44
CHAPTER 1
13
ec4bn
FUNCflONS AND MODaS
4. The graph off is given. Draw the grap
hs of the following
functions.
(a) y = f(s)
2
(h) = f(s
2)
(c) y = —2f(x)
(d) v = [(Is) + I
15. v
sin(s/2)
=
16.
V
=
—
—
—
HT
f
(c) v
=
23. v
i
is given. Use it to graph the Ibllowing
f(—s)
+ 8.v)
18.
(s(
20. v
I +
(b) v
=
(d)
=
y
s— 2!
=
=
24. v
—
2
1/’
tan.v
22. v
—
4
—-
=
—
cos
25. The city of New Orleans is locat
ed at latitude 30°N. Use Fig
f( x)
-f(--s)
¶.!_I [\.J,.J—’
LL Li L
6—7 The graph of y = x/3.v
2
is given. Use transformations to
create a function whose graph is as show
n.
—
1.5
19. v
21. v
L
functions.
(a) v = f(2s)
=
= I(.2
+ 3
-
H
5. The graph of
17. y
ure 9 to find a function that models the num
ber of hours of
daylight at New Orleans as a function of
the time of year. To
check the accuracy of your model, use the
fact that on March 31
the sun rises at 5:51 AM and sets at 6:18
PM in New Orleans.
25. A variable star is one whose brightness
alternately increases
and decreases. For the most visible varia
ble star, Delta Cephei,
the time between periods of maximum brigh
tness is 5.4 days,
the average brightness (or magnitude) of
the star is 4.0, and its
brightness varies by ±0.35 magnitude. Find
a function that
models the brightness of Delta C’ephei as
a function of time.
27. (a) How is the graph of v
=f(jxl) related to the graph off?
(b) Sketch the graph of y = sin s
(c) Sketch the graph of v
=
2
v=3
s—v
28. Use the given graph off to sketc
h the graph of v
= I/f(s).
Which features of f are the most impo
rtant in sketching
r
I/f(s)? Explain how they are used.
0
V
6.
7.
yA
-1U
0
—1
8. (a) How is the graph of v = 2 sins relat
ed to the graph of
v = sin x? Use your answer and Figure 6 to
sketch the
graph of v= 2sinx.
(h) How is the graph of ‘3 = i +
related to the graph of
s = I? Use your answer and Figure 4(a) to
sketch the
graph of v
I +
9—24 Graph the function by hand, not
by plotting points, but by
starting with the graph of one of the stand
ard functions given in Sec
tion 1.2, and then applying the appropriate
transformations.
9. v= —.r
3
10. v = I
—
11.
v
=
(s + I)
13. v= 1 + 2cos.v
12. v
=
2
—
4s + 3
14. v4sin3s
7
—2.5
29—30 Find (a)
domains.
29. f(s)
30. f(s)
f+
g, (b)
,
2
s + 2x
.f
()
—
g. (c) fg. and (d) f/f, and state their
2
3v
Lv
q(s)
=
NJ
0
I
—
—
I
31—36 Find the functions (a) fog, (h)
g
of,
and their domains.
31. f(s)
I.
—
32. f(s)
s
33. f(s)
I
—
—
34.fc)/,
2,
3s,
g(s)
g(s)
g(x)
=
=
=
2s + I
2 + 3s + 4
v
cos s
9
=
./iEi
(c)
f of, and (d) gog
Ho
35. f(s)
+
5
36. f(.v)
=
37—40 Find
pC)
I
fog
=
sin 2s
Ii.
2s
39. f(s)
‘s—3.
I,
—
h(s)=s— I
gC)2s,
38. f(s)
=
v + 2
45
‘
37.f(s)s+l,
40. f(s)
=
NEW FUNCTIONS FROM OED FUNCTIONS
52. Use the given graphs off and p to estimate the value of
5. Use these estimates to
f(g(s)) for .v = —5, —4, —3
sketch a rough graph of fo p.
s+l
p(s)
--,
SECTION 1.3
tans,
gC)
g(s)
.2
=
=
h(s)
I
=
s
s’ ± 2
h(s)
2
u
—
53. A stone is dropped into a lake, creating a circular ripple that
travels outward at a speed of 60 cm/s.
(a) Express the radius r of this circle as a function of the
time ((in seconds).
(b) tf A is the area of this circle as a function of the radius, find
A r and interpret it.
h(s)
g(s)
=
41—46 Express the function in the form
Jo j.
°
(2s +
41. F(s)
43. F(s)
2)4
42. F(s)
‘t
=
44. G(s)
‘
l+’s
()
47—49 Express the function in the form
fo q
u(t)
=
=
I
49. H(s)
=
sec(t)
—
=
l+s
I + tan 1
°
48. H(s)
3’
47. H(s)
6.
=
50. Use the table to evaluate each expression.
(a) f(q( I))
(h) g(f( I))
(e) (p of)(3)
(d) p(p(l))
F:
s
2
cos
tan
46.
45.
=
/2 ± s
(c) f(fW)
(f) (fop)(6)
I
I
.r
I
2
3
4
5
6
f(s)
3
I
4
2
2
5
g(s)
6
2
I
2
51. Use the given graphs of f. and g to evaluate each expression,
or explain why it is undefined.
(c) (Jo g)(0)
(a) f(g(2))
(b) g(f(0))
(f) (fof)(4)
(e) (go g)(—
(d) (p of)(6)
)
2
El I
I
54. A spherical balloon is being inflated and the radius of the bal
loon is increasing at a rate of 2 cm/s.
(a) Express the radius r of the balloon as a function of the
time I (in seconds).
(h) If V is the volume of the balloon as a function of the radius,
find V o r and interpret it.
55. A ship is moving at a speed of 30 km/h parallel to a straight
shoreline. The ship is 6 km from shore and it passes a light
house at noon.
(a) Express the distance s between the lighthouse and the ship
as a function of d, the distance the ship has traveled since
noon: that is, lind f so that s =f(d).
(b) Express a’ as a function oft, the time elapsed since noon:
that is.,t-nd p so that a’ =
(c) Find Jo p. What does this function represent?
56. An airplane is flying at a speed of 350 mi/h at an altitude of
one mile and passes directly over a radar station at time I = 0.
(a) Express the horizontal distance a’ (in miles) that the plane
has flown as a function oft.
(b) Express the distance s between the plane and the radar
station as a function of a’.
(c) Use composition to express s as a function oft.
57. The Heaviside function H is defined by
H(r)
=
lo
if t<0
l
ift0
It is used in the study of electric circuits to represent the
sudden surge of electnc current, or voltage, when a switch is
instantaneously turned on.
(a) Sketch the graph of the Heaviside function.
(h) Sketch the graph of the voltage VII) in a circuit if the
switch is turned on at time I = 0 and 120 volts are applied
instantaneously to the circuit. Write a formula for V(t) in
terms of H(r).
(c) Sketch the graph of the voltage ‘(t) in a circuit if the
switch is turned on at time I = 5 seconds and 240 volts are
CHAPTER 1
46
FUNCTIONS AND MODELS
C I’
-(-1H3y
applied instantaneously to the circuit. Write a formula for
V(t) in terms of H(t). (Note that starting at t = 5 corre
sponds to a translation.)
IL
58. The Heaviside function defined in Exercise 57 can also be used
to define the ramp function s = ctH(r). which represents a
gradual increase in voltage or current in a circuit.
(a) Sketch the graph of the ramp function)’ = tH(t).
(h) Sketch the graph of the voltage V(t) in a circuit if the
switch is turned on at time r = 0 and the voltage is gradu
ally increased to 120 volts over a 60-second time interval.
Write a formula for 17(r) in terms of H(t) for r 60.
(c) Sketch the graph of the voltage V(r) 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 fonnLlla for V(r) in terms of H(t) for t
32.
59. Let
f and g be linear functions
with equations f(s) = ,n
x + h
1
. Is Jo g also a linear function? If so,
2
ins + b
and g(x) =
what is the slope of its graph?
I
1)
60. If you invest s dollars at 4% interest compounded annually,
then the amount A(s) of the investment after one year is
A(s) = l.04s. Find A °A, A A oA, and A cA oA oA. What
do these compositions represent? Find a formula for the com
position of ii copies of A.
61. (a) If y(v)
= 4v
2 -I- 4x + 7, find a function
A. (Think about what operations you
would have to perform on the formula for g to end up with
the formula for Ii.)
(b) If f(s) = 3s + 5 and h(s) = 3s + 3.v + 2, find a function
g such that Jo g = Ii.
=
2.v + I and Ii(v)
f such that Jo g
62. If f(s)
=
of=
A.
O
s +
=
4 and h(s)
=
4s
—
I, find a function p such that
63. Suppose p is an even function and let h =j’o p. Is Ii always an
even function?
64. Suppose p is an odd function and let A =fo p. Is A always an
odd function? What if f is odd? What if f is even?
Graphrng CalcWators and Computers
(a,d)
I
=
y=d
a
(a,c)
(b,d)
I
y
=
C
=
(b,c)
[a, b] X [c, ci] = {(x, v)
FIGURE 1
The viewing rectangle [a, 1] by
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 com
plicated 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 fLinc
tion 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 s-values to range from a mini
mum value of X,nin = a to a maximum valLie of Xmas = band the y-values to range from
a minimum of Yinin = c to a maximum of Ymax = d, then the visible portion of the graph
lies in the rectangle
[c, d]
a
x
b, c
v
d}
shown in Figure 1. We refer to this rectangle as the [a, b] by [c, d] viewing rectangle.
The machine draws the graph of a function f much as you would. It plots points of the
form (x,f(x)) for a certain number of equally spaced values of x between a and b. If an
s-value is not in the domain of f, or if f(s) lies outside the viewing rectangle, it moves on
to the next s-value. The machine connects each point to the preceding plotted point to form
a representation of the graph of f.
EXAMPLE 1 Choosing a good viewing rectangle Draw the graph of the function
f(s) = .s2 + 3 in each of the following viewing rectangles.
(a) [—2, 2] by [—2, 2]
(b) [—4. 4] by [—4, 4]
(c) [—10, 10] by [—5, 30]
(d) [—50, 50] by [—100, 1000]
SECTION 1.5
EXPONENTIAL FUNCTIONS
59
EXAMPLE 5 Exponential functions get big fast Use a graphing device to find the values of
x for which e’> 1,000,000.
SOLUTION In Figure 16 we graph both the function v = e’ and the horizontal line
v = 1,000,000. We see that these curves intersect when x
13.8. Thus e’ > 106 when
>
13.8.
It is perhaps surprising that the values of the exponential function have already
.v
surpassed a million when x is only 14.
1.5 X 106
15
FIGURE 16
I
1—4 Use the Law of Exponents to rewrite and simplify the
expression.
43
I
q’
(b)
1. (a)
.5
2. (a)
(h) s.s)
3
3. (a) 4
(2b)
8
b
(b)
(6y2)
4. (a)
5—”
(h)
5. (a) Write an equation that defines the exponential function
with base a > 0.
(b) What is the domain of this function?
(c) If a 0 1, what is the range of this function?
(d) Sketch the general shape of the graph of the exponential
function for each of the following cases.
(i) a> 1
(iii) 0< (1< 1
(ii) a = I
6. (a) How is the number e defined?
(b) What is an approximate value for e?
(c) What is the natural exponential function?
7—10 Graph the given functions on a common screen. 1-low are
these graphs related?
7. v
=
2’,
S.ve’,
9.
v
=
3’
v
=
e’,
v’e’,
y
=
10’,
v
=
5’,
v
i=8’,
=
20’
10. v
=
()‘,
v
0.6’,
=
0.3’,
v
=
=
102
12. v
13. v
=
26
14. v
15. y
=
I
16. v
—
e’
0.1’
=
(0.5) —2
=
2(1
—
e)
= e’, write the equation of the
graph that results from
(a) shifting 2 units downward
(b) shifting 2 units to the right
(c) reflecting about the .r-axis
(d) reflecting about the v-axis
(e) reflecting about the .v-axis and then about the v-axis
17. Starting with the graph of y
18. Starting with the graph of v = e’, find the equation of the
graph that results from
(a) reflecting about the line v = 4
(b) reflecting about the line x = 2
19—20 Find the domain of each function.
19. (a) f(s)
=
g(t)
I
l—e
=
1 + s
-
sin(e’)
(j)’
Graphing calculator or computer with graphing 6oftware required
v
11. v
yS’
v=
=
11—16 Make a rough sketch of the graph of the function. Do not
use a calculator. Just use the graphs given in Figures 3 and 13
and, if necessary, the transformations of Section 1.3.
20. (a)
V
0.96,
1. Homework Hints available in TEC
(b) f(s)
(h) g(t)
=
=
/l
—
2’
CHAPTER 1
60
FUNCTIONS AND MODELS
21—22 Find the exponential function f(s)
is given.
=
Ca’ whose graph
21.
3D. A bacterial culture starts with 500 bacteria and doubles in
size every half hour.
(a) How many bacteria are there after 3 hours?
(b) How many bacteria are there alter I hours’?
(c) How many bacteria are there alier 40 minutes?
(d) Graph the population function and estimate the time for
the population to reach 100,000.
3.24i
-v
0
31. The half-life of bismuth-210, 2
°Bi, is 5 days.
(a) If a sample has a mass of 200 mg, find the amount
remaining after 15 days.
(b) Find the amount remaining after r days.
(c) Estimate the amount remaining after 3 weeks.
(d) Use a graph to estimate the time required for the mass to
be reduced to 1 mg.
22.
14
(1.
3
23. If f(s)
=
(d) Graph the population function and estimate the time for
the population to reach 50,000.
32. An isotope of sodium, 24
Na. has a half-life of 15 hours.
A sample of this isotope has mass 2 g.
(a) Find the amount remaining after 60 hours.
(b) Find the amount remaining after t hours.
(c) Estimate the amount remaining alter 4 days.
(d) Use a graph to estimate the time required for the mass to
be reduced to 0.01 g.
5’, show that
f(s + h) —f(x)
=
5(5
—
I)
24. Suppose you are offered ajob that lasts one month. Which
of the following methods of payment do you prefer’?
1. One million dollars at the end of the month.
11. One cent on the first day of the month, two cents on the
second day, four cents on the third day, and, in general,
2” cents on the oth day.
25. Suppose the graphs of f() = .v
2 and 9
C) = 2 are drawn
on a coordinate grid where the unit of measurement is
1 inch. Show that, at a distance 2 ft to the right of the origin,
the height of the graph off is 48 ft but the height of the
graph of g is about 265 mi.
26. Compare the functions f(x) = x
5 and g(s) = 5’ by graphing
both functions in several viewing rectangles. Find all points
of intersection of the graphs correct to one decimal place.
Which function grows more rapidly when s is large?
and g(s) = a’ by graph
27. Compare the functions f(s) =
ing both f and p in several viewing rectangles. When does
the graph of p finally surpass the graph off?
33. Use a graphing calculator with exponential regression capa
bility to model the population of the world with the data
from 1950 to 2000 in Table I on page 56. Use the model to
estimate the population in 1993 and to predict the population
in the year 2010.
34. The table gives the population of the United States, in mil
lions, for the years 1900—2000. Use a graphing calculator
with exponential regression capability to model the US pop
ulation since 1900. Use the model to estimate the population
in 1925 and to predict the population in the years 2010 and
2020.
Year
Population
29. Under ideal conditions a certain bacteria population is
known to double every three hours. Suppose that there are
initially 100 bacteria.
(a) What is the size of the population after 15 hours’?
(b) What is the size of the population after t hours?
(c) Estimate the size of the population after 20 hours.
Population
76
1960
179
1910
1920
1930
1940
1950
92
106
123
131
150
1970
1980
1990
2000
203
227
250
281
35. If you graph the function
f(.v)
28. Use a graph to estimate the values of x such that
e’ > 1,000,000,000.
Year
1900
youll see that
I
’
1
e’
l+e
—
=
f, appears
to be an odd function. Prove it.
36. Graph several members of the family of functions
f(s)
=
I +
’
5
ie
where > 0. How does the graph change when b changes?
How does it change when a changes?