Solutions to Selected Exercises

Transcription

1 59 Solutions to Selected Exercises Chapter Section.. a. f 0 b. Tons of garbage per week is produced by a city with a population of 5,000.. a. In 995 there are 0 ducks in the lake b. In 000 there are 0 ducks in the late 5. a,b, d, e 7. a, b 9. a, b, d f, f. b. b, c, e, f g, g 9. f 5, f f f f 0 f f DNE - - -/. / / 5. a. -6 b a. 5 b. 9. a. iii b. viii c. I d. ii e. vi f. iv g. v h. vii. a. iv b. ii c. v d. I e. vi f. iii. ( x ) ( y 9) 6 5. (a) (b) (c) 5 height height of head postage age time weight 7a. t b. a c. r d. L: (c, t) and K: (a, p)

2 50 Section.. D: [-5, ) R: [0,]. D: t 8 R: gt D: [0,] R: [-, 0] 7. [, ) 9. (,] (. (, ),,, (, ),,.,6) 6, f f 0 f f if 5. f x if x 7. f x 9. f x if x x 6 x if if if x x x x 5 if x 0 x if x

3 5 Section.. a) 6 million dollars per year b) million dollars per year b+. 5. h h h 9. x h. Increasing:.5,. Decreasing:,.5,. Increasing:,,. Decreasing:,, 5. Increasing, concave up 7. Decreasing, concave down 9. Decreasing, concave up. Increasing, concave down. Concave up,. Concave down, 5. Concave down,, 7. Local minimum at (, -). Inflection points at (0,5) and (, -). Increasing on,,. Decreasing Concave up, 0,. Concave down 0, 9. Local minimum at (-, -) Decreasing, Increasing, Concave up,. Inflection point at (, ). Local minimums at (-.5, -7.66) and (.0, -.0) Local maximum at (-0.89, 5.979) Inflection points at (-, -) and (, -5).5, , Increasing Decreasing, ,.0 Concave up,, Concave down,

4 5 Section.. f ( g(0)) 6. g ( f (0)) 57. f ( g(0)). g ( f (0)) x 7. f g x 7 6 g f x x. f gx x 5. 5 f g x x 7. f g h x x 6 6 g f x x 5 g f x x 9a. 0,, b.,, c. 0,. b a. r V t 5. g x x, f x x t b f x, g x x 5 x 9. f x x, g x x, or f x x, gx x a. f f x a ax b b a x ab b b. g x 6x or g x x 6 a. C f s s s 0 60 b. C g h 70 60h 0 60h c. v C m m m Section.5. Horizontal shift right 9 units. Horizontal shift left units

5 5 5. Vertical shift up 5 units 7. Vertical shift down units 9. Horizontal shift right units, Vertical shift up units x. f x x. f x 5. g x f x hx f x, y x 7. y x 9. y x. a. f x 6 x x b. f x 5. y x 6 7. y x 9a. Even b. Neither c. Odd. Reflect f(x) about the x-axis. Vertically stretch y values by

6 5 5. Horizontally compress x values by /5 7. Horizontally stretch x values by 9. Reflect f(x) about the y-axis and vertically stretch y values by f x x f x x 55. f x 5 x Horizontal shift left unit, vertical stretch y values by, vertical shift down 5 units becomes 59. Horizontal shift right units, vertical stretch y values by, reflect over x axis, vertically shift up units. becomes 6. Vertically compress y values by ½ becomes

7 55 6. Horizontally stretch x values by, vertical shift down units becomes 65. Reflected over the y axis, horizontally shift right units ax x becomes 67. This function is increasing on (, ) and decreasing on (, ) 69. This function is decreasing on (,) 7. This function is concave down on (, ) and concave up on (, ) 7. This function is concave up everywhere 75. f x 77. f x 79. f x 8. f x 8. f x 85. f x 87. y x 89. y x 9. y x 9. y x 97. y x 95. y x 99a. Domain :.5 x 6 d. Range : 9 y 7

8 56 Section ½ 7a. b. c. d. 9a. 0 b. 7 c. d.. x 7 6 f (x ) 6 9. f x x 5. f x x 7. f x x 7, f x x 7 9. Restricted domain x 0, f x x 5. Restricted domain a. b. g f x x 5 5 x f g x x 5 5 x c. This means that they are inverse functions (of each other) x 7 Chapter Section.. Pt 700t Dt 0 t 5. Mn 0 n 7. Increasing 9. Decreasing. Decreasing. Increasing 5. Decreasing mph (or 0.05 miles per hour toward her home) 7. Population is decreasing by 00 people per year 9. Monthly charge in dollars has an initial base charge of $, and increases by $0.0 for each minute talked. Terry started at an elevation of,000 ft and is descending by 70ft per second.. y x 5. yx 7. y x 5

9 57 9. y.5x. y x. y x 5. Pn 0.00n 7. The st, rd & th tables are linear: respectively. g ( x) x 5. f ( x) 5x 5. k ( x) x a. C F b. F C c. 9. F Section.. E. D 5. B a. g x x b. ¾ c. -5/ 5. y

10 58 7. x Vertical Intercept Horizontal Intercept 9. (0,) (,0). (0,-5) (5/, 0). (0,) (-0,0) 5. Line : m 0 Line : m 0 Parallel 7. Line : m Line : m Neither 9. Line : m Line : m Perpendicular. y 5x. y t 5. (-,) 7. (., 0) 9. Plan B saves money if the miles are 9 Section. a. 696 people b. years c. 7 people per year d. 05 people e. Pt 05 7 a. C x 0.5x 0 t f. 9 people. b. The flat monthly fee is $0 and there is an additional $0.5 fee for each additional minute used c. $.05 5a. Pt 90t 70 b. 660 moose 7a. Rt 6. t b. 5.5 billion cubic feet c. During the year More than minutes. More than $,857. worth of jewelry. 0.0 square units 5. 6 square units b 7. A m 9a. Hawaii b. $80,60 c. During the year miles

11 59 Section.. y.97x.59, r y 0.90x 6.0, r situps 9. D. A. Yes, trend appears linear because r =0.99 and will exceed 5% near the end of the year 09. Section.5. y x. y x x or x. x or x x or x Horizontal Intercepts Vertical Intercept 7. (-6, 0 ) and (, 0) (0, -8) 9. none (0, -7). x or (, ). x 5, x or (, ] [5, ) x or (, )

12 50 Chapter Section.. As x, f ( x) As x, f ( x). As x, f ( x) As x, f x 5. As x, f x As x, f x 7. As x, f x As x, f ( x) 9. 7 th Degree, Leading coefficient. nd Degree, Leading coefficient -. th Degree, Leading coefficient - 5. rd Degree, Leading coefficient 6 7. As x, f x As x, f x 9. As x, f ( x) As x, f ( x). intercepts: 5, turning points: Horizontal Intercepts (,0), (-, 0), (, 0) Vertical Intercept (0, ). Horizontal Intercepts (/, 0) (-/, 0) Vertical Intercept (0, ) Section.. f x x. f x x 7 5. f x x Vertex Vertical Intercept Horizontal Intercepts.5, 0.5 (0,) (-, 0) (-, 0) , 8.5 (0,) (0.8, 0) (.56,0). 0.75,.5 (0,-) (0.9, 0) (.09, 0). f x x 6 5. x x 8 f 7. b = and c = -9 f x x 7a. m b ft c seconds 9a. ft b. ft c ft..9 in by.9 in 5 f x 9 x 9. f x x x. f x x x ft by 5..6 cm 7. $ ft

13 5 Section. C(t) C, t, intercepts intercepts. (0,8) (,0), (-,0), (6,0). (0,0) (0,0), (,0), (-,0) 5. (0,0) (0,0), (,0), (,0) 7. (-.66, 0) (.66, 0) (5,0) t, h t t, h t t, p t t, p t 9. As. As (, ).,,. [.5,6] 5.,, 7. [, ], 9.,, (, ). y x x x. y ( x ) x x y x x x y x x x x y x x y x x x 6 x 5. y 5 x x ( x ) ( x ) y. y 6 x x x 5. Base.58, Height.6. y x x x

14 5 Section.. D. A Vertical Asymptotes Horizontal Asymptote Vertical y- Intercept Horizontal x- intercept 5. x y (0,-/) (/, 0) 7. x y 0 (0,) DNE 9. x,. x, hole at x. y (0, 5/6) (-/, 0), (5,0) y (0,) (-, 0) x none y=x (oblique) (0, ¼) (-, 0), (/, 0) 5. x 0, y 0 DNE (-, 0), (/, 0) 7. x, y (0, -5/6) (, 0), (-, 0), (5, 0)

15 x x 9. y x5 x5. y 7. y x x 7 x xx 6x x x x x x x xx xx x xx xx x x x x 7 x x6. y x x5 5. y 9. y. y. y 5. y 7. y 9. a. C( n) b. C ( 0).% c. 80 ml d. as n, C 0 0 n Section.5. Domain, Inverse f x x. Domain,0 Inverse 5. Domain, Inverse f x f x x x x 9 7. f x 9. f x x 9. f x 8x 7x. f x x x 5x 5. f x mph x mph.. feet

16 5 Chapter Section.. Linear. Exponential 5. Neither 7. Pt, Fox. $ y y x x 7. y 9. y = x. y mg 5..9%; $55, $, Annual $75.8 Quarterly $7,69.6 Monthly $7,96.7 Continuously $7,50...0%. 7. years 5a. wt..06 t b. $. c. Below what the model predicts $5.70 Section.. B. A 5. E 7. D 9. C.. 5. x 7. y 9.. As x f x. As x f x 5. As x f x As x f x 7. As x f x As x f x y x. y x x 9. x x y (). y () x y 5. y. 7 x Section.. m q. 7. e n c a w 9. log ( y) a ln h. log( b) 5. b 5.0 t v x. log ( k) d k / ½ e c

17 log log5 log8 log 5 5. ln log log log log log.0 5. log.0 5 log log ln t 57. f t 00e f t 0e 6. f t t t f t 65. During the year During the year hours 7..5 years t Section.. log. log 7 5. log 5 7. ln x. log 7. 9 log 7 9. log6 x x x 5. xz log y ln a ln b 5ln c 7. 5log x log y 9log z 9.. log x log( y ). ln y ln y ln y 8 5. logx logy 7. x x t 7.9. x 7 5. x x.6 9. x x x 6.87 or x x x 0 Section.5. Domain: : x 5 V. x 5. Domain: x x 5. Domain: x x

18 56 7. Domain: x 0 x y log x log 9.. y log x. log y log x log y log x 5 log 5 Section.6. f t t. mg will remain after.098 minutes. f t t. f mg 5. r = Initial mass: mg. After days: mg 7. f t t. Half-life = minutes 9. f t a t. 60% (0.60a) would remain after.8 years. Pt t (t in minutes). After hours = 000. After 00 minutes = 59. a) 60.5 (about 6) b) 5.67 minutes c) 0. d) minutes 5..9 years hours t 9. T t a). deg b).7 minutes

19 57. a) b) 00 c) d) 7. years. log x 0.5. x = log x.5. x =.6 Whisper Vacuum Jet times more intense. MMS magnitude a) about 6067 b). hours c) No, because.077 e d) Anja s data predicts a continuous growth rate of 0.6, which is much smaller than the rate you calculated. Our model would overestimate the number of cells. 5. a) The curve that increases rapidly at first is M(p) b) H(00) = c) Myoglobin: M(0) = Hemoglobin: H(0) = 0. d) At 0 torrs: At 0 torrs: At 60 torrs: a) Ct.056 t 0.066t, or C t e b) Volume of one cell: days Efficiency seems to be maximized at about 8 torr cm, so will need about Ct after 7. hours cells for a volume of cm. 6

20 58 Section.7. log f ( x) log. x log. log f ( x) log0. x. 5. x x y e e e 0.68(.687) x 7. x x x y (0. ) 9. y x.. Expenditures are approximately $05 5. y x y 7.9(0.78) r 0.806, y 0.9x 7.89, r Using the better function, we predict electricity will be.57 cents per kwh x

21 59 Chapter 5 Section x y 5. x 7 y 9 7. x y (0, 5) and (0, 5). ( , ) 5. (-.075,.85) miles Section π miles 7. 8π cm miles cm rad/min 60.5 RPM in/sec, π/ rad/sec,.5 RPM 9. 75,98. mm/min =.57 m/sec. Angular speed: π/ rad/hr. Linear speed: 06.7 miles/hr

22 50 Section 5.. a. III b. II a. reference: 5. Quadrant III. sin 5. cos 5 b. reference: 60. Quadrant IV. sin 00. cos 00 c. reference: 5. Quadrant II. sin 5. cos d. reference: 0. Quadrant III. sin 0. cos 0 5. a. reference:. Quadrant III. sin.. a. 7 b. reference:. Quadrant III. sin c. reference:. Quadrant IV. sin. d. reference:. Quadrant II. sin. sin b. sin 6 c. sin d. sin 5 0 cos cos 6 cos 0 cos 5 5 cos 7 cos 6 5 cos cos 5. a. 7. a. 5 b. 00 c. 0 d. 5 b. 80 c. 0 d. e. 5 e (-.9, -9.6)

23 5 Section 5. tan,. sec, csc, cot. sec, csc, tan, 5. sec, csc, tan, cot cot 7. a. sec5 b. csc0 c. tan 60. d. cot cos, sec, csc, tan, cot sin, csc, sec, tan, cot 5. sin, cos, sec, csc, cot a. sin(0.5) = 0.9 cos(0.5) = tan(0.5) = 0.5 b. sin() = cos() = tan() =.578 c. sin(70 ) = cos(70 ) = 0.0 tan(70 ) =.775 d. sin(8 ) = cos(8 ) = 0.50 tan(8 ) = sec( t ) 9. tan( t ). tan( t ). cot( ) t 5. sec( t ) Section 5.5 sin. sec 5 A,cosA, tana 5 A,cscA,cotA 5. c, b 7, B a 5.7, c.57, A 8 7. a 9.06, b.6, B ft ft ft feet ft

24 5 Chapter 6 Section Amp:. Period=. Midline: y= -. f t t sin cos t cos t 6. Amp:. Period=. Midline: y= -. f t sin t 7. Amp:. Period= π. Midline: y=. f t 8. Amp:. Period= π. Midline: y= -. f t 9. Amp:. Period= 5. Midline: y=. f t 0. Amp:. Period=. Midline: y= -. cos t 5 f t sin t. Amp:, Period =, Shift: left, Midline: y = 5. Amp:, Period =, Shift: right, Midline: y = 7. Amp:, Period =, Shift: 7 right, Midline: y =

25 5. Amp: 5, Period =, Shift: left, Midline: y = Amp:, Period =, Shift: 6 left, Midline: y = - 6. Amp: 8, Period = 7, Shift: left, Midline: y = 6 7. f x sin x 5 8. f x sin x 9. f x cos x 5 0. f x cos x. D( t) 50 7sin t. D( t) 68 sin t. a. Amp:.5. Midline: y =.5. Period: 0 b. ht.5cos t.5 5 h meters c a. Amp: 7.5. Midline: y = 0.5. Period: 8 b. ht 7.5cos t 0. 5 h meters c. 8 Section 6.. II. I 5. Period:. Horizontal shift: 8 right 7. Period: 8. Horizontal shift: left 9. Period: 6. Horizontal shift: left

26 f x sec x 9. f x csc x. tan x.5. x 5. cscx 5 7. csc x sec Section x x 9x

27 55 Section 6.. 5, 7 5., 7 9. k, k, where k is an integer. 7, k k, where k is an integer k, k, where k is an integer k, 7 k, where k is an integer 5 7. k, k, where k is an integer k, k, where k is an integer. 8k, where k is an integer. k, 5 k, where k is an integer , , , , , , , , ,.69 Section 6.5. c 89, A = , B =.005. b 76, A = 7.88, B = sin 5. yx x π 7. Dt 50 cos t 5 t 9. a. Pt 9 5cos degrees months 6 b. Pt 9 5cos ( t )

28 56 Chapter 7 Section 7.. 7, , 5. 8k, and 0 8k, where k is an integer 7. 5 k and 7 k, where k is an integer k and k, where k is an integer..8 k and.9978 k, where k is an integer.,, 0.6, ,.55,.97, ,,, 9.,,, ,.958,.5, 5.00., 7, ,, 7..8, ,.98, 0.7, , ,,,,, ,,, , Section ,,,,, 9. 0,,.,

29 57 9. sin x cos x. cos x sin x. sect 5. tan x 7. 8 cos 5x cos 7x 9. sin 8x sin x. cos 5tcos t. sin 5xcos x 5. a b. 0.7 k and 9. k, where k is an integer 0.67 k, where k is an integer. 7 7 k, 7 7 k, k, and. 7 k, k, and k k, where k is an integer 5. sin( x 5.00) or sin( x 0.988) 7. 9sin(x ) , ,.858. tan 6t, where k is an integer Section 7.. a. 7 b. c. 7. cos cos( ) 7. cos 8x 9. sin 6x. 0,,.89, ,.9,.87, 5.555

30 ,,, a ,,,,,,0,, cos 0x 8 8. cos6x cosx cos x cos x cos xcos x 5. a. b. 7 c. 7 7 Section 7. sin 6. y x. Amplitude: 8, Period: second, Frequency: Hz (cycles per second) 0 P t 9cos t t t 7. Pt cos t t 9. Dt cos(6 t). Dt t cos 8 t. a. IV b. III 5. y x 6 5sin x 7. y sin x 7 x 9. y 8 cos x

31 59 Chapter 8 Section = 68, a =.7, c = 0.8. = 8.096, =.90, c = 6.9. Not possible. 5. = 6., = 7.657, c = 57.8 OR =5.757, =., c = c.066, 5.55, a.69, 7.57, ft 9. Distance to A: ft. Distance to shore: 5.69 ft m feet 5..6 km,.79 km ft miles cm. 7.7

32 550 Section ,., 5., 7. (0,) 9.,. (.8,.78). 5, ,.59 7., ,.057. r sec. 5. r sin x y y sin r cos r cos. y7x. x 5. x y x cos sin 7. A 9. C. E. C 5. D 7. F

33 Section 8.. i. 5. i 7. 8 i 9. i. 8i. 0 0i 5. 0i i. 5 i i i cos sin i.8.78i 9.. i e i 7. e 9. i e 0.50i e 5..67i e 9. 5 i 5 i i e i. e 5. 0e e 6e 7 i e i i.80i 6.086i 55. 0e i i , i, i, i, i 67., i, i,, i, i

34 55 Section 8..,. The vectors do not need to start at the same point 5. v u 7., , 5.. Magnitude:, Direction: 90. Magnitude: 7.80, Direction: Magnitude:.6, Direction: Magnitude: 5.85, Direction: Magnitude: 7., Direction: 6.0. u v,, u v, 8, u v,..65 miles, 7.76 deg N of E 5. 7 miles. 0.8 miles 7. Fnet, 9. Distance:.868. Direction: 86.7 North of West, or.56 West of North..9 degrees. 659 km/hr.. degrees 5. (0.08, 8.60) degrees, relative to the car s forward direction

35 55 Section 8.5. C. E 5. F x(t) y(t).. y x 5. y 5 x 7. x e or y 5ln 9.. y x. 5. x t y t t t 7. x y y y x x y 5 x t log t t. y t t

36 x t y t x t y t x t y t. yx cos sin t t t t cost 6sin t x x x t t. y t t xt t 5. yt 5 t xt cost 9. y t sin t xt 0sin t 8sin t 5. y t 5 0 cos t 8cos t 5