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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 Infini... |
, 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... |
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 Secti... |
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 Infin... |
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) ... |
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 2... |
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 coe... |
......... 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 t... |
........ 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. ... |
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... |
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... |
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... |
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 o... |
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 b... |
.............. 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................. |
............ 1033 Glossary.................................................... 1035 Selected Answers............................................ 1054 Index........................................................ 1148 968 Appendix ALGEBRA REVIEW This appendix reviews the fundamental algebraic facts that are used frequen... |
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... |
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. I... |
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 r... |
. 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–... |
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 ... |
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 ... |
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 ac... |
. 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. 4... |
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 onc... |
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 the... |
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... |
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 t... |
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 ... |
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 ... |
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 ■ A... |
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 ■ Multipl... |
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... |
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 ... |
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 an... |
, 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 Midp... |
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) t... |
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, Co... |
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 po... |
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.... |
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... |
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... |
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... |
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,...... |
. 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. ■ Exam... |
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 thir... |
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 ... |
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... |
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 f... |
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... |
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... |
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 ... |
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 b... |
, 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 othe... |
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... |
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 ... |
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... |
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 ... |
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 ... |
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 i... |
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 y... |
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 poi... |
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 graph... |
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... |
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... |
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-... |
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 tab... |
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... |
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 val... |
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 frequenc... |
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... |
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... |
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 Th... |
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 va... |
. 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 la... |
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 :Fo... |
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 us... |
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 a... |
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, ... |
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... |
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 betw... |
. 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 s... |
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 p... |
(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 ascendin... |
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 pl... |
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... |
. 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, geometri... |
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... |
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 sys... |
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 n... |
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 t... |
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... |
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, ... |
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 ... |
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 distrib... |
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) abbre... |
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 param... |
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 horizo... |
, 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... |
. 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, an... |
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