Clulus Cht Sht Limits Dfiitios Pris Dfiitio : W sy lim f L if Limit t Ifiity : W sy lim f L if w for vry ε > 0 thr is δ > 0 suh tht mk f ( ) s los to L s w wt y whvr 0 < < δ th f L < ε. tkig lrg ough positiv. Workig Dfiitio : W sy lim f L if w mk f ( ) s los to L s w wt y tkig suffiitly los to (o ithr si of ) without lttig. Right h limit : lim + f L. This hs th sm fiitio s th limit pt it rquirs >. Lft h limit : lim f L. This hs th Thr is similr fiitio for lim sm fiitio s th limit pt it rquirs gtiv. <. Rltioship tw th limit o-si limits lim f L lim f lim f L lim f lim f L + + lim f lim f lim f ( ) Dos Not Eist + Assum lim f lim g. lim f lim f. lim f ± g lim f ± lim g. lim f g lim f lim g Proprtis oth ist is y umr th, 4. pt w rquir lrg gtiv. f L Ifiit Limit : W sy lim f if w mk f ( ) ritrrily lrg ( positiv) y tkig suffiitly los to (o ithr si of ) without lttig. Thr is similr fiitio for lim f pt w mk f ( ) ritrrily lrg f lim f lim g lim g lim 5. lim f lim f 6. lim f lim f Bsi Limit Evlutios t ± Not : sg ( ) if > 0 sg ( ) if < 0.. lim & lim 0 5. v : lim. lim l & lim l 0. If r > 0 th lim 0 r r 4. If r > 0 is rl for gtiv th lim 0 r ± f L provi g 6. o : lim & lim lim 0 7. v : lim sg ± 8. o : lim + + + sg 9. o : lim sg + + + + + + Visit http://tutoril.mth.lmr.u for omplt st of Clulus ots. 005 Pul Dwkis
Clulus Cht Sht Evlutio Thiqus Cotiuous Futios L Hospitl s Rul If f is otiuous t th lim f f f 0 f ± If lim or lim th, g 0 g ± Cotiuous Futios Compositio f f f ( ) is otiuous t lim g th lim lim is umr, or g g lim f ( g ) f ( lim g ) f ( Polyomils t Ifiity ) p ( ) q( ) r polyomils. To omput Ftor Cl p + 4 ( )( + 6) lim ftor lrgst powr of out of oth lim lim ± q ( ) p ( ) q( ) th omput limit. + 6 8 lim 4 4 4 4 ( ) Rtioliz Numrtor/Domitor lim lim lim 5 5 5 ( ) + lim lim 9 9 8 8 Piwis Futio + + 5 if < 9 lim lim lim g whr g 9 9 ( 8)( + ) ( + 9)( + ) if Comput two o si limits, lim g ( 8)( 6) lim + 5 9 08 lim g lim 7 + + Comi Rtiol Eprssios ( + h) O si limits r iffrt so lim g lim lim h0 h + h h0 h ( + h) os t ist. If th two o si limits h h qul th lim g woul hv ist lim lim h0 h ( + h) h0 ( + h) h th sm vlu. Som Cotiuous Futios Prtil list of otiuous futios th vlus of for whih thy r otiuous.. Polyomils for ll. 7. os( ) si ( ) for ll.. Rtiol futio, pt for s tht giv ivisio y zro. 8. t ( ) s( ) provi. ( o) for ll. π π π π,,,,, 4. ( v) for ll 0. 9. ot 5. for ll. ( ) s( ) provi 6. l for > 0., π, π,0, π, π, Itrmit Vlu Thorm Suppos tht f ( ) is otiuous o [, ] lt M y umr tw f ( ) Th thr ists umr suh tht < < f ( ) M. f. Visit http://tutoril.mth.lmr.u for omplt st of Clulus ots. 005 Pul Dwkis
Clulus Cht Sht Drivtivs Dfiitio Nottio f ( + h) f If y f th th rivtiv is fi to f lim. h0 h If y f th ll of th followig r quivlt ottios for th rivtiv. f y f y f Df If y f ll of th followig r quivlt ottios for rivtiv vlut t. f y f y Df If y f th,. m f is th slop of th tgt li to y f t th qutio of th tgt li t is y f + f. giv y If f ( ) Itrprttio of th Drivtiv f is th isttous rt of. hg of f ( ) t.. If f ( ) is th positio of ojt t tim th f is th vloity of th ojt t. Bsi Proprtis Formuls g r iffrtil futios (th rivtiv ists), r y rl umrs,. ( f) f. ( f ± g) f ± g. ( f g) f g f g + Prout Rul f f g f g 4. Quotit Rul g g Powr Rul ( f ( g )) f g g This is th Chi Rul 5. ( ) 0 6. 7. ( ) ( si ) ( os ) os si ( t ) s s s t Commo Drivtivs ( s ) s ot ( ot ) s ( si ) ( os ) ( t ) + ( ) l ( ) ( ) ( l ), > 0 ( l ), 0 ( log ), > 0 l Visit http://tutoril.mth.lmr.u for omplt st of Clulus ots. 005 Pul Dwkis
Clulus Cht Sht Chi Rul Vrits Th hi rul ppli to som spifi futios.. ( f ) f f 5. ( os f ) f si f f f. ( ) f 6. ( t f ) f s f f. ( l f ) 7. s [ ] s t f f 4. ( si f ) f os f 8. ( t f ) + f ( f ) f [ f ] [ f ] Highr Orr Drivtivs Th So Drivtiv is ot s Th th Drivtiv is ot s ( ) f ( f f is fi s ) f f is fi s f ( f ( )), i.. th rivtiv of th ( ) ( ) f ( f ), i.. th rivtiv of first rivtiv, f. th (-) st rivtiv, f. Impliit Diffrtitio 9y Fi y if + y si ( y) +. Rmmr y y hr, so prouts/quotits of y will us th prout/quotit rul rivtivs of y will us th hi rul. Th trik is to iffrtit s orml vry tim you iffrtit y you tk o y (from th hi rul). Aftr iffrtitig solv for y. ( y ) y y y ( y) y y y 9y 9 + + os + y + y + y y y y + y 9y 9y 9 os y 9 os y y y 9 9 Critil Poits is ritil poit of f. f ( ) 0 or. f ( ) Irsig/Drsig Cov Up/Cov Dow os t ist. provi ithr Irsig/Drsig f > for ll i itrvl I th. If 0 f ( ) is irsig o th itrvl I.. If f 0 f ( ) is rsig o th itrvl I.. If f 0 f ( ) is ostt o th itrvl I. < for ll i itrvl I th for ll i itrvl I th 9y y y 9y 9 os ( y) Cov Up/Cov Dow. If f > 0 for ll i itrvl I th f ( ) is ov up o th itrvl I.. If f 0 f ( ) is ov ow o th itrvl I. < for ll i itrvl I th Ifltio Poits ovity hgs t is ifltio poit of f. if th Visit http://tutoril.mth.lmr.u for omplt st of Clulus ots. 005 Pul Dwkis
Asolut Etrm. is solut mimum of f ( ) if f ( ) f for ll i th omi. is solut miimum of f ( ). if f ( ) f for ll i th omi. Frmt s Thorm f hs rltiv (or lol) trm t If, th is ritil poit of Clulus Cht Sht f. Etrm Vlu Thorm f is otiuous o th los itrvl If [, ] th thr ist umrs so tht,,. f ( ) is th s. m. i., [, ],. f is th s. mi. i [, ]. Fiig Asolut Etrm To fi th solut trm of th otiuous futio f ( ) o th itrvl [, ] us th followig pross.,.. Fi ll ritil poits of f ( ) i [ ]. Evlut f ( ) t ll poits fou i Stp.. Evlut f ( ) f ( ). 4. Itify th s. m. (lrgst futio vlu) th s. mi.(smllst futio vlu) from th vlutios i Stps &. Etrm Rltiv (lol) Etrm. is rltiv (or lol) mimum of f ( ) if f ( ) f for ll r.. is rltiv (or lol) miimum of f f for ll r. f ( ) if st Drivtiv Tst If f th is is ritil poit of. rl. m. of f ( ) if f 0 of f 0. rl. mi. of f ( ) if f 0 of f 0. ot rltiv trm of f ( ) if f > to th lft < to th right of. < to th lft > to th right of. is th sm sig o oth sis of. Drivtiv Tst If M Vlu Thorm, is ritil poit of f ( ) suh tht f ( ) 0 th. is rltiv mimum of f ( ) if f ( ) 0. is rltiv miimum of f ( ) if f ( ) 0 <. >.. my rltiv mimum, rltiv f. miimum, or ithr if 0 Fiig Rltiv Etrm /or Clssify Critil Poits. Fi ll ritil poits of f ( ).. Us th st rivtiv tst or th rivtiv tst o h ritil poit. If f ( ) is otiuous o th los itrvl [ ] iffrtil o th op itrvl (, ) f f th thr is umr < < suh tht f ( ). Nwto s Mtho If is th th guss for th root/solutio of f ( ) 0 th (+) st guss is provi f ( ) ists. f + f ( ) ( ) Visit http://tutoril.mth.lmr.u for omplt st of Clulus ots. 005 Pul Dwkis
Clulus Cht Sht Rlt Rts Skth pitur itify kow/ukow qutitis. Writ ow qutio rltig qutitis iffrtit with rspt to t usig impliit iffrtitio (i.. o rivtiv vry tim you iffrtit futio of t). Plug i kow qutitis solv for th ukow qutity. E. A 5 foot lr is rstig gist wll. Th ottom is iitilly 0 ft wy is ig push towrs th wll t 4 ft/s. How fst is th top movig ftr s? E. Two popl r 50 ft prt wh o strts wlkig orth. Th glθ hgs t 0.0 r/mi. At wht rt is th ist tw thm hgig wh θ 0.5 r? is gtiv us is rsig. Usig Pythgor Thorm iffrtitig, + y 5 + y y 0 0 7 Aftr s w hv so y 5 7 76. Plug i solv for y. 7 7( 4 ) + 76 y 0 y ft/s 4 76 4 W hv θ 0.0 r/mi. wt to fi. W us vrious trig fs ut sist is, sθ sθ tθ θ 50 50 W kowθ 0.05 so plug i θ solv. s( 0.5) t ( 0.5)( 0.0) 50 0. ft/s Rmmr to hv lultor i ris! Optimiztio Skth pitur if, writ ow qutio to optimiz ostrit. Solv ostrit for o of th two vrils plug ito first qutio. Fi ritil poits of qutio i rg of vrils vrify tht thy r mi/m s. E. W r losig rtgulr fil with E. Dtrmi poit(s) o y + tht r 500 ft of f mtril o si of th losst to (0,). fil is uilig. Dtrmi imsios tht will mimiz th los r. Mimiz A y sujt to ostrit of + y 500. Solv ostrit for plug ito r. A y( 500 y) 500 y 500y y Diffrtit fi ritil poit(s). A 500 4y y 5 By riv. tst this is rl. m. so is th swr whr ftr. Filly, fi. 500 5 50 Th imsios r th 50 5. Miimiz f ( 0) ( y ) ostrit is + th y +. Solv ostrit for plug ito th futio. y f y + y + y y y+ Diffrtit fi ritil poit(s). f y y By th rivtiv tst this is rl. mi. so ll w to o is fi vlu(s). ± Th poits r th (, ) (, ). Visit http://tutoril.mth.lmr.u for omplt st of Clulus ots. 005 Pul Dwkis
Clulus Cht Sht Itgrls Dfiitios Dfiit Itgrl: Suppos f ( ) is otiuous Ati-Drivtiv : A ti-rivtiv of f ( ) o [, ]. Divi [, ] ito suitrvls of is futio, F( ), suh tht F f. * with Δ hoos i from h itrvl. Ifiit Itgrl : f F + * Th f ( ) lim f ( i ) Δ. whr F( ) is ti-rivtiv of f ( ). i Fumtl Thorm of Clulus f is otiuous o [, ] th Vrits of Prt I : u g f ( t) t is lso otiuous o [, ] f () t t u f u g f t t f. f () t t v f v v f is otiuous o[, ], F( ) is u f () t t u f u v f v F f ) Prt I : If () Prt II : ti-rivtiv of f ( )(i.. th f F F. ± ± ± ± f g f g f g f g f ( ) 0 f f If f g o th If f 0 o th 0 Proprtis f f [ ] [ v ], is ostt f f, is ostt f g f f f t t f f If m f M o th m ( ) f( ) M( ) + + k k+ + +, + l + + l l uu ul ( u) u+ u u u + Commo Itgrls osuu si u+ si uu osu+ s uu t u+ sut uu su+ su ot uu su + s uu ot u+ t uu l su + suu l su+ t u + u u u + + u si + t u u Visit http://tutoril.mth.lmr.u for omplt st of Clulus ots. 005 Pul Dwkis
Clulus Cht Sht Str Itgrtio Thiqus Not tht t my shools ll ut th Sustitutio Rul t to tught i Clulus II lss. u Sustitutio : Th sustitutio u g will ovrt u g. For ifiit itgrls rop th limits of itgrtio. E. 5 os u u u u :: u 8 Itgrtio y Prts : uv uv vu g f ( g ) g f ( u) u usig g 5 8 5 si ( u) ( si ( 8) si ( ) ) 8 5 5 os os u uv uv vu. Choos u v from itgrl omput u y iffrtitig u omput v usig v v. E. 5 E. l u v u v u l v u v + + 5 l l l ( ) 5 5 5 5l 5 l Prouts (som) Quotits of Trig Futios m m For si os w hv th followig : For t s w hv th followig :. o. Strip si out ovrt rst to osis usig si os, th us th sustitutio u os.. m o. Strip osi out ovrt rst to sis usig os si, th us th sustitutio u si.. m oth o. Us ithr. or. 4. m oth v. Us oul gl /or hlf gl formuls to ru th itgrl ito form tht itgrt.. o. Strip tgt st out ovrt th rst to sts usig t s, th us th sustitutio u s.. m v. Strip sts out ovrt rst to tgts usig s + t, th us th sustitutio u t.. o m v. Us ithr. or. 4. v m o. Eh itgrl will lt with iffrtly. si si os os os si os Trig Formuls :, ( + ), ( ) E. t s 5 4 ( s ) s t s 4 ( u ) u u ( u s) 5 4 t s t s t s s s + 7 5 7 5 E. si5 os 5 4 si si si (si ) si os os os (os ) si os ( u) u u4 u + u u ( os ) u u s + l os os + Visit http://tutoril.mth.lmr.u for omplt st of Clulus ots. 005 Pul Dwkis
Clulus Cht Sht Trig Sustitutios : If th itgrl otis th followig root us th giv sustitutio formul to ovrt ito itgrl ivolvig trig futios. siθ os θ si θ sθ t θ s θ + tθ s θ + t θ 6 E. 49 siθ osθ θ 4 4si 4os os 4 9 θ θ θ Rll. Bus w hv ifiit itgrl w ll ssum positiv rop solut vlu rs. If w h fiit itgrl w to omput θ s rmov solut vlu rs s o tht, if 0 if < 0 I this s w hv 4 9 osθ. 6 4 si θ θ si θ 9 ( os ) ( os ) θ θ θ s θ otθ + Us Right Trigl Trig to go k to s. From sustitutio w hv siθ so, From this w s tht 49 otθ 6 4 4 9 49. So, + Prtil Frtios : If itgrtig P whr th gr of Q P is smllr th th gr of Q( ). Ftor omitor s ompltly s possil fi th prtil frtio ompositio of th rtiol prssio. Itgrt th prtil frtio ompositio (P.F.D.). For h ftor i th omitor w gt trm(s) i th ompositio orig to th followig tl. Ftor i Q( ) Trm i P.F.D Ftor i Q + A + A+ B + + ( + ) k ( + + ) + + k Trm i P.F.D A A Ak + + + + + + A + B A k + Bk + + + + + + k k 7+ ( )( + 4) E. 7+ 4 + 6 ( )( 4) + + + 4 + + 4 6 + 4 + 4 ( ) 4l + l + 4 + 8t Hr is prtil frtio form romi. A ( + 4 + 4 + 4 7+ A B+ C + 4) + ( B+ C ) ( ) + St umrtors qul ollt lik trms. 7 + A+ B + C B + 4A C St offiits qul to gt systm solv to gt ostts. A+ B 7 C B 4A C 0 A 4 B C 6 A ltrt mtho tht somtims works to fi ostts. Strt with sttig umrtors qul i prvious mpl : 7 + A( + 4) + ( B+ C) ( ). Chos i vlus of plug i. For mpl if w gt 0 5A whih givs A 4. This wo t lwys work sily. Visit http://tutoril.mth.lmr.u for omplt st of Clulus ots. 005 Pul Dwkis
f Clulus Cht Sht Applitios of Itgrls Nt Ar : ( ) rprsts th t r tw f th -is with r ov -is positiv r low -is gtiv. Ar Btw Curvs : Th grl formuls for th two mi ss for h r, y f A f y A y uppr futio lowr futio & right futio lft futio If th urvs itrst th th r of h portio must fou iiviully. Hr r som skths of oupl possil situtios formuls for oupl of possil ss. A f ( y) g( y) y + A f g A f g g f Volums of Rvolutio : Th two mi formuls r V A V A y y. Hr is som grl iformtio out h mtho of omputig som mpls. Rigs Cylirs A π ( outr rius) ( ir rius) A π ( rius) ( with / hight) Limits: /y of right/ot rig to /y of lft/top rig f, f y, Horz. Ais us g( ), A( ). Vrt. Ais us g( y ), A( y ) y. Limits : /y of ir yl. to /y of outr yl. f y, f, Horz. Ais us g( y ), A( y ) y. Vrt. Ais us g( ), A( ). E. Ais : y > 0 E. Ais : y 0 E. Ais : y > 0 E. Ais : y 0 outr rius : f ir rius : g outr rius: + g ir rius: + f rius : y with : f ( y) g( y) rius : + y with : f ( y) g( y) Ths r oly fw ss for horizotl is of rottio. If is of rottio is th -is us th y 0 s with 0. For vrtil is of rottio ( > 0 0 ) itrhg y to gt pproprit formuls. Visit http://tutoril.mth.lmr.u for omplt st of Clulus ots. 005 Pul Dwkis
Work : If for of F movs ojt, th work o is W i Clulus Cht Sht F Avrg Futio Vlu : Th vrg vlu of f ( ) o is fvg f Ar Lgth Surf Ar : Not tht this is oft Cl II topi. Th thr si formuls r, L s SA π y s (rott out -is) SA π s (rott out y-is) whr s is pt upo th form of th futio ig work with s follows. y ( ) s + if y f, s + y if f y, y y y () () s + t if f t, y g t, t t t r s r + θ if r f θ, θ With surf r you my hv to sustitut i for th or y pig o your hoi of s to mth th iffrtil i th s. With prmtri polr you will lwys to sustitut. Impropr Itgrl A impropr itgrl is itgrl with o or mor ifiit limits /or isotiuous itgrs. Itgrl is ll ovrgt if th limit ists hs fiit vlu ivrgt if th limit os t ist or hs ifiit vlu. This is typilly Cl II topi. Ifiit Limit. f ( ) lim t f ( ) t. lim f f tt. f ( ) f ( ) f ( ) + provi BOTH itgrls r ovrgt. Disotiuous Itgr t f lim f f lim f. Disot. t :. Disot. t :. Disotiuity t + t t θ < < : + t f f f provi oth r ovrgt. Compriso Tst for Impropr Itgrls : If f g 0 o [, ) th,. If ov. th ov.. If ivg. th f Usful ft : If > 0 th For giv itgrl f ( ) ivi [, ] g p ovrgs if g p > ivrgs for p. f ivg. Approimtig Dfiit Itgrls (must v for Simpso s Rul) fi Δ ito suitrvls [, ], [, ],, [ ] 0 with 0, * * * Mipoit Rul : Δ ( ) + + + ( ) th, f f f f *, i i, i Δ f f 0 + f ++ f + + f + f Δ f f 0 + 4f + f + + f + 4f + f is mipoit [ ] Trpzoi Rul : Simpso s Rul : Visit http://tutoril.mth.lmr.u for omplt st of Clulus ots. 005 Pul Dwkis