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Unformatted text preview: 20 CHAPTER1 FUNCTlONS AND LIMITS 7—10 Determine whether the curve is the graph of a function of x.
If it is, state the domain and range of the function. 7. y 8.F M “ll
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+ 11. The graph shown gives the weight of a certain person as a
function of age. Describe in words how this person’s weight
varies over time. What do you think happened when this
person was 30 years old? 200
weight 150
d (poun s) .100
50 1 1 t «1 1 l t + 0 10 20 30 40 50 60 70 age (years) 12. The graph shows the height of the water in a bathtub as a
function of time. Give a verbal description of what you think
happened. height
(inches)
15 10
5 0 5 10 15 time
(min) 13. ‘You put some ice cubes in a glass, ﬁll the glass with cold
water, and then let the glass sit on a table. Describe how the
temperature of the water changes as time passes. Then sketch a
rough graph of the temperature of the water as a function of the
elapsed time. 14. Three runners compete in a lOOmeter race. The graph depicts
the distance run as a function of time for each runner. Describe in words What the graph tells you about this race. Who won the
race? Did each runner ﬁnish the race? y (In) 20 2(5) 15. The graph shows the power consumption for a clay in Septem
ber in San Francisco. (P is measured in megawatts; t is mea
sured in hours starting at midnight.) (a) What was the power consumption at 6 AM? At 6 PM?
(b) When was the power consumption the lowest? When was it
the highest? Do these times seem reasonable? P+— 1 f
300 a 1 L r f“
7' ‘ A
600 j i
.l 4
400 at
J t .7.
200 l i .p .
1 7'
0 3 6 9 12 15 18 21 ? Paciﬁc Gas & Electric 16. Sketch a rough graph of the number of hours of daylight as a
function of the time of year. 1']. Sketch a rough graph of the outdoor temperature as a function
of time during a typical spring day. 18. Sketch a rough graph of the market value of a new car as a
function of time for a period of 20 years. Assume the car is
well maintained. 19. Sketch the graph of the amount of a particular brand of coffee
sold by a store as a function of the price of the coffee. 20. You place a frozen pie in an oven and bake it for an hour. Then
you take it out and let it cool before eating it. Describe how the
temperature of the pie changes as time passes. Then sketch a
rough graph of the temperature of the pie as a function of time. 21. A homeowner mows the lawn every Wednesday afternoon.
Sketch a rough graph of the height of the grass as a function of
time over the course of a four—week period. 22. An airplane takes off from an airport and lands an hour later at
another airport, 400 miles away. If t represents the time in min—
utes since the plane has left the terminal building, let x(t) be the horizontal distance traveled and y(t) be the altitude of the plane. SECTION 1.1 FOUR WAYS TO REPRESENT A FUNCTION 21 (a) Sketch a possible graph of x(t). (b) Sketch a possible graph of y(t). (c) Sketch a possible graph of the ground speed.
(d) Sketch a possible graph of the vertical velocity. 23. The number N (in millions) of US cellular phone subscribers is
shown in the table. (Midyear estimates are given.) z‘ 1996 1998 2000 ‘i 2002 2004 2006 N '44 69 109 141 182 l 233 ‘ (a) Use the data to sketch a rough graph of N as a function of t. (b) Use your graph to estimate the number of cellphone sub
scribers at midyear in 20071 and 2005. 24. Temperature readings T (in °F) were recorded every two hours
from midnight to 2:00 PM in Phoenix on September 10, 2008.
The time t was measured in hours from midnight. t02l_468101214 T 82 75 74 75 84 90 L 93 94 (a) Use the readings to sketch a rough graph of T as a function
of t. (b) Use your graph to estimate the temperature at 9:00 AM. 25. If f(x) = 3x2 — x + 2, ﬁnd f(2), f(—2). f(a). f(—a).
N + 1), 2f(a), fan). ﬂaz). [f(a)]2, and f(a + h). 26. A spherical balloon with radius r inches has volume D
V(r) = §7rr3. Find a function that represents the amount of air
required to inﬂate the balloon from a radius of r inches to a
radius of r + 1 inches. 27—30 Evaluate the difference quotient for the given function.
Simplify your answer. f(3 + h) —f(3) Z7.f(x)=4+3xx2, h f(a + h) —f(a) 28. f(x) = x3, h
29. re) = hf“: :1; (a)
m M = j: + 3 f(x) ~ fa) + l ’ x — 1
31—37 Find the domain of the function. x+4 2x3—5 x2+x—6 34. g(t) =1/3 — t — d2 +t 31. f(x) = 32. f (x) = x2—9 3?. f(t) = €/2t — 1 l u + 1
35. h(x) ﬁ— 36. = 1
1 + u+l
31. F(p) = 1/2 — y; Y 38. Find the domain and range and sketch the graph of the function h(x) = 1/4  x2_ '
39—50 Find the domain and sketch the graph of the function.
39. f(x) = 2 — 0.4x 4D. F(x) =x2  2): +1 41. f(t) = 2x + t2 42. H(t) = 42?:
43. g(x)=«/x—5 44. F(x)=2x+ ll
45. G(x) = Ex—lx—l 4B. g(x) = Ix]  x x+2 ifx<0
l—x ifxBO 47 f (x) __ 3—%x ifx$2
48'f(x)_{2x—51fx>2 x+2 ifx<—l 49. = ﬁx) 1:2 ifx>—l
x+9 ifx<—3 50.f(x)= —2x if [xf$3
—6 ifx>3 51~56 Find an expression for the function whose graph is the
given curve. 51. The line segment joining the points (1, —3) and (5, 7) ' 52. The line segment joining the points (—5, 10) and (7, ~10) 53. The bottom half of the parabola x + (y — l)2 = O 54. The top half of the circle 2:2 + (y — 2)2 = 4 55. r 3,— T A! I 56. ' 4_ yr I’— . l RV» 57—61 Find a formula for the described function and state its
domain. 57. A rectangle has perimeter 20 m. Express the area of the rect~
angle as a function of the length of one of its sides. 22 58. 59. 60.
61. 62. 63. E4. 65. 66. CHAPTER 1 FUNCTIONS AND LIMITS A rectangle has area 16 m2. Express the perimeter of the rect—
angle as a function of the length of one of its sides. Express the area of an equilateral triangle as a function of the
length of a side. Express the surface area of a cube as a function of its volume. An open rectangular box with volume 2 In3 has a square base. Express the surface area of the box as a function of the length
.of a side of the base. A Norman window has the shape of a rectangle surmounted by
a semicircle. If the perimeter of the window is 30 ft, express
the area A of the window as a function of the width x of the
window. .. 16% A box with an open top is to be constructed from a rectangular
piece of cardboard with dimensions 12 in. by 20 in. by cutting
out equal squares of side x at each corner and then folding up
the sides as in the ﬁgure. Express the volume V of the box as a
function of x. A cell phone plan has a basic charge of $35 a month. The plan
includes 400 free minutes and charges 10 cents for each addi—
tional minute of usage. Write the monthly cost C as a function
of the number x of minutes used and graph C as a function of x
for 0 S x S 600. In a certain state the maximum speed permitted on freeways is
65 mi/ h and the minimum speed is '40 mi/ h. The ﬁne for vio
lating'these limits is $15 for every mile per hour above the imaximum speed or below the minimum speed. Express the amount of the ﬁne F as a function of the driving speed )6 and
graph F(x) for O S x S 100. An electricity company charges its customers a base rate of
$10 a month, plus 6 cents per kilowatt—hour (kWh) for the ﬁrst
1200 kWh and 7’cents per kWh for all usage over 1200 kWh.
Express the monthly cost E as a function of the amount .7: of
electricity used. Then graph the function E for 0 S x S 2000. 77. = 1 + 3x2 — x4 67. In a certain country, income tax is assessed as follows. There is
no tax on income up to $10,000. Any income over $10,000 is
taxed at a rate of 10%, up to an income of $20,000. Any income
over $20,000 is taxed at 15%. _ (a) Sketch the graph of the tax rate R as a function of the
income I. . (b) How much tax is assessed on an income of $14,000?
On $26,000? (0) Sketch the graph of the total assessed tax T as a function of
the income I. 58. The functions in Example 10 and Exercise 67 are called step
functions because their graphs look like stairs. Give two other
examples of step functions that arise in everyday life. 69—70 Graphs of f and g are shown. Decide whether each function
is even, odd, or neither. Explain your reasoning. 59. 70. y 71. (a) If the point (5, 3) is on the graph of an even function, what
other point must also be on the graph?
(b) If the point (5, 3) is on the graph of an odd function, what
other point must also be on the graph? 72. A function f has domain [—5, 5] and a portion of its graph is
shown.
(a) Complete the graph of f if it is known that f is even.
(b) Complete the graph of f if it is known that f is odd. 73~7B Determine whether f is even, odd, or neither. If you have a
graphing calculator, use it to check your answer visually. Z 73. f(x) = x, + 1 74 f(x) = x4: 1
75.f(x)=x:1 7s.f(x)=xx1 78. f(x)=1+ 3x3  x5 SECTION 1.2 MATHEMATICAL MODELS: A CATALOG OF ESSENTIAL FUNCTIONS Exercises 33 12 Classify each function as a power function, root function,
polynomial (state its degree), rational function, algebraic function,
trigonometric function, exponential function, or logarithmic
function. 1. (a)f(x)=10g2x (b) g(x) = v;
3
(c) h(x) = 1 _ x2 ((1) u(t) = 1 — 1.11 + 2.54!2 8‘
(e) 0(1) = 5’ (f) 16(0) = sin6 cos26
2(a)y=77" (b)y=x" (c) y = x2(2 ~ x3) (d) y = tant — cos t s I V563  l = f = _
(e) y 1 + s ( ) y 1 + {fac—
3—4 Match each equation with its graph. Explain your choices. 9 (Don’t use a computer or graphing calculator.) 3 (a)y=x2 (b) y=x5 (c) y =x8 4.(a)y=3x (b>y=3*
(c) y = x3 (d) y T, 3/; 12. 5. (a) Find an equation for the family of linear functions with
slope 2 and sketch several members of the family.
(b) Find an equation for the family of linear functions such that
f (2) = 1 and sketch several members of the family.
(0) Which function belongs to both families? E3 Graphing calculator or computer required 7. 10. 11. 13. 14. B. What do all members of the family of linear functions f (x) = 1 + m(x + 3) have in common? Sketch several mem—
bers of the family. What do all members of the family of linear functions
f (x) = c — x have in common? Sketch several members of
the family. Find expressions for the quadratic functions whose graphs are
shown. Find an expression for a cubic function f if f (1) = 6 and f(—1)=f(0) =f(2) =0 Recent studies indicate that the average surface tempera— ture of the earth has been rising steadily. Some scientists have modeled the temperature by the linear function T = 0.02t + 8.50, where T is temperature in °C and t represents years since 1900. (a) What do the slope and T~intercept represent? (b) Use the equation to predict the average global surface
temperature in 2100. If the recommended adult dosage for a drug is D (in mg), then
to determine the appropriate dosage c for a child of age a,
pharmacists use the equation 0 = 0.0417D(a + 1). Suppose
the dosage for an adult is 200 mg. (a) Find the slope of the graph of c. What does it represent?
(b) What is the dosage for a newborn? The manager of a weekend ﬂea market knows from past expe—
rience that if he Charges x dollars for a rental space at the mar
ket, then the number y of spaces he can rent is given by the
equation y = 200  4x. (a) Sketch a graph of this linear function. (Remember that the
rental charge per space and the number of spaces rented
can’t be negative qtlantities.) , V (b) What do the slope, the y—intercept, and the x—intercept of
the graph represent? The relationship between the Fahrenheit (F) and Celsius (C) temperature scales is given by the linear function F = §C + 32. (a) Sketch a graph of this function. (b) What is the slope of the graph and what does it represent?
What is the F intercept and what does it represent? Jason leaves Detroit at 2:00 PM and drives at a constant speed
west along I—96. He passes Ann Arbor, 40 mi from Detroit, at
2:50 PM. (a) Express the distance traveled in terms of the time elapsed. 1. Homework Hints available at stewartcalculuscom “ ‘eu—NW ‘4“ 42 CHAPTER1 FUNCTIONS AND LIMITS Merehes 1 suppose the'gTaPh 0f f is gtVen Write equations for the graphs 6—7 The graph of y = «3x — x2 is given. Use transformations to
that are Obtained from the graph 0f f as fOHOWS create a function whose graph is as shown. (a) Shift 3 units upward. (b) Shift 3 units downward.
(c) Shift 3 units to the right. (d) Shift 3 units to the left. 3’ A
(e) Reﬂect about the xaxis. (f) Reﬂect about the yaxis. _ r'——“2
(g) Stretch vertically by a factor of 3. 1‘5 " y _ 3x _ x
(h) Shrink vertically by a factor of 3. f E
2. Explain how each graph is obtained from the graph of y = f (x). —0 3 x
, (a)y=f(x)+3 (b)y=f(x+8)
‘ (c) y = 2W) . (d) y = mm 5.
(e) y = —f<x) — 1 (f) y = 8f(%x)
 3. The graph of y = f (x) is given. Match each equation with its
' graph and give reasons for your choices.
(a)y=f(x—4) (b) y=f(x)+3
(c) y = %f(x) (d) y = —f(x + 4)
(6) y = 2f(x + 6) ® 8. (a) How is the graph of y = 2 sinx 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 = l + J); related to the graph of
y = #7 Use your answer and Figure 4(a) to sketch the
graphofy =1 + 9~24 Graph the function by hand, not by plotting points, but by
starting with the graph of one of the standard functions given in Sec—
tion 172, and then applying the appropriate transformations. 1 9.y= 111)):(115—1)3
4. The graph of f is given. Draw the graphs of the following 3‘ + 2
functions.
11. =—3x 12. =x2+6x+4
<a>y=f(x)—2 (b) y=f(x2) y f y
(c) y = —2f(x) (d) y = f(§x) + 1 13. y = 1/x — 2 — 1 14. y = 4sin 3x
3’ A ‘1 . 2
2 15. y = smex) 16. y = —x— — 2
__ , 17.y=%(1~cosx) 18.y=l—21/x+3
0 1 x ‘
1S!.y=l—2x—x2 20;y=lx—2
5. The graph of f is given. Use it to graph the following 21' y __. l x _ 2' 22' y = i tan(x _ 1)
functions. ‘ 4 4
z = i
(3)3} f(2x) (my “236) 23_y=I\/;—1 24.y=]cos'rrx{
(c) y =f(x) (d) y = f(—x) 
_y.T
25. The city of New Orleans is located at latitude 30°N. Use Fig
' 7 1 ure 9 to ﬁnd a function that models the number of hours of
0 /————> daylight at New Orleans as a function of the time of year. To
$.41 \/ x check the accuracy of your model, use the fact that on March 31
L ' I the sun rises at 5:51 AM and sets at 6:18 PM in New Orleans. 1. Homework Hints available at stewartcalculuscom 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 :035 magnitude. Find a function that
models the brightness of Delta Cephei as a function of time. 27. (a) How is the graph of y = f x related to the graph ‘of f?
(b) Sketch the graph of y = sin [x (c) Sketch the graph of y = «Ix 23. Use the given graph of f to sketch the graph of y = l/ f (x). Which features of f are the most important in sketching
y = 1/ f (x)? Explain how they are used. 29—30 Find (a) f + g, (b) f — g, (c) fg, and (d) f/g and state their
domains. 29."if(x) = x3 + 2x2, g(x) = 3x2 — 1 30. f(x) = V3 — x, g(x) = «Dc2 —1 31—36 Find the functions (a) f° g, Ch) 9 of, (c) fof, and (d) g 0 g
and their domains. 31. f(x) = x2 — 1, g(x) = 2x +1
32. f(x) = x  2, g(x) = x2 + 3x +n4
33., f(x) = l — 3x, g(x) = cosx 34. an = ﬁ. 90:) = J31 — x l x+1
35‘ = +__ =
f(x) x 3, go) H2 x
1+x’ 36. f (x) = g(x) = sin 2x 37—40 Find f0 g 0 h.
37. f(x) = 3x — 2, g(x) = sin x, 4106) = x2
38. f(x) = Ix — 41, g(x) = 2", h(x) = x5 39f(x)=x/x— 3, g(x)=x2, h(x) =x3 +2 x . htx)=e/£
xl finfa) = tan x, go.) = SECTION 1.3 NEW FUNCTIONS FROM OLD FUNCTIONS 43 41—46 Express the function in the form f O g.
41. F(x) = (2x + xz)4 42. F(x) = coszx 43. F(x) = 1 jg; 44. G(x) = .311 :x tant
1+ tant 45. v(t) = sec(t2) tan(t2) 46. 140?) = 47—49 Express the function in the form f 0 g 0 h. 47. R(x) = ./\/“ — 1 43. Hot) = .8/2 + m
49. H(x) = sec4(\/'x_) 50. Use the table to evaluate each expression. (a) f(g(1)) (b) 9(f(1))
(d) {MOD (6) (9 °f)(3) (C) f(f(1))
(f) (f° 9X6) x l 2 i 3 4 5 6
f(x) 3 1 4 2 2 5
g(x) 6 3 2 1 i 2 J 3 1 51. Use the given graphs of f and g to evaluate each expression, or explain why it is undeﬁned. (a) f(9(2)) (b) 9(f(0))
(d) (9°f)(6) (e) (9°9)(r2)
f (C) (f°g)(0)
(f) (f °f)(4) l 52. Use the given graphs of f and g to estimate the value of
f(g(x)) for x = —5, 4, —3, . . . , 5...Use these estimates to
sketch a rough graph of f o g. T RY 44 CHAPTER 1 FUNCTIONS AND LIMITS
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 t (in seconds).
(b) If A is the area of this circle as a function of the radius, ﬁnd
A 0 r and interpret it. 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 t (in seconds). .
(b) If V is the volume of the balloon as a function of the radius,
ﬁnd V 0 r and interpret it. “I 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, ﬁnd f so that s =f(d). (b) Express d as a function of t, the time elapsed since noon;
that is, ﬁnd 9 so that d = 90‘). i (c) Find f 0 9. 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 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. 57. The Heaviside function H is deﬁned by 0 ift<0
Ht:
0 {1 ift>0 It is used in the study of electric cifcuits 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 V(t) 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 V(t) in
terms of H (t). The Tangent and Velocity Problems 58. 59. 60. 61. 62. 63. B4. (c) Sketch the graph of the voltage V(t) 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
V(t) in terms of H(t). (Note that starting at t = 5 corre
sponds to a translation.) The Heaviside function deﬁned in Exercise 57 can also be used
to deﬁne the ramp function y = ctH(t), which represents a
gradual increase in voltage or current in a circuit. (a) Sketch the graph of the ramp function y = tH(t). (b) Sketch the graph of the voltage V0) in a circuit if the
switch is turned on at time t = 0 and the voltage is gradu—
ally increased to 120 volts over a 60—second time interval.
Write a formula for V(t) in terms of H(t) for t < 60. (0) Sketch the graph of the voltage V(t) 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 V(t) in terms of H (t) for
t S 32. Let f and g be linear functions with equations f (x) = mlx + In
and g(x) = mgx + b2. Is f 0 9 also a linear function? If so,
what is the slope of its graph? If you invest x dollars at 4% interest compounded annually,
then the amount A(x) of the investment after one year is
A(x)=1.04x.FindA 0A, A DA 0A, andA C>A C’A 0A. What
do these compositions represent? Find a formula for the com—
position of 11 copies of A. (a) If g(x) = 2x + l and h(x) = 4x2 + 4x + 7, ﬁnd a function
f such that f 0 g = h. (Think about what operations you
would have to perform on the formula for g to end up with
the formula for h.) (b) If f(x) = 3x + 5 and h(x) = 3x? + 3x + 2, ﬁnd a function
9 such that f0 g = h. Iff(x) = x + 4 and h(x) = 4x — 1, ﬁnd a function 9 such that
g of: 11. Suppose g is an even...
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 Calculus

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