Copyright 2012 Pearson Education, Inc. Publishing as Prentice Hall.

Transcription

1 Chapr Rviw 0 6. ( a a ln a. This will qual a if an onl if ln a, or a. + k an (ln + c. Thrfor, a an valu of, whr h wo curvs inrsc, h wo angn lins will b prpnicular. 6. (a Sinc h lin passs hrough h origin an has slop, is quaion is. (b Th graph of ln lis blow h graph of h lin for all posiiv. Thrfor, ln < for all posiiv. (c Mulipling b, ln< or ln <. ( Eponniaing boh sis of ln <, w ln hav <, or < for all posiiv.. (a v ( ( sin + cos (sin + cos Th paricl is a rs whn v( 0, which rquirs ha sin + cos 0, or quivalnl ha an. Th onl valus of in h inrval [0, ] saisfing his coniion ar 7 an. (b From par (a w hav ( (sin+ cos, ( (sin+ cos + (cos sin cos. Subsiuion givs: A ( + ( ( A( cos + (sin+ cos sin A cos + cos 0 A + 0 A Chapr Rviw Erciss (pp ( ( 7 ( L o s ha <. Thrfor, is biggr. Quick Quiz Scions... A; ( ( f + + f ( 0 (. B; h slop of g a (, is h rciprocal of h slop of f a (,. g ( f ( +. C; (sin ( ( (... (an( sc ( ( sc ( (sin ((sin (sin (sin sin cos (ln(csc (csc csc ( cscco csc co Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

2 06 Chapr Rviw s. cos ( sin ( ( sin ( 6. s co csc csc csc + cos / (( + cos / ( + cos ( + cos sin + cos 7. ( + ( ( + ( + ( + + ( ( r 9. sc( + θ θ θ sc( + θ an ( + θ( sc( + θ an ( + θ 0.. r θ θ an ( θ θ an ( an ( θ an ( θ sc ( θ ( θ an ( sc ( θ θ θ θ ( csc ( ( cscco ( + (csc ( csc co + csc ln, > 0 ln ( + ( ( ( ( ( + ( ( + + ln ln ( ( ( ln (sin (sin sin cos sin co, for valus of in h inrvals ( k, ( k +, whr k is vn. r ln ( cos cos cos cos for < < cos r θ θ θ θ θ log ( θ ln θ ln ln θ θ ( Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

3 Chapr Rviw s log ( 7 ( 7 ( 7 ln, > 7 ( 7 ln s 0. ( 8 8 (ln 8 ( 8 ln 8. Us logarihmic iffrniaion. ln ln ln ln ( ln (ln (ln ln (ln ln ln ln ln ln. ( + + [( ] ( ( ( ( (ln + ( ( ( ( ( + + ( + ( ( ln + ( / ( + ( [( + ( ln + ] / ( + ( ( ln+ + ln + / ( + ( ( ln+ ln + / ( + Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

4 08 Chapr Rviw Alrna luion, using logarihmic iffrniaion: ( + ln ln( + ln ( ln + ln ln + ln + ln ln ( + ln [ln + ln + ln ln ( + ] 0+ + ln ( + + ln + ( + ln an an an an + u u sin u ( u ( u u u u u u sc ln ( + (sc ( + sc u [( + co ] ( + ( + (co ( + ( + + co + u Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

5 z cos z z z z 7. ( ( z + (cos z( ( z z z + cos z + z z cos z z z Chapr Rviw ( csc ( ( + csc ( csc + ( csc + 9. csc (sc (sc sc sc sc an sc an sc an sc an sin cos cos sin cos cos sin sin for 0 < Alrna mho: On h omain 0 <, w ma rwri h funcion as follows: csc (sc sc (sc cos (cos Thrfor, for 0 <. Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

6 0 Chapr Rviw No ha h rivaiv iss a 0 onl bcaus his is an npoin of h givn omain; h wo-si rivaiv of csc (sc os no is a his poin sinθ cos + sin θ ( cos θ(cos θ ( + sin θ(sin θ cos θ ( cos θ + sinθ cosθ cos θ sinθ sin θ cos θ ( cos θ + sinθ cosθ sinθ cos θ ( cos θ r θ θ θ. Sinc ln is fin for all 0 an (, h funcion is iffrniabl for all 0.. Sinc sin( is fin for all ral an cos(, h funcion is iffrniabl for all ral.. Sinc is fin for all < an + / ( + ( ( ( + ( +, which is fin onl for <, h funcion is iffrniabl for all <. / ( +. Sinc is fin for all 0 an (( ( (, ( h funcion is iffrniabl for all 0.. Us implici iffrniaion. + + ( + ( + ( ( + ( ( ( + ( Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

7 Chapr Rviw 6. Us implici iffrniaion. / 6 / + 0 / 6 / ( + ( 0 ( / / Us implici iffrniaion. / ( ( ( ( / / Alrna mho: ln ln( [ln + ln ] 0 ln + ln Us implici iffrniaion. + + ( + ( ( ( ( + ( ( + ( ( sinc +. ( ( ( ( ( ( ( ( ( + ( + ( ( ( ( ( [( ( + ( ( ] ( ( + ( + + Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

8 Chapr Rviw.. + cos ( + ( ( cos + sin ( + sin sin + sin + ( + ( cos ( sin ( 6 ( + ( + ( cos ( sin sin + ( + ( + cos+ sin ( + / / + / / ( + ( ( / / + 0 / / / / / ( ( / / ( ( / / / / ( / / / / +. No ha h h, 8h,... an 0h rivaivs of sin ccl back o sin. B h chain rul, ach rivaiv gnras anohr facor of 8. Thus ( 8 ( 8 0 ( 8 sin sin. 0 ( A, an (. ( (a Tangn: ( + or (b Normal: ( + or + (an sc A, an an sc ( 8. (a Tangn: + 8 (b Normal: ( 8 Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

9 Chapr Rviw 7. Us implici iffrniaion. + 9 ( + ( ( Slop a (, : ( (a Tangn: 9 ( + or + (b Normal: ( + or 8. Us implici iffrniaion. + 6 ( 6 ( + ( + ( + ( ( 0 9. Slop a (, : (a Tangn: ( + or + 6 (b Normal: ( + or sin an cos A, w hav sin, cos, an an. Th quaion of h angn lin is ( + (, or cos co sin A, w hav cos, sin, an co. Th quaion of h angn lin is + +, or +. sc sc sc an an sin A, w hav sc, an an 6, sin ( 6. Th quaion of h angn lin is 0 0 ( +, or. + cos sin A, w hav cos, + sin, an. (a ( ( + cos + +. sin lim f ( lim (sin a+ b cos b 0 0 an lim f( lim ( +. Thus lim f( f( 0 if an onl if b. 0 acos a bsin, < 0 (b f (, > 0 Th slops mach a 0 if an onl if a. (c No; alhough h slops mach, h funcion is no coninuous. Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

10 Chapr Rviw. (a Th funcion is coninuous for all valus of m, bcaus h righ-han limi as 0 is qual o f ( 0 0 for an valu of m. (b Th lf-han rivaiv a 0is cos( 0, an h righ-han rivaiv a 0 is m, in orr for h funcion o b iffrniabl a 0, m mus b.. No ha 7 7 f( ( ( /, 7 / f ( (, which is fin if an 7 onl if. Thus h answrs ar (a For all (b A (c Nowhr 6. (a For all (b Nowhr (c Nowhr 7. Th iniviual funcions ar coninuous an iffrniabl on [, ]. A w hav lim f( lim + an lim f( lim ( + f(, f is + + coninuous a. Howvr,, < f ( +, h slop is, > coming from h lf an coming from h righ. Thus f is no iffrniabl a. Th answrs ar: (a [, (, ] (b A (c Nowhr 8. Th iniviual funcions ar coninuous an iffrniabl on [, ]. A 0 w hav lim f( lim sin 0 an 0 0 lim f( lim ( + 0 f( 0, f is coninuous a 0. Al, cos, < 0 f (, h slop is +, > 0 coming from h lf an coming from h righ. Thus f is iffrniabl a 0. Th answrs ar (a [, ] (b Nowhr (c Nowhr 9. (a Sinc sin > 0 for all, (b (c ( 60. (a (b ( ( sin sin, cos. + 7 ln( + 7 ln + ln( ln + 7 +,. s an(an (, s s (sin(sin, s ln ln( + 7 ln( +, ( + + ( ( + ( ( + + ( (,. Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

11 Chapr Rviw (c ( s sin (cos s. cos (cos, s ( 6 / ( 6 /, s 6 / (a Th lin passs hrough (,, which mus b h poin of angnc. Th slop of h angn lin is, h slop of h normal lin is. Th quaion of h normal lin is (. (b Th angn lin o f a (, has slop, h angn lin o f a (, has slop. Th quaion of h lin is (. f ( (c Whn, an f ( f (. Th lin has quaion (. 6. (a Th lin passs hrough (,, which mus b h poin of angnc. Th slop of h angn lin is, h slop of h normal lin is. Th quaion of h normal lin is (. (b Th angn lin o g a (, has slop, h angn lin o g a (, has slop. Th quaion of h lin is (. (c Whn, g( g( an g ( g (. Th lin has quaion (. 6. Firs, no ha ln ln( + + ln( ln( + 7. Thn , an ( + ( ( 7 6. Firs, no ha ln ( + ln( +. Thn 6. (a ln( (, an ( ln(. f( or f( + C (b f( or f( C 66. (a (c f( or f( C ( f( or f( or f( C + D ( f( sin or f( cos or f( C sin + D cos f( f ( + f( A, h rivaiv is f ( + f( + (. 0 Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

12 6 Chapr Rviw (b f ( f( f ( f( f( A 0, h rivaiv is f ( 0. f ( (a f f f ( A 0, h rivaiv is f (( 0 f( 0 ( ( ( ( ( ( (. f ( (c f( f ( A, h rivaiv is f ( f (. 0 ( f ( an f ( an ( sc A 0, h rivaiv is f ( an 0( sc 0 f ( ( (. f( ( + cos ( + cos ( f ( ( f( ( sin ( + cos A 0, h rivaiv is ( + cos 0( f ( 0 ( f( 0( sin 0 ( + cos 0 f ( 0. (f [ 0sin f ( ] 0 sin ( ( ( 0 ( cos f f + f 0 f( f ( sin + f ( cos A, h rivaiv is 0 f( f (sin + f (cos 0( ( + ( ( 0. (b f ( g ( f g g + g f f f g f g + g f ( ( ( ( ( ( [ ] ( ( ( ( ( ( A 0, h rivaiv is f( 0 g ( 0 f( 0 g ( 0 + g( 0 f ( 0 [ ] [ ] [ ]. ( ( ( ( + ( ( (c g ( f ( g ( f ( f ( A, h rivaiv is g ( f( f ( g ( 0 f ( ( ( 8. ( f ( g ( f ( g ( g ( A, h rivaiv is f '( g( g ( f ( g ( ( (. ( f ( g ( f ( g ( g ( A 0, h rivaiv is f ( g( g ( f ( g ( ( (. (f g ( + f ( g ( + f( ( + f( g ( + f( ( + f ( A 0, h rivaiv is g ( 0+ f( 0[ + f ( 0] g ( 0 [ + ( ] (( Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

13 Chapr Rviw w w r s r s sin ( r 8sin s+ r s 6 cos( r 8cos s+ r 6 A s 0, w hav r 8sin 0+ an 6 w s ( cos 8cos 0+ 6 cos0 8cos Solving θ + θ for, w hav θ θ θ, an w ma wri: θ r r θ θ / r ( θ + 7 ( θ θ θ θ / r ( θ + 7 ( θ ( θ + θ r / / θθ ( + 7 θ ( θ ( θ ( θ θ A 0, w ma lv θ + θ o obain θ, an / / r ( ( (. ( (a On possibl answr: ( 0cos +, ( 0 (b s( 0 0cos (c Farhs lf: Whn cos +, w hav s( 0. Farhs righ: Whn cos +, w hav s( 0. ( Sinc cos 0, h paricl firs rachs h origin a. Th vloci is givn b v ( 0sin +, h vloci a is 0sin 0, an h sp a is 0 0. Th acclraion is givn b a ( 0cos +, h acclraion a is 0cos (a 8+ 8( + + 0, +. Tangn lins ar + horizonal whr 0, which is whr + 0, in which cas. Ling in h quaion of h hprbola, w obain + 8( + ( + 0. Solving ils ±. Rcalling ha, w g h poins A(, ; B(,. (b Again, +. Tangn lins ar + vrical whr is unfin, which is whr + 0, in which cas. Ling in h quaion of h hprbola, w obain + 8( + ( + 0. Solving ils ±0.. Rcalling ha, w g h poins C( 0., ; D(0.,. 7. (a ( + + 0,. Tangn lins ar horizonal whr 0, which is whr 0, in which cas. Ling in h quaion of h llips, w obain ( + ( 0. Solving ils ±. Rcalling ha, w g h poins A(, ; B (,. Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

14 8 Chapr Rviw (b Again,. Tangn lins ar vrical whr is unfin, which is whr 0, in which cas. Ling in h quaion of h llips, w obain ( + ( 0. Solving ils ±. Rcalling ha, w g h poins C(, ; D(,. 7. (a ( + + 0, +. Th angn lin is vrical whr is unfin, which is whr 0, in which cas. Ling in h quaion of h parabola, w obain ( + 8. Solving ils. Rcalling ha, w g h poin A(,. (b Again, +. Th angn lin is horizonal whr 0, which is whr + 0, in which cas. Ling in h quaion of h parabola, w obain ( ( 8. ils. Rcalling ha, w g h poin B(,. + Solving 7. Ling 0 in h quaion of h parabola, w g 8, which has luions ±. Ling 0 in h quaion of h parabola, w g 8, which has luions ±. Implicil iffrniaing, ( + + 0, +. A -inrcp ( 0, h slop is A -inrcp ( 0, h slop is A -inrcp (, 0 + h slop is 0 ( ( + + A -inrcp (, 0 0 ( +. ( 0 h slop is 7. Evr sinui wih ampliu an prio is h graph of m quaion of h form sin( + C + D. Th slop a an is 6cos( + C. Sinc h maimum valu of cosin is, h maimum slop is Evr sinui wih ampliu A an prio p is h graph of m quaion of h form Asin + k + D. p Th slop a an is A cos k. p + p Sinc h maimum valu of cosin is, h maimum slop is A. p 77. L f( sin( sin. Thn f ( cos( sin ( sin cos( sin ( cos. This rivaiv is zro whn cos( sin 0 (which w n no lv or whn cos, which occurs a k for ingrs k. For ach of hs valus, f( f( k sin( k sin k sin( k 0 0. Thus, f( f ( 0 for k, which mans ha h graph has a horizonal angn a ach of hs valus of. 78. (a (b 00 P( 0 sun + 00 lim P ( lim h h suns Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

15 Chapr Rviw 9 (c P ( 00( ( + P ( ( ( A graph of h rivaiv ( shows a maimum valu a, a which poin P ( 0. Th spra of h isas is gras a, whn h ra is 0 suns/a. Th maimum ra occurs a, an his ra is P ( 0 + ( 0 suns pr a. 79. Diffrniaing implicil, + ( + + 0, (a A (, w g Us implici iffrniaion. ( ( ( 0 ( ( ( ( ' ( (sinc h givn quaion is A (,,. ( k 8. (a g ( k + f (, g ( 0 k +. k g ( k + f (, g k ( 0. (b ( ( ( ( ( A (, an rcalling ha w g ( ( , ( ( ( 8. (a (b h ( bsin( b f( + f ( cos( b, h ( 0 b sin( 0 + cos( 0. No ha h(0 cos(0 f(0, h angn lin has quaion ( 0. (b ( + ( + ( ( ( ( + Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.

16 0 Chapr Rviw (c ( + (. ; + 7. ;. 7. ( m / ( ( 0 s qual o 0 (mulipl boh sis 0 b 0 (mulipl boh sis b 0 ln ln 0 0 Th angn lin is horizonal a (a (b > < < < < < 0 Domain of f (, f ( ( (c Domain of f { an Domain of f} Domain of f (, ( ( ( ( ( f ( ( + ( + ( ( + < ± 0 for ( (Th numraor an nominaor ar clarl boh posiiv. Thrfor, f ( < 0 for all (,. Coprigh 0 Parn Eucaion, Inc. Publishing as Prnic Hall.