 ϩ 3 is ͓3, ϱ͒. This means that the graph of f lies entirely outside the viewing rectangle ͓Ϫ2, 2͔ by ͓Ϫ2, 2͔.
The graphs for the viewing rectangles in parts (b), (c), and (d) are also shown in
Figure 2. Observe that we get a more complete picture in parts (c) and (d), but in part (d)
it is not clear that the y-intercept is 3.

30

1000

_4

4

_4 (b) ͓_4, 4͔ by ͓_4, 4͔ FIGURE 2 Graphs of ƒ=≈+3

_10

10

_5

(c) ͓_10, 10͔ by ͓_5, 30͔

_50

50

_100

(d) ͓_50, 50͔ by ͓_100, 1000͔

We see from Example 1 that the choice of a viewing rectangle can make a big difference in the appearance of a graph. Sometimes it’s necessary to change to a larger viewing rectangle to obtain a more complete picture, a more global view, of the graph. In the next example we see that knowledge of the domain and range of a function sometimes provides us with enough information to select a good viewing rectangle.
EXAMPLE 2 Determine an appropriate viewing rectangle for the function f ͑x͒ ෇ s8 Ϫ 2x 2 and use it to graph f . SOLUTION The expression for f ͑x͒ is deﬁned when
8 Ϫ 2x 2 ജ 0 &? 2x 2 ഛ 8 &? x 2 ഛ 4
&? Խ x Խ ഛ 2 &? Ϫ2 ഛ x ഛ 2

50 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS
4

_3

3

_1 FIGURE 3

5

_5

5

_5 FIGURE 4

20

Therefore, the domain of f is the interval ͓Ϫ2, 2͔. Also,
0 ഛ s8 Ϫ 2x 2 ഛ s8 ෇ 2 s2 Ϸ 2.83
so the range of f is the interval [0, 2s2].
We choose the viewing rectangle so that the x-interval is somewhat larger than the domain and the y-interval is larger than the range. Taking the viewing rectangle to be ͓Ϫ3, 3͔ by ͓Ϫ1, 4͔, we get the graph shown in Figure 3.

EXAMPLE 3 Graph the function y ෇ x 3 Ϫ 150x.
SOLUTION Here the domain is ‫ޒ‬, the set of all real numbers. That doesn’t help us choose a viewing rectangle. Let’s experiment. If we start with the viewing rectangle ͓Ϫ5, 5͔ by ͓Ϫ5, 5͔, we get the graph in Figure 4. It appears blank, but actually the graph is so nearly vertical that it blends in with the y-axis.
If we change the viewing rectangle to ͓Ϫ20, 20͔ by ͓Ϫ20, 20͔, we get the picture shown in Figure 5(a). The graph appears to consist of vertical lines, but we know that can’t be correct. If we look carefully while the graph is being drawn, we see that the graph leaves the screen and reappears during the graphing process. This indicates that we need to see more in the vertical direction, so we change the viewing rectangle to ͓Ϫ20, 20͔ by ͓Ϫ500, 500͔. The resulting graph is shown in Figure 5(b). It still doesn’t quite reveal all the main features of the function, so we try ͓Ϫ20, 20͔ by ͓Ϫ1000, 1000͔ in Figure 5(c). Now we are more conﬁdent that we have arrived at an appropriate viewing rectangle. In Chapter 4 we will be able to see that the graph shown in Figure 5(c) does indeed reveal all the main features of the function.

500

1000

_20

20

_20

20

_20

20

_20 (a)
FIGURE 5 ƒ=˛-150x

_500 ( b)

_1000 (c)

EXAMPLE 4 Graph the function f ͑x͒ ෇ sin 50x in an appropriate viewing rectangle.
SOLUTION Figure 6(a) shows the graph of f produced by a graphing calculator using the viewing rectangle ͓Ϫ12, 12͔ by ͓Ϫ1.5, 1.5͔. At ﬁrst glance the graph appears to be reasonable. But if we change the viewing rectangle to the ones shown in the following parts of Figure 6, the graphs look very different. Something strange is happening.
In order to explain the big differences in appearance of these graphs and to ﬁnd an appropriate viewing rectangle, we need to ﬁnd the period of the function y ෇ sin 50x. We know that the function y ෇ sin x has period 2␲ and the graph of y ෇ sin 50x is compressed horizontally by a factor of 50, so the period of y ෇ sin 50x is
2␲ ෇ ␲ Ϸ 0.126 50 25

SECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS ❙❙❙❙ 51

1.5

1.5

_12

12

_10

10

|||| The appearance of the graphs in Figure 6

_1.5

_1.5

depends on the machine used. The graphs you

get with your own graphing device might not

(a)

(b)

look like these ﬁgures, but they will also be

quite inaccurate.

1.5

1.5

_9

9

_6

6

FIGURE 6

Graphs of ƒ=sin 50x

_1.5

_1.5

in four viewing rectangles

(c)

(d)

This suggests that we should deal only with small values of x in order to show just a few

1.5

oscillations of the graph. If we choose the viewing rectangle ͓Ϫ0.25, 0.25͔ by ͓Ϫ1.5, 1.5͔,

we get the graph shown in Figure 7.

Now we see what went wrong in Figure 6. The oscillations of y ෇ sin 50x are so rapid

that when the calculator plots points and joins them, it misses most of the maximum and

_.25

.25 minimum points and therefore gives a very misleading impression of the graph.

_ 1.5
FIGURE 7 ƒ=sin 50x

We have seen that the use of an inappropriate viewing rectangle can give a misleading impression of the graph of a function. In Examples 1 and 3 we solved the problem by changing to a larger viewing rectangle. In Example 4 we had to make the viewing rectangle smaller. In the next example we look at a function for which there is no single viewing rectangle that reveals the true shape of the graph.

EXAMPLE 5

Graph the function

f

͑x͒

෇

sin

x

ϩ

1 100

cos

100x.

SOLUTION Figure 8 shows the graph of f produced by a graphing calculator with viewing

rectangle ͓Ϫ6.5, 6.5͔ by ͓Ϫ1.5, 1.5͔. It looks much like the graph of y ෇ sin x, but per-

haps with some bumps attached. If we zoom in to the viewing rectangle ͓Ϫ0.1, 0.1͔ by

͓Ϫ0.1, 0.1͔, we can see much more clearly the shape of these bumps in Figure 9. The

reason

for

this

behavior

is

that

the

second

term,

1 100

cos

100x,

is

very

small

in

comparison

with the ﬁrst term, sin x. Thus, we really need two graphs to see the true nature of this

function.

1.5

0.1

_6.5

6.5

_0.1

0.1

_1.5 FIGURE 8

_0.1 FIGURE 9

52 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS

|||| Another way to avoid the extraneous line is to change the graphing mode on the calculator so that the dots are not connected. Alternatively, we could zoom in using the Zoom Decimal mode.

EXAMPLE 6

Draw the graph of the function y ෇

1 1Ϫx.

SOLUTION Figure 10(a) shows the graph produced by a graphing calculator with viewing rectangle ͓Ϫ9, 9͔ by ͓Ϫ9, 9͔. In connecting successive points on the graph, the calculator produced a steep line segment from the top to the bottom of the screen. That
line segment is not truly part of the graph. Notice that the domain of the function
Խ y ෇ 1͑͞1 Ϫ x͒ is ͕x x 1͖. We can eliminate the extraneous near-vertical line by exper-
imenting with a change of scale. When we change to the smaller viewing rectangle ͓Ϫ4.7, 4.7͔ by ͓Ϫ4.7, 4.7͔ on this particular calculator, we obtain the much better graph in Figure 10(b).

9

4.7

_9

9

_4.7

4.7

FIGURE 10

1

_9

_4.7

y=1-x

(a)

(b)

EXAMPLE 7 Graph the function y ෇ s3 x.

SOLUTION Some graphing devices display the graph shown in Figure 11, whereas others produce a graph like that in Figure 12. We know from Section 1.2 (Figure 13) that the graph in Figure 12 is correct, so what happened in Figure 11? The explanation is that some machines compute the cube root of x using a logarithm, which is not deﬁned if x is negative, so only the right half of the graph is produced.

2

2

_3

3

_3

3

_2

_2

FIGURE 11

FIGURE 12

You should experiment with your own machine to see which of these two graphs is

produced. If you get the graph in Figure 11, you can obtain the correct picture by graph-

ing the function

f ͑x͒ ෇

x
ԽxԽ

Խ Խ ؒ x 1͞3

Notice that this function is equal to s3 x (except when x ෇ 0).

To understand how the expression for a function relates to its graph, it’s helpful to graph a family of functions, that is, a collection of functions whose equations are related. In the next example we graph members of a family of cubic polynomials.

SECTION 1.4 GRAPHING CALCULATORS AND COMPUTERS ❙❙❙❙ 53
EXAMPLE 8 Graph the function y ෇ x 3 ϩ cx for various values of the number c. How does the graph change when c is changed?
SOLUTION Figure 13 shows the graphs of y ෇ x 3 ϩ cx for c ෇ 2, 1, 0, Ϫ1, and Ϫ2. We see that, for positive values of c, the graph increases from left to right with no maximum or minimum points (peaks or valleys). When c ෇ 0, the curve is ﬂat at the origin. When c is negative, the curve has a maximum point and a minimum point. As c decreases, the maximum point becomes higher and the minimum point lower.

(a) y=˛+2x

(b) y=˛+x

(c) y=˛

(d) y=˛-x

(e) y=˛-2x

FIGURE 13 Several members of the family of functions y=˛+cx, all graphed in the viewing rectangle ͓_2, 2͔ by ͓_2.5, 2.5͔
_5

EXAMPLE 9 Find the solution of the equation cos x ෇ x correct to two decimal places.

SOLUTION The solutions of the equation cos x ෇ x are the x-coordinates of the points of intersection of the curves y ෇ cos x and y ෇ x. From Figure 14(a) we see that there is only one solution and it lies between 0 and 1. Zooming in to the viewing rectangle ͓0, 1͔ by ͓0, 1͔, we see from Figure 14(b) that the root lies between 0.7 and 0.8. So we zoom in further to the viewing rectangle ͓0.7, 0.8͔ by ͓0.7, 0.8͔ in Figure 14(c). By moving the cursor to the intersection point of the two curves, or by inspection and the fact that the
x-scale is 0.01, we see that the root of the equation is about 0.74. (Many calculators have
a built-in intersection feature.)

1.5

1

0.8

y=x y=Ł x
5

y=Ł x y=x

y=x y=Ł x

FIGURE 14 Locating the roots of cos x=x

_1.5
(a) ͓_5, 5͔ by ͓_1.5, 1.5͔ x-scale=1

1

0.8

0

0.7

(b) ͓0, 1͔ by ͓0, 1͔ x-scale=0.1

(c) ͓0.7, 0.8͔ by ͓0.7, 0.8͔ x-scale=0.01

|||| 1.4 ; Exercises

1. Use a graphing calculator or computer to determine which of

the given viewing rectangles produces the most appropriate

graph of the function f ͑x͒ ෇ x 4 ϩ 2.

(a) ͓Ϫ2, 2͔ by ͓Ϫ2, 2͔

(b) ͓0, 4͔ by ͓0, 4͔

(c) ͓Ϫ4, 4͔ by ͓Ϫ4, 4͔

(d) ͓Ϫ8, 8͔ by ͓Ϫ4, 40͔

(e) ͓Ϫ40, 40͔ by ͓Ϫ80, 800͔

2. Use a graphing calculator or computer to determine which of
the given viewing rectangles produces the most appropriate graph of the function f ͑x͒ ෇ x 2 ϩ 7x ϩ 6. (a) ͓Ϫ5, 5͔ by ͓Ϫ5, 5͔ (b) ͓0, 10͔ by ͓Ϫ20, 100͔ (c) ͓Ϫ15, 8͔ by ͓Ϫ20, 100͔ (d) ͓Ϫ10, 3͔ by ͓Ϫ100, 20͔

54 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS

3. Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f ͑x͒ ෇ 10 ϩ 25x Ϫ x 3. (a) ͓Ϫ4, 4͔ by ͓Ϫ4, 4͔ (b) ͓Ϫ10, 10͔ by ͓Ϫ10, 10͔ (c) ͓Ϫ20, 20͔ by ͓Ϫ100, 100͔ (d) ͓Ϫ100, 100͔ by ͓Ϫ200, 200͔
4. Use a graphing calculator or computer to determine which of the given viewing rectangles produces the most appropriate graph of the function f ͑x͒ ෇ s8x Ϫ x 2. (a) ͓Ϫ4, 4͔ by ͓Ϫ4, 4͔ (b) ͓Ϫ5, 5͔ by ͓0, 100͔ (c) ͓Ϫ10, 10͔ by ͓Ϫ10, 40͔ (d) ͓Ϫ2, 10͔ by ͓Ϫ2, 6͔

5–18 |||| Determine an appropriate viewing rectangle for the given function and use it to draw the graph.

5. f ͑x͒ ෇ 5 ϩ 20x Ϫ x 2

6. f ͑x͒ ෇ x 3 ϩ 30x 2 ϩ 200x

7. f ͑x͒ ෇ 0.01x 3 Ϫ x 2 ϩ 5

8. f ͑x͒ ෇ x͑x ϩ 6͒͑x Ϫ 9͒

9. f ͑x͒ ෇ s4 81 Ϫ x 4

10. f ͑x͒ ෇ s0.1x ϩ 20

11. f ͑x͒ ෇ x 2 ϩ 100 x

x 12. f ͑x͒ ෇ x 2 ϩ 100

13. f ͑x͒ ෇ cos 100x

14. f ͑x͒ ෇ 3 sin 120x

15. f ͑x͒ ෇ sin͑x͞40͒

16. y ෇ tan 25x

17. y ෇ 3cos͑x2͒

18. y ෇ x 2 ϩ 0.02 sin 50x

s

s

s

s

s

s

s

s

s

s

s

s

19. Graph the ellipse 4x 2 ϩ 2y 2 ෇ 1 by graphing the functions whose graphs are the upper and lower halves of the ellipse.
20. Graph the hyperbola y 2 Ϫ 9x 2 ෇ 1 by graphing the functions whose graphs are the upper and lower branches of the hyperbola.

21–23 |||| Find all solutions of the equation correct to two decimal places.

21. x 3 Ϫ 9x 2 Ϫ 4 ෇ 0

22. x 3 ෇ 4x Ϫ 1

23. x 2 ෇ sin x

s

s

s

s

s

s

s

s

s

s

s

s

24. We saw in Example 9 that the equation cos x ෇ x has exactly one solution. (a) Use a graph to show that the equation cos x ෇ 0.3x has three solutions and ﬁnd their values correct to two decimal places. (b) Find an approximate value of m such that the equation cos x ෇ mx has exactly two solutions.
25. Use graphs to determine which of the functions f ͑x͒ ෇ 10x 2 and t͑x͒ ෇ x 3͞10 is eventually larger (that is, larger when x is very large).

26. Use graphs to determine which of the functions f ͑x͒ ෇ x 4 Ϫ 100x 3 and t͑x͒ ෇ x 3 is eventually larger.
Խ Խ 27. For what values of x is it true that sin x Ϫ x Ͻ 0.1?
28. Graph the polynomials P͑x͒ ෇ 3x 5 Ϫ 5x 3 ϩ 2x and Q͑x͒ ෇ 3x 5 on the same screen, ﬁrst using the viewing rectangle ͓Ϫ2, 2͔ by [Ϫ2, 2] and then changing to ͓Ϫ10, 10͔ by ͓Ϫ10,000, 10,000͔. What do you observe from these graphs?

29. In this exercise we consider the family of root functions f ͑x͒ ෇ sn x, where n is a positive integer. (a) Graph the functions y ෇ sx, y ෇ s4 x, and y ෇ s6 x on the same screen using the viewing rectangle ͓Ϫ1, 4͔ by ͓Ϫ1, 3͔. (b) Graph the functions y ෇ x, y ෇ s3 x, and y ෇ s5 x on the same screen using the viewing rectangle ͓Ϫ3, 3͔ by ͓Ϫ2, 2͔. (See Example 7.) (c) Graph the functions y ෇ sx, y ෇ s3 x, y ෇ s4 x, and y ෇ s5 x on the same screen using the viewing rectangle ͓Ϫ1, 3͔ by ͓Ϫ1, 2͔. (d) What conclusions can you make from these graphs?

30. In this exercise we consider the family of functions f ͑x͒ ෇ 1͞x n, where n is a positive integer. (a) Graph the functions y ෇ 1͞x and y ෇ 1͞x 3 on the same screen using the viewing rectangle ͓Ϫ3, 3͔ by ͓Ϫ3, 3͔. (b) Graph the functions y ෇ 1͞x 2 and y ෇ 1͞x 4 on the same screen using the same viewing rectangle as in part (a).
(c) Graph all of the functions in parts (a) and (b) on the same screen using the viewing rectangle ͓Ϫ1, 3͔ by ͓Ϫ1, 3͔.
(d) What conclusions can you make from these graphs?

31. Graph the function f ͑x͒ ෇ x 4 ϩ cx 2 ϩ x for several values of c. How does the graph change when c changes?

32. Graph the function f ͑x͒ ෇ s1 ϩ cx 2 for various values of c. Describe how changing the value of c affects the graph.

33. Graph the function y ෇ x n2Ϫx, x ജ 0, for n ෇ 1, 2, 3, 4, 5, and 6. How does the graph change as n increases?

34. The curves with equations

y

෇

ԽxԽ
sc Ϫ x 2

are called bullet-nose curves. Graph some of these curves to see why. What happens as c increases?

35. What happens to the graph of the equation y 2 ෇ cx 3 ϩ x 2 as c varies?

36. This exercise explores the effect of the inner function t on a composite function y ෇ f ͑ t͑x͒͒.
(a) Graph the function y ෇ sin(sx ) using the viewing rect-
angle ͓0, 400͔ by ͓Ϫ1.5, 1.5͔. How does this graph differ from the graph of the sine function? (b) Graph the function y ෇ sin͑x 2 ͒ using the viewing rectangle ͓Ϫ5, 5͔ by ͓Ϫ1.5, 1.5͔. How does this graph differ from the graph of the sine function?

37. The ﬁgure shows the graphs of y ෇ sin 96x and y ෇ sin 2x as displayed by a TI-83 graphing calculator.

SECTION 1.5 EXPONENTIAL FUNCTIONS ❙❙❙❙ 55
explain its appearance, we replot the curve in dot mode in the second graph.

0

2π 0

2π

y=sin 96x

y=sin 2x

The ﬁrst graph is inaccurate. Explain why the two graphs appear identical. [Hint: The TI-83’s graphing window is 95 pixels wide. What speciﬁc points does the calculator plot?]
38. The ﬁrst graph in the ﬁgure is that of y ෇ sin 45x as displayed by a TI-83 graphing calculator. It is inaccurate and so, to help

0

2π 0

2π

What two sine curves does the calculator appear to be plotting? Show that each point on the graph of y ෇ sin 45x that the TI-83 chooses to plot is in fact on one of these two curves. (The TI-83’s graphing window is 95 pixels wide.)

|||| 1.5 Exponential Functions

The function f ͑x͒ ෇ 2x is called an exponential function because the variable, x, is the exponent. It should not be confused with the power function t͑x͒ ෇ x 2, in which the variable is the base.
In general, an exponential function is a function of the form
f ͑x͒ ෇ a x

where a is a positive constant. Let’s recall what this means. If x ෇ n, a positive integer, then an ෇ a ؒ a ؒ и и и ؒ a
n factors
If x ෇ 0, then a 0 ෇ 1, and if x ෇ Ϫn, where n is a positive integer, then

y

1

01

x

FIGURE 1 Representation of y=2®, x rational

a Ϫn

෇

1 an

If x is a rational number, x ෇ p͞q, where p and q are integers and q Ͼ 0, then

( ) a x ෇ a p͞q ෇ sq a p ෇ sq a p
But what is the meaning of a x if x is an irrational number? For instance, what is meant by 2s3 or 5␲ ?
To help us answer this question we ﬁrst look at the graph of the function y ෇ 2x, where x is rational. A representation of this graph is shown in Figure 1. We want to enlarge the domain of y ෇ 2x to include both rational and irrational numbers.
There are holes in the graph in Figure 1 corresponding to irrational values of x. We want to ﬁll in the holes by deﬁning f ͑x͒ ෇ 2x, where x ʦ ‫ޒ‬, so that f is an increasing function. In particular, since the irrational number s3 satisﬁes
1.7 Ͻ s3 Ͻ 1.8

56 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS

|||| A proof of this fact is given in J. Marsden and A. Weinstein, Calculus Unlimited (Menlo Park, CA: Benjamin/Cummings, 1980).
y

1

01

x

FIGURE 2 y=2®, x real

we must have

21.7 Ͻ 2s3 Ͻ 21.8

and we know what 21.7 and 21.8 mean because 1.7 and 1.8 are rational numbers. Similarly, if we use better approximations for s3, we obtain better approximations for 2s3:

1.73 Ͻ s3 Ͻ 1.74

?

21.73 Ͻ 2s3 Ͻ 21.74

1.732 Ͻ s3 Ͻ 1.733 ? 21.732 Ͻ 2s3 Ͻ 21.733

1.7320 Ͻ s3 Ͻ 1.7321 ? 21.7320 Ͻ 2s3 Ͻ 21.7321

1.73205 Ͻ s3 Ͻ 1.73206 ? 21.73205 Ͻ 2s3 Ͻ 21.73206

.

.

.

.

.

.

.

.

.

.

.

.

It can be shown that there is exactly one number that is greater than all of the numbers

21.7, 21.73, 21.732, 2 , 1.7320 2 , 1.73205 . . .

and less than all of the numbers

21.8, 21.74, 21.733, 2 , 1.7321 2 , 1.73206 . . .
We deﬁne 2s3 to be this number. Using the preceding approximation process we can compute it correct to six decimal places:

2s3 Ϸ 3.321997

Similarly, we can deﬁne 2x (or a x, if a Ͼ 0) where x is any irrational number. Figure 2
shows how all the holes in Figure 1 have been ﬁlled to complete the graph of the function f ͑x͒ ෇ 2x, x ʦ ‫ޒ‬.
The graphs of members of the family of functions y ෇ a x are shown in Figure 3 for various values of the base a. Notice that all of these graphs pass through the same point ͑0, 1͒ because a 0 ෇ 1 for a 0. Notice also that as the base a gets larger, the exponential function grows more rapidly (for x Ͼ 0).

”21 ’®

” 41 ’®

y 10®

4®

2®

1.5®

|||| If 0 Ͻ a Ͻ 1, then a x approaches 0 as x

becomes large. If a Ͼ 1, then a x approaches 0

as x decreases through negative values. In both

cases the x-axis is a horizontal asymptote.

1®

These matters are discussed in Section 2.6.

FIGURE 3

0

1

x

You can see from Figure 3 that there are basically three kinds of exponential functions y ෇ a x. If 0 Ͻ a Ͻ 1, the exponential function decreases; if a ෇ 1, it is a constant; and if a Ͼ 1, it increases. These three cases are illustrated in Figure 4. Observe that if a 1,

SECTION 1.5 EXPONENTIAL FUNCTIONS ❙❙❙❙ 57

then the exponential function y ෇ a x has domain ‫ ޒ‬and range ͑0, ϱ͒. Notice also that, since ͑1͞a͒x ෇ 1͞a x ෇ a Ϫx, the graph of y ෇ ͑1͞a͒x is just the reﬂection of the graph of y ෇ a x about the y-axis.

y

y

y

(0, 1)

0

x

(a) y=a®, 0<a<1 FIGURE 4

1

0

x

(b) y=1®

(0, 1)

0

x

(c) y=a®, a>1

One reason for the importance of the exponential function lies in the following properties. If x and y are rational numbers, then these laws are well known from elementary algebra. It can be proved that they remain true for arbitrary real numbers x and y.

|||| In Section 5.6 we will present a deﬁnition of the exponential function that will enable us to give an easy proof of the Laws of Exponents.

Laws of Exponents If a and b are positive numbers and x and y are any real numbers,

then

1. a xϩy ෇ a xa y

2.

a xϪy

෇

ax ay

3. ͑a x ͒y ෇ a xy

4. ͑ab͒x ෇ a xb x

|||| For a review of reﬂecting and shifting graphs, see Section 1.3.

EXAMPLE 1 Sketch the graph of the function y ෇ 3 Ϫ 2x and determine its domain and range.
SOLUTION First we reﬂect the graph of y ෇ 2x (shown in Figure 2) about the x-axis to get the graph of y ෇ Ϫ2x in Figure 5(b). Then we shift the graph of y ෇ Ϫ2x upward 3 units to obtain the graph of y ෇ 3 Ϫ 2x in Figure 5(c). The domain is ‫ ޒ‬and the range is ͑Ϫϱ, 3͒.

y

y

y

y=3

2 1

0

x

0

x

_1

0

x

FIGURE 5

(a) y=2®

(b) y=_2®

(c) y=3-2®

EXAMPLE 2 Use a graphing device to compare the exponential function f ͑x͒ ෇ 2x and the power function t͑x͒ ෇ x 2. Which function grows more quickly when x is large?
SOLUTION Figure 6 shows both functions graphed in the viewing rectangle ͓Ϫ2, 6͔ by ͓0, 40͔. We see that the graphs intersect three times, but for x Ͼ 4 the graph of

58 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS

stays above the graph of

. Figure 7 gives a more global view and

shows that for large values of x, the exponential function

grows far more rapidly

than the power function

.

Ϫ

Applications of Exponential Functions

The exponential function occurs very frequently in mathematical models of nature and

society. Here we indicate brieﬂy how it arises in the description of population growth

and radioactive decay. In later chapters we will pursue these and other applications in

greater detail.

First we consider a population of bacteria in a homogeneous nutrient medium. Suppose

that by sampling the population at certain intervals it is determined that the population

doubles every hour. If the number of bacteria at time t is , where t is measured in hours,

and the initial population is

, then we have

p͑3͒ ෇ 2p͑2͒ ෇ 2Ϫ⌻ϩѥ␾ЉϬϾϭϮЈϫ␮գՌ␰րϯЊ⌺␦␽␸␶␺ 3ϫϪ
It seems from this pattern that, in general, p͑t͒ ෇ 2t ϫ 1000 ෇ ͑1000͒2t
This population function is a constant multiple of the exponential function y ෇ 2t, so it exhibits the rapid growth that we observed in Figures 2 and 7. Under ideal conditions (unlimited space and nutrition and freedom from disease) this exponential growth is typical of what actually occurs in nature.
What about the human population? Table 1 shows data for the population of the world in the 20th century and Figure 8 shows the corresponding scatter plot.
P
6x10 '

1900 1920 1940 1960 1980 2000 t FIGURE 8 Scatter plot for world population growth

74 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS

EXAMPLE 14 Simplify the expression cos͑tanϪ1x͒.
SOLUTION 1 Let y ෇ tanϪ1x. Then tan y ෇ x and Ϫ␲͞2 Ͻ y Ͻ ␲͞2. We want to ﬁnd cos y but, since tan y is known, it is easier to ﬁnd sec y ﬁrst:

sec2 y ෇ 1 ϩ tan2 y ෇ 1 ϩ x 2

sec y ෇ s1 ϩ x 2

͑since sec y Ͼ 0 for Ϫ␲͞2 Ͻ y Ͻ ␲͞2͒

œ„1+ „„„≈„
y 1
FIGURE 24
y
π 2
0

_

π 2

FIGURE 25 y=tan–! x=arctan x

y

0

π

_1

FIGURE 26 y=sec x

x

Thus

cos͑tanϪ1x͒ ෇ cos y ෇ 1 ෇ 1 sec y s1 ϩ x 2

SOLUTION 2 Instead of using trigonometric identities as in Solution 1, it is perhaps easier to use a diagram. If y ෇ tanϪ1x, then tan y ෇ x, and we can read from Figure 24 (which illustrates the case y Ͼ 0) that

cos͑tanϪ1x͒ ෇ cos y ෇ 1 s1 ϩ x 2

The inverse tangent function, tanϪ1 ෇ arctan, has domain ‫ ޒ‬and range ͑Ϫ␲͞2, ␲͞2͒.

Its graph is shown in Figure 25.

x

We know that the lines x ෇ Ϯ␲͞2 are vertical asymptotes of the graph of tan. Since the

graph of tanϪ1 is obtained by reﬂecting the graph of the restricted tangent function about

the line y ෇ x, it follows that the lines y ෇ ␲͞2 and y ෇ Ϫ␲͞2 are horizontal asymptotes

of the graph of tanϪ1.

The remaining inverse trigonometric functions are not used as frequently and are sum-

marized here.

Խ Խ 11 y ෇ cscϪ1x ͑ x ജ 1͒ &? csc y ෇ x and y ʦ ͑0, ␲͞2͔ ʜ ͑␲, 3␲͞2͔

Խ Խ y ෇ secϪ1x ͑ x ജ 1͒ &? sec y ෇ x and y ʦ ͓0, ␲͞2͒ ʜ ͓␲, 3␲͞2͒

2π

x

y ෇ cotϪ1x ͑x ʦ ‫ ?& ͒ޒ‬cot y ෇ x and y ʦ ͑0, ␲͒

The choice of intervals for y in the deﬁnitions of cscϪ1 and secϪ1 is not universally agreed upon. For instance, some authors use y ʦ ͓0, ␲͞2͒ ʜ ͑␲͞2, ␲͔ in the deﬁnition of secϪ1. [You can see from the graph of the secant function in Figure 26 that both this choice
and the one in (11) will work.]

|||| 1.6 Exercises
1. (a) What is a one-to-one function? (b) How can you tell from the graph of a function whether it is one-to-one?
2. (a) Suppose f is a one-to-one function with domain A and range B. How is the inverse function f Ϫ1 deﬁned? What is the domain of f Ϫ1? What is the range of f Ϫ1?
(b) If you are given a formula for f , how do you ﬁnd a formula for f Ϫ1?

(c) If you are given the graph of f , how do you ﬁnd the graph of f Ϫ1?

3–14 |||| A function is given by a table of values, a graph, a formula, or a verbal description. Determine whether it is one-to-one.

3.

x

1

2

3

4

5

6

f ͑x͒ 1.5 2.0 3.6 5.3 2.8 2.0

SECTION 1.6 INVERSE FUNCTIONS AND LOGARITHMS ❙❙❙❙ 75

4.

x

1

2

3

4

5

6

f ͑x͒ 1

2

4

8

16 32

21.

The formula C

෇

5 9

͑F

Ϫ

32͒,

where F

ജ

Ϫ459.67, expresses

the Celsius temperature C as a function of the Fahrenheit

temperature F. Find a formula for the inverse function and

interpret it. What is the domain of the inverse function?

5.

y

6.

y

22. In the theory of relativity, the mass of a particle with speed v

is

m

෇

f ͑v͒

෇

s1

m0 Ϫ v 2͞c 2

x

x

where m 0 is the rest mass of the particle and c is the speed of

light in a vacuum. Find the inverse function of f and explain

its meaning.

7.

y

8.

y

x

9.

f

͑x͒

෇

1 2

͑x

ϩ

5͒

11. t͑x͒ ෇ Խ x Խ

10. f ͑x͒ ෇ 1 ϩ 4x Ϫ x 2 12. t͑x͒ ෇ sx

13. f ͑t͒ is the height of a football t seconds after kickoff.

14. f ͑t͒ is your height at age t.

s

s

s

s

s

s

s

s

s

s

s

; 15–16 |||| Use a graph to decide whether f is one-to-one.

15. f ͑x͒ ෇ x 3 Ϫ x

16. f ͑x͒ ෇ x 3 ϩ x

s

s

s

s

s

s

s

s

s

s

s

23–28 |||| Find a formula for the inverse of the function.

23. f ͑x͒ ෇ s10 Ϫ 3x

x

25. f ͑x͒ ෇ e x3

24.

f ͑x͒

෇

4x Ϫ 1 2x ϩ 3

26. y ෇ 2 x3 ϩ 3

27. y ෇ ln͑x ϩ 3͒

1 ϩ ex 28. y ෇ 1 Ϫ e x

s

s

s

s

s

s

s

s

s

s

s

s

; 29–30 |||| Find an explicit formula for f Ϫ1 and use it to graph f Ϫ1,
f , and the line y ෇ x on the same screen. To check your work, see whether the graphs of f and f Ϫ1 are reﬂections about the line.

29. f ͑x͒ ෇ 1 Ϫ 2͞x 2, x Ͼ 0

30. f ͑x͒ ෇ sx 2 ϩ 2x, x Ͼ 0

s

s

s

s

s

s

s

s

s

s

s

s

s

31. Use the given graph of f to sketch the graph of f Ϫ1.

y
1
s

17. If f is a one-to-one function such that f ͑2͒ ෇ 9, what is f Ϫ1͑9͒?
18. Let f ͑x͒ ෇ 3 ϩ x 2 ϩ tan͑␲x͞2͒, where Ϫ1 Ͻ x Ͻ 1. (a) Find f Ϫ1͑3͒. (b) Find f ͑ f Ϫ1͑5͒͒.
19. If t͑x͒ ෇ 3 ϩ x ϩ e x, ﬁnd tϪ1͑4͒.
20. The graph of f is given. (a) Why is f one-to-one? (b) State the domain and range of f Ϫ1. (c) Estimate the value of f Ϫ1͑1͒.

1

x

32. Use the given graph of f to sketch the graphs of f Ϫ1 and 1͞f . y

y 2 1
_3 _2 _1 0 _1 _2

1 2 3x

1

1

x

33. (a) How is the logarithmic function y ෇ loga x deﬁned? (b) What is the domain of this function? (c) What is the range of this function? (d) Sketch the general shape of the graph of the function y ෇ loga x if a Ͼ 1.

76 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS

34. (a) What is the natural logarithm? (b) What is the common logarithm? (c) Sketch the graphs of the natural logarithm function and the natural exponential function with a common set of axes.

53–54 |||| Solve each inequality for x.

53. (a) ex Ͻ 10

(b) ln x Ͼ Ϫ1

54. (a) 2 Ͻ ln x Ͻ 9

(b) e2Ϫ3x Ͼ 4

35–38 |||| Find the exact value of each expression.

35. (a) log2 64

(b)

log

6

1 36

36. (a) log8 2

(b) ln es2

37. (a) log10 1.25 ϩ log10 80 (b) log5 10 ϩ log5 20 Ϫ 3 log5 2

38. (a) 2͑log2 3 ϩ log2 5͒

(b) e 3 ln 2

s

s

s

s

s

s

s

s

s

s

s

s

39–41 |||| Express the given quantity as a single logarithm.

39. 2 ln 4 Ϫ ln 2

40. ln x ϩ a ln y Ϫ b ln z

41.

ln͑1

ϩ

x

2͒

ϩ

1 2

ln

x

Ϫ

ln

sin

x

s

s

s

s

s

s

s

s

s

s

s

s

42. Use Formula 10 to evaluate each logarithm correct to six deci-

mal places.

(a) log12 10

(b) log2 8.4

; 43–44 |||| Use Formula 10 to graph the given functions on a common screen. How are these graphs related?

43. y ෇ log1.5 x , y ෇ ln x, y ෇ log10 x , y ෇ log50 x 44. y ෇ ln x, y ෇ log10 x , y ෇ e x, y ෇ 10 x

s

s

s

s

s

s

s

s

s

s

s

s

45. Suppose that the graph of y ෇ log2 x is drawn on a coordinate grid where the unit of measurement is an inch. How many miles to the right of the origin do we have to move before the height of the curve reaches 3 ft?
; 46. Compare the functions f ͑x͒ ෇ x 0.1 and t͑x͒ ෇ ln x by graphing both f and t in several viewing rectangles. When does the graph of f ﬁnally surpass the graph of t?

47–48 |||| Make a rough sketch of the graph of each function. Do not use a calculator. Just use the graphs given in Figures 12 and 13 and, if necessary, the transformations of Section 1.3.

47. (a) y ෇ log10͑x ϩ 5͒ 48. (a) y ෇ ln͑Ϫx͒

(b) y ෇ Ϫln x
(b) y ෇ ln Խ x Խ

s

s

s

s

s

s

s

s

s

s

s

s

49–52 |||| Solve each equation for x.

49. (a) 2 ln x ෇ 1

(b) eϪx ෇ 5

50. (a) e2xϩ3 Ϫ 7 ෇ 0

(b) ln͑5 Ϫ 2 x͒ ෇ Ϫ3

51. (a) 2xϪ5 ෇ 3

(b) ln x ϩ ln͑x Ϫ 1͒ ෇ 1

52. (a) ln͑ln x͒ ෇ 1

(b) e ax ෇ Ce bx, where a b

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

s

55–56 |||| Find (a) the domain of f and (b) f Ϫ1 and its domain.

55. f ͑ x͒ ෇ s3 Ϫ e 2x

56. f ͑ x͒ ෇ ln͑2 ϩ ln x͒

s

s

s

s

s

s

s

s

s

s

s

s

CAS 57. Graph the function f ͑x͒ ෇ sx 3 ϩ x 2 ϩ x ϩ 1 and explain why it is one-to-one. Then use a computer algebra system to ﬁnd an explicit expression for f Ϫ1͑x͒. (Your CAS will produce three possible expressions. Explain why two of them are irrelevant in this context.)
CAS 58. (a) If t͑x͒ ෇ x 6 ϩ x 4, x ജ 0, use a computer algebra system to ﬁnd an expression for t Ϫ1͑x͒.
(b) Use the expression in part (a) to graph y ෇ t͑x͒, y ෇ x, and y ෇ t Ϫ1͑x͒ on the same screen.
59. If a bacteria population starts with 100 bacteria and doubles every three hours, then the number of bacteria after t hours is n ෇ f ͑t͒ ෇ 100 и 2t͞3. (See Exercise 25 in Section 1.5.) (a) Find the inverse of this function and explain its meaning. (b) When will the population reach 50,000?
60. When a camera ﬂash goes off, the batteries immediately begin to recharge the ﬂash’s capacitor, which stores electric charge given by Q͑t͒ ෇ Q0͑1 Ϫ e Ϫt͞a ͒
(The maximum charge capacity is Q0 and t is measured in seconds.) (a) Find the inverse of this function and explain its meaning. (b) How long does it take to recharge the capacitor to 90% of
capacity if a ෇ 2?
61. Starting with the graph of y ෇ ln x, ﬁnd the equation of the graph that results from (a) shifting 3 units upward (b) shifting 3 units to the left (c) reﬂecting about the x-axis (d) reﬂecting about the y-axis (e) reﬂecting about the line y ෇ x (f) reﬂecting about the x-axis and then about the line y ෇ x (g) reﬂecting about the y-axis and then about the line y ෇ x (h) shifting 3 units to the left and then reﬂecting about the line y ෇ x
62. (a) If we shift a curve to the left, what happens to its reﬂection about the line y ෇ x? In view of this geometric principle, ﬁnd an expression for the inverse of t͑x͒ ෇ f ͑x ϩ c͒, where f is a one-to-one function.
(b) Find an expression for the inverse of h͑x͒ ෇ f ͑cx͒, where c 0.

63–68 |||| Find the exact value of each expression.

63. (a) sinϪ1(s3͞2)

(b) cosϪ1͑Ϫ1͒

64. (a) arctan͑Ϫ1͒

(b) cscϪ1 2

65. (a) tanϪ1s3

(b) arcsin(Ϫ1͞s2 )

66. (a) secϪ1s2 67. (a) sin͑sinϪ1 0.7͒

(b) arcsin 1

ͩ ͪ (b)

tanϪ1

4␲ tan

3

68. (a) sec͑arctan 2͒

(b) cos(2 sinϪ1(153))

s

s

s

s

s

s

s

s

s

s

s

69. Prove that cos͑sinϪ1x͒ ෇ s1 Ϫ x 2.

70–72 |||| Simplify the expression. 70. tan͑sinϪ1x͒

CHAPTER 1 REVIEW ❙❙❙❙ 77

71. sin͑tanϪ1x͒

72. sin͑2 cosϪ1x͒

s

s

s

s

s

s

s

s

s

s

s

s

; 73–74 |||| Graph the given functions on the same screen. How are these graphs related?

73. y ෇ sin x, Ϫ␲͞2 ഛ x ഛ ␲͞2; y ෇ sinϪ1x; y ෇ x

74. y ෇ tan x, Ϫ␲͞2 Ͻ x Ͻ ␲͞2; y ෇ tanϪ1x; y ෇ x

s

s

s

s

s

s

s

s

s

s

s

s

75. Find the domain and range of the function
s
t͑x͒ ෇ sinϪ1͑3x ϩ 1͒
; 76. (a) Graph the function f ͑x͒ ෇ sin͑sinϪ1x͒ and explain the appearance of the graph.
(b) Graph the function t͑x͒ ෇ sinϪ1͑sin x͒. How do you explain the appearance of this graph?

|||| 1 Review

s CONCEPT CHECK s

1. (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How can you tell whether a given curve is the graph of a function?

2. Discuss four ways of representing a function. Illustrate your discussion with examples.

3. (a) What is an even function? How can you tell if a function is even by looking at its graph?
(b) What is an odd function? How can you tell if a function is odd by looking at its graph?

4. What is an increasing function?

5. What is a mathematical model?

6. Give an example of each type of function.

(a) Linear function

(b) Power function

(c) Exponential function

(d) Quadratic function

(e) Polynomial of degree 5 (f) Rational function

7. Sketch by hand, on the same axes, the graphs of the following

functions.

(a) f ͑x͒ ෇ x

(b) t͑x͒ ෇ x 2

(c) h͑x͒ ෇ x 3

(d) j͑x͒ ෇ x 4

8. Draw, by hand, a rough sketch of the graph of each function.

(a) y ෇ sin x

(b) y ෇ tan x

(c) y ෇ e x

(d) y ෇ ln x

(e) y ෇ 1͞x (g) y ෇ sx

(f) y ෇ ԽxԽ
(h) y ෇ tanϪ1 x

9. Suppose that f has domain A and t has domain B. (a) What is the domain of f ϩ t?

(b) What is the domain of f t? (c) What is the domain of f͞t?
10. How is the composite function f ‫ ؠ‬t deﬁned? What is its domain?
11. Suppose the graph of f is given. Write an equation for each of the graphs that are obtained from the graph of f as follows. (a) Shift 2 units upward. (b) Shift 2 units downward. (c) Shift 2 units to the right. (d) Shift 2 units to the left. (e) Reﬂect about the x-axis. (f) Reﬂect about the y-axis. (g) Stretch vertically by a factor of 2. (h) Shrink vertically by a factor of 2. (i) Stretch horizontally by a factor of 2. ( j) Shrink horizontally by a factor of 2.
12. (a) What is a one-to-one function? How can you tell if a function is one-to-one by looking at its graph?
(b) If f is a one-to-one function, how is its inverse function f Ϫ1 deﬁned? How do you obtain the graph of f Ϫ1 from the graph of f ?
13. (a) How is the inverse sine function f ͑x͒ ෇ sinϪ1x deﬁned? What are its domain and range?
(b) How is the inverse cosine function f ͑x͒ ෇ cosϪ1x deﬁned? What are its domain and range?
(c) How is the inverse tangent function f ͑x͒ ෇ tanϪ1x deﬁned? What are its domain and range?

78 ❙❙❙❙ CHAPTER 1 FUNCTIONS AND MODELS

s TRUE-FALSE QUIZ s

Determine whether the statement is true or false. If it is true, explain why. If it is false, explain why or give an example that disproves the statement. 1. If f is a function, then f ͑s ϩ t͒ ෇ f ͑s͒ ϩ f ͑t͒. 2. If f ͑s͒ ෇ f ͑t͒, then s ෇ t. 3. If f is a function, then f ͑3x͒ ෇ 3 f ͑x͒. 4. If x1 Ͻ x2 and f is a decreasing function, then f ͑x1 ͒ Ͼ f ͑x2 ͒. 5. A vertical line intersects the graph of a function at most once.

6. If f and t are functions, then f ‫ ؠ‬t ෇ t ‫ ؠ‬f .

7. If f is one-to-one, then f Ϫ1͑x͒ ෇

1 .

f ͑x͒

8. You can always divide by e x.

9. If 0 Ͻ a Ͻ b, then ln a Ͻ ln b.

10. If x Ͼ 0, then ͑ln x͒6 ෇ 6 ln x.

ln x

x

11. If x Ͼ 0 and a Ͼ 1, then ෇ ln .

ln a a

1. Let f be the function whose graph is given. (a) Estimate the value of f ͑2͒. (b) Estimate the values of x such that f ͑x͒ ෇ 3. (c) State the domain of f. (d) State the range of f. (e) On what interval is f increasing? (f) Is f one-to-one? Explain. (g) Is f even, odd, or neither even nor odd? Explain.

y f

1

1

x

2. The graph of t is given. (a) State the value of t͑2͒.
(b) Why is t one-to-one? (c) Estimate the value of tϪ1͑2͒. (d) Estimate the domain of tϪ1. (e) Sketch the graph of tϪ1.

y

g

s EXERCISES s

3. The distance traveled by a car is given by the values in the table.

t (seconds) 0 1

2

3

4

5

d (feet)

0 10 32 70 119 178

(a) Use the data to sketch the graph of d as a function of t. (b) Use the graph to estimate the distance traveled after
4.5 seconds.
4. Sketch a rough graph of the yield of a crop as a function of the amount of fertilizer used.

5–8 |||| Find the domain and range of the function.

5. f ͑x͒ ෇ s4 Ϫ 3x 2

6. t͑x͒ ෇ 1͑͞x ϩ 1͒

7. y ෇ 1 ϩ sin x

8. y ෇ ln ln x

s

s

s

s

s

s

s

s

s

s

s

s

9. Suppose that the graph of f is given. Describe how the graphs

of the following functions can be obtained from the graph of f.

(a) y ෇ f ͑x͒ ϩ 8

(b) y ෇ f ͑x ϩ 8͒

(c) y ෇ 1 ϩ 2 f ͑x͒

(d) y ෇ f ͑x Ϫ 2͒ Ϫ 2

(e) y ෇ Ϫf ͑x͒

(f) y ෇ f Ϫ1͑x͒

10. The graph of f is given. Draw the graphs of the following

functions.

(a) y ෇ f ͑x Ϫ 8͒

(b) y ෇ Ϫf ͑x͒

(c) y ෇ 2 Ϫ f ͑x͒

(d)

y

෇

1 2

f

͑x͒

Ϫ

1

(e) y ෇ f Ϫ1͑x͒

(f) y ෇ f Ϫ1͑x ϩ 3͒

y

1

01

x

1

01

x

CHAPTER 1 REVIEW ❙❙❙❙ 79

11–16 |||| Use transformations to sketch the graph of the function.

11. y ෇ Ϫsin 2 x

12. y ෇ 3 ln ͑x Ϫ 2͒ 13. y ෇ ͑1 ϩ e x ͒͞2

14. y ෇ 2 Ϫ sx

15.

f ͑x͒

෇

x

1 ϩ

2

ͭϪx
16. f ͑x͒ ෇ e x Ϫ 1

if x Ͻ 0 if x ജ 0

s

s

s

s

s

s

s

s

s

s

s

s

17. Determine whether f is even, odd, or neither even nor odd. (a) f ͑x͒ ෇ 2x 5 Ϫ 3x 2 ϩ 2 (b) f ͑x͒ ෇ x 3 Ϫ x 7 (c) f ͑x͒ ෇ eϪx 2
(d) f ͑x͒ ෇ 1 ϩ sin x

18. Find an expression for the function whose graph consists of the line segment from the point ͑Ϫ2, 2͒ to the point ͑Ϫ1, 0͒ together with the top half of the circle with center the origin and radius 1.
19. If f ͑x͒ ෇ ln x and t͑x͒ ෇ x 2 Ϫ 9, ﬁnd the functions f ‫ ؠ‬t, t ‫ ؠ‬f , f ‫ ؠ‬f , t ‫ ؠ‬t, and their domains.

20. Express the function F͑x͒ ෇ 1͞sx ϩ sx as a composition of three functions.

21. Life expectancy improved dramatically in the 20th century. The table gives the life expectancy at birth (in years) of males born in the United States.

Birth year
1900 1910 1920 1930 1940 1950 1960 1970 1980 1990 2000

Life expectancy
48.3 51.1 55.2 57.4 62.5 65.6 66.6 67.1 70.0 71.8 73.0

Use a scatter plot to choose an appropriate type of model. Use your model to predict the life span of a male born in the year 2010.

22. A small-appliance manufacturer ﬁnds that it costs $9000 to produce 1000 toaster ovens a week and $12,000 to produce 1500 toaster ovens a week. (a) Express the cost as a function of the number of toaster ovens produced, assuming that it is linear. Then sketch the graph. (b) What is the slope of the graph and what does it represent? (c) What is the y-intercept of the graph and what does it represent?
23. If f ͑x͒ ෇ 2x ϩ ln x, ﬁnd f Ϫ1͑2͒.

24.

Find the inverse function of

f ͑x͒ ෇

xϩ1 2x ϩ 1 .

25. Find the exact value of each expression.

(a) e 2 ln 3

(b) log10 25 ϩ log10 4

(c)

tan(arcsin

)1
2

(d) sin(cosϪ1(45))

26. Solve each equation for x. (a) e x ෇ 5 (c) e e x ෇ 2

(b) ln x ෇ 2 (d) tanϪ1x ෇ 1

27. The half-life of palladium-100, 100Pd, is four days. (So half of any given quantity of 100Pd will disintegrate in four days.) The initial mass of a sample is one gram. (a) Find the mass that remains after 16 days. (b) Find the mass m͑t͒ that remains after t days. (c) Find the inverse of this function and explain its meaning. (d) When will the mass be reduced to 0.01 g?

28. The population of a certain species in a limited environment with initial population 100 and carrying capacity 1000 is

100,000 P͑t͒ ෇ 100 ϩ 900eϪt

where t is measured in years. ; (a) Graph this function and estimate how long it takes for the
population to reach 900. (b) Find the inverse of this function and explain its meaning. (c) Use the inverse function to ﬁnd the time required for the
population to reach 900. Compare with the result of part (a).

; 29. Graph members of the family of functions f ͑x͒ ෇ ln͑x 2 Ϫ c͒ for several values of c. How does the graph change when c changes?

; 30. Graph the three functions y ෇ x a, y ෇ a x, and y ෇ loga x on the same screen for two or three values of a Ͼ 1. For large values of x, which of these functions has the largest values and which has the smallest values?

PRINCIPLES OF PROBLEM SOLVING
1 UNDERSTAND THE PROBLEM
2 THINK OF A PLAN

There are no hard and fast rules that will ensure success in solving problems. However, it is possible to outline some general steps in the problem-solving process and to give some principles that may be useful in the solution of certain problems. These steps and principles are just common sense made explicit. They have been adapted from George Polya’s book How To Solve It.
The ﬁrst step is to read the problem and make sure that you understand it clearly. Ask yourself the following questions:
What is the unknown? What are the given quantities? What are the given conditions?
For many problems it is useful to
draw a diagram
and identify the given and required quantities on the diagram. Usually it is necessary to
introduce suitable notation
In choosing symbols for the unknown quantities we often use letters such as a, b, c, m, n, x, and y, but in some cases it helps to use initials as suggestive symbols; for instance, V for volume or t for time.
Find a connection between the given information and the unknown that will enable you to calculate the unknown. It often helps to ask yourself explicitly: “How can I relate the given to the unknown?” If you don’t see a connection immediately, the following ideas may be helpful in devising a plan.
Try to Recognize Something Familiar Relate the given situation to previous knowledge. Look at the unknown and try to recall a more familiar problem that has a similar unknown.
Try to Recognize Patterns Some problems are solved by recognizing that some kind of pattern is occurring. The pattern could be geometric, or numerical, or algebraic. If you can see regularity or repetition in a problem, you might be able to guess what the continuing pattern is and then prove it.
Use Analogy Try to think of an analogous problem, that is, a similar problem, a related problem, but one that is easier than the original problem. If you can solve the similar, simpler problem, then it might give you the clues you need to solve the original, more difﬁcult problem. For instance, if a problem involves very large numbers, you could ﬁrst try a similar problem with smaller numbers. Or if the problem involves three-dimensional geometry, you could look for a similar problem in two-dimensional geometry. Or if the problem you start with is a general one, you could ﬁrst try a special case.
Introduce Something Extra It may sometimes be necessary to introduce something new, an auxiliary aid, to help make the connection between the given and the unknown. For instance, in a problem where a diagram is useful the auxiliary aid could be a new line drawn in a diagram. In a more algebraic problem it could be a new unknown that is related to the original unknown.

3 CARRY OUT THE PLAN 4 LOOK BACK

Take Cases We may sometimes have to split a problem into several cases and give a different argument for each of the cases. For instance, we often have to use this strategy in dealing with absolute value.
Work Backward Sometimes it is useful to imagine that your problem is solved and work backward, step by step, until you arrive at the given data. Then you may be able to reverse your steps and thereby construct a solution to the original problem. This procedure is commonly used in solving equations. For instance, in solving the equation 3x Ϫ 5 ෇ 7, we suppose that x is a number that satisﬁes 3x Ϫ 5 ෇ 7 and work backward. We add 5 to each side of the equation and then divide each side by 3 to get x ෇ 4. Since each of these steps can be reversed, we have solved the problem.
Establish Subgoals In a complex problem it is often useful to set subgoals (in which the desired situation is only partially fulﬁlled). If we can ﬁrst reach these subgoals, then we may be able to build on them to reach our ﬁnal goal.
Indirect Reasoning Sometimes it is appropriate to attack a problem indirectly. In using proof by contradiction to prove that P implies Q, we assume that P is true and Q is false and try to see why this can’t happen. Somehow we have to use this information and arrive at a contradiction to what we absolutely know is true.
Mathematical Induction In proving statements that involve a positive integer n, it is frequently helpful to use the following principle.
Principle of Mathematical Induction Let Sn be a statement about the positive integer n. Suppose that 1. S1 is true. 2. Skϩ1 is true whenever Sk is true.
Then Sn is true for all positive integers n.
This is reasonable because, since S1 is true, it follows from condition 2 (with k ෇ 1) that S2 is true. Then, using condition 2 with k ෇ 2, we see that S3 is true. Again using condition 2, this time with k ෇ 3, we have that S4 is true. This procedure can be followed indeﬁnitely.
In Step 2 a plan was devised. In carrying out that plan we have to check each stage of the plan and write the details that prove that each stage is correct.
Having completed our solution, it is wise to look back over it, partly to see if we have made errors in the solution and partly to see if we can think of an easier way to solve the problem. Another reason for looking back is that it will familiarize us with the method of solution and this may be useful for solving a future problem. Descartes said, “Every problem that I solved became a rule which served afterwards to solve other problems.”
These principles of problem solving are illustrated in the following examples. Before you look at the solutions, try to solve these problems yourself, referring to these Principles of Problem Solving if you get stuck. You may ﬁnd it useful to refer to this section from time to time as you solve the exercises in the remaining chapters of this book.

|||| Understand the problem |||| Draw a diagram

EXAMPLE 1 Express the hypotenuse h of a right triangle with area 25 m2 as a function of its perimeter P.
SOLUTION Let’s ﬁrst sort out the information by identifying the unknown quantity and the data:
Unknown: hypotenuse h Given quantities: perimeter P, area 25 m 2
It helps to draw a diagram and we do so in Figure 1.
h b

FIGURE 1

a

|||| Connect the given with the unknown |||| Introduce something extra

In order to connect the given quantities to the unknown, we introduce two extra variables a and b, which are the lengths of the other two sides of the triangle. This enables us to express the given condition, which is that the triangle is right-angled, by the Pythagorean Theorem:
h2 ෇ a2 ϩ b2

The other connections among the variables come by writing expressions for the area and perimeter:

25

෇

1 2

ab

P෇aϩbϩh

Since P is given, notice that we now have three equations in the three unknowns a, b, and h:

1

h2 ෇ a2 ϩ b2

2

25

෇

1 2

ab

3

P෇aϩbϩh

|||| Relate to the familiar

Although we have the correct number of equations, they are not easy to solve in a straightforward fashion. But if we use the problem-solving strategy of trying to recognize something familiar, then we can solve these equations by an easier method. Look at the right sides of Equations 1, 2, and 3. Do these expressions remind you of anything familiar? Notice that they contain the ingredients of a familiar formula:
͑a ϩ b͒2 ෇ a 2 ϩ 2ab ϩ b 2 Using this idea, we express ͑a ϩ b͒2 in two ways. From Equations 1 and 2 we have
͑a ϩ b͒2 ෇ ͑a 2 ϩ b 2 ͒ ϩ 2ab ෇ h 2 ϩ 4͑25͒
From Equation 3 we have
͑a ϩ b͒2 ෇ ͑P Ϫ h͒2 ෇ P2 Ϫ 2Ph ϩ h 2

Thus

h 2 ϩ 100 ෇ P2 Ϫ 2Ph ϩ h 2

2Ph ෇ P2 Ϫ 100

h ෇ P2 Ϫ 100 2P

This is the required expression for h as a function of P.

|||| Take cases

As the next example illustrates, it is often necessary to use the problem-solving principle of taking cases when dealing with absolute values.

Խ Խ Խ Խ EXAMPLE 2 Solve the inequality x Ϫ 3 ϩ x ϩ 2 Ͻ 11.

SOLUTION Recall the deﬁnition of absolute value:

ͭ ԽxԽ ෇

x Ϫx

if x ജ 0 if x Ͻ 0

It follows that

ͭ Խ x Ϫ 3 Խ ෇

xϪ3 Ϫ͑x Ϫ 3͒

ͭ෇

xϪ3 Ϫx ϩ 3

if x Ϫ 3 ജ 0 if x Ϫ 3 Ͻ 0
if x ജ 3 if x Ͻ 3

Similarly

ͭ Խ x ϩ 2 Խ ෇

xϩ2 Ϫ͑x ϩ 2͒

ͭ෇

xϩ2 Ϫx Ϫ 2

if x ϩ 2 ജ 0 if x ϩ 2 Ͻ 0
if x ജ Ϫ2 if x Ͻ Ϫ2

These expressions show that we must consider three cases:

x Ͻ Ϫ2

Ϫ2 ഛ x Ͻ 3

xജ3

CASE I s If x Ͻ Ϫ2, we have

Խ x Ϫ 3 Խ ϩ Խ x ϩ 2 Խ Ͻ 11
Ϫx ϩ 3 Ϫ x Ϫ 2 Ͻ 11
Ϫ2x Ͻ 10
x Ͼ Ϫ5

CASE II s If Ϫ2 ഛ x Ͻ 3, the given inequality becomes

Ϫx ϩ 3 ϩ x ϩ 2 Ͻ 11 5 Ͻ 11

(always true)

CASE III s If x ജ 3, the inequality becomes
x Ϫ 3 ϩ x ϩ 2 Ͻ 11 2x Ͻ 12 xϽ6

Combining cases I, II, and III, we see that the inequality is satisﬁed when Ϫ5 Ͻ x Ͻ 6. So the solution is the interval ͑Ϫ5, 6͒.

In the following example we ﬁrst guess the answer by looking at special cases and recognizing a pattern. Then we prove it by mathematical induction.
In using the Principle of Mathematical Induction, we follow three steps:
STEP 1 Prove that Sn is true when n ෇ 1. STEP 2 Assume that Sn is true when n ෇ k and deduce that Sn is true when n ෇ k ϩ 1. STEP 3 Conclude that Sn is true for all n by the Principle of Mathematical Induction.

|||| Analogy: Try a similar, simpler problem

EXAMPLE 3 If f0͑x͒ ෇ x͑͞x ϩ 1͒ and fnϩ1 ෇ f0 ‫ ؠ‬fn for n ෇ 0, 1, 2, . . . , ﬁnd a formula for fn͑x͒.

SOLUTION We start by ﬁnding formulas for fn͑x͒ for the special cases n ෇ 1, 2, and 3.
ͩ ͪx
f1͑x͒ ෇ ͑ f0 ‫ ؠ‬f0 ͒ ͑x͒ ෇ f0͑ f0͑x͒͒ ෇ f0 x ϩ 1

x

x

෇

xϩ1

x

x ϩ

1

ϩ

1

෇

xϩ1 2x ϩ 1 xϩ1

෇

x 2x ϩ 1

ͩ ͪx
f2͑x͒ ෇ ͑ f0 ‫ ؠ‬f1 ͒ ͑x͒ ෇ f0͑ f1͑x͒͒ ෇ f0 2x ϩ 1

x

x

෇

2x ϩ 1

x 2x ϩ

1

ϩ

1

෇

2x ϩ 1 3x ϩ 1 2x ϩ 1

෇

x 3x ϩ 1

|||| Look for a pattern

ͩ ͪx
f3͑x͒ ෇ ͑ f0 ‫ ؠ‬f2 ͒ ͑x͒ ෇ f0͑ f2͑x͒͒ ෇ f0 3x ϩ 1

x

x

෇

3x ϩ 1

x 3x ϩ

1

ϩ

1

෇

3x ϩ 1 4x ϩ 1 3x ϩ 1

෇

x 4x ϩ 1

We notice a pattern: The coefﬁcient of x in the denominator of fn͑x͒ is n ϩ 1 in the three cases we have computed. So we make the guess that, in general,

4

fn͑x͒ ෇

x ͑n ϩ 1͒x ϩ 1

To prove this, we use the Principle of Mathematical Induction. We have already veriﬁed that (4) is true for n ෇ 1. Assume that it is true for n ෇ k, that is,

x fk͑x͒ ෇ ͑k ϩ 1͒x ϩ 1

Then

ͩ ͪ x
fkϩ1͑x͒ ෇ ͑ f0 ‫ ؠ‬fk ͒ ͑x͒ ෇ f0͑ fk͑x͒͒ ෇ f0 ͑k ϩ 1͒x ϩ 1

x

x

෇

͑k ϩ 1͒x ϩ 1 ෇

͑k

ϩ

x 1͒ x

ϩ

1

ϩ

1

͑k ϩ 1͒x ϩ 1 ͑k ϩ 2͒x ϩ 1 ͑k ϩ 1͒x ϩ 1

෇

x ͑k ϩ 2͒x ϩ 1

This expression shows that (4) is true for n ෇ k ϩ 1. Therefore, by mathematical induction, it is true for all positive integers n.

PROBLEMS

1. One of the legs of a right triangle has length 4 cm. Express the length of the altitude perpendicular to the hypotenuse as a function of the length of the hypotenuse.
2. The altitude perpendicular to the hypotenuse of a right triangle is 12 cm. Express the length of the hypotenuse as a function of the perimeter.
3. Solve the equation Խ 2 x Ϫ 1 Խ Ϫ Խ x ϩ 5 Խ ෇ 3. 4. Solve the inequality Խ x Ϫ 1 Խ Ϫ Խ x Ϫ 3 Խ ജ 5.
Խ Խ Խ Խ 5. Sketch the graph of the function f ͑x͒ ෇ x 2 Ϫ 4 x ϩ 3 .
Խ Խ Խ Խ 6. Sketch the graph of the function t͑x͒ ෇ x 2 Ϫ 1 Ϫ x 2 Ϫ 4 . Խ Խ Խ Խ 7. Draw the graph of the equation x ϩ x ෇ y ϩ y .
8. Draw the graph of the equation x 4 Ϫ 4 x 2 Ϫ x 2y 2 ϩ 4y 2 ෇ 0.
Խ Խ Խ Խ 9. Sketch the region in the plane consisting of all points ͑x, y͒ such that x ϩ y ഛ 1.
10. Sketch the region in the plane consisting of all points ͑x, y͒ such that
Խx Ϫ yԽ ϩ ԽxԽ Ϫ ԽyԽ ഛ 2
11. Evaluate ͑log2 3͒͑log3 4͒͑log4 5͒ и и и ͑log31 32͒.
12. (a) Show that the function f ͑x͒ ෇ ln(x ϩ sx 2 ϩ 1 ) is an odd function.
(b) Find the inverse function of f.
13. Solve the inequality ln͑x 2 Ϫ 2 x Ϫ 2͒ ഛ 0.
14. Use indirect reasoning to prove that log2 5 is an irrational number.
15. A driver sets out on a journey. For the ﬁrst half of the distance she drives at the leisurely pace of 30 mi͞h; she drives the second half at 60 mi͞h. What is her average speed on this trip?
16. Is it true that f ‫ ͑ ؠ‬t ϩ h͒ ෇ f ‫ ؠ‬t ϩ f ‫ ؠ‬h?
17. Prove that if n is a positive integer, then 7n Ϫ 1 is divisible by 6.
18. Prove that 1 ϩ 3 ϩ 5 ϩ и и и ϩ ͑2n Ϫ 1͒ ෇ n2.
19. If f0͑x͒ ෇ x 2 and fnϩ1͑x͒ ෇ f0͑ fn͑x͒͒ for n ෇ 0, 1, 2, . . . , ﬁnd a formula for fn͑x͒.
1 20. (a) If f0͑x͒ ෇ 2 Ϫ x and fnϩ1 ෇ f0 ‫ ؠ‬fn for n ෇ 0, 1, 2, . . . , ﬁnd an expression for fn͑x͒ and
use mathematical induction to prove it.
; (b) Graph f0, f1, f2, f3 on the same screen and describe the effects of repeated composition.

The idea of a limit is illustrated by secant lines approaching a tangent line.
Limits and Derivatives

In A Preview of Calculus (page 2) we saw how the idea of a limit underlies the various branches of calculus. It is therefore appropriate to begin our study of calculus by investigating limits and their properties. The special type of limit that is used to ﬁnd tangents and velocities gives rise to the central idea in differential calculus, the derivative.

|||| 2.1 The Tangent and Velocity Problems

In this section we see how limits arise when we attempt to ﬁnd the tangent to a curve or the velocity of an object.

Locate tangents interactively and explore them numerically.
Resources / Module 1 / Tangents / What Is a Tangent?

The Tangent Problem
The word tangent is derived from the Latin word tangens, which means “touching.” Thus, a tangent to a curve is a line that touches the curve. In other words, a tangent line should have the same direction as the curve at the point of contact. How can this idea be made precise?
For a circle we could simply follow Euclid and say that a tangent is a line that intersects the circle once and only once as in Figure 1(a). For more complicated curves this deﬁnition is inadequate. Figure l(b) shows two lines l and t passing through a point P on a curve C. The line l intersects C only once, but it certainly does not look like what we think of as a tangent. The line t, on the other hand, looks like a tangent but it intersects C twice.

t P

t

C

l

FIGURE 1

(a)

(b)

y=≈

y Q{x, ≈} t

P(1, 1)

0

x

FIGURE 2

To be speciﬁc, let’s look at the problem of trying to ﬁnd a tangent line t to the parabola y ෇ x 2 in the following example.
EXAMPLE 1 Find an equation of the tangent line to the parabola y ෇ x 2 at the point P͑1, 1͒.
SOLUTION We will be able to ﬁnd an equation of the tangent line t as soon as we know its slope m. The difﬁculty is that we know only one point, P, on t, whereas we need two points to compute the slope. But observe that we can compute an approximation to m by choosing a nearby point Q͑x, x 2 ͒ on the parabola (as in Figure 2) and computing the slope mPQ of the secant line PQ.
We choose x 1 so that Q P. Then
x2 Ϫ 1 mPQ ෇ x Ϫ 1

x 2 1.5 1.1 1.01 1.001
x 0 0.5 0.9 0.99 0.999
y
0

mPQ 3 2.5 2.1 2.01 2.001
mPQ 1 1.5 1.9 1.99 1.999
Q
P

For instance, for the point Q͑1.5, 2.25͒ we have

mPQ

෇

2.25 Ϫ 1 1.5 Ϫ 1

෇

1.25 0.5

෇

2.5

The tables in the margin show the values of mPQ for several values of x close to 1. The closer Q is to P, the closer x is to 1 and, it appears from the tables, the closer mPQ is to 2. This suggests that the slope of the tangent line t should be m ෇ 2.
We say that the slope of the tangent line is the limit of the slopes of the secant lines, and we express this symbolically by writing

x2 Ϫ 1

lim mPQ ෇ m
Q lP

and

lim
x l 1

xϪ1

෇2

Assuming that the slope of the tangent line is indeed 2, we use the point-slope form of the equation of a line (see Appendix B) to write the equation of the tangent line through ͑1, 1͒ as

y Ϫ 1 ෇ 2͑x Ϫ 1͒ or y ෇ 2x Ϫ 1

Figure 3 illustrates the limiting process that occurs in this example. As Q approaches P along the parabola, the corresponding secant lines rotate about P and approach the tangent line t.

y

y

t

t

t

Q

Q

P

P

x

0

x

0

x

Q approaches P from the right

y

y

t

t

y t

Q

P

0

x

P Q

0

x

P

Q

0

x

FIGURE 3
In Module 2.1 you can see how the process in Figure 3 works for ﬁve additional functions.

Q approaches P from the left
Many functions that occur in science are not described by explicit equations; they are deﬁned by experimental data. The next example shows how to estimate the slope of the tangent line to the graph of such a function.

t

Q

0.00

100.00

0.02

81.87

0.04

67.03

0.06

54.88

0.08

44.93

0.10

36.76

EXAMPLE 2 The ﬂash unit on a camera operates by storing charge on a capacitor and releasing it suddenly when the ﬂash is set off. The data at the left describe the charge Q remaining on the capacitor (measured in microcoulombs) at time t (measured in seconds after the ﬂash goes off ). Use the data to draw the graph of this function and estimate the slope of the tangent line at the point where t ෇ 0.04. [Note: The slope of the tangent line represents the electric current ﬂowing from the capacitor to the ﬂash bulb (measured in microamperes).]
SOLUTION In Figure 4 we plot the given data and use them to sketch a curve that approximates the graph of the function.

Q 100

90

80

A

70

60

50

B

P C

FIGURE 4

0

0.02

0.04

0.06

0.08

0.1

t

R
(0.00, 100.00) (0.02, 81.87) (0.06, 54.88) (0.08, 44.93) (0.10, 36.76)

mPR
Ϫ824.25 Ϫ742.00 Ϫ607.50 Ϫ552.50 Ϫ504.50

|||| The physical meaning of the answer in Example 2 is that the electric current ﬂowing from the capacitor to the ﬂash bulb after 0.04 second is about –670 microamperes.

Given the points P͑0.04, 67.03͒ and R͑0.00, 100.00͒ on the graph, we ﬁnd that the slope of the secant line PR is

100.00 Ϫ 67.03 mPR ෇ 0.00 Ϫ 0.04 ෇ Ϫ824.25

The table at the left shows the results of similar calculations for the slopes of other secant lines. From this table we would expect the slope of the tangent line at t ෇ 0.04 to lie somewhere between Ϫ742 and Ϫ607.5. In fact, the average of the slopes of the two closest secant lines is

1 2

͑Ϫ742

Ϫ

607.5͒

෇

Ϫ674.75

So, by this method, we estimate the slope of the tangent line to be Ϫ675. Another method is to draw an approximation to the tangent line at P and measure the
sides of the triangle ABC, as in Figure 4. This gives an estimate of the slope of the tangent line as

Ϫ

Խ AB Խ Խ BC Խ

Ϸ

Ϫ

80.4 0.06

Ϫ Ϫ

53.6 0.02

෇

Ϫ670

The Velocity Problem
If you watch the speedometer of a car as you travel in city trafﬁc, you see that the needle doesn’t stay still for very long; that is, the velocity of the car is not constant. We assume from watching the speedometer that the car has a deﬁnite velocity at each moment, but how is the “instantaneous” velocity deﬁned? Let’s investigate the example of a falling ball.

90 ❙❙❙❙ CHAPTER 2 LIMITS AND DERIVATIVES

The CN Tower in Toronto is currently the tallest freestanding building in the world.

EXAMPLE 3 Suppose that a ball is dropped from the upper observation deck of the CN Tower in Toronto, 450 m above the ground. Find the velocity of the ball after 5 seconds.
SOLUTION Through experiments carried out four centuries ago, Galileo discovered that the distance fallen by any freely falling body is proportional to the square of the time it has been falling. (This model for free fall neglects air resistance.) If the distance fallen after t seconds is denoted by s͑t͒ and measured in meters, then Galileo’s law is expressed by the equation
s͑t͒ ෇ 4.9t 2

The difﬁculty in ﬁnding the velocity after 5 s is that we are dealing with a single instant of time ͑t ෇ 5͒, so no time interval is involved. However, we can approximate the desired quantity by computing the average velocity over the brief time interval of a tenth of a second from t ෇ 5 to t ෇ 5.1:

average velocity ෇ distance traveled time elapsed

෇ s͑5.1͒ Ϫ s͑5͒ 0.1

4.9͑5.1͒2 Ϫ 4.9͑5͒2

෇

෇ 49.49 m͞s

0.1

The following table shows the results of similar calculations of the average velocity over successively smaller time periods.

Time interval
5ഛtഛ6 5 ഛ t ഛ 5.1 5 ഛ t ഛ 5.05 5 ഛ t ഛ 5.01 5 ഛ t ഛ 5.001

Average velocity (m͞s)
53.9 49.49 49.245 49.049 49.0049

It appears that as we shorten the time period, the average velocity is becoming closer to 49 m͞s. The instantaneous velocity when t ෇ 5 is deﬁned to be the limiting value of these average velocities over shorter and shorter time periods that start at t ෇ 5. Thus, the (instantaneous) velocity after 5 s is

v ෇ 49 m͞s

You may have the feeling that the calculations used in solving this problem are very similar to those used earlier in this section to ﬁnd tangents. In fact, there is a close connection between the tangent problem and the problem of ﬁnding velocities. If we draw the graph of the distance function of the ball (as in Figure 5) and we consider the points P͑a, 4.9a 2 ͒ and Q͑a ϩ h, 4.9͑a ϩ h͒2 ͒ on the graph, then the slope of the secant line PQ is

mPQ

෇

4.9͑a ϩ h͒2 Ϫ 4.9a 2 ͑a ϩ h͒ Ϫ a

SECTION 2.1 THE TANGENT AND VELOCITY PROBLEMS ❙❙❙❙ 91

which is the same as the average velocity over the time interval ͓a, a ϩ h͔. Therefore, the velocity at time t ෇ a (the limit of these average velocities as h approaches 0) must be equal to the slope of the tangent line at P (the limit of the slopes of the secant lines).

s s=4.9t @

s s=4.9t @

Q

slope of secant line ϭ average velocity
P

slope of tangent P ϭ instantaneous velocity

0

a a+h

t

0

a

t

FIGURE 5

Examples 1 and 3 show that in order to solve tangent and velocity problems we must be able to ﬁnd limits. After studying methods for computing limits in the next ﬁve sections, we will return to the problems of ﬁnding tangents and velocities in Section 2.7.

|||| 2.1 Exercises

1. A tank holds 1000 gallons of water, which drains from the bottom of the tank in half an hour. The values in the table show the volume V of water remaining in the tank (in gallons) after t minutes.

t (min)

5

10

15

20 25 30

V (gal) 694 444 250 111 28 0

(a) If P is the point ͑15, 250͒ on the graph of V, ﬁnd the slopes of the secant lines PQ when Q is the point on the graph with t ෇ 5, 10, 20, 25, and 30.
(b) Estimate the slope of the tangent line at P by averaging the slopes of two secant lines.
(c) Use a graph of the function to estimate the slope of the tangent line at P. (This slope represents the rate at which the water is ﬂowing from the tank after 15 minutes.)
2. A cardiac monitor is used to measure the heart rate of a patient after surgery. It compiles the number of heartbeats after t minutes. When the data in the table are graphed, the slope of the tangent line represents the heart rate in beats per minute.

t (min) Heartbeats

36 2530

38 2661

40 2806

42 2948

44 3080

The monitor estimates this value by calculating the slope of a secant line. Use the data to estimate the patient’s heart rate

after 42 minutes using the secant line between the points with

the given values of t.

(a) t ෇ 36 and t ෇ 42

(b) t ෇ 38 and t ෇ 42

(c) t ෇ 40 and t ෇ 42

(d) t ෇ 42 and t ෇ 44

What are your conclusions?

3.

The

point

P

(1,

1 2

)

lies

on

the

curve

y

෇

x͑͞1

ϩ

x͒.

(a) If Q is the point ͑x, x͑͞1 ϩ x͒͒, use your calculator to ﬁnd

the slope of the secant line PQ (correct to six decimal

places) for the following values of x:

(i) 0.5 (iii) 0.99 (v) 1.5 (vii) 1.01

(ii) 0.9 (iv) 0.999 (vi) 1.1 (viii) 1.001

(b) Using the results of part (a), guess the value of the slope of

the

tangent

line

to

the

curve

at

P(1,

)1
2

.

(c) Using the slope from part (b), ﬁnd an equation of the

tangent

line

to

the

curve

at

P(1,

)1
2

.

4. The point P͑2, ln 2͒ lies on the curve y ෇ ln x. (a) If Q is the point ͑x, ln x͒, use your calculator to ﬁnd the slope of the secant line PQ (correct to six decimal places) for the following values of x:

(i) 1.5 (iii) 1.99 (v) 2.5 (vii) 2.01

(ii) 1.9 (iv) 1.999 (vi) 2.1 (viii) 2.001

(b) Using the results of part (a), guess the value of the slope of the tangent line to the curve at P͑2, ln 2͒.

92 ❙❙❙❙ CHAPTER 2 LIMITS AND DERIVATIVES

(c) Using the slope from part (b), ﬁnd an equation of the tangent line to the curve at P͑2, ln 2͒.
(d) Sketch the curve, two of the secant lines, and the tangent line.

5. If a ball is thrown into the air with a velocity of 40 ft͞s, its height in feet after t seconds is given by y ෇ 40t Ϫ 16t 2. (a) Find the average velocity for the time period beginning when t ෇ 2 and lasting
(i) 0.5 second (ii) 0.1 second
(iii) 0.05 second (iv) 0.01 second (b) Find the instantaneous velocity when t ෇ 2.

6. If an arrow is shot upward on the moon with a velocity of 58 m͞s, its height in meters after t seconds is given by h ෇ 58t Ϫ 0.83t 2.

(a) Find the average velocity over the given time intervals:

(i) [1, 2]

(ii) [1, 1.5]

(iii) [1, 1.1]

(iv) [1, 1.01]

(v) [1, 1.001]

(b) Find the instantaneous velocity after one second.

7. The displacement (in feet) of a certain particle moving in a straight line is given by s ෇ t 3͞6, where t is measured in

seconds.

(a) Find the average velocity over the following time periods:

(i) [1, 3]

(ii) [1, 2]

(iii) [1, 1.5]

(iv) [1, 1.1]

(b) Find the instantaneous velocity when t ෇ 1.

(c) Draw the graph of s as a function of t and draw the secant lines whose slopes are the average velocities found in part (a).
(d) Draw the tangent line whose slope is the instantaneous velocity from part (b).
8. The position of a car is given by the values in the table.

t (seconds) 0 1

23

4

5

s (feet)

0 10 32 70 119 178

(a) Find the average velocity for the time period beginning when t ෇ 2 and lasting
(i) 3 seconds (ii) 2 seconds (iii) 1 second
(b) Use the graph of s as a function of t to estimate the instantaneous velocity when t ෇ 2.
9. The point P͑1, 0͒ lies on the curve y ෇ sin͑10␲͞x͒. (a) If Q is the point ͑x, sin͑10␲͞x͒͒, ﬁnd the slope of the secant line PQ (correct to four decimal places) for x ෇ 2, 1.5, 1.4, 1.3, 1.2, 1.1, 0.5, 0.6, 0.7, 0.8, and 0.9. Do the slopes appear to be approaching a limit?
; (b) Use a graph of the curve to explain why the slopes of the secant lines in part (a) are not close to the slope of the tangent line at P.
(c) By choosing appropriate secant lines, estimate the slope of the tangent line at P.

|||| 2.2 The Limit of a Function

Having seen in the preceding section how limits arise when we want to ﬁnd the tangent to a curve or the velocity of an object, we now turn our attention to limits in general and numerical and graphical methods for computing them.
Let’s investigate the behavior of the function f deﬁned by f ͑x͒ ෇ x 2 Ϫ x ϩ 2 for values of x near 2. The following table gives values of f ͑x͒ for values of x close to 2, but not equal to 2.

x
1.0 1.5 1.8 1.9 1.95 1.99 1.995 1.999

f ͑x͒
2.000000 2.750000 3.440000 3.710000 3.852500 3.970100 3.985025 3.997001

x
3.0 2.5 2.2 2.1 2.05 2.01 2.005 2.001

f ͑x͒
8.000000 5.750000 4.640000 4.310000 4.152500 4.030100 4.015025 4.003001

From the table and the graph of f (a parabola) shown in Figure 1 we see that when x is close to 2 (on either side of 2), f ͑x͒ is close to 4. In fact, it appears that we can make the

||||
FIGURE 3
FIGURE 4 ||||
lim x 0 x ෇ 0

SECTION 2.3 CALCULATING LIMITS USING THE LIMIT LAWS 109
Some limits are best calculated by rst nding the left- and right-hand limits. The following theorem is a reminder of what we discovered in Section 2.2. It says that a two-sided limit exists if and only if both of the one-sided limits exist and are equal.

1 Theorem

lim f x ෇ L if and only if lim f x ෇ L ෇ lim f x

xa

xa

xa

When computing one-sided limits, we use the fact that the Limit Laws also hold for one-sided limits.

EXAMPLE 7 Show that lim x ෇ 0. x0
SOLUTION Recall that x ෇ x if x 0 x if x 0

Since x ෇ x for x 0, we have

lim x ෇ lim x ෇ 0

x0

x0

For x 0 we have x ෇ x and so

lim x ෇ lim x ෇ 0

x0

x0

Therefore, by Theorem 1,

lim x ෇ 0
x0

x EXAMPLE 8 Prove that lim does not exist.
x0 x

SOLUTION

lim x ෇ lim x ෇ lim 1 ෇ 1

x0 x

x0x x0

x

x

lim ෇ lim ෇ lim

x0 x

x0 x

x0

1෇ 1

Since the right- and left-hand limits are different, it follows from Theorem 1 that lim x 0 x x does not exist. The graph of the function f x ෇ x x is shown in Figure 4 and supports the one-sided limits that we found.

EXAMPLE 9 If

f x ෇ x 4 if x 4 8 2x if x 4

determine whether lim x 4 f x exists. SOLUTION Since f x ෇ x 4 for x 4, we have

lim f x ෇ lim x 4 ෇ 4 4 ෇ 0

x4

x4

110 CHAPTER 2 LIMITS AND DERIVATIVES

FIGURE 5

||||

x

x

x

Since f x ෇ 8 2x for x 4, we have

lim f x ෇ lim 8 2x ෇ 8 2 4 ෇ 0

x4

x4

The right- and left-hand limits are equal. Thus, the limit exists and lim f x ෇ 0
x4
The graph of f is shown in Figure 5.

EXAMPLE 10 The greatest integer function is de ned by x ෇ the largest integer

that is less than or equal to x. (For instance, 4 ෇ 4, 4.8 ෇ 4, ෇ 3, 2 ෇ 1,

1 2

෇

1.) Show that lim x 3 x does not exist.

SOLUTION The graph of the greatest integer function is shown in Figure 6. Since x ෇ 3 for 3 x 4, we have

lim x ෇ lim 3 ෇ 3

x3

x3

Since x ෇ 2 for 2 x 3, we have

lim x ෇ lim 2 ෇ 2

x3

x3

Because these one-sided limits are not equal, lim x 3 x does not exist by Theorem 1.

FIGURE 6 Greatest integer function

The next two theorems give two additional properties of limits. Their proofs can be found in Appendix F.

2 Theorem If f x

x when x is near a (except possibly at a) and the limits

of f and both exist as x approaches a, then

lim f x lim x

xa

xa

FIGURE 7

3 The Squeeze Theorem If f x at a) and

x h x when x is near a (except possibly

lim f x ෇ lim h x ෇ L

xa

xa

then

lim x ෇ L

xa

The Squeeze Theorem, which is sometimes called the Sandwich Theorem or the Pinching Theorem, is illustrated by Figure 7. It says that if x is squeezed between f x and h x near a, and if f and h have the same limit L at a, then is forced to have the same limit L at a.

SECTION 2.3 CALCULATING LIMITS USING THE LIMIT LAWS 111

EXAMPLE 11 Show that lim x 2 sin 1 ෇ 0.

x0

x

SOLUTION First note that we cannot use

lim x 2 sin 1 ෇ lim x 2

1 lim sin

x0

x x0

x0 x

because lim x 0 sin 1 x does not exist (see Example 4 in Section 2.2). However, since

1 1 sin 1
x

we have, as illustrated by Figure 8,

x 2 x 2 sin 1 x 2 x

sin

FIGURE 8

We know that

lim x 2 ෇ 0 and lim x 2 ෇ 0

x0

x0

Taking f x ෇ x 2, x ෇ x 2 sin 1 x , and h x ෇ x 2 in the Squeeze Theorem, we obtain

lim x 2 sin 1 ෇ 0

x0

x

2.3 Exercises

1. Given that

lim f x ෇ 3 lim x ෇ 0 lim h x ෇ 8

x a

x a

x a

nd the limits that exist. If the limit does not exist, explain

why.

(a) lim f x h x x a

(b) lim f x 2 x a

(c) lim 3 h x xa fx
(e) lim x a hx fx
(g) lim xa x

1 (d) lim
x af x x
(f ) lim x af x 2f x
(h) lim x ahx f x