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f (1) = 3; f (2) = 9 75. 20 ; f (βˆ’1) = 9 3 77. The range for this viewing window is [0, 100]. 79. The range for this viewing window is [βˆ’0.001, 0.001]. y 100 80 60 40 20 –5 –20 –40 –60 –80 –100 –10 x 5 10 –0.1 y 0.001 0.0008 0.0006 0.0004 0.0002 –0.05 –0.0002 –0.0004 –0.0006 –0.0008 –0.001 0.05 0.1 x 83. The range for...
same scale for the x-axis and y-axis for each graph. Indicate included endpoints with a solid circle and excluded endpoints with an open circle. Use an arrow to indicate βˆ’βˆž or ∞. Combine the graphs to find the graph of the piecewise function. x to nonnegative numbers and the 5. Graph each formula of the piecewise 9. (...
1; f (βˆ’2) = 0; f (βˆ’1) = 0; f (0) = 0 49. f (βˆ’1) = βˆ’4; f (0) = 6; f (2) = 20; f (4) = 34 51. f (βˆ’1) = βˆ’5; f (0) = 3; f (2) = 3; f (4) = 16 53. (βˆ’βˆž, 1)βˆͺ(1, ∞) 55. y y 104 96 88 80 72 64 56 48 40 32 24 16 8 –8 –0.5 –0.4 –0.3 –0.2 –0.1 104 96 88 80 72 64 56 48 40 32 24 16 8 x 0.1 –0.1 –8 0.1 0.2 0.3 0.4 0.5 x The viewing ...
absolute minimum at approximately (βˆ’7.5, βˆ’220) 27. a. βˆ’3,000 people per year 29. βˆ’4 31. 27 (3, βˆ’22), decreasing on (βˆ’βˆž, 3), increasing on (3, ∞) 37. local minimum: (βˆ’2, βˆ’2), decreasing on (βˆ’3, βˆ’2), increasing on (βˆ’2, ∞) minima: (βˆ’3.25, βˆ’47) and (2.1, βˆ’32), decreasing on (βˆ’βˆž, βˆ’3.25) and (βˆ’0.5, 2.1), increasing on (βˆ’3.2...
x2 βˆ’ 1 ___________ 2x ; domain: (βˆ’βˆž, 0)βˆͺ(0, ∞) (f βˆ’ g)(x) = β€” β€” (fg)(x) = x + 2; domain: (βˆ’βˆž, 0)βˆͺ(0, ∞) f ξ€’ g ξ€ͺ (x) = 4x 3 + 8x 2; domain: (βˆ’βˆž, 0)βˆͺ(0, ∞) _ 9. (f + g)(x) = 3x 2 + √ x βˆ’ 5 ; domain: [5, ∞) (f βˆ’ g)(x) = 3x 2 βˆ’ √ (fg)(x) = 3x 2 √ f ξ€’ 3x2 g ξ€ͺ (x) = _ _ x βˆ’ 5 √ b. f ( g(x)) = 18x 2 βˆ’ 60x + 51 d. ( g ∘ g)(x) ...
(x) = √ 31. Many solutions; one possible answer: f (x) = β€” 33. Many solutions; one possible answer: f (x) = 4 √ 35. Many solutions; one possible answer: f (x) = √ 37. Many solutions; one possible answer: f (x) = 39. Many solutions; one possible answer: f (x) = x 3; g(x) = β€” 3 √ x ; g(x _____ 2x βˆ’ 3 3x βˆ’ 2 ______ x + 5 ...
compression results when a constant greater than 1 multiplies the input. A vertical compression results when a constant between 0 5. For a function f, substitute and 1 multiplies the output. (βˆ’x) for (x) in f (x) and simplify. If the resulting function is the same as the original function, f (βˆ’x) = f (x), then the fun...
(x) = ∣ x + 3 ∣ βˆ’ 2 41. f (x) = βˆ’ √ 43. f (x) = βˆ’(x + 1)2 + 2 45. f (x) = √ 47. Even 51. Even of g is a vertical reflection (across the x-axis) of the graph 55. The graph of g is of f. a vertical stretch by a factor of 4 of the graph of f. βˆ’x + 1 49. Odd 53. The graph β€” 1 _ 57. The graph of g is a horizontal compressi...
a –8 –7 –6 –5 –4 –3 –2 2 1.9 1.8 1.7 1.6 1.5 1.4 1.3 1.2 1 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0.2 0 0.5 1 1.5 2 2.5 3 3.5 4 x Section 1.6 79. 41. 43. y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 g 21 3 4 5 6 7 8 x –5 –4 –3 –2 y 5 4 3 2 1 –1 –1 –2 –3 –4 –5 21 3 4 5 x –6 –5 –4 –3 –2 g 21 1 –1 –2 –3 –4 –5 –6 –7 –8 45. 471 –1 –2 –3 –4 ...
6, 0) and (4, 0) 29. ( βˆ’βˆž, βˆ’ 8)βˆͺ(12, ∞) 27. (0, βˆ’7); no x-intercepts. ξ€² βˆͺ[6, ∞) 35. ξ€’ βˆ’βˆž, βˆ’ 8, 4 ξ€² 33. ξ€’ βˆ’βˆž, βˆ’ 8 31. ξ€° βˆ’ 4 ξ€² βˆͺ[16, ∞) _ _ _ 3 3 3 11 17.  _ 5 11. {1, 11},, 37. y 39. y 49. 51. 321 4 5 x –7 –6 –5 –4 –3 –2 y –1 –1 –2 –3 –4 –5 –6 –7 21 1 –1 –2 –3 –4 –5 –6 –5 –4 –3 –2 53. range: [0, 20] y 55. y f –100 –75 ...
βˆ’1(x) = x βˆ’ 3 9. f βˆ’1(x) = 2 βˆ’ x 13. Domain of f (x): [βˆ’7,∞); f βˆ’1 (x) = √ 5. y = f βˆ’1(x) 11. f βˆ’1(x) = βˆ’ 2x _ x βˆ’ 1 x βˆ’ 7 β€” 15. Domain of f (x): [0, ∞); f βˆ’1 (x) = √ 17. f ( g(x)) = x and g( f (x)) = x 21. One-to-one 23. Not one-to-one β€” x + 5 19. One-to-one 25. 3 27. 2 33. 6 37. 0 39. 1 31. [2, 10] 35. βˆ’4 41. x f βˆ’1...
οΏ½), decreasing on (βˆ’βˆž, 2) 29. Increasing on (βˆ’3, 1), constant on (βˆ’βˆž, βˆ’3) and (1, ∞) 31. Local minimum: (βˆ’2, βˆ’3); local maximum: (1, 3) 33. Absolute maximum: 10 35. ( f ∘ g )(x) = 17 βˆ’ 18x, ( g ∘ f )(x) = βˆ’7 βˆ’18x 37. ( f ∘ g )(x) = √ ______ )(x) = 1 _ β€” x + 2 √ 39. (f ∘ g )(x) = = 1 + x _____ 1 + 4x ξ€ͺ βˆͺ ξ€’ βˆ’ 1 ; Domain:...
71. ξ€’ βˆ’ 5, 3 ξ€ͺ _ 3 73. f βˆ’1(x) = 75. f βˆ’1(x) = √ x βˆ’ 9 _ 10 β€” x βˆ’ 1 –5 –4 –3 –2 5 4 3 2 1 –1 –1 –2 –3 –4 –5 1 2 3 4 5 x Chapter 1 practice test 1. Relation is a function parabola and the graph fails the horizontal line test. β€” 9. βˆ’2(a + b) + 1; b β‰  a 7. 2a2 βˆ’ a 11. √ 2 3. βˆ’16 5. The graph is a 21. (βˆ’βˆž, βˆ’2)βˆͺ(βˆ’2, 6)βˆͺ(6,...
= βˆ’2x + 3 47. Linear, g(x) = βˆ’3x + 5 51. Linear, g(x) = βˆ’ 25 __ 2 55. f (x) = βˆ’58x + 17.3 x + 6 642 8 10 x 57. y 30,000 25,000 20,000 15,000 10,000 5,000 βˆ’10 βˆ’4βˆ’6βˆ’8 βˆ’2 βˆ’5,000 βˆ’10,000 βˆ’15,000 βˆ’20,000 βˆ’25,000 βˆ’30,000 61. y 30 20 10 –0.1 –0.05 0.05 0.1 x –10 –20 –30 59. a. a = 11,900, b = 1001.1 b. q(p) = 1000p βˆ’ 100 x βˆ’...
, perpendicular 2 23. Line 1: m = βˆ’2, Line 2: m = βˆ’2, parallel 25. g(x) = 3x βˆ’ 3 5 31. ξ€’ βˆ’ 17 ξ€ͺ _ _, 5 3 27. p(t) = βˆ’ 1 __ t + 2 29. (βˆ’2, 1) 3 35. C 13. (βˆ’2, 0), (0, 4) 17. (8, 0), (0, 28) 33. F 37. A 39. –6 –5 –4 –3 –2 43. –6 –5 –4 –3 –2 47. –6 –5 –4 –3 –2 51. –6 –5 –4 –3 –2 y 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 y 6 5 4 ...
.4 8 59. a. g(x) = 0.75x βˆ’ 5.5 b. 0.75 c. (0, βˆ’5.5) 11. No 61. y = 3 63. x = βˆ’3 65. no point of intersection 67. (2, 7) 69. (βˆ’10, βˆ’5) 71. y = 100x βˆ’ 98 73. x < 1999 _ 201, x > 1999 _ 201 75. Greater than 3,000 texts 250 200 150 100 50 0 Section 2.3 19. W(t) = 0.5t + 7.5 7. 20.012 square units 25. C(t) = 12,025 βˆ’ 205t 5...
t) = 190t + 4,360 b. 6,640 moose cubic feet c. During the year 2017 133 minutes 57. More than $66,666.67 in sales 55. More than $42,857.14 worth of jewelry 51. a. R(t)= βˆ’2.1t + 16 b. 5.5 billion 41. During the year 1933 31. y = βˆ’2t + 180 53. More than Section 2.4 1. When our model no longer applies, after some value in...
Line 1: m βˆ’2, Line 2: m = βˆ’2, parallel 19. y = βˆ’0.2x + 21 21. 7. 3 y 23. More than 250 25. 118,000 27. y = βˆ’300x + 11,500 29. a. 800 b. 100 students per year c. P(t) = 100t + 1700 31. 18,500 33. y = $91, 625 –6 –5 –4 –3 –2 6 5 4 3 2 1 –1 –1 –2 –3 –4 –5 –6 321 4 5 6 x 35. Extrapolation y 37. y 120 100 80 60 40 20,900 6...
βˆ’4 βˆ’5 15. 321 4 5 r βˆ’2βˆ’3βˆ’4βˆ’5 i 5 4 3 2 1 βˆ’1βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 321 4 5 r 17. 8 βˆ’ i 19. βˆ’11 + 4i 21. 2 βˆ’ 5i 25. βˆ’16 + 32i 27. βˆ’4 βˆ’ 7i 29. 25 23. 6 + 15i 2 __ 31. 2 βˆ’ i 3 β€” 2 __ + 33. 4 βˆ’ 6i 35. 5 11 __ i 5 41. 1 43. βˆ’1 45. 128i β€” 3 39. 1 + i √ 37. 15i 6 471 + 2 2 9 9 __ __ 55. βˆ’2i βˆ’ 57. i 2 2 Section 3.2 37 ξ€ͺ _ 12 βˆ’ 37 _ 12...
= x 2 βˆ’ 4x + 4 297 6 ___ __ 49 49 25. Domain: (βˆ’βˆž, ∞); range: [βˆ’12, ∞) 51. f (x) = βˆ’x 2 + 1 29.  3i √ 49. f (x) = 2, βˆ’2i √ 3, βˆ’3i √ 60 __ 49 ξ€Ά 23. Domain: (βˆ’βˆž, ∞); ODD ANSWERS C-9 51. y-intercept: (0, 0); x-intercepts: (0, 0) and (2, 0); as x β†’ βˆ’βˆž, f (x ) β†’ ∞, as x β†’ ∞, f (x) β†’ ∞ 53. y-intercept: (0, 0); x-intercepts...
8 5 _ symmetry: x =, intercept: 4 (0, βˆ’8) βˆ’10 βˆ’5 y 12 8 4 βˆ’4 βˆ’8 βˆ’12 βˆ’16 βˆ’20 βˆ’24 5 10 x y = f (x) 59. f (x) = x2 βˆ’ 4x + 1 61. f (x) = βˆ’2x 2 + 8x βˆ’ 1 1 7 __ __ 63. f (x) = x 2 βˆ’ 3x + 2 2 65. f (x) = x 2 + 1 67. f (x) = 2 βˆ’ x 2 69. f (x) = 2x 2 71. The graph is shifted up or down (a vertical shift). 73. 50 feet 75. Domai...
0), (βˆ’2, 0), and (3, 0) 27. y-intercept is (0, βˆ’16), x-intercepts are (2, 0), and (βˆ’2, 0) 29. y-intercept is (0, 0), x-intercepts are (0, 0), (4, 0), and (βˆ’2, 0) 31. 3 33. 5 least possible degree: 3 degree: 2 45. Yes, 0 turning points, least possible degree: 1 47. As x β†’ βˆ’βˆž, f (x ) β†’ ∞, as x β†’ ∞, f (x) β†’ ∞ 37. 5 39. Y...
+ 1 67. V(m ) = 8m 3 + 36m 2 + 54m + 27 69. V(x ) = 4x 3 βˆ’ 32x 2 + 64x Section 3.4 9. (3, 0), (βˆ’1, 0), (0, 0) 7. (βˆ’2, 0), (3, 0), (βˆ’5, 0) 3. If we evaluate the function at a and at b and 1. The x-intercept is where the graph of the function crosses the x-axis, and the zero of the function is the input value for which ...
-intercept: (0, 4); as x β†’ βˆ’βˆž, g (x) β†’ βˆ’βˆž, as x β†’ ∞, g (x) β†’ ∞ 45. x-intercept: (3, 0) with multiplicity 3, (2, 0) with multiplicity 2; y-intercept: (0, βˆ’108); as x β†’ βˆ’βˆž, k (x) β†’ βˆ’βˆž, as x β†’ ∞, k (x) β†’ ∞ k(x) g(x) 20 16 12 8 4 βˆ’1βˆ’2βˆ’3βˆ’4βˆ’5 βˆ’4 βˆ’ 24 12 βˆ’1βˆ’2βˆ’3βˆ’4βˆ’5βˆ’6 βˆ’12 βˆ’24 βˆ’36 βˆ’48 βˆ’60 βˆ’72 βˆ’84 βˆ’96 βˆ’108 βˆ’120 47. x-intercepts:...
βˆ’ 1)(x 2 + 2x + 4) 49. Quotient: 4x 2 + 8x + 16, remainder: βˆ’1 51. Quotient 53. Quotient is x 3 βˆ’ 2x 2 + is 3x 2 + 3x + 5, remainder: 0 4x βˆ’ 8, remainder: βˆ’6 55 57 47. (x βˆ’ 5)(x 2 + x + 1) 61. 1 + 59 63. x 2 + ix βˆ’ 1 + 1 βˆ’ i _ x βˆ’ i 65. 2x 2 + 3 67. 2x + 3 69. x + 2 71. x βˆ’ 3 73. 3x 2 βˆ’ 2 Section 3.6 1. The theorem ca...
0.62); local min: (0.58, βˆ’1.38) 69. Global min: (βˆ’0.63, βˆ’0.47) 71. Global min: (0.75, βˆ’1.11) 73. f (x) = (x βˆ’ 500)2(x + 200) 75. f (x) = 4x 3 βˆ’ 36x 2 + 80x βˆ’5βˆ’6 βˆ’4 77. f (x) = 4x 3 βˆ’ 36x 2 + 60x + 100 1 _ Ο€ (9x 3 + 45x 2 + 72x + 36) 79. f (x) = Section 3.5 1. The binomial is a factor of the polynomial. 3. x + 6 +, quot...
βˆ’3 βˆ’6 βˆ’9 βˆ’12 βˆ’15, Β±1, Β±5, Β± 5 57. Β± 1 _ _ 2 2 59. Β±1 61. 1, 3 2, βˆ’ 3 1 _ _ 63. 2, 4 2 4 _ 67. f (x) = (x 3 + x 2 βˆ’ x βˆ’ 1) 9 69. f (x) = βˆ’ 1 __ (4x 3 βˆ’ x) 5 71. 8 by 4 by 6 inches 5 _ 65. 4 73. 5.5 by 4.5 by 3.5 inches 77. Radius: 6 meters; height: 2 meters meters, height: 4.5 meters 75. 8 by 5 by 3 inches 79. Radius: ...
f (x) β†’ 2 27. Local behavior: x β†’ 6+, f (x) β†’ βˆ’βˆž, x β†’ 6βˆ’, f (x) β†’ ∞ End behavior: x β†’ ±∞, f (x) β†’ βˆ’2 βˆ’ + 1 1 _ _ 29. Local behavior: x β†’ βˆ’, f (x) β†’ βˆ’βˆž,, f (x) β†’ ∞, x) β†’ ∞, x β†’ βˆ’ 5 _ _, f (x) β†’ βˆ’βˆž 2 2 1 _ End behavior: x β†’ ±∞, f (x) β†’ 3 33. y = 2x 31. y = 2x + 4 35. Vertical asymptote at x = 0, horizontal asymptote at ...
. Vertical asymptote at x = 4; slant asymptote at 1 y = 2x + 9; (βˆ’1, 0), ξ€’, 0 ξ€ͺ, _ 2 1 ξ€ͺ ξ€’ 0, _ 4 h(x) 50 40 30 20 10 βˆ’50 βˆ’40 βˆ’30 βˆ’20 βˆ’10 βˆ’10 βˆ’20 βˆ’30 βˆ’40 βˆ’50 y = 2x + 9 x = 4 10 20 30 40 50 45. Vertical asymptote at x = βˆ’1; horizontal asymptote at y = 1; (βˆ’3, 0), (0, 3) a(x) x =βˆ’1 15 12 9 6 3 βˆ’15 βˆ’6βˆ’9βˆ’12 βˆ’3βˆ’3 βˆ’6 βˆ’9 βˆ’12...
–4 –6 –8 –10 42 6 8 10 x –10 –8 –6 –4 10 8 6 4 2 –2–2 –4 –6 –8 –10 42 6 8 10 x 21 3 4 5 x βˆ’1βˆ’2βˆ’3βˆ’4βˆ’5 βˆ’1 βˆ’2 βˆ’3 45. [βˆ’4, 2) βˆͺ [5, ∞) y 5 4 3 2 1 16 14 12 8 4 84 12 14 16 βˆ’16 βˆ’14 βˆ’4βˆ’8βˆ’12 βˆ’4 βˆ’8 βˆ’12 βˆ’14 βˆ’16 47. (βˆ’2, 0), (4, 2), (22, 3) y C-12 67. Vertical asymptote at x = βˆ’4; horizontal asymptote at y = 2 x y x y βˆ’4.1 82 1...
. f βˆ’1(x) = √ 3 β€” 4 βˆ’ x 17. f βˆ’1(x) =, [0, ∞) 19. f βˆ’1(x) = (x βˆ’ 9)2 + 4 __________ 4, [9, ∞) √ x 2 βˆ’ 1 ______ 2 x βˆ’ 9 _____ ξ€ͺ 2 7x βˆ’ 3 ______ 27. f βˆ’1(x) = 1 βˆ’ x 3 23. f βˆ’1(x) = 2 βˆ’ 8x ______ x 5x βˆ’ 4 ______ 4x + 3 31. f βˆ’1(x) = √ β€” x + 6 + 3 21. f βˆ’1(x) = ξ€’ 25. f βˆ’1(x) = 29. f βˆ’1(x) = √ 33. f βˆ’1(x β€” 10 8 6 4 2 βˆ’10 βˆ’2...
2 –1 βˆ’8 βˆ’16 βˆ’24 βˆ’32 βˆ’40 59. r(V) =, 5.53 seconds b2 + 4x √ _ 55. f βˆ’1(x) =, β‰ˆ 3.63 feet 61. n(C) = β€” 3V _ 4Ο€ ___ V _ 6Ο€ 53. f βˆ’1(x) = βˆ’ b _ + 2 ________ 57. t(h) = √ 200 βˆ’ h _ 4.9 √ 63. r(V) = √ Section 3.9 1. The graph will have the appearance of a power function. 5. y = 5x 2 3. No. Multiple variables may jointly vary...
2)2 βˆ’9; vertex: (2, βˆ’9); intercepts: (βˆ’1, 0), (5, 0), (0, βˆ’5) f(x) βˆ’2βˆ’3βˆ’4βˆ’5 10 8 6 4 2 βˆ’1βˆ’2 βˆ’4 βˆ’6 βˆ’8 βˆ’10 321 4 5 6 7 x 5. {2 + i, 2 βˆ’ i} 3 _ 25 9. f (x) = (x + 2)2 + 3 11. 300 meters by 150 meters, the longer side parallel to the river 13. Yes; degree: 5, leading coefficient: 4 15. Yes; degree: 4; leading coefficient:...
47. f βˆ’1(x) =, x β‰₯ βˆ’3 49. y = 64 51. y = 72 (x + 3)2 βˆ’ 5 __ 4 53. β‰ˆ 148.5 pounds Chapter 3 practice test 1. 20 βˆ’ 10i 3. {2 + 3i, 2 βˆ’ 3i} 5. As x β†’ βˆ’βˆž, f (x) β†’ βˆ’βˆž, as x β†’ ∞, f (x) β†’ ∞ 7. f (x) = (x + 1)2 βˆ’ 9, vertex: (βˆ’1, βˆ’9), intercepts: (2, 0), (βˆ’4, 0)(0, βˆ’8) y 9. 60,000 square feet 11. 0 with multiplicity 4, 3 with ...
a proportional rate. the charge decreases by a constant amount each visit, so the statement represents a linear function. 11. After 20 years forest A will have 43 more trees than forest B. 13. Answers will vary. Sample response: For a number of years, the population of forest A will increasingly exceed forest B, but b...
. 47,622 foxes 67. $82,247.78; $449.75 63. 1.39%; $155,368.09 x = a(eβˆ’n)x = a(e)βˆ’nx. = a(bβˆ’1)x = a ((en)βˆ’1 ) 65. $35,838.76 Section 4.2 1. An asymptote is a line that the graph of a function approaches, as x either increases or decreases without bound. The horizontal asymptote of an exponential function tells us the li...
4(2)x + 2 27. Horizontal asymptote: h(x) = 3; domain: all real numbers; range: all real numbers strictly greater than 3. h(x1βˆ’2βˆ’3βˆ’4βˆ’5 21 3 4 5 x 29. As x β†’ ∞, f (x) β†’ βˆ’βˆž; as x β†’ βˆ’βˆž, f (x) β†’ βˆ’1 31. As x β†’ ∞, f (x) β†’ 2; as x β†’ βˆ’βˆž, f (x) β†’ ∞ 33. f (x) = 4x βˆ’ 3 35. f (x) = 4x βˆ’ 5 37. f (x) = 4βˆ’x 39. y = βˆ’2x + 3 41. y = βˆ’2(...
be applied to solve for x. 5. The natural logarithm is a special case of the logarithm with base b in that the natural log always has base e. Rather than notating the natural logarithm as loge notation used is ln(x). 7. ac = b 15. e n = w 21. logn(103) = 4 1 _ 27. x = 8 35. x = e2 45. 4 55. β‰ˆ 2.708 defined value for x...
y-axis 5. No. A horizontal asymptote would will affect its domain. suggest a limit on the range, and the range of any logarithmic function in general form is all real numbers. 1 7. Domain: ξ€’ βˆ’βˆž, ξ€ͺ ; range: (βˆ’βˆž, ∞) _ 2 9. Domain: ξ€’ βˆ’ 17, ∞ ξ€ͺ ; range: (βˆ’βˆž, ∞ ) _ 4 11. Domain: (5, ∞); vertical asymptote: x = 5, ∞ ξ€ͺ ; ver...
10 x βˆ’5 βˆ’4 g(x) = log (x) l 2 43. f(x) 5 4 3 2 1 21 3 4 5 x βˆ’3βˆ’4βˆ’5βˆ’6βˆ’7βˆ’8 βˆ’3 βˆ’2 βˆ’1βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 x 4 5 21 3 g(x) = ln(x) y 5 4 3 2 1 βˆ’2 βˆ’1βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 x 21 47. f (x) = log2(βˆ’(x βˆ’ 1)) 49. f (x) = 3log4(x + 2) 51. x = 2 53. x β‰ˆ 2.303 55. x β‰ˆ βˆ’0.472 45. g(x) 5 4 3 2 1 21 3 4 x βˆ’3βˆ’4βˆ’5βˆ’6 βˆ’2 βˆ’1βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 C-15 57. The g...
21. ln(2x7) 17. 3 2 1 _ n logb(x). n ) = 9. ln(7xy) 14 _ 3 _ 23. log ξ€’ xz3 ξ€ͺ _ β€” y √ 1 _ 27. log11(5) = b 25. log7(15) = 6 29. log11 ξ€’ _ 11 35. β‰ˆ 0.93913 ln(15) _ ln(7) ξ€ͺ = or 37. β‰ˆ βˆ’2.23266 33. β‰ˆ 2.81359 39. x = 4, By the quotient rule: log6(x + 2) βˆ’ log6(x βˆ’ 3) = log6 ξ€’ Rewriting as an exponential equation and solvi...
13. No solution 17. k = βˆ’ ln(38) _ 3 19. x = 6 _ 9. b = 7. n = βˆ’1 5 17 15. p = log ξ€’ ξ€ͺ βˆ’ 7 _ 8 38 ln ξ€’ ξ€ͺ βˆ’ 8 _ 3 __ 9 21. x = ln(12) ODD ANSWERS C-16 23. x = 25. No solution 27. x = ln(3) y 67. About 5 years 3 ln ξ€’ ξ€ͺ βˆ’ 3 _ 5 __ 8 1 _ 100 29. 10βˆ’2 = 31. n = 49 33. k = 1 _ 36 35. x = 9 βˆ’ e _ 8 37. n = 1 43. x = Β± 10 _ 3...
of the initial amount of that 3. Doubling time is a measure substance or quantity to decay. of growth and is thus associated with exponential growth models. The doubling time of a substance or quantity is the amount of time it takes for the initial amount of that substance or quantity 5. An order of magnitude is the n...
number b such that b β‰  1. Then, ln (y) = ln (b x) ln (y) = x ln (b) e ln(y) = e xln(b) y = e xln(b) log ξ€’ S _ ξ€ͺ S0 M = log ξ€’ S _ ξ€ͺ S0 2 = ξ€’ S _ ξ€ͺ S0 S010 3M 2 = S 29. A = 125e (βˆ’0.3567t); A β‰ˆ 43mg 33. f (t) = 250e βˆ’0.00914t; half-life: about 76 minutes 35. r β‰ˆ βˆ’ 0.0667; hourly decay rate: about 6.67% 37. f (t) = 1350 ...
. B y 23. About 38 wolves 25. About 8.7 years 27. f (x) = 776.682 (1.426)x 29. y C-17 43. f (10) β‰ˆ 2.3 45. When f (x) = 8, x β‰ˆ 0.82 47. f (x) = 25.081 __ 1 + 3.182eβˆ’0.545x 49. About 25 41. y 10 51. y 140 130 120 110 100 90 80 70 60 50 40 30 20 10 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 53. 0 y 140 130 120 110 100 ...
. a = ln (4) + 8 ________ 10 23. No solution 25. x = ln(9) β€” 27. x = Β± 3 √ 3 ____ 2 29. f (t) = 112eβˆ’0.019792t; half-life: about 35 days 31. T(t) = 36 eβˆ’0.025131t + 35; T(60) β‰ˆ 43Β° F 33. Logarithmic C-18 11. g(x) = 7(6.5)βˆ’x; y-intercept: (0, 7); domain: all real numbers; range: all real numbers greater than 0. 2 _ 15. ...
y 40 38 36 34 32 30 28 26 24 22 20 18 16 14 12 10 8 6 4 2 63. y = 4(0.2)x; y = 4e–1.609438x 65. About 7.2 days 67. Logarithmic y = 16.68718 βˆ’ 9.71860ln(x 10 11 x 35. Exponential; y = 15.10062(1.24621)x y 150 140 130 120 110 100 90 80 70 60 50 40 30 20 10 10 x 37. Logistic; y = 18.41659 __ 1 + 7.54644 eβˆ’0.68375x y 20 1...
. 13. 15. 17. 240Β° 19. 4Ο€ ___ 3 21. 2Ο€ ___ 3 35. βˆ’3Ο€ radians β‰ˆ 12.72 cm2 27. 20Β° 23. 7Ο€ ___ β‰ˆ 11.00 in2 2 81Ο€ ____ 25. 20 Ο€ __ 29. 60Β° 31. βˆ’75Β° 33. radians 2 radians 39. 37. Ο€ radians 25Ο€ ___ 9 43. 47. 104.7198 cm2 5Ο€ ___ 6 5.02Ο€ _____ 3 41. 21Ο€ ___ 10 49. 0.7697 in2 51. 250Β° 53. 320Β° β‰ˆ 6.60 meters 45. β‰ˆ 5.26 miles β‰ˆ 8...
οΏ½ = 47. 3 3 Ο€ 7Ο€, Quadrant IV, sin ξ€’ ___ __ ξ€ͺ = βˆ’ 49. 4 4 β€” √ 15 ____ 3 ξ€ͺ 4 1 __, cos t = βˆ’ 57. (βˆ’2.778, 15.757) 59. [βˆ’1, 1] 61. sin t = 2 β€” √ 3 ____ 2 3Ο€ Ο€, Quadrant II, sin ξ€’ ___ __ ξ€ͺ = 45. 4 4 55. ξ€’ βˆ’10, 10 √ β€” √ 77 ____ 9 65. sin t = 63. sin t = βˆ’ β€” √ 2 ____ 2, cos t = βˆ’ 53. βˆ’ 51. β€” 67. sin t = βˆ’ β€” √ 2 ____ 2 β€” √ 2...
and 5. The outputs of tangent and β€” β€” 3 7. 9. √ β€” 2 √ 3 _____ 3 19. βˆ’ 2 √ √ 3 3 ____ ____ 3 3 β€” 27. βˆ’2 29. βˆ’ √ 3 ____ 3 17. β€” 11. √ β€” 2 13. 1 15. 2 21. √ β€” 3 23. βˆ’ √ β€” 2 25. βˆ’1 31. 2 33. 35. βˆ’2 37. βˆ’1 β€” √ 3 ____ 3 ODD ANSWERS C-20 β€” β€”, sec t = βˆ’ 3, csc t = βˆ’ 3 √ 39. sin t = βˆ’ 2 √ 2 2 ____ ____, 4 3 2, cot t = tan t = ...
11. b = β€” 20 √ 3 ______ 3, c = β€” 40 √ 3 ______ 3 13. a = 10,000, c = 10,000.5 15. b = 17. β€” 5 √ 29 _____ 29 5 __ 19. 2 21. β€” √ 29 ____ 2 β€” 5 √ 3 ____ 3, c = 23. β€” 10 √ 3 _____ 3 β€” 5 √ 41 _____ 41 29. c = 14, b = 7 √ β€” 3 33. b = 9.9970, c = 12.2041 27. 5 __ 25. 4 β€” √ 41 ____ 4 31. a = 15, b = 15 35. a = 2.0838, b = 11....
3 Ο€ __ 21 _____ __ 23. a = 2 2 17. 15. √ 19. ChapteR 6 Section 6.1 1. The sine and cosine functions have the property that f (x + P) = f (x) for a certain P. This means that the function 3. The absolute values repeat for every P units on the x-axis. value of the constant A (amplitude) increases the total range and the...
t 0.5 1 1.5 2 t 0 βˆ’2βˆ’1.5βˆ’1βˆ’0.5 βˆ’1 βˆ’2 βˆ’3 βˆ’4 ODD ANSWERS Ο€ _ 15. Amplitude: 3; period: ; 4 midline: y = 5; maximum: y = 8 occurs at x = 0.12; minimum: y = 2 occurs at x = 0.516; horizontal shift: βˆ’4; vertical translation: 5; for one period, the graph starts at 0 Ο€ _. and ends at 16 Ο€ 8 3Ο€ 16 Ο€ 4 x Ο€βˆ’ 4 3Ο€βˆ’ 16 Ο€βˆ’ 8 0 Ο€βˆ’ ...
οΏ½s bounds decrease as |x| grows. There appears to be a horizontal asymptote at y = 0. Ο€ 2 Ο€ 3Ο€ 2Ο€ 2 x 3Ο€βˆ’βˆ’ 2Ο€ 2 βˆ’Ο€ βˆ’2 Ο€ βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 y 5 4 3 2 1 Ο€ 2Ο€ 3Ο€ 4Ο€ 5Ο€ x βˆ’4Ο€ 0 βˆ’5Ο€ βˆ’3Ο€βˆ’2Ο€βˆ’Ο€ βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 5Ο€βˆ’βˆ’2Ο€ 4Ο€βˆ’3 2Ο€βˆ’3 βˆ’Ο€ 3 5 4 3 2 1 0 Ο€βˆ’ 3 βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 βˆ’3Ο€ 8Ο€βˆ’ 7Ο€βˆ’3 3 Ο€ 3 2Ο€ 3 Ο€ 4Ο€ 3 5Ο€ 3 2Ο€ 7Ο€ 3 8Ο€ 3 3Ο€ t Section 6.2 21....
3; 2Ο€ x ξ€ͺ + 3 _ 5 23. Amplitude: 2, midline: y = βˆ’3; period: 4; equation: Ο€ f (x) = 2sin ξ€’ _ x ξ€ͺ βˆ’3 2 period: 5; equation: f (x) = βˆ’2cos ξ€’ 27. Amplitude: 4, midline: y = 0; period: 2; equation: Ο€ f (x) = βˆ’4cos ξ€’ Ο€ ξ€’ x βˆ’ _ ξ€ͺ ξ€ͺ 2 period: 2; equation: f (x) = 2cos(Ο€x) + 1 Ο€ Ο€ 33. sin ξ€’ _ _ ξ€ͺ = 1 35. 2 2 with respect to t...
οΏ½ tan 39. f (x) = csc(2x) 41. f (x) = csc(4x) 43. f (x) = 2csc x 1 _ 45. f (x) = tan(100Ο€x) 2 f (x) 47. 8 6 4 2 5Ο€βˆ’ 12 Ο€ 2 Ο€ 3Ο€ 2 2Ο€ x Ο€βˆ’ 4 Ο€βˆ’ Ο€βˆ’ 6 3 0 Ο€βˆ’ 12 βˆ’2 βˆ’4 βˆ’6 βˆ’8 Ο€ 12 Ο€ 6 Ο€ 4 Ο€ 3 5Ο€ 12 x 8 6 4 2 f(x) 16 12 8 4 βˆ’2Ο€ βˆ’ 3Ο€ 2 βˆ’Ο€ 0 Ο€βˆ’ 2 βˆ’8 βˆ’12 βˆ’16 Ο€ 4 Ο€ 2 3Ο€ 4 Ο€ x 49. f(x) βˆ’Ο€ 8 6 4 2 3Ο€βˆ’ 4 2 0 Ο€βˆ’ Ο€βˆ’ 4 βˆ’2 βˆ’4 βˆ’6 βˆ’8 per...
βˆ’ 7Ο€ 4 53. y 4 3 2 1 Ο€ 2 Ο€ 3Ο€ 2 5Ο€ 2Ο€ 2 x 5Ο€βˆ’ 2 βˆ’2Ο€ 3Ο€βˆ’ 2 βˆ’Ο€ 0 Ο€βˆ’ 2 βˆ’1 βˆ’2 βˆ’3 βˆ’4 ; period: 2Ο€ ; asymptotes: x = Ο€ 7 _ _ + Ο€k, 35. Stretching factor: 5 4 where k is an integer f (x) Ο€ 55. a. f(x) 7Ο€ βˆ’4 5Ο€ βˆ’4 βˆ’ 7 6 5 4 3 2 1 3Ο€ βˆ’4 0 Ο€ 4 βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 βˆ’6 βˆ’7 Ο€ 4 3Ο€ 4 5Ο€ 4 7Ο€ 4 9Ο€ 4 x 16 14 12 10 βˆ’ 2 Ο€βˆ’ 3 Ο€βˆ’ βˆ’2 6 βˆ’4 βˆ’6 βˆ’8 ...
4 βˆ’5 βˆ’6 βˆ’7 7. Amplitude: 6; period: is 2Ο€ _ ; 3 midline: y = βˆ’1; no asymptotes f(x1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 βˆ’6 βˆ’7 2Ο€ Ο€ βˆ’ βˆ’ 9 3 Ο€ 9 2Ο€ 9 Ο€ 3 4Ο€ 9 5Ο€ 9 2Ο€ 3 x βˆ’ 2Ο€ 3 5Ο€ βˆ’9 4Ο€ 9 βˆ’ 9. Stretching factor: none; period: Ο€; midline: y = βˆ’4; Ο€ _ asymptotes: x = + Ο€k, 2 where k is an integer f(x) 10 8 6 4 2 x Ο€ 2 Ο€ βˆ’Ο€ Ο€βˆ’ 2 βˆ’2 βˆ’4 βˆ’6 βˆ’8 βˆ’10 1...
ξ€° βˆ’ Ο€ Ο€ ξ€² ; thus, this _ _, 2 2 interval is the range of the inverse function of y = sin x, f (x) = sinβˆ’1 x. The function y = cos x is one-to-one on [0, Ο€]; thus, this interval is the range of the inverse function of Ο€ _ y = cos x, f (x) = cosβˆ’1 x. 3. is the radian measure of an 6 Ο€ angle between βˆ’ Ο€ _ _ 5. In order f...
with the horizontal is 60 degrees. Ο€ βˆ’1 1 x βˆ’Ο€ Chapter 6 Review exercises 1. Amplitude: 3; period: is 2Ο€; midline: y = 3; no asymptotes f (x) 6 5 4 3 2 1 Ο€βˆ’ 2 βˆ’2 βˆ’3 βˆ’4 βˆ’5 βˆ’6 βˆ’2Ο€ 3Ο€βˆ’ 2 βˆ’Ο€ x Ο€ 2 Ο€ 3Ο€ 2 2Ο€ ODD ANSWERS C-24 15. Amplitude: none; period: no phase shift; asymptotes: Ο€ _ x = k, where k is an integer 5 f(x) 2Ο€...
βˆ’ 2 Ο€βˆ’ 3 41. The graphs appear to be identical. y 1 0.8 0.6 0.4 0.2 βˆ’0.2 0 βˆ’0.2 βˆ’0.4 βˆ’0.6 βˆ’0.8 βˆ’1 βˆ’1 βˆ’0.6 0.2 0.6 1 x βˆ’1 βˆ’0.6 y 1 0.8 0.6 0.4 0.2 0 βˆ’0.2 βˆ’0.2 βˆ’0.4 βˆ’0.6 βˆ’0.8 βˆ’1 0.2 0.6 1 x 9Ο€βˆ’ 4 βˆ’2Ο€ 7Ο€βˆ’ 4 3Ο€βˆ’ 2 5Ο€βˆ’ 4 βˆ’Ο€ 3Ο€βˆ’ 4 4 3 2 1 7Ο€βˆ’ βˆ’3Ο€ 2 5Ο€βˆ’ 2 βˆ’2Ο€ 3Ο€βˆ’ 2 βˆ’Ο€ Ο€ βˆ’ 2 βˆ’1 βˆ’2 βˆ’3 βˆ’4 Ο€ 2 Ο€ 3Ο€ 2 2Ο€ 5Ο€ 2 3Ο€ 7Ο€ 2 4Ο€ 9Ο€ 2 5Ο€ 2 ...
βˆ’6; midline: y = βˆ’3 19. D(t) = 68 βˆ’ 12sin ξ€’ Ο€ _ 21. Period: ; horizontal 6 shift: βˆ’7 23. f (x) = sec(Ο€x); period: 2; phase shift: 0 25. 4 Ο€ _ 12 x ξ€ͺ Chapter 6 practice test 1. Amplitude: 0.5; period: 2Ο€ ; midline: y = 0 3. Amplitude: 5; period: 2Ο€ ; midline: y = 0 y y 0.5 0.25 0 Ο€βˆ’ βˆ’Ο€ 2 βˆ’0.25 βˆ’0.5 βˆ’2Ο€ 3Ο€βˆ’ 2 Ο€ 2 Ο€ 3Ο€ 2...
Ο€ x βˆ’2Ο€ 3Ο€βˆ’ 2 βˆ’Ο€ Ο€βˆ’ 2 βˆ’0.5 βˆ’1 βˆ’1.5 βˆ’2 Ο€ _ 37. 3 Ο€ _ 39. 2 x + 1 _ x 41. √ β€” 1 βˆ’ (1 βˆ’ 2x)2 1 β€” 43. _ 1 + x 4 √ 49. 0.07 radians 45. csc t = 47. False ChapteR 7 Section 7.1 1. All three functions, F, G, and H, are even. This is because F(βˆ’x) = sin(βˆ’x)sin(βˆ’x) = (βˆ’sin x)(βˆ’sin x) = sin 2 x = F(x), G(βˆ’x) = cos(βˆ’x)cos(βˆ’x) = c...
2 ΞΈ + cos 2 ΞΈ)+ cos 2 ΞΈ = 3 + cos 2 ΞΈ Section 7.2 1. The cofunction identities apply to complementary angles. Viewing the two acute angles of a right triangle, if one of those Ο€ _ βˆ’ x. Then angles measures x, the second angle measures 2 Ο€ βˆ’ x ξ€ͺ. The same holds for the other cofunction sin x = cos ξ€’ _ 2 identities. The...
cos x) β€” √ 2 _ 2 y 1 0.8 0.6 0.4 0.2 Ο€βˆ’ Ο€βˆ’ 4 2 βˆ’0.4 βˆ’0.6 βˆ’0.8 βˆ’1 βˆ’2Ο€ 7Ο€βˆ’ 4 3Ο€βˆ’ 2 5Ο€βˆ’ 4 βˆ’Ο€ 3Ο€βˆ’ 4 Ο€ 4 Ο€ 2 3Ο€ 4 Ο€ 5Ο€ 4 3Ο€ 2 7Ο€ 4 2Ο€ x 33. They are the same. g(x) = sin(9x) βˆ’ cos(3x) sin(6x) 35. They are the different, try 37. They are the same. 39. They are the different, try g(ΞΈ) = 41. They are different, try g(x) = 2ta...
a. β€” √ 1 3 __ ____ c. βˆ’ √ b. βˆ’ 2 2 β€” 3 b. 31 _ 32 β€” 5 2 √ _ 5 9. cos ΞΈ = βˆ’, sin ΞΈ = sec ΞΈ = βˆ’, cot ΞΈ = βˆ’ βˆ’ √ __ 3 2 15. β€” 5 √ _ 5 1 _, tan ΞΈ = βˆ’, csc ΞΈ = √ 2 β€” 5, Ο€ ξ€ͺ 11. 2sin ξ€’ __ 2 β€” β€” 2 βˆ’ √ √ __ 2 13. 2 17. 2 + √ β€” 3 19. βˆ’1 βˆ’ √ β€” 2 21. a. 23. a. b. βˆ’ 3 __ c. βˆ’ 2 β€” 2 √ 13 _ 13 β€” 3 √ 13 _ 13 β€” 6 √ √ 10 _ _ 4 4 β€” 13 1...
) ___ 4(cos(2x) + 1) 53. 4sin x cos x (cos 2 x βˆ’ sin 2 x) 51. (1 + cos(4x)) sin x __ 2 55. 2tan x _ = 1 + tan 2 x 2sin x _ cos x _ sin 2 x _ 1 + cos 2 x = 2sin x _ cos x __ cos 2 x + sin 2 x __ cos 2 x = cos 2 x _ 1 2sin x _ cos x. sin(2x) _ cos(2x) = tan(2x) 57. 2sin x cos x __ = 2cos 2 x βˆ’ 1 = 2sin x cos x = sin(2x) ...
13. 2cos(7x) 11. 2cos(5t)cos t 15. 2cos(6x)cos(3x) 1 1 1 3 βˆ’ 2) ( √ 3 ) (1 + √ ( √ _ _ _ 17. 19. 21. 4 4 4 1 _ 23. cos(80Β°) βˆ’ cos(120Β°) (sin(221Β°) + sin(205Β°)) 25. 2 β€” 27. √ 2 cos(31Β°) 31. 2sin(βˆ’1.5Β°)cos(0.5Β°) 33. 2sin(7x) βˆ’ 2sin x = 2sin(4x + 3x) βˆ’ 2sin(4x βˆ’ 3x) = = 2(sin(4x)cos(3x) + sin(3x)cos(4x)) βˆ’ 2(sin(4x)cos(3...
+ cos y. Make a substitution and let x = Ξ± + Ξ² and let y = Ξ± βˆ’ Ξ², so cos x + cos y becomes cos(Ξ± + Ξ²) + cos(Ξ± βˆ’ Ξ²) = 53. βˆ’sin(14x) 51. 2cos(2x) = cos Ξ± cos Ξ² βˆ’ sin Ξ± sin Ξ² + cos Ξ± cosΞ² + sin Ξ± sin Ξ² = 2cos Ξ± cos Ξ² Since x = Ξ± + Ξ² and y = Ξ± βˆ’ Ξ², we can solve for Ξ± and Ξ² in terms of x and y and substitute in for 2cos Ξ± ...
equation has no solution. 7Ο€ _ 4 29Ο€ _ 18 19. 7. 37 ___ 6 17. 15. 29. 0, Ο€ Ο€ _ 9., 4 5Ο€ _ 4 Ο€ _, 18 3Ο€ _, 4 7Ο€ _, 6 13Ο€ ____, 12, 5Ο€ ____, 4 25Ο€ _, 18 29 ___, 6 3Ο€ Ο€ _ _ 11., 4 4 17Ο€ _, 18 25 ___, 6 Ο€ 5Ο€ 2Ο€ _ _ _ 5., 4 3 3 Ο€ 7Ο€ 11Ο€ 13Ο€ 5Ο€ _ _ _ _ _ 13.,,, 4 4 18 18 6 3Ο€ 19Ο€ 11Ο€ 5Ο€ 17 21Ο€ 13 5 1 ___ ____ ____ ___ ___ _...
οΏ½ _ ξ€’ √ 2 1 Ο€ + tanβˆ’1 ξ€’ _ ξ€’ √ 2 49. There are no solutions. 2Ο€ 3Ο€ 5Ο€ 7Ο€ 4Ο€ Ο€ _ _ _ _ _ _ 53. 0, 55 3Ο€ ξ€ͺ,, Ο€ βˆ’ sinβˆ’1 ξ€’ ξ€ͺ, 57. sinβˆ’ ξ€ͺ ξ€ͺ, 2Ο€ βˆ’ cosβˆ’1 ξ€’ βˆ’ 59. cosβˆ’ 5Ο€ ξ€ͺ, ξ€ͺ, 2Ο€ βˆ’ cosβˆ’1 ξ€’ βˆ’, cosβˆ’1 ξ€’ βˆ’ _ _ _ _ 61. 4 4 3 3 3 3 2 2 ξ€ͺ ξ€ͺ, 2Ο€ βˆ’ cosβˆ’1 ξ€’ ξ€ͺ, cosβˆ’1 ξ€’ βˆ’ 63. cosβˆ’1 ξ€’ ξ€ͺ, 2Ο€ βˆ’ cosβˆ’ ξ€ͺ ξ€ͺ, 2Ο€ βˆ’ cosβˆ’1 ξ€’ __ ξ€’ 1 βˆ’ √ 3 1 29 βˆ’ 5 ξ€ͺ ...
ξ€’ ξ€ͺ, ξ€ͺ, Ο€ βˆ’ sinβˆ’1 ξ€’ _ _ _ 4 4 2 Ο€ 3Ο€ Ο€ _ _ _ 85. 87. There are no solutions. 89. 0,, Ο€,, 2 2 2 3Ο€ _ 2 91. There are no solutions. 93. 7.2Β° 95. 5.7Β° 97. 82.4Β° 99. 31.0Β° 103. 59.0Β° 101. 88.7Β° 105. 36.9Β° Section 7.6 1. Physical behavior should be periodic, or cyclical. cumulative rainfall is always increasing, a sinusoida...
2 x Ο€ 53. y = 3 (2)xcos ξ€’ _ x ξ€ͺ + 1 2 Ο€ 51. y = 4(βˆ’2)x + 8sin ξ€’ _ x ξ€ͺ 2 47. 234.3 miles, at 72.2Β° 43. 15.4 seconds Chapter 7 Review exercises 1. sinβˆ’1 ξ€’ 7Ο€ ___, 6 3. β€” β€” β€” 3 3 3 √ √ √ ξ€ͺ, 2Ο€ βˆ’ sinβˆ’1 ξ€’ ξ€ͺ, Ο€ + sinβˆ’1 ξ€’ ξ€ͺ, Ο€ βˆ’ sinβˆ’1 ξ€’ _ _ _ 3 3 3 11Ο€ 1 1 ξ€ͺ 7. 1 ξ€ͺ, Ο€ βˆ’ sinβˆ’1 ξ€’ 5. sinβˆ’1 ξ€’ ____ _ _ 4 4 6 β€” √ 2 ____ 13. 2 9. Y...
2 x, 10 β€” √ 3 _ 3 21. βˆ’ 7 _, 25 25. 19. 23. √ 24 _ 7 β€”, βˆ’ 2 ξ€ͺ β€” (2)sin 2 x = cot x βˆ’ cos x _ sin x = βˆ’2sin x cos x + cot x = βˆ’sin (2x) + cot x 29. 10sin x βˆ’ 5sin(3x) + sin(5x) ___ 8(cos(2x) + 1) 1 __ (sin(6x) + sin(12x)) 35. 2, Ο€ 39. 43. 7Ο€ ___ 4 3Ο€ ___, 4 5Ο€ Ο€ ___ __ 41. 0,, 6 6 47. 0.2527, 2.8889, 4.7124 51. 3sin ξ€’ ...
cos x = sec x = sec x 1 __ 60, frequency: 60 Hz 1 __ 21. Amplitude:, period 4 23. Amplitude: 8, fast period: 1 ___ 500, slow frequency: 10 Hz period: 1 __ 10 cos (4Ο€t), 31 second, fast frequency: 500 Hz, slow 25. D(t) = 20 (0.9086)t ChapteR 8 Section 8.1 11. b β‰ˆ 3.78 7. Ξ² = 72Β°, a β‰ˆ 12.0, 5. A triangle with two 9. Ξ³ =...
.6 km 67. 371 ft 69. 5,936 ft 71. 24.1 ft 73. 19,056 ft 2 75. 445,624 square miles 77. 8.65 ft 2 25. A β‰ˆ 47.8Β° or Aβ€² β‰ˆ 132.2Β° 33. 12.2 47. 42.0 55. AB β‰ˆ 2.8 39. x = 76.9Β°or x = 103.1Β° 53. AD β‰ˆ 13.8 59. 51.4 feet 41. 110.6Β° 29. 370.9 45. 57.1 31. 12.3 35. 16.0 Section 8.2 9. 34.7 7. 11.3 11. 26.7 17. 95.5Β° 19. 26.9Β° 13....
the coordinate plane, there is one representation, but for each point in the polar plane, there are infinite representations. 3. Determine ΞΈ for the point, then move r units from the pole to plot the point. If r is negative, move r units from the pole in the opposite direction but along the same angle. The point is a ...
3y + x = 6; line 33. y = 3; line 37. x 2 + y 2 = 4; circle 3Ο€ ξ€ͺ 41. ξ€’ 3, ___ 4 45. 43. (5, Ο€) 47. 39. x βˆ’ 5y = 3; line 35. xy = 4; hyperbola βˆ’4βˆ’6βˆ’8 69. (1.618, βˆ’1.176) 71. (10.630, 131.186Β°) 73. (2, 3.14) or (2, Ο€) 75. A vertical line with a units left of the y-axis. 77. A horizontal line with a units below the x-axis...
limaΓ§on 45. (ΞΈ from 0 to 8) 47. (ΞΈ from βˆ’Ο€ to Ο€) (ΞΈ from 0 to 2Ο€) (ΞΈ from 0 to 2Ο€ 11 0.5 1 1.5 2 0.5 1 1.5 2 25. One-loop/dimpled limaΓ§on (ΞΈ from 0 to 2Ο€) 27. Inner loop/two-loop limaΓ§on 49. (ΞΈ from 0 to 2Ο€) 51. (ΞΈ from 0 to 3Ο€) 1 2 3 2 4 6 8 10 1 3 5 7 9 1 2 3 4 53. (ΞΈ from 0 to 2Ο€) 29. Inner loop/two-loop limaΓ§on 31...
69. ξ€’ 0, ___ __ 2 2 ξ€ͺ and at ΞΈ = 7Ο€ 5Ο€ ___ ___ 4 4 since r is squared 3Ο€ ___, 4 Section 8.5 β€” βˆ’1 1. a is the real part, b is the imaginary part, and i = √ 3. Polar form converts the real and imaginary part of the complex number in polar form using x = r cos ΞΈ and y = r sin ΞΈ. 5. zn = rn(cos (nΞΈ) + i sin (nΞΈ)) It is us...
Imaginary 2βˆ’3βˆ’4βˆ’5βˆ’6 βˆ’1 βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 βˆ’6 1 2 3 4 5 6 Real βˆ’2βˆ’3βˆ’4βˆ’5βˆ’6 βˆ’1 βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 βˆ’6 1 2 3 4 5 6 Real 55. Plot of 1 βˆ’ 4i in the complex plane (1 along the real axis, βˆ’4 along the imaginary axis). 59. βˆ’2 + 3.46i 61. βˆ’4.33 βˆ’ 2.50i 57. 3.61eβˆ’0.59i Section 8.6 1 βˆ’ y _____ 5 7. y = βˆ’2 + 2x 3. Choose one equation to 1....
at t = 2 45. 1 t x βˆ’3 y 1 2 0 7 3 5 17 C-31 47. Answers may vary: x(t) = t βˆ’ 1 y(t) = t 2 x(t) = t + 1 y(t) = (t + 2)2 49. Answers may vary:,  and   x(t) = t y(t) = t 2 βˆ’ 4t + 4 and  x(t) = t + 2 y(t) = t 2 Section 8.7 3. The arrows show the orientation, the direction 1. Plotting points with the orientation arrow ...
40 βˆ’60 βˆ’80 βˆ’100 βˆ’120 βˆ’140 1000 2000 3000 4000 5000 x βˆ’15 βˆ’10 (t from βˆ’1 to 5) y 3.0 2.5 2.0 1.5 1.0 0.5 βˆ’1 βˆ’0.5 βˆ’1.0 βˆ’1.5 βˆ’2.0 βˆ’2.5 βˆ’3.0 y 30 25 20 15 10 5 βˆ’5 βˆ’5 βˆ’10 βˆ’15 βˆ’20 βˆ’25 βˆ’30 31. y 33. y (t from 0 to 1000) βˆ’1000 βˆ’800 βˆ’600 βˆ’400 35 30 25 20 15 10 5 βˆ’200 βˆ’5 βˆ’10 βˆ’15 βˆ’20 βˆ’25 x 200 35. βˆ’3 βˆ’2 βˆ’1 y 3 2 1 βˆ’1 βˆ’2 βˆ’3 1 x 2 ...
1 y 3 2.5 2 1.5 1 0.5 βˆ’2βˆ’3βˆ’4βˆ’5 βˆ’1 βˆ’0.5 βˆ’1 βˆ’1.5 βˆ’2 βˆ’2.5 βˆ’3 y 1 0.5 (t from βˆ’4Ο€ to 6Ο€) βˆ’10 βˆ’5 5 10 15 20 x βˆ’0.5 βˆ’1 βˆ’1.5 βˆ’2 2 61. The y-intercept changes. x x 63. y(x) = βˆ’16 ξ€’ + 20 ξ€’ _ _ ξ€ͺ ξ€ͺ 15 15 65.  x(t) = 64cos(52Β°) y(t) = βˆ’16t2 + 64tsin(52Β°) 67. Approximately 3.2 seconds 69. 1.6 seconds 39. There will be 100 back-an...
βˆ’ 2 √ 29 _ 29 229 15 √ _ j 229 β€” 29 i + 5 √ _ j 29 β€” β€” 10 10 7. 〈7, βˆ’ 5βŒͺ 15. βˆ’7i βˆ’ 3j 9. Not equal 17. βˆ’6i βˆ’ 2j 229 2 √ _ 229 21. βˆ’10i βˆ’ 4j 13. Equal 27. βˆ’ 25. βˆ’ i + β€” β€” 3. C = 120Β°, a = 23.1, c = 34.1 1. Not possible 5. Distance of the plane from point A: 2.2 km, elevation of the plane: 1.6 km 7. B = 71.0Β°, C = 55.0Β°...
. Parallel: 16.28, perpendicular: 47.28 pounds 77. 19.35 pounds, 51.65Β° from the horizontal 79. 5.1583 pounds, 75.8Β° from the horizontal 69. 4.424Β° 71. (0.081, 8.602 __ 33. 2.3 + 1.9i 31. cis ξ€’ βˆ’ 29. 5 ξ€ͺ 3 3Ο€ 4Ο€ 37. 3cis ξ€’ ξ€ͺ 39. 25cis ξ€’ ξ€ͺ ___ ___ 2 3 43. Ο€ ξ€ͺ 35. 60cis ξ€’ __ 2 3Ο€ ξ€ͺ, 5cis ξ€’ 41. 5cis ξ€’ 7Ο€ ___ ___ ξ€ͺ 4 4 1 _...
5 √ 3 _____ i 2 β€” 2 cis(18Β°), 2 √ 15. 4cis (21Β°) 19. y = 2(x βˆ’ 1)2 17. 2 √ 21. y 6 5 4 3 2 1 23. βˆ’4i βˆ’ 15j 25. β€” 2 √ 13 ______ 13 i + β€” 3 √ 13 ______ j 13 (ΞΈ from 0 to 2Ο€) 1 2 3 4 5 6 x βˆ’2βˆ’3βˆ’4βˆ’5βˆ’6 βˆ’1 βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 βˆ’6 ChapteR 9 Section 9.1 1. No, you can either have zero, one, or infinitely many. Examine 3. This means...
$350,000 the first account, $10,500 in the second account tops: 45, Low-tops: 15 more information. 73. $12,500 in 75. High- 77. Infinitely many solutions. We need β€” 3 ξ€ͺ Section 9.2 1. No, there can be only one, zero, or infinitely many solutions. 3. Not necessarily. There could be zero, one, or infinitely many solutio...
οΏ½s share was $22.05. more information. 61. The BMW was $49,636, the Jeep was $42,636, and adults the Toyota was $47,727. 63. $400,000 in the account that pays 3% interest, $500,000 in the account that pays 4% interest, and 65. The United $100,000 in the account that pays 2% interest. States consumed 26.3%, Japan 7.1%, ...
2 β€” 5 7. (0, βˆ’3), (3, 0) 11. (βˆ’3, 0), (3, 0) 15. ξ€’ βˆ’ __________ 19. ξ€’ βˆ’ √ 1 β€” __ ( √ 2 ) ξ€ͺ 21. (5, 0) 29. No solutions exist 23. (0, 0) 25. (3, 0) 27. No solutions exist 31. ξ€’ βˆ’ 33. (2, 0) β€” √ 2 ____, βˆ’ 2 β€” ξ€ͺ, ξ€’ βˆ’ √ 2 ____ 2 35. (βˆ’ √ β€” √ 2 ____, 2 β€” ξ€ͺ, ξ€’ √ 2 ____ 2 7, βˆ’3), (βˆ’ √ β€” β€” β€” √ 2 ____, βˆ’ 2 7, 3), ( √ β€” ξ€ͺ √ 2 _...
___ ξ€’ 2 √ ____ 35 ξ€ͺ ___ 29 70 ____ 383 51. No solution exists 53. x = 0, y > 0 and 0 < x < 1, √ 57. 2–20 computers β€” 1 _ x x < y < 55. 12,288 43. 47. Section 9.4 C-35 1. No, a quotient of polynomials can only be decomposed if the denominator can be factored. For example, cannot be 1 _ x 2 + 1 7. βˆ’ + + + βˆ’ + + 15 19. 1...
_____ x βˆ’ 1 4 _____ x βˆ’ 1 1 ___________ βˆ’ x 2 + 3x + 25 4x __ x 2 βˆ’ 6x + 36 2 _______ (x βˆ’ 7) 2 x + 6 ______ x 2 + 1 1 _____ x + 6 41. βˆ’ 49. 45. 33. 37. 47. 51. 39. 35. 31. + βˆ’ + βˆ’ βˆ’ βˆ’ + 1 ______ 2x βˆ’ 3 4x + 3 _______ (x 2 + 1) 2 3x _____________ (x 2 + 3x + 25) 2 2x + 3 _______ (x + 2) 2 x ________ + 8(x 2 + 4) 9 ___...
28 βˆ’72 βˆ’360 βˆ’20 βˆ’12 βˆ’116 ξ€² 17. ξ€° 21. ξ€° 1,800 1,200 1,300 800 1,400 600 700 400 2,100 60 41 2 βˆ’16 120 βˆ’216 ξ€² ξ€² 23. ξ€° 19. ξ€° 20 102 ξ€² 28 28 βˆ’68 24 136 βˆ’54 βˆ’12 64 βˆ’57 30 128 ξ€² 25. Undefined; dimensions do not match. ξ€² 29. ξ€° βˆ’8 41 βˆ’3 40 βˆ’15 βˆ’14 4 27 42 27. ξ€° 31. ξ€° 33. Undefined; inner dimensions do not match. βˆ’350 1,050 350...
. No. A matrix with 0 entries for an entire row would have either zero or infinitely many solutions. 7. ξ€° 0 16 9 βˆ’1 | 4 2 ξ€² 9. ξ€° 1 5 8 12 3 0 3 4 9 | 19 4 βˆ’7 ξ€² 11. βˆ’2x + 5y = 5 6x βˆ’ 18y = 26 15. 4x + 5y βˆ’ 2z = 12 y + 58z = 2 3x + 2y = 13 13. βˆ’x βˆ’ 9y + 4z = 53 8x + 5y + 7z = 80 17. No solutions 19. (βˆ’1, βˆ’2) 8x + 7y βˆ’ 3z...
οΏ½. The inverse is formula. found with the following calculation: 1 0 0 1 0 1 0 βˆ’1 1 ξ€° __________ βˆ’1 0 0(0) βˆ’ 1(1) ξ€² = I 1 0 5. Yes. Consider the matrix ξ€° ξ€² = ξ€° ξ€². 9. AB = BA = ξ€° ξ€² = I 13. 1 0 17. There is no inverse Aβˆ’1 = 21. 15. 1 ___ 17. AB = BA = ξ€° 11. AB = BA = ξ€° βˆ’2 7 1 ξ€° ξ€² ___ 9 3 69 βˆ’5 5 βˆ’3 ξ€° 20 βˆ’3 12 1 βˆ’1 4 18 6...
and Albert ate 3 61. 124 oranges, 10 lemons, 8 pomegranates Section 9.8 51. x βˆ’ 3z = 7 y + 2z = βˆ’ 5 with infinite solutions C-37 53. ξ€° βˆ’2 2 1 7 2 βˆ’8 5 0 19 βˆ’10 22 3 | ξ€² 55. ξ€° 1 0 3 βˆ’1 4 0 0 1 2 | 12 0 βˆ’7 ξ€² 59. No solutions exist 57. No solutions exist ξ€² 63. No inverse exists 1 2 7 ξ€° __ 61. 6 1 8 67. (βˆ’1, 0.2, 0.3) 1 _...
300 almonds, Chapter 9 Review exercises 3. (βˆ’2, 3) 1. No 9. (300, 60) 5. (4, βˆ’1) 11. (10, βˆ’10, 10) 17. ξ€’ x, 7. No solutions exist 13. No solutions exist 14x 8x ____ ___ ξ€ͺ, 5 5 23. No solution 25. No solution 19. 11, 17, 33 15. (βˆ’1, βˆ’ 2, 3) 21. (2, βˆ’ 3), (3, 2) 272βˆ’3βˆ’4βˆ’5 βˆ’1 βˆ’1 βˆ’2 βˆ’3 βˆ’4 βˆ’5 29. y 5 4 3 2 1 βˆ’2βˆ’3 βˆ’1 βˆ’1 βˆ’2 ...
29. 32 or more cell phones per day 1 ____ 100 ChapteR 10 Section 10.1 1. An ellipse is the set of all points in the plane the sum of whose distances from two fixed points, called the foci, is a constant. 3. This special case would be a circle. 5. It is symmetric about the x-axis, y-axis, and the origin. 7. Yes; + x 2 ...
19 ), (x βˆ’ 3) ), (3 βˆ’ 3 √ (y βˆ’ 5)); endpoints of = 1; β€” β€” 2 ); foci: (7, 5), (βˆ’1, 5) endpoints of major axis: (3 + 3 √ minor axis: (3, 5 + √ 2 ), (3, 5 βˆ’ √ β€” β€” (x + 5)2 _______ 52 (y βˆ’ 2)2 _______ 22 β€” β€” + 25. 21, 2) = 1; endpoints of major axis: (0, 2), (βˆ’10, 2); 23. endpoints of minor axis: (βˆ’5, 4), (βˆ’5, 0); foci: (...
. The circle has only one focus, which coincides with the center. y 10 7.5 5 2.5 2.5 5 7.5 10 x βˆ’10 βˆ’7.5 βˆ’5 βˆ’2.5 βˆ’2.5 βˆ’5 βˆ’7.5 βˆ’10 41. Center: (βˆ’4, 5); vertices: (βˆ’2, 5), (βˆ’6, 5), (βˆ’4, 6), (βˆ’4, 4); 3, 5) foci: (βˆ’4 + √ y 3, 5), (βˆ’4 βˆ’ √ β€” β€” 10 7.5 5 2.5 2.5 5 7.5 x βˆ’10 βˆ’5βˆ’7.5 βˆ’2.5 βˆ’2.5 βˆ’5 βˆ’7.5 39. Center: (1, 1); vertices...