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Unformatted text preview: 1.1 l Preliminaries 13 The term I)2  4ac under the square root sign in the quadratic formula is called
the discriminant. If the discriminant is nonnegative, the two solutions of the corre
sponding quadratic equation are real. (When the discriminant is equal to 0, the two
solutions are identical.) If the discriminant is negative, the two solutions are complex conjugates of each other. Without solving what can you say about the solution? Solution 2xz—3x+7=0 We compute the discriminant . b2 — 4ac = {—3)2 — (4)(2)(7) = 9 — 56 = 47 < 0 Since the discriminant is negative, the equation 2x2 — 3x + 7 = 0 has two complex
solutions, which are conjugates of each other. I I 1.1.1 1. Find the two numbers that have distance 3 from —1 by (a)
measuring the distances on the realnumber line and (b) solving
an appropriate equation involving an absolute value. 2. Find all pairwise distances between the numbers —5, 2, and
7 by (a) measuring the distances on the realnumber line and
(b) computing the distances by using absolute values. 3. Solve the following equations: (a)l2x—4=6 (b)x3l=2 (c) 2x+3l=5 (d) l7—3x=—2 4. Solve the following equations: (a) l2x+4l=5x—2] (h) [53u=3+2u
(c) l4+él=l§t2l (d) l2s—3l=l7—SI
5. Solve the following inequalities: (a) [Sat2154 (b) ll—3xl>8
(c) 7x+423 (d) 6~5xl<7
6. Solve the following inequalities:
(a)2x+3l<6 (b)3—4x22
(e)lX+5lsl (d)I7ZXI<0
I 1.1.2 In Problem 7—42, determine the equation of the line that satisﬁes
the stated requirements Put the equation in standard form. 7. The line passing through (2, 4) with slope % 8. The line passing through (1, —2) with slope 2 9. The line passing through (0. —2) with slope —3
10. The line passing through («3, 5) with slope 1/2
11. The line passing through (—2, —3) and (1. 4)
12. The line passing through (—1. 4) and (2. —%
13. The line passing through (0. 4) and (3, 0)
14. The line passing through (1. —l) and (4, 5)
15. The horizontal line through (3. g) 16. The horizontal line through (0, —1) 17. The vertical line through (—1, g) 18. The vertical line through (2, —3) 19. The line with slope 3 and yintercept (0. 2) Section 1.1 Problems . 20. The line with slope —1 and yintercept (0, ——3)
21. The line with slope 1/2 and yintercept (0, 2) 22. The line with slope —1/3 and yintercept (0, —1)
23. The line with slope —2 and xintercept (1. 0) 24. The line with slope 1 and xintercept (2, 0) 25. The line with slope —1/4 and xintercept (3. 0)
26 The line with slope 1/5 and xintercept (—1/2. 0)
27 The line passing through (2, —3) and parallel to x + 2y — 4 = 0
28. The line passing through (1, 2) and parallel to x  3y — 6 = 0
29. The line passing through (—1, ~1) and parallel to the line
passing through (0, 1) and (3, 0) 30. The line passing through (2, —3) and parallel to the line
passing through (0, l) and (2, 1) 31. The line passing through (1. 4) and perpendicular to
2y  5x + 7 = 0
32. The line passing through (—1. —1) and perpendicular to xy+3=0 v 33. The line passing through (5. l) and perpendicular to the line
passing through (—2, 1) and (1. 2) 34. The line passing through (4. —1) and perpendicular to the line
passing through (—2. 0) and (1. l) 35. The line passing through (4. 2) and parallel to the horizontal
line passing through (1, —2) 36. The line passing through (—1. 5) and parallel to the horizontal
line passing through (2. —1) 37. The line passing through (I. 1) and parallel to the vertical
line passing through (2. ——1) 38. The line passing through (3, 1) and parallel to the vertical line
passing through (—1, 2) 39. The line passing through (1, —3) and perpendicular to the
horizontal line passing through (—l, —1) 14 Chapterl 1 Preview and Review 40. The line passing through (4.2) and perpendicular to the
horizontal line passing through (3, 1) 41. The line passing through (7,3) and perpendicular to the
vertical line passing through (—2, 4) 42. The line passing through («2.5) and perpendicular to the
vertical line passing through (1, 4) 43. To convert a length measured in feet to a length measured
in centimeters, we use the facts that a length measured in feet
is proportional to a length measured in centimeters and that 1 it
corresponds to 30.5 cm. If x denotes the length measured in ft and
y denotes the length measured in cm, then y = 305x (2) Explain how to use this relationship. (b) Use the relationship to convert the following measurements
into centimeters:
(i) 6ft (ii) 3ft,2in (iii) 1ft,7in (c) Use the relationship to convert the following measurements
into ft:
(i) 173 cm (ii) 75 cm (iii) 48 cm 44. (a) To convert the weight of an object from kilograms (kg)
to pounds (lb), you use the facts that a weight measured in
kilograms is proportional to a weight measured in pounds and that
1 kg corresponds to 2.20 lb. Find an equation that relates weight
measured in kilograms to weight measured in pounds (1)) Use your answer in (a) to convert the following measure
ments:
(i) 63 lb (ii) 1501b (lil) 2.5 kg (iv) 140 kg 45. Assume that the distance a car travels is proportional to the
time it takes to cover the distance. Find an equation that relates
distance and time if it takes the car 15 min to travel 10 mi. What
is the constant of proportionality if distance is measured in miles
and time is measured in hours? 46. Assume that the number of seeds a plant produces is
proportional to its aboveground biomass. Find an equation that
relates number of seeds and aboveground biomass if a plant that
weighs 217 g has 17 seeds. 47. Experimental study plots are often squares of length 1 m. [f
1 ft corresponds to 0.305 m, compute the area of a square plot of
length 1 m in ftz. 48. Large areas are often measured in hectares (ha) or in acres.
If 1 ha = 10,000 m2 and 1 acre = 4046.86 m2, how many acres is 1
hectare? 49. To convert the volume of a liquid measured in ounces to a
volume measured in liters, we use the fact that 1 liter equals 33.81
ounces Denote by x the volume measured in ounces and by y the
volume measured in liters. Assume a linear relationship between
these two units of measurements. (I) Find the equation relating x and y. (b) A typical soda can contains 12 ounces of liquid. How many
liters is this? 50. To convert a distance measured in miles to a distance
measured in kilometers, we use the fact that 1 mile equals 1.609
kilometers. Denote by x the distance measured in miles and by y
the distance measured in kilometers. Assume a linear relationship
between these two units of measurements. (in) Find an equation relating .t and y. (b) The distance between Minneapolis and Madison is 261 miles.
How many kilometers is this? 51. Car speed in many countries is measured in kilometers per
hour. In the United States, car speed is measured in miles per hour.
To convert between these units, use the fact that 1 mile equals
1.609 kilometers. (a) The speed limit on many US. highways is 55 miles per hour.
Convert this number into kilometers per hour. (b) The recommended speed limit on German highways is 130
kilometers per hour. Convert this number into miles per hour. To measure temperature, three scales are commonly used:
Fahrenheit, Celsius, and Kelvin. These scales are linearly related.
We discuss these scales in Problems 52 and 53. 52. (a) The Celsius scale is devised so that 0°C is the freezing
point of water (at 1 atmosphere of pressure) and 100°C is the
boiling point of water (at 1 atmosphere of pressure). If you are
more familiar with the Fahrenheit scale, then you know that water
freezes at 32°F and boils at 212°F. Find a linear equation that relates temperature measured in degrees Celsius and temperature
measured in degrees Fahrenheit. (b) The normal body temperature in humans ranges from 97.6“F
to 99.6°F. Convert this temperature range into degrees Celsius 53. (a) The Kelvin (K) scale is an absolute scale of temperature.
The zero point of the scale (0 K) denotes absolute zero, the coldest
possible temperature; that is, no body can have a temperature
below 0 K. It has been determined experimentally that 0 K
corresponds to —273.15°C. If 1 K denotes the same temperature difference as 1°C, ﬁnd an equation that relates the Kelvin and
Celsius scales. (b) Pure nitrogen and pure oxygen can be produced cheaply by
first liquefying puriﬁed air and then allowing the temperature
of the liquid air to rise slowly. Since nitrogen and oxygen
have different boiling points, they are distilled at different
temperatures The boiling point of nitrogen is 77.4 K and
of oxygen is 90.2 K. Convert each of these boilingpoint
temperatures into Celsius. If you solved Problem 52(a), convert
the boiling—point temperatures into Fahrenheit as well. Consider the two techniques described for distilling nitrogen and oxygen.
Which element gets distilled ﬁrst? 54. Use the following steps to show that if two nonvertical lines (1
and 12 with slopes m, and m;, respectively, are perpendicular, then
mm; = —1: Assume that m1 < Oand M2 > O. (it) Use a graph to show that if 91 and 92 are the respective angles
of inclination of the lines 11 and [2, then 91 = 91 + ’2'. (The angle of inclination of a line is the angle 0 e [0, 7r) between the line and
the positively directed xaxis.) (b) Use the fact that tan(7r — x) = — tanx to show that m; =
tan 61 and m; = tan 02. (c) Use the fact that tan(§ — 'x) = cotx and cot(x) =  cotx
to show that m1 = — cot62. (d) From the latter equation, deduce the truth of the claim set
forth at the beginning of this problem. I 1.1.3 55. Find the equation of a circle with center («1, 4) and radius 3.
56. Find the equation of a circle with center (2, 3) and radius 4. 57. (2) Find the equation of a circle with center (2, 5) and radius
3. (b) Where does the circle intersect the yaxis?
(c) Does the circle intersect the .raxis? Explain. 58. (1:) Find all possible radii of a circle centered at (3. 6) so that
the circle intersects only one axis (b) Find all possible radii of a circle centered at (3. 6) so that the
circle intersects both axes. 59. Find the center and the radius of the circle given by the
equation
(x2)1+y2 =16 60. Find the center and the radius of the circle given by the
equaﬁon
(x+1)2+(y—3)2=9 61. Find the center and the radius of the circle given by the
equation
0=x2+y2—4x+2y—— 11 (To do this, you must complete the squares.)
62. Find the center and the radius of the circle given by the
equation
x2+y2+Zt—4y+1 =0
(To do this, you must complete the squares.)
l 1.1.4
63. (I) Convert 75° to radian measure.
(b) Convert 1%}! to degree measure. 64. (2) Convert —15° to radian measure.
(h) Convert %n to degree measure. 65. Evaluate the following expressions without using a calculator:
in) sin(57") (b) costég) (c) tan(§) 66. Evaluate the following expressions without using a calculator:
in) sin(’7") (b) cost—”7") (c) taut?) 67. (I) Find the values of at e [0, 211') that satisfy 1
sinot = ——/5
2
(b) Find the values of a e [0. 271) that satisfy tanot = ﬁ 68. (it) Find the values of a e [0. 27:) that satisfy cosa = —%\/E (b) Find the values ofa E [0, 271') that satisfy secot = 2 69. Show that the identity 1+ tan26 == sec29
follows from sin20 +c0829 =1
70. Show that the identity 1+ c0129 = 0589 follows from
sin2 0 + cos2 0 =1
71. Solve 2cos 9 sin 0 = sin 0 on [0, 271').
72. Solve seczx = ﬁtanx + 1 on [0, 7r).
1 1.1.5
73. Evaluate the following exponential expressions: 15 1.1 I Preliminaries 2 1/2 t ZAl
(a) 4342” 00 93%”; (c) 5—5;.— 74. Evaluate the following exponential expressions: 3 _ 1 3
(a) (2424”)2 (b) (”5:17;”) (0 (33:3) 75. Which real number it satisﬁes (a) log‘x :2 —2? (b) logmx = ~37 (c) logmx = 2?
76. Which real number x satisﬁes (a) logmx = —4? (b) logmx = 2? (c) log,x = 3? 77. Which real number .1: satisﬁes
(a) log1 ,2 32 = x? (b) log”J 81 = x? (c)
78. Which real number x satisﬁes (a) log. 64 = x? (b) 1081/5 625 = x? (c) log10 0.001 = x? loglo 10, 000 = x? 79. Simplify the following expressions: (a) 411% (b) log.(x2 —4) (c) 103,4”l
80. Simplify the following expressions: '
(3) ~01} (b) ln‘zﬁfl (c) tog3 321‘+1
81. Solve for x. (0) e3"'1 =2 (b) e2x =10 (c) 1"!“ =10
82. Solve for x. ' (a) 3" = 81 (b) 97'"+1 = 27 (c) 105‘ = 1000
83. Solve for x. (a) ln(x — 3) = 5 (b) ln(x +2) + ln(x — 2) =1
(c) 1033.12 — log32x = 2 84. Solve for x. (a) ln(2x — 3) = 0 (b) log2(1  x) = 3 (c) lnx3 —2lnx =1 l1.1.6 In Problems 8592, simplify each expression and write it in the
standard form a + bi.
85. (3 — 2i) — (—2 + 51')
87. (4 — 21') + (9 + 4i)
89. 3(5+3i) 90. (2—3i)(5+2i) 91. (6—i)(6+i) 92. (—4—3i)(4+2i) In Problems 93—98, let z = 3 — 2i, u = —4 + 3i, 0 = 3 + 5i, and
w = 1 — i. Compute the following expressions: 93.2 94.z+u 95.z+v
96. 17:13 97. W 98. 72' 99. Ifz=a+bi.ﬁndz+fandzE. 100. If 2 = a + bi. ﬁnd 2. Use your answer to compute (—2). and
compare your answer with z. 86. (7+i)—4
88. (6—4i)+(2+5i) In Problems [01—106, solve each quadratic equation in M complex
number system. 101. 2x2—3x+2=0
103. ~x2+x+2=0
105. 4x2—3x+l=0 102. 3x2—2x+1=0
104. —2x2+x+3=0
106. —2x2+4x——3=0 In Problems 107—112, ﬁrst determine whether the solutions of each quadratic equation are real or complex without solving the
equation. Then solve the equation. 107. 3x2—4x—7=0 103.3x2—4x+7=0
109. —x1+2x—1=0 110. 4x2—x+1=0
111.3x2—5x+6=0 112. —x2+7x——2=0
113. Showﬁ=e v & it
CJD’W// 34 Chapterl I Preview and Review W compare 7!“ f(x) = 3sin (Ex) and g(x) = sinx Solution The amplitude of f (x) is 3, whereas the amplitude of g(x) is 1. The period p of f (x)
satisﬁes %p = 2:: or p = 8, whereas the period of g(x) is lit. Graphs of f (x) and g(x) are shown in Figure 1.40. I Figure 1.40 The graphs of y = 35in(%x) and g(x) = aim:
in Example 16. ' Remark. A number is called algebraic if it is the solution of a polynomial equation
with rational coefﬁcients. For instance, ﬂ is algebraic, as it satisﬁes the equation x2 — 2 = 0. Numbers that are not algebraic are called transcendental. For instance,
71' and e are transcendental. A similar distinction is made for functions. We call a function y = f (x) algebraic
if it is the solution of an equation of the form 3.06))” + ‘   + P1(x)y + Po(x) = 0 in which the coefﬁcients are polynomial functions in x with rational coefficients For
instance, the function y = 1 / (1 + x) is algebraic, as it satisﬁes the equation (x +
1)y — 1 = 0. Here, P1 (x) = x + 1 and Po(x) = —1. Other examples of algebraic functions are polynomial functions with rational coefﬁcients and rational functions
with rational coefﬁcients. Functions that are not algebraic are called transcendental. All the trigonometric, exponential, and logarithmic functions that we introduced in this section are
transcendental functions Section 1.2 Problems l 1.2.1 (b) Are the functions In Problems [—4, state the range forrhe given functions. Graph each f (x) __ x2 — 1 x ié 1 function. _ x — 1 ' 1. f(x)=xz,xeR 2. f(x)=x2,xe[0.1] and 3. f(x)=x2,—1<x_<_0 4.f(x)=x2,%<x<% g(x)=x+1, XER equal? 5. (a) Show that, forx #1, 6. (a) Show that
x2_l~x+1 2X*1l= 2(x——1)forx21
x—~1_‘ 2(1—x) forxgl (1)) Are the functions 2—21. forOsxsl f“): 2x—2 forlsxsZ and
80:) = ZIX 1l. x E [0.2] equal? In Problems 7—12, sketch the graph of each ﬁmcrion and decide in
each case whether the function is (i) even, (ii) odd, or (iii) does not
show any obvious symmetry. Then use the criteria in Subsection
1.2.1 to check your answers. 7. f(x)=2x 8. f(x)=3.t:2
9. for) = 3xl 10. f(x) = 2x + 1
ll. f(x) = —lx 12. f(x) = 3x3
13. Suppose that
f(x) = x2. x e R
and
g(x) =3+x, x e R
(a) Show that
(f og)(x) = (3+x)2. x E R
(b) Show that
(gaf)(X) =3+x2, x e R
14. Suppose that
f(x) = x3, x e R
and
g(x)=l—x. xER
(a) Show that
(ng)(X)=(1—x)3. x e R
(b) Show that
(80f)(x) =1—x3, x e R
15. Suppose that
f(x)=1—x2, x eR
and
g(x) = 2x. x 2 0
(it) Find
(f 0 slot)
together with its domain.
(1)) Find
(8 ° f)(X)
together with its domain.
16. Suppose that
1
f0?) = m. x 7‘5 ‘1
and
g(x) = 2x2, 1: e R (8) Find (f c>g)(x). (b) Find (3 o f)(X).
In both (a) and (b). ﬁnd the domain. 1.2 I Elementary Functions 35 17. Suppose that f(x) = 3x2. .1: 3 3
and .906) = J}.
Find (f o g)(x) together with its domain.
18. Suppose that x20 f(X) = x‘. g(x)=./x+1. x23 Find (f o g)(x) together with its domain. 19. Suppose that f(x) = x2,x 2 0, and g(x) = .fx‘,x 2 0.
Typically. f o g ;é g o f , but this is an example in which the order
of composition does not matter. Show that f o g = g o f. 20. Suppose thatf(x) = x‘,x 2 0. Findg(x) so that fog = gof. l 1.2.2 21. Use a graphing calculator to graph [(x) = x2, x 2 0, and
g(x) = x4, x 2 0, together. For which values of x is f (x) > g(x),
and for which is f (x) < g(x)? 22. Use a graphing calculator to graph f (x) = x3, x 2 O, and
g(x) = x5,x 2 0, together. When is f(x) > g(x), and when is
f (x) < g(x)? 23. Graph y = x". x 2 0, for n = 1, 2. 3, and 4 in one coordinate
system. Where do the curves intersect? 24. (in) Graph f(x) = x,x 2 O, and g(x) 7— x2,x 2 0, together,
in one coordinate system. (b) For which values of x is f (x) 2 g(x), and for which values of
x is f(x) 5 g(x)? 25. (3) Graph f (x) = x2 and g(x) = x3 forx 2 0, together, in
one coordinate system. (b) Show algebraically that x23 and x2 2 Jr3
for 0 5 x 5 1.
(c) Show algebraically that
x2 S x3
for): 2 1. I 26. Show algebraically that if n 2 m, x"5x"' forOstl and x" 2x" forx 21 27. (a) Show that y = x2, x e R. is an even function. (b) Show that y = x3. x e R, is an odd function; 28. Show that (a) y = x", x e R. is an even function when n is an even integer.
(b) y = x". x E R, is an odd function when n is an odd integer. 29. In Example 5 of this section, we considered the chemical
reaction A+B~>AB Assume that initially only A and B are in the reaction vessel and
that the initial concentrations are a = [A] = 3 and b = [B] = 4. _—_——.—_.______________________—————————______________ 36 Chapterl I Preview and Review . (a) We found that the reaction rate R(x), where x is the
concentration of AB, is given by R(x) = k(a  x)(b — x) where a is the initial concentration of A, b is the initial
concentration of B, and k is the constant of proportionality.
Suppose that the reaction rate R(x) is equal to 9 when the
concentration of AB is x = 1. Use this relationship to ﬁnd the
reaction rate R(x). (b) Determine the appropriate domain of R(x), and use a
graphing calculator to sketch the graph of R(x). 30. An autocatalytic reaction uses its resulting product for the
formation of a new product, as in the reaction A+X—>X if we assume that this reaction occurs in a closed vessel, then the
reaction rate is' given by R(x) = kx(a — x) for 0 5 x _<_ a. where a is the initial concentration of A and x is
the concentration of X.
(a) Show that R(x) is a polynomial and determine its degree. (b) Graph R(x) for k = 2 and a = 6. Find the value of x at which
the reaction rate is maximal. 31. Suppose that a beetle walks up. a tree along a straight line at a
constant speed of 1 meter per hour. What distance will the beetle
have covered after 1 hour, 2 hours, and 3 hours? Write an equation
that expresses the distance (in meters) as a function of the time (in
hours), and show that this function is a polynomial of degree 1. 32. Suppose that a fungal disease originates in the middle of an
orchard, initially affecting only one tree. The disease spreads out
radially at a constant speed of 10 feet per day. What area will be
affected after 2 days, 4 days, and 8 days? Write an equation that
expresses the affected area as a function of time, measured in days,
and show that this function is a polynomial of degree 2. I 1.2.3 In Problems 33—36, for each function, ﬁnd the largest possible ' domain and determine the range. 1 2:: 33. f(x)=‘1—:‘; 34. f(x)=m
x—Z 35f(x)=;:— 36.f(x)=x2+1 31. Compare y = é and y = :1, for x > 0 by graphing the
two functions. Where do the curves intersect? Which function is
greater for small values of x? for large values of x? 38. Let n and m be two...
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