6 Solutons to Selected Eercses Chapter Secton.. a. f ( 0) b. Tons of garbage per week s produced by a cty wth a populaton of,000.. a. In 99 there are 0 ducks n the lake b. In 000 there are 0 ducks n the late. a,b, d, e 7. a, b 9. a, b, d f, f. b. b, c, e, f. ( ) ( ) 7. g ( ), g ( ) 9. f ( ), f ( ) f ( ) f ( ) f ( 0) f ( ) f ( ). 8 6 0. 9 8. - 0-7...7.6 9. - -6-6 - 0. DNE - - -/. / /. a. -6 b.-6 7. a. b. 9. a. b. v c. I d. e. v f. v g. v h. v. a. v b. c. v d. I e. v f.. ( ) ( y 9) 6. (a) (b) (c) heght heght of head postage age tme weght 7a. t b. a c. r d. L: (c, t) and K: (a, p)
6 Secton.. D: [-, ) R: [0,]. D: t 8 < R: g ( t ) 6 < 8. D: [0,] R: [-, 0] 7. [, ) 9. (,] (. (, ),,, (, ),,.,6) ( 6, ). [ ) ( ) 7. ( ) ( ) f ( ) f ( 0) f ( ) f ( ) 9. - 6 0. - - 7. - 6. f ( ) f < 7. f ( ) 9. f ( ) f f f f 6 < < 0 0 f 0 f > 0... Secton.. a) 6 mllon dollars per year b) mllon dollars per year.. 6
6 7. 7 9. 7. b.. 7. 9 9h h h 69 9. h.,,.,. Increasng: ( ). Decreasng: ( ) ( ). Increasng: (, ) (,). Decreasng: (,) (, ). Increasng, concave up 7. Decreasng, concave down 9. Decreasng, concave up. Increasng, concave down. Concave up (,). Concave down (, ). Concave down (, ) (, ) 7. Local mnmum at (, -). Inflecton ponts at (0,) and (, -)., Increasng on (, ) Concave up ( 0) (, ) 9. Local mnmum at (-, -), Decreasng ( ) Increasng (, ) Concave up (, ). Decreasng ( ),. Concave down ( 0,). Inflecton pont at (, ). Local mnmums at (-., -7.66) and (.0, -.0) Local mamum at (-0.89,.979) Inflecton ponts at (-, -) and (, -)., 0.89.0, Increasng ( ) ( ) Decreasng (,.) ( 0.89,.0) Concave up (, ) (, ) Concave down (,)
6 Secton.. f ( g(0)) 6. g ( f (0)) 7. f ( g(0)). g ( f (0)). 7. 9 9.. 7. 0. 7. 9. ( ) 7. f g ( ) ( ) ( ) 7 6 g f. f ( g( ) ) ( ). ( ) ( ) f g 7. ( ) ( ) ( ( )) ( ) f g h 6 6 ( ) g f ( ) g f ( ) 9. b a. r V ( t) ( t) 0 0 b..609n. ( 0, ).,, (, ) 7. [,) (, ) 9. g ( ), f ( ). ( ) ( ) π f, g. f ( ), g ( ), or f ( ), g( ) a. ( ) ( ) ( ) ( ) ( ) f f a a b b a ab b b. g ( ) 6 or g ( ) 8 6 8 6 6 ( ) 7a. C f ( s) s 70 60 s 0 60 ( ) b. C g ( h) ( h) ( h) 70 60 0 60 ( ) c. v C ( m) 80 70m 600 0 m
6 Secton.. Horzontal shft rght 9 unts. Horzontal shft left unts. Vertcal shft up unts 7. Vertcal shft down unts 9. Horzontal shft rght unts, Vertcal shft up unts. f ( ). f ( ). g ( ) f ( ) h( ) f ( ), 7. 9.... y 7. y 9. y. a. f ( ) 6 b. f ( ). y ( ) 6 7. y 9a. Even b. Nether c. Odd
66. Reflect f() about the -as. Vertcally stretch y values by. Horzontally compress values by / 7. Horzontally stretch values by 9. Reflect f() about the y-as and vertcally stretch y values by. ( ) f. f ( ) ( ). f ( ( ) ) ( ( ) ) 7. Horzontal shft left unt, vertcal stretch y values by, vertcal shft down unts becomes 9. Horzontal shft rght unts, vertcal stretch y values by, reflect over as, vertcally shft up unts. becomes 6. Vertcally compress y values by ½
67 becomes 6. Horzontally stretch values by, vertcal shft down unts becomes 6. Reflected over the y as, horzontally shft rght unts a ( ) ( ) becomes 67. Ths functon s ncreasng on (, ) and decreasng on (, ) 69. Ths functon s decreasng on (,) 7. Ths functon s concave down on (, ) and concave up on (, ) 7. Ths functon s concave up everywhere 7. f ( ) 77. f ( ) 79. f ( ) 8. f 8. f ( ) 8. f ( ) 87. ( ) y 89. ( ) y 9. y ( )
68 9. y ( ) 9. y 97. y ( ) ( ) f 99. f ( ) f > f < 0. f ( ) ( ) f f > 0a. Doman :. 6 d. Range : 9 y 7 Secton.6. 6. -. ½ 7a. b. c. d. 9a. 0 b. 7 c. d.. 7 6 f ( ) 6 9. f ( ). f ( ) 7. f ( ) 9. Restrcted doman ( ) 7, f 7. Restrcted doman ( ) a. ( ) ( ) ( ) 0, f b. g ( f ( ) ) f g c. Ths means that they are nverse functons (of each other) 7
69 Chapter Secton.. P ( t ) 700t 000. D ( t) 0 t. M ( n) 0 n 7. Increasng 9. Decreasng. Decreasng. Increasng. Decreasng 7. 9.... - 0.0 mph (or 0.0 mles per hour toward her home) 7. Populaton s decreasng by 00 people per year 9. Monthly charge n dollars has an ntal base charge of $, and ncreases by $0.0 for each mnute talked. Terry started at an elevaton of,000 ft and s descendng by 70ft per second.. y. y 7. y 9. y.. y. y. P ( n) 0.00n 7. The st, rd & th tables are lnear: respectvely. g ( ). f ( ). k ( ) 60 9a. C F b. 9 9 9 F C c. 9. F Secton.. E. D. B 7. 9...
60. 7. 9... a. g ( ) ( ) b. ¾ c. -/. y 7. Vertcal Intercept Horzontal Intercept 9. (0,) (,0). (0,-) (/, 0). (0,) (-0,0). Lne : m 0 Lne : m 0 Parallel 7. Lne : m Lne : m Nether 9. Lne : m Lne : m Perpendcular. y. y t. (-,) 7. (., 0) 9. Plan B saves money f the mles are > 9. f ( ) f f f < < Secton. a. 696 people b. years c. 7 people per year d. 0 people e. P ( t) 0 7 t f. 9 people.
6 a. C ( ) 0. 0 b. The flat monthly fee s $0 and there s an addtonal $0. fee for each addtonal mnute used c. $.0 a. P ( t) 90t 70 b. 660 moose 7a. R ( t ) 6. t b.. bllon cubc feet c. Durng the year 07 9. More than mnutes. More than $,87. worth of jewelry. 0.0 square unts. 6 square unts b 7. A m 9a. Hawa b. $80,60 c. Durng the year 9. 6. mles Secton.. 60 0 0 0 0 0 0 0 0 0 0 0 0 60. y.97.9, r 0.967. y 0.90 6.0, r 0.968 7. 7.8 7 stups 9. D. A. Yes, trend appears lnear because r 0.99 and wll eceed % near the end of the year 09. Secton.. y. y. 7. 9.
6 9. or. or. or Horzontal Intercepts Vertcal Intercept 7. (-6, 0 ) and (, 0) (0, -8) 9. none (0, -7). < < or (, )., or (, ] [, ). < < or (, ) Chapter Secton.. As, f ( ) As, f ( ). As, f ( ) As, f ( ). As, f ( ) As, f ( ) 7. As, f ( ) As, f ( ) 9. 7 th Degree, Leadng coeffcent. nd Degree, Leadng coeffcent -. th Degree, Leadng coeffcent -. rd Degree, Leadng coeffcent 6 7. As, f ( ) As, f ( ) 9. As, f ( ) As, f ( ). ntercepts:, turnng ponts:.. 7. 9.. Horzontal Intercepts (,0), (-, 0), (, 0) Vertcal Intercept (0, ). Horzontal Intercepts (/, 0) (-/, 0) Vertcal Intercept (0, ) Secton. f. f ( ) ( ). f ( ) ( ) 7. ( ) ( ) Verte Vertcal Intercept Horzontal Intercepts., 0. (0,) (-, 0) (-, 0) 7. ( ) 9. (., 8.) (0,) (0.8, 0) (.6,0). ( 0.7,. ) (0,-) (0.9, 0) (.09, 0)
6. f ( ) ( 6). ( ) ( ) 8 f 7. b and c -9 f 7a. m b. 909.6 ft c. 7.7 seconds 9a. ft b. ft c. 7.97 ft..9 n by.9 n f 9 9. f ( ) ( )( ). f ( ) ( )( ). ( ) ( ). ( ) ( ). ft by..6 cm 7. $0.70 8 ft Secton. C(t) C, t, ntercepts ntercepts. (0,8) (,0), (-,0), (6,0). (0,0) (0,0), (,0), (-,0). (0,0) (0,0), (,0), (,0) 7. (-.66, 0) (.66, 0) (,0) t, h t t, h t t, p t t, p t 9. As ( ) ( ). As ( ) ( ).. 7. 9. (, ). (, ) (,)
6. [.,6]. (,] [, ) 7. [, ] [, ) 9. (, ) (, ) (, ). y ( )( )( ). y ( ) ( ) ( ) y y y y 6. y ( ) ( ) 7. ( )( )( ) 9. ( ) ( ) y. ( )( )( )( ) y 6. Base.8, Heght.6. y ( )( )( ). ( ) ( ) 7. ( )( )( ) 9. ( )( )( ) ( ) Secton.. ( )( ). ( )( 8) ( 7). 9 9 7 8 ( ) 8 7. ( ) ( )( ) 9. ( ) ( )( ). ( 8) ( )( ) 0 8 8. ( 8 ) (8 ) 0. ( ) ( ) 0 7. ( ) 9. ( 6 9) ( )( ) 0. 6 6 ( )( )( ). ( ). 6 ( )( )( ) 7. 8 6 9 ( )
6 Secton.. All of the real zeros le n the nterval [ 7,7] - Possble ratonal zeros are ±, ±, ±. All of the real zeros le n the nterval [,] - Possble ratonal zeros are ±, ±, ±, ±, ± 6, ±. All of the real zeros le n the nterval [ 8,8] - Possble ratonal zeros are ±, ± 7 7. All of the real zeros le n the nterval [,] - Possble ratonal zeros are ±, 7 ±, 7 ±, 7 9. All of the real zeros le n the nterval, - Possble ratonal zeros are.,.. 7 (mult. ) 7. ±, ±, (each has mult. ) ±,, (mult. ), (mult. ), (mult. ), ± (each has mult. ) 7 9.,. 0, 69 ± (each has mult. ) 6 6 ± (each has mult. ) 8. ± (each has mult. ). ± (each has mult. ) 7., (each has mult. ) 9., ± (each has mult. ). (mult. ), 6 (mult. ) Secton.6.. 0 ±, ±, ±, ±, ± 0 7 0 ±, ±, ±, ±, ± 0. 7. 8 9.. 8. 0 0. 0
66 7. 0 9... 8. )) ( ))( ( ( ) ( f. Zeros: ± 7. f 9 9 0 ) (. Zeros: 9 ± 9. f ) ( ) )( ( 6 6 ) ( Zeros:, ±. ( ) ) )( )( ( ) ( ) ( f. Zeros:, ±. 9 9 ) ( 9 7 ) ( f Zeros: 9, ±. ( ) ) )( ( ) ( f Zeros:, ± 7. ( ) ) )( )( )( ( 9 ) )( 8 ( 9 7 ) ( f Zeros:,, ± 9. ( ) f ) ( ) ( 8 ) ( Zeros:, ±. ( )( ) ( )( ) ) )( ( 0 9 ) ( f Zeros:, ± ± Secton.7. D. A Vertcal Asymptotes Horzontal Asymptote Vertcal y- Intercept Horzontal - ntercept
67. y (0,-/) (/, 0) 7. y 0 (0,) DNE 9. y (0, /6) (-/, 0), (,0),., hole at y (0,) (-, 0). none (0, ¼) (-, 0), (/, 0) y (oblque). 0, y 0 DNE (-, 0), (/, 0) 7., y (0, -/6) (, 0), (-, 0), (, 0). 7. 9....
68 7. 9.. 7... y y y 0( )( ) ( )( ) ( ) ( ) 7( ) ( )( ) 6( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( ) ( )( ) ( )( ) ( )( ) ( )( ) 7 6. y. y 9. y y. y y 7. y 9. y. y. y. a. C( n) b. C ( 0).% c. 80 ml d. as n, C 0 0 n Secton.8. Doman (, ) Inverse ( ) f. Doman (,0) Inverse ( ) f. Doman (, ) Inverse f ( ) ( ) 9 7. f ( ) 9. f ( ) 9 7. f ( ). f ( ) 9..07 mph.. feet. f ( ) 8 7. 6.7 mph
69 Chapter Secton.. Lnear. Eponental. Nether 7. P( t ),000 (.08) t 9. 76 Fo. $76.70. y 6( ). y ( ) 000 0. 7. y ( ) 9. y.9( 0.699). y ( ) 6 6 8.. mg..9%; $,68.09 7. $,8. 9. Annual $7.8 Quarterly $7, 69.6 Monthly $7, 96.7 Contnuously $7,0...0%. 7. years a. w( t ) (.)(.06) t b. $. c. Below what the model predcts $.70 Secton.. B. A. E 7. D 9. C... 7. y 9.. As f ( ). As f ( ) As f ( ). As f ( ) 7. As f ( ) As f ( ) y. y 9. y (). y () y. y. ( ) 7 Secton.. m q. 7. e n c a w 9. log ( y) a ln h. log( b). ( ) b.0 t v. log ( k) d k 7. 9 9. /8. 000. e. 7. - 9. ½ c
660.. -. - log( ) 7. -.98 9..708..697 log( ) log( 8) log. ln.9. ( 7) log( ) 0.67 7..078 log 7 9.. ( ) 8 log log ( ).9 log (.0). log(.0) log 8 0.678 log 8. 6. f ( t ) 0(.068) t 6. ( ) ( ). ln. 0. 0.09t 7. f ( t) 00e 0.09 9. f ( t) 0e f t 0 0.98807 t 6. Durng the year 0 67. Durng the year 07 69. hours 7.. years t Secton.. log ( ). log ( 7). log ( ) 7. ( ) ln. log ( ) 7. ( ) ( ) 9 log 7 9. log( 6 ). z log y ln a ln b ln c 7. log ( ) log ( y) 9 log ( z) 9. ( ) ( ) ( ). log ( ) log( y ). ln ( y) ( ln ( y) ln ( y) ) 8. log( ) log( y) 7. 0.77 9. 6. 9. t 7.9. 7. 0. 7..6 9. 0.8..889.. 6.87or 0. 87..09 7. 0 Secton.. Doman: : > V. A. @. Doman: < V.A. @. Doman: > V.A. @
66 7. Doman: < 0 V.A. @ 0 9.... 7. y log( ( ) ) log 9. ( ). y log ( ). log ( ) y log log ( ) ( ) ( ) y log log ( ( )) Secton.6. f ( t ) ( 0.99) t. mg wll reman after.098 mnutes. f ( t ) 00( 0.9996) t. ( ) f 000 9. mg. r -0.068. Intal mass: 9.908 mg. After days: 0.068 mg 7. f ( t ) 0( 0.9909) t. Half-lfe 7.86 mnutes 9. f ( t) a( ) 0.999879 t. 60% (0.60a) would reman after.8 years. P( t ) 00(.07) t (t n mnutes). After hours 000. After 00 mnutes 9. a) 60. (about 6) b).67 mnutes c) 0. d) 06.96 mnutes..9 years 7..9 hours T t 90 0.9966 7. a). deg b).7 mnutes 9. ( ) ( ) t
66. a) b) 00 c) 69.87 d) 7. years. log ( ) 0.. 0.6. ( ) log...6 Whsper Vacuum Jet 7. 0-0 0-9 0-8 0-7 0-6 0-0 - 0-0 - 0-9. 609.7 tmes more ntense. MMS magntude.87 0 0 0 0 0.697 0. 90. a) about 6067 b). hours c) No, because ( ).077 e d) Anja s data predcts a contnuous growth rate of 0.6, whch s much smaller than the rate 0.90 you calculated. Our model would overestmate the number of cells.. a) The curve that ncreases rapdly at frst s M(p) b) H(00) 0.977 c) Myoglobn: M(0) 0.9. Hemoglobn: H(0) 0. d) At 0 torrs: 0.68. At 0 torrs: 0.060. At 60 torrs: 0.07 7. a) C ( t ).06 t 0.066t, or C ( t ) e b) Volume of one cell: ( ) 6.9099 0 9..699 days Effcency seems to be mamzed at about 8 torr 0 0.6 0 π 7 cm, so wll need about cells for a volume of cm. ( ) 6 C t.9099 0 after 7. hours
66 Secton.7. log ( f ( ) ) log(.) log( ). log ( f ( ) ) log( 0.).. y e e e 0.68(.687) 7. y 0 0 0 0.0(0.) 9. y 776.68(.6).. Ependtures are appromately $0. y ( ) y 7.9(0.78) 7.99.06 r 0.806, y 0.9 7.89, r 0.87. Usng the better functon, we predct electrcty wll be.7 cents per kwh
66 Chapter Secton.. 0. ( ) ( y ). ( 7 ) ( y ) 9 7. ( ) ( y ) 8 0 8 8 9.. (0, ) and (0, ). (.60786, 7.6887). (-.07,.8) 7. 9.87 mles Secton. 70 0 7-00.. π. 0 7. 9.. 8 π 9 π.. mles 7. 8π cm 9..796 mles. 8.679..7 cm. 960 rad/mn 60. RPM 7..09 n/sec, π/ rad/sec,. RPM 9. 7,98. mm/mn.7 m/sec. Angular speed: π/ rad/hr. Lnear speed: 06.7 mles/hr
66 Secton.. a. III b. II.. 7. 7 9. a. reference:. Quadrant III. sn ( ). cos ( ) b. reference: 60. Quadrant IV. sn ( 00 ). cos ( 00 ) c. reference:. Quadrant II. sn ( ). cos ( ) 8 d. reference: 0. Quadrant III. sn ( 0 ). cos ( 0 ) π π. a. reference:. Quadrant III. sn. π 7π b. reference:. Quadrant III. sn. 6 6 π π c. reference:. Quadrant IV. sn. π π d. reference:. Quadrant II. sn. π. a. sn π b. sn 6 π c. sn d. sn ( ) 0 π cos π cos 6 π cos 0 cos π ( π ) π cos 7π cos 6 π cos π cos. a. π 7. a. π b. 00 c. 0 d. π b. 80 c. 0 d. π e. e. 9. (-.9, -9.6)
666 Secton.. sec( θ ), csc( θ ), tan ( θ ), ( ) cot θ. sec( θ ), csc( θ ), tan ( θ ), ( ). sec( θ ), csc( θ ), tan ( θ ), cot ( ) cot θ θ 7. a. sec( ) b. csc( 0 ) c. tan ( 60 ). d. ( ) cot 7 7 7 9. cos ( θ ), sec( θ ), csc( θ ), tan ( θ ), cot ( ) 7 7 θ 7. sn ( θ ), csc( θ ), sec( θ ), tan ( θ ), cot ( ). sn ( θ ), cos ( θ ), sec( θ ), csc( θ ), cot ( ) θ. a. sn(0.) 0.9 cos(0.) 0.9888 tan(0.) 0. b. sn() -0.768 cos() -0.66 tan().78 c. sn(70 ) 0.997 cos(70 ) 0.0 tan(70 ).77 d. sn(8 ) -0.97 cos(8 ) 0.0 tan(8 ) -. 7. sec( t ) 9. tan( t ). tan( t ). cot( ) t. ( sec( t )) θ Secton. sn ( A),cos( A), tan( A). sec ( A ),csc( A),cot( A). c, b 7, B 60. a.7, c.7, A 8 7. a 9.06, b.6, B 9..987 ft. 86.698 ft. 60.069 ft. 660. feet 7. 8.0 ft 9..07. 86.668
667 Chapter 6 Secton 6...... Amp:. Perod. Mdlne: y -. f ( t ) ( πt ) sn π cos t cos t 6. Amp:. Perod. Mdlne: y -. f ( t) sn t 7. Amp:. Perod π. Mdlne: y. f ( t) 8. Amp:. Perod π. Mdlne: y -. f ( t) ( ) 9. Amp:. Perod. Mdlne: y. f ( t) 0. Amp:. Perod. Mdlne: y -. ( ) π cos t π f t sn t π. Amp:, Perod, Shft: left, Mdlne: y. Amp:, Perod, Shft: rght, Mdlne: y 7. Amp:, Perod π, Shft: 7 rght, Mdlne: y
668. Amp:, Perod π, Shft: left, Mdlne: y -. Amp:, Perod, Shft: 6 left, Mdlne: y - 6. Amp: 8, Perod, Shft: left, Mdlne: y 6 7 π 7. f ( ) sn ( ) π 8. f ( ) sn ( ) π 9. f ( ) cos ( ) π 0. f ( ) cos ( ) π. D( t) 0 7 sn t π. D( t) 68 sn t. a. Amp:.. Mdlne: y.. Perod: 0 π b. h( t ).cos t. h meters c. ( ) 6. a. Amp: 7.. Mdlne: y 0.. Perod: 8 π b. h( t) 7.cos t 0. h meters c. ( ) 8 Secton 6.. II. I. Perod: π. Horzontal shft: 8 rght 7. Perod: 8. Horzontal shft: left 9. Perod: 6. Horzontal shft: left
669... π 7. f ( ) sec π 9. f ( ) csc. tan ( ).. ( ). csc( ) 7. csc( ) sec Secton 6. π π π... 6 7. π π π 9....98. -0.97 7..7 9. π 7.. 9. π. 6 9 0 7. 7
670 Secton 6.. π, 7 π π π., π 7π 9. π k, π k, where k s an nteger. 7 π, π π k π k, where k s an nteger 6 6. π π k, π π k, where k s an nteger 8 8. π π k, 7π π k, where k s an nteger π π 7. π k, π k, where k s an nteger 6 6 9. π π k, π π k, where k s an nteger. 8k, where k s an nteger. k, k, where k s an nteger 6 6. π 7. π π,. 0.7,.868 7..760,.66 9..,.00. 0.78,.09. 0.089, 0.7. 0.78,.6 7. 0.99,.0709 9..077,.69 Secton 6.. c 89, A 7.996, B.00. b 76, A 7.88, B 6.89 π 6sn. y ( ) ( ) π 7. D ( t ) 0 cos ( t ) π t 6 9. a. P ( t ) 9 cos. 7 degrees. 8..808697 7..0 months π 6 b. P ( t) 9 cos ( t )
67 Chapter 7 Secton 7.. 7 π, π 6 6. π π,. 8k, and 0 8k, where k s an nteger 7. π kπ and 7 π kπ, where k s an nteger 9. 0.9 0k and 8.66 0k, where k s an nteger π..8 k π and.9978 k, where k s an nteger π π.,, 0.6,.98 7.. 0.06,.,.97,.67 π π π π 7π π 0, π,, 9.,,, 6 6 6 6..8,.98,.,.00. π, 7π, π 6 6 π π. π,, 7..8,.60 9..0,.98, 0.7,.60..0, 6.0. π π π π 0,,, π,,. π π π π 7.,,, 6 6 π π., Secton 7... 6 6. 7. π π π 7π 0,,, π,, 9. 0, π,.,.0 6 6 9. sn ( ) cos ( ). cos( ) sn ( )
67. sec( t ). tan ( ) ( ) 7. 8 cos ( ) cos ( 7) 9. sn ( 8) sn ( ). cos( t ) cos ( t ). sn ( ) cos ( ). a. 7. b. π 0.7 k and 9. π k, where k s an nteger π 0.67 k, where k s an nteger π π. 7 7 k, π π 7 7 k π π, k, and π. 7 π π π π k, π k, and k π k, where k s an nteger. sn(.00) or sn( 0.988) 7. 9sn( 0.80 ) 9. 0.68,.8. 0.78,.88. tan ( 6t ), where k s an nteger Secton 7.. a. 7 b. c. 7. cos( 6 ). cos( ) 7. cos( 8 ) 9. sn ( 6 ). 0, π,.89,.86. 0.797,.9,.87,.
67. π π π π,,, 6 6 7. a. π π 8π 0π π 6π π,,,,,,0,, π 9 9 9 9 9 9 9. ( ) cos 0 8 8. cos( 6) cos( ) 6 6 6 6. cos ( ) cos ( ) cos( ) cos ( ). a. b. 7 c. 7 7 Secton 7. π sn 6. y ( ). Ampltude: 8, Perod: second, Frequency: Hz (cycles per second) π 0 P t 9 cos t t 60 6. ( ) π 6 t 7. P ( t ) cos t 900(.07) t 9. D( t) 0( 0.8) cos(6 πt). D ( t) 7( 0.9) t cos ( 8π t). a. IV b. III. y ( ) π 6 sn π 7. y sn 7 π 9. y 8 cos
67 Chapter 8 Secton 8. 60 0.6 70 0..0. 9.09 6 0. 6.9 6. 6 9.08 6.06. 7. 0 6.668.0 9. β 68, a.7, c 0.8. β 8.096, γ.90, c 6.9. Not possble.. β 6., γ 7.67, c 7.8 OR β.77, γ., c 97.8 60 0 8 76.0.898.980 7. 9. 0.0 0.60 6.870. c.066, α., β 86.. a.69, β 7.7, γ.. 77.6 7. 978. ft 9. Dstance to A: 6.8 ft. Dstance to shore:.69 ft. 9.0 m. 7.877 feet..6 km,.79 km 7. 77.96 ft 9. 7.9 mles. 6.7 cm. 7.7
67 Secton 8.. 7 7,. (, ). (, ) 7. (0,) 9.,. (.8,.78). (, 0.6 ). (,.9 ) 7. (,. ) 9. ( 69,.07 ). r sec( θ ).. r sn ( θ ) 9. 7. y y sn r cos r. y 7.. ( θ ) ( θ ) cos ( θ ) ( θ ) sn ( θ ) ( cos ) y 7. A 9. C. E. C. D 7. F 9.... 7. 9.
676 6. 6. 6. Secton 8.... 7. 8 9.. 8. 0 0. 0 7. 0 9... 8. 7. cos sn.8.78 9. ( ) ( )... 0 6e 7. π 9. e. 7.. 7 6e 0.0 e..67 e 9. π π e π. e. 0e e 6e 7π π e π.80 6.086. 0e 7. 9. 096 6. 0.788.90 6..77 0. 6..9, 0..09, 0.99 0.67, 0.99 0.67, 0.. 09 67.,,,,,
677 Secton 8..,. The vectors do not need to start at the same pont. v u 7., 9. 6.8,.. Magntude:, Drecton: 90. Magntude: 7.80, Drecton: 9.806. Magntude:.6, Drecton:. 7. Magntude:.8, Drecton: 9.80 9. Magntude: 7., Drecton: 6.0. u v,, u v, 8, u v,..6 mles, 7.76 deg N of E. 7 mles. 0.8 mles 7. F net, 9. Dstance:.868. Drecton: 86.7 North of West, or.6 West of North..9 degrees. 69 km/hr.. degrees. (0.08, 8.60) 7..80 degrees, relatve to the car s forward drecton
678 Secton 8.. 6 0 cos(7 ). 9. ( 0)( ) ()(0) 0. ( )( 0) ()() 0 7. cos 90 9. ()() ()( ) cos ( ). ()(8) ()() cos 0 8 (8)() ( )( ). 6. ( ). ( )( k ) (7)() 0, k - ( 6)() (0)( ) 7.,.6,0. 8 ( ) 9. The vectors are, and,. The acute angle between the vectors s.09.. pounds. 0 cos(0 ),0sn(0 ) 0, 0, so.796 ft-lbs. 0 0 cos( ) 0. 77 ft-lbs Secton 8.6. C. E. F 7. 9. (t) y(t)
679.. y. y y y y 7. e or y ln 9. y. y.. 9.. 7. ( ) t ( ) t t y t ( ) cos ( t) ( ) sn ( t) t y t t y t ( ) t ( ) t ( ) cos( t ) ( ) 6sn ( t) t y t. y ( ) 6 0 ( ) log ( ) ( ) t t t t 7.. y t ( t) t. y ( t) t ( t) t. y( t) t ( t) cos ( t ) 9. y ( t) sn ( t ) π ( t ) 0sn t 8sn ( πt). π y t 0 cos t 8 cos πt ( ) ( )
680 Chapter 9 Secton 9.. D. B. Vertces (0,±), mnor as endponts (±,0), major length 0, mnor length 7. Vertces (±,0), mnor as endponts (0,±), major length, mnor length 9. Vertces (±,0), mnor as endponts (0,±), major length 0, mnor length. Vertces (0,±), mnor as endponts (±,0), major length 8, mnor length 6. Vertces ( 0, ± ), mnor as endponts (,0) length ±, major length 6, mnor
68 y y y. 7. 9. 6 0 9 9. B. C. F 7. G 9. Center (,-), vertces (6,-) and (-,-), mnor as endponts (,0) and (,-), major length 0, mnor length. Center (-,), vertces (-,8) and (-,-), mnor as endponts (-,) and (-,), major length 0, mnor length. Center (-,0), vertces (-,) and (-,-), mnor as endponts (-,0) and (,0), major length 8, mnor length. Center (-,-), vertces (,-) and (-,-), mnor as endponts (-,0) and (-,-), major length 8, mnor length
68 7. Center (,-), vertces (,) and (,-7), mnor as endponts (6,-) and (-,-), major length, mnor length 8 ( ) y 9. ( ) ( ) ( y ). 6 6..08 feet. 7 feet 7. 6 feet 9. (±,0). (-6,6) and (-6,-). 6. y y. 9 6 7. y ( ) y 6. 6 8 6 y 9. 9 ( ) ( y ) 6. y 0 6 ( ) ( ) y 67. ( ) ( ) y 69. ( ) ( y ) 7. 89 0 y y 7.. feet 7. 77. 860.60 868. 9 79. The center s at (0,0). Snce a > b, the ellpse s horzontal. Let (c,0) be the focus on the postve -as. Let (c, h) be the endpont n Quadrant of the latus rectum passng through (c,0).
68 The dstance between the focus and latus rectum endpont can be found by substtutng (c,0) and (c,h) nto the dstance formula h ( ) ( y y ) ( ) ( 0) whch yelds h c c h h. So h s half the latus rectum dstance. Substtutng (c,h) nto the ellpse equaton to fnd h gves c a h. Solve for h yelds b c a c a c b b h b b b b a a a a a a b dstance of the latus rectum s h. a. so b b h a a. The Secton 9.. B. D. Vertces (±,0), transverse length, asymptotes y ±/,
68 7. Vertces (0, ±), transverse length, asymptotes y ±/, 9. Vertces (±,0), transverse length 6, asymptotes y ±/,. Vertces (0, ±), transverse length 8, asymptotes y ±/. Vertces (±,0), transverse length, asymptotes y ±, y y y. 7. 9.. 9 6 6 9 6. C. H 7. B 9. A y 6 6
68. Center (,-), vertces (6,-) and (-,-), transverse length 0, asymptotes y ±/(-)-. Center (-,), vertces (-,) and (-,-), transverse length 6, asymptotes y ±(). Center (,0), vertces (,0) and (-,0), transverse length, asymptotes y ±(-) 7. Center (-,), vertces (-,) and (-,0), transverse length, asymptotes y ±/() 9. Center (-,), vertces (0,) and (-,), transverse length, asymptotes y ±/()
686 ( ) ( ) y. ( y ) ( ). 9 6. Center (0,0), vertces (±/,0), transverse length /, asymptotes y ± 7. Center (-,), vertces (-,/) and (-,/), transverse length, asymptotes y ± / ( ) 9. Foc (0,±). Foc (,6) and (-,6). Foc (-,6) and (-,-). y 7. 6 9 y 9. y 6. 6 y 6 6 ( ) ( ) y 6. ( ) ( y ) 6. 6 9 y 69. 900 00.66 67. y 900 600
687 7. y 0 697 7. y 0 n the form can be put n the form y. y showng they are conjugate. y 0 can be put 7. 77. No matter the value of k, the foc are at ( ± 6,0) Secton 9.. C. A. Verte: (0,0). As of symmetry: y 0. Drectr: -. Focus: (,0) 7. Verte: (0,0). As of symmetry: 0. Drectr: y -/. Focus: (0,/) 9. Verte: (0,0). As of symmetry: y 0. Drectr: /6. Focus: (/6,0). Verte: (,-). As of symmetry:. Drectr: y -. Focus: (,). Verte: (-,). As of symmetry: -. Drectr: y. Focus: (-,). ( y ) ( ) 7. ( y ) ( ) 9. ( y ). At the focus, (0,).. feet above the verte.. 0. ft 7.,,,. (,8 ), (,8) 9. (, ), (, ), (, ), (, ).,,,,,,,. (-6.0766, 9.788007) (-6.0, 9.8)
688 Secton 9.. e. Drectr:. Hyperbola.. e /. Drectr: y -/. Ellpse.. e. Drectr: -/. Parabola. 7. e /7. Drectr:. Ellpse. 9. 0 r. cos( θ ) r, or sn( θ ) r sn( θ ). r sn( θ ). Hyperbola. Vertces at (-9,0) and (-,0) Center at (-6,0). a. c 6, so b 7 ( 6) y 9 7 7. Ellpse. Vertces at (0,) and (0,-6) Center at (0,-.). a., c., b 8 (.) y 8 0. 9. Parabola. Verte at (,0). p. y ( ). a) y L F d(q,f ) Q(,y) d(q,f ) (c,0) F (a,0) L p
689 b) d ( Q, L ) ( p) p, d( Q, L ) p c) d ( Q F ) ed( Q, L ) e( ). d( Q F ) ed( Q, L ) e( p ), p d) d( Q F ) d( Q, F ) e( p) e( p ) ep,,, a constant. e) At Q (a, 0), d ( Q, F ) a ( c) a c, and d( Q F ) a c d( Q F ) d( Q, F ) ( a c) ( a c) a, Combnng wth the result above, f) d( Q, F ) a c, and d( Q, L ) p a d( Q, F ) a c e, so e. d( Q, L ) p a ep a, so a c e( p a). Usng the result from (e), a a c e a e a c a ea c e a,, so a p. e
690