text stringlengths 71 10k |
|---|
. Therefore g # 1, and the result follows. • • • dl 1 1 e. Subgroups of finite index In Section 7.6b, page 228, we proved that a subgroup of index n in a group generated by r elements is generated by nr elements. We shall now find the rank of a subgroup of index n in a free group. To find the rank of the subgroup we us... |
e group of Problem 8.30, page 257.) Let G = la, b, e, d; [a, el, [a, dj, [b, el, [b, dll. Prove G is the direct product of two free groups each of rank 2. Prove that if G = lXI, x2, ••• ; 2x2 = xl, ... , (i + l)xi+ I = Xi' ••• 1, then G is isomorphic to the additive group of rationals. (Hint: Note that G has the additi... |
ays difficult to say what the group of a presentation is or what its properties are. Indeed, there is not even a general and effective procedure for deciding whether the group of any given presenta tion is of order 1. All this is connected with the word problem, which is, roughly speaking, to determine in a finite numb... |
omposition, 17 definition, 11, 13 domain, 12, 13 equality, 13 matching, 14 one-to-one, 14 onto, 14 restriction of, 14 Matching, 14 Matrices, 81 Matrices, groups of, 81 Maximal independent set, 193 Maximal set, 194 Metacyclic, 243 Mixed group, 188 Mobius transformations, 77 Monomorphism, 42 Multiplication in a set, 2 Mu... |
now going going to apply the machinery of bounded cohomology to understand actions on S1. Recall that the central extension 0 Z Homeo+ Z (R) Homeo+(S1) 0 defines the Euler class e ∈ H 2(Homeo+(S1), Z). We have also shown that there is a representative cocycle c(f, g) taking the values in {0, 1}, defined by f ◦ g ◦ Tc(f,... |
ct, i.e. every point is an accumulation point, and K is totally disconnected. We definitely have K = S1 and K is infinite. It is also minimal and invariant. Let x ∈ S1. We want to show that the closure of its orbit contains K. Since K is minimal, it suffices to show that h(Γ)x contains a point in K. If x ∈ K, then we are d... |
e can pick different injective resolutions to compute group cohomology depending on the scenario, and often this can be helpful. It also allows us to extend group cohomology to allow non-trivial coefficients. Thus, we would like to develop a similar theory for bounded cohomology. We begin by specifying the category we are... |
D MODEL Using the model The examples we study in this section are simple, but the methods are far reaching. This will become more apparent as we explore the applications of algebra in the Focus on Modeling sections that follow each chapter starting with Chapter 1. ■ Using Algebra Models We begin our study of modeling b... |
Numbers Let’s review the types of numbers that make up the real number system. We start with the natural numbers: 1, 2, 3, 4, . . . The integers consist of the natural numbers together with their negatives and 0: . . . , 3, 2, 1, 0, 1, 2, 3, 4, . . . We construct the rational numbers by taking ratios of integers. Thus,... |
mon denominator Property ; ; Property Property 19–22 ■ Rewrite the expression using the given property of real numbers. 19. Commutative Property of Addition, x 3 P. ▼ CONCE PTS 1. Give an example of each of the following: (a) A natural number (b) An integer that is not a natural number (c) A rational number that is not... |
01 0 0.0001 0.001 | _3 |=3 | 5 | =5 _3 0 5 Graph each set. (a) 2, 7 3 1, 3 2 1 ▼ SO LUTI O N (a) The intersection of two intervals consists of the numbers that are in both intervals. 1, 3 1 2, 7 (b) 2 4 3 4 Therefore, 1, 3 1 2 2 This set is illustrated in Figure 7. 1 x 3 and 2 x 7 2 x 3 2, 3 6 6 3 2 (b) The union of tw... |
nd n are integers. LAWS OF EXPONENTS Law 1. 2. 3. 4. 5. aman amn am an am amn n amn 1 2 ab 1 2 a b b a n anbn an bn n Example 32 # 35 325 37 35 32 352 33 5 32 5 310 32 2 3 # 4 2 32 # 42 32 3 42 4 b 2 2 a 1 1 # Description To multiply two powers of the same number, add the exponents. To divide two powers of the same num... |
ut 0.0000000000004 cm. (c) A drop of water contains more than 33 billion billion molecules. 90. (a) The distance from the earth to the sun is about 93 million miles. (b) The mass of an oxygen molecule is about 0.000000000000000000000053 g. (c) The mass of the earth is about 5,970,000,000,000,000,000,000,000 kg. 3 2 47–... |
isites E X AM P L E 6 | Rationalizing Denominators Rationalize the denominator in each fraction. (a) 2 13 (b) 1 23 5 (c) 1 25 5x2 ▼ SO LUTI O N This equals 1. (a) 2 13 # 13 13 2 13 213 3 1 23 5 23 25 5 1 25 x2 25 x3 x # 23 52 23 52 # 25 x3 25 x3 (b) (c) 1 23 5 1 25 x2 Multiply by 13 13 13 # 13 3 Multiply by 23 52 23 52... |
terms, the Inner terms, and the Last terms. (a b)(c d) ac ad bc bd L F O I In general, we can multiply two algebraic expressions by using the Distributive Prop- erty and the Laws of Exponents. E X AM P L E 2 | Multiplying Binomials Using FOIL 2x 1 1 2 1 3x 5 2 6x2 10x 3x 5 L I O F 6x2 7x 5 Distributive Property Combine... |
E X AM P L E 1 | Factoring Out Common Factors Factor each expression. (a) 3x2 6x (b) 8x4y2 6x3y3 2xy4 ▼ SO LUTI O N (a) The greatest common factor of the terms 3x2 and 6x is 3x, so we have (b) We note that 3x2 6x 3x x 2 1 2 8, 6, and 2 have the greatest common factor 2 x4, x3, and x have the greatest common factor x y... |
2 50. 4x 2 25 52. 4t 2 9s 2 54. x 2 10x 25 56. r2 6rs 9s2 1 1 x b 58. a 2 ✎ 57 a2 1 2 1 b2 4 a2 1 60. 2 1 62. x 3 27 64. x 6 64 66. 27a 3 b 6 68. 3x 3 27x 70. x 3 3x 2 x 3 72. 18y 3x 2 2xy4 74. y 3 y 2 y 1 76. 3x 3 5x 2 6x 10 x 2 1 80 ✎ ✎ x2 1 2 2 1 9 x2 1 x2 59. 1 61. t 3 1 63. 8x 3 125 65. x 6 8y 3 67. x 3 2x 2 x 69... |
is effective because, by Product Formula 1 in Section P.6, the product of the denominator and its conjugate radical does not contain a radical: A B2C, A B2C. A B2C 1 2 1 A B2C 2 A2 B2C E X AM P L E 10 | Rationalizing the Denominator Rationalize the denominator: 1 1 22 SE CTIO N P.8 | Rational Expressions 51 ▼ SO LUTI ... |
s been taking the tablets for x days. (b) How many tablets are left after 30 days? (c) How many days will it take for her to run out of tablets? 2. Alonzo's Delivery is having a sale on calzones. Each calzone costs $2, and there is a $3 delivery charge for phone-in orders. (a) Find a formula for the total cost C of ord... |
ebraic problem the aid could be a new unknown that relates to the original unknown. ■ TAKE CASES You might sometimes have to split a problem into several cases and give a different argument for each case. For instance, we often have to use this strategy in dealing with absolute value. ■ WORK BACKWARD Sometimes it is us... |
Bhaskara had determined that great misfortune would befall his daughter if she married at any time other than at a certain hour of a certain day. On her wedding day, as she was anxiously watching the water clock, a pearl fell unnoticed from her headdress. It stopped the flow of water in the clock, causing her to miss th... |
or many generations it was considered the best way to develop logical reasoning. Abraham Lincoln, for instance, studied the Elements as a way to sharpen his mind. The ing by this that mathematics does not respect wealth or social status. Euclid was revered in his own time and was referred to by the title “The Geometer”... |
you to set up equations, we note them as we work each example in this section. GUIDELINES FOR MODELING WITH EQUATIONS 1. Identify the Variable. Identify the quantity that the problem asks you to find. This quantity can usually be determined by a careful reading of the question that is posed at the end of the problem. T... |
95 gal Amounts are equal. ✔ ✎ Practice what you’ve learned: Do Exercise 45. ▲ 80 CHAPTER 1 | Equations and Inequalities ■ Problems About the Time Needed to Do a Job When solving a problem that involves determining how long it takes several workers to complete a job, we use the fact that if a person or machine takes H ... |
e is the pasture? 37. Dimensions of a Lot A square plot of land has a building 60 ft long and 40 ft wide at one corner. The rest of the land outside the building forms a parking lot. If the parking lot has area 12,000 ft 2, what are the dimensions of the entire plot of land? 38. Dimensions of a Lot A half-acre building... |
, see page 63) and the Chinese mathematician Chang Ch’iu-Chien (6th century A.D.) also studied and wrote about equations. Of course, equations continue to be important today. 1. Solve the Babylonian problem and show that their answer is correct. 2. Solve the Egyptian problem and show that their answer is correct. 3. Th... |
ation has no real ✎ Practice what you’ve learned: Do Exercises 65, 67, and 69. ▲ ■ Modeling with Quadratic Equations Let’s look at some real-life problems that can be modeled by quadratic equations. The principles discussed in Section 1.2 for setting up equations as models are useful here as well. E X AM P L E 5 | Dime... |
rom 288 ft above the ground, how long does it take to reach ground level? 96. A ball is dropped from the top of a building 96 ft tall. (a) How long will it take to fall half the distance to ground level? (b) How long will it take to fall to ground level? 97–98 ■ Falling-Body Problems Use the formula h 16t 2 √0t discuss... |
Dividing Complex Numbers Express the following in the form a bi. (a) 3 5i 1 2i (b) 7 3i 4i ▼ SO LUTI O N We multiply both the numerator and denominator by the complex conjugate of the denominator to make the new denominator a real number. (a) The complex conjugate of 1 2i is 1 2i 1 2i . 3 5i 1 2i 3 5i 1 2i b a 1 2i 1 2... |
re each side Expand LHS Add 2 + x Factor Zero-Product Property 1 4 x Solve and x 1 are only potential solutions. We must check them The values to see whether they satisfy the original equation. From Check Your Answers we see that x is a solution but x 1 is not. The only solution is ✎ Practice what you’ve learned: Do Ex... |
x 2/3 6 0 ✎ ✎ 4 40. x 4 5x 2 4 0 42. x 6 2x 3 3 0 44. x 8 15x 4 16 46. 1x 3 14 x 4 0 x 1 x 4 2 1/2 5 7/3 x 1 1 x 4 3/2 2 4/3 x 1 1 x 4 5/2 0 2 1/3 0 2 1 2 48. 49. x 3/2 8x 1/2 16x1/2 0 50. x 1/2 3x1/2 10x3/2 51. x 1/2 3x 1/3 3x 1/6 9 x 51x 6 0 52. 2 1 1 2 53. 55. 57 1x 5 x 5 2 1x 3 x 3 x 1 54. 4x4 16x2 4 0 56. 23 4x 2 ... |
t is graphed in Figure 1. ✎ Practice what you’ve learned: Do Exercise 19. B A 2 . In other words, the solution of the ▲ 114 CHAPTER 1 | Equations and Inequalities E X AM P L E 2 | Solving a Pair of Simultaneous Inequalities Solve the inequalities 4 3x 2 13. 0 2 5 FIGURE 2 ▼ SO LUTI O N The solution set consists of all ... |
ratures does this correspond to on the Fahrenheit scale? 30 °C and ▼ SO LUTI O N (F ) is given by the equation terms of inequalities, we have C 5 91 The relationship between degrees Celsius (C ) and degrees Fahrenheit . Expressing the statement on the bottle in F 32 2 So the corresponding Fahrenheit temperatures satisf... |
et is the open interval 3, 7 . 1 2 ▼ SO LUTI O N 2 Geometrically, the solution set consists of all numbers x whose distance from 5 is less than 2. From Figure 4 we see that this is the interval 3, 7 1 . 2 ✎ Practice what you’ve learned: Do Exercise 27. E X AM P L E 4 | Solving an Absolute Value Inequality Solve the ine... |
4 ■ Find all real and imaginary solutions of the equation. 67. x 2 16 0 69. x 2 6x 10 0 71. x 4 256 0 2 4x 68. x 2 12 70. 2x 2 3x 2 0 72. x 3 2x 2 4x 8 0 74. x 3 125 2x 1 73. x 2 1 2 75–88 ■ Solve the inequality. Express the solution using interval notation, and graph the solution set on the real number line. 75. 3x 2 ... |
400 miles? 800 miles? (c) At what mileage do the two plans cost the same? 3. Cost and Revenue A tire company determines that to manufacture a certain type of tire, it costs $8000 to set up the production process. Each tire that is produced costs $22 in material and labor. The company sells this tire to wholesale distr... |
een the points A 2, 5 1 2 and B 1 4, 1 . 2 ▼ SO LUTI O N Using the Distance Formula, we have d(A, B)Å6.32 A 222 2 6 2 1 5 2 2 1 2 24 36 240 6.32 1 1 2 3 4 5 x B(4, _1) See Figure 5. ✎ Practice what you’ve learned: Do Exercise 25(b). FIGURE 5 y 8 6 4 2 0 _2 FIGURE 6 Q(8, 9) A(5, 3) P(1, _2) 8 x E X AM P L E 3 | Applying... |
in 20 blood samples taken from expectant mothers, yielding the data shown in the table (enzyme levels in milligrams per deciliter). Sample 1 2 3 4 5 6 7 8 9 10 A 1.3 2.6 0.9 3.5 2.4 1.7 4.0 3.2 1.3 1.4 B 1.7 6.8 0.6 2.4 3.8 3.3 6.7 4.3 8.4 5.8 C 49 22 53 15 25 30 12 17 45 47 Sample 11 12 13 14 15 16 17 18 19 20 A 2.2 1... |
e complete the square within each grouping. That is, we complete the square for x 2 2x by adding 2 1 , and we complete the square for y 2 6y by adding 2x x 2 2x y2 6y y2 6y Group terms Complete the square by adding 1 and 9 to each side Factor and simplify r 13 Comparing this equation with the standard equation of a cir... |
nequalities Graphically 2.3 LEARNING OBJECTIVES After completing this section, you will be able to: ■ Use a graphing calculator to graph equations ■ Solve equations graphically ■ Solve inequalities graphically In Chapter 1 we solved equations and inequalities algebraically. In the preceding section we learned how to sk... |
He also did pioneering work on artificial intelligence and computer models of biological processes. At the age of 42 Turing died of poisoning after eating an apple that had mysteriously been laced with cyanide. SE CTI ON 2. 3 | Graphing Calculators: Solving Equations and Inequalities Graphically 161 The Quadratic Formul... |
(in dollars) generated by producing x cooktops per month is given by the equation ▼ DISCOVE RY • DISCUSSION • WRITI NG 74. Misleading Graphs Write a short essay describing different ways in which a graphing calculator might give a misleading graph of an equation. 75. Algebraic and Graphical Solution Methods Write a sho... |
and Graphs y 1 0 (0, _4) 2x-3y-12=0 x 1 3 2 FIGURE 12 y D E F A FIGURE 13 l¤ l⁄ B C x This equation is in the form y mx b, so the slope is b 4. To sketch the graph, we plot the y-intercept and then move 3 units to the right and 2 units up as shown in Figure 12. ✎ Practice what you’ve learned: Do Exercise 53. and the y... |
6 2 1, 2 2 2, 5 , Q 3, 3 2 4, 3 , Q 1 2 15. y 3 1 0 _2 1 3 5 x 2 16. 17. 18. y 3 _3 0 2 x y 1 0 _3 _3 SECTION 2.4 | Lines 177 1, 7 1 2 ; parallel to the line passing through 2, 5 1 2 and 37. Through 2, 1 1 2 38. Through through 2, 11 1 and 1, 1 1 2 39. (a) Sketch the line with slope . 2, 1 3 2 (b) Find an equation for... |
ctly proportional to x, or simply y is proportional to x. The constant k is called the constant of proportionality. Recall that the graph of an equation of the form y mx b is a line with slope m and y-intercept b. So the graph of an equation y kx that describes direct variation is a line with slope k and y-intercept 0 ... |
way? 36. Stopping Distance The stopping distance D of a car after the brakes have been applied varies directly as the square of the speed s. A certain car traveling at 50 mi/h can stop in 240 ft. What is the maximum speed it can be traveling if it needs to stop in 160 ft? 37. A Jet of Water The power P of a jet of wate... |
ter and radius, and sketch its graph. 11. x 2 y 2 2x 6y 9 0 12. 2x 2 2y 2 2x 8y 13. x 2 y 2 72 12x 14. x 2 y 2 6x 10y 34 0 1 2 15–24 ■ Sketch the graph of the equation by making a table and plotting points. 15. y 2 3x 17. x 3y 21 16. 2x y 1 0 18. x 2y 12 19. x 2 y 7 1 21. y 16 x 2 23. x 1y 20. x 4 y 5 0 22. 8x y 2 0 24... |
atching TV, the higher the BMI. So although we cannot fit a line exactly through the data points, a line like the one in Figure 1(a) shows the general trend of the data. BMI 30 20 10 BMI 30 20 10 a) Line of best fit (b) Line fits data exactly FIGURE 1 Fitting lines to data is one of the most important tools available to r... |
s lie close to a line. In the second plot, there is still a linear trend but the points are more scattered. In the third plot there doesn’t seem to be any trend at all, linear or otherwise. y r=0.98 y r=0.84 y r=0.09 x x x FIGURE 9 A graphing calculator can give us a regression line for each of these scatter plots. But... |
A D. Kahanamoku, USA D. Kahanamoku, USA J. Weissmuller, USA J. Weissmuller, USA Y. Miyazaki, Japan F. Csik, Hungary 1908 1912 1920 1924 1928 1932 1936 1948 W. Ris, USA C. Scholes, USA 1952 J. Henricks, Australia 1956 J. Devitt, Australia 1960 D. Schollander, USA 1964 1968 M. Wenden, Australia 1972 M. Spitz, USA 1976 J.... |
4. Thus, numbers x, we have x 2 4 4, so for every output of f we have f 4, q the range of f is d) A machine diagram for f is shown in Figure 5. ✎ Practice what you’ve learned: Do Exercises 9, 13, 17, and 43. ▲ x input 3 _2 square and add 4 square and add 4 square and add 4 x2+4 output 13 8 FIGURE 5 Machine diagram ■ E... |
f x 1 2 x 2 1 212 CHAPTER 3 | Functions 2x 2 3x 4x 2 ; 1 1 , f 1 12 , 21. g g 22. h h f f f f f f 23. 24. 25. 26 27–30 ■ Evaluate the piecewise defined function at the indicated values. ✎ 27. f f 28. f f 29 2x 2x x 1 3 2 B , f 1 30. f x 1 2 • 3x x 1 x 2 2 2 1 0 if x 0 if if x 2 if if x 1 if 1 x 1 if x 1 1 2 , f 0 , f 2... |
use the methods of Section 2.3 2 to graph functions on a graphing calculator. x 1 E X AM P L E 2 | Graphing a Function with a Graphing Calculator Use a graphing calculator to graph the function f rectangle. 1 x 2 x3 8x 2 in an appropriate viewing To graph the function f x3 8x 2, we graph the equation ▼ SO LUTI O N 1 y ... |
0 2, 10 4 10, 10 3 3 3 3 5, 5 3 by 4 by 4 10, 10 3 5, 20 3 4 100, 100 by a) (b) (c) (d) x h 1 (a) (b) (c) (d) x k 1 (a) (b) (c) (d) 4 10, 10 3 4 4 100, 100 3 4 4 4 x 2 x 20 2, 2 5, 5 by 3 10, 10 by 7, 7 25, 20 3 10, 10 by 4 by 4 x 3 5x 4 2, 2 2, 2 by 4 3 4 10, 10 3, 3 by 3 4 10, 5 3, 3 by 3 4 10, 10 by 4 4 10, 10 4 4 3... |
N 3. 3 | Getting Information from the Graph of a Function 227 Getting Information from the Graph of a Function 3.3 LEARNING OBJECTIVES After completing this section, you will be able to: ■ Find function values from a graph ■ Find the domain and range of a function from a graph ■ Find where a function is increasing or ... |
. From the graph of f we see that one local minimum 1 2 value of f is is . and that this value occurs when x 1. To find a function value f a from the graph of f, we find the 1 2 height of the graph above the x-axis at x . From the graph of f we see that f 3 1 2 . 2. The domain of the function f is all the -values of the ... |
erpret average rate of change in real-world situations ■ Recognize that a function with constant average rate of change is linear Functions are often used to model changing quantities. In this section we learn how to find the rate at which the values of a function change as the input variable changes. ■ Average Rate of ... |
and 412 s? (c) Calculate the man’s speed for each lap. Is he slowing down, speeding up, or neither? Time (s) Distance (m) 32 68 108 152 203 263 335 412 200 400 600 800 1000 1200 1400 1600 242 CHAPTER 3 | Functions 27. CD Player Sales The table shows the number of CD players sold in a small electronics store in the year... |
Then we shift the resulting graph shown in Figure 3. y 4 0 f (x) = x – 3 + 4 (3, 4 FIGURE 3 ✎ Practice what you’ve learned: Do Exercise 27. ▲ 1 x and y f y f ■ Reflecting Graphs Suppose we know the graph of . How do we use it to obtain the graphs of y f is simply the negative of the y-coordinate of the corresponding po... |
x x2 x2 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. ; shift 3 units to the right and shift upward 1 unit ; shift 4 units to the left and shift downward x 0 0 ; reflect in the y-axis and shift upward 1 unit ; shift 2 units to the left and reflect in the x-axis x f 2 units, and shift 3 units to the right 1 2 ; stretch vertica... |
2 and g x 2 1 2 (a) Find the functions f g (b) Find and and 1 2 1 2 1 2 1 2 and their domains. In Example 3, f is the rule “square” and g is the rule “subtract 3.” The function f g first subtracts 3 and then squares; the function g f first squares and then subtracts 3. ▼ SO LUTI O N (a) We have f g 1 x 2 2 1 and 22 22 1 ... |
so, what is its slope? 69. Solving an Equation for an Unknown Function Suppose that 2x 1 4x2 4x Find a function f such that . (Think about what operations you would have to perform on the formula for g to end up with the formula for h.) Now suppose that 3x 5 3x2 3x 2 f h x x 1 1 2 2 Use the same sort of reasoning to fi... |
E X AM P L E 5 | Verifying That Two Functions Are Inverses x1/3 are inverses of each other. Show that x3 and g x x f 1 2 1 2 ▼ SO LUTI O N Note that the domain and range of both f and g is . We have g f x 1 g 1 x f 22 g f x 3 2 1 x 1/3 1 x 3 1/3 x 3 x 2 x 1/3 1 1 1 22 2 So by the Property of Inverse Functions, f and g... |
nt on all its new cars. In addition, the manufacturer offers a $1000 rebate on the purchase of a new car. Let x represent the sticker price of the car. (a) Suppose only the 15% discount applies. Find a function f that models the purchase price of the car as a function of the sticker price x. (b) Suppose only the $1000 ... |
4 3 4 10, 10 by 4 3 10, 40 by 3 4 100, 100 by 3 4, 4 3 10, 10 4 3 10, 10 3 4 100, 100 3 4 4 1 2 47–50 ■ Draw the graph of the function in an appropriate viewing rectangle. 47. f 48. f x x 1 1 2 2 x2 25x 173 1.1x3 9.6x2 1.4x 3.2 276 CHAPTER 3 | Functions 49. f 50. f x x 1 1 2 2 x 2x2 16 51. Find, approximately, the dom... |
solve this inequality graphically, as shown in Figure 3.) The steps in Example 1 are typical of how we model with functions. They are summa- rized in the following box. 400 0 FIGURE 1 400 y=15x£ y=90 0 15x £=90 FIGURE 2 400 0 y=15x£ y=60 15x £≥60 FIGURE 3 3 3 3 282 Focus on Modeling GUIDELINES FOR MODELING WITH FUNCTIO... |
is 19 and whose product is as large as possible. (a) Experiment with the problem by making a table like the one following, showing the product of different pairs of numbers that add up to 19. On the basis of the evidence in your table, estimate the answer to the problem. Modeling with Functions 287 First number Second... |
es move.” Galileo revolutionized science by expressing scientific principles in the language of mathematics. He said, “The great book of nature is written in mathematical symbols.” 2 1 2 (b) The standard form tells us that we get the graph of f by taking the parabola y x 2, shifting it to the right 3 units, stretching i... |
on whose graph is a parabola with vertex and that passes through the point 4, 16 . 2 1 44. Find a function whose graph is a parabola with vertex and that passes through the point 1, 8 1 . 2 45–48 ■ Find the domain and range of the function. 1, 2 2 3, 4 2 1 1 45. 47. f f x x 1 1 2 2 x2 4x 3 2x2 6x 7 46. 48. f f x x 1 1 ... |
in the positive direction” xq means “x becomes large in the negative direction” MATHEMATICS IN THE MODERN WORLD Splines A spline is a long strip of wood that is curved while held fixed at certain points. In the old days shipbuilders used splines to create the curved shape of a boat’s hull. Splines are also used to make... |
. P x 2 1 ▼ SO LUTI O N (a) To find the zeros, we factor completely. P x 1 2 2x 4 x 3 3x 2 x 2 2x 2 x 3 2x 3 x 2 1 Thus, the zeros are x 0, (b) The x-intercepts are x 0, P make a table of values of to the right and left of) successive zeros. 1 2 1 x 3 2 x x Factor x 2 Factor quadratic 2 x 1 2 1 , and x 1. , and x 1. Th... |
how many local maxima and minima it has. 59. y 2x 2 3x 5 60. y x 3 12x 61. y x 3 x 2 x 62. y 6x 3 3x 1 63. y x 4 5x 2 4 64. y 1.2x 5 3.75x 4 7x 3 15x 2 18x 65. y x 2 5 32 1 2 67. y x 8 3x 4 x 66. 68 17x 2 7 y 1 69–74 ■ Graph the family of polynomials in the same viewing rectangle, using the given values of c. Explain h... |
inder is 4. Thus 2x 3 7x 2 5 ✎ Practice what you’ve learned: Do Exercise 31. x 3 2 1 2x 2 x 3 1 4 2 ▲ ■ The Remainder and Factor Theorems The next theorem shows how synthetic division can be used to evaluate polynomials easily. REMAINDER THEOREM If the polynomial P x 1 2 is divided by x c, then the remainder is the val... |
emaining zeros. E X AM P L E 2 | Finding Rational Zeros Factor the polynomial P x 2x 3 x 2 13x 6 , and find all its zeros. 1 2 ▼ SO LUTI O N By the Rational Zeros Theorem the rational zeros of P are of the form possible rational zero of P factor of constant term factor of leading coefficient The constant term is 6 and th... |
s. We see that 3 is a lower bound for the solutions. _3 2 _20 FIGURE 3 y 3x 4 4x 3 7x 2 2x 3 Volume of a cylinder: V pr 2h Volume of a sphere: V 4 3 pr 3 150 0 50 FIGURE 5 y 4 3 px 3 4px 2 3 and y 100 3 3 4 7 2 15 24 8 26 3 78 75 Entries alternate in sign. 9 3 5 4 Thus, all the solutions lie between 3 and 2. So the vie... |
equation with rational coefficients can be written as x 3 ax 2 bx c 0 (a) Show that if we replace x by X a /3 and simplify, we end up with an equation that doesn’t have an X 2 term, that is, an equation of the form X 3 pX q 0 This is called a depressed cubic, because we have “depressed” the quadratic term. (b) Use the p... |
as ✎ Practice what you’ve learned: Do Exercise 19 ▲ ■ Zeros and Their Multiplicities In the Complete Factorization Theorem the numbers c1, c2, . . . , cn are the zeros of P. These zeros need not all be different. If the factor x c appears k times in the complete factor, then we say that c is a zero of multiplicity k (... |
has zeros 5, 3, and 1 x 3 3 2 1 x 2 has degree 2 2 1 . The zero 5 has mul- tiplicity , and the zero 3 has multiplicity . 2. (a) If a is a zero of the polynomial P, then must be a factor of P(x). (b) If a is a zero of multiplicity m of the polynomial P, then must be a factor of P(x) when we factor P completely. 3. A pol... |
2x2 4x 5 x 1 The line x 1 is a vertical asymptote because the denominator of r is zero when x 1. To see what the graph of r looks like near the vertical asymptote, we make tables of values for x-values to the left and to the right of 1. From the tables shown below we see that x r 2 1 2 1 2 y q as x 1 and y q as x 1 x ... |
orithm to express the function in the form P / ax where the degree of R is less than the degree of Q and a 0. This means that as x q, approaches the graph of the line y ax b. In this situation we say that y ax b is a slant asymptote, or an oblique asymptote. , so for large values of the graph of y r 0 / AM P L E 8 | A ... |
with vertical asymptotes x 1 and x 1, horizontal asymptote y 0, and x-intercept 4. 90. A Rational Function with No Asymptote Explain how you 87. The Doppler Effect As a train moves toward an observer can tell (without graphing it) that the function (see the figure), the pitch of its whistle sounds higher to the observer... |
one is thrown upward from the top of a building. Its height (in feet) above the ground after t seconds is given by the funct tion 2 the stone reach? . What maximum height does 16t 2 48t 32 h 1 8. The profit P (in dollars) generated by selling x units of a cer- tain commodity is given by the function 1500 12x 0.004x 2 P ... |
tree, record the age of a fish. The table gives the lengths of rock bass caught at different ages, as determined by the otoliths. Scientists have proposed a cubic polynomial to model this data. (a) Use a graphing calculator to find the cubic polynomial of best fit for the data. (b) Make a scatter plot of the data and grap... |
R 0.4807498 32 9 32/3 0.4807 3p 31.544 (a) (b) (c 12 P^3 ENTER A f 312 4.7288 (d) ✎ Practice what you’ve learned: Do Exercise 5. 21^3 ENTER B 31.5442807 4.7288043 ▲ ■ Graphs of Exponential Functions We first graph exponential functions by plotting points. We will see that the graphs of such functions have an easily reco... |
r a third time period Notice that this is an exponential function with base 1 i. A P 1 i 1 k 2 If the annual interest rate is r and if interest is compounded n times per year, then in each time period the interest rate is i r/n, and there are nt time periods in t years. This leads to the following formula for the amoun... |
population behaves according to the logistic growth model n t 1 2 5600 0.5 27.5e0.044t where t is measured in years. (a) Find the initial bird population. (b) Draw a graph of the function n (c) What size does the population approach as time t . 2 1 goes on? 65. World Population The relative growth rate of world popula-... |
fact that y a x (for a 1) is a very rapidly 0, q 2 . loga x f is obtained by reflecting the graph of and range 1 ax x x 2 1 1 f ax 386 CHAPTER 5 | Exponential and Logarithmic Functions Arrow notation is explained on page 345. increasing function for x 0 implies that y loga x is a very slowly increasing function for x 1... |
s the interval 3, 3 we choose the viewing rectangle 4 from it we see that the lines x 2 and x 2 are vertical asymptotes. , so . The graph is shown in Figure 10, and 3, 3 4 3 by 2, 2 The function has a local maximum point to the right of x 1 and a local minimum point to the left of x 1. By zooming in and tracing along t... |
IO N 5.3 | Laws of Logarithms 395 ▼ P RO O F We make use of the property logaax x from Section 5.2. Law 1 Let loga A u and loga B √ . When written in exponential form, these equations become Thus loga1 Law 2 Using Law 1, we have au A and a√ B loga1 loga1 2 2 u √ loga A loga B au√ aua√ AB 2 loga A loga c a A B b B d log... |
ecies and habitat. (a) Solve the equation for S. 10 ✎ 69. Wealth Distribution Vilfredo Pareto (1848–1923) 400 CHAPTER 5 | Exponential and Logarithmic Functions (b) Use part (a) to show that if k 3, then doubling the area increases the number of species eightfold. 71. Magnitude of Stars The magnitude M of a star is a me... |
terms to one side of the equation: x 1 log x 2 log 1 2 1 1 0 2 Then we graph 6 y log x 2 log x 1 1 2 2 1 1 as in Figure 2. The solutions are the x-intercepts of the graph. Thus, the only solution is x 3. ✎ Practice what you’ve learned: Do Exercise 49. ▲ In Example 9 it’s not possible to isolate x algebraically, so we ... |
heric Pressure Atmospheric pressure P (in kilopascals, kPa) at altitude h (in kilometers, km) is governed by the formula ln a P P0 b h k where k 7 and P0 (a) Solve the equation for P. (b) Use part (a) to find the pressure P at an altitude of 4 km. 100 kPa are constants. 84. Cooling an Engine Suppose you’re driving your ... |
will be 12 n 1 2 4100e0.55 1 12 2 3 million Twelve years from now, t 8 12 20 and 50e0.55t n t 1 2 20 n 1 2 50e0.55 1 20 2 2,993,707 We estimate that the rabbit population on the island 12 years from now will be about 3 million. ✎ Practice what you’ve learned: Do Exercise 5. ▲ Can the rabbit population in Example 4(b) a... |
matoes Oranges Apples Limes Battery acid pH 10.5 8.0–8.4 7.3–7.5 7.0–8.5 6.9–7.9 6.4–6.8 5.1–5.7 4.1–4.4 3.0–4.0 2.9–3.3 1.3–2.0 1.0 (a) The hydrogen ion concentration of a sample of human blood was measured to be Find the pH and classify the blood as acidic or basic. H 3.16 108 M. 3 4 (b) The most acidic rainfall ever... |
aining after t years. (b) How much of the sample will remain after 4000 years? (c) After how long will only 18 mg of the sample remain? ✎ 15. Radioactive Cesium The half-life of cesium-137 is 30 years. Suppose we have a 10-g sample. (a) Find a function that models the mass remaining after t years. (b) How much of the s... |
.1 ■ Evaluate exponential functions ■ Graph exponential functions ■ Evaluate and graph the natural exponential function ■ Find compound interest ■ Find continuously compounded interest Review Exercises 1–4 5–8, 73–74, 80 4, 13–14 89–90 89(d), 90(c), 92 Section 5.2 ■ Evaluate logarithmic functions ■ Change between logar... |
rthquake? 428 ● CUMUL ATIVE REVIE W TEST | CHAPTERS 3, 4, and 5 1. Let f x x2 4x (a) The domain of f 1 2 and (b) The domain of g 1x 4 . Find each of the followingcd) (e) The average rate of change of g between x 5 and x 21 (f) 12 1 , f 6 12 , , g f, f 1 (g) The inverse of g 1 2 2 1 1 2 2 2. Let f x 1 2 (a) Evaluate 4 x... |
ln a n ln x Property of ln Property of ln To see that ln y is a linear function of ln x, let Y ln y, X ln x, and A ln a; then Y nX A We apply this technique to the planetary data ln d, ln T 1 so a power model seems appropriate. in Table 2 to obtain the points 1 in Table 4. The scatter plot in Figure 7 shows that the da... |
. lead emissions into the environment in millions of metric tons for 1970–1992. (a) Find an exponential model for these data. (Use t 0 for the year 1970.) (b) Find a fourth-degree polynomial model for these data. (c) Which of these curves gives a better model for the data? Use graphs of the two models to decide. (d) Us... |
at the point ✎ Practice what you’ve learned: Do Exercise 15. . Figure 4 shows that the graphs of the equations in the system inter. ▲ 2 1 E X AM P L E 4 | Elimination Method Find all solutions of the system. 3x 2 2y 26 5x 2 7y 3 Equation 1 Equation 2 ▼ SO LUTI O N We choose to eliminate the x-term, so we multiply the ... |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.