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�smaller and smaller negative” in the preceding definition. These definitions are informal because such phrases as “arbitrarily close” have not been precisely defined. Rigorous definitions, similar to those in Section 14.3 for ordinary limits, are discussed in Exercises 49–50. Horizontal Asymptotes and Limits at Infinity Limits as x approaches infinity or negative infinity correspond to horizontal asymptotes of the graph of the function. 952 Chapter 14 Limits and Continuity Horizontal Asymptotes y L The line function f if either is a horizontal asymptote of the graph of the xSqq f(x) L lim or xSqq f(x) L. lim Example 3 Limits at Infinity Discuss the behavior of 1 approaches negative infinity. f x 1 x 2 as x approaches infinity and as x Solution 1 When x is a very large positive number, 1 x is a positive number that is very close to 0. Similarly, when x is a very small negative number—which 20 20 is large in absolute value—such as 5,000,000, that is very close to 0. These facts suggest that tote, or 1 x y 0 is a negative number is a horizontal asymp- 1 Figure 14.5-5 as confirmed in Figure 14.5-5. ■ xSq f lim x 1 2 0 and xSq f lim x 1 2 0, Example 4 Limits at Positive and Negative Infinity Discuss the behavior of as x approaches negative infinity. x f 1 2 x3 10x 5 as x approaches infinity and Solution As shown in Figure 14.5-6, approaches infinity or as x approaches negative infinity. does not approach a single value as x x f 2 1 100,000,000,000 5,000 0 0 0 0 5,000 Figure 14.5-6 100,000,000,000 Section 14.5 Limits Involving Infinity 953 Thus, xSq f lim x and xSq f lim 1 However, the situations are often described by writing 2 1 2 x, as defined in Section 14.1, do not exist. xSq f lim x 2 1 q and xSq f lim x 1 2 q ■ In fact, no polynomial graph has a horizontal asymptote. That is, no polynomial function has a limit as x approaches infinity or negative infinity. Limit of a Constant Function The limits of constant functions are easily found. Consider, for example |
, As x approaches infinity or negative infinity, the the function corresponding value of and 5. xSq f lim 5 is always the number 5, so A similar argument works for any constant function. xSq f lim 5 Limit of a Constant If c is a constant, then xSqq c c lim and lim xSqq c c. Properties of Limits Infinite limits have the same useful properties that ordinary limits have. For instance, suppose that as x approaches infinity, the values of a function f approach a number L, and the values of a function g approach a number M. Then it is plausible that the values of approach L M, L M, x fg and so forth. Similar remarks 1 apply when x approaches negative infinity. the values of approach f g 21 21 x 2 1 2 Properties of Limits at Infinity If f and g are functions and L and M are numbers such that xSqq f(x) L lim and xSqq g(x) M, lim then 1. 2. 3. 4. 5. xSqq( f g)(x) lim lim xSqq( f g)(x) lim lim xSqq( fg)(x) lim lim xSqq lim a f gb (x) lim xSqq xSqq( f(x) g(x)) lim xSqq( f(x) g(x)) lim xSqq( f (x) g(x)) lim xSqq f(x) B A A xSqq f(x) lim f(x) L g(x)b M xSqq g(x) lim f(x) 0 xSqq f(x) lim xSqq f(x) lim xSqq g(x) lim B M 0 for all large x provided a, 2f(x) 2L, lim xSqq xSqq g(x) L M xSqq g(x) L M L M Properties 1–5 also hold with for all small x. erty 0 5, f x 1 2 q in place of q, provided that for prop- 954 Chapter 14 Limits and Continuity Limit of c x n If c is a constant, then Property 3 and Example 3 show that lim xSq c x lim xSqa c 1 xb lim xSq c b Q lim xSq a 1 xb c 0 0. Repeatedly using Property 3 with this fact and Example |
3, note that the result holds for any integer n 2. p 1 xb lim xSq c x 1 x 1 x c xn lim xSqa lim xSq 0 0 0 p 0 0 c xba lim xSq a 1 xba lim xSq 1 xb p lim xSq a 1 xb A similar argument works with and produces this useful result, which is essentially a formal statement of half of the Big-Little Concept discussed in Section 4.4. in place of q q Limit Theorem If c is a constant, then for each positive integer n, lim xSq c x n 0 and xSq lim c x n 0 The Limit Theorem and the limit properties now make it possible to determine the limit, if it exists, of any rational function as x approaches infinity or negative infinity. Example 5 End Behavior of a Rational Function 5 Describe the end behavior of 3x2 2x 1 2x2 4x 5 f x 2 1 and justify your 10 conclusion. 20 Solution 5 Figure 14.5-7 f If you graph 3x2 2x 1 2x2 4x 5 that there appears to be a horizontal asymptote close to in Figure 14.5-7. This can be confirmed algebraically by computing y 1.5, to the right of the y-axis you will see as shown x, 2 1 lim xSq 3x2 2x 1 2x2 4x 5. Property 4 cannot be used directly because neither the numerator nor denominator have a finite limit as x approaches infinity, as discussed in Example 4. To rewrite the expression in an equivalent form, divide both Section 14.5 Limits Involving Infinity 955 numerator and denominator by the highest power of x that appears, namely x2. Dividing both by x2 is the same as multiplying by form of 1, so the value of the fraction is not changed. 1 x2 1 x2, a lim xSq 3x2 2x 1 2x2 4x 5 lim xSq 1 x2 5 x2 3x2 2x 1 x2 2x2 4x 5 x2 2x x2 4x x2 1 x2 5 x2 1 3x2 x2 2x2 x2 3 2 x 2 4 x 3 2 x x2b lim xSq lim xSq lim xSqa lim xSqa 2 4 x 5 x2b lim x |
Sq 3 lim xSq 2 x lim xSq lim xSq 2 lim xSq 4 x lim xSq 1 x2 5 x2 3 lim xSq 2 x lim xSq 1 x2 5 x2 lim xSq 4 2 lim x xSq 3 0 0 2 0 0 3 2 property 4 property 1, 2 limit of constant limit theorem ■ A slight variation on the last example can be used to compute certain limits involving square roots. Example 6 Limits at Infinity Find each limit. a. lim xSq 23x2 1 2x 3 b. lim xSq 23x2 1 2x 3 956 Chapter 14 Limits and Continuity Solution a. Only positive values of x need to be considered when finding the limit as x approaches infinity. When x is positive, Therefore, 2x2 x. 23x2 1 2x 3 lim xSq lim xSq lim xSq lim xSq lim xSq B 23x2 1 x 2x 3 x 23x2 1 2x2 2x 3 x 3x2 1 x2 2x 3 x 3 1 B x2 2 3 x 3 1 x2 2 3 xb 3 1 lim xSqB lim xSqa B lim xSqa x2b lim xSqa 2 3 xb lim xSq B 3 lim xSq 1 x2 3 x 2 lim xSq lim xSq 23 0 2 0 multiply by 1 x 1 x 2x2 x, for x 7 0 2a 2b a b B property 4 property 5 property 1 constant limit and limit theorem 23 2 b. To compute the limit as x approaches negative infinity, you need only consider negative values of x and use the fact that when x is negative, Then an argument similar to the one in part a shows that x 2x2. 2 2 For instance, 2 24. 2 2 1 lim xSq 23x2 1 2x 3 23 2 ■ Section 14.5 Limits Involving Infinity 957 Although the properties of limits and some algebraic ingenuity can often by used to compute limits, as in the preceding examples, more sophisticated techniques are needed to determine certain limits. This is the case, for example, with the proof that n lim nSqa 1 1 nb exists and is the number e. Exercises 14.5 In Exercises 1–8, use a calculator to estimate the limit. 10. 2x2 |
1 x 1 1 2D 2x2 x 1 x D 1. 2. lim xSq lim xSq C C 3. lim xSq x 4 3 2 3 x x3 5. lim xSq sin 1 x 7. lim xSq ln x x 8. lim xSq 1 5 1.1 1 x 20 2 4. lim xSq 5 4 x x 2x x 5 4 6. lim xSq sin x x 11. In Exercises 9–14, list the vertical asymptotes of the graph, if any exist. Then use the graph of the function to find xSq f(x) lim and xSq f(x). lim 9. y 3 2 1 −10 −5 5 10 x 12. y y y 3 2 1 −20 −10 −1 3 2 1 −20 −1 −2 −3 15 10 5 10 20 30 20 40 60 x x x −40 −20 −5 20 40 60 −10 −15 958 13. 14. Chapter 14 Limits and Continuity 15 10 5 −20 −10 −5 −10 −15 y y 12 8 4 −20 −4 −8 −60 −40 x x 10 20 30 20 40 60 In Exercises 15–20, use the limit theorem and the properties of limits to find the horizontal asymptotes of the graph of the given function. 15. f x 1 2 3x2 5 4x2 6x 2 16. g x 2 1 x2 x2 2x 1 17. h x 2 1 2x2 6x 1 2 x x2 18. x k 1 2 3x x2 4 2x x3 x2 19. f x 1 2 3x4 2x3 5x2 x 1 7x3 4x2 6x 12 20. g x 2 1 2x5 x3 2x 9 5 x5 In Exercises 21–39, use the limit theorem and the properties of limits to find the limit. 21. lim xSq 1 x 3 x 2 21 2x2 x 1 2 22. 1 lim xSq 3x 2 2x 1 21 3x2 2x 5 2 23. lim xSq 3x 1 x2b a 24. lim xSq 1 3x2 1 2 2 25. lim xSq 3x x 2 a 2x x 1b 26. lim xSq a x x |
2 1 2x2 x3 xb 27. lim xSq 2x 2x2 2x 29. lim xSq 3x 2 22x2 1 28. lim xSq x 2x2 1 30. lim xSq 3x 2 22x2 1 31. lim xSq 22x2 1 3x 5 32. lim xSq 22x2 1 3x 5 33. lim xSq 23x2 3 x 3 34. lim xSq 23x2 2x 2x 1 35. lim xSq x2 2x 1 2x4 2x 36. lim xSq 2x6 x2 2x3 Hint: Rationalize the denominator. 37. lim xSq 38. lim xSq 1 2x 1 2x 2x 2 2x 3 39. lim xSq A 2x2 1 x Hint: Multiply by B 2x2 1 x 2x2 1 x. In Exercises 40–42, find the limit by adapting the hint from Exercise 39. 40. 41. 42. lim xSq lim xSq A lim xSq x 2x2 4 A 2x2 1 2x2 1 B B 2x2 5x 5 x 1 A B 43. A free-falling body has two forces acting on it: gravity, which causes the body to speed up as it falls, and air resistance, which causes the body to slow down. Assuming that a free-falling body has an initial velocity of zero and that its velocity is proportional to the force due to air resistance, the velocity of a falling object can be written as a function of the amount of elapsed time. velocity f t 2 1 mg 1 e k m t R k Q The variable m represents the mass of the body, g is acceleration due to gravity ( 32 feet per second per second), and k is the contant of proportionality. For a fall with a parachute, k 1.6m; without a parachute, k 0.18m. When the air resistance has built until it nearly balances the gravitational force, the body speeds up very little. Upon reaching this condition, the body continues to move downward with a constant maximum speed called terminal velocity. Find the terminal velocity of falling bodies with a parachute and without a parachute by finding the limit as t approaches infinity. In Exercises 44–45, find the limit. 44. Critical Thinking lim xSq |
x x 0 0 45. Critical Thinking lim xSq 46. Critical Thinking Let function and find denote the greatest integer a. 3 lim xSq x x 4 b. lim xSq 3 x x 4 47. Critical Thinking Find lim xSq 4x 3x 2x 2x 1. Section 14.5 Limits Involving Infinity 959 48. Critical Thinking Let f x 1 2 be a nonzero polynomial x g with leading coefficient a, and let nonzero polynomial with leading coefficient c. Prove that be a 2 1 a. if degree f b. if degree f x 1 2 x 2 1 6 degree g x 1 2, then lim xSq 0. degree g x 2 1, then lim xSq f g a c. c. if degree f x 1 2 7 degree g x 1 2, then lim xSq does not exist Formal definitions of limits at infinity and negative infinity are given in Exercises 49 and 50. Adapt the discussion in Section 14.3 to explain how these definitions are derived from the informal definitions in this section. 49. Critical Thinking Let f be a function and L be a real number. Then the statement that for each positive number real number k that depends on with the following property: lim xSq e, e 2 1 x means f there is a positive L If x 7 k, then 0 f x 2 1 L 0 6 e Hint: Concentrate on the second part of the informal definition. The number k measures “large enough,” that is, how large the values of x must be in order to guarantee that you want to L. is as close as x f 2 1 50. Critical Thinking Let f be a function and L be a real means number. Then the statement that for each positive number negative real number n that depends on with the following property: L x 2 there is a e xSq f lim e, 1 If x 6 n, then 14 R E V I E W Important Concepts Section 14.1 Section 14.2 Section 14.2.A Section 14.3 Section 14.4 Section 14.5 Limit notation............................ 910 Informal definition of limit.................. 911 Limits and function values......... |
......... 912 Nonexistence of limits..................... 914 Limit of a constant........................ 918 Limit of the identity function................ 918 Properties of limits........................ 919 Limits of polynomial functions.............. 920 Limits of rational functions................. 921 The limit theorem......................... 922 Limit of a function from the right............ 924 Limit of a function from the left.............. 924 Computing one-sided limits................. 925 Two-sided limits.......................... 926 The formal definition of limit................ 931 Proving limit properties.................... 933 x c Continuity at........................ 936 Continuity on an interval................... 942 Properties of continuous functions............ 943 Continuity of composite functions............ 944 The Intermediate Value Theorem............. 944 Infinite limits............................ 948 Vertical asymptotes................ |
........ 950 Limit of a function as x approaches infinity or negative infinity........................ 951 Horizontal asymptotes..................... 952 Properties of limits at infinity............... 953 960 Chapter Review 961 Review Exercises In Exercises 1–2, use a calculator to estimate the limit. Section 14.1 1. lim xS0 3x sin x x 2. lim xSp 2 1 sin x 1 cos 2x In Exercises 3–4, use the graph of the function to determine the limit. 3. lim xS2 f x 1 2 y 3 2 1 −1 −1 −2 −3 −3 −2 x 1 2 3 4. lim xS1 f x 1 2 y 3 2 1 −1 −1 −2 −3 −3 −2 x 1 2 3 Section 14.2 In Exercises 5–6, assume that lim xS3 f(x) 5 and g(x) 2. lim xS3 5. Find lim xS3 2f 2 x 2 4 x 2 1 6. Find lim xS3 2f f 1 2 x 1 x 2 2g g x 1 1 2 x 2 In Exercises 7–10, find the limit if it exists. If the limit does not exist, explain why. 7. lim xS1 x2 1 x2 3x 2 9. lim xS0 21 x 1 x 8. lim xS2 x2 x 6 x2 x 2 10. lim xS2 x2 2x 3 x2 6x 9 962 Chapter Review 11. If f x 1 2 x2 1, find lim hS0. 12. If f x 1 2 3x 2 and c is a constant, find lim hS0. Section 14.2.A 13. 0 lim xS5 x 5 x 5 0 14. lim xS7 A 27 x2 6x 2 B Section 14.3 In Exercises 15–16, use the formal definition of limit to prove the statement. 15. lim xS3 1 2x 1 7 2 16. lim xS2 1 2 a x 3 b 4 Section 14.4 In |
Exercises 17–18, determine whether the function whose graph is given is continuous at x 3 x 2. and 17. 18. y y 3 2 1 −1 −1 −2 −3 3 2 1 −1 −1 −2 −3 −3 −2 −3 −2 19. Show that f x 1 2 x2 x 6 x2 9 a. continuous at has the given traits. b. discontinuous at x 3 Chapter Review 963 20. Is the function given by f x 1 2 3x 2 if x 3 10 x if x 7 3 e continuous at x 3? Justify your answer. Section 14.5 In Exercises 21–22, find the vertical asymptotes of the graph of the given function, and state whether the graph moves upward or downward on each side of each asymptote. 21. f x 1 2 x2 1 x2 x 2 22. g x 2 1 x2 1 x2 3x 2 In Exercises 23–26, find the limit. 23. lim xSq 2x3 3x2 5x 1 4x3 2x2 x 10 24. lim xSq 4 3x 2x2 x3 2x 5 25. lim xSq a 2x 1 x 3 4x 1 3x b 26. lim xSq 23x2 2 4x 1 In Exercises 27–28, find the horizontal asymptotes of the graph of the given function algebraically, and verify your results graphically with a calculator. 27. f x 1 2 x2 x 7 2x2 5x 7 28. f x 1 2 x 9 24x2 3x 2 C H A P T E R 14 Riemann Sums Calculus deals with rates of change, such as the speed of a car, and problems such as the following: If you know the continuously changing speed of a car at any instant, can you determine how far the car has traveled? A special case of this question will be answered in this section by using Riemann sums. Example 1 Total Distance Given Velocities Suppose a racecar is moving with increasing velocity. The velocity of the car at various times is given in the table. Estimate the total distance traveled in the 4-second interval. Solution Because the velocity is increasing, the car has gone at least 14 feet during the first second, at least 29 feet during the second, at least 61 feet during the third second, |
and at least 128 feet during the fourth second. During the four-second interval, the car has traveled at least 14 29 61 128 232 feet underestimate Therefore, 232 feet is an underestimate of the total distance traveled. An overestimate can be found by noting that the car travels no more than 29 feet in the first second, no more than 61 feet in the next second, no more than 128 in the third second, and no more than 268 feet in the last second. Altogether, the car traveled no more than 29 61 128 268 486 feet overestimate Therefore, the total distance traveled is between 232 feet and 486 feet. ■ The lower and upper estimates can be represented on a graph, where the velocity is shown as a smooth curve passing through each point given in the table, and the estimates of the distance traveled each second are represented by the area of rectangles. See Figure 14.C-1. The darker rectangles represent the underestimate for each second and the darker and lighter rectangles stacked together represent the overestimate. Because the time interval between each measurement is 1 second, each rectangle is 1 unit wide. Each height corresponds to how far the car could have traveled during each time interval. Therefore, the areas of the darker rectangles are 14, 29, 61, and 128, and the sum of the areas represents the total underestimate of 232. Time (sec) Velocity (ft/sec) 0 1 2 3 4 14 29 61 128 268 velocity 250 200 150 100 50 time 0 1 2 3 4 Figure 14.C-1 964 Similarly, the sum of the areas of the darker and lighter rectangles, which represents the overestimate, is 486. There is a difference of feet between the estimates. This difference can also be found by adding the areas of the lighter rectangles. 486 232 254 A Better Estimate Time (sec) Velocity (ft/sec) To get a better estimate of how far the car traveled during the 4-second interval, the velocity is measured for each half second, as shown in the table. The graph shown in Figure 14.C-2 displays the new data. 0 0.5 1 1.5 2 2.5 3 3.5 4 14 20 29 42 61 88 128 185 268 velocity Overestimate 250 200 150 100 50 time 0 1 2 3 4 Figure 14.C-2 In the first half second, the car travels at least 14 1 2b a 7 feet and at most 1 2b 10 20 a for both the underestimate |
and the overestimate as feet. The distance traveled in each half second is calculated vi where val and represents the velocity as measured at one end of the time inter¢t represents length of each the time interval. vi1 ¢t 2 Underestimate 14 a 1 2b 1 2b 20 a 1 2b 1 2b 29 a 1 2b 88 a 1 2b 128 a 1 2b 29 a 1 2b 128 a 42 a 1 2b 185 a 1 2b 42 a 1 2b 61 a 1 2b 185 a 1 2b 61 a 1 2b 20 a 88 a 1 2b 283.5 feet 268 a 1 2b 410.5 feet The difference between the estimates is again shown as the area of the lighter rectangles. This difference is 410.5 283.5 127. Notice that the difference between the better estimates is half what it was in Example 1. By halving the intervals of measurement, the difference between the estimates is halved. Similarly, if the interval of measurement was given for every tenth of a second, the estimates would differ by 254 25.4 feet, and if the interval of measurement was every thousandth of a second, the difference between the estimates would be 254 0.254. 0.001 0.1 2 1 1 2 965 Example 2 Accuracy of Estimates How frequently must the velocity be measured to ensure that the estimated distances traveled by the race car are within 5 feet of each other? Solution The difference between the velocity at the beginning and end of the meas¢t, urements is then the difference between the estimates is For the differences of the estimates to be within 5 feet, If the time between each measurement is 268 14 254. 254 ¢t. 2 1 254 ¢t 2 1 6 5 or ¢t 6 0.0196850394 seconds. ■ As the length of the intervals of measurement become smaller and smaller, the underestimate and overestimate approach the same number—the area under the curve—which represents the total distance traveled. Riemann Sums. Suppose that velocity is given as an increasing function of time: 2 To find the total distance traveled by a moving object over the time individe the interval into n equally spaced times terval a 6 t 6 b, 1 t v f t0 a, t1, p, tn b, where each time interval is ¢t b a n in duration. ti2, The velocity at the beginning or end of each time |
interval is given by so the estimated distance traveled during each interval is velocity times the length of each time interval. f 1 ¢t f ti2 Both the underestimate and the overestimate of the total distance traveled can be written as the sum of the individual distances. The t02 underestimate sum begins with and the overestimate sum begins with distance at each ¢t ¢t ¢t f f 1 1 t12 1 f t12 1 ¢t f t22 1 ¢t p f tn2 1 f t02 1 ¢t f t12 1 ¢t p f 1 tn12 n f ¢t a i1 n1 ¢t a i0 t22 ¢t f 1 ¢t ti2 1 overestimate underestimate ¢t f ti2 1 ¢t p f ¢t tn2 1 lim nSq f t12 1 n nSq a i1 lim ¢t f ti2 1 As Examples 1 and 2 show, these estimates will be very close to each other when n is very large and the correspoding is very small. In fact, the actual total distance can be found by taking the limit of one of these sums as n gets very large without bound, written n S q. ¢t velocity f(t4) = f(b) Total distance traveled f(t0) = f(a) f(t3) f(t2) f(t1) t0 = a t1 t2 t3 t4 = b Figure 14.C-3 966 This limit can be interpreted geometrically as the area between the graph of f and the horizontal axis from t a to t b. The sum a Riemann sum. The limit of the Riemann sums n a i1 ¢t ˛f ti2 1 is called n lim nSq a i1 ¢t f ti2 1 is called the definite integral of f from a to b and is denoted by b dt. f t 1 2 a Definite integrals are studied fully in calculus. They have a wide variety of other applications, including determining the lengths of curves and the volumes of irregularly shaped solids, finding the amount of work done by a force, and making sophisticated probability calculations. Exercises 1. A driver slams on the brakes and comes to a stop 3. a. For the diagram below, estimate the shaded in five seconds. The |
following velocities are recorded after the brakes are applied. area with an error of at most 0.1. b. How can the shaded area be approximated to any desired degree of accuracy? Time (sec) 0 1 2 3 4 Velocity (ft/sec) 102 70 46 29 12 5 0 a. Find an upper and lower estimate of the distance traveled by the car after the brakes were applied. b. Sketch the graph of velocity versus time, and show the upper and lower estimates and the difference between them. c. How often would the velocity need to be measured to assure that the estimates differ by less than 5 feet? by less than 1 foot? 2. Use the grid to estimate the area of the region x ± 4. bounded by the curve, the horizontal axis, and the lines that are within 4 square units of one another. Explain your procedure. Get an upper and a lower estimate y 4 2 −4 −2 0 2 4 t y − x2 3 y = e −1 0 t 1 4. Estimate the total distance an object travels t 10 between represents the velocity v of the object in ft/sec. if the graph below t 0 and 10 t t cos t 5. Suppose the velocity of an object is given by 0 t 1.5. v Estimate the distance traveled during the 1.5-second interval, accurate to one decimal place. for 1 2 6. A snail is crawling at a velocity given by v 1 t 2 where Estimate the distance that the snail crawls during the second hour. hours and v is in feet per hour. 1 t, t 2 1 967 APPENDIX Algebra Review............................................... 969 A.1 Integral Exponents...................................... 969 A.2 Arithmetic of Algebraic Expressions......................... 973 A.3 Factoring................................ |
.............. 978 A.4 Fractional Expressions.................................... 981 A.5 The Coordinate Plane.................................... 987 Advanced Topics............................................. 994 B.1 The Binomial Theorem................................... 994 B.2 Mathematical Induction.................................. 1002 Geometry Review........................................... 1011 G.1 Geometry Concepts..................................... 1011 Technology.................................................. 1018 T.1 Graphs and Tables...................................... 1018 T.2 Lists, Statistics, Plots, and Regression....................... 1027 T.3 Programs................................. |
............ 1033 Glossary.................................................... 1035 Selected Answers............................................ 1054 Index........................................................ 1148 968 Appendix ALGEBRA REVIEW This appendix reviews the fundamental algebraic facts that are used frequently in this book. You must be able to handle these algebraic manipulations in order to succeed in this course and in calculus. A.1 Integral Exponents Exponents provide a convenient shorthand for certain products. If c is a c3 real number, then denotes cc and denotes ccc. More generally, for any positive integer n c2 cn denotes the product ccc p c (n factors). In this notation c1 is just c, so we usually omit the exponent 1. Example 1 34 3 3 3 3 81 2 1 and 5 2 1 2 For every positive integer n, Example 2 2 21 21 0n 0 2 p 2 21 21 0 0. 2 32. 2 To find 2.4 1 9 2 use the ^ (or ) key on your calculator:* or ab xy 2.4 ^ 9 ENTER* which produces the (approximate) answer 2641.80754. ■ ■ Because exponents are just shorthand for multiplication, it is easy to determine the rules they obey. For instance, c3c5 1 ccccccc cccc 21 ccccc ccc 2 ccccccc cccc c8, that is, c3c5 c35. ccc c3, that is, c7 c4 c74. c7 c4 *The ENTER key is labeled EXE on Casio calculators. Section A.1 Integral Exponents 969 Similar arguments work in the general case: To multiply cm by cn, add the exponents: |
cmcn cmn. To divide cm by cn, subtract the exponents: cm cn cmn. Example 3 42 47 427 49 and 28 23 283 25. ■ The notation as follows: cn can be extended to the cases when n is zero or negative If If c 0, c 0 then c0 is defined to be the number 1. and n is a positive integer, then cn is defined to be the number 1 cn. 00 and negative powers of 0 are not defined (negative powers Note that of 0 would involve division by 0). The reason for choosing these definifor nonzero c is that the multiplication and division rules for tions of exponents remain valid. For instance, n c c5 c0 c5 1 c5, so that c5c0 c50. 7 c7 1 c0, so that c7c 7 c77. c7c 1 c7b a Example 4 6 1 216 3 1 2 2 63 12 3,201,969.857.* and 0.287 1 5 1 2 1 1 2 1 32. 5 2 A calculator shows that ■ If c and d are nonzero real numbers and m and n are integers (positive, negative, or zero), then we have these Exponent Laws 1. 3. 5. cmcn cmn (cm)n cmn cn n c dn dR Q 2. 4. 6. cm n cm cn (cd)n cndn cn 1 cn 970 Algebra Review * means “approximately equal to.” Example 5 Here are examples of each of the six exponent laws. 1. 2. 3. p5p2 p52 p3 1 p3. x9 x4 x94 x5. 3 5 1 2 2 51 3 2 5 2 6. 4. 5. 6. 5 25x5 32x5. 2x 1 2 10 7 3b a 710 310. 1 5 x 1 1 x5b a x5. ■ The exponent laws can often be used to simplify complicated expressions. Example 6 CAUTION 5 is not the same as Part 4 of Example 5 5 32x5 2x 2x 2 1 2x 5. shows that 1 and not 2x 5. 2 1 1 a. b. c. 2x2y3z 2 4 24 c Law (4) x2 4 2 1 1 y3 2 4z4 16x8y12z4. c Law ( |
3) 4 r6 s4. s2 1 2 2 r6s c Law (3) 3s2 r 2 2 3 2 r 1 2 c Law (4) x5y6 3 x2 y 2 1 c x5y6 2 2y2 x2 2 2 1 2 c Law (3) Law (4) 2 x5 y2 1 x2y 1 Law (3) Law (2) x5y6 x4y2 c x54y62 xy4. c ■ ■ It is usually more efficient to use the exponent laws with the negative exponents rather than first converting to positive exponents. If positive exponents are required, the conversion can be made in the last step. Example 7 Simplify and express without negative exponents 2 a 1 3b a b2c3 5 1 2 2c 2 2. Solution 2 a 1 3b a b2c3 5 2 2 2c 2 1 a 2 b2 2 1 3 2 a 1 2 2 2 c3 1 5 b 2 2c 2 2b a 6b a 4c 6 10c 1 c Law (4) c Law (3) 6 1 4 2b 10 1 61 a4b6c 2c 7 a4b6 c7. 2 a c Law (2) Section A.1 Integral Exponents ■ 971 1 1 1 1, Since be equal to 1. Every odd power of 1 1 1 any even power of 1 12, will 2 for instance Consequently, for every positive number c such as is equal to 2 1 1 4 2 1; 21 1 1. 1, 1 1 1 5 or c)n [(1)c]n (1)ncn cn cn e if n is even. if n is odd Example 8 3 1 2 4 34 81 and 5 1 2 3 53 125. ■ CAUTION 4, 12 1 Be careful with negative bases. For instance, if you want to compute ^ 4 ENTER the calculator will interpret this as a negative answer. To get the correct answer, you must key in the parentheses: which is a positive number, but you key in 12 2 1 and produce 124 1 2 2 ( ( ) 12 ) ^ 4 ENTER. Exercises A.1 In Exercises 1–18, evaluate the expression. 19. x2 x3 x5 20. y y4 y6 2. 62 3 2 4 4 2 2 1 1 4. 1 2 4 2 6. |
8. 22 42 4b 10. 2 5 7b a 2 2 7b a 12. 33 3 7 14. 16. 1 33 3 2 1 43 5 2 2 42 5 1 1. 3. 2 6 2 1 5 4 3 5. 1 32 23 1 2 2 2 22 1 1 7. 9. 3 5 4 b a 3 1 3b a 3 2 3b a 11. 24 27 2 2 2 2 2 1 22 3 3 32 2 3 13. 15. 17. 1 23 1 4 2 18. 32 1 3 a 1 3 2 b In Exercises 19–38, simplify the expression. Each letter represents a nonzero real number and should appear at most once in your answer. 972 Algebra Review 21. 23. 25. 27. 29. 1 1 1 1 1 31. a 33. 35. 37. 1 1 1 0.03 y2 y7 2 33x 2x2 2 3x2y 2 2 a2 7a 21 21 3a3 2 3w 4w 2 2 21 3 2w 1 2 2b3a3 1 2 2x 2 3a4 3 2y 1 2 9x 3 4x 2 1 2 1 2 2 0 2x2y 3xy 2 2 1 22. 24. 26. 28. 30. 32. 34. 36. 38. 1 1 1 1 1 1 1 z3 z5 1.3 2 3y3 2 2xy3 b3 21 45y2 3 2 b2 3b 2 21 2 2 1 5d 2 4 2d 3d 1 2 1 3 c4d5c 2x 2 2 1 2 2y 2 1 3y2 2 3 3x 2 2y3 3 1 2 3x 2y4 0 2 In Exercises 39–42, express the given number as a power of 2. 39. 41. 1 1 64 2 2 24 16 2 3 2 40. 3 1 8b a 42 16 b In Exercises 43–60, simplify and write the given expression without negative exponents. All letters represent nonzero real numbers. 43. x4 3 x2 1 x3 2 45. 2 e6 c4b a 3 c3 e b a 47. 2 ab2c3d4 abc2d b a 49. 2 a6 4 b b a 51. 53. a a 2 c5 3 b d 3 3x y2 b 2 x 2y3b a 55 |
. 1 1 3b2c a 2c3 ab 2 1 3 2 2 2 57. 1d c 1 2 44. 46. 4 z2 t3 b a 2 x7 y6b a 5 z3 t b a 4 y2 x b a 3x 48. 1 2 y2 2 1 2xy2 3x2 2 3 2 1 2 2 2 b 50. x y a 52. 54. 1 1 b a x 2y a 2 2y x b 2 5u2v 2uv2 b a 3uv 2u2v b a 3 56. 1 2cd2e 3de 5c 1 1 3 2 2 2 58. 3 1 x2y 1 2 3 4 2 b2 a2 59. a2 1 1 a a 3 2 60. 2 2 a b In Exercises 61–66, determine the sign of the given number without calculating the product. 2.6 3 2 1 4.3 2 2 6.7 5 2 61. 63. 65. 66.1 45.8 3 2 7 2 1 4.6 2 1 7.9 6 7.2 2 8.5 1 9 2 1 4 2 62. 64. 1 1 7 2.5 3 2 2 1 1269 4.1 2 4 2 In Exercises 67–72, r, s, and t are positive integers and a, b, and c are nonzero real numbers. Simplify and write the given expression without negative exponents. 67. r 3 sr 3 t c 6b s 2 70. 1 68. 71. 1 t1 4 4 2t 2 69. t a6 4 b b a rbs c s ctb t r 2 2 1 1 72. 1 1 s arb btcr t 2 s 2 In Exercises 73–80, give an example to show that the statement may be false for some numbers. 73. ar br 75. arbs ab 1 r 2 a b 1 rs 2 74. aras ars 76. r cr c 77. r s c cr cs 79. a 2 1 2 a2 78. 80. 1 1 a 1 b 1 2 21 ab 1 a b 2 21 ab A.2 Arithmetic of Algebraic Expressions Expressions such as b 3c2, 3x2 5x 4, 2x3 z, x3 4xy p x2 xy are called algebraic expressions. Each expression represents a number that is obtained |
by performing various algebraic operations (such as addition or taking roots) on one or more numbers, some of which may be denoted by letters. A letter that denotes a particular real number is called a constant; its value remains unchanged throughout the discussion. For example, the Greek. letter Sometimes a constant is a fixed but unspecified real number, as in “an angle of k degrees” or “a triangle with base of length b.” has long been used to denote the number 3.14159 p p A letter that can represent any real number is called a variable. In for example, the variable x can be any real the expression 2x 5, Section A.2 Arithmetic of Algebraic Expressions 973 x 3, then 2x 5 2 3 5 11. If x 1 2, then If number. 2x 5 2 1 2 5 6, and so on.* Constants are usually denoted by letters near the beginning of the alphabet and variables by letters near the end of the alphabet. Consequently, it is understood that c and d in expressions such as are constants and x and y are variables. cy2 dy, cx d and The usual rules of arithmetic are valid for algebraic expressions: Commutative Laws: a b b a and ab ba Associative Laws: a b c a 1 Distributive Laws: 2 b c 1 2 and ab 1 2 c a bc 1 2 b c 2 a 1 ab ac and b c 1 2 a ba ca. Example 1 Use the distributive law to combine like terms; for instance, 3x 5x 4x 3 5 4 x 12x. 2 1 In practice, you do the middle part in your head and simply write 3x 5x 4x 12x. ■ Example 2 In more complicated expressions, eliminate parentheses, use the commutative law to group like terms together, and then combine them. a2b 31c 5ab 71c ■ B 7a2b a2b 31c 5ab 71c 7a2b a2b 7a2b 31c 71c 5ab ⎧⎪⎨⎪⎩ ⎧⎪⎪⎪⎨⎪⎪⎪⎩ 41c 8a2b 5ab. Combine like terms: A B A Regroup: ■ CAUTION Be careful when parentheses are preceded by a minus sign: b 3 means, so that by the |
distributive law, b 3 b 3 Here’s the reason: b 3. and not 1 1 b 3 2 1 Similarly, 2 7 y 1 1 21. 1 2 3 b 3. 2 2 1 1 21 b 3 2 974 Algebra Review *We assume any conditions on the constants and variables necessary to guarantee that an z 0 algebraic expression does represent a real number. For instance, in and in we assume c 0. we assume 1z 1 c The examples in the Caution Box illustrate the following. Rules for Eliminating Parentheses Parentheses preceded by a plus sign (or no sign) may be deleted. Parentheses preceded by a minus sign may be deleted if the sign of every term within the parentheses is changed. The usual method of multiplying algebraic expressions is to use the distributive laws repeatedly, as shown in the following examples. The net result is to multiply every term in the first sum by every term in the second sum. Example 3, we first apply the distributive law, 3y2 7y 4 2 as a single number: 1 To compute treating y 2 21 3y2 7y 4 1 y 2 Distributive law: 21 1 2 3y2 7y 4 2 1 2 3y2 7y 4 3y2 7y 4 y 2 3y3 7y2 4y 6y2 14y 8 3y3 7y2 6y2 4y 14y 8 3y3 8. ⎧⎪⎪⎨⎪⎪⎩ ⎧⎪⎪⎨⎪⎪⎩ 13y2 18y 1 2 Regroup: Combine like terms: Example 4 1 2x 5y We follow the same procedure with 3x 4y 3x 4y 2x 3x 4y 1 5y 2x 3x 2x 4y 6x2 8xy 15xy 20y2 6x2 20y2. 7xy 2x 5y 5y ⎧⎪⎪⎨⎪⎪⎩ 21 21 1 1 2 1 2 2 3x 4y : 2 2 3x 5y 1 2 4y ■ ■ Observe the pattern in the second line of Example 4 and its relationship to the terms being multiplied: 2x 5y 3x 4y 2 21 1 2x 3x 2x 4y > > 5y 3x > 2 5 |
y 4y > 2 1 1 First terms 3x 4y 2x 5y 2x 5y 2x 5y 1 1 1 21 21 21 3x 4y 3x 4y Outside terms 2 2 2 Inside terms Last terms Section A.2 Arithmetic of Algebraic Expressions 975 CAUTION The FOIL method can be used only when multiplying two expressions that each have two terms. This pattern is easy to remember by using the acronym FOIL (First, Outside, Inside, Last). The FOIL method makes it easy to find products such as this one mentally, without the necessity of writing out the intermediate steps. Example 5 3x 2 x 5 2 21 1 3x2 15x 2x 10 3x2 17x 10. c c c c First Outside Inside Last ■ Exercises A.2 In Exercises 1–54, perform the indicated operations and simplify your answer. 1. x 7x 2. 5w 7w 3w 3. 6a2b 8b a2 2 1 4. 6x31t 7x31t 15x31t 1 3 1 A 1 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 16. x2 2x 1 1 2 u3 u 2 3 2 x3 3x2 4 2 1 T u4 2u3 5 u 2 R Q 3 u3 u 2 2 1 T u4 2u3 5 u 2 R Q u4 u4 S S 1 1 6a2b 3a1c 5ab1c B A 4z 6z2w 2 1 2 z3w2 6ab2 3ab 6ab1c B A 8 6z2w zw3 4z3w2 2 x5y 2x 3xy3 2 9x x3 1 1 x 1y z 2 3 1 2x3 2 1 x 1y z 1 4 2x x5y 2xy3 xy 1 1 2 2 x2 xy 1y z x 1 2 x2 3xy 2 x2 2 2x 1 x2y 2 xy 6xy2 1 2 4ax 2a2y 2ay 17. 3ax 19. 6z3 1 2z 5 1 2 4a 6b 2a2b 2 x 2 2 21 21. 3ab 1 x 1 23. 1 976 Algebra Review 15. 5y 1 21 3y2 1 2 2 18. 20. 22. 24 |
. x2 3xy 2y2 2x 1 3x2 1 3ay 21 x 2 1 1 12x6 7x5 2 4ay 5y 2 2x 5 2 21 B 2 2y 2 2 3w 1 2 a 2 21 y 8 2 2 3x y 21 2 y 6 21 w 2 21 ab 1 y 8 21 3x y x 6 2 2 2 2x 3y 2 2s2 9y 21 4x3 5y2 2s2 9y 2 4x3 5y2 2 21 26. 28. 30. 32. 34. 36. 38. 40. 42 50. 3y 1 y 2 3y 1 2 21 25. 27. 29. 31. 33. 35. 37. 39. 41. 43. 44. 45. 46. 47. 48. 49. 51. 52. 53. 54 2x 4 x 3 2 21 y 4 2 2x 5 21 3y2 4 21 x 4 2 2 4a 5b 2 2 y 3 21 3x 7 y 3 x 4 21 4a 5b y 11 21 2 2 2 5x b 2 4x3 y4 2 2 3x2 2y4 2 2 2c2 3c 1 2 y2 3y 1 2 2x2 xy y2 21 3w2 4w 3 2 x2 2xy 3y2 2 4x 2 21 x 2 21 3y 2 2 x 3 2 y 2 21 2y x 21 3x 2y 2 3x y 2 y x 21 2 c 2 21 2y 3 x 2y 21 5w 6 21 5x 2y 21 3x 1 2x 1 x 1 21 y 2 21 x 4y 21 2x y 21 In Exercises 55–64, find the coefficient of in the given product. Avoid doing any more multiplying than necessary. x2 x2 1 21 1 13 x x 1 2 13 x A BA x2 x 1 1 21 B x 1 55. 57. 59. 61. 62. 63. 64. 1 1 1 1 1 1 1 x2 3x 1 21 x3 2x 6 2x 3 2 x2 1 2 21 x 2 3 2 x2 x 1 x2 x 1 2 21 56. 58. 60. 2x2 1 2x 1 21 2x2 1 2 21 x2 3x 2 1 2x 2 1 4x2 x 1 2 2 In Exercises 65–70, perform the indicated multiplication |
and simplify your answer if possible. 65. 66. 67. 69. A A A A 1x 5 1x 5 B BA 21x 12y 21x 12y BA B 68. 3 1y 2 1 13x 2 BA x 13 B 70. A 7w 12x A 2y 13 BA 2 B 15y 1 B In Exercises 71–76, compute the product and arrange the terms of your answer according to decreasing powers of x, with each power of x appearing at most once. Example: (ax b)(4x c) 4ax2 (4b ac)x bc. 71. 73. 75. 1 1 1 ax b 3x 2 2 21 ax b bx a 21 x b 2 x c 21 x a 21 72. 74. 76. 2 4x c dx c 21 3rx 1 rx 1 2dx c 2 4x r 2 21 3cx d 1 1 21 79. 81. 82. 1 1 1 xm 2 21 2xn 5 xn 3 2 x3n 4xn 1 21 80. yr 1 ys 4 2 21 1 2 3y 2k yk 1 yk 3 2 21 In Exercises 83–92, find a numerical example to show that the given statement is false. Then find the mistake in the statement and correct it. 2 b 5, is false Example: The statement The when mistake is the sign on the 2. The correct statement is (b 2) b 2 but (b 2) b 2. (5 2) 7 5 2 3. since 83. 84. 85. 87. 89. 91. 92. 1 1 1 x 3y 4 y 2 3y 2 3 1 x 2 3y 4 1 x y 2 2 x y2 2 7xy 7x 7y 21 2 1 y y y y3 x 3 a b x 2 2 21 a2 b2 21 x2 5x 6 a3 b3 2 3 2x3 86. 88. 90. 1 1 1 2x 2 x y a b 2 2 x2 y2 2 2 a2 b2 In Exercises 93 and 94, explain algebraically why each of these parlor tricks always works. 93. Critical Thinking Write down a nonzero number. Add 1 to it and square the result. Subtract 1 from the original number and square the result. Subtract this second square from the first one. Divide by the number with which you started. The answer is |
4. 2 94. Critical Thinking Write down a positive number. In Exercises 77–82, assume that all exponents are nonnegative integers and find the product. Example: 2xk(3x xn1) (2xk)(3x) (2xk)(xn1) 77. 3r343t 6xk1 2xkn1. 78. 2xn 8xk 2 21 1 Add 4 to it. Multiply the result by the original number. Add 4 to this result and then take the square root. Subtract the number with which you started. The answer is 2. 95. Critical Thinking Invent a similar parlor trick in which the answer is always the number with which you started. Section A.2 Arithmetic of Algebraic Expressions 977 A.3 Factoring Factoring is the reverse of multiplication: We begin with a product and find the factors that multiply together to produce this product. Factoring skills are necessary to simplify expressions, to do arithmetic with fractional expressions, and to solve equations and inequalities. The first general rule for factoring is Common Factors If there is a common factor in every term of the expression, factor out the common factor of highest degree. Example 1 4x6 8x, for example, each term contains a factor of In 4x6 8x 4x x5 2 x3y2 2xy3 3x2y4 so that Similarly, the common factor of highest degree in xy2. 2 is 4x, 1 and x3y2 2xy3 3x2y4 xy2 x2 2y 3xy2. 2 1 ■ You can greatly increase your factoring proficiency by learning to recognize multiplication patterns that appear frequently. Here are the most common ones. Quadratic Factoring Patterns Difference of Squares Perfect Squares u2 v2 (u v)(u v) u2 2uv v2 (u v)2 u2 2uv v2 (u v)2 Example 2 a. b. c. 2, 3y x2 9y2 x2 9y2 y2 7 y2 36r2 64s2 can be written x 3y x 3y 21 2 17 B A A 2 8s 6r 1 2 1 2 3r 4s 2 2 x2. 2 y 17 2 1 3r 4s 1 1 2 1 2 1 y 17 BA 6r 8s 21 4 1 2.* B |
6r 8s 2 3r 4s 21 a difference of squares. Therefore, 3r 4s. 2 ■ *When a polynomial has integer coefficients, we normally look only for factors with integer coefficients. But when it is easy to find other factors, as here, we shall do so. 978 Algebra Review Example 3 Since the first and last terms of to use the perfect square pattern with 4x2 36x 81 4x2 36x 81 u 2x 2 36x 92 2 2 2x 9 92 2x 2x 2 1 1 2 are perfect squares, we try and v 9: 2x 9 2. 2 1 ■ Cubic Factoring Patterns Difference of Cubes Sum of Cubes Perfect Cubes u3 v3 (u v)(u2 uv v2) u3 v3 (u v)(u2 uv v2) u3 3u2v 3uv2 v3 (u v)3 u3 3u2v 3uv2 v3 (u v)3 Example 4 a. b. c. x 5 x 5 x3 125 x3 53 x3 8y3 x3 1 1 3 x3 12x2 48x 64 x3 12x2 48x 43 21 21 x 2y x 2y x2 x 2y 1 x2 2xy 4y2 x2 5x 52 2 x2 5x 25 2 2 3 21 2y 1 1. 1 2 2 2 4 2y. 2 x3 3x2 4 3x 42 43 x 4 3. 1 2 ■ When none of the multiplication patterns applies, use trial and error to factor quadratic polynomials. If a quadratic has two first-degree factors, then the factors must be of the form for some constants a, b, c, d. The product of such factors is ax b cx d cx d ax b and 1 21 2 acx2 adx bcx bd acx2 x bd. ad bc 1 2 Note that ac is the coefficient of and bd is the constant term of the product polynomial. This pattern can be used to factor quadratics by reversing the FOIL process. x2 Example 5 x2 9x 18 If ficient of only integer factors of 1). The only possibilities for b and d are then we must have a ± 1 and cx d ax b (constant term). Thus, factors as 1 bd 18 ) and x 2 21, 2 ac |
1 c ± 1 (coef(the ± 1, ± 18 or ± 2, ± 9 or ± 3, ± 6. We mentally try the various possibilities, using FOIL as our guide. For x 9 x 2. example, we try 2 so this prodThe sum of the outside and inside terms is and check this factorization: 1 9x 2x 11x, b 2, d 9 21 Section A.3 Factoring 979 uct can’t be d 6 x2 9x 18. By trying other possibilities we find that leads to the correct factorization: x2 9x 18 x 3 b 3, x 6. 2 ■ 21 1 Example 6 6x2 11x 4 To factor c whose product is 6, the coefficient of product is the constant term 4. Some possibilities are cx d, 2 x2, ax b as 21 1 we must find numbers a and and numbers b and d whose ac 6 a c ±1 ±6 ±2 ±3 ±3 ± 2 ±6 ±1 bd 4 b d ±1 ±4 ±2 ±2 ±4 ±1 Trial and error shows that 2x 1 3x 4 2 21 1 6x2 11x 4. ■ Occasionally the patterns above can be used to factor expressions involving larger exponents than 2. Example 7 a. b. x6 y6 x8 1 1 Example 8 x3 y3 x3 y3 2 1 2 x3 2 1 x y 1 x4 2 y3 1 x2 xy y2 x4 1 x4 1 x4 1 21 2 1 2 21 x y 21 x4 1 x2 1 x2 1 2 21 21 21 21 21 1 1 1 2 x2 xy y2. 2 21 x2 1 x 1 2 21 x 1. 2 To factor x4 2x2 3, Then, x4 2x2 3 1 x2 u x2. let 2 2x2 3 u2 2u 3 2 u 1 21 1 x2 1 21 1 x2 1 BA A u 3 2 x2 3 2 x 13 ■ ■ ■ x 13 BA. B ■ ■ Example 9 can be factored by regrouping and using the distrib- 3x3 3x2 2x 2 utive law to factor out a common factor: 3x2 3x3 3x2 2x 2 1 2 2 1 x 1 1 3x2 2 2 1 2 x 1 21 1 x 1. 2 2 ■ ■ ■ |
980 Algebra Review Exercises A.3 In Exercises 1–58, factor the expression. 37. x3 125 38. y3 64 1. x2 4 3. 9y2 25 2. x2 6x 9 4. y2 4y 4 39. x3 6x2 12x 8 40. y3 3y2 3y 1 41. 8 x3 42. z3 9z2 27z 27 5. 81x2 36x 4 6. 4x2 12x 9 43. x3 15x2 75x 125 7. 5 x2 8. 1 36u2 9. 49 28z 4z2 10. 25u2 20uv 4v2 44. 27 t3 46. x3 1 11. x4 y4 13. x2 x 6 15. z2 4z 3 12. x2 1 9 14. y2 11y 30 16. x2 8x 15 17. y2 5y 36 18. z2 9z 14 19. x2 6x 9 20. 4y2 81 21. x2 7x 10 22. w2 6w 16 23. x2 11x 18 24. x2 3xy 28y2 25. 3x2 4x 1 26. 4y2 4y 1 27. 2z2 11z 12 28. 10x2 17x 3 29. 9x2 72x 30. 4x2 4x 3 31. 10x2 8x 2 32. 7z2 23z 6 33. 8u2 6u 9 34. 2y2 4y 2 35. 4x2 20xy 25y2 36. 63u2 46uv 8v2 45. x3 1 47. 8x3 y3 49. x6 64 51. y4 7y2 10 48. 50. 3 1 x 1 2 1 x5 8x2 52. z4 5z2 6 53. 81 y4 54. x6 16x3 64 55. z6 1 56. y6 26y3 27 57. x4 2x2y 3y2 58. x8 17x4 16 In Exercises 59–64, factor by regrouping and using the distributive law (as in Example 9). 59. x2 yz xz xy 60. x6 2x4 8x2 16 61. a3 2b2 2a2b ab 62. u2v 2w2 |
2uvw uw 63. x3 4x2 8x 32 64. z8 5z7 2z 10 65. Critical Thinking Show that there do not exist real numbers c and d such that x c. x d 1 21 2 x2 1 A.4 Fractional Expressions Quotients of algebraic expressions are called fractional expressions. A quotient of two polynomials is sometimes called a rational expression. The basic rules for dealing with fractional expressions are essentially the same as those for ordinary numerical fractions. For instance, the “cross products” are equal: 2 6 4 3. In the general case we have 2 4 3 6 and Section A.4 Fractional Expressions 981 Properties of Fractions 1. Equality rule: a b c d exactly when ad bc. * 2. Cancellation property: If k 0, then ka kb a b. The cancellation property follows directly from the equality rule because ka b kb a. 1 2 1 2 Example 1 Here are examples of the two properties: x x 1 x2 2x 1. 2. x2 2x x2 x 2 x4 1 x2 1 1 because the cross products are equal: 1 x2 1 21 x2 1 21 x2 1 1 2 x 1 x3 x2 2x 2 x2 1 1 x2 1. 2 x2 x 2 1 x. 2 ■ A fraction is in lowest terms if its numerator (top) and denominator (bottom) have no common factors except To express a fraction in lowest terms, factor numerator and denominator and cancel common factors. ± 1. Example 2 x2 x 6 x2 3x 21 21 1 1 x 3 x 1. ■ To add two fractions with the same denominator, simply add the numera b ators as in ordinary arithmetic: Subtraction is done a c b c b. similarly. Example 3 7x2 2 x2 3 4x2 2x 5 x2 3 7x2 2 2 1 4x2 2x 5 1 x2 3 2 7x2 2 4x2 2x 5 x2 3 3x2 2x 7 x2 3. ■ *Throughout this section we assume that all denominators are nonzero. 982 Algebra Review To add or subtract fractions with different denominators, you must first find a common denominator. One common denominator for a/b and c/d is the product of the two denominators bd because |
both fractions can be expressed with this denominator: a b ad bd and c d bc bd. Consequently, a b c d ad bd bc bd ad bc bd and a b c d ad bd bc bd ad bc bd. Example 4 2x 1 3x x2 2 x 1 x2 2 3x 1 x 1 3x 1 x2 2 3x 2 2 2 1 x 1 2 21 x 1 2x 1 3x 2x 1 1 1 21 2 x 1 2 x 1 1 3x 2x2 x 1 3x3 6x 3x2 3x 3x3 2x2 5x 1 3x2 3x. 1 2 ■ Although the product of the denominators can always be used as a common denominator, it’s often more efficient to use the least common denominator. The least common denominator can be found by factoring each denominator completely (with integer coefficients) and then taking the product of the highest power of each of the distinct factors. Example 5 1 120 1 In the sum 100 120 23 3 5. The distinct factors are 2, 3, 5. The highest exponent of 2 in either denominator is 3, the highest of 3 is 1, and the highest of 5 is 2. So the least common denominator is the denominators are 23 3 52 600 100 22 52 and, 1 100 1 120 6 600 5 600 11 600. ■ Example 6 To find the least common denominator of 3x 7 x4 x3, factor each of the denominators completely: 1 x2 2x 1, 5x x2 x, and x 1 1 2 2, x x 1 1 2, x3 x 1, 2 1 Section A.4 Fractional Expressions 983 x 1, The distinct factors are nator is determined by the highest power of each factor: x 1. and x, The least common denomi- x3 ■ To express one of several fractions in terms of the least common denominator, multiply its numerator and denominator by those factors in the common denominator that don’t appear in the denominator of the fraction. Example 7 The preceding example shows the least common denominator (LCD) of 1 x 1 2, 2 x 1 5x x 1 2 1 3x 7 x 1 x3, and 1 2 2 1 1 x 1 5x x 1 1 1 x 1 5x x 1 x 1 3x 7 x 1 x3 |
1 2 x 1 2 3x 7 x 1 x3 1 1 1 2 to be x3 x 1 2 2 1 x 1. Therefore, 2 2 2 x3 x3 x2 x2 21 21 x3 2 x 1 2 2 x 1 x3 1 x 1 5x3 1 x 1 x3 1 3x 7 x3 21. 21 x 1 Example 8 To find 1 z 3z z 1 z2 z 1 1 2 1 z 3z z 1 z2 z 1 1 2 2 z3 z 1 we use the LCD z 2 z 1 2 3z2 3z2 z 1 z3 1 1 z 1 2 z z2 2z 1 3z3 3z2 z3 z 1 2 1 2z3 4z2 2z ■ Multiplication of fractions is easy: Multiply corresponding numerators and denominators, then simplify your answer. Example 9 x2 1 x2 2 3x 4 x 1 1 1 3x 4 x2 1 21 x 1 x2 2 21 1 x 1 x 1 2 2 3x 4 21 x2 2 21 x 1 21 2 984 Algebra Review 1 2 1 x 1 3x 4 21 x2 2 2. ■ Division of fractions is given by the rule: Invert the divisor and multiply: a b c d a b d c ad bc. Example 10 x2 x 2 x2 6x 9 x2 1 x 3 x 3 x2 1 x2 x 2 x2 6x 9 x 2 x 1 2 21 21 1 1 x 3 x 1 x 1 21 1 2 2. 2 ■ Division problems can also be written as fractions. For instance, means 8 2 8 2 4. Similarly, the compound fraction a b c d means a b c d. So, the basic rule for simplifying compound fractions is: Invert the denominator and multiply it by the numerator. Example 11 16y2z 8yz2 yz 6y3z3 16y2z 8yz2 6y3z3 yz 16 6 y5z4 8y2z3 2 6 y52z43 12y3z. y2 y 2 y3 y y2 y 2 1 y3 y y 2 1 y2 21 a. b. 2 y3 y y2 y 2 y 1 y2 1 2 y 2 1 1 y 2 21 y2 1. 2 ■ Exercises A.4 In Exercises 1–10, express the |
fraction in lowest terms. 1. 4. 63 49 x2 4 x 2 2. 5. 121 33 x2 x 2 x2 2x 1 3. 6. 13 27 22 10 6 4 11 12 z 1 z3 1 7. a2 b2 a3 b3 8. x4 3x2 x3 x c 9. 1 x2 cx c2 21 x4 c3x 2 10. x4 y4 x2 y2 1 21 x2 xy 2 Section A.4 Fractional Expressions 985 In Exercises 11–28, perform the indicated operations. In Exercises 43–60, compute the quotient and express in lowest terms. 5 6 3c e 13. 19 7 a 1 2b 1 3 16. r s s t t r 2a b2 3a b3 19. 1 x 1 1 x 11. 14. 17. 3 7 1 a b c 2 5 2a b c b 12. 15. 18. 7 8 c d a b 20. 1 2x 1 1 2x 1 21 x2 8x 16 22. 1 x 1 xy 1 xy2 23. 1 x 1 3x 4 24. 3 x 1 4 x 1 25. 1 x y x y x3 y3 26. 27. 28 21 x 2 2 2 1 x 1 4x 1 x 2 3 2 21 x y x2 xy x y 2 2 21 1 x 1 3 1 2 6x 2 x 1 2 1 4 3 2 x2 y2 1 2 2 In Exercises 29–42, express in lowest terms. 29. 3 4 12 5 10 9 31. 3a2c 4ac 8ac3 9a2c4 7x 11y 66y2 14x3 30. 10 45 6 14 1 2 32. 6x2y 2x y 21xy 34. ab c2 cd a2b ad bc2 43. 5 12 4 14 46. 3x2y 2 xy 2 1 3xyz x2y 44. 47. 100 52 27 26 x 3 x 4 2x x 4 45. uv v2w uv u2v 48 x2 2x x2 4 49. x y x 2y x y 2 a xy b 50. u3 v3 u2 v2 u2 uv v2 u v 51. c d 2 2 1 c2 d2 cd 52 53. 1 x2 1 x |
1 y2 55. 3 6 y 1 1 y 1 1 3x 5 6x2 1 4y 1 y 1 x2 57. 59 54. 56. 58. 60. 33. 35. 37. 39. 40. 41. 42. 3x 9 2x 8x2 x2 9 36. 4x 16 3x 15 2x 10 x 4 5y 25 3 y2 y2 25 38. 6x 12 6x 8x2 x 2 u u 1 u2 1 u2 t2 t 6 t2 6t 9 t2 4t 5 t2 25 2u2 uv v2 4u2 4uv v2 8u2 6uv 9v2 4u2 9v2 2x2 3xy 2y2 6x2 5xy 4y2 6x2 6xy x2 xy 2y2 In Exercises 61–67, find a numerical example to show that the given statement is false. Then find the mistake in the statement and correct it. 61. 1 a 1 b 1 a b 62. x2 x2 x6 1 x3 63. 2 1 1a 1bb a 1 a b 64. r s r t 1 s t 65. u v v u 1 66. 1 x 1 y 1 xy 1x 1y 67. A 1 1x 1y B x y 986 Algebra Review A.5 The Coordinate Plane The Distance Formula We shall often identify a point with its coordinates and refer, for example, to the point (2, 3). When dealing with several points simultaneously, it is customary to label the coordinates of the first point the sec1 x3, y32 and so on.* Once the plane is ond point coordinatized, it’s easy to compute the distance between any two points: the third point x1, y12 x2, y22,,, 1 1 The Distance Formula The distance between points (x1, y1) x2)2 (y1 and y2)2. (x2, y2) is 2(x1 y Before proving the distance formula, we shall see how it is used. −2 2 (−1, −3) −2 −4 10 Figure A.5-1 x Example 1 To find the distance between the points 1, 3 A.5–1, substitute and formula. x1, y12 for 1 2 1 1 1, 3 1 2, 4 2 for 2 and |
1 x2, y22 1 2, 4 in Figure in the distance 2 (2, −4) Distance formula: Substitute: Simplify: 1 2 x22 distance 2 x1 1 2 2 2 3 2 2 2 19 1 110 2 1 1 1 2 1 y22 y1 3 1 2 1 3 4 2 4 2 22 The order in which the points are used in the distance formula doesn’t make a difference. If we substitute for x2, y22 1, 3 x1, y12 2, 4 and for, 1 2 1 2 1 1 we get the same answer 232 1 2 110. 1 2 4 1 2 ■ CAUTION cannot be 2a 2 4b 2 simplified. In particular, it is not equal to a 2b. Example 2 To find the distance from (a, b) to b for numbers, substitute a for tance formula: x1, 2a, b 1 y1,, 2a for 2 where a and b are fixed real x2, in the disfor and b y2 2 x1 1 x22 2 y1 1 y22 1 a 2a 2 2 2 2 a 2 1 2a2 4b2 22 2 2a2 2 2b 1 2 2 ■ x1 * “ ” is read “x-one” or “x-sub-one”; it is a single symbol denoting the first coordinate of the first point, just as c denotes the first coordinate of (c, d). Analogous remarks apply to y1, x2, and so on. Section A.5 The Coordinate Plane 987 Proof of the Distance Formula and Q in the plane. We must find length d of line segment PQ. Figure A.5-2 shows typical points P y y1 y2 ⎜y1 − y2 ⎜ P(x1, y1) d R x1 Q(x2, y2) x x2 ⎜x1 − x2 ⎜ Figure A.5-2 x2 to As shown in Figure A.5-2, the length of RQ is the same as the distance from x1 Similarly, the length y2 0 of PR is the same as the distance from y1. According to the Pythagorean Theorem* the length d of PQ is given by: on the x-axis (number line), namely, y2. to on the y-axis |
, namely, 0 x2 0 x1 y1 x1 Since Length PQ 2 d2 0 c2 length PR 1 y2 0 y1 0 c2 0, 2 y22 Since the length d is nonnegative, we must have d 2 y22 length RQ 2 2 x2 0 (because x22 c2 0 d2 2 2 y1 y1 x1 x1 x22 The distance formula can be used to prove the following useful fact (see Exercise 54). this equation becomes: The Midpoint Formula The midpoint of the line segment from y2 2 x2 2 y1 x1, a b (x1, y1) to (x2, y2) is y (−1, 4) 2 −1 Example 3 (1, 5 2) (3, 1) x To find the midpoint of the segment joining 1 3, x1 mula in the box with x2 2 x2 1 3 2 1, y2 2 4, x1 y1 y1,, b a a 4 1 2 3 as shown in Figure A.5-3. 1, 4 and 2 y2 and (3, 1), use the for 1. The midpoint is 5 2b 1, a b ■ Figure A.5-3 *See the Geometry Review Appendix. 988 Algebra Review y y = x2 − 2x − 1 4 3 2 1 (−1, 2) x 1 2 3 (1.5, −1.75) −2 −1 (0, −1) −2 Figure A.5-4 (x, y) r (c, d) Figure A.5-5 Circle Equation Graphs A graph is a set of points in the plane. Some graphs are based on data points. Other graphs arise from equations, as follows. A solution of an equation in variables x and y is a pair of numbers such that the substitution of the first number for x and the second for y produces a true statement. For instance, 5x 7y 1 because 3, 2 1 2 5 3 7 is a solution of 2 1 1 is not a solution because 2 2 2, 3 2 1 The graph of an and equation in two variables is the set of points in the plane whose coordinates are solutions of the equation. Thus the graph is a geometric picture of the solutions. 5 2 1 7 3 1. Example 4 y x2 2x 1 The graph of is shown in Figure A.5- |
4. You can readily verify that each of the points whose coordinates are labeled is a soluis a solution because tion of the equation. For instance, 1 02 2 0, 1 1. 0 1 2 1 2 ■ Circles If (c, d) is a point in the plane and r a positive number, then the circle with center (c, d) and radius r consists of all points (x, y) that lie r units from (c, d), as shown in Figure A.5-5. According to the distance formula, the statement that “the distance from (x, y) to (c, d) is r units” is equivalent to: Squaring both sides shows that (x, y) satisfies this equation r2 1 Reversing the procedure shows that any solution (x, y) of this equation is a point on the circle. Therefore, 2 1 2 The circle with center (c, d) and radius r is the graph of (x c)2 (y d)2 r 2. x c 2 y d 2 r2 We say that ter (c, d) and radius r. If the center is at the origin, then the equation has a simpler form: is the equation of the circle with cenand 0, 0 c Circle at the Origin The circle with center (0, 0) and radius r is the graph of x2 y2 r2. Section A.5 The Coordinate Plane 989 Example 5 a. Letting r 1 shows that the graph of is the circle of radius 1 centered at the origin, as shown in Figure A.5-6. This circle is called the unit circle. x2 y2 1 b. The circle with center 3, 2 1 and radius 2, shown in Figure A.5-7, is 2 the graph of the equation x 11 2 3 1 22 1 y 2 2 2 22 or equivalently. ■ y 2 1 −2 −1 −1 −2 x 1 2 (−3, 2) 2 y 4 3 2 1 x Figure A.5-6 Figure A.5-7 −5 −4 −3 −2 −1 1 Example 6 Find the equation of the circle with center (2, 4). 1 3, 1 2 that passes through x Solution We must first find the radius. Since (2, 4) is on the circle, the radius is the distance from (2, 4) to 1 2 3 as shown in Figure A. |
5-8, namely, 2 11 25 126 3 22 The equation of the circle with center at 3, 1 and radius 26 2 x2 6x 9 y2 2y 1 26 x2 y2 6x 2y 16 0. 1 1 22 1 y 1 2 2 1 1 126 2 B 126 is ■ The equation of any circle can always be written in the form x2 y2 Bx Cy D 0 for some constants B, C, D, as in Example 6 (where D 16 determined. C 2, Conversely, the graph of such an equation can always be B 6,. 2 y (2, 4) 26 (3, −1) Figure A.5-8 990 Algebra Review Example 7 To find the graph of sides by 3 and rewrite the equation as 3x2 3y2 12x 30y 45 0, we divide both x2 4x 1 2 1 y2 10y 2 15. Next we complete the square in both expressions in parentheses (see page x2 4x, we add 4 (the square of half the 91). To complete the square in coefficient of x) and to complete the square in we add 25 (why?). In order to have an equivalent equation we must add these numbers to both sides: y2 10y x2 4x 4 1 1 x 2 y2 10y 25 y 5 2 15 4 25 2 2 14 2 1 2 2 1 Since radius 14 1 114. 114 2, 2 this is the equation of the circle with center (2, 5) and ■ Exercises A.5 In Exercises 1–8, find the distance between the two points and the midpoint of the segment joining them. In Exercises 11–14, find the equation of the circle with given center and radius r. 1. 3. 5. 7. 3, 5 1 1, 5 1 12, 7 2 2, 1 2 13, 2 B a, b, 2 1 1 b, a 2 2. 4. 6. 8. 1, 5 2 3, 2 1 1, 2, 4 2 2, 3 2 1 1, 15, 1 A, A B 2 12, 13 s, t, 2 1 1 0, 0 2 9. According to the Information Technology Industry Council, there were about 12 million personal computers sold in the United States in 1992 and about 36 million in 1998. a. Represent the data graphically by two points. b. Find the midpoint of the line segment |
joining these points. c. How might this midpoint be interpreted? What assumptions, if any, are needed to make this interpretation? 10. A standard baseball diamond (which is actually a square) is shown in the figure at right. Suppose it is placed on a coordinate plane with home plate at the origin, first base on the positive x-axis, and third base on the positive y-axis. The unit of measurement is feet. a. Find the coordinates of first, second, and third base. b. If the left fielder is at the point (50, 325), how far is he from first base? c. How far is the left fielder in part b from the right fielder, who is at the point (280, 20)? 11. 13. 1 1 3, 4 ; r 2 2 ; r 12 0, 0 2 12. 14. 1 1 2, 2 2 B In Exercises 15–18, sketch the graph of the equation. 15. 16. 17. 18 y2 2nd base 3rd base 90 ft 90 ft Pitcher's mound 60.5 ft 1st base 90 ft 90 ft Home plate Section A.5 The Coordinate Plane 991 In Exercises 19–24, find the center and radius of the circle whose equation is given. 19. x2 y2 8x 6y 15 0 20. 15x2 15y2 10 21. x2 y2 6x 4y 15 0 22. x2 y2 10x 75 0 23. x2 y2 25x 10y 12 24. 3x2 3y2 12x 12 18y In Exercises 25–27, show that the three points are the vertices of a right triangle, and state the length of the hypotenuse. [You may assume that a triangle with sides of lengths a, b, c is a right triangle with hypotenuse c provided that a2 b2 c2. ] 25. 1 1, 1, 1 0, 0 2 12 2 2, 2, 1 2 12 2 2 12 2 b, 26. a, 0, b a 0, 0, 1 2 27. 3, 2 1 0, 4, 1 2, 2 1 2, 3 2 28. What is the perimeter of the triangle with vertices (1, 1), (5, 4), and 1 2, 5? 2 40. Find the three points that divide the line segment 4, 7 to 1 2 10, 9 into four parts |
of equal 2 from 1 length. 41. Find all points P on the x-axis that are 5 units from (3, 4). Hint: P must have coordinates (x, 0) for some x and the distance from P to (3, 4) is 5. 42. Find all points on the y-axis that are 8 units from 2, 4. 2 1 43. Find all points with first coordinate 3 that are 6 units from 1 2, 5. 2 44. Find all points with second coordinate 1 that are 4 units from (2, 3). 45. Find a number x such that (0, 0), (3, 2), and (x, 0) are the vertices of an isosceles triangle, neither of whose two equal sides lie on the x-axis. 46. Do Exercise 45 if one of the two equal sides lies on the positive x-axis. 47. Show that the midpoint M of the hypotenuse of a right triangle is equidistant from the vertices of the triangle. Hint: Place the triangle in the first quadrant of the plane, with right angle at the origin so that the situation looks like the figure. In Exercises 29–36, find the equation of the circle. 29. Center (2, 2); passes through the origin. y (0, r) 30. Center 1, 3 ; 2 1 passes through 31. Center (1, 2); intersects x-axis at 32. Center (3, 1); diameter 2. 4, 2. 2 and 3. 1 1 33. Center 5, 4 1 the x-axis. 34. Center 2, 6 1 ; tangent (touching at one point) to ; tangent to the y-axis. 2 2 35. Endpoints of diameter are (3, 3) and 1 36. Endpoints of diameter are 3, 5 1 and 1 2 37. One diagonal of a square has endpoints 1, 1. 2 7, 5. 2 3, 1 1 Find the endpoints of the other 2, 4 and 1 diagonal.. 2 38. Find the vertices of all possible squares with this property: Two of the vertices are (2, 1) and (2, 5). Hint: There are three such squares. 39. Do Exercise 38 with (c, d) and (c, k) in place of (2, 1) and (2, 5). 992 Algebra Review |
M x (s, 0) 48. Show that the diagonals of a parallelogram bisect each other. Hint: Place the parallelogram in the first quadrant with a vertex at the origin and one side along the x-axis, so that the situation looks like the figure. 2 y (a, b) c (a + c, b) c (c, 0) x 49. Show that the diagonals of a rectangle have the same length. Hint: Place the rectangle in the first quadrant of the plane and label its vertices appropriately, as in Exercises 47–48. 50. If the diagonals of a parallelogram have the same length, show that the parallelogram is actually a rectangle. Hint: See Exercise 48. 51. Critical Thinking For each nonzero real number k, the graph of all possible such circles. 1 2 x k 2 y2 k2 is a circle. Describe 52. Critical Thinking Suppose every point in the coordinate plane is moved 5 units straight up. a. To what point does each of these points go: (2, 2), (5, 0), (5, 5), (4, 1)? 0, 5, b. Which points go to each of the points in 1 2 part a? c. To what point does (a, b) go? a, b 5 d. To what point does 2 1 4a, b e. What point goes to? 1 f. What points go to themselves? 2 go? 53. Critical Thinking Let (c, d) be any point in the c 0. Prove that (c, d) and plane with 2 lie on the same straight line through the origin, on opposite sides of the origin, the same distance from the origin. Hint: Find the midpoint of the line segment joining (c, d) and c, d. 1 c, d 1 2 54. Critical Thinking Proof of the Midpoint Formula Let P and Q be the points x2, y22 respectively and let M be the point with coordinates x1, y12 and 1 1 x1 a x2 2, y1 y2 2. b Use the distance formula to compute the following: a. the distance d from P to Q; b. the distance from M to P; d1 c. the distance from M to Q. d2 d2. d. Verify that d1 d2 e. Show that d1 |
1 2 Hint: Verify that 1 2 d. and d2 d1 d. d f. Explain why parts d and e show that M is the midpoint of PQ. Section A.5 The Coordinate Plane 993 ADVANCED TOPICS B.1 The Binomial Theorem The Binomial Theorem provides a formula for calculating the product x y for any positive integer n. Before we state the theorem, some pre1 liminaries are needed. 2 n Let n be a positive integer. The symbol product of all the integers from 1 to n. For example, n! (read n factorial) denotes the 2! 1 2 2, 3! 1 2 3 6, 4! 1 2 3 4 24, 5! 1 2 3 4 5 120, 10 10 3,628,800. In general, we have this result: n Factorial Let n be a positive integer. Then p n! 1 2 3 4 (n 2)(n 1)n. 0! is defined to be the number 1. Learn to use your calculator to compute factorials. You will find! in the PROB (or PRB) submenu of the MATH or OPTN menu. Calculator Exploration 15! is such a large number your calculator will switch to scientific notation to express it. Many calculators cannot compute factorials larger than 69! If yours does compute larger ones, what is the largest factorial that you can compute without getting an error message? 994 Advanced Topics If r and n are integers with 0 r n, then Binomial Coefficients Either of the symbols n r b a or nCr denotes the number n! r!(n r)!. n r b a is called a binomial coefficient. For example, 5C3 4C2 a 5 3b 4 2b a 5! 5 3 3! 1 4! 4 2 2! 1! 2! 2 5! 3!2! 4! 2!2 21 21 1 6. 10 nCr Binomial coefficients can be computed on a calculator by using Comb in the PROB (or PRB) submenu of the MATH or OPTN menu. or Calculator Exploration Compute 56C47 56 47b a. Although calculators cannot compute 475!, they can compute many binomial coefficients, such as 475 400b a, because most of the factors cancel out (as in the previous example). Check yours. Will it also compute 475 20 b a? The preceding examples illustrate a fact whose |
proof will be omitted: Every binomial coefficient is an integer. Furthermore, for every nonnegative integer n, n 0b a 1 and n nb a 1 because n 0b a n nb a n! n 0 n! n n 2 0! 1 n! 1 n! 0!n! n! n!! 1 and n! n!0! n! n! 1.! 2 If we list the binomial coefficients for each value of n in this manner, we find that they form a rectangular array. Section B.1 The Binomial Theorem 995 o 3 0b a 4 0b a 0 0b a 2 1b a 4 2b a 1 1b a 3 2b a 1 0b a 3 1b a 2 0b a 4 1b a 2 2b a 4 3b a 3 3b a 4 4b a ∞ Calculating each binomial coefficient, we obtain the following array of numbers: row 0 row 1 row 2 row 3 row 4 1 † ∞ This array is called Pascal’s triangle. Its pattern is easy to remember. Each entry (except the 1’s at the beginning or end of a row) is the sum of the two closest entries in the row above it. In the fourth row, for instance, 6 is the sum of the two 3’s above it, and each 4 is the sum of the 1 and 3 above it. See Exercise 47 for a proof. In order to develop a formula for calculating we first calculate these products for small values of n to see if we can find some kind of pattern: 1 2 x y n, (*) 1x 1y 1x2 2xy 1y2 1x3 3x2y 3xy2 1y3 1x4 4x3y 6x2y2 4xy3 1y4 One pattern is immediately obvious: the coefficients here (shown in color) for example, this are the top part of Pascal’s triangle! In the case means that the coefficients are the numbers n 4, 1 4 6 4 1 4 0b a, 4 1b a, 4 2b a, 4 3b, a 4 4b. a 996 Advanced Topics If this pattern holds for larger n, then the coefficients in the expansion of x y are n 1 2 n 0b a, n 1b a, n 2b a, n 3b a, p, n n1b a, n |
nb a. As for the xy-terms associated with each of these coefficients, look at the pattern in (*) above: the exponent of x goes down by 1 and the exponent of y goes up by 1 as you go from term to term, which suggests that the terms of the expansion of (without the coefficients) are: x y n 1 2 xn, xn1y, xn2y2, xn3y3,..., xyn1, yn. Combining the patterns of coefficients and xy-terms and using the fact that n 0b a 1 and the expansion of n nb a x y 1 1 n. 2 suggests that the following result is true about The Binomial Theorem For each positive integer n, (x y)n xn n 1b a xn1y xn2 y2 n 2b a xn3y3 p n 3b a n n 1b a xyn1 yn. Using summation notation and the fact that the Binomial Theorem compactly as n 0b a 1 n nb a, we can write x y 2 1 n n a j0 a n jb xnjy j. The Binomial Theorem will be proved in Section B.2 by means of mathematical induction. We shall assume its truth for now and illustrate some of its uses. Example 1 Expand x y 8. 2 1 Solution We apply the Binomial Theorem in the case n 8: x y 1 2 8 x8 8 1b a x7y a 8 2b 8 4b a x6y2 x4y4 8 3b a 8 5b a x 5y3 x3y5 8 6b a x2y6 8 7b a xy7 y8. The coefficients can be computed individually by hand or by using (or COMB) on a calculator; for instance, nCr 8C2 8 2b a 8! 2!6! 28 or 8C3 8 3b a 8! 3!5! 56. Section B.1 The Binomial Theorem 997 Alternatively, you can display all the coefficients at once by making a table 8Cx, of values for the function as shown in Figure B.1-1 at left. x f 1 2 Substituting these values in the preceding expansion, we have x y 8 x8 8x7y 28x6y2 1 2 Example 2 Expand 1 z 6 |
. 2 1 Solution Note that y z, 1 z 1 n 6: and Figure B.1-1 56x5y3 70x4y4 56x3y5 28x2y6 8xy7 y8. ■ z 2 1 and apply the Binomial Theorem with x 1, 1 z 2 1 6 16 6 1b a z 15 1 2 6 2b a z 14 1 2 2 6 3b a 13 1 z 3 2 1 6 1b a z 6 2b a z2 6 3b a z3 12 6 4b a 6 4b a 4 z 2 1 6 5b z4 6 5b a z5 z6 1 6z 15z2 20z3 15z4 6z5 z6. ■ Example 3 2 x The unemployment rate in the United States can be modeled by.0051x4.1813x3 2.2202x2 10.8212x 12.1394 f 1 where x 4 corresponds to 1994.* Write and simplify the rule of a function g(x) that provides the same information as f but has x 0 corresponding to 1994. 4 x 14 1 2 *Source: Bureau of Labor Statistics Solution f The graph of g will be the graph of f shifted 4 units to the left, which x g means that 2 1 x4.0051 2 1 x4 4x3.0051 x 4. 2 1 4.1813 210.8212 12.1394 x4 1813 32.2202 x4 1 2 1 2 4x 6x2 4 4 4 1 2 1 2 1 2 3x x3 3x2 4 4 1 2 1 3 x2 2x 2.2202 2 4 4 2 1 3 x 4 10.8212 12.1394 2 1 2 2 1 4 x4 0051x4.099x3.534x2.455x 5.92 g x 1 2 ■ 998 Advanced Topics Example 4 Show that 1.001 1 2 1000 7 2 without using a calculator. Solution 1.001 and apply the Binomial Theorem with x 1, We write 1.001 as y.001, and n 1000: 1000 1.001 1 2 1.001 1 11000 1000 2 1000 1 b a 1 1000 a 1999 1.001 2 other positive terms.001 other positive terms. But 1000 1 b 1000! 1!999! 1,000 and 2 other positive terms. a.001 1 1 |
2 1 b 1 1000 999! 1000. 999! 1000 1 1 1.001 1 2 Therefore, 1 b 1 other positive terms 1000 a.001 2 2 Hence, 1.001 1 2 1000 7 2. ■ n. Sometimes we need to know only one term in the expansion of If you examine the expansion given by the Binomial Theorem, you will see that in the second term y has exponent 1, in the third term y has exponent 2, and so on. Thus, 1 2 x y Properties of the Binomial Expansion In the binomial expansion of (x y)n, The exponent of y is always one less than the number of the term. Furthermore, in each of the middle terms of the expansion, The coefficient of the term containing yr is n. r b a The sum of the x exponent and the y exponent is n. For instance, in the ninth term of the expansion of x y 13, 2 1 y has expo- nent 8, the coefficient is 13 8 b a, and x must have exponent 5 (since 8 5 13 2. Thus, the ninth term is 13 8 b a x5y8. Section B.1 The Binomial Theorem 999 Example 5 Find the ninth term of the expansion of 2x2 a 13 41y 16 b. Solution We shall use the Binomial Theorem with n 13 and with 2x2 in place of x and in place of y. The remarks on the previous page show that the 41y 16 ninth term is Since 41y y 13 8 b 1 a 2x2 5 2 a 1 4 and 41y 16 b 8 13 8 b 1 a 2x2 5 2 a 8 41y 16 b. 1 2, 1 4 16 1312 3 1 2 2 we can simplify as follows 25 x2 B 2 y2 34 24 34 x10y2 13 12 11 10 9 5 4 3 2 13 8 b 25x10 34 x10y2 13 8 b a 13 8 b a 286 9 x10y2. ■ 20. a 1c 1 1cb 7 22. 1 3x 2 x2 13 13 1 B i2 1 where 19. 1 c 10 2 1 21. 23. 24. 25. x 3 x 1 1 13 A 13 1 A 1 i 6, 2 1 26. The median income of U.S. households (in thousands of dollars) from 1992 through 2002 can be modeled by x f 1 2.001x4.003x3.211 |
x2.322x 30.345 2 x 12 1 2 where x 2 corresponds to 1992.* Write and simplify the rule of a function g(x) that provides the same information as f but has x 0 corresponding to 2000. *Source: U.S. Census Bureau Exercises B.1 In Exercises 1–10, evaluate the expression. 1. 6! 2. 11! 8! 3. 12! 9!3! 4. 9! 8! 7! 5. 7. 8. 9. 5 3b a 5 2b a 6 a 3b 6. 12 11b a 11 10b a 7 0 b a 6 0b a 6 0b a 6 1b a 6 1b a 6 2b a 6 2b a 6 3b a 6 3b a 6 4b a 6 4b a 6 5b a 6 5b a 6 6b a 6 6b a 100 96 b a 10. 75 72b a In Exercises 11–16, expand the expression. 11. 14 12. 15. a b 7 2 2x y2 1 1 5 2 13. 16. a b 5 2 3u v3 1 1 6 2 In Exercises 17–26, use the Binomial Theorem to expand and (where possible) simplify the expression. 17. A 1x 1 6 B 18. A 2 1y 5 B 1000 Advanced Topics In Exercises 27–32, find the indicated term of the expansion of the given expression. 27. third, 29. fifth, 1 5 2 x y 1 c d 28. fourth 30. third, 1 31. fourth, 7 2 u u 2 b a 32. fifth, 1x 12 A B 7 33. Find the coefficient of 9. 2x y2 34. Find the coefficient of 10. x3 3y 2 2 1 1 x5y8 in the expansion of x12y6 in the expansion of 35. Find the coefficient of 1 x3 in the expansion of 2x 1 x2b a 6. 36. Find the constant term in the expansion of y 1 2yb a 10. 37. a. Verify that 9 1b a 9 and 9 8b a 9. b. Prove that for each positive integer n, n 1b a n and a when n n 1b n 9 n. and Note: Part a is just the case n 1 8. 38. a. Verify that 7 2b a b. Let |
r and n be integers with 7. 5b a 0 r n. Prove n rb that a case when n n rb a n 7 and r 2.. Note: Part a is just the 39. Prove that for any positive integer n, 2n n 0b a n 1b a n 2b a p n nb a. Hint: 2 1 1. 40. Prove that for any positive integer n, n 0b a n 1b a n 2b a n 3b a n 4b a p 1 k 2 1 n kb a p 1 n 2 1 n nb a 0. 42. a. Use DeMoivre’s Theorem on page 441 to find cos u i sin u 4. 2 1 b. Use the fact that the two expressions obtained in part a and in Exercise 41 must be equal to express cos 4 and sin cos in terms of sin and 4u u. u u 43. a. Let f be the function given by f x5. Let x 1 2 h be a nonzero number and compute f (but leave all binomial x h f x 2 1 1 coefficients in the form 2 5 rb a here and below). b. Use part a to show that h is a factor of and find 1 x h 2 h c. If h is very close to 0, find a simple approximation of the quantity f 1 See part b 44. Do Exercise 43 with x5. x f 45. Do Exercise 43 with x5. x f 1 1 2 2 x8 in place of x12 in place of f x 1 2 f x 2 1 46. Let n be a fixed positive integer. Do Exercise 43 in place of x5. xn with x x f f 1 2 1 2 47. Let r and n be integers such that a. Verify that b. Verify that c. Prove that 1 1 2 2!! 0 r n 1b 1b a Hint: Write out the terms on the 2 4! 2 4 for any! r n 1. left side and use parts a and b to express each of them as a fraction with denominator r 1 1 simplify the numerator, and compare the result Then add these two fractions, n r!.! 1 2 2 with n 1 r 1b a. d. Use part c to explain why each entry in Pascal’s triangle (except the 1’s at the beginning or end of a row) |
is the sum of the two closest entries in the row above it. 48. a. Find these numbers and write them one below 112, the next: 113, 110, 114. 111, b. Compare the list in part a with rows 0 to 4 of Pascal’s triangle. What’s the explanation? and row 5 of c. What can be said about 115 41. Use the Binomial Theorem with to find cos u i sin u x sin u where 4 1 2 y cos u i2 1. and Pascal’s triangle? d. Calculate all integer powers of 101 from 1010 1018, list the results one under the other, and compare the list with rows 0 to 8 of Pascal’s triangle. What’s the explanation? What happens with 1019? to Section B.1 The Binomial Theorem 1001 B.2 Mathematical Induction Mathematical induction is a method of proof that can be used to prove a wide variety of mathematical facts, including the Binomial Theorem, DeMoivre’s Theorem, and statements such as: The sum of the first n positive integers is the number 2n 7 n for every positive integer n. For each positive integer n, 4 is a factor of 7n 3n. n n 1 2 1 2. All of the preceding statements have a common property. For example, a statement such as The sum of the first n positive integers is the number n(n 1) 2 or, in symbols is really an infinite sequence of statements, one for each possible value of n: n 1: n 2: n 3 and so on. Obviously, there isn’t time enough to verify every one of the statements on this list, one at a time. But we can find a workable method of proof by examining how each statement on the list is related to the next statement on the list. For example, for n 50, the statement is 1 2 3 p 50 50 51 1 2 2. At the moment, we don’t know whether or not this statement is true. But just suppose that it were true. What could then be said about the next statement, the one for n 51: 1 2 3 p 50 51 51 52 1 2 2? Well, if it is true that 1 2 3 p 50 50 51 1 2 2 then adding 51 to both sides and simplifying the right side would yield these equalities: 1002 Advanced Topics 1 2 3 p 50 |
51 50 51 1 2 3 p 50 51 50 2 50 1 51 2 1 51 2 2 2 2 1 51 2 1 2 3 p 50 51 1 51 2 1 2 3 p 50 51 51 2 2 51 1 2 51 1 2 50 2 2 52 1 2. 2 Since this last equality is just the original statement for clude that n 51, we con- If the statement is true for n 50, then it is also true for n 51. We have not proved that the statement actually is true for that if it is, then it is also true for n 51. n 50, but only We claim that this same conditional relationship holds for any two consecutive values of n. In other words, we claim that for any positive integer k, 1 If the statement is true for n k 1. n k, then it is also true for The proof of this claim is the same argument used earlier (with k and k 1 in place of 50 and 51): If it is true that Original statement for n k ] then adding these equalities: k 1 to both sides and simplifying the right side produces 21 Original statement for n k 1 ] 1 is valid for each positive integer k. We We have proved that claim have not proved that the original statement is true for any value of n, but n k, only that if it is true for Applyk 1, 2, 3,..., we see that a recursive pattern emerges. ing this fact when Beginning with the smallest positive integer, 1, then it is also true for n k 1. Section B.2 Mathematical Induction 1003 ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪...2 ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ If the statement is true for n 1 1 2; If the statement is true for n 2 1 3; If the statement is true for n 3 1 4; If the statement is true for n 50 1 51; If the statement is true for n 51 1 52;... and so on. n 1, then it is also true for n 2, then it is also true for n 3, then it is also true for n 50, then it is also true for n 51, then it is also true for 2 n 1 in a position /2. We are finally, the statement is true |
for n 4, for and hence for the original statement is true for every positive integer n. the original statement: n 1 since 2. Since and hence it must also be true for and so on, for every value of n. Therefore, Now apply in turn each of the propositions on list n 2, to prove Obviously, it is true for 2 1 n 3, /2. 2 The preceding proof is an illustration of the following principle: Suppose there is given a statement involving the positive integer n and that: (i) The statement is true for n 1. n k (ii) If the statement is true for (where k is any Principle of Mathematical Induction positive integer), then the statement is also true for n k 1. Then the statement is true for every positive integer n. Property (i) is simply a statement of fact. To verify that it holds, you must This is usually easy, as in the prove the given statement is true for preceding example. n 1. Property (ii) is a conditional property. It does not assert that the given statement is true for then it is also true for So to verify that property (ii) holds, you need only prove this conditional proposition: but only that if it is true for n k 1. n k, n k, If the statement is true for n k, then it is also true for n k 1. In order to prove this, or any conditional proposition, you must proceed as in the previous example: Assume the “if” part and use this assumption to prove the “then” part. As we saw earlier, the same argument will usually work for any possible k. Once this conditional proposition has been proved, you can use it together with property (i) to conclude that the 1004 Advanced Topics given statement is necessarily true for every n, just as in the preceding example. Thus proof by mathematical induction reduces to two steps: Step 1 Prove that the given statement is true for n 1. Step 2 Let k be a positive integer. Assume that the given statement is true for n k. Use this assumption to prove that the statement is true for n k 1. Step 2 may be performed before step 1 if you wish. Step 2 is sometimes referred to as the inductive step. The assumption that the given statein this inductive step is called the induction ment is true for hypothesis. n k Example 1 Prove that 2n 7 n for every positive integer n. Solution Here the statement involving |
n is 2n 7 n. Step 1 When n 1, we have the statement 21 7 1. This is obviously true. Step 2 Let k be any positive integer. We assume that the statement is true for n k, 2k 7 k. We shall use this assumption to 2k1 7 k 1. prove that the statement is true for that is, we assume that n k 1, that is, that We begin with the induction hypothesis:* of this inequality by 2 yields: 2k 7 k. Multiplying both sides 3 2 2k 7 2k 2k1 7 2k. Since k is a positive integer, we know that of the inequality we have k 1, k 1. Adding k to each side k k k 1 2k k 1. Combining this result with inequality 3, we see that 2k1 7 2k k 1. *This is the point at which you usually must do some work. Remember that what follows is the “finished proof.” It does not include all the thought, scratch work, false starts, and so on that were done before this proof was actually found. Section B.2 Mathematical Induction 1005 2k1 7 k 1. ThereThe first and last terms of this inequality show that n k 1. This argument works for any fore, the statement is true for positive integer k. Thus, we have completed the inductive step. By the Principle of Mathematical Induction, we conclude that for every positive integer n. 2n 7 n ■ Example 2 Simple arithmetic shows that 72 32 49 9 40 4 10 and 73 33 343 27 316 4 79. In each case, 4 is a factor. These examples suggest that 7n 3n. This conjecture can be proved by induction as follows. For each positive integer n, 4 is a factor of Step 1 n 1, When 71 31 4 4 1, the statement is “4 is a factor of the statement is true for n 1. 71 31.” Since Step 2 7k1 3k1. 7k 3k 4D. n k, Let us denote the other factor by D, so We must use this assumpthat is, that 4 is a Let k be a positive integer and assume that the statement is true for 7k 3k. that is, that 4 is a factor of that the induction hypothesis is: tion to prove that the statement is true for factor of 7k1 3k1 7k1 7 3k |
7 3k 3k1 7k 3k 1 2 7 3 4D 1 2 4 3k 4D 2 7D 3k. [Induction hypothesis] 7 3 4 [ [Factor out 4] 7 7 7 4 7 3k 7 3k 0 ] Here is the proof: 7 3 3k n k 1, [Factor] [Since 3k ] 2 2 7k1 3k1. From this last line, we see that 4 is a factor of Thus, the staten k 1, and the inductive step is complete. Therefore, ment is true for by the Principle of Mathematical Induction the conjecture is actually true for every positive integer n. 1 1 1 1 2 ■ Another example of mathematical induction, the proof of the Binomial Theorem, is given at the end of this section. Sometimes a statement involving the integer n may be false for and (possibly) other small values of n, but true for all values of n beyond a n 1, 2n 7 n2 particular number. For instance, the statement and all larger values of n. A variation on 2, 3, 4. But it is true for the Principle of Mathematical Induction can be used to prove this fact and similar statements. See Exercise 28 for details. is false for n 5 n 1 1006 Advanced Topics A Common Mistake with Induction It is sometimes tempting to omit step 2 of an inductive proof when the given statement can easily be verified for small values of n, especially if a clear pattern seems to be developing. As the next example shows, however, omitting step 2 may lead to error. Example 3 7 1 1 2 is said to be prime if its only positive integer factors are An integer itself and 1. For instance, 11 is prime since its only positive integer factors are 11 and 1. But 15 is not prime because it has factors other than 15 and 1 (namely, 3 and 5). For each positive integer n, consider the number n2 n 11. f n 2 1 You can readily verify that 11, f 2 1 f 13, f 1 2 1 2 3 2 1 17, f 4 1 2 23, f 31 5 1 2 and that each of these numbers is prime. Furthermore, there is a clear pattern: The first two numbers (11 and 13) differ by 2; the next two (13 and 17) differ by 4; the next two (17 and 23) differ by 6; and so on. On the basis of this evidence |
, we might conjecture: For each positive integer n, the number 2 We have seen that this conjecture is true for nately, however, it is false for some values of n. For instance, when 2, 3, 4, 5. Unfortun 11, 1 n2 n 11 n 1, is prime. n f 112 11 11 112 121. 11 f 1 2 But 121 is obviously not prime since it has a factor other than 121 and 1, namely, 11. You can verify that the statement is also false for but true for n 13. n 12 ■ In the preceding example, the proposition n k, k 1 11. If the statement is true for then it is true for n k 1 k 10 and is false when If you were not aware of this and tried to complete step 2 of an inductive proof, you would not have been able to find a valid proof for it. Of course, the fact that you can’t find a proof of a proposition doesn’t always mean that no proof exists. But when you are unable to complete step 2, you are warned that there is a possibility that the given statement may be false for some values of n. This warning should prevent you from drawing any wrong conclusions. Proof of the Binomial Theorem We shall use induction to prove that for every positive integer n, x y 1 2 n xn n 1b a n 2b a xn1y xn2y2 n 3b a xn3y3 p n n 1b a xyn1 yn. Section B.2 Mathematical Induction 1007 This theorem was discussed and its notation explained in Section B.1. Step 1 n 1, When equation, and the statement reads true. there are only two terms on the right side of the preceding This is certainly x1 y1. x y 1 2 1 Step 2 Let k be any positive integer and assume that the theorem is true for that is, that x y xk2 y2 p k xk xk1y k 1b a k 2b a 1 2 n k, k rb a xkryr p k k 1b a xyk1 yk. [On the right side of this equation, we have included a typical middle term k rb a xkryr. The sum of the exponents is k, and the bottom part of the binomial coefficient is the same as the y exponent.] We shall use this assumption to prove that the theorem |
is true for that is, that k 1 1 b a xk1y2 p k1 xk1 n k 1, xky 1b a xkryr1 p k 1 k b a xyk yk1. We have simplified some of the terms on the right side; for instance, k 1 But this is the correct state- and 1 1 n k 1: ment for, term is and the bottom part of each binomial coefficient is the same as the y exponent. and so on; the sum of the exponents of each middle The coefficients of the middle terms are k 1 1 b a,, 2 In order to prove the theorem for binomial coefficients: For any integers r and k with we shall need this fact about 0 r 6 k, n k 1, 4 k r 1b a k rb a k 1 r 1b a. A proof of this fact is outlined in Exercise 47 on page 1001. n k 1, k1 we first note that x y x y To prove the theorem for x y k. 1 2 1 Applying the induction hypothesis to 2 21 x y 1 we see that k, 2 x y 1 2 k1 x y 1 2 xk xk2y2 p a xk1y k k 2b 1b a k 2 yr1 p r 1b xk1y p yk yxk xk r1 a 1 k rb a xkryr xyk1 yk k k 1b xk1y p yk. a k 1b a 1008 Advanced Topics xxk k 1b a Next we multiply out the right-hand side. Remember that multiplying by x increases the x exponent by 1 and multiplying by y increases the y exponent by 1. k1 xk1 x y 1 2 k 1b a xky k 2b a k r 1b a xky k 1b a xk1y2 k 2b a xkr1yr k rb xk1y2 p a xkryr1 p k k 1b a k xk2y3 p rb a xkryr1 x2 yk1 xyk xyk yk1 1 k xk r 1b a 1xky k 1b a r1 2 yr2 p k k 1b a xk1y2 p k 2b a xkryr1 p 1 k 1b a k k 1b a xyk yk |
1. xk1 k r 1b a k rb a 4 Now apply statement to each of the coefficients of the middle terms. k 2b a k 1, 1 b a k 1b a k 1 2 b a. and so on. Then For instance, with r 1, statement 4 shows that Similarly, with k 1b a the expression above for r 0, 1 x y a k1 x y 1 2 k1 xk1 k k 0b a 1b becomes k 1 2 b a xk1y2 p 1 k 1 1 b a 2 xky k 1 r 1b a xkryr1 p xyk yk1. k 1 k b a n k 1, Since this last statement says the theorem is true for the inductive step is complete. By the Principle of Mathematical Induction the theorem is true for every positive integer n. Exercises B.2 In Exercises 1–18, use mathematical induction to prove that each of the given statements is true for every positive integer n. 1. 1 2 22 23 24 p 2n1 2n 1 6. 7. 1 2 1 4 1 8 p 1 2n 1 1 2n 32 33 34 p 3n1 3n 1 2 8. a 1 1 1 1 1ba 1 1 3b 2ba p 1 n 1 n 1 1 1 nb a n n 1 2 n 1 2. 3. 4. 1 3 5 7 p n2 2 4 6 8 p 2n n2 n 2n 1 2 1 5. 12 22 32 p n2 n 1 n 1 2n 1 2 21 6 9. n 2 7 n 11. 3n 3n 13. 3n 7 n 1 10. 2n 2 7 n 12. 3n 1 2n 14. n 3 2b a 7 n 15. 3 is a factor of 22n1 1 Section B.2 Mathematical Induction 1009 16. 5 is a factor of 24n2 1 17. 64 is a factor of 32n2 8n 9 18. 64 is a factor of 9n 8n 1 19. Let c and d be fixed real numbers. Prove that c c d 2 1 p 1 c 2d c 3d 2c 3 n 1 1 2 d 2 4 20. Let r be a fixed real number with r 1. Prove that 1 r r2 r3 p rn1 rn 1 r 1. Remember: reduces to 1 r |
0; r0 1. so when n 1 the left side State the two steps necessary to use this principle to prove that a given statement is true for all (See discussion on page 1005.) n q. In Exercises 29–34, use the Extended Principle of Mathematical Induction (Exercise 28) to prove the given statement. 29. 2n 4 7 n for every n 5. (Use 5 for q here.) 30. Let r be a fixed real number with r 7 1. Then for every integer n 2. (Use 2 n 7 1 nr 1 r 2 1 for q here.) 31. n2 7 n for all n 2 32. 2n 7 n2 for all n 5 as a 33. 3n 7 2n 10n for all n 4 21. a. Write each of x2 y2, product of x y b. Make a conjecture as to how x y x3 y3, and another factor. xn yn and another written as a product of factor. Use induction to prove your conjecture. x4 y4 can be and 22. Let x2 12; x1 32 22 12; 6 2 x3 xn 22 12; for every positive integer n. and so on. Prove that In Exercises 23–27, if the given statement is true, prove it. If it is false, give a counterexample. 23. Every odd positive integer is prime. 24. The number n2 n 17 is prime for every positive integer n. 25. n 1 1 2 2 7 n2 1 for every positive integer n. 26. 3 is a factor of the number n3 n 3 for every positive integer n. 27. 4 is a factor of the number n4 n 4 for every positive integer n. 28. Let q be a fixed integer. Suppose a statement involving the integer n has these two properties: i. The statement is true for ii. If the statement is true for n q. n k (where k is any integer with also true for k q 2 n k 1., then the statement is Then we claim that the statement is true for every integer n greater than or equal to q. a. Give an informal explanation that shows why this claim should be valid. Note that when q 1, Mathematical Induction. this claim is precisely the Principle of b. The claim made before part a will be called the Extended Principle of Mathematical Induction. 1010 Advanced Topics 34. |
2n 6 n! for all n 4 35. Let n be a positive integer. Suppose that there are three pegs and on one of them n rings are stacked, with each ring being smaller in diameter than the one below it (see the figure). We want to transfer the stack of rings to another peg according to these rules: (i) Only one ring may be moved at a time; (ii) a ring can be moved to any peg, provided it is never placed on top of a smaller ring; (iii) the final order of the rings on the new peg must be the same as the original order on the first peg. a. What is the smallest possible number of moves when n 2? n 3? n 4? b. Make a conjecture as to the smallest possible number of moves required for any n. Prove your conjecture by induction. 36. The basic formula for compound interest was discussed in Chapter 5. Prove by induction that the formula is valid whenever x is a positive integer. [Note: P and r are assumed to be constant.] 37. Use induction to prove DeMoivre’s Theorem: For any complex number any positive integer n, z r cos u i sin u 1 and 2 zn rn cos nu 2 1 3 i sin nu. 2 4 1 GEOMETRY REVIEW G.1 Geometry Concepts An angle consists of two half-lines that begin at the same point P, as in Figure G.1-1. The point P is called the vertex of the angle and the halflines the sides of the angle. M θ P a. Q P b. Figure G.1-1 An angle may be labeled by a Greek letter, such as angle in Figure G.1-1a, or by listing three points (a point on one side, the vertex, a point on the other side), such as angle QPM in Figure G.1-1b. u In order to measure the size of an angle, we must assign a number to each angle. Here is the classical method for doing this: 1. Construct a circle whose center is the vertex of the angle. 2. Divide the circumference of the circle into 360 equal parts (called degrees) by marking 360 points on the circumference, beginning with the point where one side of the angle intersects the circle. Label these points and so on. 3. The label of the point where the second side of the angle intersects 1°, 2°, |
3°, 0°, the circle is the degree measure of the angle. For example, Figure G.1-2 on the next page shows an angle of measure 25 degrees (in symbols, u and an angle of measure 135°. 25° b 2 Section G.1 Geometry Concepts 1011 30˚ 20˚ 10˚ 0˚ θ 1 2 0˚ 130˚ 140˚ 9 0 ˚ ˚ 80˚ 0 0 1 0 ˚ 1 1 70˚60˚50˚ 4 0 Figure G.1-2 An acute angle is an angle whose measure is strictly between such as angle 90°. and 90°, in Figure G.1-2. A right angle is an angle that measures 90° An obtuse angle is an angle whose measure is strictly between 180°, in Figure G.1-2. such as angle and 0° b u A triangle has three sides (straight line segments) and three angles, formed at the points where the various sides meet. When angles are measured in degrees, the sum of the measures of all three angles of a triangle is always 180. For instance, see Figure G.1-3. A B 60° 90° 45° 30° C C 90° 45° A B Figure G.1-3 17° 143° 20° 49° 71° 60° A right triangle is a triangle, one of whose angles is a right angle, such as the first two triangles shown in Figure G.1-3. The side of a right triangle that lies opposite the right angle is called the hypotenuse. In each of the right triangles in Figure G.1-3, side AC is the hypotenuse. Pythagorean Theorem If the sides of a right triangle have lengths a and b and the hypotenuse has length c, then c2 a2 b2. a c b 1012 Geometry Review Example 1 Consider the right triangle with sides of lengths 5 and 12, as shown in Figure G.1-4. 5 c 12 Figure G.1-4 According to the Pythagorean Theorem the length c of the hypotenuse 169 132, satisfies the equation: we see that c must be 13. c2 52 122 25 144 169. Since ■ Theorem I If two angles of a triangle are equal, then the two sides opposite these angles have the same length. Example 2 Suppose the hypotenuse of the right |
triangle shown in Figure G.1-5 has length 1 and that angles B and C measure each. 45° B x A 45° 1 45° C Figure G.1-5 Then by Theorem I, sides AB and AC have the same length. If x is the length of side AB, then by the Pythagorean Theorem: x2 x2 12 2x2 1 x2 1 2 x 1 2 1 12 B 12 2. Section G.1 Geometry Concepts 1013 (We ignore the other solution of this equation, namely, x A 1 2, since x represents a length here and thus must be nonnegative.) Therefore, the sides of a 90°45°45° triangle with hypotenuse 1 are each of length 12 2. ■ In a right triangle that has an angle of 30 side opposite the hypotenuse. angle is one-half the length of the 30, the length of the Example 3 Suppose that in the right triangle shown in Figure G.1-6 angle B is and the length of hypotenuse BC is 2. 30° C A 2 x 30° B Figure G.1-6 angle, namely, side AC, has length 30° By Theorem II the side opposite the 1. If x denotes the length of side AB, then by the Pythagorean Theorem: 12 x2 22 x2 3 x 13. ■ Example 4 The right triangle shown in Figure G.1-7 has a 30° angle at C, and side AC has length 13 2. A 3 2 B 30° C x Figure G.1-7 Theorem II 1014 Geometry Review Let x denote the length of the hypotenuse BC. By Theorem II, side AB has length 1 2 x. By the Pythagorean Theorem: 2 1 2 a x b a x2 4 2 x2 x2 13 2 b 3 4 3 3 4 4 x2 1 x 1. x2 Therefore, the triangle has hypotenuse of length 1 and sides of lengths 1 2 13 2 and. ■ Two triangles, as in Figure G.1-8, are said to be similar if their corre. sponding angles are equal (that is, 2 Thus, similar triangles have the same shape but not necessarily the same size. A D; B E; C F and F C A B D E Figure G.1-8 Section G.1 Geometry Concepts 1015 Theorem III 1016 Geometry Review Suppose triangle ABC |
with sides a, b, c is similar to triangle C F DEF with sides d, e, f (that is, A D; B E; ). Then a d b e c f. These equalities are equivalent to. The equivalence of the equalities in the conclusion of the theorem is easily verified. For example, since we have a d b e ae db. Dividing both sides of this equation by be yields: ae be a b db be d e. The other equivalences are proved similarly. Example 5 Suppose the triangles in Figure G.1-9 are similar and that the sides have the lengths indicated 10 F Figure G.1-9 Then by Theorem III, In other words, so that length AC length DF length BC length EF 18 s 3 10 3s 1018 s 10 3 b a 18. Similarly, by Theorem III, length AB length DE length BC length EF so that 3 1 r 10 3r 10 r 10 3. Therefore the sides of triangle DEF are of lengths 10, 10 3, and 10 3 18. ■ Section G.1 Geometry Concepts 1017 Technology This appendix describes calculator features used throughout the book. The first section focuses on graphing functions and sequences; the second section presents procedures for creating lists, statistics, statistical plots, and regression equations. Students who are unfamiliar with a graphing calculator should complete the entire appendix; all students may use it as a reference to specific features seen in the text. T.1 Graphs and Tables Graphing calculators have a MODE or SET UP feature that allows you to set different modes for number format, type of angle measure, graph type, and drawing mode. Select the following modes by using the procedures explained in detail below for TI-84 Plus/TI-83 and Casio 9850GB-Plus. Number format: Float or Standard Angle measure: Degree Graph type: Function or Rect Drawing mode: Connected or Connect TI-84/TI-83 Press MODE, then use the up/down arrow keys to move from one row to another and the left/right arrow keys to move to a desired selection. When the cursor is on a desired selection, press ENTER to choose that option. After all selections have been made, press 2nd QUIT to return to the home screen. Casio 9850 From the MAIN MENU, select RUN EXE. Then press SHIFT SET UP. Use the up/down arrow key to select the different mode types. Use the |
F1 through F6 buttons to select the desired setting for each mode. 1018 Technology Appendix Solutions and Graphs The graph of an equation in two variables is the set of points in the plane whose coordinates are solutions of the equation. Thus, the graph is a geometric picture of the solutions. y x2 consists of all points where x is a real numThe graph of ber. A table of values for this function is shown below along with the graphical representation of the points. See Figure T1-1a. The points suggest that the graph looks like Figure T1-1b, which is obtained by connecting the plotted points and extending the graph., 1 2 x, x2 x 2 1.5 1 0.5 0 0.5 1 1.5 2 y x2 4 2.25 1 0.25 0 0.25 1 2.25 2 0 −1−1 x 1 2 −2 0 −1−1 x 1 2 Figure T1-1a Figure T1-1b Using Technology to Graph Functions Technology improves graphing speed and accuracy by plotting a large number of points quickly. A graphing calculator graphs in the same way you would graph by hand, but it uses many more points. Viewing Windows The first step in graphing with technology is to choose a preliminary viewing window, which is the portion of the coordinate plane that will appear on the screen. Using an inappropriate viewing window may display no portion of a graph or may misrepresent its important characteristics. The viewing window for the graph in Figure T1-2a is the rectangular x, y region indicated by the dashed blue lines. It includes all points 2 1 4 x 5 To display whose coordinates satisfy this viewing window on a calculator, press the WINDOW (or SHIFT V-Window) key, and enter the appropriate numbers, as shown in Figure T1-2b for the TI-84/TI-83. Other calculators are similar. The settings Xscl 1 put the tick marks 1 unit apart on the respective axes. This is usually the best setting for small viewing windows but not for large ones. 3 y 6. Yscl 1 and and Technology Appendix 1019 (−4, 6) −4 −2 (−4, −3) y 6 4 2 −1 −3 (5, 6) x (5, −3) 1 3 5 Figure T1-2a Figure T1-2b Resolution Xres sets the pixel resolution for function graphs. When Xres is 1, functions |
are evaluated and graphed at each pixel (point) along the x-axis. At 10, functions are evaluated and graphed at every 10th pixel along the x-axis. Some calculators do not have a Xres setting, and on those that do, it should normally be set at 1. Graphing Functions The following example outlines the procedure for graphing an equation. Example 1 Using Technology to Graph a Function Use technology to graph the relation 2u3 8u 2v 4 0. Solution Step 1 Choose a preliminary viewing window. If it is unknown where the graph lies in the plane, start with a viewing. This window setting is window with called the standard window on most calculators. The window may be adjusted to fit the functions after graphing. 10 x 10 10 y 10 and Figure T1-3a To display the viewing window editor on a calculator, press WINDOW (or SHIFT V-Window) and enter the appropriate numbers, as shown in Figure T1-3a. Xscl and Yscl determine where the tick marks will be displayed on the axes. Setting both to 1 is usually best for small viewing windows but not for large ones. Step 2 Solve for the output variable, if necessary. Calculators can only graph functions in a form where the output variable is expressed as a function of the input variable. In this example, assume that the output variable is v and solve the given equation for v: 1020 Technology Appendix 2u3 8u 2v 4 0 2v 2u3 8u 4 v u3 4u 2 Rearrange terms. Divide by.2 Figure T1-3b 10 10 10 10 Figure T1-3c 6 −6 −3 10 10 10 Figure T1-4, replace v Because the calculator graphs functions of the form with y and replace u with x. The function can be represented as the following. x 1 2 y f y x3 4x 2 Step 3 Enter the function by selecting the following. TI-84/TI-83 Casio 9850 Y GRAPH from the MAIN MENU Key in the function, as shown in Figure T1-3b. Display the graph by pressing GRAPH (or DRAW). The graph of the function is shown in Figure T1-3c. Step 4 If necessary, adjust the viewing window for a better view. Notice that the point where the graph crosses the y-axis is not clear in the standard window. Changing the viewing window and displaying |
the graph again shows that the graph crosses the y-axis at 2. Figure T1-3d shows the graph after TRACE—which is discussed below—was pressed and the arrow keys were used to move the cursor to the y-intercept. ■ 3 Graphing Relations Some relations must be written as two separate functions before they can be graphed. The method used to graph the function in Example 1 can be used to graph any equation that can be solved for y. Figure T1-3d Example 2 Graphing a Relation Graph the relation given by the equation 12x2 4y2 16x 12 0. 10 Solution Solve the equation for y: 12x2 4y2 16x 12 0 4y2 12x2 16x 12 y2 3x2 4x 3 y ± 23x2 4x 3 The graph is shown in Figure T1-4, where the upper portion of the graph and lower portion. y 23x2 4x 3 y 23x2 4x 3 represents represents the ■ Other Graphing Calculator Features Trace The trace feature allows you to display the coordinates of points on a graph of a function by using the left/right arrow keys, as illustrated in Technology Appendix 1021 Technology Tip When you get a blank screen, press TRACE and use the left/right arrow keys. The coordinates of the points on the graph will be displayed at the bottom of the screen, even though they are not in the current window. Use these coordinates as a guide for selecting a viewing window in which the graph does appear. Figure T1-3d. The TRACE feature is located on the different calculators as follows. TI-84/TI-83 Casio 9850 Trace is on the keyboard. Trace is the 2nd function above F1. Zoom Using the ZOOM feature of a graphing calculator will change the viewing window. ZOOM IN may be used to obtain better approximations for the coordinates of a point, while ZOOM OUT may be used to view a larger portion of the graph. TI-84/TI-83 Casio 9850 ZOOM is on the keyboard. ZOOM is the 2nd function above F2. To use the ZOOM feature, select Zoom In (or IN) from the ZOOM menu, move the cursor to the desired point, and press ENTER. The coordinates of the cursor’s position are shown at the bottom of the screen. Repeatedly using |
Zoom In will give better approximations of the coordinates of points. The scaling factors used to Zoom In or Zoom Out may be adjusted on most calculators. To set the ZOOM factors, look for Set Factors or FACT in the Zoom menu or in the MEMORY submenu of ZOOM of TI-84/TI-83. Maximums and Minimums There are several ways to approximate the coordinates of a high point or a low point: using the TRACE feature as discussed in Example 1, using the ZOOM feature, and using the maximum finder. The maximum/ minimum finder automatically finds the highest/lowest point with the calculator’s greatest degree of accuracy. On some calculators, left and right bounds and an initial guess must be indicated. On other calculators, the cursor must be placed near the maximum point. The maximum finder is referenced by the term in the second column of the chart following this paragraph; it can be found in the menu referred to in the third column. The minimum finder works in a similar way. Minimum finders are found in the same menu. Model TI-84/TI-83 Casio 9850 Reference maximum MAX Menu(s) CALC (2nd TRACE) G-Solv 1022 Technology Appendix Technology Tip To use a square viewing window, use ZSquare, or SQR in the ZOOM menu. This will adjust the x scale while keeping the y scale fixed. Technology Tip For a decimal window that is also square, use ZDecimal in the TI-84/TI-83 ZOOM menu and INIT in the Casio 9850 V-Window menu. Graphing Exploration In a standard viewing window, graph the relation 12x2 4y2 16x 12 0, which was given in Example 2. Use TRACE, ZOOM, and the maximum and minimum finders to find the coordinates of the maximum and minimum points of the relation. Use the up/down arrow keys to switch between the functions that represent the relation. Compare the values you get from each feature and determine which is most accurate, which is fastest, and which is easiest to use. The relative maximum point’s exact coordinates are and the relative minimum point’s exact coordinates are,,. Square Windows Another useful screen is one in which one-unit segments on the x-axis are the same one-unit segments on the y-axis. This type of window is called a square window. Because |
calculator screens are wider than they are high, the y-axis in a square window must be shorter than the x-axis. On many calculators, such as the TI-84/TI-83, the ratio of height to width is about 2 3 when square. Therefore, a square window with could have 6 y 7 height-to-width ratio. Check your calculator’s height-to-width ratio by looking at the scales after producing a square window. A square window should be used to display circles and perpendicular lines. 20 2 10 x 10 2 : 3, so an x-axis with 20 units will have on its y-axis about, on a screen that has a 0 y 13 13 2 3 1 units, or Graphing Exploration and y 2x 2 y 0.5x are perpendicular because the The lines product of their slopes is. Graph both in the standard viewing window, and then graph them in a square window. Describe the different appearances of the lines in both window types. 1 Decimal Window When using the TRACE feature, a calculator typically displays points as 2.34042519, for example, rather than as 2.3 or 2.34. A screen in which the points represent values with one or two decimal places is called a decimal window. A decimal window is appropriate when accuracy to one or two decimal places is desired. Technology Appendix 1023 Technology Tip The screen widths, in pixels, of commonly used calculators: TI-84/TI-83 and Casio 9800: 95 TI-86, Sharp 9600, Casio 9850: 127 HP-38: 131 TI-89: 159 TI-92: 239 Figure T1-5a Choosing a viewing window carefully can make the trace feature much more convenient, as the following Exploration demonstrates. Graphing Exploration • Graph y 5 9 1 x 32 2, which relates the temperature x in degrees Fahrenheit and the temperature y in degrees Celsius. • Use a viewing window with 40 y 40 and 0 x k, where k is one less than the number of pixels, or points, on your calculator screen divided by 10. On a TI-84/TI-83, for example, k 95 1. (See Technology Tip at left.) 9.4 10 • Use the TRACE feature to determine the Celsius temperatures corresponding to 20°F and 77°F. • Set the window to 32 k 20 x 32 k 20, where k is the same as above. This is a decimal window |
with (32, 0) at its center. • Graph the equation again, and use TRACE to determine the Celsius equivalent of 33.8°F. Function Tables Table of Function Values The table feature of a calculator is a convenient way to display points and evaluate functions. By setting the initial input value and the increment value, a table of values may be displayed for functions stored in the memory, as shown in Figure T1-5a. If more than one function is in the function memory, the output values for each are stored in separate columns of a table. To access the table setup screen, select the following. TI-84/TI-83 Casio 9850 TBLSET on the keyboard RANG in the TABLE menu The increment is labeled Tbl on TI-84/TI-83 and Pitch on Casio 9850. y x3 4x 2 Figure T1-5b shows a table of values for the function which is stored as Y1., ¢ Figure T1-5b Select the following to view the table of values. TI-84/TI-83 Casio 9850 TABLE on the keyboard Press TABL. Graphs On and Off An equation stays in the equation memory until you delete it. If there are several equations in memory and you want to graph only some of them, turn those equations “on’’ and the others “off.’’ On most calculators an equation is “on’’ if its equal sign is shaded and “off’’ if its equal sign is clear. 1024 Technology Appendix Graphing Conventions 1. Unless directed otherwise, use a calculator for graphing. 2. Complete graphs are required unless a viewing window is specified or the context of a problem indicates that a partial graph is acceptable. 3. If the directions say “obtain the graph,” “find the graph,” or “graph the equation,” you need not actually draw the graph on paper. For review purposes, however, it may be helpful to record the viewing window used. 4. The directions “sketch the graph’’ mean to draw the graph on paper, indicating the scale on each axis. This may involve simply copying the display that is on the calculator screen, or it may require graphing if the calculator display is misleading. Calculator Exploration 1. Tick Marks a. Set Xscl 1 so that adjacent tick marks on |
the x-axis are one unit apart. Find the largest range of x values such that the tick marks on the x-axis are clearly distinguishable and appear to be equally spaced. b. Do part a with y in place of x. 2. Viewing Window Look in the ZOOM menu to find out how many built-in viewing windows your calculator has. Take a look at each one. 3. Maximum/Minimum Finders Use your minimum finder to approximate the x-coordinates of the lowest point on the graph of 3 y 8 y x3 2x 5. in the window with 0 x 5 and The correct answer is x 2 3 A 0.81649658. How good is your approximation? 4. Square Windows Find a square viewing window on your cal- culator that has 4 x 4. 5. Dot Graphing Mode To see the points plotted by a calculator without the connecting segments, set your calculator to DOT mode, then graph Try some other equations as well. y 0.5x3 2x2 1 in the standard window. Technology Appendix 1025 Sequence Graphing Many calculators can produce a table of values and graph functions that are defined by recursive and nonrecursive sequences. In a recursive sequence, the nth term is defined in relation to a previous term or terms, such as n 2 3u u. In a nonrecursive 2n 5. sequence, the nth term is a function of the variable n, such as or To enter, display a table of values for, and graph a sequence, select the following. In general, first set the mode to sequence and enter the sequence rule. • Set the window parameters, set the drawing mode (dot or connected), and graph the sequence. • Choose the table settings, and display the table. TI-84/TI-83 MODE Y access the sequence editor screen SEQ nMin u(n) u(nMin) the minimum n value evaluated the rule of the sequence the value of the sequence at the minimum n value (recursive sequence only) WINDOW nMin the smallest n value evaluated; must be an integer the largest n value evaluated nMax PlotStart first term plotted PlotStep incremental n value (graphing only); designates which points are plotted; does not affect sequence evaluation Xmin, Xmax, Xscl, Ymin, Ymax, Yscl same as function graphing values displays the graph of sequences stored in memory table settings TblStart smallest |
n value ¢ Tbl increment GRAPH TBLSET TABLE displays a table of values of sequences stored in memory Casio 9850 RECUR TYPE a general term of the sequence { n a a linear recursion between two terms linear recursion between three terms un } n1 n2 RANG displays table range settings Start starting value of n End ending value of n a, b value of first term 1 1 (the value of n increments by 1) TABL displays a table of values G-PLT displays a graph of the data in the table Press EXIT to return to the table and EXIT again to return to the sequence editor. NOTE In the TI-84/ TI-83, nMax, PlotStart, and PlotStep must be integers greater than 1. NOTE In the TI-84/ TI-83, both Indpnt (independent) and Depend (dependent) values may be set by the user or the calculator. Input can be set Sharp 9600 by the user or the calculator. Auto: Calculator enters values. Ask: User enters values. 1026 Technology Appendix T.2 Lists, Statistics, Plots, and Regression NOTE Words printed in all caps refer to buttons to be pressed or menu choices to be selected. For example, STAT EDIT directs you to press the STAT button and then to select EDIT from the options shown. Most calculators allow you to store one or more lists that can be used with statistical operations and graphs. Although the procedures needed to create the statistical values seen in this book are shown below, refer to your calculator’s manual for a complete guide. This section outlines the procedures to create the following items: • statistical lists for 1-variable and 2-variable data • statistics for 1-variable and 2-variable data • histograms and box plots for 1-variable data • scatter plots of 2-variable data • regression equations of 2-variable data Refer to Sections 1.5, 4.3.A, 5.7, and 13.1 for examples that involve lists and regression. Lists Call up the list editor to enter data into lists by using the commands below. TI-84/TI-83 STAT Lists are L1, L2,..., L6. EDIT Casio 9850 STAT Lists are List 1, List 2,..., List 6. Press ENTER (or EXE) after each entry in a list to proceed to the next position. Use the left/right arrow keys to move among lists and |
the up/down arrow keys to move within a list. One-Variable Statistics Data for 1-variable statistics can be entered into the list editor in two ways: • Each value is entered into a single list. • Each value is entered into one list and its frequency into the corresponding position of a second list to create a frequency table. The symbols commonly used to represent the statistics for 1-variable data are as follows: n x x sn1 Sx, sx x sn, ©x ©x2 minX Q1 Med Q3 maxX sample number mean standard deviation from the sample population standard deviation from the sample sum of the data sum of squares of the data smallest value of the data value of the first quartile median of the data value of the third quartile largest value of the data Technology Appendix 1027 NOTE Directions for Casio 9850 begin from the main menu screen. To compute 1-variable statistics, use the following procedures. TI-84/TI-83 STAT CALC 1: 1-Var Stats • If the data is contained in one list, enter the list name by pressing the list name shown above the number keys 1 through 6. • If the data was entered as a frequency table, enter the list name of the data, press [, ] (comma), and enter the name of the list that contains the frequency. Casio 9850 STAT EXE CALC SET • If the data is contained in one list: 1. Set XList to the list that contains the data by pressing the key below the appropriate list name. 2. Use the down arrow key to highlight 1Var Freq and set 1Var Freq to 1 by pressing F1. • If the data is contained in a frequency table. 1. Set 1Var XList to the list that contains the data by pressing the key below the appropriate list name. 2. Set 1Var Freq to the list that contains the frequency of the data. Press EXIT to return to the list editor. Press 1VAR to display the statistics. Histograms and Box Plots (1-Variable Graphs) A histogram is a bar graph that displays 1-variable data values on the x-axis and represents each corresponding frequency as the height of the box above the x-axis. A box plot and a modified box plot display the values Q1 to Q3 as a box with a vertical line at the median. A box plot displays the values of xMin to Q1 and of Q3 to xMax as |
whiskers at either end of the box, and a modified box plot displays outliers (values at least 1.5 * (Q3–Q1) below Q1 and above Q3) as points beyond the whiskers. To create a graph of 1-variable data, enter the data into a single list or into a frequency table, and use the procedures to create a histogram or a box plot. TI-84/TI-83 • STAT PLOT • Highlight On and press ENTER. Plot1 • Highlight the desired graph type by using the arrow keys, and press ENTER. • Use the down arrow key to select Xlist. Enter the name of the list that contains the data by using the keys L1 through L6, which are the 2nd functions above the number keys 1 through 6. • Move the cursor to Freq and enter 1 if the data is contained in a single list, or enter the name of the list that contains the frequency. 1028 Technology Appendix • If a modified box plot is selected, select the type of symbol that will denote outliers. • Set the window values for xMin and xMax, and if creating a histogram, set the yMin and yMax values. Box plots ignore yMin and yMax values. • GRAPH Casio 9850 STAT EXE • GRPH SET • Select the StatGraph area desired by choosing GPH1, GPH2, or GPH3. • Use the down arrow key to highlight Graph Type. Press to display the graph type options. Choose Hist, Box (MedBox), or Box (MeanBox). MedBox shows the distribution of the data items that are grouped within Q1, Med, and Q3. MeanBox shows the distribution of the data around the mean when there is a large number of data items, and a vertical line is drawn at the mean. • Select XList and enter the name of the list that contains the data. • Select Frequency and enter 1 if the data is in one list or the name of the list that contains the frequency of the data. • Choose the graph color. • Press EXIT. • Press GPH1 (GRPH2 or GRPH3) to display the StatGraph area. Press EXIT to return to the previous screen. Two-Variable Statistics and Graphs Two-variable statistics for both the x-variable and the y-variable include all the statistics listed for 1-variable statistics; that is, the mean of the y- |
variable data is denoted as, the minimum value of the y-variable data is denoted by minY or yMin, and so forth. Additionally, xy, the sum of the product of corresponding data pairs, is also given. g y To compute 2-variable statistics, use the following procedure after entering the data pairs into two lists. TI-84/TI-83 STAT CALC 2–Var Stats • Enter the list names that contain the data pairs separated by a comma, and press ENTER. The first named list is the x-list, and the second named list is the y-list. Casio 9850 STAT EXE CALC SET • Use the down arrow key to select 2Var XList, and enter the name of the list that contains the x-data. Similarly, enter the name of the list that contains the y-data. 2Var Freq specifies the list where paired-variable frequency values are located. Enter 1 if the data pairs are entered separately in the XList, and set 2Var Freq to 1 if the data pairs are entered separately in the XList and the Ylist. • EXIT 2VAR Technology Appendix 1029 Scatter Plots Scatter plots graph the data points from Xlist and Ylist as coordinate pairs. The general procedure for graphing 2-variable data contained in two lists is as follows. • Define the statistical plot. • Define the viewing window. • Display the graph. TI-84/TI-83 • STAT PLOT, select the desired stat plot editor, highlight On, and press ENTER. • Select the icon that represents the type of graph desired and press ENTER. • Enter the names of the lists that contain the x-data and the y-data. Choose the graph type and Mark symbol. Option GS.D. represents a scatter plot. • GRAPH Casio 9850 STAT EXE GPH SET • Select the graph number: GPH1, GPH2, or GPH3. • Highlight Graph Type and choose Scat. • Enter the names of the lists that contain the x-data and the y-data, enter the frequency of the data pairs, choose the mark type and the graph color. EXIT • V-Window, enter the viewing window values, EXIT • GPH, select the graph number that contains the desired settings Regression Equations Most graphing calculators can compute the following types of regression equations. Regression Type Model linear quadratic cubic |
quartic logarithmic exponential power y ax b y ax2 bx c y ax3 bx2 cx d y ax4 bx3 cx2 dx e y a ln x y aebx y axn y c 1 ae y a sin bx bx c 1 d 2 logistic sinusoidal 1030 Technology Appendix Reference LinReg or LinearReg QuadReg CubicReg QuartReg LnReg or LogReg ExpReg PwrReg or PowerReg Logistic or LogisticReg SinReg The procedure to compute regression equations follows. Enter the data into two lists. TI-84/TI-83 STAT CALC Select the regression model by using the up/down arrow keys. Enter the names of the lists separated by a comma by using the L1 through L6 keys, which are the 2nd functions for the numerals 1 through 6. The default is L1, L2. Casio 9850 While viewing the lists, press CALC REG. Select the regression model by pressing the corresponding F key. Additional models are accessed by pressing F6. Displaying a Regression Equation’s Graph Set the bounds of the display window by selecting the following. TI-84/TI-83 WINDOW V-Window Casio 9850 TI-84/TI-83 There are two methods for entering a regression equation into the Y editor for display: automatic and manual. Note that each time a regression equation is found, the contents of RegEQ are overwritten with the new regression equation. • Automatic The calculator can automatically place the regression equation into the Y editor as Y3 when computing a regression equation. The following entry places the linear regression equation for the lists L1 and L2 into Y3 in the Y editor. LinReg L1,L2,Y3 • Manual Whenever a regression model is found, the calculator places the regression equation into the variable RegEQ (or RegEqn), which can be entered into the Y editor. Press VARS Statistics EQ RegEQ from the Y editor screen. A regression equation is displayed using the procedure to graph any type of equation. Casio 9850 • While viewing the data lists, press GRPH. • Select GPH1, GPH2, or GPH3 to view the desired graph type. (Press SET to change the options.) • Choose the type of regression model desired (X represents linear regression.) COPY lets you store the displayed regression equation as a function in the Y editor. |
Use the up/down arrow keys to select the desired Y variable. DRAW graphs the displayed regression equation Technology Appendix 1031 Regression Coefficients r2 R2 r, the correlation When some regression models are created, values of ), the coefficient of determination, are computed coefficient, and (or and stored as values. Values of and are computed for the following regression models: linear, logarithmic, power, and exponential. The value is computed for the following regression models: quadratic, cubic, of and quartic. No correlation coefficient or coefficient of determination is given for logistic and sinusoidal regression models. R2 r2 r TI-84/TI-83 r r2, and are displayed when a R2 The variables, regression equation is computed by executing the DiagnosticOn instruction, which is found in the CATALOG menu. When DiagnosticOff is set, the values for, are not displayed., and R2 r2 r Casio 9850 Neither form of a coefficient of determination is computed on a Casio. Scatter Plots of Residuals Residuals, also called relative error, are stored as a list of values in a variable named Resid or RelErr on some calculators. Scatter plots of the residuals can be graphed but may not be visible in the same viewing window as the data. Adjust the window to view the scatter plot of the residuals. TI-84/TI-83 From a STAT PLOT menu set the following: ON Type Scatter Plot XList Ylist Mark Ln (the list that contains the input data) RESID (which is found in LIST NAMES) as desired GRAPH Casio 9850 Residuals are not automatically computed when a regression equation is found. 1032 Technology Appendix T.3 Programs The programs listed here are of two types: programs to give older calculators some of the features that are built-in on newer ones and programs to do specific tasks discussed in this book (such as synthetic division). Each program is preceded by a Description, which describes, in general terms, how the program operates and what it does. Some programs require that certain things be done before the program is run (such as entering a function in the function memory); these requirements are listed as Preliminaries. Occasionally, italic remarks appear in brackets after a program step; they are not part of the program, but are intended to provide assistance when you are entering the program into your calculator |
. A remark such as “[MATH NUM menu]” means that the symbols or commands needed for that step of the program are in the NUM submenu of the MATH menu. Fraction Conversion (Built-in on TI-82/83/84/85/86) Description: Enter a repeating decimal; the program converts it into a fraction. The denominator is displayed on the last line and the numerator on the line above. Casio 9850 Fix 7 ”N ”? S N 0 S D Lbl 1 D 1 S D N D TI-82/83/84 Quadratic Formula (Built-in on other calculators) [MATH NUM menu] Rnd 2 [MATH NUM menu] 0 1 Goto 1 S N [DISP menu] 2 1 Frac Ans Ans.5 1 Norm (Int N) D Description: Enter ax2 bx c 0; the coefficients of the program finds all real solutions. the quadratic equation :ClrHome [Optional] :Disp ”AX2 BX C 0” [Optional] :Prompt A :Prompt B :Prompt C B2 4AC : 1 :If S 6 0 2 S S :Goto 1 B 1S B 1S /2A 2 /2A 2 1 1 :Disp :Disp :Stop :Lbl 1 :Disp “NO REAL ROOTS” Technology Appendix 1033 Synthetic Division (Built-in on TI-89/92) Preliminaries: Enter the coefficients of the dividend (in order of decreasing powers of x, putting in zeros for missing coefficients) as list L1 (or List 1). If the coefficients are 1, 2, 3, for example, key in {1, 2, 3,} The symbols { } are on the keyboard or in the LIST menu. and store it in The list name is on the keyboard of TI-82/83 and Sharp 9600. On TI-85/86 and HP-38, type in L1. On Casio 9850, use “List” in the LIST submenu of the OPTN menu to type in List 1. L1. L1 F x 1 2 x a and enter a. The proDescription: Write the divisor in the form x, gram displays the degree of the quotient (in order of decreasing powers of x), and the remainder. If the program pauses before |
it has displayed all these items, you can use the arrow keys to scroll through the display; then press ENTER (or OK) to continue. the coefficients of Q Q x 2 1 2 1 TI-82/83/84/85/86 :ClrHome [ClLCD on TI-85/86] :Disp “DIVISOR IS X A ” :Prompt A S L2 [See Preliminaries for how to enter list names] :L1 S N :dim L1 3 :For (K, 2, N) L11 : 1 :End A L21 on TI-85/86] S L21 dimL L1 K 1 K K 22 2 2 [LIST OPS menu] S S dim L2 :round(L2(N),9) N 1 : 1 :Disp “DEGREE OF QUOTIENT” 2 R [MATH NUM menu] 1 :Disp dim L2 :Disp “COEFFICIENTS” :Pause L2 :Disp “REMAINDER” :Disp R Casio 9850 [OPTN MATH NUM menu] Rnd Ans S R [See Preliminaries for how to enter list names] Norm [SETUP DISP menu] [OPTN LIST menu] Seq(List 2 [X], X, 1, N 1, 1) S List 2 “DEGREE OF QUOTIENT” [OPTN LIST menu] A List 2 K 1 3 4 2 S List 2 K 4 3 dim List 2 1 “COEFFICIENTS” List 2 “REMAINDER” R X A ” “DIVISOR IS A ”? S A “ List 1 S List 2 List 1 S N dim 2 S K Lbl 1 List Goto 1 Fix 9 K 4 [SETUP DISP menu] List 2 [N] 1034 Technology Appendix Glossary A absolute value (of a complex number) a bi 2a2 b2 (p. 638) 0 0 absolute value of a number For any real number c, if braic definition). number line (geometric definition). (p. 107, 108) and if (alge, then is the distance from c to 0 on the c c c 6 0 c c 0, then c c 0 0 0 0 0 0 absolute-value inequalities For a positive number k and any real number |
r, r k r k (p. 129) k is equivalent to is equivalent to r k k and or r k. r 0 0 0 0 acute angle an angle with a degree measure of less than (p. 414) 90° addition and subtraction identities trigonometric identities involving a function of the sum or difference of two angle measures (p. 582) adjacency matrix a matrix used to represent the connections between vertices in a directed network (p. 809) adjacent side (of a right triangle) abbreviated adj, the side of a given acute angle in a right triangle that is not the hypotenuse (p. 415) ambiguous case When the measures of two sides of a triangle and the angle opposite one of them are known, there may be one, two, or no triangles that satisfy the given measures. (p. 627) amplitude The amplitude of a sinusoidal function is one-half of the difference between the maximum and minimum function values and is always positive. (p. 497) See also sinusoidal function. analytic geometry the study of geometric properties of objects using a coordinate system (p. 691) angle a figure formed by two rays and a common endpoint (p. 413) angle of depression the angle formed by a horizontal line and a line below it (p. 427) angle of elevation the angle formed by a horizontal line and a line above it (p. 426) angle of inclination If L is a nonhorizontal straight line in a coordinate plane, then the angle of inclination of L is the angle formed by the part of L above the x-axis and the x-axis in the positive direction. (p. 589) Angle of Inclination Theorem If L is a nonvertical line with angle of inclination (p. 589) then tan slope of L. u u, u Angle Theorem If is the angle between nonzero u v u v cos u vectors u and v, then cos u u v u v. (p. 672) and angular speed a measure of speed of a point rotating at a constant rate around the center of a circle, given as the angle through which the point rotates over time (p. 440) approach infinity Output values of a function that get larger and larger without bound as input values increase are said to approach infinity. (p. 201) arc an unbroken part of a circle (p. 434) arc length the length of an arc, which is |
equal to the radius times the radian measure of the central angle of the arc (p. 435, 439) arccosine function the inverse cosine function, denoted by arccos x (p. 533) x g 1 2 arcsine function the inverse sine function, denoted by arcsin x (p. 530) x g 1 2 arctangent function the inverse tangent function, denoted by arctan x (p. 535) g x 1 2 area of a triangle formula The area of any triangle ABC in standard notation is 1 2 ab sin C. (p. 632) Glossary 1035 argument (p. 639) the angle u in a trigonometric expression arithmetic progression See arithmetic sequence. arithmetic sequence a sequence in which the difference between each term and the preceding term is always constant (p. 22) asymptotes of a hyperbola two lines intersecting at the center of a hyperbola which the hyperbola approaches but never touches (p. 701) augmented matrix a matrix in which each row represents an equation of a system and contains the coefficients of the variables in the equation (p. 795) average See mean. average rate of change For any function f, the average rate of change of with respect to x as x x f 1 2 changes from a to b is the value change in f x 1 change in p. 216) axes (singular: axis) nate system (p. 5) the number lines in a coordi- axis of a parabola the line through the focus of a parabola perpendicular to the directrix of the parabola (p. 709) B bar graph a visual display of qualitative data in which categories are displayed on the horizontal axis and frequencies or relative frequencies on the vertical axis (p. 845) Basic Properties of Logarithms only for for all real k; and logb 1 0 blogb x x x 7 0; for all and logb b 1; x 7 0. logb x is defined logb bk k (p. 364, 372) bell curve See normal curve. Bernoulli experiment See binomial experiment. If c is a number far from 0, then Big-Little Concept 1 c is a number close to 0. Conversely, if c is a number close to 0, then 1 c is a number far from 0. (p. 280) binomial distribution the probability distribution for a binomial |
experiment (p. 888) binomial experiment A probability experiment that can be described in terms of just two outcomes is a binomial experiment, also known as a Bernoulli experiment. It must meet the following conditions: the experiment consists of n trials whose outcomes are either 1036 Glossary successes or failures, and the trials are identical and independent with a constant probability of success, p, and a constant probability of failure, q 1 p. (p. 885) bounds test a test used to determine the lower and upper bounds for the real zeros of a polynomial function (p. 256) C Cartesian coordinate system a two-dimensional coordinate system that corresponds ordered pairs of real numbers with locations in a coordinate plane (p. 5) center of a hyperbola the midpoint of the segment that has the foci of the hyperbola as endpoints (p. 701) center of an ellipse the midpoint of the segment that has the foci of the ellipse as endpoints (p. 692) central angle an angle whose vertex is at the center of a circle (p. 434) central tendency a value that is used to represent the center of an entire data set (p. 853) change in x the horizontal distance moved from one point to another point in a coordinate plane; sometimes denoted and read “delta x” (p. 31) ¢ x, change in y the vertical distance moved from one point to another point in a coordinate plane; sometimes denoted and read “delta y” (p. 31) ¢ y, Change-of-Base Formula For any positive number logb x ln x ln b logb x log x log b (p. 374) and x,. closed interval an interval of numbers in which both endpoints of the interval are included in the set; denoted with two brackets (p. 118) coefficient nomial (p. 239) the numerical factor of a term in a poly- coefficient of determination a statistical measure, that is the proportion of variation often denoted by in y that can be attributed to a linear relationship between x and y in a data set (p. 47) r2, Cofunction Identities relate the sine, secant, and tangent functions to the cosine, cosecant, and cotangent functions, respectively (p. 586) trigonometric identities that combination an arrangement of objects in |
which order is not important; a collection of objects (p. 880) common difference the constant number, usually denoted by d, that is the difference between each term and the preceding term in an arithmetic sequence (p. 22) common logarithm (of x) at the number x, denoted log x (p. 356) the value of g x 1 2 log x common logarithmic function the inverse of the log x exponential function (p. 356) denoted 10x, x x g f 1 1 2 2 common ratio the constant value, denoted by r, given by the quotient of consecutive terms in a geometric sequence (p. 58) complement of an event are not contained in the event (p. 866) the set of all outcomes that complete graph a graph that shows all of its important features, including all peaks and valleys, points where it touches an axis, and suggests the general shape of the portions of the graph that are not in view (p. 82) completely factored (over the set of real numbers) a polynomial written as the product of irreducible factors with real coefficients (p. 253) completing the square a process used to change an x2 bx expression of the form into a perfect square by adding a suitable constant (p. 92) complex number system the number system that consists of real and nonreal numbers (p. 293) complex numbers numbers of the form where a is a real number and bi is an imaginary number (p. 294) a bi, complex plane a coordinate plane with the horizontal axis labeled for real numbers and the vertical axis labeled for imaginary numbers (p. 301, 638) components (of a vector) u the vector point at the origin and terminal point at (a, b) (p. 655) the numbers a and b in where u is a vector with initial a, b, I H composite functions the composite function of f and g is 2 read “g circle f” or “f followed by g.” (p. 193) If f and g are functions, then g f x g 21 x f 1 1 1 concave up a description of the way a curve bends if for any two points in a given interval that lie on the curve, the segment that connects them is above the curve (p. 154) concavity a description of the way that a curve bends, such as concave up or concave down (p. 154) |
conic section a curve that is formed by the intersection of a plane and a double-napped right circular cone (p. 691); Let L be a fixed line, called a directrix, P a fixed point not on L, and e a positive constant. The set of all points X in the plane such that distance between X and the fixed point XP XL distance between X and the fixed line conic section with P as one of its foci. (p. 747) e is a conjugate The conjugate of the complex number a bi a bi, is is the number (p. 296, 309) and the conjugate of a bi. a bi conjugate pairs (p. 296) complex numbers a bi and a bi Conjugate Zero Theorem For every polynomial function f, if the complex number z is a zero of f, then its conjugate, is also a zero of f. (p. 309) z, consistent system a system of equations with at least one solution (p. 781) constant function A function is said to be constant on an interval if its graph is a horizontal line over the interval; that is, if its output values are always constant as the input values are increasing. (p. 152) constant polynomial a polynomial that consists of only a constant term (p. 240) constant term the coefficient is written in the form (p. 239) in a polynomial that anxn an1xn1 p a1x a0 a0, 22 constraints that exist in linear programming (p. 829) restrictions, represented by inequalities, composition of functions a way of combining functions in which the output of one function is used as the input of another function (p. 193) compound inequality an inequality that compares more than two quantities and contains more than one inequality symbol (p. 118) compound interest This interest on an investment is compounded, or becomes part of the investment, at a particular interest rate per time period. If P dollars is invested at annual interest rate r per time period (expressed as a decimal), then the amount A after t 1 r time periods is A P (p. 345) t. 1 2 concave down a description of the way a curve bends if for any two points in a given interval that lie on the curve, the segment that connects them is below the curve (p. 154) continuous compounding This compound interest is compounded infinitely many times during a time period |
. If P dollars is invested at annual interest rate r and compounded continuously, then the amount A after t years is A Pert. (p. 348) continuous function a function whose graph is an unbroken curve with no jumps, gaps, or holes (p. 261, 939) S2, S1, S3, p convergent series a geometric series in which the terms of the sequence of partial sums get closer and closer to a particular real number S in such Sk a way that the partial sum when k is large enough; a geometric series with com6 1 mon ratio r such that is arbitrarily close to S (p. 77) r 0 0 coordinate plane See Cartesian coordinate system. Glossary 1037 coordinate system a system of locating points in a plane or in space by using ordered pairs or ordered triples, respectively, of real numbers (p. 5) any point of a feasible region where at corner point least two of the graphs of the constraints intersect (p. 829) correlation coefficient a statistical measure, often denoted by r, of how well the least squares regression line fits the data points that it models (p. 47) cosecant ratio For a given acute angle u triangle, the cosecant of is written as csc and is equal to the reciprocal of the sine ratio of the given angle. (p. 416) u in a right u u cosine ratio For a given acute angle in a right triu u is written as cos and is equal to angle, the cosine of the ratio of the adjacent side length to the length of the hypotenuse. (p. 416) cotangent ratio For a given acute angle u triangle, the cotangent of equal to the reciprocal of the tangent ratio of the given angle. (p. 416) is written as cot and is u in a right u coterminal angles angles formed by different rotations that have the same initial and terminal sides (p. 434) cubic function a third-degree polynomial function (p. 240) cycle (of a periodic function) a portion of the graph of a periodic function in which the function goes through one period (p. 493) D data information gathered in a statistical experiment (p. 843) decreasing function A function is said to be decreasing on an interval if its graph always falls as you move from left to right over the interval; that is, its output values are always decreasing as the |
input values are increasing. (p. 152) degenerate conic section a point, line, or intersecting lines formed by the intersection of a plane and a double-napped right circular cone (p. 691) degree (measure) a unit of angle measure that of a circle, denoted with the degree symbol equals 1 360 (p. 414) ° 2 1 degree (of a polynomial) the exponent of the highest power of the variable that appears with a nonzero coefficient in a polynomial (p. 240) 1038 Glossary delta See change in x or change in y. DeMoivre’s Theorem For any complex number, z r zn rn and for any positive integer n, cos n u i sin n u cos u i sin u (p. 644)., 1 2 1 denominator below the fraction bar (p. 4) 2 the expression in a fraction that lies deviation (of a data value) value from the mean of the data set (p. 857) the difference of a data x difference function For any functions 1 their difference function is the new function f g x (p. 191) g f x x f. 1 21 2 1 2 1 2 and g x, 2 1 2 difference quotient the quantity a function f (p. 219 for differential calculus a method of calculating the changes in one variable produced by changes in a related variable (p. 138) dimensions of a matrix used to indicate the number of rows and columns in a matrix (Example: an m n matrix has m rows and n columns) (p. 804) directed network a finite set of connected points in which permissible directions of travel between the points are indicated (p. 809) direction (of a vector) line segment representing a vector positive x-axis (p. 662) the angle that the directed a, b I H makes with the directrix of a parabola the line in the formation of a parabola such that its distance from any point on the parabola is equal to the distance from that point to the focus of the parabola (p. 709) discontinuous function a function that has one or more jumps, gaps, or holes (p. 937) the expression discriminant formula, used to determine the number of real solu(p. 93) tions of a quadratic equation ax2 bx c 0 in the quadratic b2 4ac. |
(p. 108) distance (between real numbers) The distance on the number line between real numbers c and d is c d 0 distance difference the constant difference between the distances from each focus of a hyperbola to a point on the hyperbola (p. 700) 0 distribution an arrangement of numerical data in order (usually ascending) (p. 846) divergent series a geometric series that is not convergent; a geometric series with common ratio r such that (p. 77) 1 r 0 0 f x h Division Algorithm If a polynomial by a nonzero polynomial q x polynomial 2 h 0 x x f that 2 x r has a degree less than the degree of the divisor, 2 1 x h (p. 243). 2 1 x is divided 2 then there is a quotient such or r and a remainder polynomial q x where either DMS form a form of degree measure expression which includes degrees, minutes, and seconds (p. 414) domain the set of first numbers in the ordered pairs of a relation (p. 6, 142) domain convention Unless information to the contrary is given, the domain of a function f includes every real number input for which the function rule produces a real number output. (p. 145) dot product a real number produced by multiplying corresponding components of two vectors and adding the products (p. 670) Double-Angle Identities involving a function of an angle multiplied by 2 (p. 593) trigonometric identities E eccentricity (of an ellipse or hyperbola) e distance between the foci where, distance between the vertices e 7 1 all ellipses and for all hyperbolas (p. 745) the ratio 0 6 e 6 1 for elementary row operations operations used on an augmented matrix that produce an augmented matrix of an equivalent system (p. 796) eliminating the parameter expressing a curve that is given by parametric equations as part of the graph of an equation in x and y (p. 757) ellipse For any points P and Q in the plane and any number k greater than the distance from P to Q, an ellipse, with foci P and Q, is the set of all points (x, y) such that the sum of the distance from (x, y) to P and the distance from (x, y) to Q is k. (p. 692) a conic section with eccentricity |
between 0 and 1, not inclusive (p. 747) empirical rule a rule that describes the areas under the normal curve over intervals of one, two, and three standard deviations on either side of the mean in terms of percentages of the number of data values (p. 892) 0 end behavior the far left and far right of the coordinate plane when x the shape of the graph of a function at is large (p. 262, 287) 0 endpoints (of an interval) set of numbers that can be expressed as an interval from c to d (p. 118) the numbers c and d in a equal matrices same dimensions and equal corresponding entries (p. 804) two or more matrices that have the equivalent equations solutions (p. 81) equivalent inequalities same solutions (p. 119) equations that have the same inequalities that have the equivalent systems same solutions (p. 795) systems of equations with the equivalent vectors vectors, such as u and v, that have the same magnitude and direction, denoted u v (p. 654) Euler’s Formula the identity (p. 688) eix cos x i sin x even function a function f for which for all x in its domain, its graph symmetric with respect to the y-axis (p. 188, 482) x f 2 2 1 1 x f any outcome or collection of outcomes in the event sample space of an experiment (p. 865) eventually fixed point orbit of c for a given function eventually produces constant output values (p. 202) the number c for which the eventually periodic point the number c for which the orbit of c for a given function eventually produces repeating output values (p. 202) expected value of a random variable the average value of the outcome values for a random variable (p. 869) experiment one or more observable outcomes (p. 864) in probability, any process that generates explicit form of a geometric sequence In a geometric sequence n 1. for all with common ratio r, un6 5 (p. 60) rn1u1 un explicit form of an arithmetic sequence In an arithmetic sequence d un with common difference d, n 2. un6 5 for all n 1 (p. 23) u1 1 2 exponential decay decay that can be represented by x f a function of the form 1 quantity at time x, P is the initial quantity when and decreases when x increases by 1 (p. 351) is the factor by which the quantity 0 6 a |
a three-dimensional coordi- first octant nate system in which all coordinates are positive (p. 790) first quartile See quartiles. five-number summary (of a data set) ing list of values: minimum, first quartile, second quartile, third quartile, and maximum (p. 861) the follow- fixed point (of an orbit) the number c for which the orbit of c for a given function produces constant output values (p. 202) focal axis of a hyperbola the line through the foci of a hyperbola (p. 701) foci (singular, focus) of a hyperbola the points in the formation of hyperbola such that the difference of the distances from each focus to a point on the hyperbola is constant (p. 701) foci (singular, focus) of an ellipse the points in the formation of an ellipse such that the sum of the distances from each focus to a point on the ellipse is constant (p. 692) focus of a parabola the point in the formation of a parabola such that its distance from any point on the 1040 Glossary function a special type of relation in which each member of the domain corresponds to one and only one member of the range (p. 7) A function consists of a set of inputs called the domain, a rule by which each input determines one and only one output, and a set of outputs called the range. (p. 142) function notation There is a customary method of denoting a function in abbreviated form. If f denotes a function and a denotes a number in the domain, then f(a) denotes the output of the function f produced by input a. (p. 9) function rule a set of operations that defines a function (p. 7) Fundamental Counting Principle In a set of k experiments, if the first experiment has n1 p, outcomes, the second has the kth has nk of outcomes for all k experiments is (p. 879) and outcomes, then the total number outcomes, n2 n1 n2 p nk. Fundamental Theorem of Algebra Every nonconstant polynomial has a zero in the complex number system. (p. 307) Fundamental Theorem of Linear Programming The maximum or minimum of the objective function (if it exists) always occurs at one or more of the corner points of the feasible region. (p |
. 829) G Gauss-Jordan elimination the method of using elementary row operations on an augmented matrix to produce a matrix in reduced row-echelon form that represents an equivalent system (p. 797) general form (of a line) a linear equation in the where A and B are not both form equal to zero (p. 39) Ax By C 0, geometric progression See geometric sequence. geometric sequence a sequence in which terms are formed by multiplying a preceding term by a nonzero constant (p. 58) graph a visual display of a set of points (p. 30) graph of an equation the set of points whose coordinates are solutions of the equation (p. 30) the calculation performed by graphical zero finder a graphics calculator in which the x-intercepts of the graph of a function are identified; labeled ROOT, ZERO, or X-INCPT in the TI-83 and Sharp 9600 CALC menu, the Casio 9850 G-SOLVE menu, the MATH submenu of TI-86/89 GRAPH menu, and the FCN submenu of the HP-38 PLOT menu (p. 84) greatest integer function a piecewise-defined function denoted as that converts a real number 4 x into the largest integer that is less than or equal to x (p. 147) x x f 2 1 3 H Half-Angle Identities involving a function of an angle divided by 2 (p. 596) trigonometric identities half-open interval an interval of numbers in which one endpoint of the interval is included in the set and the other endpoint of the interval is not included; denoted with one bracket and one parenthesis, respectively (p. 118) Heron’s Formula The area of any triangle ABC in s a standard notation is where s 1 a b c 2 (p. 633) s b s c 1s 21 21., 1 2 1 2 Hertz the unit of measure for the frequency of a sound wave, where one Hertz is one cycle per second (p. 559) histogram a display of quantitative data in which the data is divided into classes of equal size and displayed along the horizontal axis and frequencies or relative frequencies on the vertical axis (p. 848) f x g h rational function hole in a graph A point is omitted in the graph of a function and is not contained by an asymptote. For any x 1 x 1 duces both a zero |
numerator and a zero denominator, if the multiplicity of d as a zero of the related function g is greater than or equal to the multiplicity of d as a zero of the related function h, then the graph of f has a hole at and number d that pro- x d. (p. 283) 2 2 1 2 horizontal asymptote a horizontal line that the graph of a function approaches but never touches or crosses as gets large (p. 284, 952) x 0 0 horizontal compression For any positive number c 7 1, cx is the graph of f com1 pressed horizontally, toward the y-axis, by a factor the graph of y f 2 of 1 c. (p. 179) horizontal line a line that has a slope of zero and an equation of the form (p. 37) where b is the y-intercept y b, horizontal shifts For any positive number c, the y f graph of the left, and the graph of shifted c units to the right. (p. 175) is the graph of f shifted c units to is the graph of horizontal stretch For any positive number the graph of c 6 1, is the graph of f stretched hori- y f 2 zontally, away from the y-axis, by a factor of cx 1 1 c. (p. 179) horizontal line test A function f is one-to-one if and only if no horizontal line intersects the graph of f more than once. (p. 209) hyperbola For any points P and Q in the plane and any positive number k, a hyperbola with foci P and Q is the set of all points (x, y) such that the absolute value of the difference of the distance from (x, y) to P and the distance from (x, y) to Q is k. (p. 700) a conic section with eccentricity greater than 1 (p. 700) hypotenuse abbreviated hyp, the side of a right triangle that is across from the right angle (p. 415 cos 1 tan t cos t, 2 (p. 459) identities involving sin tan sin t, p t 1 2 identity an equation that is true for all values of the variable for which every term of the equation is defined (p. 454) n n identity matrix The on the diagonal from the top left to the bottom right and 0s in all other entries. For any AIn matrix, denoted InA |
A. matrix A, (p. 815) n n In, has 1s imaginary axis plane where each imaginary number sponds to the point (0, b) (p. 638) the vertical axis in the complex corre- 0 bi imaginary numbers a number of the form bi, where b is a real number and i is the imaginary unit (p. 294) The number. (p. 294, 297) i 11 inconsistent system a system of equations with no solutions (p. 781) Glossary 1041 increasing function A function is said to be increasing on an interval if its graph always rises as you move from left to right over the interval, that is, if its output values are always increasing as the input values are increasing. (p. 152) independent events rence or non-occurrence of one event has no effect on the probability of the other event (p. 867) two events such that the occur- infinite geometric series p an6 a1 where 5 with common ratio r (p. 77) a3 a2, the infinite series is a geometric sequence infinite limit a limit of infinity as x approaches some constant c; corresponds to a vertical asymptote (p. 949) infinite sequence a sequence with an infinite number of terms (p. 13) infinite series continues without end, or an infinite sequence; an a1 expression of the form the sum of terms of a sequence that in which p a3 a2, an is q a real number; also denoted by an (p. 76) a n1 inflection point tion changes concavity (p. 154, 266) a point where the graph of a func- initial point (of a vector) that extends from point P to point Q (p. 653) the point P in a vector initial side the starting position of a ray that is rotated around its vertex (p. 433) input (of a relation) numbers in the ordered pairs of a relation (p. 7) the values denoted by the first instantaneous rate of change the rate of change of a function at a particular point (p. 234) integers numbers and their opposites: 3, the set of numbers that consists of whole p, 0, 1, 2, (p. 3) 1, 2, 3, p one number c between a and b such that (p. 944) f c 1 2 k. interquartile range a measure of variability resistant to outliers; the difference between |
the first and third quartiles (p. 860) intersection method a method of solving an equax tion of the form and x y2 and finding the x-coordinate of each point of intersection (p. 82) on the same screen of a graphics calculator by graphing g g f y1 interval (of numbers) between two fixed numbers (p. 118) the set of all numbers lying c 6 d: interval notation There is a customary method of denoting an interval of numbers. For real numbers c and d with that that that such that (c, d) denotes all real numbers x such [c, d) denotes all real numbers x such and (c, d] denotes all real numbers x c x d; c 6 x 6 d; c x 6 d; [c, d] denotes all real numbers x such c 6 x d. (p. 118) interval of convergence the set of values of x for which an infinite series converges to a function (p. 520) inverse cosine function the inverse of the cosine function with a domain restricted to by (p. 533) cos 1, 1 1 x denoted x g, 3 4 1 2 inverse function an inverse relation that is a function (p. 205) inverse of a matrix For an inverse of A is an AB In such that 1 In AA A and n n and 1A In. BA In, (p. 816) matrix A, the A 1, matrix B, also denoted or equivalently, n n inverse relation the result of exchanging the input and output values of a function or relation (p. 205) inverse sine function the inverse of the sine func- tion with a domain restricted to p 2, p 2 T S, denoted by integral calculus a method of calculating quantities such as distance, area, and volume (p. 138) sin 1 x (p. 530) g x 1 2 intercepts (of a rational function) rational function, f, has a y-intercept, it occurs at f(0), and the x-intercepts occur at the numbers that are zeros of the numerator and not zeros of the denominator. (p. 279) If the graph of a intercepts (of polynomial functions) The graph of a polynomial function of degree n has one y-intercept, which is equal to the constant term, and at most n x-intercepts. (p. 264) interest |
a fee paid for the use of borrowed money; calculated as a percentage of the principal (p. 100) Intermediate Value Theorem If the function f is continuous on the closed interval [a, b] and k is any number between then there exists at least and b a f f, 1 2 1 2 1042 Glossary inverse tangent function the inverse of the tangent function with a domain restricted to by g x 1 2 tan 1 x (p. 535) p 2, p 2 T S, denoted invertible matrix an exists an inverse matrix (p. 815) n n matrix for which there irrational number not be expressed as a fraction of integers (p. 4) the set of real numbers that can- irreducible (polynomial) be written as the product of polynomials of lesser degree (p. 253) a polynomial that cannot iterations (of a function) tions of a function with itself (p. 199) the repeated composi- K kth partial sum the sum of the first k terms of a where k is a positive integer (p. 26) sequence, un6 5 L Law of Cosines For any triangle ABC in a2 b2 c2 2bc cos A, standard notation, b2 a2 c2 2ac cos B, (p. 617) and c2 a2 b2 2ab cos C. Law of Sines For any triangle ABC in standard nota- tion, a sin A b sin B c sin C. (p. 625) Laws of Exponents For any nonnegative real numbers c and d and any rational numbers r and s, crcs crs, c 0 cr dr 2 2 1 r 1 c cr cr cs d 0 r crdr, crs s crs, (p. 330) c 0 c db and cd cr leading coefficient highest power of the variable in a polynomial (p. 240) the nonzero coefficient of the least-squares regression line the one and only one line for which the sum of the squares of the residuals for a set of data is as small as possible (p. 47) length (of a vector) point Q in a vector that extends from point P to point Q, denoted the distance from point P to! PQ (p. 653) limit a number (or infinity) that a function value approaches but never reaches as the domain values of that function approach a particular value or infinity (p. |
909, 931) limit at infinity a real number limit as x gets large or small without bound; corresponds to a horizontal asymptote (p. 951) 1 x g f (p. 922, 954) Limit Theorem If f and g are functions that have limits as x approaches c and x then lim f lim g 2 1 xSc xSc linear combination The vector be a linear combination of i and j, where j are unit vectors. (p. 662) v ai bj i is said to and 1, 0 I x c, for all, 1 H I linear function a first-degree polynomial function (p. 240) linear programming a process that involves finding the maximum or minimum output of a linear function, called the objective function, subject to certain restrictions called constraints (p. 829) linear regression the computational process for finding the least-squares regression line for a set of data (p. 47) linear speed a measure of speed of a point rotating at a constant rate around the center of a circle, given as the distance that the point travels over time (p. 440) linear system a system of equations in which all equations are linear (p. 779) local extremum (plural: extrema) maximum or a local minimum (p. 266) either a local local maximum (plural: local maxima) A function is said to have a local maximum of graph of f has a peak at the point 1 for all x near c. (p. 153) f x c that is, f 1 c, f at ; if the 6 f c 2 c 22 x c 1 1 2 1 2 local minimum (plural: local minima) A function is said to have a local minimum of if the graph of f has a valley at the point for all x near d. (p. 153) f x d that is, ; d f 2 1 d, f at d 7 f 22 x d 1 1 1 2 1 2 logarithm to the base b (of x) g at the number x, denoted logb x x the value of logb x (p. 371) 1 2 logarithmic function to the base b the inverse of x f the exponential function x g 2 1 b 7 1 bx,, where b is a fixed positive number and logb x (p. 371) denoted 2 1 limit of a constant (p. 918) If |
d is a constant, then lim d d. xSc logarithmic model represent the trend in a data set (p. 389) a logarithmic function used to limit of a constant at infinity If c is a constant, xSqlim c c. then xSqlim c c (p. 953) and is a polyno- 1 f c x x f limit of a polynomial function If mial function and c is any real number, then lim f xSc limit of a rational function If function and c is any real number such that defined, then (p. 920) (p. 921 is a rational f is c 1 2 2 1 limit of the identity function For every real number c, (p. 918) 2 1 lim x c. xSc lim f xSc logarithmic scale a scale of numbers, such as the Richter scale, that is determined by a logarithmic function to measure logarithmic growth, which is very gradual and slow (p. 368) logistic model y a 1 be kx, a logistic function of the form where a, b, and k are constant, used to represent the trend in a data set (p. 389) lower bound (for the real zeros of a polynomial function) a polynomial function r and s are real numbers and the number r such that all the real zeros of are between r and s, where (p. 255) r 6 s x f 1 2 Glossary 1043 M Multiplication Principle See Fundamental Counting Principle. magnitude (of a vector) The length of a vector v v 2a2 b2. (p. 653) is a, b I H major axis of an ellipse the segment connecting the vertices of an ellipse (p. 692) Mandelbrot Set that the orbit of 0 under the function does not approach infinity (p. 304) z2 c f x 1 2 the set of complex numbers c such mathematical model a mathematical description or structure, such as an equation or graph, which illustrates a relationship between real-world quantities and which is often used to predict the likely value of an unknown quantity (p. 43) matrix an array of numbers often used to represent a system of equations (p. 795) multiplicity (of a zero) If occurs m times in the complete factorization of a polynomial, then a is called a zero |
with multiplicity m of the related polynomial function. (p. 265) is a factor that x a mutually exclusive events two events in a sample space that do not have outcomes in common (p. 866) N natural logarithm (of x) the number x, denoted ln x (p. 358) the value of g x 1 2 ln x at natural logarithmic function the inverse of the nat ln x x f ural exponential function (p. 358) denoted ex, x g 1 2 2 1 matrix addition addition of corresponding entries of matrices that have the same dimensions (p. 804) natural numbers The set of numbers that consists of p counting numbers: 1, 2, 3, (p. 3) matrix equation a matrix equation in the form AX B that represents a system of equations, where A contains the coefficients of the equations in the system, X contains the variables of the system, and B contains the constants of the equations (p. 814) matrix multiplication a method of multiplying two compatible matrices to produce a product matrix (p. 806) matrix subtraction subtraction of corresponding entries of matrices that have the same dimensions (p. 804) mean a measure of central tendency—also known as the average, denoted by and read “x bar”—that is calculated by adding the data values and then dividing the sum by the number of data values (p. 853) x mean of a random variable See expected value of a random variable. median a measure of central tendency that is, or indicates, the middle of a data set when the data values are arranged in ascending order (p. 855) minor axis of an ellipse the segment through the center of the ellipse, perpendicular to the major axis, and with points of the ellipse as endpoints (p. 692) Negative Angle Identities sin t, sin tan t t cos 1 tan t t cos t, 2 (p. 459) 1 1 2 2 negative correlation the relationship between two real-world quantities when the slope of the leastsquares regression line that represents the relationship is negative, that is, as one quantity increases, the other quantity decreases (p. 52) no correlation the relationship between two realworld quantities when the correlation coefficient, r, for a least-squares regression line that represents the relationship is close to zero; no apparent trend in the data (p. 52) nonlinear |
system a system of equations in which at least one equation is nonlinear (p. 779) nonnegative integers (p. 3) 1, 2, 3, p the set of whole numbers: 0, nonsingular matrix See invertible matrix. norm (of a vector) See magnitude. normal curve the graph of a probability density function that corresponds to a normal distribution: bell-shaped and symmetric about the mean, with the x-axis as a horizontal asymptote (p. 889) minute a unit of degree measure equal to degree (p. 414) 1 60 of a normal distribution a distribution of data that varies about the mean in such a way that the graph of its probability density function is a normal curve (p. 889) mode a measure of central tendency that is given by the data value that occurs most frequently in a data set (p. 855) modulus See absolute value of a complex number. 1044 Glossary nth root of a complex number The nth root of a bi zn a bi. is any of the n solutions of the equation (p. 645) nth roots For any real number c and any positive or2n integer n, the nth root of c is denoted by either c and is defined to be the solution of 1 c n odd or the nonnegative solution of even and nonnegative. (p. 328) xn c when n is xn c when n is numerator above the fraction bar (p. 4) the expression in a fraction that lies numerical derivative A calculator term for the instantaneous rate of change at a given input value; denoted nDeriv, nDer, d/dx, dY/dX, or (p. 237) 0. O objective function a linear function of which a minimum or maximum is obtained in linear programming (p. 829) oblique asymptote See slant asymptote. oblique triangle a triangle that does not contain a right angle (p. 617) octant one of eight regions into which a threedimensional coordinate system is divided by the intersection of the three coordinate planes (p. 790) odd function a function f for which for all x in its domain, its graph symmetric with respect to the origin (p. 189, 483) f 1 2 x f x 1 2 one-stage path (of a network) a direct path from one vertex to another in a directed network (p. |
809) one-to-one function a function f in which a b; f relation is a function (p. 208) only when f b a 2 1 1 2 a function whose inverse open interval ther endpoint of the interval is included in the set; denoted with two parentheses (p. 118) an interval of numbers in which nei- opposite side (of a right triangle) abbreviated opp, the side of a right triangle that is across from a given acute angle of the triangle (p. 415) the sequence of output values orbit (of a number) produced by iterating a given function with that number; the orbit of a number c for a given function is c, f (p. 200 ordered pair A pair of real numbers in parentheses, separated by a comma, is used to locate or represent a point in a coordinate plane. The first number represents the horizontal distance from the origin and the second number represents the vertical distance from the x-axis. (p. 5) ordered set of numbers a set of numbers such that for any two numbers a and b in the set, exactly one of a 7 b the following statements is true: (p. 118) a 6 b, a b, or orientation (of a parametric curve) that a parametric curve is traced out (p. 757) the direction origin the point of intersection of the axes in a coordinate system (p. 5) origin symmetry A graph is symmetric with respect x, y to the origin if whenever x, y is also on it. (p. 187) 1 orthogonal vectors perpendicular vectors; vectors u and v such that is on the graph, then u v 0 (p. 673) 2 2 1 outlier from the general trend of the distribution (p. 847) a data value that shows a strong deviation output (of a relation) second numbers in the ordered pairs of a relation (p. 7) the values denoted by the P parabola the shape of the graph of a quadratic function (p. 165) For any line L in the plane and any point P not on line L, a parabola with focus P and directrix L is the set of points such that the distance from X to P is equal to the distance from X to line L. (p. 709) a conic section with eccentricity equal to 1 (p. 709) parabolic asymptote a parabolic curve that the graph of a |
function approaches as gets large (p. 286) x 0 In a plane, these lines have the same parallel lines slope. All vertical lines are also parallel. (p. 38) 0 parallel vectors vectors that are scalar multiples of each other (p. 671) the third variable used as input for the parameter two functions that form a pair of parametric equations (p. 157, 785) parameterization a pair of parametric equations that describe a given curve (p. 755) parametric equations a pair of continuous functions that define the x- and y-coordinates of points in a coordinate plane in terms of a third variable, the parameter (p. 157) parametric graphing graphing parametric equations (p. 157) parent function a function with a certain shape that has the simplest algebraic rule for that shape (p. 172) partial fraction decomposition See partial fractions. partial fractions When a fraction is decomposed (broken down) and written as the sum of fractions, the terms of the sum are called partial fractions, and the sum is called the partial fraction decomposition of the original fraction. (p. 838) Glossary 1045 partial sums of a geometric sequence For each positive integer k, the kth partial sum of a geometric sequence with common ratio r 1 is un6 5 1 rk u1a 1 r b k a n1 un. (p. 61) partial sums of an arithmetic sequence For each positive integer k, the kth partial sum of an arithmetic sequence 5 ku1 un6 k with common difference d is k a n1 k 1 2 k 2 1 or un u1 d 2 1 uk2 k a n1 un. (p. 27) period (of a function) the smallest value of k in a function f for which there exists some constant k such that f (p. 457, 498) See also sinusoidal function. for every number t in the domain of t k f t f 2 2 1 1 periodic function a nonconstant function that repeats its values at regular intervals; a function f for which there exists some constant k such that f f 2 (p. 457) for every number t in the domain of f t k t 1 2 1 periodic orbit duces repeating output values (p. 202) an orbit for a given function that pro- periodic point (of an orbit) the orbit of c for a given function produces repeating output values (p. 202) the number c for which |
periodicity identities cos t cos t ± 2p, sin t sin t ± 2p, 1 t ± p 2 (p. 458) tan t tan 1 2 1 2 permutation an arrangement of objects in a specific order (p. 880) In a plane, two lines are per- perpendicular lines pendicular when their slopes are negative reciprocals 1 (having a product of ). Vertical lines and horizontal lines are perpendicular to each other. (p. 38) phase shift a number representing the horizontal translation of a sinusoidal graph (p. 502) See also sinusoidal function. pie chart a visual display of qualitative data in which categories are displayed in sectors of a circle, where the central angle of each sector is the product of (p. 845) the relative frequency of that category and 360° piecewise-defined function a function whose rule includes several formulas, each of which is applied to certain values of the domain, as specified in the definition of the function (p. 146) plane curve the set of all points (x, y) such that x f t 1 functions of t on an interval I (p. 755) and that f and g are continuous y g and t 1 2 2 point-slope form a linear equation in the form y y1 x1, y12 1 m is a point on the line (p. 36) where m represents the slope and x x12, 1 1046 Glossary polar axis from the pole in a polar coordinate system (p. 734) the horizontal ray extending to the right r, u polar coordinates coordinates polar coordinate system, where r gives the distance from u the point to the pole, and with the polar axis as its initial side and the segment from the pole to the point as its terminal side (p. 734) is the measure of the angle of a point in the 1 2 polar form of a complex number For the complex number where the polar form is a r cos u, cos u i sin u b r sin u. r 1 and a bi, a bi (p. 639) r,, 2 0 0 pole the origin of a polar coordinate system (p. 734) an algebraic expression that can be anxn an1xn1 p polynomial written in the form a3x3 a2x2 a1x a0, ger, x is a variable, and each of stant (p. 239) where n is a nonnegative intea1, p |
, is a con- a0, an polynomial equation (of degree n) an equation that can be written in the form 0, a1x a0 variable, and each of where n is a nonnegative integer, x is a an is a constant (p. 94) anxn an1xn1 p a1, p, a0, polynomial form of a quadratic function a quadwhere a, x f ratic function in the form 2 b, and c are real numbers and ax2 bx c, a 0 (p. 164) 1 polynomial function a function whose rule is given by a polynomial (p. 240) polynomial model represent the trend in a data set (p. 273) a polynomial function used to population a group of individuals or objects studied in a statistical experiment (p. 843) positive correlation the relationship between two real-world quantities when the slope of the leastsquares regression line that represents the relationship is positive, that is, as one quantity increases, the other quantity increases (p. 52) positive integers 3, (p. 3) p the set of natural numbers: 1, 2, Power Law of Logarithms For all positive b and v, vk and all k, and (p. 366, 373) b 1, k logb v. logb1 2 1 2 y axr, power model where a and r are constant, used to represent the trend in a data set (p. 389) a power function of the form If both sides of an equation are Power Principal raised to the same positive integer power, then every solution of the original equation is a solution of the derived equation. However, the derived equation may have solutions that are not solutions of the original equation. (p. 112) power-reducing identities that relate second-degree expressions to first-degree expressions (p. 595) trigonometric identities prime number an integer greater than 1 whose only factors are itself and 1 (p. 20) principal borrowed (p. 100) an amount of money that is deposited or a number from 0 to 1 (or probability (of an event) 0% to 100%) that indicates how likely an event is to occur; calculated by dividing the number of elements in the event by the number of elements in the sample space (p. 865) probability density function A function with the property that the area under the graph corresponds to a probability distribution. (p |
. 871) probability distribution a table that describes the rule of a function P(E) that gives the probability of an event, where the domain of the function is the sample space and the range of the function is the closed interval [0, 1] (p. 865) probability of a binomial experiment P(r successes in n trials) success, and (p. 886) where p is the probability of is the probability of failure. nCrprqnr, q 1 p probability of a complement probability p, then the complement of the event has probability If an event E has 1 p. (p. 866) quadratic equation an equation that can be written where a, b, and c are real in the form constants and ax2 bx c 0, a 0 (p. 88) quadratic formula The solutions of a quadratic equation ax2 bx c 0 are x (p. 92) b ± 2b2 4ac 2a. quadratic function a function whose rule is a second-degree polynomial (p. 163, 240) qualitative data data that is categorical in nature, such as “liberal,” “moderate,” and “conservative” (p. 843) quantitative data numerical data (p. 843) quartic function a fourth-degree polynomial function (p. 240) quartiles These values divide a data set into fourths. The median, or second quartile divides the data into a lower half and an upper half; the first quartile Q1 is the Q3 median of the lower half; and the third quartile is the median of the upper half. (p. 860) Q2, quotient function For any functions x 2 their quotient function is the new function f 1 and g x, 2 1 f gb1 p. 192) and g x, 2 1 2 quotient identities tan t sin t cos t and cot t cos t sin t product function For any functions x their product function is the new function. (p. 192) f g fg 21 1 Product Law of Logarithms For all positive b, v, and w, and (p. 365, 373) logb v logb w. b 1, logb1 vw 2 product-to-sum trigonometric identities involving the product of two functions (p. 599)! OQ, projection vector A vector called the |
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