Solutions to Selected Exercises

Transcription

1 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 ( ) DNE - - -/. / /. a. -6 b 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)

2 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 ( ) 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

3 b h h h 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 (, -)., , Increasng ( ) ( ) Decreasng (,.) ( 0.89,.0) Concave up (, ) (, ) Concave down (,)

4 6 Secton.. f ( g(0)) 6. g ( f (0)) 7. f ( g(0)). g ( f (0)) ( ) 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 ( ) ( ) 7a. C f ( s) s s 0 60 ( ) b. C g ( h) ( h) ( h) ( ) c. v C ( m) 80 70m m

5 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 ( ), y 7. y 9. y. a. f ( ) 6 b. f ( ). y ( ) 6 7. y 9a. Even b. Nether c. Odd

6 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

7 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 ( )

8 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 ½ 7a. b. c. d. 9a. 0 b. 7 c. d 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

9 69 Chapter Secton.. P ( t ) 700t 000. D ( t) 0 t. M ( n) 0 n 7. Increasng 9. Decreasng. Decreasng. Increasng. Decreasng 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 F C c. 9. F Secton.. E. D. B

10 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.

11 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 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 mles Secton y.97.9, r y , r 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

12 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: 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)

13 6. f ( ) ( 6). ( ) ( ) 8 f 7. b and c -9 f 7a. m b ft c. 7.7 seconds 9a. ft b. ft c ft..9 n by.9 n f 9 9. f ( ) ( )( ). f ( ) ( )( ). ( ) ( ). ( ) ( ). ft by..6 cm 7. $ 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 ( ) ( ) (, ). (, ) (,)

14 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) ( ) 8 7. ( ) ( )( ) 9. ( ) ( )( ). ( 8) ( )( ) ( 8 ) (8 ) 0. ( ) ( ) 0 7. ( ) 9. ( 6 9) ( )( ) ( )( )( ). ( ). 6 ( )( )( ) ( )

15 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 ±, ±, ±, ±, ± ±, ±, ±, ±, ±

16 )) ( ))( ( ( ) ( f. Zeros: ± 7. f ) (. 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

17 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)

18 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 ( ) mph.. feet. f ( ) mph

19 69 Chapter Secton.. Lnear. Eponental. Nether 7. P( t ),000 (.08) t Fo. $ y 6( ). y ( ) y ( ) 9. y.9( 0.699). y ( ) mg..9%; $, $,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 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 / e ½ c

20 log( ) log( ) log( 8) log. ln.9. ( 7) log( ) log ( ) 8 log log ( ).9 log (.0). log(.0) log log f ( t ) 0(.068) t 6. ( ) ( ). ln t 7. f ( t) 00e f ( t) 0e f t t 6. Durng the year Durng the year 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) t or Secton.. Doman: : > V. Doman: < Doman: >

21 66 7. Doman: < 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( ) t. ( ) f mg. r Intal mass: mg. After days: mg 7. f ( t ) 0( ) t. Half-lfe 7.86 mnutes 9. f ( t) a( ) 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) mnutes..9 years 7..9 hours T t a). deg b).7 mnutes 9. ( ) ( ) t

22 66. a) b) 00 c) d) 7. years. log ( ) ( ) log...6 Whsper Vacuum Jet tmes more ntense. MMS magntude 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) c) Myoglobn: M(0) 0.9. Hemoglobn: H(0) 0. d) At 0 torrs: At 0 torrs: At 60 torrs: a) C ( t ).06 t 0.066t, or C ( t ) e b) Volume of one cell: ( ) days Effcency seems to be mamzed at about 8 torr π 7 cm, so wll need about cells for a volume of cm. ( ) 6 C t after 7. hours

23 66 Secton.7. log ( f ( ) ) log(.) log( ). log ( f ( ) ) log( 0.).. y e e e 0.68(.687) 7. y (0.) 9. y (.6).. Ependtures are appromately $0. y ( ) y 7.9(0.78) r 0.806, y , r Usng the better functon, we predct electrcty wll be.7 cents per kwh

24 66 Chapter Secton.. 0. ( ) ( y ). ( 7 ) ( y ) 9 7. ( ) ( y ) (0, ) and (0, ). (.60786, ). (-.07,.8) mles Secton π π 9 π.. mles 7. 8π cm mles cm. 960 rad/mn 60. RPM 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

25 66 Secton.. a. III b. II 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)

26 666 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 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.) tan(0.) 0. b. sn() cos() tan().78 c. sn(70 ) cos(70 ) 0.0 tan(70 ).77 d. sn(8 ) cos(8 ) 0.0 tan(8 ) 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 ft ft ft feet ft

27 667 Chapter 6 Secton 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

28 668. 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

29 π 7. f ( ) sec π 9. f ( ) csc. tan ( ).. ( ). csc( ) 7. csc( ) sec Secton 6. π π π π π π π π

30 670 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 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, , , , , , , ,.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 months π 6 b. P ( t) 9 cos ( t )

31 67 Chapter 7 Secton 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 k and k, where k s an nteger π..8 k π and.9978 k, where k s an nteger π π.,, 0.6, ,.,.97,.67 π π π π 7π π 0, π,, 9.,,, ,.98,.,.00. π, 7π, π 6 6 π π. π,, 7..8, ,.98, 0.7,.60..0, 6.0. π π π π 0,,, π,,. π π π π 7.,,, 6 6 π π., Secton π π π 7π 0,,, π,, 9. 0, π,., sn ( ) cos ( ). cos( ) sn ( )

32 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 ) , ,.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, ,.9,.87,.

33 67. π π π π,,, a. π π 8π 0π π 6π π,,,,,,0,, π ( ) cos cos( 6) cos( ) 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 ( ) 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

34 67 Chapter 8 Secton 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 c.066, α., β 86.. a.69, β 7.7, γ ft 9. Dstance to A: 6.8 ft. Dstance to shore:.69 ft. 9.0 m feet..6 km,.79 km ft mles. 6.7 cm. 7.7

35 67 Secton ,. (, ). (, ) 7. (0,) 9.,. (.8,.78). (, 0.6 ). (,.9 ) 7. (,. ) 9. ( 69,.07 ). r sec( θ ).. r sn ( θ ) 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

36 Secton cos sn ( ) ( ) e 7. π 9. e e 0.0 e..67 e 9. π π e π. e. 0e e 6e 7π π e π e , 0..09, , , ,,,,,

37 677 Secton 8..,. The vectors do not need to start at the same pont. v u 7., ,.. Magntude:, Drecton: 90. Magntude: 7.80, Drecton: Magntude:.6, Drecton:. 7. Magntude:.8, Drecton: 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) degrees, relatve to the car s forward drecton

38 678 Secton cos(7 ). 9. ( 0)( ) ()(0) 0. ( )( 0) ()() 0 7. cos ()() ()( ) 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( ) ft-lbs Secton 8.6. C. E. F (t) y(t)

39 679.. y. y y y y 7. e or y ln 9. y. y ( ) 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 ( ) ( )

40 680 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

41 68 y y y 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

42 68 7. Center (,-), vertces (,) and (,-7), mnor as endponts (6,-) and (-,-), major length, mnor length 8 ( ) y 9. ( ) ( ) ( y ) feet. 7 feet 7. 6 feet 9. (±,0). (-6,6) and (-6,-). 6. y y y ( ) y y 9. 9 ( ) ( y ) 6. y 0 6 ( ) ( ) y 67. ( ) ( ) y 69. ( ) ( y ) y y 7.. feet 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).

43 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 ±/,

44 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 C. H 7. B 9. A y 6 6

45 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 ±/()

46 686 ( ) ( ) y. ( y ) ( ) 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 y 9. y 6. 6 y 6 6 ( ) ( ) y 6. ( ) ( y ) y y

47 y y 0 n the form can be put n the form y. y showng they are conjugate. y 0 can be put 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. (, ), (, ), (, ), (, ).,,,,,,,. ( , ) (-6.0, 9.8)

48 688 Secton 9.. e. Drectr:. Hyperbola.. e /. Drectr: y -/. Ellpse.. e. Drectr: -/. Parabola. 7. e /7. Drectr:. Ellpse 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 Ellpse. Vertces at (0,) and (0,-6) Center at (0,-.). a., c., b 8 (.) y 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

49 689 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

50 690