ma1012_hw14_201012

ma1012_hw14_201012 - d t dy 7 150 5 y œ 7Èx bChapterb...

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Unformatted text preview: d) t dy 7 150 5. y œ 7Èx bChapterb 6)"Î# Ê È) œ ˆ (x b seccˆ ‰ œ 7 6 œ 7(x 3 Differentiation 6) "Î# tan tan ‰ " b È tan ˆ " ‰ sec È) tan È) œ Šsec È)‹ ” dxÈ # c ) • 2Èx b 6 ) #) #) Thomas’ Calculus 11wth Eœ d , Chapter ) ˆ " ‰ (tan xb sec ) dition ˆ ) tan " ‰ sec 3 d d " " # dr 32. k(x) )b tan ))c" Ê d) œ (sec ) bœ x#)sec ˆdd)‰(sec )kb tan )x#œdxsecsec b tanb) secœ xsec†) dx "a)tanb)) œ) x# sec ˆ x ‰ tan ˆ x ‰ † dx ˆ " ‰ b 2x sec ˆ " ‰ tan )c# " Ê (x) dy ) x x x (sec ) x (sec ) b "Î# odd problems))c"Î# Section œ "c  ­3 ˆ 1) 6. y œ c2Èx # 11: 2‰ 2(x 3 "(‰ Ȉ Ê ‰ dx dœÈ 1(x " ‰ 1) c ˆ 7 c only " c ˆc œ ˆ"‰ xc1 ˆ"‰ cÈ " c (1) † ˆ t b 1 t d t œ x sec x tan x † t b 1 x ctbdt2x sec ‰ x œ 2x sec x c sec x tan ˆ x ‰ t œ cos Š Èt b 1 ‹ † dt Š Èt b 1 ‹ œ cos Š Èt, 23, †29, 33, 39, 45, 49, and 65a Section 5: 19 b 1 ‹ ˆÈ t b 1 ‰ Section 6: 19, 25, 29, 35, 39, 45 (2 points), 57, and 61 (4 pSection 3.6 Implicit Differentiation oints) t dy (1 b cos ))(cos )) c (sin ))(csin )) œ 2(t b 5)ct"Î# sin ) ‰#b%Ê f"((2x b25)c$Î# ) 2 † d ˆ b c$Î# 2 sin 2t1 w d d œ cos 7.È ydx ‹(2x#b)1)œ5‹% œÊ dxtbœxc‹#cos œ bb1 dx a† ) ‰œ c(2xsin )5) # œ dx sin ) Š 2(t b c xb 12 2 b x cosc# x ŠÊttbdy33. xf()d1)asin1 8+Š)2(t=25 †pdx )) # bÈttˆx1b coscosc#d) b b cosc‰ x † 1db (x)) † œTotal: ˆ+ b cos sin 1) roblems ‹ b ax Š x 1 b cos ) cos (1 b cos )) 1 dx 1 d dy % (2ˆc)) acos )dy cos )d (cosb x)‰ bsin ))c# x) b 1) sin 2 cosc$ x † b sin ) dy (2 cos (cos b 2 sin ) x dx (sin x)‰ b 2x sin x œ x 2: 2x dx b 2y dx c dx œ 1 c 2y Sb tep œ œ (1 (1 cos )) dx 8. y œ (" c 6x)#Î$# Ê bdy œ 2 (1 c 6x)c"Î$(1c6) œ c4(1 c b cosc"Î$ ( b cos )) 6x) )) $ 3 sin t 2x d ˆ sin ‰Answers:dy t dx( ˆ sin t c sin t cos sin ‰ # xb b ‰ sin% x tb xa ˆc2 cosc xb ‰ct cos x)b b ‰ c# x x cos ccsc ˆ t † dt t œ tep csc ˆ dxt (2x b 2y tc 1) œ " c 2y S a 3: c$ .1 Chapter 3x c"1 c x y c# d cos x b 2x sin% x b 2x sin x 4: ˆ 1 bdcosb‰cosc#2y w t) c t) sin t tep cos dx t w 2y 1 34. S# œ "Î#sin t œ 2x bÊcg (t)# œ c ˆ 1 b cos t ‰ † dt ˆ 1 b cos t ‰ œ c (1#b cos t)c"Î# t)(csin #(sin(" b cos t)(cosb 1 g(t) † (sin # c"Î# "Î# 2x 9. y œ x27. b b Ê y œ x † " ax b 1b sin #xd b ax# Section " œ ax d ChainaRulexandt) Parametric Equations ax b 1 a b sin x b b "b œ È † cos (1t c 2) t† b ( 1t c 2) 1 b t † 3.5 The 1b 29. b d # x b1 œ 2 sin (1t c 2) † dt sin (1t cacsin t c cos t c1t cb 2) c 2) œ 2 sin ( cos t " dt "œ x d d ˆ" xd d 33. #x cos t c 2) cos$ x Ê yw œ 31. asinc&(1 b$b sinc& x † œ# 1 b ‰ c 3 dy acos$#xdyc cos$ x † dx ˆ 3 ‰ xb cos t) b dx x dx y c 3x $ tc 3 22.x dx xy "Î#t œ 1 Ê 3x c y c x dx c 3y " † d c 0 c " dy t)c c œ# c dy x c' & 19. ˆx cÈcb y œ (3wc t)"Î#ˆ Ê ‰dp c " b$Î# ˆc"Î#œ dt #Ê a3y"Î# xb dx œ y "Î# 3x 2 Ê" dx œ " dt # œ ax x cos xb b asincœxb ap #œ" ‰ bcc a aÊcos# xb (†csin" x)b #œ a#1(3$cbt)#3dx b ax(3b 1t)cœ c " (3 c # b 1bc$Î# È3 c#t b x#3y c x œ c b 1 3 3 3 y œ x c ax b cos x a x‰ 10. y x x x b b b † acx b "b # dy 2 È È 3dr y œ $ 2x œ (2x)"Î$ Ê dy œ " (2x)c#Î$ † 2 œ 3x b . 4. y )œ %) b5x œ (5x)"Î% Ê dx œ (cot csc )) dx 3 ) cotdr )c" Ê È œ c(csc ) ‰ ˆcot")‰c# dd) (cscathematics csc ) cotb cotcsc ) œ csc4. ) berivatives Part II ) ) " ‰ ) b cot )) œ È ) " )) # "b d) )‹ ˆ sec (csc‹ Š c ) b tan ˆM Šsec È) tan I: Homework 1(csc D cot )) ) È‹ d) œ Šsec ) ) œ sin a) b (csin 2x ) d) Ê)) bœ (2)) acos a) bb † d) a) b (2 cos x 1. y œ ) dx 4 1 c sin 3) t ‰ (cos 2)# acos a)# bb (2)) œ c2 sin a)# b sin (#)) b 2) cos (2)) cos a)# b # 2. y œ x Ê dx œc5x " 4 (5x)c$Î% † 5 œ 5 4x ) 161 149 ax b 1 b d d # # # # " " 35. cos# x sin Ê dr x cos d x sinc& œ (2r #œ sin a)# bxcos (2))t)$ d (1 œ sin a) b (csin 2)) (2)) b cos (2)) acos a) bb † d) a) b sec 1t)x† c (sec 1t) x b xsec 1t)(secœ ctancos: †xdt d) t) œ 21 sec# 1t tan 1td) 1t # 3c 1 # dt 23. x (x in y)# b cx 2) c y # "Î# c"Î# sin a)# b sin a# ) # c"Î# cos 20. qœ sÈa) c(r# sin a2r)(2) # b (cos 2)) acos " ) 2rbc r#)bœ c2 dr % 2r c r# b (#)" a2r c cos (2))(2 c a) bœ " c r œ 2r œ c rb Ê dq œ # a b (2) †da œ # b 2) rdrb 2r) 11. c" ( S#tep 1: Ê ’ds œ 7 y) Š1 c dy ‹“ dr (x c y)# (2x)c# 2x c 2y dyc$ œ )c$Î% Ê d) œ c 3 )c(Î% È2r c r s œ Èt œ t#Î( x# dt c2 tc&Î( 12. r œ È) 2(x d b œd8 4 dy " 7 y2)( b ˆ4 c " ‰ d Ê dx œ 21 (3x c 2)' † dx b c dx c& (csin 2t) "d (2t) dx ˆ4 c dx cos c& † dt (1 b cos 2t) œ c4(1(3xcos2) b3.61) ˆ4 c †#x ‰ †œ (1 bsin 2t x ‰ 2t) # Sectiondy(c Implicit Differentiation # " 161 È t œ c4(1 b #x 2t) dt # " cos 2t) dr ‰ 36. c#œ Šsec È)‹# (x c"y) Ê b 2y œ sec2x c 2x#d c) y)ˆc 2x(x c y) ˆ d Šsec 4) tan È)‹4Š " ‹ rˆ œ Step"2: c2x tan ˆ dy dy d ds Š œ È)‹ ˆ sec ‰ c " ‰ b tan ‰ (x #Î$ c 2)' c dx " )' † 3 b (c1) ˆ4 y œ "sinsˆœ x4 ‰ sinc3t b 54 cos 5t cos)ˆdt dxb345)cos 3t †† ˆdt (3t) (2t 54 )(csin 5t) †œ c 4 (2t b cos &Î$c 1 ˆ#È) b 5)c#Î$ ‰ 1œ 1 1 13. c 21. ‰ (2t3b 5) (3x‰ Ê# )dt œŠÊ (2t 1 c#Î$ ‰ c 2 ‰ b b 5)c&Î$ † 2 )dt (5t) œ 1 5)c3t cos sin 5t (2t #x 3 3 Rule and Parametric Equations dy x 4c ‹ 151 y dy cc (x c y) ˆ x œ c c ˆ Section 3.5 #The Chain c‰ y) c$ 4 3: dy c œ c21 1 b cot ˆ t ‰‰Step d cosbdxÈsin‰‰#œ"c2 b 2ydcot ˆ t"‰1c$ †È c y) È (x c d ˆ td‰È œ ˆc 2y œ †1 ( ˆ1seccot)2x 5t) ˆ ‰ b 1 " tan 2x ‰‰secx(x)csc# ˆ t ) ‰ † Šsec )‹ tan È) tan ˆ ‰ c sec ˆ ‰ " 3t c ˆ t x dx b œc) sec c x(x c y) c È c y) d ˆ# x c1 c x(xcy) c (x c#y) œ dt x a# c x b xy c x b 2xy c y b tan ” • dt # dt # ) ) ) c d 2x c1 1 dy # (x ) #È ) Step 4: œ 1) œ " c 2y % z œ cos ˆ(" c 6t)#Î$ ‰œ# "Î# 2x (xcdy bˆ(" c$6t)#Î$ ‰ †y 2 (1(xc y) d c"Î$ œ ') œ 4(1 c " 6t)y "Î$ sin ˆ(1c"Î# #Î$ ‰ # d # c c c"Î# 6t) b dx Ê dz Ê 4 y) 2 (c t# b x y cx a1 c c 6t) sin at b † at b‰ dyy œ a1 b cos at c%b dt œ csin 2y" a‰ b cos at# bbx c † ˆc1 " ˆ 2 14. 2‰ 3 2 c$ c% cos"a‰ ˆb œ ‰$ b cos at# bb 47. a ˆb ˆ b 8 x b 1‰ Ê dxsœ c3(5 3ct ‰ b c 2xc2) tb cÊ x ds #œ1cos ˆ 31t ‰œ d ˆ 3c‰dtc 1 bˆ 3ˆ x‰ † dx ˆ 311‰ œ 31 cos ˆ 31t ‰ c 3 sin ˆ 31t ‰ 6(5 1t 2x) sinc 1t b #t 1 2x) b ( b 1 y xy œ dt 22. œ sin ˆ # œ x cos ˆ 33x ‰ 8dt dt 1 c x # # dt È # 2dÈ# 2 # 162 œ ChaptertcosDifferentiationb †t 2t œ #c † tdt t at #b 3 ax#y ccdqb yasin at# b 12 sin x"Î# " ˆ t b 1‰ t b 1 (1)ct † dt ferentiation Chain Rule1sinaŠ b 1t ‹ tÊb31t ‰Equations ‹ †149 d t cˆ1 b Š x b 1‹ ion 3.5 The 37. q œ andÈ3 b 1c sin dt "Î# cos Š Èt b 1 " dt È1 bb 1 ‹c"Î# Š"Ètc"Î# † Š Èt cos at bœ cos ‹ œ 3 # cosParametric œ c" c" t# # ˆÈ t b 1 ‰ 15. f(x) œ É1 2 Èx œ ˆ1 c x"Î# ‰ Ê f w (x) œ # ˆ1 c x"Î# ‰ d ˆc # x b 1 ‰ œ œ dyx dc ȉ Chapter # .5 3 4É ˆ 4 É c È x‹ È c 5)) † (dy œ (2t c c 4b $Î% t c c dt d (2t c ‰ c dt cos (2t ) 5) œ cos # "‰ # w œ cos (cos ˆ 34."5)) † 2Ȉ "œ 3 c#Î$#b tdy 1 c Ê "dr# c )c"Î# sinyˆ "3x b )dy"Î%(2t c 5) œ )c"Î#Šb1)c"Î$ b x c"Î% x 1 c x dtc y œ x# dy bsec# dy b sec Ê ‰a† d# ax xÈœ)x(cos (2t ‰ tan ˆ "dy †œd )c "Î$ 5)) † sec ˆÊ dr dy dy) ) ‰ x k (x) c "dx 3y24.r (3xy b"x 3y 6y bb 3 2(3xytcˆ3x † Š3xxdx tb 3y xc2(t b6 c t Ê x 2(3xy b 7)(3x) c 6 œ c6y(3xy b 7) y b 7) Ê ‰ 2 œc ‹ œ 1) dx "Î# œ c c 7) t dx c t)cxdx œ 0 os Š #dx ‹ † Êdx œ sec 1 xd) cos Š dx dx3yŠ xb 2x ‹ œ Š d)t b 2 ‹ cos Š t ‹ dx) b csc dx c t) cc csc c" t b sin (2tœ c5)) (3 dy 23. rœ 2(csc ) "t É ))È% Ê1 dr "œ œ (csc )ÈtÉc‹))c# bd1)(csc$ )É cotb 1) œ csc ) cot Èt bÉ) œ È ) (cot ) b csc )) d ˆ È È 2(t " db) %† ˆc " ‰ b#2x sec ˆ " ‰ œœ 4 % cˆŒ‰ cotb ˆ "9 tandˆ dy œ 4 cos Œ b 11$b Ètd) † d Œ b1 2(t Èt9 œ 4 ) b cot )) 1 b (csc )†b cot )) ‰ 48.19.œ dy3 (c x b cb 1) t † dÊ ) dt c y 2x sec t b 1 sec 9 [6x(3xy b 7) c 6] œ c6y(3xy bb cot dycœ c dt † dx 7) )œ (csc 7y 1 (x b x $ Ê dx œ (4x b 3) sin 3)(x1c"Î$cx ‰ dx (x xb 1) b (x b 1) (4)(4x b 3) y(3xy b(4x b 3) 3xy bcos Œ c%Î$ t9 É È † dt 1 b Èt‰ 1) x %Î$ # Ê2xc"Î#" b 1‰ 21 (r) ˆ )) csc ) Ê [cos (r))]gwr b œ dr ‰2œ2xc"Î#Ê 1) cos (r)c1)xcr cos c$)c2xc"Î# b 1‰ r cos xc)$Î# c r 1b t ˆ 0 7) cdr [dx ‰ dxœ x(3xy b c$Î# œ (r ˆ 3x y c dr œ c 7) c 1 7x 16. g(x) œ 2 Ê (x) ) c † ( )] œ 35. bœ ) b cot ) Êb )Ê ( b 1)c% (1) b (xsin .(r c$ (4)(4x b $ dq œ ˆ ˆ # b ˆ 3 % b t sin t%) d 16(4x b 3)$ (x b 1) 3 t c sin t d) ) cos (r)) œ ), dy dˆc t ‰) 3) t #c3)(x sin ˆ5 sin ˆ t ‰‰ 231)csc "Î#ˆ# t ‰t‰È 3) (4) " c c3(4x‰dsin 5‰ (x dˆˆ1)‰d† b ˆˆ ‰ t cos # sin t dc # ‰ t 38. c # (4xsin ˆsin dy9c È œ ccsc )(cos t † dt Œ sin œ " ‰ # cos œ)cot3) ‰3t bb 7)Êœsin ˆ5 rb cos È 9 ) (sin 3 ‰t ) œ b)) 0 œ dt† wœ a2r c r bsin ) #3 a2r†d q r 2sinbŒ(21c‰2r)sin )dtcos(1 É1 b )3 tˆ t ) ccos ))(csin ‰)dt 3ccsc ˆ t ‰‰ ˆ dt 5cos É t ˆ œ Á 2y b f dr dx 2 b c () œ 3(4x )b ‹“ b116(x )b 1)dd) œ(rb 2x )b(4x 12 cos ) †2r c rb 1) c (x c 1) )) 2 1 ct dy 3) ˆ (xcos y) †(2x) 1 cos c1)œ dx œ c # "Î$ os ˆ 3 ‰‰ yÈ x(xb" †2 t2y dy " (x 2b (1 b cosœ w b ) " dy cos " 1) d 17.) (cos )25.29œœÉ1b 1ÈÊ)) œ dx)cbÉt b(xdrt )œ (sec )hb)1) œ))c(1dx œ y(x)bb tan † (csin 2))tan†)2 œ c 2 (sin ) (tan ) b sec )) 2))c#Î$ h( ) ) œ r$ . 1 b cos t(2Ètan ) œ cos È)1) Ê (x( tan Ê # b (sec 2))c#Î$ )) œ sec ) b sec ) 3 sec 2))(1 b cos (1 Ê x 24. c(sec ) b œ (sec ) b tan )) 3 sin (2 b 1) 2 sin ) dy )b œ dy sin ) d d) (sec ) b tan )) # d 2y œ 2x c)39.# (x c sinc(2x(x c Ê dy œ 2)sin (1t 2) d d ) 2x œ (1 b # 1 4 c Ê dt sin ( t 2) dr [ sin ( r c 2) † r b csc † dt dr dx 4b csin 5t) b cos(5t) 334œ y) cos))) t sin y) (# dx (1 † dt 36. œy1 rcossec c) œ r2)#Section 3.5drThe c # ) †œ rsin )1dand Parametric Equations c ) 153 (1tœ c rsincsc )) 3t c 1 ) 5t (c& r) d) c csc # dt Rule drcÊ œ) 2csin1tc )] œcos (1t # 2) Ê d) c 2) b r b Chain b ' ' dy cos. b cot " 5 c" 1 c œ dc) c tan ) $ % ab 2yd5xb tan% c x(x c y)1x1bb(cty)#2) ‰w‘ # (1t <1 2) tan b ‰‘ <4 tan$ 1)(2x d dtan# (2) (2x2 c in tancˆ$ cos 3 c b # c 5x 3 t# † dy x c< b Ê ˆ t œ # # sec< 5)y " (6) ax t c 5xb (2x5c"5) % ˆ atx" # # "Î% b (cˆ ) ‰w "c#5) 5 ˆ w t ‰‘ ## 3 y) œ 3 1 18. c1 )26. ‘yœ œ() sb 5))&Î%Ê # y xœ)3 ˆ1 b(sinÊc3x ‰b 2xy bˆ(1#b œ 1 c ycos œ ax#5)(sin ydw œ 5))"Î% c ˆ2xy " ‰# †ywdœ 1 c 3x c 2xy œ 2x k( dx#œ x œdtˆ1 b 1 xdÊ kwœ ) œ c y‰ (ˆ b 5))œ c cos x y 5) œÊ 1#ww () b b ‰ b () b 1 c"3x c 1 b Ê † y ˆ3‰ ) † x 1 b x ‰ 4 y Ê ˆc x 1† dx ˆ1 b x ‰ ) 49.‰ (sin x b(xx Ê 1 b ( œ x 4 ) 1#x dt 1 39. dt y‰ x x dx x x b 1 c# c$ w1x ‰ ac c5x t c1 t x(x # 1 (x ˆ y) d ‰ œ c # b xyt c2 c‰ b ‰ x b&gc t(xœ y) d # b1 bxcosc‰ c†y)d xˆ 1c 1cos t% 1xt a1 c# 1sin $c x t† (sin t)(# sint t) c (" b cos t)(cos t) y) (t) x c ˆ dy b 2xy c c b xx y # ‰‘ (1 tan # sec " " anc y) 1#2yÊ 5)dx 1# ‰tˆ†2y"# ‘x#œdt<1## sin‰t %œ ˆ#1#sec "<bˆcos wt)"1b‰ b ˆ 6ˆ‰## (sin t) " ‰% œ 6 ˆ1%b " ‰d b 6 x 1 b d ‰# œ c6 d 1 bd" ‰ c# b 1d b " ‰x an ˆ ˆ4b ‰ sec œ sin 37.sec œ cy b3 tanœˆ1 2x b42yy ˆœ ‰ dy 2yy‰d‘œ c2x Ê dy œ yw œ c ˆ; now to find# ˆ y , ayw b œ d c (2x c ˆ œ w (x 1 c y(x 4 y) x4 x .œ x sin x b x ‰c y c x ‰ 0 Êx Ê 25.45bˆc œ 1ˆ2 b x cosx #c x Êy dx œ x1 ˆ1 bsin xb b sin x † dx axx# byb x xd acos x xb bdxx ˆ x x † dx Š x y ‹ x cos (x) a "x œ x # 1 x tbc xy 1 31t #19. x31t b xy# œ 6:# 31t Ê 3dy œ (2 1t dy t) † d dy dx1t) 1 (2 sec dx t)(sec 1t tan 1t) †dx (1t) œdx 1 sec# 1 t tandx t " ‰ d ˆ # y‰ 1 40. y 1 tan 1 t" 1 dt sin sec 1 sec (sec 1 yœ ˆ ‰ sec # c sin b cos t †ˆdt ‰ œ27.œx32œ6# ˆ1ˆb Ê c1œ a2 ‰ ˆ#3yb‰dx y(1) dtdx 2œ sec y ‰œ cosand ydw (1) œ 1 ‰Ê tangent line2is œ1 # sec 1 # 1 Ê œxcos4y # x ‰ ˆ d 2(sin 2; ‰ b 2x Ê % x b x ˆˆ 42 cos2 $ x † (cos x) b cosdt # x $ c œ slope of tangent iscybthus, ‹ sin œ tan c œ c y b x x) x Šc x # 4 œ x ˆ sin 1)x by x=1 d "Î# ‰ dx y Èx‰ b 7 Ê Sw (x) 49. 2dxxˆtan ˆ2xbxy2x‹ b Š ˆ2x"Î#dy b cos b 0 (cos (7t) sin (7t)) ˆ2dy œ 3 c1 b cos#tep 1:# x ycos (7t)(csin (7t))(7)xœ 2y7s‰1†bdy(x)#1(7t)xd#0 d y œ yww œ cy cx œ cy ca"cy b œ c" # œc h œ † Šww # dy c$ y †dx ‰ b tan † c cince yw † ‹ œ Ê Ê œ y( b œ y y y y y dt 6 2 œ 1(x (7t)d œ x#y 4dx1xybcos xb b 2x sin% dx bdx a ac2 cosc$ xbdxcsin x)b b cosc# x en by yd c "Î# c 1) Ê a œsin x # c% 1 dy " x x 2c d 8 sin 2t c& d ( # "Î# dr ‰b 41. 2 a"Î# ‰(2 cos 2t)dy(2) x cosÈ% #b#tan a Èc ‰ œ $ dt (1 ˆ dy" x b c ˆ2 dy "Î# c& Ê tan y # b ) cos 2È a † œ dyb4(1#È cos 2x d œ x 2 dy Ê1‰s†) dx ˆs1x ‰ )#dyb50.65)œ d(1dy )Èsecc"ˆxxb) 2xfunctioncscdd)b )# Šx dycÈx‹sec‰ x2is b1Ê È(xy)Èxcos 2t)csc(csin c y† dt (2t) œ (1 b cos 2t) œˆ2x a and(csinˆxy ).4ˆ)#dy b bbcos secant‰w2xyacxbb†2xˆsin‰bx#2t) c#†yxc"Î# bccos1 ‰ c tan4(1#bc‰ cy # (xy) 2t) ) c c (xy) x cos b x b cos 2#È 2t) œ the minimum # in $ 2xyÊ Ê c dt œ c can have in c "  xÊ œ ˆ x csc# x‰c x œ ˆ# 28.smallestdx Ê 2(3xy dx 7)(3x) y bœ x y dec † Š3x Step3y‹ œ 61 valuex‰ the2:y œœ cot axy thedx œ y sin cˆ1 c 6 dy œ c6y(3xy b 7) dxx # c x œ xx dx x sin 3xy b 7) 4 b dx dx # dx b ) dx dx (cot csc b (2) "Î$ dy (2) of)(cos 2)and that # bcotb cot)dyww #Î$œ1sin )Êsec# csc1)))b 2# yc"Î$sec#)ˆc"Î# )Êdy„ 2cyc"Î$(xy) 1xc 2 "xccy"Î# Ê yw œ dy œ c x a) sin #) (csc w cot ) a œ 38. )3:)#Î$b csca )# 2 csc a1##b)2bx2xy b " ) cos (2 1x Ê#dy < y csc c$ "‰ Ê "Î$ 0. y y(3xy ) È c 4x œˆ 0 œ c ˆy‰ ; "1 ue b ybis 1 )) costep occursdyœxcb.6(cscb)œ3c‰("Î$‰cby32xcœ ‰dx)œcos‰a(cbdx ˆ11c<Èx‘ ˆ b ˆ‘ 3 x x œ ‰“ when3 ’œbc b cot ) # ˆ Ê 1 S (cscx Ê b))<x b#x2xy#7)(xy)‘ˆ)c3xy c <csc#$Î# b x ‘ Ê c$œ 3 œ sec‰ œ4 cœ c dx dy dx œ # y c # t c œ œ dy 7y 1 Chapter ccotcsc‰‰1 # x$Êc cyœw7x 2(xy)b cot4ˆ& t ‰‰ 2) † dcˆ" bb cot ˆ‘t ‰‰# œ c2 ˆ1 b cot ˆ t ‰‰c$ †yˆccsc# ˆ t x ‰ † d ˆ t ‰ b" 6] œ c6y(3xy b 7) 42. y œœ 1 b & b 7) c Ê # < dx dx x ˆ ˆ d" & 26. ydx dy sin xc x cos 1 Êy c c x 1d asinc #xb b sin x x1 ††csc (xy) ‰ cc3y d ab $ ˆxb c cos$ x † d ˆ x ‰ # ‰ dt # œ " x(3xy ˆ c x 3x y œ " x dx ˆ x Œ x cos Step 4: "#x "c" c2xy" y# 3 È "‰cdt †ˆc c" ˆ dxy b y ‰ˆc$x " ‰ " dt c" ˆˆc 3 yÈ# ‰‰c$ x<dx9" yc"Î# x ‰# È ‰ dx ‘3 œ ˆxc$Î# ˆ # ‰È t ‰ 2xy x y dr 19. " acyw (0) cagain, (0)œ a2bc& tangent ‰ c xœ“sin 2x x the c x) c cos$ x 3 ˆ È 1c œ dx # s b ' 2 c xˆ ‰ secx b tan " to‹ 3 a3‹# x 1 sin x c# x w ‰ sin 2x) Ê Šsec Ècosˆ2x # ’csc5ˆx in)œ‰x1cosywwbxb ŠsinÊ 3x) ˆc È)xyŠ aÈœ cos#at b (coriginbis yaœ 2x; b ˆ " ‰ 1 c x b 1 ‰c Ê d œ y œ 2 )‹Differentiating b tan ) œ secÊ œ œx ) œ dy xx x 3# ) 3 (x b 1) c (x c 1) ˆ t "‰ ‰ ˆ dy c" c y sec (xy) # œ (x 2 29. Êcos1ˆ x) y(x 'œ w c$ "Î$ c " cos#&(xy)c y y " tangent y $ sec# ˆ dy x c b cot"c sec#Î$ Ê c b "Î$0 c%Î$ Š cos x dy ‹ œ" cos sec ) " sec ˆ x tanÊ) œ bc"Î%Ê)bd"tan1‰inÊc1))!(tan xb1b""csincÈx tand ˆ œ b#xˆdx ˆ xÈ0toÊ œ3csin(xy) at the c1 c yis ‰ c"Î$ yw œ ctanœbdxy5œ(xy)b xy‰(0) )œ sec")) sec "x)œ 1 x " Êsec c‰ cx" c$ x xc " x ‰ dx œ origin sec (xy) Ê dx œ x sec (xy) c b csin 1) " (sec"Î# b# ) ) b $ 1) 25.tan ))x#È3 x (sec y bŠc)#)x dy y ‰ sœ sin ˆœ c Èxcos tan È3 tan # bb ‹ # 3xdy b 3y x #‰ Š # dyÈ #‹ c(x) b œc Ê 3y " (sec È x #) ) ‰ b b t20.ˆ " ‰ b y$ b ) dx ) sec )# œ #tan Ê 3xsec È)#‹#” œ 18y b 18x #x 1 a• 3 c 18xb # œ 18y c 3x# Ê dy œ 6y c x ˆ ) " È an ) x x x 18xy ) œ Š b 3y dx dx y dx y c 6x y c ) c # x.# The tangents are perpendicular to each other at È) ccos (xy) c ) the d product ofdx their slopes is œ sec 1(xy) c y œ ccos (xy) c x# the origin since œ d x 43. y .œx sin acos (2t c 5)b x # dy c" cos (cosx #c 5)) † dt cos w(2t cc 3x œ 2xy (cos (2t c 5))c#(csin (2t c 5)) † dt (2t c 5) Ê dt œ w (2t 5) c cos † 29b x#"yw œ 1 c yw Ê ax b 1b y œ 1dy 3x7 c 2xy Ê yd œ 1 ( # 1 .œ x c y Ê 3x r b 2xy œ " r (3x c 2) b ˆ4 c w " ‰ " Ê # cœ (3x c 2)' " (3x c 2) 1b (c1) ˆ4 cˆ "(x‰ ‰ Š x yd ‹ˆ4 c " ‰ d cos dr 27. y 51. ) # œ # , y: c csc cb xc 21 #x s (r)) Ê d)21. wc cos (r)))y#œcx)cot2x Ê(2t cw 5))(sinb 29c 5))dxÈ c ‰1)(3) œb 1 ; then yww x cy1) (xÊ1)y wwœ y cx 3 1) † dx(3x c 1) † dx csc (3x c 1)) œ 39.(r yyœ 21 cos (cos 1) Ê y #œ (2t csc d (3x 2x b 2 œ x c † dx # (3x b cx b y œ # 2 (csc w 2 (3x œ c œc dq (mx) # t y 2xy bcos œx bÊ cw (0)2yy œ 2x0 b 1 (1)ʆ dt # œ b 12y x ‰ Ê y3 œ c " dy d y y sin† d ax Êb x)‹d†mcoscbx9 2‹ ycosy #'ŠÊ td 1(x) † Èt œ m; ty cy b"x‰ dy mÊd (cosw' c x)m cos ˆy#c 1 Ê dy œ yy c 1 œ30.35. #t 7sin œ cos x † œ m‹ (cos y) dy" œyˆ ˆt sin ˆ (mx) œ xy cos ˆ % œ cos Š " œ m‰ bdy yc ‰ Section 3.6 Implicitdx csc (3x c 1) dx ndt x dx b t b 1 dx a Šœ tcb(bdyc 2) cÈtdx cc1)(3x cˆdxxt b 1‰ (3x cœ (3x c 2) c 1)) œ Differentiation (3x163x È dt xÈ b 1 3x (3x † 3 b1(csc 4 b y 1) 1)( " 1) 1) 2 cot coscc 21y # " " S (0)# œc d 2x cscww 2yy‹c 3 b 1) 1 Èdycot x dx † dx (3x " # yw (0) œ c dy cos tep 1:84 Š#.3 œChapterb 2y‰ dx œ 1, b dx x Š4 c ‹ y b m (x Differentiation ybd dy x dx y ‰ œ sec44.xÊ.œ ySinceœ ˆ ‰ Ê dy œ the ˆ5 sin t ‰‰ d 5 sin ˆ t ‰‰ œ at the origin. m m † c 2dxtt Êx)dx b coscty œ ycos cos dx5t sin ˆ†t ‰c tm 2 œ c csintangentˆlines† areˆperpendicularcsin ˆ5 sin ˆ t ‰‰ ˆ5 cos ˆ t ‰‰ † d ˆ t ‰ 1 dx (cos ˆ 39 2(t b 1) c 1 t b dt 3 3 3 dt 3 ‹ ‹ Ê c w œ tm ‹ Š 2(t 5 xx " œ 3Š t value‹ cos Š and "dy 3 is #dt y cos tsin x)b $ œ cos Š 2x cos (mx).1) The largest b¹ b1 b dr dy œ 2y c(mx)bb csc2 xÈ31.1 ÊœcŠtaninœc1sin xy wdyœˆ9y ’cos(mx)‰can‰attainsec1“at ‰sin Š yÊœthe†œ cx ˆ value Êx ‰ tan ˆ x ‰‰ ˆ " ‰ œ 2 sec# ˆ x ‰ tan ˆ x ‰ sin œ r œ c# ) Ê yœœ b ‹ˆ b5y c ˆ2(t‰Ê cos œ 3 ˆ x"Èt † 1dx ¹ " † dy 1 x x œ 0 ‹ † dy largest x c y ˆ ˆ ,  ­y y1‰1) x b3 ‰Š3 ‹ ˆ 3 œ y dx 3 Ê1œ 1) w ˆ ‰ 3 c )] 4yb cos # c52.45.sin9c"rsb#csc‰))1;Ê "3dxc( "2 ˆsect#c1y‰dyb (c1) 23 1) w œ ˆ b% Ê " ww 4 dx w 2 sec" dy ‰ 2 sec ˆ 3 c" 3 c% ww 3 " 2 c#$ w 3 dx œ sin y3 % 60. w y km cos $ , 2xr b c 3 d ˆ x$ ˆ 3 ˆ an attain is y mx Ÿ k w 1 c . dy 40. y#k)‹k œ3yx 2x)c(mx)8 ˆ 2 m ‰ c#2 c (xy)(†c y km c (c2) yy 8 ˆ 2 œ ‰ ‰dx x ‰ œ 1) ; 2x) 2y#dy b œ c‰ ˆ x 1) ‰ c x osy # xccsc#kmk becausey yœ 2x œ dy c 2ycsck# œ kb k1kcosÊœkdx #œmk Êdyœ(2ykb Also, 2 œ csin 1 mb 1cœ (ycy 6(5 c then y c x (y bb 1 † y œ (xy)Šx 28.b d c (5 c dx b x bÊ(xy) † y œ cy csc 2y c3(5y 2x) 2) œcb Ê xyb y yÊ1 dx 57. w 2 " c" dx y "x "c xd " x¸ t y  ­ œ t ¸" . % dt yw œ c "ˆ cos‰ m ‰ Êœk‰ ’c y¸ csc##ˆ 2‹ m1‰sin cosmt‹ b x“ˆœ‰cy kmk dx œ # Ÿ" value w can œ ccsc#m sin t yˆ† dt ˆ sin dxck(dyˆbyc<oscossinb‰1)ˆ t"Š Ê ¸sin y ‰œmyww œ Ê%c" Ê wthe largestb sin Š ‹y& x œ attain wisc cos¸Š ‹ b xy œ c m cŠŠ ˆbb‹¸$c ¸ y "¸ cos y # 1) (y w ‘‰ Ÿ cos " sin Š (u)m # 27 dy (1) < w (g(1)) ‰ wd œ 1t# y dx cc tanmx b# ‘ x 8 y bt dy Ê < dy ‹ œ ; 1t# c† œ Š ‹f(u) œ (< 2x) xÊ 2xd <csc#y)td bdx53. wtbg(x)(5œŠÈcsc$ʈ t1ta"‰# b Ê(cx dx # 2y b œ 1 and g‘(1) m<1 bdy;‹ œ ub ): 1 yÊ cy ‹ œ 5u c now todfind # at("ßd‘a)yand$œœc63b2):‹(xy) œxcœœ;##yÈdt2x b ):1m œ (y8b2(x‰c): ydt 1œy; 27tan% ߈c# ‰b œ 3 <1 fb8tan% ˆy1t# ‰‘Ê 4ftan$ ˆ 1t#œ †f dt tan ˆ5; ‰‘ ,x d 45. 61.("ß 1c1)xc %w (xy)‘(x)27c x Š$ßœ 3 g(1) tan ˆ1)(3ß # y) d #c 8 (3 y 1t # ‘ m œ 27 œ c xß " b csc g œ & œ (x c " dx w "! Ê dx (xy) b cy Š y c dx dx sin (mx)ˆ x ‰ c 3œ m 61. (mx) <Ê y (0) œ m Ê slope of curve at"the origin is m. Also, sin (mx) completes cos x † dx Ê y dx acos xb c cos x † wdx ˆ 3 ‰ w # # 5 w % #ˆ %ˆ t $t #t herefore, (f g) t ‘curve $ ˆ t dy (mx) t d " œ œ1 dy eriodsyona# x) b c 2) † d tœ c 2 t 1x 2)tan 2ˆ 1the(1 f wtan # 1# ‰ sec œ 15a‰ †b"Î#1b c<dy b tan dy 1#"‰‘ <tandy ˆ ‰ sec ˆ 1# ‰‘ [0 b Therefore < b ‰ y (1)‰œ < (g(1)) † g (1) c † c1œ‘ " (x c1 xs# xx# csin cy 21].ac" $ xsin "2x cslope # b sin(x c y) y †œ sin1t c2xat †the#origin y) the same asœ cnumber Èy t 2yœb cbcinb(ß b (x c y)dtȈ 3 1the" # bœ of b "Î# t c 2)Êcosw ( œ "2y2)x bdty(bb#(x c‰ is Ê d ¹dy the 1 1#" ( y" 1t œ 41. 2 b y ( ‰ x c y %Ê y# c$y œ 1 c y Ê yw#ˆ% # y b b 1 2) c1 Ê cos d # œ a dta œ 2 s dx (1 0) yw œ 32. 62. cos Š bw k3) (x b c2y Êk y dyœ dx(4x b 3) (c1) ydy† dxœ "%c†y secŠdxb$1) 54dx3œ 2bb$2 dx yÊ1 ;3)$ †can(4x b 3) y œ (a) y ‹ œ 2x b1) dy yÊ ’csin Š y ‹ a†x y(c3)(x by) “ b cos (xy Š † 2y œ y b 1)‹ œ dy bb we d ‹ È b ß(x $ 4 c (4)(4x 5 4 w y y È È 29. y (4x c (b)(xy) periods 2) completes on‹ œ20 ]. Inx(4 2) œ(xy) forœ (2 4)1 m,5we can think of “compressing" the graph of 1 dx b(ost$c (xy)d Šy b x dy [0ß 1 Êy particular, dx large c y sec# (xy) Ê dx œ1) x secdx(xy) csec# it sec# 4 and cdx œ sc 1 x dx " " " " wc c# w $ dy gives "c%x)c" œb 1) ifferentiate cos2y $Ê$ completed 3b11b cos# œ dalso3)$ (4) ww c3(4x" bc$Î# the (cc% b 4b 1(7t)ww # (cos u c$ y œ ’ in (1b $( equation"yw% (1) "Î# (x [01)c$], again † x) 2 y œ yw sin # y # andw b 1c"Î# ; ‰ b 1 c ( " 1 ‹ b (7t) b 1)‹ dyc “ on ‰ß 1 (7t) # (1 cos (7t)( ˆ1) œ 3)% œ sinœ 1 x horizontally 54. dg(x)sœcby3)theperiods ŠÊ ˆdt 2c(1 cc Êœdycbutœ b2find Ê: g(the slope of(x ‰ c1)cy b cos#f(u) œœ 0 (7t) sin (7t)) which œ4x Šmorec#3)(x d g y(x)c œ œ 2bx)21((4)(4x tocincreases cc (7t))(7) y g b7ˆ1) œ 16(4x y d 3) (x b 1) 46. y 6 È "Î$ cos ‹ È54 6 (1 1) dx dy 2 (xy) "Î$ y œ ccos (xy) c y dx (4x b 3) 3ˆ y ‰4ß x (c) Š cc w sin Š y7) b 2y cos Š y ‹ c # ‹ " " wœ w w w x xÊ yd œ dx d 3ph at the origin. x œ cÊ f (u)ccc3(4x ; f 3) b 16(x b 1)d œœ 4;#therefore, (f ‰ g) (c1) œ f (g(c1))g (c1) œ 4 † y œ sec1)cœ x Ê b w (g(c1)) œ f w ˆ " ‰1 (4x b 3) (4x b Œ # 4 œ1 ' (2 d 9y b tan 1 † ysec 1t) † 2) b (c1) ˆ4 cˆ(x"b"Î# t)(secˆ1t c #"1‰w † dt (1t) œ#2d ysec(x1t1) 1ty (3x c dt (sec 1t) œ (2 #x ‰ 1 † ud‰ 4 tan"x t) # c$Î# " " c ww ww dx dx Êy b 1 y œ # cy d y Ê dx œ y œ œ œˆ 2x Œc y# œb y Ê3 x b y b xyw c 2yyw œ 0 Ê (x c 2y)yw œ c2x c y Ê yw œ ay b yb 1b 2y ay b 1b c x 9 1 ˆ 2x ‰ dy "Î# ; # 1 b È y‰ È dy 1 " 64. " 2)'t,dy y) dy "œ y 0 Ÿ33. 63. (cos "Î#c x) dx #œ "y)c"Î#5Êx"dyc"Î# cosdrc xt),cyÊw sin#2y1È “ œ0 Ÿ34t Ÿˆ1 udr # c 2È r œ c È r# ˆ 1u ‰ ˆ 1 ‰ c œ dx 2t, b x tdx Ê 0 r y œc1 dy w c 168. dx cos†(y cc 0 " œ dr ’ c5x t), c1 Ê 1 ‰ œ w È (cos )œ r g(1) d) 5 2x y x Šsin ‹ 55. c& d w5Èx Ê Ê (x) œ ' b # œ œc& 1y œand g (1) œ# c f(u) #È)cot d) Ê f2(u) œ ccsc Ÿ g(x) b 1 "7 g # d Êline (2xyc 5)2t)(6)daxccœr; Ê8(2xœ 75) b a" c 5xb')(c1)(2x ) 5)c# (2) œsin3 œ †7 (x) ( 2)5xb& sinœc x c x 2t 4 4 tangent 2t) œ (2x b5) 2t) œ the tangent œ 30. # dt (1 (2 œ a c 5xb # b dx Ê c 10 10 œ cc(1 b cos lineym†œ#y kc 3)cos 4 xÊ c4(1Èx cos 2t) is(c c (2t) # (1 b cos 2t) 4 10 # slope of the x y # dt 2t b sin# 2t œ 1 Ê [email protected] w w x bc1# œ 1 1u y Ê cos# (1 c t) b 4 dt # (1 c t) œ 1 sin cy d 1 1 #ˆ & w f w (5) œ c w csc# ˆ 1 ‰ œ c 1 ; therefore, (f ‰ g)w (1) œw f w (g(1))gw (1) y c ww † 5 w w œ 10 c 2) Ê cœ ac c 5x b œ normal line is y c42.œxy b6cxcsc1 5xb ‰y Êbf y(g(1)) 29Ê 0 #Ê y# 10b 2yy # ! cy Ê66. (x b 2y) œ cy Ê y œ (xb2y) ; dxœœ y10 # 3 c 4 (x 6# c Ê xy 2 (2x c b b2yy7 œ x b xyœ 1, y œ 10 y c4 œ y œ " 10 dy x 4 5) dy a 7x dy 7 " " 65. 1 y b 4 2 ( 1‰$ y 2 c b “c ‹ % œ " 2c y $ d (Ê2)dy Š yˆ‹ 2bc1) cot†ˆ=t ‰cœbsin c12x)cotc tˆcc‰ ˆ cb’1‰Ê cotyˆ’1 ‰‰c$(x† ˆccsc# ˆ(xt ‰2y) †y(x ˆ t2y) b y(x b 2y) c 2y d c’cos œ c † ˆ1 b ˆc xdx‰‰(x4$6(5d Š yy(1 †cdx )ˆ ‰x xœb 2y) ˆ(x b “ b t b 2 Š y2y) ‹“ ‰ c dt b# ‰ † dt ˆb bb 2y # ‰ (xdxx 2 1 y2y) 2y)y dt 8 x # # xœ œ œ# y c (x b 1)wŠ yy 1 ‹ 2y) (x b c (x b 1)y 2x b 2yy d 0 Ê yw(x b cy x ; cy(x b 2y) " 25 Ê œ ‹ b x“œ cy Ê 31. œ (a) xFalse2Èx‰ b œ Ê hw (x) œ x d ˆtan ˆ2x"Î# ‰‰ b tan(b) "Î# ‰ †2y) (x) b 0 œ 67. œ œ c y œ ˆ2xTrue d h(x) tan ˆ 7 ...
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This note was uploaded on 12/26/2010 for the course MA 1012 taught by Professor Franciscoperez during the Fall '10 term at ITESM.

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