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projection of u on v, determined by constructing a segment from the terminal point of a vector u perpendicular to another vector v at a point Q on the vector, where point O is the initial point of both vectors, denoted projv u. (p. 674) Pythagorean identities sin2 t cos2 t 1 (p. 456) the identity and the identities de... |
; for any positive real number c and rational number with positive denominator, (p. 330) t k c 1 k ct 2 1 1 c k t 2 1 or c ct 2k c t. B A t k k 2k t rational function a function whose rule is the quotient of two polynomials, defined only for input values that produce a nonzero denominator (p. 278) rational number the s... |
first nonzero entry in each nonzero row is a 1 (called leading 1); any column containing a leading 1 has zeros in all other entries; and each leading 1 appears to the right of leading 1s in any preceding row. (p. 797) reference angle the positive acute angle formed by the terminal side of x-axis (p. 449) in standard p... |
vectors u and v, u v u v. 0 secant line (of a function) the straight line determined by two points on the graph of a function; the slope of the secant line joining points b, f 1 rate of change of the function from a to b (p. 218) on the graph of a function, equal to the average and a, f 22 22 b a 1 1 1 u secant ratio ... |
where is the period, a cos 2 c b a sin bt c bt c d, d x a 1 1 1 0 0 is the phase shift, and d is the vertical shift. (p. 548) skewed distribution a type of distribution in which the right or left side of its display indicate frequencies that are much greater than those of the other side (p. 846) slant asymptote a nonv... |
zero real number p, h, k 2 1 x h. 1 720 or 2 1 2 2 2 1 1 (p. 710, standard equation of an ellipse For any point in the plane and real numbers a and b with x h a2 x h b2 y k b2 y k a2 h, k a 7 b 7 0, 1. 1 1 1 or p. 693, 720) standard form (of a line) used to display the equation of a line without fractions, a linear equ... |
customary method of denoting the sum of terms by using the Greek letter Sigma c2 p cm ) as follows: (p. 25) c1 c3 ck © ( m a k1 supplementary angle identity For any acute angle u, (p. 628) sin u sin 180° u. 1 2 symmetric distribution a type of distribution in which the right and left sides of its display indicate freq... |
the two-stage path (of a network) a path in a directed network from one vertex to another with exactly one intermediate vertex (p. 810) U uniform distribution a type of distribution in which all of the data values have approximately the same frequency; its display is level (p. 846) unit circle the circle of radius 1 c... |
x of 1 c is the graph of f shifted and the graph of downward c units. (p. 174) See also sinusoidal function. y f c x 1 2 2 vertical stretch For any positive number graph of cally, away from the x-axis, by a factor of c. (p. 179) the is the graph of f stretched verti, vertical line test A graph in a coordinate plane re... |
constant polynomial 0 (p. 240) y-axis often the name of the vertical axis of a coordinate plane with the positive direction up and the negative direction down (p. 5) Zero Product Property If a product of real numbers ab 0, is zero, then at least one of the factors is zero; if (p. 89) then b 0. a 0 or y-axis symmetry A... |
. 2 P 1 4, 2 1 b. y (–5, 4) ; 2 (4, 1) x 2 (3, –2) (–2, –3) Chapter 1 1. Section 1.1, page 10 3, 3 ; 2 1 F ; 0, 2 1 2 6, 3 B 1 0, 0 A E 1.5, 3 G ; 3. P 1 2 1 ; 2 2, 0 C ; 2 1 2 7. y 500 400 300 200 100 2 x 0 987654321 10 9. a. About $0.94 in 1987 and $1.19 in 1995. b. About 26.6% c. In the first third of 1985 and from ... |
. 3 e. 1 4 2 c. f. 1 Section 1.2, page 19 1. 20 0 3. 20 0 5. u1 6 and un un1 2 10 0 −10 10 10 10 7. u1 6 and un un1 5 30 0 10 0 4 2 4 3 11 2 11 3 25 2 25 3 53 2 53 3 109 400; 0.8un1; un u1 u2 u3 u4 u5 u1 u2 u3 u4 u5 u0 9. 11. 13. 15. For 2 rays: 6; un u4 1; u2 un1 for 3 rays: n 1 for 3; u3 n 3 for 4 rays: 325 0 25 17. ... |
3. Slope, 2; y-intercept, b 5 5. Slope, 3 7 ; y-intercept, b 11 7 7. Slope, 5 2 15. y 9. Slope, 4 11. t 22 13. t 12 5 10 L P(1, C) x (0, 0) (1, 0) Slope of 10 9. 224 17. 30 3 2 1 3 n 1 2 4 2; 0 5 7. 45 11. 13. 87 15. 21 4 3 2n 19. un1 un arithmetic with 1 2 d 2 21. un un1 5 3n 2 d 3 2 c 2n 2 d 2 2n 4 7n 10 3n 2 15 2 a... |
. $375,000; $60,000 y 5x 150 x 5, y 125 pounds x 7, y 185 pounds 50x 110,000 22x 110,000 59. a. c ft ; 2 1 6 ft 2 r 72x b. x 2 d. x 5000 1 61. a. 63. a. y 8.50x 50,000 x 10 b. $11, $9.50, $9 per hat b. x 30 15. No High School Diploma 12.31x 238 y2 1. a. Section 1.5, page 53 x 5 y 3 4 4 Sum of squares 3 Model B still ha... |
; geometric with r 5 20 23. u5 1; un 1 1 n164 2 4n2 1 n1 1 2 4n5 25. u5 1 16 ; un 1 4n3 27. u5 8 25 ; un 2n2 5n3 29. 254 31. 4921 19,683 33. 665 8 100 0 0 Loan data Grant/work-study data c. 1983 40 21. a. 35. a. Since for all n, the ratio r is un1 un 1.71 1 1.71 1.191n1 1.191n 1 2 2 1.71 the sequence is geometric. b. $... |
66 98.92 69.17 29.75 The finite differences show that the data is not linear. 20.79 20.10 0.69 34.89 25.23 9.66 69.17 48.55 20.62 93. Second method is better. Answers to Selected Exercises 1059 Chapter 1 can do calculus, page 79 1. 1 4. 2 3 2. Diverges 5. 500 0.6 833 1 3 7. 4 212 8. Diverges 10. 13. 2 9 8428 99 11. 14.... |
B or A 1 15 2 B 29. x 2 ± 13 31. x 3 ± 12 33. No real number solutions 35. 39. ± 12 x 1 2 4 ± 16 5 u 47. x 3 or 6 51. x 5 or 3 2 37. x 2 ± 13 2 41. 2 43. 2 45. 1 49. x 1 ± 12 2 53. No real number solutions 55. No real number solutions 57. 61. 65. x 1.824 or 0.470 y ± 1 or ± 16 y ± 2 or ± 1 12 59. 63. 67. x 13.79 x ± 1... |
43210 Since 3w 6 or 10, dividing by 3, w 2, 3 5. 0 2x Since 2x 1 or 9, dividing by 2, x 1 2, 9 2. 3x 7. 0 2 1 2 0 5 or 0 3x 2 5 0 −7 −6 −5 −4 −3 −2 −1 0 1 2 3 Since 3x 7 or 3, dividing by 9. x 6 or 3 15. x 3 2 3, x 7 3 1., 13. x 2 11. x 3 2 17. x 5 or 1 or 3 or 1 19. x 1 or 4 or 5 133 2 or 5 133 2 21. For any real numb... |
T 23. 1, q 1 2 29. q, a 4 7b 35. 5, q 3 2 13. 19. 3, 14 2, q 2 2 1 1 25. (2, 4) 31. 37., q 7 b 17 S x 6 b c a 15. 21. 27. 33. 8, q 2 q, 8 5 T 3 a 3, 1, 5 2b 1 8b S S between Joan’s actual weight and her ideal weight is x 120. 39. c 6 x 6 a c 41. 1 x 3 Answers to Selected Exercises 1061 43. x 9 121 2 x 1 133 2 or or x ... |
; silver, 8 11 oz 21. 9.6 ft 23. 4 ft 27. b 1 2 2 1 29. x 1 2 or 11 2 31. 33. 0 4 3x 1 0 3x 1 4 or 3x 1 4 3x 3 3x 5 x 5 3 x 1 2x2 x 2 0 Squaring both sides, x2 x 2 0 x 1 0 2 x 2, 1 x 2 21 1 Both of these check in the original equation. x2 6x 8 x 1 0 35. Set the numerator equal to 0. or x 1 4 or x 5 4 1 x 2 x2 6x 8 0 x ... |
.5 yd 2 yd 3 yd 2.5 yd 2.5 yd 3 yd 2 yd 3.5 yd 1.5 yd 4 yd 1 yd area 4 yd2 5.25 yd2 6 yd2 6.25 yd2 6 yd2 5.25 yd2 4 yd2 2.25 yd2 4.5 yd 5 yd 5.5 yd 0.5 yd nr — nr — 6 nr b. x y 1 2 3 4 5 5.92 5.66 5.20 4.47 3.32 area 5.92 11.32 15.6 17.88 16.6 — 6 c. 20 c. 8 0 0 d. The maximum area 6.25 yd2 appears to occur when the ba... |
x. 13. 23 1 2.73 15. 322 3 22 1 1.69 17. 4 19. 34 3 21. 59 12 a k 2 2 1 a k 2 23. 25. 1 1 2 x 2 27. 8 33. t2 1 2 1 2 6 4x x2 1 2 x 29. 1 35. 1 2 x 2 1 s2 2s s 1 31. 1 37. 3 2 4 39. 2x h 1 41. 1 2x h 2x 43. All real numbers 45. All real numbers d. A maximum area of 2 square units appears to occur at x 2. Analytical and... |
inputs is temperatures of gas. The set of outputs is pressures of gas. The function rule is the formula P k T. 5. y is a function of x. 7. y is not a function of x. 9. y is a function of x. 1064 Answers to Selected Exercises d t 2 1 2.25 5 t 3 4b a 8.5 8.5 3 b. The domain is t such that t 2..5 2.5 6 t 4 0 t 4. of annu... |
and 0.6, 0.6 1. 2 1 0.6, 0.6 10 2 37. a. 10 10 10 b. This function is increasing over the interval and decreasing over the interval c. There is a local minimum at the point (1, 0). d. This function is concave up over the interval q, 1. 2 1 1, q 2 1 1 q, q. 2 e. There is no point of inflection. 39. a. 10 10 10 10 b. Th... |
range: all real numbers 61. Many correct answers, including 6 4 2 −2 −1 2 4 −2 −4 −6 63. Entire graph: 75 69. 10 2 10 near the origin: 10 2 10 65. Entire graph: 60 16 near the origin: 2 62 2 32 5 2 2 10 10 10 10 x t, y t4 3t3 t2 71. 10 10 10 x t4 3t2 5, y t Section 3.3, page 170 3. (1, 2), downward 5. 1. (5, 2), upwar... |
1 2 h x 2 1 x3 x 4 3 11. 5 3 10 13. 10 10 10 21. Vertex is 0.5, 24.5 24. 1. 2 y-intercept is The x-intercepts are 3 and 4. 10 10 10 10 23 x, h h x 2 1 23 x 1 x 2 5 15. 5 5 30 2x2 14x 20 2 Answers to Selected Exercises 5 g x 1 2 x 4 3 1, g x 1 2 x 4 3 x 23. g 1 1068 17. 10 35. 3 10 10 5 5 10 19 21. 5 5 5 5 h x 2 1 23. ... |
origin. However, it is symmetric with respect to the point (0, 2). 5. y 23 x x2 7. Yes 13. No Symmetric with respect to the origin. 9. Yes 15. Yes 11. Yes 17. Yes 19. Origin 21. Origin 23. y-axis 25. Odd 27. Even 29. Even 31. Even 33. Neither 35. 39. Many correct graphs, including the one shown here: (−7, f(−7)) (−5, ... |
3 3x 2 Answers to Selected Exercises 1071 7. fg 1 2 1 f g b1 21 x 1 x2 1 21 b1 x 2 a 2 21 21 x 1 2 2x2 1 2x 1 2x2 1 2x 1 B x2 1 x 1 2x 1 x 1 x2 1 B 2x 1 2x2 1 1 x 1 B 9. Domain of fg: all real numbers except 2; domain of f g : all real numbers except 2 11. Domain of fg: all real numbers; domain of f g : all real number... |
x3 3, y2 51. 53. 55. 2x 1, y2 y1 21 15,600 19.5n n gives the unit price as a function of n, the number of telephones produced. 57. V 256pt3 3 ; 17,157.28 cm3 59. s 10t 3 61. One such function is f x 1 x ˛. x 1 2 Section 3.5.A, page 203 1. 2, 2.16, 2.3328, 2.5194, 2.7210, 2.9387, 3.1737, 3.4276 3. 0.2, 0.64, 0.9216, 0.... |
4 3t2 t x1 y1 x2 y2 x 11. x 5y2 4, y ± x 4 5 B Section 3.6, page 212 1. y 4 2 3 6 1 f(y) 1 2 3 4 5 9. g 13. g 17. g 21 23. No 3 5 x B 2 1 x 3 B 15. g x 1 2 19. x x2 7 4, 1 x 0 1 2y2 1 ˛, y ± B 2 1 x 2x 5x 1 1 x 25. Yes 27. Yes 29. No Answers to Selected Exercises 1073 31. 10 −5 5 49. 1 g f 25 x x 21 25 x5 x A B 25 x A ... |
1500 55.5 9. 7. a. 2 2x h 17. b. 11. 92.5 1 b. 438 ties/mo d. 563 ties/mo f. h. ties/mo ties/mo 750 375 c. 462.5 13. 1.5858 19. 2t h 8000 15. 1 21. 2pr ph −5 37. One restricted function is x 39. One restricted function is that x h ; 2 1 2 h inverse function x 1 2 with x. x 0 0 x g 2 1 x2 x 0 (so x h 1 1x. with 2 x 0; ... |
17 ft/sec. (10, 800), namely, 29. a. From day 0 until any day up to day 94, the average growth rate is positive. b. From day 0 to day 95 c. 27, meaning that the population is decreasing at a rate of 27 chipmunks per day d. 20, 10, and 0 chipmunks per day Chapter 3 Review, page 226 11 3 7 2t 9 1 2b 2 2 7 5 3 3 x3 4 2 2 ... |
y-intercept is The x-intercepts are 0.35, 4.2025 2 4.08. 2.4. and 1.7. 39. Compress the graph of g toward the x-axis by a factor of 0.25, then shift the graph vertically 2 units upward. 41. Shift the graph of g horizontally 7 units to the right; then stretch it away from the x-axis by a factor of 3; then reflect it ac... |
maximum height of the ball will be 81.25 feet. 3. s t 1 4. 14 2 16t2 300; 5. 96 0.111111 feet per second 6. 3 7. 2a 8. instantaneous rate of t 4: tangent line at change 16 y 16t 76 ; equation of 100 −2 8 −100 77. 79 21 21 0.25 4 3 g f x 2 21 1.5 x 1.5 1.5 x 6 x 6 6 x 7x x 3 1 1 2 1 4x 6 2 0.25x 1.5 2x 1 x 3 b a 2x 1 x... |
a. 83. 6 b. 5 8 85. 3 87. 2x h 11. quotient remainder 7 3 2 3 2 9. No 11. Degree 3, no; degree 4, no; degree 5, yes 15. 2 1 1 quotient remainder 17. Quotient 19. Quotient 21. Quotient 23. No 29. 222, 1 30 41. No 35. 4 8 4 6 2 4 x3 4x2 4x 6; 2 8 6 7 12 19 19 3x3 3x2 5x 11; x2 2x 6; 5x2 5x 5; remainder remainder 0 remai... |
5; 19. Lower 17. 21 2 1 1 2 x2 3 11. 15. x 5, x2 3 x3 2, or 3 x 2 21 2 1 1 upper 2 2 21. Lower 7; upper 3 23. x 1, 2, or 1 2 25. x 1, 1 2, or 1 3 1078 Answers to Selected Exercises 27. x 2 or 5 ± 137 2 31. x 1, 5, or ± 13 29. 33. x 1 2 x 1 3 or ± 12 or ± 13 or 1.8393 35. x 2.2470 or 0.5550 or 0.8019 or 50 37. a. The o... |
Left half: 250,000; 90 x 120 37. 39. Overall: y-axis: and 3 x 3 0.1 x 0.2 20 y 30 50,000 y and and 2 x 3 15,000 y 5000 20 y 20; and near 4.997 y 5.001 and 41. a. The graph of a cubic polynomial (degree 3) has at most 3 1 2 local extrema. When is large, x 0 0 ax3, the graph resembles the graph of that is, one end shoot... |
, 0) 1 2 x (1.45, −0.37) 57. a. The solutions are zeros of 2 1 x and 3.99 y 4.01 4 0.01x3 0.06x2 0.12x 0.08. g This polynomial has degree 3 and hence has at most 3 zeros. 1 x 3 b. c. Suppose f(x) has degree n. If the graph of f(x) y k had a horizontal segment lying on the line k x for some constant k, then the equation... |
. All real numbers except 5. All real numbers except 5 2 3 15 12, 7. Vertical asymptotes x 1 x 0; 9. Hole at vertical asymptote 11. Hole at x 2; vertical asymptotes 13. any window with and 3 15 12 and 1, and x 6 x 1 x 2 115 x 110 31 x 35 any window with any window with 40 x 42 y 3; y 1; y 5 2 ; 15. 17. 19. Asymptote: y... |
ote y 0 x 5, x 1 37. y x −1 −2 −3 −4 vertical asymptote horizontal asymptote y 4 x 0 39. y 4 2 −2 1 3 5 −2 −4 vertical asymptotes horizontal asymptote x 1, x 5 y 1 y 41. 4 2 −2 −4 −6 −4 −2 vertical asymptotes x 3 hole at horizontal asymptote y 0 x 4, x 5 1082 Answers to Selected Exercises vertical asymptote x 2 x 3 1 −... |
near the origin: 0.02 y 0.02 65. Overall: 16 y 8 12 y 8 20 y 20; and 2.5 x 1 and hidden b. 20 e. x p 1 2 4x 10 x 3 f. Shift the graph of f(x) horizontally s units (to 0 0 s 6 0 ); stretch (or r to the right if (away from s 7 0; r the left if shrink) the graph by a factor of 7 1, the x-axis if 0 6 1 0 6 r ); also if th... |
. y 2.60; $2.60. v 50u u 50 b. v 50 0 0 100,000 the average cost can never be below Answers to Selected Exercises 1083 c. 100 Section 4.5.A, page 306 15. 4 17. i 19. i 21. i Section 4.6, page 313 1. 3. 5. 1 1 f 3 0.39.4521 0 0 ; 2 2 2 02 0.4521. 0.5 i; 0.3; f 2 1 0.4521 2 0.5 0.5i; f 2 0 2 1 d 1.5207 0.25 1.5i ; 1.2 0.... |
B 23. Many correct answers, including 21 25. Many correct answers, including 21 1 2 f 27 2x 21 x 4 1 2 x 3 3 2 1 21 2 x 12i B BA 313 2 i b 50 0 35,000 d. If the object is close, a small change in u leads to a large change in v. However, when u is large, a small change in u leads to nearly no change in v, so that u may... |
last row, 1 alternating signs. Therefore, for the real zeros. 1 is divided by 5 20, has is a lower bound x 1 5 5 1 23. rational zeros: and 4; irrational zero: 1.328 is a zero of multiplicity 2; 4 is a zero of is a zero of multiplicity 1; 3 is a 3 multiplicity 1; zero of multiplicity 1 3 2 i 25. z a bi c di 2 z w. and ... |
lynomial with real coefficients, then g11 degree g11 p degree g21 is a polynomial with real coefficients f(z) can be factored as each and degree 1 or 2. The rules of polynomial multiplication show that the degree of f(z) is the degree sum: gk1 z g31 z. degree 2, then this last sum is an even number. But f(z) has odd de... |
3 ± 131i 10 51. 55. 57. 59. 2 A 3 1 23i or x 2 A 3 x 2 or i, i, 2, 1 or i or i or 1 23i 61. Many correct answers, including x4 2x3 2x2 x f 1 2 63. a fixed orbit of one point: 65. 67. 69. 1 1 1 21 x 1 x 1 21 x2 1 21 x 2 x 2 21 x2 1 x 3 ; 2 x2 1 1 ; 2 x i ; 2 1 21., 3 a 5 x 1 21 x 1 x i 4 5b x 2 21 x 2 21 x i 1 21 21. x... |
left, then 2 units down. Section 5.2, page 343 21. 23. 25. 27. 29. 1 2 x : B 20 x 2 : C; h 1 and and and and 2 x : D : A; k 1 0 y 1 10 y 10 0 y 1 0 y 10 2 2 xh 2 h 1. Shift the graph of h vertically 5 units downward. 3. Stretch the graph of h vertically by a factor of 3. 31. Neither 35. When x is large, 5. Shift the g... |
Not at the right side of the viewing window; x f121 2 Section 5.3, page 353 1. Annually: $1469.33; quarterly: $1485.95; monthly: $1489.85; weekly: $1491.37 3. $585.83 5. $610.40 7. $639.76 Answers to Selected Exercises 1087 9. $563.75 11. $582.02 49. y 13. About $3325.29 15. About $3359.59 17. About $6351.16 19. About... |
x y 35. x2 37. 1 41. They are exactly the same. 1 2 1, q 31. 931 2 39. q, 0 1 2 f(x) = log(x − 31 −2 51. y h(x) = −2 log 1 −2 x x 53. 55. 57. 0 x 9.4 asymptote at 10 x 10 0 x 20 and 6 y 6 and x 1 ) (vertical and 3 y 3 59. 0.5493 63. a. ln 1 3 h h 2 ln 3 6 y 3 61. 0.2386 b. h 2.2 65. a. About: 17.67, 11.90, 9.01, 6.12,... |
False; the graph of the left side differs from the graph of the right side. 35. Answers may vary: log 3 log 2 1.585 and 3 2b log a b e 0.1761 thus log 3 log 2 log 3 2b a 39. A 3, B 2 41. 2 37. 43. Approximately 2.54 45. 20 decibels 4 69. Horizontal shift of units to the right, then 3 compress horizontally by a factor ... |
. log10u 2 log100u logbx 1 2 b3 2v logbA g f x 2 1 is false. x 2 1 logbv 3 logb2v logbb3 ; hence B only when x b32v. x 0.123, so the statement y 6 4 2 (−0.3679, 0.3195) −6 −4 −2 −2 −4 h(x) = x log x2 Hole at (0, 0) x 2 4 6 (0.3679, −0.3195) Section 5.6, page 386 1. x 4 3. x 1 9 5. x 1 2 or 3 7. x 2 or 1 2 9. x ln 5 ln ... |
.36 years b. In the year 2027 b. t 0.182 75. a. There are 20 bacteria at the beginning and 2500 three hours later. b. ln 2 ln 5 0.43 77. a. At the outbreak: 200 people; after 3 weeks: about 2718 people b. In about 6.09 weeks k 0.229, c 83.3 b. 12.43 weeks 79. a. Section 5.7, page 396 1. Cubic, exponential, logistic 3. ... |
the number of kids home schooled by the quadratic model will exceed the number of kids in the world. The logistic model, on the other hand, gives us a maximum that can never be exceeded. y 17.5945 13.4239 ln x 27. a. b. 77.4 years c. 2012 29. a. 85 0 0 b. y 10.48 1.16x 1 2 13 Answers to Selected Exercises 1091 c.-d. 2... |
. 33. r2 1 1 2 31. u v ln e7.118 1234 39. Undefined 35. 41. Reflection across the y-axis, horizontal translation of 4 units to the right; Domain: all real numbers 6 4; Range: all real numbers 43. Vertical stretch by a factor of 3, vertical translation of 5 units downward; Domain: all positive real numbers; Range: all r... |
1. 7. 13. 17. 19. 22 3 12. y 1 9 a ln 3 b1 x 2 1 9 2 5 −1 −5 Chapter 6 Section 6.1, page 419 1. 5. 9. 11. 13. 15. 19. 23. 27. 47.26° 23°9¿36– sin u 3. 15.4125° 4°12¿27– ˛, tan u 7., cos u 3 2 11 3 sin u ˛, sec u ˛, tan u 211 211 3 ˛, csc u A cot u 3 22 7 ˛, cos u 2 27 27 ˛, sec u 2 ˛, csc u m ˛, tan u h d d ˛, csc u m... |
5 sin 72° 3.3, 24.8, c 6 24.1 tan 14° b 5 cos 65° 11.8 25. About 48.59° 27. About 48.19° 29. 31. 33. 35. A 33.7°, C 56.3° A 44.4°, C 45.6° A 48.2°, C 41.8° A 60.8°, C 29.2° 37. a. b. 23.18 feet. 6.21 feet. 39. 460.2 ft 41. 8598.3 ft 43. No 45. Approximately 263.44 feet 47. 351.1 m 53. a. 56.7 ft 55. 173.2 mi 49. 10.1 f... |
.69 mph 83. 15.92 ft 85. approximately 8.6 miles Section 6.4, page 452 sin t 7 1., cos t 2 3. sin t 5. sin t 253 6 261 10 2103, cos t, cos t 253 5 261 23 2103, tan t 7 2, tan t 6 5, tan t 10 23 7. 9. sin t 1 25 sin t 4 5, cos t 2 25, tan t 4 3, cos t 3 5, tan t 1 2 21. a. terminal side is in the third quadrant. sin 9.5... |
b a 23 2, tan 23p 6 b a 1 23 23 3 39. sin 19p 3 b a 23 2, cos 19p 3 b a 1 2, tan 19p 3 b a 23 41. sin 15p 4 b a 22 2, cos 15p 4 b a 22 2, tan 15p 4 b a 1 43. sin 5p 6 b a 1 2, cos 5p 6 b a 23 2, tan 5p 6 b a 1 23 23 3 45. 47. 49. 55. 57. 59. is undefined and tan u sin u 1, cos u 0, sin u 0, cos u 1, and tan u 0 22 2 1... |
. The boat has moved about 95.3 feet. 19. 255° 21. p 5 23. ˛ 3p 4 25. 16p 3 27. 2 revolutions per minute 29. 37. 3 5 23 3 31. 0 39. 2 33. 41. ˛ 1 2 23 2 43. quadrants 2 and 3 45. 9 4 35. 23 61. r cos t, r sin t 1 2 63. Domain: all real numbers with q, 1 65. Domain: all real numbers with Range: 1, q ´ 2 1 1 2 Range: all... |
3p 3p 2, 5p 6 t 6 3p 2 4 p 2 7p 4 and 9. 1 13. 1 3p 2π 45. 47. 19. all values on the interval p, 2p 3 4 except 3p 2 21. t p 4 2np or t 3p 4 2np, where n is any −2π 2np or t 4p 3 2np, where n is any 2np or t 5p 3 2np, where n is any 49. d integer t 2p 3 integer t 4p 3 integer t p 6 integer t p 6 integer t 3p 4 integer ... |
units. 2 1 3. The graph of csc t f t 2 1 t m 4 csc shifted 4 units up. t 1 2 1 2 is the graph of 5. The graph of p 1 2 t 2 1 sec t 1 is the graph of sec t g t 2 1 compressed vertically by a factor of 1 2 and shifted up 1 unit. 7. The graph of t sec t g shifted 8 units down. 2 1 t q sec 8 reflected across the vertical ... |
is the graph of f reflected across the y-axis; amplitude: none; period: p. 25. g is the graph of f horizontally compressed by a factor of 5 8 ˛; amplitude: 1; period: 5p 4 ˛. 27. g is the graph of f vertically stretched by a factor of 3; amplitude: 3; period: 2p. 29. g is the graph of f vertically compressed by a fact... |
t 0.2 9 t 2p 9 b 10t p 2 b 31. a. f b. g t t 2 2 1 1 12 sin 12 cos 10t a 2t p 2 b a 8t p a 2 b 35. a. f t 2 1 1 2 sin 8t b. g t 2 1 1 2 cos 37. a. b. 39. a. f f f b sin 4 cos 2 sin 2 cos 3p 41. 12 −12 π 6 1 − π 6 43. 4 0 −4 45. d 47. b 49. f 51. 55. 59 sin t 2 sin pt 3 2 2 5 sin 5t 1 53. 57.8 sin 4t 3 2 sin 4t 2 61. l... |
horizontal line is the same as 55. Not an identity 57. Possibly an identity 59. 61. A 3.8332, b 4, c 1.4572 A 5.3852, b 1, c 1.1903 63. All waves in the graph of g are of equal height, which is not the case with the graph of f. It cannot be constructed from a sine curve through translations, stretches, or contractions... |
cos t 0 ± 5p 2 ± 3p 2 at these (four,...,, Chapter 7 Review, page 517 1. (c) 3. 5. 7. 0 n2p, p 3 np, where n is an integer where n is an integer 9. The graph of g is the graph of f reflected across the horizontal axis and compressed horizontally 1 2 ˛. domain: all real numbers except −2π where n is an integer; range: ... |
1.5708 2kp or 4.7124 2kp or or 1.8256 2kp or 2.8867 2kp or or or x 5.0671 2kp 3.8212 2kp 5.6766 2kp sin x sin x 1 on the interval from 0 only when x p 2. Since sin x has period obtained by adding or subtracting integer all other solutions are 2p, from p 2, that is,, 2p multiples of 2p 5p p 2 2 13p 2 7p 2 p 2 p 2 p 2 3... |
2 15. 19. 23. 25. 27. 33. 17. 25. 27. 35. f(x) = cos −1 (x + 1) f(x) = sin x x 2π −2 −1 2 0 −2 Answers to Selected Exercises 1103 49. y 2π 3 π 3 x −2π −π π 2π −2π 3 51. a. u sin 1 40 x b a 53. y csc x b. 9.2° 5 −5 π 2 y csc x is one-to-one and has an − π 2 The graph of inverse. y csc 1 x − π 2 5 −5 55. a. Let 1 b. Let... |
5.3004 x p 6,,, 39. 5p 4 3p 2 x 0.8481, 1.7682, 2.2935, 4.9098 x 0.8213, 2.3203 x 0.3649, 1.2059, 3.5065, 4.3475 x 1.0591, 2.8679, 4.2007, 6.0095 x p 7p 4 4 x p 4 5p 4 3p 4 53, 5p 6, 3p 2, 5p 4 kp 6 1.2682, 0.7446, 0.2210, 1.7918 5p 4 1 4 t tan 6 1. 3. 5 125 sin pt 5 b cos 20pt 216 sin2 a 1 6 sin pt 2 b a 7. h t 2 1 2... |
.6180 2kp or or or 4.7124 2kp 25 1 tan 3 5b a 3 kp 3 t. In the first 2 seconds the solutions are 0.1801, 1.2273. 5 −1 −1 e. About 12.2 This model provides a much better fit. Section 8.4.A, page 562 1. 1 0.01 0 −1 y sin 1 294 2px 2 3. 1 0 −1 y sin 1 440 2px 2 9. p 3 19. π 2 0 − π 2 21. 25. 27. 2 515 x p 6 x 4p 9 11. p 3... |
. Possibly an identity 5. b 7. e 9. tan x cos x sin x cos xb a cos x sin x 11. cos x sec x cos x 1 cos xb a 1 13. tan x csc x sin x cos xb a 1 sin xb a 1 cos x sec x 15. tan x sec x sin x cos x 1 cos x sin x 1 cos x 1 21 17. 1 cos x 19. Not an identity x 2 x 2 21. sin x cos x sin 1 cos 1 x 1 2 23. cot x cos 1 x sin 1 2... |
: 1 cos x sin2x 1 cos x cos x cos2x 1 cos x 1 sin2x 1 cos x 1 sin2x 1 1 cos x 2 cos x cos x 1 cos x 1 1 cos x 2 cos x 51. Conjecture: tan x: Proof: sec x csc x sin x cos x 1 sin x sec x cos x csc x cot x 2 sin x 21 sin x csc x 2 cot x 2 cos x sec x 1 cos x 53. 55. 57. 59. 1 sin x cot x 2 cos x sin x 1 sin x sin x cot x... |
y 2 tan x tan x cot x 1 1 2 63. 65. tan y cot y 2 cos2x sin2y 1 1 1 1 1 1 cos y sin x 1 sin2x 2 cos y sin x 21 cos x sin y 1 cos2y 2 1 cos x sin y 21 cos2y sin2x cos y sin x cos y sin x cos y sin x cos x sin y cos y sin x cos x sin y 21 21 21 2 2 2 2 2 cos y sin x cos x sin y Section 9.2, page 587 1. 16 12 4 7. 2 23 9... |
x y 2 sin x sin sin x sin y 1 2 1 x y 1 1 cos x cos y sin x sin y 1 2 cos x cos y 1 cos 2 cos2x cos2y sin2x sin2y x y cos 2 sin x cos y cos x sin x sin y cos y 2 1 1 1 sin x sin y 2 2 2 cos x cos y sin x sin y sin x cos y cot x tan y cos x cos y sin x sin y 2 3. 2 23 5. 2 23 59. Not an identity Answers to Selected Exe... |
x cos 3x tan x sin 4x sin 6x cos 4x cos 6x cot x 1 1 1 cos x x 2 2R Q 2 cos 2x sin 1 2 cos 2x cos 1 x 2 x 2 2 1 cos x sin x cos x 2 sin 5x cos 1 2 sin 5x sin x 2 x 1 cos x sin x 2 2 sin a 2 sin x y cos 2 R x y 2 sin cot x y 2 a b 69. sin x sin y cos x cos y x y cos 2 Q x y 2 Q 1 cos x sin x sin R 71. a. and part (a), ... |
p 8, 15p 8 15. cos x sec x cos x 1 1 cos2 x cos x sin x cos x cos x sin2 x cos x sin x tan x sin x 75. 2 sin2 1 a 2 a sin2 1 2 1 2 1 cos cos a 1 cos u 2 cos u cos a a 1 cos 2 1 2 Q u R ¢ 2 b Section 9.4, page 608 1. no solution 5. 9. 13. 15. 19. 23. 27. 31. 35 3p 4 x 3p 4 x 5p 12 x p 3 x p 4 3p 2 3p 2 3p 4 3p 4,, 5p 4 ... |
2y 1 cos2x sin2y 4 3 1 2 2 cos x cos y sin x sin y 4 21. a. 3 5 23. 120 169 b. 117 44 c. 44 125 25. 142 212 10 29. 1.23 radians 2 2 sin2 x tan x 27. 31. 33. 1 1 1 1 2 sin2 x tan x 2 sin2 x cos x sin x cos x 1 cos 2x tan x 2 sin2 x sin x cos x 2 sin x cos x 2 cos x 2 cos3 x 2 cos x 2 cos x sin2 x sin 2x sin x 1 1 2 1 co... |
2°, 124.8°, A1 A2 19. C1 C2 104.8°, 35.2°, 14.1; 8.4 c1 c2 B2 21. No solution 65.8°, 9.8°, 23. a1 10.3; 58.2°, B1 A1 A2 2.1 a2 C 72°, b 14.7, c 15.2 a 9.8, B 23.3°, C 81.7° A 18.6°, B 39.6°, C 121.9° c 13.9, A 60.1°, B 72.9° C 39.8°, A 77.7°, a 18.9 25. 27. 29. 31. 33. 15. 31.4 114.2°, 35. No solution 37. 6.5 39. About... |
unit increase in x when x p 2 d. 13 2 ; sin x is increasing approximately 0.8660 per unit increase in x when x p 6. 2. sin x 3. a. 0; cos x is not changing per unit increase in x x 0. when 12 2 ; b. c. cos x is decreasing approximately 0.7071 units per unit increase in x when 1; cos x is decreasing 1 unit per unit inc... |
21i 37. 3 2 a cos p 4 i sin p 4 b 322 a 4 b 322 a 4 b i 39. 222 cos a 7p 12 i sin 7p 12 b real 41. cos p 2 i sin p 2 43. 12 cos a 2p 3 i sin 2p 3 b 45. 222 cos a 19p 12 i sin 19p 12 b 47. The polar form of i is 1 cos 90° i sin 90° 1. Hence, 2 by the Polar Multiplication Rule i sin zi r 1 u 90° cos u 90°. 1 1 1 2 You c... |
sin 7p 5 b, cos a 19. cos a cos a 9p 5 p 10 9p 10 i sin 9p 5 b i sin p 10b, cos a p 2 i sin p 2 b, i sin 9p 10 b, cos a 13p 10 i sin 13p 10 b, cos a 17p 10 i sin 17p 10 b 21. 24 2 cos a p 8 i sin p 8 b, 24 2 cos a 9p 8 i sin 9p 8 b 1 2 i or 23 2 1 2 i or 23 2 1 2 i or 23. x 23 2 1 2 23 2 i or i or i 23 2 23 2 a 24 2 2... |
5, 2 5 i 9. h u v 8, H 3 ; I 3u 2v 3 422, 3u 2v u v 1 322 ; I 9 822, H 3 422, H 2 922 I u v ;, 13 i 9 4 h, 7 ; i, 24 i 2 h 3u 2v 17 2 u v 2130 v w 1022 2 0 w 2u 1 v 225 7 v 2 3 u3 u2 u4 v c, d 2 I H v r s rc, rd u1 v 1 r s 1v 1 6 1 2 3, 9 I H c, d H c, d 2H I c, d c, 9 I H 0 0, 2 r v sv d 2 1 I H1 I sc, sd I H v, 0v 0... |
° v 5213, u 213.7° i 8 7 31. j 2113 2113 1 210 i 3 210 j 33. Direction: 82.5°; magnitude: 9.52 lb 35. Direction: 37. u sin 1 a 18.4°; 894.8 1500 b 36.6° magnitude: 80.4 kg 39. Parallel to plane: 68.4 lb; perpendicular to plane: 187.9 lb 41. 1931.85 pounds 43. Ground speed: 401.1 mph; course: 154.3° 45. Ground speed: 44... |
°, b 86.9 B 81.8°, C 38.2°, c 2.5 B 98.2°, C 21.8°, c 1.5 a 41.6; C 75°, c 54.1 and 19. 147.4 21. 13.4 km 23. Joe is 217.9 m from the pole and Alice is 240 m from the pole. 25. a. 3617.65 ft b. 4018.71 ft c. 3642.19 ft 27. 10 31. 210 220 29. 37.95 33. The graph is a circle of radius 2 centered at the origin. 35. 2 cos ... |
ians 17. 8p 19. 223p 23. approximately 1,507,964 sq. ft. x2 a2 y2 a2 25. If a b, then 1. Multiplying both sides a2 gives x2 y2 a2, by radius a with center at the origin. the equation of a circle of Chapter 10 can do calculus, page 689 1. 1 cos 2 3 1 1 ; 0.4161 0.9093i 2. i sin i sin 3. cos.9900 0.1411i 27. As b gets la... |
�x ± 2 b close to, but not equal to, 0 when b is large). are not horizontal (their slopes, are, 27. 8000 −3 3 −200 3800 −2 213 and 0, foci are at 0, a y ± a b x; 6 b y 2 3 x a and y 2 3 are 213 6 b −8000 ; asymptotes 29. y x 2 x 31. The distance between the vertices is 2a. One point on the hyperbola is the vertex at (a... |
14-1147 9/21/05 2:03 PM Page 1117 31. 9 −18 18 5. 7. u 53.13°; x 3 5 u 36.87°; ; y 4 5 v; y 3 5 33. −15 10 −10 10 −10 35. Ellipse; 37. Hyperbola; 6 x 3 and 7 x 13 2 y 4 and 3 y 9 39. Parabola; 41. Ellipse; 1 x 8 1.5 x 1.5 and 3 y 3 1 y 1 and 43. Hyperbola; 15 x 15 and 10 y 10 45. Parabola; 19 x 2 and 1 y 13 47. Hyperbo... |
B cos 2u 0. A C B cot 2u cos2 u sin2 u C A B¿ 1 1 2 2 and sin 2u B cos 2u sin 2u B cos 2u. C A sin u cos u is the coefficient of uv 1 2 d. If 11. a. From Exercise 9 (a) we have B¿ 2 4A¿C¿ 1 2 2 2 4 C cos2 u 2 1 2 1 1 2 B2 C A 1 C A cos u sin u 2cos2 u sin2 u 4B 2 A2 C2 B2 2 2 1 cos2 u sin2 u 2 cos2 u A cos2 u B cos u ... |
3p 2 b 5, 2p, 1 2 1, 7p 6 b a 3, 2p 3 b 6, p 3 b a 5, 3p 2 1, a,, R 5, p, S 1 7p 6 b, 7, a 2 5p 3 b, V 7, 0 1 2 or 6, a 5, p, 1 11p 6 b, and others 2 1, 13p 6 b a,, and 6, p 6 b, 323 3 a 2 225, 1.1071 2 b A B 9. 23 2 a, 1 2 b a 231.25, 2.6779 11. B 15. A Answers to Selected Exercises 1117 25. θ = π 2 −1 −0.5 0.5 1 θ =... |
� = π 2 0.75 39. −1 −0.75 θ = 3π 2 41. θ = π 2 2 1.5 1 0.5 −0.5 −1 −1.5 −2 0.2 0.4 0.6 0.8 1 θ = 0 θ = 3π 2 43 45. θ = π 2 0.8 0.6 0.4 0.2 −0.2 −0.2 0.2 0.4 θ = 0 θ = 3π 2 47. a. 3 −4 4 θ = 0 1 −3 Answers to Selected Exercises 1119 b. 3 19. a. 4 6 6 4 b. 215 4, 210 4, 22 4 c. The smaller the eccentricity, the closer th... |
15 10 5 ( 3 2, )π 2 5 10 )3π 2 (−10, )π (2, 2 −10 −5 −5 5 10 33. r 6 1 cos u 37. r 3 1 2 cos u 41. r 3 1 sin u 45. r 2 1 2 cos u Section 11.7, page 763 35. r 39. r 16 5 3 sin u 8 1 4 cos u 43. 47. r 2 2 cos u r 3 107 1 cos u 2 1 and 4, 7 2, 5 25. Both give a straight line segment between Q P 1 equations in (a) move fr... |
67 ft x2 y2 9 23. 16x2 9y2 144 Answers to Selected Exercises 1121 80° 60° 40° 20° 43. a. 200 0 0 b. c. 40° 200 45° 40° 0 0 An angle of distance. c. 6 0 0 12 The particles do not collide; they are closest when 8 t 1.13. d. 0 −1 2 350 350 45° seems to result in the longest d is smallest when t 1.1322. Section 11.7A, page... |
real number 2 −5 10 −10 15 15. x 2 cos t 1, y 3 sin t 5 0 t 2p 2 1 21. x 2 t 2 1 2 2, y t 1 any real number t 2 7 −3 −2 13 23. x 4 tan t 1, y 5 cos t 3 1 0 t 2p 2 9 −18 18 10 −15 Answers to Selected Exercises 1123 1; 0, ± 25 y2 x2 1 4 points A Shown is the graph on the window 6 y 6. This is a hyperbola with foci at th... |
0), vertices at 6, 0, foci at the points and asymptotes 252, 0 2 1 the points (6, 0) and 252, 0 A y ± 2 and 1 B 3 ˛x. 2 37. Hyperbola 39. Ellipse 41. 43. 45. 47. 9 x 9 and 6 y 6 15 x 10 and 10 y 20 6 x 6 and 4 y 4 23 x 1 2 2 23 u 1 2 2 u y v v 10 49. 45° 51. θ = π 2 −10 10 −10 25. This is a parabola with vertex at the... |
from 2 2 ˛x 3 x 5 cos t 3, y 5 sin t 5, 0 t 2p 2 p to 2p. 83. numbers 74 and 75 11 −1 9 −9 63. 2 ˛, 323 3 2 b a 67. Hyperbola 65. Eccentricity 2 3 B 0.8165 81. 85. θ = π Pole (−2, π) (4, 0) θ = 0 1126 Answers to Selected Exercises 87. x cos t 2, y 2 sin t 3, 0 t 2p 95. x 212 tan t 2, y 2 sec t 3, 0 t 2p −9 6 −6 9 −9 9... |
000 37. a. Electric: solar: b. Electric: c. Costs same in fourteenth year; electric; solar $14,570 $6800; solar: 39. a. b. c. y 7.50x 5000 y 8.20x 130,000 ≈ (7143, 58564) 0 0 15,000 3. 50, 2, −3) y y Costs equal at approximately 7143 cases. d. The company should buy from the supplier any number of cases less than 7143 ... |
t is any real 25. No solutions x t 2, for any real number t y t 1, y 0, z 0 number x 1, y 2 z t, x 0, x 1, x 3, x 3 4 ˛, 23. 27. 29. 31. 33. 35. y 1, z 3, y 1, z 2, w 2 w 5. y 10 3 ˛, z 5 2 Answers to Selected Exercises 1129 37. 10 quarters; 28 dimes; 14 nickels 39. $3000 from her friend; $6000 from the bank; $1000 fr... |
. AB defined, 11. AB defined, 13. 3 2 0 8 11 10 ¢ 17 17 3 33 19 5 19. § AB BA not defined ≥ BA defined, 2 2 BA not defined 1 3 2 1 6 5 15. £ ≥ ¥ ; BA 4 24 9 2 21. AB 1130 ¢ £ 8 2 2 ≤ 3 21 24 8 6 15 ; BA ¢ 19 10 ≤ 0 9 2 0 8 0 0 Answers to Selected Exercises ≥ £ ≥ 5910aans_1114-1147 9/21/05 2:03 PM Page 1131 Section 12.4... |
1 625a 125b 25c 5d e 0 16a 8b 4c 2d e 3 a b c d e 5 16a 8b 4c 2d e 4 10,000a 1000b 100c 10d e c 1 ; y e x 4e x 1 47. A 0, F, C 0, D 0, E 0, and F t, 2 b 4, B 1 12 where t is any real number. The equation is t 12 which reduces to xy t 0, xy 12; hyperbola l 10,128.2, 49. h 224.4, b 2339.7 Section 12.5, page 824 ≥ 1. x 3... |
19.3201 x 2.1407, x 4.8093, y 7.7374 or y 11.7195 x 3.8371, y 7.7796 x 1.4873, y 0.0480 y 1.4873 x 0.0480, or x 1.4873, y 0.0480 r 5 29. center (0, 0); 31. center (7.5, 12.5); r 12.75 27. 33. (440.2, 38205.5) and (1893.1, 81794.5). 37. 35. Two possible boxes: one is 2 by 2 by 4 m and the other is approximately 3.123 b... |
10 11 z t,, for any real number t; consistent 2 3 4 1 9 ¢ 4 7 3 ≤ 29. Not defined 33 46 85,,, £ z 21 34 x 4, y 3, z 2 ≤ ¢ x 1 y 14 85 85 y 5x 2 2x 1 x 3, y 9 or x 1, y 1 x 1 27, y 1 27 x 1 27, y 1 27 x 1.692, y 3.136 or or x 1.812, y 2.717 47. maximum is 150 at (5, 0); minimum is 20 at (0, 2) 49. minimum of 97.5 pound... |
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