Functions, Limit, And Continuity

Transcription

1 Fucios, Limi d coiuiy of fucio Fucios, Limi, Ad Coiuiy. Defiiio of fucio A fucio is rule of correspodece h ssocies wih ech ojec i oe se clled f from secod se. The se of ll vlues so oied is he domi, sigle vlue clled he rge of he fucio. f Domi Rge For rel fucio f we c defie s follow f : y f c e clled he idepede vrile d y he depede vrile. The domi of he fucio f, commoly deoed y D is defied y {, such h } Df y y f Emple:. f is defied for D f f is defied for d.. f. Hece, { } f is defied for. Composiio of Fucios f f ( ) g( f ) g f g. Hece, D (, ] [, + ) If f wors o o produce f ( ) d g wors o f o produce f g f, we sy h we hve composed g wih f. The resulig fucio, clled he composiio of g wih f, is deoed y g f. Thus, g f g f Emple: Give he fucio f ( ), g. Fid g f d f g

2 Fucios, Limi d coiuiy of fucio Soluio f g f ( g ) p + 5 s composie fucio g f ( g f ) g f Emple: Wrie he fucio 5 Soluio: The mos ovious wy o decompose p is o wrie p g f f + 5, where g, d. Iverse Fucios Iverse Fucio Le f e fucio wih domi D d rge R. The he fucio domi R d rge D is he iverse of f if f ( f ) for ll i D d f f y y for ll y i R. ( ) Emple: Le f. Fid Soluio: To fid filly solve for y. f, le y f f if i eiss. f wih, he ierchge he d y vriles, d y, he y, implyig y ( ) +, hece f ( + ) Crieri For Eisece of A Iverse f A fucio f will hve iverse f o he iervl I whe here is ecly oe umer i he domi ssocied wih ech umer i he rge. Th is, f eiss if f ( ) d f ( ) re equl oly whe. A fucio wih his propery is sid o e oe-o-oe fucio. Horizol Lie Tes A fucio f hs iverse iff o horizol lie iersecs he grph of y f more h oe poi. A fucio is clled o e sricly moooic o he iervl I if i is sricly icresig or sricly decresig o h iervl. Sricly icresig o I: For, Isuch h < f < f Sricly decresig o I: For, Isuch h < f ( ) > f ( )

3 Fucios, Limi d coiuiy of fucio Theorem Le f e fucio h is sricly moooic o iervl I. The is moooic o I. Grph of f If f eiss, is grph my e oied y reflecig he grph of f i he lie y.. Iverse Trigoomeric Fucios Iverse Sie Fucio π π y si si y d y The fucio si is someimes wrie s rcsi. Iverse Tge Fucio π π y d < < The fucio is someimes wrie s rc Defiiio of Iverse Trigoomeric Fucio Iverse Fucio Domi Rge y si π π y y cos y π y < <+ π π < y < y csc or π π y, y π y sec or y π, y y co < <+ < y < π Emple: Evlue he give fucio f eiss d. si Soluio:. si π. cos c.. cos π π c. 6

4 Fucios, Limi d coiuiy of fucio Iverse Trigoomeric Ideiies Iversio Formuls si si for y y π π y for ll si si for π π ( y) y for < y< Emple: Evlue he give fucios. si ( si.5). si ( si.5) Soluio:. si ( si.5).5 ecuse.5 π π. si ( si.5).5, ecuse.5 Emple: For, show h. si si. cos( si ) Some oher Ideiies π si + cos π + co π sec + csc 5. Hyperolic Fucios d Their Iverses 5. Defiiio The hyperolic sie d hyperolic cosie fucio, deoed respecively y sih d cosh, re defied y e e e + e sih d cosh The oher hyperolic fucio, hyperolic ge, hyperolic coge, hyperolic sec d hyperolic cosec re defied i erms of sih d cosh s follows sih e e cosh e + e h coh cosh e + e sih e e sec h csc h cosh e + e sih e e 5. Hyperolic Ideiies /. cosh sih /. h sech /. coh csc h

5 Fucios, Limi d coiuiy of fucio /. sih + y sih cosh y+ cosh sih y /. cosh + y cosh cosh y+ sih sih y 5/. cosh + sih e 5/. cosh sih e 6/.sih sih cosh 6/. cosh cosh + sih 7/. cosh sih + 7/. cosh cosh 8/. cosh cosh 8/. sih 9/. 9/. sih sih y sih cosh y cosh sih y cosh y cosh cosh y sih sih y 5. Iverse Hyperolic Fucios The Hyperolic iverses h re impor d o e sudied here re he iverse hyperolic si e, he iverse hyperolic cosie, d iverse hyperolic ge. These fucios re y sih (or y Arcsih ), y cosh (or y Arccosh ) d y h (or y Arch ) re he iverses of y sih, y cosh d y h respecively. Theorem ih l ( ) ii/. cosh l ( ) i/. s + + ( for y rel umer) + ( ) + iii/. h l 6 Limis Defiiio: To sy h lim correspodig f c L δ > such h < c < δ f L ( < < ) mes h for ech give ε > ( o mer how smll) here is < ε. Emple: /.Prove h ( ) f L < ε provided h < c < δ ; h is lim 7 5 Righ-hd Limi d Lef-Hd Limi By lim /. lim 5 f Awe me h f is defied i some ope iervl (, ) c d f proches A s pproches hrough vlues less h, h is, s pproches 5

6 Fucios, Limi d coiuiy of fucio from he lef. Similrly, lim f A mes h f is defied i some ope iervl + ( d, ) d f pproches A s pproches from he righ. If f is defied i ier vl o he lef of d i iervl o he righ of, he lim f Aiff lim f Ad lim f A + Limi Theorems Le e posiive ieger, e cos, d f d g e fucios h hve limis c. The /. lim c /. lim /. lim f lim f c /. lim f c c ± g lim f ± lim g c c 5/. lim f g lim f c lim c ( c ) g f lim f c 6/. lim,lim g( ) c g lim g c c 7/. lim f lim f c c if defied. 7. Coiuiy of Fucios Coiuiy Poi Le f e defied o ope iervl coiig c. We sy h f is coiuous c if lim f f ( c ). c si, Emple: f ( ) A he poi, f is defied d f ( ) si lim f ( ) lim lim f f. Thus f is coiuous he poi We see h Emple: Show h f is discoiuous he poi +, > f +, <, A he poi f he fucio is defied, h is 6

7 Fucios, Limi d coiuiy of fucio lim f lim lim f lim + We see h lim f lim f. The lim f f + He ce f is discoiuous he poi Defiiio Coiuiy o Iervl The fucio f is righ coiuous if lim f if lim f f. + f d lef coiuous We sy f is coiuous o ope iervl if i is coiuous ech poi of h iervl. I is coiuous o he clos ed iervl [, ] if i is coiuous o(, ), righ coiuous, d lef coiuous. E mple: Show h f 9 is coiuous o he closed iervl [,] Soluio: We see h he dom i of (,)we hve f ( c) lim f lim 9 9 c c c So f is coiuous o,. Also d lim lim 9 f ( ) f lim f lim 9 f ( ) + + So f is coiuous o [,]. Eercises Give ϕ + 5, deermie ϕ. f is he iervl[,]. For c i he iervl f ( α ) If f ( α ) ( α ), verify h f ( α ). f ( α ) f l d ϕ, deermie ( f ϕ ), ( f ϕ )( ) Give d ( ϕ f )( ). Fid he domi of he followig fucios. y d. y l. f c. y l + ( l ) e. y rcsi ( 5) f. l( ) y g. y si 7

8 Fucios, Limi d coiuiy of fucio 5 f +. f 5 If f ( ), show h. f ( ) f ( ) f 6 If f, show h + d f f ( ) 7 If f, he show h f ( ) f ( ) f f ( + h) f ( ) 8 Compue i he followig cses: h. f whe d + h f whe d + h. c. f 9 Prove h whe d + h - + (. sih l + + ), +. h l, ( < < ) Prove h ( ).. si cos f f cos si ( + ) ( + ) 5 sec( ) + c. d. si cos 9 e. ( si ) f. si ( ) + ( ) g. cos si h. ( ) y Prove h y + + y if π π < + y< d use he fc o prove h π. +. π + 7 Compue cos si + cos,si si + cos Prove h f + is coiuous ( ) Prove h f is coiuous. 5 Ivesige he coiuiy of ech of he followig fucios h e idiced pois: f ( ) 8

9 Fucios, Limi d coiuiy of fucio si,. f he poi,. f he poi 8, c. f he poi, 6 Fid vlue for he cos, if possile, h will me he fucio coiuous 7,. f, >., f +, > s: Fid he pois of discoiuiy, if y, of he fucio +, f,< <, s: discoiuous 8 If he fucio 6, f c, is coiuous, wh is he vlue of c? s: 8 9 For wh vlue o f is he followig coiuous fucio? ,if d f if s: 8 f such h Le, < f c+ d, + 8, > Deermie c d d so h s: d, c f is coiuous (everywhere). 9

10 Fucios, Limi d coiuiy of fucio Deermie if he followig fucio is coiuous. f 5,, Deermie if he followig fucio is coiuous -. f +, 6, > Deermie if he followig fucio is coiuous 6, < f, + >, +. Deermie if he fucio h is coiuous For wh vlues of is he fucio f coiuous?

11 Differeiio Differeiio Defiiio A fucio f is sid o e differeile if d oly if f ( + h) f lim h h eiss. If his limi eiss, i is clled he derivive of f d is deoed y f f ( + h) f f lim h h Emple f, f? Soluio f + h f + h + h+ h + h h h h The f ( + h) f f lim lim( + h) h h h Emple Fid f ( ) if f Soluio f i geerl We c firs fid f + h f + h f lim lim h h h h h h lim lim( h) h h h d he susiue for f ( ) ( ) We c lso evlue f ( ) more direcly ( + h) ( ). Hece, f + h f h h f ( ) lim lim lim h h h h h h +, Emple Fid f ( ) if f +, < < Emple Fid he derivive of f 9 The process of fidig derivive is clled differeiio. I he cse where he idepede vrile is i is deoed y he symol d f d f wih respec o red he derivive of

12 Differeiio d f f d dy If he depede vrile y f, he we wrie f d Rules for Differeiig Fucios Assume h u, v, d w re differeile fucios of d h c d m re coss d ( c ) (The derivive of cos is zero) d d du ( cu) c d d d m m ( ) m (Power Rule) d d du dv ( u± v± w) ± ± dw (sum/differece rule) d d d d d du 5 ( uv) v + u dv (Produc Rule) d d d du dv v u d u 6 d d (Quoie Rule) d v v du d 7 d, u (Reciprocl Rule) d u u The Chi Rule If we ow he derivives of f d g, how c we use his iformio o fid he derivive of he composiio f g? The ey o solvig his prolem is o iroduce depede vriles y f g f g d u g ) ( f ( u) So h y. We use he uow derivives dy du f ( u) d g du d o fid he uow derivive dy d f ( g ) d d Theorem (The Chi Rule) If g is differeile he poi d f is differeile he poi g he he composiio f g is differeile he poi. Moreover, if y f ( g ) d u g he y f ( u) d dy dy du d du d Emple Fid dy y cos d if

13 Differeiio Soluio Le u so h y cosu, he y chi rule dy dy du d [ cosu] d ( siu) ( ) si d du d du d I geerl, if ( ) f g is composie fucio i which he iside fucio g d he ouside fucio f re differeile, he d f ( g ) f ( g ) g d + Emple Fid he derivive of Soluio By usig he chi rule, we oi d + + d + d d d + Le clcule d d d ( ) ( + ) ( + ) ( d ) + d d ( ) ( + ) 5 d ( ) Hece d + + d ( + ) 5 d d ( ) ( ) Derivives of Trigoomeric d Hyperolic Fucios d d ( si ) cos 5 ( sec ) sec d d d d ( cos ) si 6 ( csc ) csc co d d d d ( ) sec 7 ( cosh ) sih d d d d ( co ) csc 8 ( sih ) cosh d d si cos N.B: co sec cos si cos d csc si Proof sih cosh Recll h lim d lim h h h h From he defiiio of derivive, sec: sec d csc: cosec

14 Differeiio [ ] d si + h si si cos h+ cos si hsi si lim lim d h h h h cosh si h si h cos h lim si cos lim cos si h + h h h h h Sice lim( si ) si d lim( cos ) h h cos, d si h cos h [ si ] cos lim si lim cos () si ( ) cos d h h h h Thus, we hve show h d [ si ] cos d The derivive of cos is oied similrly: d cos( + h) cos cos cos hsi si hcos [ cos ] lim lim d h h h h cosh si h lim cos si h h h cosh sih cos lim si lim h h h h cos si si Thus, we hve show h d [ cos ] si d Emple Fid f if f Soluio d d f [ ] + d d sec + si Emple Fid dy d if y + cos Soluio d d dy d d + cos cos si si d + cos + cos ( + cos) [ si] si [ + cos] cos + cos + si cos + + cos ( + cos) ( + cos) ( ) 5 Derivives of Fucios o Represeed Eplicily 5- Implici differeiio Cosider he equio y. Oe wy o oi dy d is o wrie he equio s from which i follows h dy d d d y

15 Differeiio Aoher wy is o differeie oh sides d d ( y) () d d d d ( y) + y d d dy + y d dy y d dy Sice y, y d This secod mehod of oiig derivives is clled implici differeiio. Emple By implici differeiio fid dy d if 5y + si y Soluio Differeiig oh sides wih respec o d reig d reig y s differeile fucio of, we oi. d d ( 5y + si y) ( ) d d d d 5 ( y ) + ( siy) d d dy dy 5 y + ( cosy) d d dy dy y + cos y d d dy d y + cos y Emple Fid dy d if 7 + y+ Emple Fid d y d if y 9 5- Derivive of he Iverse Fucios dy Le y f e fucio whose iverse is f ( y). The d d dy Emple Fid he derivive of y rcsi. Soluio d d We hve y rcsi si y d hece ( si y) cos y. The dy dy dy d ( rcsi ) d d d cos y cos ( rcsi ) dy Emple Fid he derivive of y rccos d y rc. 5- Derivives of fucios Represeed Prmericlly If fucio y is reled o vrile y mes of prmeer 5

16 Differeiio The or, i oher oio, Emple Fid dy d if cos y si Soluio d dy We fid si, cos. Hece d d Emple Fid dy d if y ϕ y ψ dy d ( ) () dy d d d y y dy cos co. d si 6 Logrihmic Differeiio Tig he derivives of some compliced fucios c e simplified y usig logrihms. This is clled logrihmic differeiio. 5 Emple Differeie he fucio y + Soluio Tig logrihms of oh sides we oi 5 l y l + 5 l l l l y + Differeie oh sides wih respec o o ge y 5 ( + ) 5 + y y y + Sovig for y 5 y y ( ) + + We c lso use logrihmic differeiio o differeie fucios i he form v y u Emple Differeie y, y, y where is cos. 6

17 Differeiio 7 Higher Order Derivives 7- Defiiio of Higher Order derivives A derivive of he secod order, or he secod derivive, of fucio derivive of is derivive; h is The secod derivive my e deoed s y, or y y d y d, or f Geerlly, he h derivive of fucio y f (-). For he h derivive we use he oio ( ) d y y, or y f is he is he derivive of he derivive of order d, or ( f ) ( ) Emple Fid he secod derivive of he fucio y l ( ) Soluio y, y 7- Higher-Order Derivives of fucios represeed Prmericlly If ϕ ( ) y ψ () dy d y he he derivive y, y,... c successively e clculed y he formuls d d y ( y ) y, y ( y ) d so forh. For he secod derivive we hve he formul y y y ( ) cos Emple Fid y if. Aswer: y si si. 8 Differeil 8- Firs-Order Differeil y f The differeil of fucio is he pricipl pr of i icreme, which is lier relive o he icreme Δ d of he idepede vrile. The differeil of fucio is equl o he produc of i derivive y he differeil of he idepede vrile dy y d whece dy y d 7

18 Differeiio 8- Properies of Differeil dc, c is cos d cu cdu d( u± v) du± dv d ( uv) udv + vdu u vdu udv d v v df u f u du 5 6 v 8- Approimio y Differeil For he fucio y f, y dy; h is ( +Δ ) Δ+ f f f 8- Higher-Order Differeil If y f d is he idepede vrile, he d y y d d y y d + Δ Δ whece Δ f ( ) f f ( d y y ) ( d) 9 Theorems Relive o Derivive 9- Rolle s Theorem If f is coiuous o he iervl [, ], differeile every ierior poi of he iervl d f ( ) f ( ), he here eis les poi ξ, < ξ < where f ( ξ ). Proof If f is coiuous o he iervl [, ], he i is o he iervl relive mimum vlue M d miimum vlue m. If mm, he f is cos, sy, f m, implyig h f ( ξ ). If m M, we suppose h M > d f is he mimum vlue M ξ, h is f ( ξ ) M, ξ,. If f ( ξ ) is he upper oud of f, he f ( ξ + h) f ( ξ) d herefore, f ( ξ + h) f ( ξ) f ( ξ + h) f ( ξ), h > lim h h h f ( ξ + h) f ( ξ) f ( ξ + h) f ( ξ), h < lim h h h Hece f ξ. Emple Cosider he fucio f ( ) si The fucio is oh coiuous d differeile everywhere, hece i is coiuous o [ ] Moreover f si, f π siπ, π d differeile o (, ). so h f sisfies he hypoheses of Rolle s heorem o he iervl [, π ]. Sice f ( c ) cosc ). Rolle s heorem gurees h here is les oe poi i (, π such h cos c 8

19 Differeiio which yields wo vlues for c, mely c π d c π Emple Verify h he hypoheses of Rolle s heorem is sisfied o he give iervl d fid ll vlues of c h sisfy he coclusio of he heorem f, [, ] Soluio,, O he iervl [ ] f is coiuous d i is differeile o + f ( ) ( ) ( ) f ξ ξ ξ + which hs he roos ξ, ξ + Hece ξ sisfies he heorem. 9- Me-Vlue Theorem Le f e differeile o ( ), d coiuous o[, ]. The here is les oe poi ξ i (, )such h f ( ) f ( ) f ( ξ )( ). Proof f ( ) f ( ) The slope of g is Q f ( ) sice g( ) psses hrough he poi (, f ( )), he he equio of he lie is defied y f g f ( ) Q( ) ( ) f g ( ) he g f ( ) + Q( ). Le F( ) which is coiuous o[, ] F( ). By Rolle's Theorem, ξ (, ) such h F ( ξ ). f ( ) f ( ) F f f ( ) f F f g f f + Q f f Hece we oi he fucio, differeile o ( ), d F f ( ) f ( ) F ( ξ) f ( ξ) f ( ) f ( ) he f ( ξ ). Hece f ( ) f ( ) f ( ξ )( ) Emple Le f +. Show h f is sisfies he hypoheses of he Me-Vlue Theorem o he iervl[, ] d fid ll vlues of ξ i his iervl whose eisece is gureed y he heorem. 9

20 Differeiio Soluio f is polyomil, f is coiuous d differeile everywhere, hece is Becuse coiuous o [, ] d differeile o (, ) Theorem re sisfied wih d. Bu f f f f 9,, f ( c) c f. Thus, he hypoheses of he Me-Vlue so h he equio f ( ) f ( ) f ( ξ ) ξ 7 which hs wo soluios ξ 7 d ξ 7 So ξ 7is he umer whose eisece is gureed y he Me-Vlue Theorem. 9- Cuchy's Theorem Le f d g e coiuous d differeile fucio over he iervl [, ] d g over[, ]. The here eiss ierior poi ξ o he iervl [, ] such h f ( ) f ( ) f ( ξ ) g( ) g( ) g ( ξ ) Proof f ( ) f ( ) Le defie Q y Q g ( ) g ( ) Noice h g( ) g( ) sice if o, g( ) g( ) he y Rolle's Theorem, g poi ierior o[, ]. I cordics o he codiio of he heorem. Le form fucio F f f ( ) Q g g( ), which sisfies he codiio of he Rolle's Theorem, he here eiss umer ξ, < ξ <, such h F ( ξ ). Sice f ( ξ ) F f Qg, he F ( ξ) f ( ξ) Qg ( ξ) Q. Hece g ( ξ ) f ( ) f ( ) f ( ξ ) g( ) g( ) g ( ξ ) 9- L' Hopil's Rule Cosider he fucio F f g, where oh f ( ) d g( ) whe. The, for y > here eiss vlue ξ, < ξ < such h f f ( ) f ( ξ ) g g( ) g ( ξ ) f f ( ξ ) or g g ( ξ ) Now s, ξ, herefore whe he limi eiss f f lim lim g ξ g This resul is ow s l' Hopil's Rule d is usully wrie s

21 Differeiio f f lim lim g g cos Emple Evlue lim L'Hopil's Rule c sill e pplied i cses where f d g whe, simply y wriig f f lim lim g g Now f d g s d he rule pplies. Therefore, f g g f L lim lim lim g f g f f g g lim L lim g f f Hece f f lim lim g g f d g oh ed o zero, or oh ed o ifiiy s ed o ifiiy Similrly, if he rule pplies. By wriig u ( ) f f u lim lim lim f g ( u) g u g u u u u u f lim{ f ( u) g ( u) } lim u g If, fer oe pplicio of l' Hopil's rule he limi is sill ideermie, he process c e repeed uil deermie form is reched. Emple Evlue si (i) lim (s: ) (ii) lim e (s: ) Tylor's Theorem for Fucios of Oe Vrile Suppose h he fucio y f hs ( + )h order derivive i he eighorhood of he poi. We will fid he polyomil of order mos such h P f, P f, P f,, P f The sough-for polyomil is of he form P C + C + C + C + + C P Le clcule he h derivive of P C+ C( ) + C( ) + + C( ) P C + 6C( ) + + ( ) C( ) P 6C + C ( ) + + ( )( ) C ( )

22 Differeiio ( ) P C The we c oi f C f ( ) C f C ( f ) ( ) ( )( ) C d hece C f ( ) C f ( ) C f ( ) C f ( ) ( C ) f ( )! Therefore, we oi ( ) P f ( ) + f ( ) + f ( ) + f ( ) + + f! Le e he differece ewee he fucio P ; h is, The R ( ) f d he polyomil R f P + f P R + ( ) R Q( ) ( +! ) Q ( ) f f + f + f ( ) + f ( ) + + f + R!!!! is clled he remider d is defied y R where is he fucio o e defied. Now we hve + ( ) ( ) ( ) ( ) ( ) f f + f + f + f + + f ( ) + Q!!!! ( + )! we will fid Q. Cosider uiliry fucio F(), < < which is defie s ( ) ( ) ( ) () ( ) ( + ) + F () f f () f () f () f Q!!! By compuig F d simplifyig, we oi ()

23 Differeiio ( ) () ( ) F () f () + f () f () + f () f!! ( ) ( + ) () () ( + ) () ( ) + f () + f f + Q!!! +! ( ) F () f +!! Q We c see h he fucio F sisfies he codiio of Rolle's Theorem, he here eiss umerξ, d hus, () < ξ < such h F ( ξ ). The ( ξ) ( + f ) ( ξ ) ( ξ) + Q!! + Q f ξ ( ) ( + ) + ( + ) ( ξ ) R f! which is clled Lgrge formul for he remider. Sice ξ is ewee d we c wrie i i he form ξ + θ( ) where θ is ewee d ; h is < θ <. The he remider c e wrie s + ( ) ( + ) R f + θ ( ) (! ) + The formul ( ) ( ) f f ( ) + f ( ) + f ( ) + f ( ) +!!! + ( ) ( ) ( ) ( + ) + f ( ) + f + θ( ), θ! ( + )! < < is clled Tylor Formul for he fucio f ( ). If, i his formul, we oi + ( ( ) ( ) ( ) ) f f + f + f + + f ( ) + f!!!! ( + ) ( + ) ( θ ) which is ow s Mcluri Formul. Emple: Use Mcluri Formul o epd he fucios e,si, d cos. Eercises Eercise hrough, use defiiio of derivive Give y f + 58, fid Δy d Δy Δs chges () from o +Δ. () d o.8. Fid Δy Δ, give y. Fid lso he vlue of Δy Δwhe (), (), (c). Fid he derivive of y f d. Fid he derivive of f +

24 5 Differeie () y, () y + 6 Fid dy d, give y y Fid he derivive of he followig fucios 7 f co 8 y co 9 f si si + cos y si cos ( + ) y y si cos f rc + rc co 7 y e 5 y l 6 y e rcsi 7 y sih 8 y h 9 y cosh y si sih y h cosh y Derivive of composie fucio f ( + 5 ) y 5 f ( si) 5 6 y y co 8 y si + cos 9 y csc + sec y + si y y cos cos α y si ( 5+ ) + + cos y cos 5 f si si + φ y si y si y cos e ( ) l ( l ) 9 y l + 7 y l y ( l ) + l ( ) y l + + l ( + ) y + Differeiio y l + e l + e + 5 y si ( ) y rcsi + rccos si y rccos y y rcsi + y + rcsi y + rcsi 5 y l ( + ) + 5 y l + f rcsi + l 5

25 Differeiio 55 y cosh l 56 y + f. h 57 Give he fucio f e, deermie f 58 Give he fucio f ( ) +, clcule he epressio f ( ) ( ) f ( f ( ) 59 Give f, g l ( ), clcule g ( ) 6 Show h he fucio y e sisfies he equio y ( ) y 6 Show h he fucio 6 Show h he fucio y Logrihmic Differeiio y y y y ( + ) ( + ) ( + ) ( ) y e, sisfies he equio ( ) l y y + ), sisfies he equio y y ( y l ) y y y si y + 7 y ( rc ) 67 y dy y is he fucio of d deermied i prmeric form. Fid y d 7 78 y cos 7 y si rccos + 8 y rcsi y e 8 + y e + 76 ( ) y + 8 Compue dy d for π ( si ), if y ( cos) 8 Show h he fucio y give i he prmeric form y he equios + y + sisfies he equio 5

26 Differeiio dy dy y + d d dy Fid he derivive y of he implici fucio y d cos y + y 89 e + y 8 y y y + 87 y rc y 88 + y y 9 l + e c 9 9 y + y crc y Fid he derivives y of specified fucios y he idiced pois 9 ( + y) 7 ( y ) for d y 9 y ye + e + for d y 95 y + l y for d y 96 Fid y ( of he fucio y si 97 Show h he fucio y e cos sisfied he differeil equio y + y 98 Fid he h derivives of he fucios ) y ) y c) y + d) y l ( + ) + e) y f) y l + ) g) y e ( d y 99 I he followig prolem fid d l rc ) ) y y l ( + ) l e e) f) y e y Use L Hopil Rule o fid he limis ) lim π si e) lim π 5 i) lim + cosh ) lim cos l si m f) lim l si j) lim rcsi c) y g) lim ) lim y cos d) y si π si c) lim d) lim si π co π h) lim l l si lim l) cos π Fid he pproime vlues of he followigs usig he formul f +Δ f Δ + f ) cos6 ) l.9 c) º d) rc.5 e). e 6

27 Differeiio Approime he fucios ) f + for. ) y e for.5 c) f + u, fid du. fid d y. y rccos, fid d y. 5 y si l, fid d y. 6 f for. o he iervls d sisfies he Rolle heorem. Fid he pproprie vlues of ξ. 7 Tes wheher he Me-Vlue heorem holds for he fucio f o he iervl [,] d fid he pproprie vlue of ξ. 8 ) For he fucio f + d g. Tes wheher he Cuchy heorem holds o he iervl[, ] d fid ξ. ) do he sme wih respec o f si d g cos 9 Verify he followig y Tylor s formul ) e e !! ) ( ) ( ). si si + cos si cos +!! c) ( ) cos cos si! d) l ( + ) l Epd l i powers of ( ) o four erms. π Epd i powers of o hree erms π Epd si i powers of + o four erms. 6 7

28 Idefiie Iegrl Idefiie Iegrl Aiderivive or Idefiie Iegrl Prolem: Give fucio f ( ), fid fucio F whose derivive is equl o f ( ); h is F f. Defiiio We cll he fucio F( ) iderivive of he fucio f o he iervl [, ] if F f, [, ]. Defiiio We cll idefiie iegrl of he fucio f, which is deoed y f d, ll he epressios of he form F + Cwhere F( ) is primiive of f ( ). Hece, y he defiiio we hve f d F + C C is clled he cos of iegrio. I is irry cos. From he defiiio we oi. If F f, he ( f d) ( F C + ) f. d( f d) f d. df F + C Tle of Iegrls r+ r. d + Cr, r + d. l C + d rc + C rc co + C, + d. l + C, ( ) + d + 5. l + C, ( ) 6. d l ± + ± + C 7. d rcsi C rccos C, ( ) 8. d sih + c + 9. d cosh + c. d + C, ( > ) l. ed e + C.

29 . si d cos + C. cos d si + C. sih d cosh + c 5. cosh sih + c d 6. h c cosh + d 7. C cos + d 8. coh c sih + d 9. co C si + d. l C l csc co si + d π. l + + C l + sec + C cos. Some Properies of Idefiie Iegrls Lieriy. f + f + + f f d+ f d+ + f d. If is cos, he f d f d Moreover,. If f d F + C, he f ( ) d F ( ) + C. If f d F + C, he f ( + ) d F( + ) + C 5. If f d F + C, he f ( + ) d F ( + ) + C Emple. ( si+ 5 ) d s: + cos+ + C. + + d s: C 9 d. s: l + + C +. cos 7d s: si ( 7 ) + c 7 5. si ( 5) d s: cos ( 5 ) + c + C Idefiie Iegrl Iegrio By Susiuio. Chge of Vrile i Idefiie Iegrl Puig ϕ ( ) where is ew vrile d ϕ is coiuously differeile fucio, we oi

30 ϕ ϕ Idefiie Iegrl f d f d () The emp is mde o choose he fucio ϕ i such wy h he righ side of () ecomes more coveie for iegrio. Emple Evlue he iegrl I d Soluio Puig, whece + d dd. Hece, d + d + d + 5 Someimes susiuio of he form u ϕ he iegrd f do he form c re used. Suppose we succeeded i rsormig f d g( u) du where u ϕ. If g( u) duis ow, h is, g( u) du F( u) +, he f d F ϕ + c Emple Evlue () d () e d 5 Soluio Puig u 5; du 5 d; d du, we oi () 5 d du u c u c. Trigoomeric Susiuios ) If he iegrl cois he rdicl, we pu si ; whece cos ) If he iegrl cois he rdicl, we pu secwhece ) If he iegrl cois he rdicl +, we pu whece + sec We summrize i he he rigoomeric susiuio i he le elow. Epressio i he iegrd Susiuio Ideiies eeded si si cos + + sec sec sec

31 Emple Soluio Le siθ, Evlue I d π π θ θ d cosθdθ cosθdθ cosθdθ dθ I si θ cos θ si θ cosθ si θ csc co θdθ θ C C + + Idefiie Iegrl Emple I d + Soluio π π θ, < θ < I sec θdθ θ + d sec θ dθ sec θ d θ + C secθdθ l secθ + θ + C l + secθ + θ + Emple 5 Soluio l + + l + l C C Evlue 5 d Le 5secθ d secθ θ or d 5sec d dθ θ θ θ Thus, 5 5sec θ 5 d ( 5sec ) d θ θ θ 5secθ 5θ 5sec d 5 5secθ 5 We oi θ. Hece 5 Emple 6 Evlue θ θ θ θ θ 5 sec θ dθ 5θ 5θ + C + d 5 5 5sec + 5 d θ d C 5 5

32 Idefiie Iegrl 5. Iegrio y Prs Suppose h u d v re differeile fucio of, he d ( uv) udv + vdu By iegrig, we oi uv udv + vdu Or udv uv vdu Emple. si d (leu ) s: cos + si + C. rc d (le u rc ) s: rc l + + C. ed (le u ) s: e ( + ) + C si cos si. ( + 75) cosd s: ( + 7 5) + ( + 7) + C 6 Sdrd Iegrls Coiig Qudric Triomil m + m + 6. Iegrls of he form d or d + + c where c< + + c We proceed he clculio y compleig squre he riomil d he use he pproprie formuls or susiuios. Emple d. s: + C + 5 d +. s: rc + C d s: l c d s: l ( + 5) + rc + C d s: l C + + d 6. Iegrls of he Form ( m + ) + + c By mes of he iverse susiuio m + hese iegrls re reduced o iegrls of he form 6.. Emple Evlue 6. Iegrls of he Form d. As: + + ( ) + + cd ( ) + + l + By ig he perfec squre ou of he qudric riomil, he give iegrl is reduced o oe of he followig wo sic iegrls 5

33 Idefiie Iegrl ) ) Emple d + rcsi + c; > + d + + l c; > Evlue d 7 Iegrio of Riol Fucios 7. The Udeermied Coefficies Iegrio of riol fucio, fer ig ou he whole pr, reduces o iegrio of he proper riol frcio P () Q where P d Q re iegrl polyomils, d he degree of he umeror P() is lower h h of he deomior Q(). If ( ) α ( ) Q l where,, l re rel disic roos of he polyomil Q(), d α,, λ re roo mulipliciies, he decomposiio of () i o pril frcio is jusified: P A A Aα L L Lλ α λ Q l l l () where A, A,, Aα,, L, L,, Lλ re coefficies o e deermied. Emple Fid d ) I As: + l C ) I d As: + l l + C If he polyomil Q() hs comple roos ± iof mulipliciy, he pril frcios of he form A + B A + B + + () + p+ q ( + p + q) will eer io he epsio (). Here, + p + q ( + i) ( i) d A, B,, A, B re udeermied coefficies. For, he frcio () is iegred direcly; for >, we use reducio mehod; here i is firs dvisle o represe he qudric p p riomil + p + q i he form + + q p d me he susiuio + z. Emple Fid + d As: 7. The Osrogrdsy Mehod ( + + ) λ ( ) + C 6

34 Idefiie Iegrl If Q() hs muliple roos, he P X Y d + Q Q Q d () where Q is he grees commo divisor of he polyomil Q() d i derivive Q ; : Q Q Q X() d Y() re polyomils wih udeermied coefficies, whose degrees re, Q. respecively, less y uiy h hose of Q d The udeermied coefficies of he polyomils X() d Y() re compued y differeiig he ideiy (). Emple Fid d I Soluio d A + B + C D + E + F d + ( ) Differeiig his ideiy, we ge ( A + B)( ) ( A + B + C) D + E + F + or ( A B)( ) ( A B C) ( D E F )( ) Equig he coefficies of he respecive degrees of, we will hve D ; E A ; F B ; D+ C ; E+ A ; B+ F whece A ; B ; C ; D ; E ; F d, cosequely, d d (5) To compue he iegrl o he righ of (5), we decompose he frcio L M + N we will fid L, M, N. Therefore, d d + l l + d + + C d d l + + C 9 8 Iegrio of ceri Irriol fucios 7

35 Idefiie Iegrl p p q q 8. Iegrls of he ype R + +,,, dwhere R is riol fucio c + d c + d + d p, q, p, q, re ieger umers. We use he susiuio z where is he c + d les commo muliple (lcm) of q, q, Emple Evlue d Soluio le z, he d z dz, d hece d z dz z dz z dz ( z ) l z C z z z + z ( ) l( ) C d Emple Evlue swer: l + + C + m 8. Iegrls of differeil iomils ( + ) p d where m, d p re riol umers. m If + is ieger, le r + z s where s is he deomior of he frcio p s m If + + p is ieger, le + z s d Emple Evlue ( + ) Soluio d We hve m + d +. We see h m,, r, s d ( + ), ieger. The ssume + z, he Hece, Emple z zdz, d d + z z z zdz d ( + ) ( z ) z Wor ou ( z ) dz ( z z ) C C + d C 8

36 Idefiie Iegrl 8. Iegrl of he Form where Pu P P + + c is polyomil of degree d Q + + c + λ () d () P d + + c + + c is polyomil of degree wih udeermied coefficies re λ is where Q umer. The coefficies of he polyomil differeiig ideiy (). Emple 5 Soluio whece Muliplyig y Fid + d Q d he umer λ re foud y + d + d d ( A + B + C + D) + + λ + + ( ) + A B C D ( A + B + C) λ + d equig he coefficies of ideicl degrees of, we oi A ; B ; C ; D ; λ Hece, + + d + l( + + ) + C 8. Iegrl of he form d () ( α ) + + c They re reduced o iegrls of he form () y he susiuio α d Emple 6 Fid 5 9 A Ceri Trigoomeric Iegrls 9. Iegrl of he Form si d d cos d If is odd posiive ieger, use he ideiy Emple Fid Soluio 5 si d d d 5 si si si si + cos 9

37 ( cos ) sid ( cos cos ) sid ( cos cos ) d( cos ) cos + cos cos + C 5 cos If is eve, use hlf-gled ideiies si d Emple Fid Soluio cos d Idefiie Iegrl + cos cos + cos cos d d ( cos cos ) cos ( ) ( cos ) + + d d d d cos cos 8d + d + d m Type: ( si cos d) + si + si + C 8 If eiher m or is odd posiive ieger d oher epoe is y umer, we fcor ou si or cos d use he ideiy si + cos Emple Fid Soluio si cos d ( ) ( si cos d cos cos si d cos cos ) d ( cos ) cos cos sec + C sec + C If oh m d re eve posiive iegers, we use hlf-gle ideiies o reduce he degree of he iegrd. Emple Fid Soluio si si cos d cos + cos cos d d ( cos cos cos ) 8 + d + cos ( + cos ) ( si ) cos 8 d

38 Idefiie Iegrl cos ( cos ) ( si ) cos d cos si cos 8 + d d cos d ( ) si d ( si ) si + si + C Iegrl of he Form si m cos d, si msi d, cos m cos d To hdle hese iegrls, we use he produc ideiies /. si m cos si ( m + ) + si ( m ) Emple 5 Soluio m m m /. si si cos( + ) cos( ) m m m /. cos cos cos( + ) + cos( ) Fid si cosd si cosd si 5 si si 5 ( 5 ) si + d d cos 5+ cos + C 9. Iegrls of he Form or co We use he formul Emple 6 Evlue Soluio d m m d d where m is posiive umer sec or co csc d d d d + + C ( sec ) ( sec ) Iegrls of he ypes R (si,cos ) d where R is riol fucio. We c use he susiuio si + + d Emple Clcule + si + cos, cos, d d hece we hve d +

39 Idefiie Iegrl Soluio Le, he we oi d d I + l + + C l + C R si, cos R si, cos is verified, he we c me he If he equliy susiuio. Ad hece we hve si, cos d + + d rc, d. + d Emple Clcule I + si Soluio d Le,si, d, he + + d d I rc ( ) + C rc ( ) + C + ( + ) + + Iegrio of Hyperolic Fucios Iegrio of hyperolic fucios is compleely logous o he iegrio of rigoomeric fucio. The followig sic formuls should e rememered ) cosh sih ) sih ( cosh ) ) cosh ( cosh + ) ) cosh sih sih Emple Fid cosh d Soluio cosh d ( cosh + ) d sih + + C Emple Fid ) sih si d cosh d ) ) sih cosh cosh Trigoomeric d Hyperolic Susiuios for Fidig Iegrls of he Form R (, + + c ) d () where R is riol fucio. Trsformig he qudric riomil + + c io sum or differece of squres, he iegrl () ecomes reducile o oe of he followig ypes of iegrls R z, m z dz R z, m + z dz R z, z m dz ) ) The ler iegrls re, respecively, e y mes of susiuios ) z msi or z mh d

40 Idefiie Iegrl ) z m or z msi ) z msec or z mcosh Emple d fid I Soluio We hve Puig + z, we he hve d sec zdz d ( ) ( ) d sec zdz cos z I dz + C C zsec z si z si z + Emple Soluio We hve Fid + + d Puig + sih d d cosh d we oi sih cosh cosh sih cosh cosh 8 8 I d d d cosh cosh cosh sih cosh 8 8 d + + C 8 8 Sice sih +,cosh + + d l l we filly hve I l Eercises Usig sic formuls o evlue iegrls. ( ) d d ( + )( + ) d ( + ) d d + +. d 6. d + d 5. ed d rcsi +. d 7. d

41 Idefiie Iegrl d d.. d ( + ) l + + si cos si.. ( e e ) d + cos d si cos 5. d 5. e + e d cos si + si 6. d ( ) cos 6. d 7. ( sih5 cosh5) d ( ) 7. + e d 8. d d 9. d 8 9. d d. d d. e l. d si e. cos d. e e d. d d. si ( l ) + d cos.. 5 d si sec d 5. + cos sid 5. rc + l d d + rc e + l ( + ) + rc d d sec ( + ) d 8. d d d d. d 5. si e + d rccos. cos 5. d. si ( ) d 5. Applyig he ideced susiuios, fid he followig iegrls d ), c) 7 5 d,5 d d d) ), l, + + e +

42 Idefiie Iegrl cos d e), si + si Applyig he suile susiuio, compue he followig iegrls ( rcsi ) + 5. d 6. d, + d 5. + d 6. + d 55. d 6. +, e e + l d 56. d 6. l d e d + X e + d si d rcsi cos d, +5 d 66. Fid he iegrl y pplyig he susiuio si 67. Fid he iegrl + d y pplyig he susiuio sih By usig he fomul of iegrio y prs 68. l d 77. d si d d 7. si d 7. cos d 7. d e 7. d 75. l d 76. l d Iegrio ivolvig qudric riomil epressio d d d d rcsi d d 79. l ( + + ) d 8. si 8. e si d 8. cos d si l d ( rcsi ) d d d + 5 d d 9. d 5

43 Idefiie Iegrl 9. 6 d d 95. d 5 + d 96. Fid he Iegrls 8. d + + ( d d ( )( + )( ) ) d 5 + Osrogrdsy s Mehod 7 +. d d As: d + + d ( ) d + d ( + )( + + 5) d rc l C + + ( 8 ) ( ) d As: l ( ) ( + ) ( ) + + ( ) d + ( ) As: ( + )( + ) ( + ) ( + ) d + + rc + C + + rc + C d + 6. As: l + C d ( + + ) As: ( + ) 8( + + ) rc C ( + + ) ( + ) d + 8. As: rc ( + ) C d As: C rc

44 Idefiie Iegrl p p q q Compue iegrls of he form R + +,,, d c + d c + d d. d d 6. d + ( + ) + d d 7. d. ( + ) +. d ( ) Iegrio of iomil differeils d d + d 5 + d 5 + Trigoomeric Iegrls. cos d si d si si si cos 5 d cos d cos d 5 cos d si si d si cos Iegrl ( si,cos ) d R d d + 5cos d si + cos cos d + cos. si cos d d. 6 cos d. si cos d. si cos d. 5 si 5. si cos5d 6. sisi5d 7. cos si d 8. si si d cos + cos d siωsi ( ω+ ϕ) d 8 si + 7cos d cos + si + si d cos 7

45 + 6. d Iegrios of hyperolic fucio 6. sih d cosh d sih sih cosh cosh d d Iegrl (, + + ) R c d d + d + d d + d Idefiie Iegrl d 66. sih cosh 67. h d d 68. sih + cosh d d d + 7

46 Defiie Iegrl Defiiio Iegrl. Riem Sum Le f e fucio defied over he close iervl wih < < < e rirry priio i suiervl. We clled he Riem Sum of he fucio f over[, ] he sum of he form S i f ξ Δ i where i ξi i, Δ i i i, i,,...,. i y f ξ ξ. Defiie Iegrl The limi of he sum S whe he umer of he suiervl pproches ifiiy d h he lrges Δ i pproches zero is clled defiie iegrl of he fucio f wih he upper limi d lower limi. lim f ( ξ ) f d i Δ i m Δi i or equivlely lim f ( i) Δ i f + i d If he fucio f is coiuous o [, ] e iegrle o[, ]. or if he limi eiss, he fucio is sid o If is i he domi of f, we defied f d d If f is iegrle o[, ] f d f d. defie Emple Fid he Riem Sum for he fucio [ ] S, he we f + over he iervl, y dividig io equl suiervls, d he fid he limi lim S. Soluio 9 9i Δ i ξ i i + iδ i +

47 Defiie Iegrl d hece f he 9i 9i ξ i ( ξ ) S f Δ i i i 9i 9 + i i i i ( ) ( ) lim S lim Emple ( ) 8 d Soluio Divide he iervl [, ] io equl suiervls. Hece we oi Δ i. I ech suiervl[, i i i], choose ξi such h ξ i +Δ i i + i f ( ξi) Δ i + 8 i i 8i 6i + 8 i 6i i 6 + i 6i 8i + i 6 8 ( ) + ( ) 6 ( + ) 8 ( + )( + )

48 Defiie Iegrl ( 8) lim ( ξi) d f Δ i i 8 lim Suiervl propery If f is iergrle o iervl coiig he pois,, d c, he + c c f d f d f d o mer wh he order of,, d c.. The firs Fudmel Theorem of Clculus Theorem A Firs Fudmel heorem of Clculus, d le e vrile poi i (, ), Le f e coiuous o he closed iervl [ ] he d d () f d f y f f Proof For (, ) we defie F f ( ) d, he d f () d F d F( + h) F lim h h + h lim f () d f () d h h + h lim f () d h h + h Bu f () d represes he re ouded y -is he curve + h hf ; h is f () d hf + h, which is pproime o d f () d lim hf f d. h h + h f ewee d. So,

49 Defiie Iegrl d Emple d d Emple d d cosd cosd d d d cosd cos d d Emple Fid ( ) d d Soluio Le u du d hece d d u ( ) d ( ) d d d d u d ( ) d u du d u 6 Theorem B Compriso Propery, If f d g re iegrle o [ ] d if f g for ll i [, ] f d g d, he Proof Over he iervl[, ], le here e rirry priio < < <. Le ξi e smple poi o he i h suiervl[, i i], he we coclude h f ( ξi) g( ξi) f ξ Δ g ξ Δ i i i i ( ξ ) ( ξ ) f Δ g Δ i i i i i i lim f i i g i i i f d g d ( ξ ) lim ( ξ ) Δ Δ Theorem C Boudedess Propery If f is iegrle o [, ] d m f M for ll i[, ], he Proof m f d M Le h( ) m, [, ], he h f, [, ] Hece, ( ) h d f d m f d. i y M m y f

50 Defiie Iegrl By similr wy, le g M, [ ] f g, [, ] f ( d ) g d f d M ( ) Therefore m ( ) f( d ) M( ),, he. Secod Fudmel Theorem of Clculus d Me vlue heorem For Iegrls Secod Fudmel Theorem of Clculus Le f e iegrle o [, ] d F e y primiive of f o[, ], he f ( d ) F ( ) F ( ) I is lso ow s Newo-Leiiz Formul. For coveiece we iroduce specil symol for F( ) F( ) y wriig F ( ) F( ) F or F ( ) F( ) F Emple Emple d π π si cos d 9 si 8 8 Me Vlue Theorem for Iegrl If f is coiuous o[, ], here is umer c ewee d such h Proof Le, () ( ) f d f c F f d By Me vlue heorem for derivive, we oi F F F c () ( ) f d f c () ( ) f d f c f () dis clled he me vlue, or verge vlue of f o [, ] f c Emple Fid he verge vlue of f o he iervl [, ] Soluio 7 f ve f d d 5

51 Defiie Iegrl Emple Fid he verge vlue of f ( ) cos o he iervl [,π ] 5. Chge of vrile i defiie iegrl If f is coiuous over he close iervl, if ϕ ( ) is coiuous d is derivive is ϕ ( ) over he iervlα β, where ϕ ( α ) d ϕ ( β ) d if f ϕ ( ) is defied e coiuous over he iervlα β, he ϕ ϕ β f d f d α Emple Fid d ( > ) Soluio π Le si, d cos, rcsi, α rcsi d β rcsi. he we oi π d si si cos d Emple Evlue Emple Evlue π π si cos d si d π π π ( cosd ) si le (swer: l + 6. Iegrio y prs If he fucios u d d ) l e d le e z (swer: π ) v( ) re coiuous differeile over [, ] u v d u v v u d π π Emple Evlue cos d (swer: ) e + Emple Evlue e d(swer: ) 8 π Emple Evlue e si d (swer: ( e π + )), we hve 7. Improper Iegrl Improper iegrls refer o hose ivolvig i he cse where he iervl of iegrio is ifiie d lso i he cse where f (he iegrd) is uouded fiie umer of pois o he iervl of iegrio. 7. Improper Iegrl wih Ifiie Limis of Iegrio Le e fied umer d ssume h f N deiss for ll N. The if 6

52 Defiie Iegrl N f deiss, we defie he improper iegrl + f lim N + lim N N + + f d f d dy The improper iegrl is sid o e coverge if his limi is fiie umer d o e diverge oherwise. + d Emple Evlue I Soluio + N d d N lim lim lim N + + N + N + N Thus, he improper iegrl coverges d hs he vlue. Emple Evlue + d p + e d Le e fied umer d ssume f lim f deiss we defie he improper iegrl The improper iegrl f deiss for ll <. The if lim f d f d dis sid o e coverge if his limi is fiie umer d o diverge oherwise. If oh f dad f + d coverge for some umer, he improper iegrl of f o he eire -is is defied y f d f d f d d d Emple Evlue + (swer:π ) (swer:π ) Improper Iegrls wih Uouded Iegrds If f is uouded d f + deiss for ll such h <, he lim f d f d + 7

53 Defiie Iegrl If he limi eiss (s fiie umer), we sy h he improper iegrl coverge; oherwise, he improper iegrl diverges. Similrly, if f is uouded d f deiss for ll such h <, he If f is uouded c where lim f d f d < < f dd f c he improper iegrl oh coverge, he f c + f d d f d c c y y f y g A c d We sy h he iegrl o he lef diverges if eiher or oh of he iegrls o he righ diverge. d d Emple Fid ( ) Noe: + +. For f g d if g dcoverges, he f d +, if coverge d + f d g d Emple Ivesige he covergece of diverges.. For + d ( + e ), if f g d if f Emple Ivesige he covergece of d. If f dis coverge he f solue coverge. + si Emple Ivesige he covergece of d 8 Are Bewee Two Curves y f y g 8. Are Bewee d If f d g re coiuous fucios o he iervl[, ], d if f g for ll i [, ], he he re of he regio ouded ove y y f, elow y y g, o he lef y lie, d o he righ y he lie is defied y A f g d ddiverges, he + g d dis lso coverge, specificlly 8

54 Defiie Iegrl Emple Fid he re of regio ouded ove y y + 6, ouded elow y y, d ouded o he sides y he lies d. s: Emple Fid he re of he regio eclosed ewee he curves y y d 8. Are Bewee v( y) d w( y) If w d v re coiuous fucios d if w( y ) v ( y) for ll y i [ cd, ], he he re of he regio ouded o he lef y v( y) y w( y) y c, o he righ, elow y, d ove y y d is defied y d c A w y v y dy d c y d y Emple Fid he re of he regio eclosed y 9 wih respec o y. (s: ) Emple Fid he re of he regio eclosed y he curves iegrig /. wih respec o 8. Are i Polr Coordies /. wih respec o y v( y ) w( y) y, iegrig d y y A θ β θ α β r ρ θ A ρ θ α dθ r ρ ( θ ) r ρ θ A θ β θ α β ( ) ρ θ ρ θ A α dθ Emple Clcule he re eclosed y he crdioid r cosθ (swer: π ) Emple Fid he re of regio h is iside he crdioid r + cosθ d ouside he circle r 6 (swer: 8 π ). 9 Volume of Solid 9. Volume By Cross Secios Perpediculr To The X-Ais Le S e solid ouded y wo prllel ples perpediculr o he -is d. If, for ech i he iervl[, ], he cross-seciol re of S 9

55 Defiie Iegrl perpediculr o he -is iegrle, is defied y A, he he volume of he s s is A is solid, provided V A ( ) d A 9. Volume By Cross Secios Perpediculrr To The Y-Ais S e solid ouded y wo prllel ples perpediculr oo he y-is y c d y d. If, for ech y i he iervl[ c,d, he cross-seciol re of S perpediculr y y Emple Derive he formul for he volume of righ pyrmid whose liude is h d whose se is squre wih sides of legh. h. 9. Volumes of Solids Of Revoluio f R c ] o he y-is is A y, he he volume of he solid, provided A y is iegrle, is defied y d V A( y) dy d Emple curve y.. Volumes y Diss Perpediculr To he -Ais V π f d Fid he volume of he solid h is oied whe he regio uder he 5π over he iervl [, ] is revolved ou he -is.( s: ) Emple Derive he formul for he volume off sphere of rdius r. (s: r.. Volumes y Wshers Perpediculr o he -Ais Suppose h f d g re oegive coiuous fucios such h g f for. Le L R e he regio eclosed ewee he grphs of hese fucios d lies d. Whe his regio is revolved ou he -is, i geeres solid whose volumes is defied y π ) V ( π f ( ) g( ) ) d

56 Defiie Iegrl Emple Fid he volume of he solid geered whe he regio ewee he grphs of f + d g over he iervl [, ] is revolved ou he -is. As: 69 π..c Volumes By Diss Perpediculr To he y-is V d c π u dy..d Volumes By Wshers Perpediculr To y-is d c ( ) V π u y v y..e Cylidricl Shells Ceered o he y-is Le R e he ple regio ouded ove y coiuous curve, elow y he -is, d o he lef d righ respecively y he lies d. The he volume of he solid geered y revolvig R ou he y-is is give y V π f d dy y f Emple 5 Fid he volume of he solid geered whe he regio eclosed ewee y,, d he -is revolved ou he y-is. Soluio Sice f,,, he he volume of he solid is 5 π π V π d π d π [ ] Emple 6 Fid he volume of he solid geered whe he regio R i he firs qudr eclosed ewee y d y is revolved ou he y-is. (Aswer: π 6 ) Legh of Ple Curve, o is defied y If f is smooh fucio o [ ], he he rc legh L of he curve y f dy d d L + f d + Similrly, for curve epressed i he form g( y) where g is coiuous o [ cd, ], he rc legh L from y co y d defied y d d d L + g ( y) dy + dy dy c c

57 Defiie Iegrl Emple Fid he rc legh of f from (,) o (, ) Soluio f f + f + + The he rc legh is defied y L + d + + l l( + 5) If he curve is give i polr coordie sysem r ρ ( θ), α θ β he he rc legh of he curve is defied y β β dr ρ ( θ) ρ ( θ) θ d dθ α α θ L + d r + Emple Fid he circumferece of he circle or rdius. Soluio As polr equio his circle is deoed y r The he rc legh is π π, θ π π L dθ dθ θ π Emple Fid he legh of he crdioid r cosθ If he curve is defied y he prmeric equio, y y, [, ] legh of he curve is Emple Soluio () () L + y d Fid he circumferece of he circle of he rdius r, he he Prmeric form, he circle is defied y ( ) rcos, y( ) rsi wih [, π ] he π π L r cos + r si d rd π r Emple 5 Fid he rc legh of he sroid, cos, y si.(s6). Are of Surfce of Revoluio Le f e smooh, oegive fucio o[, ]. The he surfce re S geered y f ewee ou -is is y revolvig he porio of he curve d

58 Defiie Iegrl S π f + f d For curve epressed i he form g( y) where g is coiuous o [, ] g( y) for c y d curve from d d, he surfce re S geered y revolvig he porio of he y co y d ou he y-is is give y d S π g y + g y c Emple Fid he surfce re geered y revolvig he curve y, ou he -is. Soluio f f. Thus, S π + d π d π Emple Fid he surfce re geered y revolvig he curve y, y ou he y-is. Soluio y g( y) y. Thus, g ( y) y, he π S π y + ydy y + ydy π π ( + y ) ( 7 ) 6 9 Eercises Wor ou he followig iegrls d 8. l d + rc + π. si cos d π. sec θdθ 5. 9 ( ) ( ) d π dy

59 Defiie Iegrl π d π 6. si + 7. d l + d π 8. rc e e + e π π 9. si d 6 π π. cos d 8 e d. l ( + l ) e d. ( + l) d 9. l z π. dz 8 z + 6 d π 5. 5 d π π 7. si d e d 8. l e l Fid he derivive of he followig fucios 9. F l d, As: l. + d, As: +. F e d, As: e + e. F cos( ) d, As: cos cos + Wor ou he followig iegrls d.. e d + d π 5., ( > ) +

60 Defiie Iegrl 6. d π 7. l d + d 8. π d 9. 9 d. e l l d π. + d π. e + e d. l l Compue he improper iegrls (or prove heir divergece) d. 5. e d, > + d l 7. d d ( + ) d d. d. +.. e d e d rc. d d d

61 Defiie Iegrl lp 7. For p, is d coverge? (Hi: l p p for e ) d d 8. For wh vlues of re he iegrls d l l 9. For wh vlues of is he iegrl 5. Show h 5. Show h 5. Show h d,( ( ) f d f d if e π π e d e d d π d rccos 5. ( si ) ( cos ) si d f d f d < ) coverge? coverge? f is eve d f d if f is odd. 5. The Lplce Trsformio of he fucio f is defied y he improper iegrl + s L { } () F s f e f d. Show h for cos (wih s > ). L { e }. L {} c. {} s s s e. L { si } s Fid he firs qudr re uder he curve s + L d. L { cos } y e (swer: ) 56. Le R e he regio i he firs qudr uder y 9 d o he righ of. Fid he volume geered y revolvig R ou he -is. (swer: 8π ) 57. Derive formul V d rdius of se r. π r hfor he volume of righ circulr coe of heigh h 58. Le R e he regio ove he curve y uder he lie y d ewee d. Fid he volume geered y revolvig R ou ). -is, ). ou y-is. (swer: ). 6 7 π, ). 5 π ) s 59. Fid he re of he regio ewee y d he lies y d y 6. Fid he re of he regio ouded y he curve y si, y cos d d π (swer: ) 6

62 Defiie Iegrl 6. Fid he re of he regio ouded y prols y (Aswer: 9) d y Fid he re of he regio ouded y he prol y + d he lie y 8. (swer: 5 ) 6 6. Fid he re of he regio ouded y he prols y d y. (Aswer: ) 6. Fid he rc legh of he curve 6. Fid he rc legh of he curve 6. Fid he rc legh of he curve y + from o (s: 8 6 ) + y from o 8 (s: 9) 6y + from o (s: 7 ) π 65. Fid he re iside he crdioid r + cosθ d ouside r (s: + ) 66. Fid he re iside he circle r siθ d ouside he crdioid r cosθ 67. Fid he volume geered y revolvig he ellipse Aswer: π y + ou -is. 7

63 Noro Uiversiy Ifiie Series Ifiie Series. SEQUENCES AND THEIR LIMITS Sequeces A sequece { } is fucio whose domi is se of oegive iegers d whose rge is he suse of rel umer. The fuciol vlue,, re clled erms of he sequece d is clled he h erm, or geerl erm of he sequece. Limi of he sequece If he erms of he sequece pproch he umer L s icreses wihou oud, we sy h he sequece coverges o he limi L d wrie L lim + Coverge sequece The sequece { } coverges o he umer L, d we wrie L lim if for every ε >, here is ieger N such h sequece diverges. lim f Limi Theorem for Sequeces If lim Ld lim M, he. Lieriy Rule: lim ( ). Produc Rule: ( ) LM r + s rl + sm L < ε wheever > N. Oherwise, he lim L. Quoie Rule: lim provided M M m. Roo Rule: lim m L provided m is defied for ll d m L eiss. Emple: Fid he limi of he coverge sequeces /. / c/.{ } + Limi of sequece from he limi of coiuous fucio The sequece{ }, le f e coiuous fucio such h f ( ) for,,, If eiss d lim f L Emple: Give h he, he sequece { } e coverge d lim L. coverges, evlue lim e

64 Noro Uiversiy Ifiie Series Bouded, Moooic Sequeces Nme Codiio Sricly icresig < < < < < Icresig Sricly decresig > > > > > Decresig Bouded ove y M M for,,,... Bouded elow y m m for,,,... Bouded If i is ouded oh ove d elow. INFINITE SERIES; GEOMETRIC SERIES A ifiie series is epressio of he form d he h pril sum of he series is The series is sid o coverge wih sum I his cse, we wrie S S if he sequece of pril sums { s } coverges o S. lim S S If he sequece { S } does o coverge, he series diverges d hs o sum. Emple: Show h he series coverges d fid is sum. + Soluio: We hve + +. The S lim S lim + Emple: Prove h he series coverge d fid is sum.. + c. ( ) Geomeric Series A geomeric series is ifiie series i which he rio of successive erm i he series is cos. If his cos rio is r, he he series hs he form r + r + r + r + + r +,

65 Noro Uiversiy Ifiie Series Geomeric Theorem The geomeric series r wih diverges if r d coverges if r < wih sum r r Proof: The h pril sum of he geomeric series is S + r + r + + r. The, rs r + r + r + + r rs S r ( r ) S, r r If r > r lim S If r < r lims r THE INTEGRAL TEST, p-series Diverge Tes If lim, he he series mus diverge. Proof: Suppose he sequece of pril sums { S } coverges wih sum L, so h lim S L. The we lso hve lim S L. We hve S S, d he i follows h lim lim ( S S ) L L We see h if coverges, he lim. Thus, if lim, he diverges. Emple: Diverges sice + + lim ( ) + The Iegrl Tes If f ( ) for,,,... where f is posiive coiuous d decresig fucio o f for he Ad f d eiher oh coverge or oh diverge. Emple: Tes he series for covergece Soluio: We hve f is posiive, coiuous d decresig for. d lim d lim[ l ], implyig h d diverges.

66 Noro Uiversiy Ifiie Series Hece diverges. Emple: Ivesige he followig series for coverge e e e e p-series A series of he form p p p p where p is posiive cos, is clled p-series. Noe: The hrmoic series is he cse where p. Ad Theorem, he p-series es The p-series p coverges if p > d diverges if p. Proof: Le f p p f he f if p p < p > Hec e f p is coiuous, posii ve d decre sig d p >. For p, he series is hrmoic, h is i diverges For p > d p we hve:, p > l lim, < p < Th is, his improper iegrl coverges if p > d diverges if < p < For p, he series ecomes For p <, we hve lim p, so he series diverges y he covergece es. Hece, p-series coverges oly whe p >. Emple: Tes ech of he followig series for covergece.. e Soluio:.. So p > d he series coverges.. We hve coverges, ecuse i is geomeric series wih r <. e e diverges ecuse i is p-series wih p < Hece e diverges. p d p im d p p p

67 Noro Uiversiy Ifiie Series. COMPARISON TEST coverges. Direc Compriso Tes Suppose cfor ll N for some N. If c coverges, he lso Le d for ll N for some N. If d diverges, he lso diverges. Emple: Tes he seri es for covergece. + Soluio: We hve + > > for. The < <. Sice coverges, + i implies h coverges. + Emple: Tes for covergece he followig series..! Limi Compriso Tes Suppose > d > for ll sufficiely lrge d h lim L where L is fiie d posiive( < L < ). The d eiher oh coverge or oh diverge. Emple: Tes he series for covergece. 5 Soluio: We see h is coverge series for i is he geomeric series wih r <. Moreover lim 5 5 Hece is coverge oo. 5 The zero-ifiiy limi compriso es Suppose > d > for ll sufficie lrge. If lim d coverges, he coverges If lim d diverges, he diverges. 5. THE RATIO TEST AND THE ROOT TEST Theorem: Give he series w ih >, suppose h lim + L 5