{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Chapter1 - 1.1 l Preliminaries 13 The term I)2 4ac under...

Info iconThis preview shows pages 1–19. Sign up to view the full content.

View Full Document Right Arrow Icon
Background image of page 1

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 2
Background image of page 3

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 4
Background image of page 5

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 6
Background image of page 7

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 8
Background image of page 9

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 10
Background image of page 11

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 12
Background image of page 13

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 14
Background image of page 15

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 16
Background image of page 17

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
Background image of page 18
Background image of page 19
This is the end of the preview. Sign up to access the rest of the document.

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 real-number 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 real-number line and (b) computing the distances by using absolute values. 3. Solve the following equations: (a)l2x—4|=6 (b)|x-3l=2 (c) |2x+3l=5 (d) l7—3x|=—2 4. Solve the following equations: (a) l2x+4l=|5x—2] (h) [5-3u|=|3+2u| (c) l4+él=l§t-2l (d) l2s—3l=l7—SI 5. Solve the following inequalities: (a) [Sat-2154 (b) ll—3xl>8 (c) |7x+4|23 (d) |6~5xl<7 6. Solve the following inequalities: (a)|2x+3l<6 (b)|3—4x|22 (e)lX+5lsl (d)I7-ZXI<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 y-intercept (0. 2) Section 1.1 Problems . 20. The line with slope —1 and y-intercept (0, ——3) 21. The line with slope 1/2 and y-intercept (0, 2) 22. The line with slope —1/3 and y-intercept (0, —-1) 23. The line with slope —2 and x-intercept (1. 0) 24. The line with slope 1 and x-intercept (-2, 0) 25. The line with slope —1/4 and x-intercept (3. 0) 26 The line with slope 1/5 and x-intercept (—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 x-y+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 boiling-point 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 x-axis.) (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 y-axis? (c) Does the circle intersect the .r-axis? 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 (x-2)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 ZA-l (a) 434-2” 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) e-2x =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 85-92, 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+fandz-E. 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 2|X*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 35-f(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...
View Full Document

{[ snackBarMessage ]}