Clulus Chet Sheet Limits Deiitios Preise Deiitio : We sy lim ( ) L i or every e > 0 there is > 0 suh tht wheever 0 L < e. < < the Workig Deiitio : We sy lim L i we mke ( ) s lose to L s we wt y tkig suiietly lose to (o either sie o ) without lettig. Right h limit : lim + L. This hs the sme eiitio s the limit eept it requires >. Let h limit : lim L. This hs the Limit t Iiity : We sy lim sme eiitio s the limit eept it requires egtive. <. Reltioship etwee the limit oesie limits lim ( ) L i lim ( ) lim ( ) L lim ( ) lim ( ) L + + lim lim i lim + Assume lim lim g. limè ( ) lim ( ) Î. limè ( ) ± g( ) lim ( ) ± lim g( ) Î. limè ( ) g( ) lim ( ) lim g( ) Î Note : sg( ) i > 0. lim e & lim e 0. liml ( ) & lim l ( ) 0 + L i we mke ( ) s lose to L s we wt y tkig lrge eough positive. There is similr eiitio or lim eept we require lrge egtive. Iiite Limit : We sy lim L i we mke ( ) ritrrily lrge ( positive) y tkig suiietly lose to (o either sie o ) without lettig. There is similr eiitio or lim eept we mke ( ) ritrrily lrge i lim Does Not Eist Properties oth eist is y umer the, 4. ( ) È lim lim Í Îg lim g. limè ( ) lim ( ) È Î Î 6. limè ( ) lim ( ) Î Bsi Limit Evlutios t ± sg i < 0.. I r > 0 the lim r 0 r 4. I r > 0 is rel or egtive the lim r 0. eve : lim ± 6. o : lim & lim L ( ) provie g lim 0 7. eve : lim L sg ± + + + 8. o : lim L sg + + + 9. o : lim L sg + + + Visit http://tutoril.mth.lmr.eu or omplete set o Clulus otes. 00 Pul Dwkis
Clulus Chet Sheet Cotiuous Futios I ( ) is otiuous t the Cotiuous Futios Compositio ( ) is otiuous t lim g ( ) the ( ) lim lim g g Ftor Cel + 4 + 6 lim lim + 6 8 lim 4 Rtiolize Numertor/Deomitor + lim lim 9 9 8 8 + 9 lim lim 9 9 8 + + 9 + 8 6 08 Comie Rtiol Epressios Ê ˆ Ê ( + h) ˆ lim Á lim h0 h h h0 Ë + há ( h) Ë + Ê h ˆ lim lim h0 há ( h) h0 Ë + ( + h) Evlutio Tehiques L Hospitl s Rule lim ( ) 0 I lim or lim g 0 g lim lim g g ± ± the, is umer, or Polyomils t Iiity q re polyomils. To ompute p( ) p( ) lim ± q( ) o oth p( ) tor lrgest power o i q( ) out q the ompute limit. 4 4 4 lim lim lim ( ) Pieewise Futio Ï + i < lim g( ) where g( ) Ì Ó i Compute two oe sie limits, lim g lim + 9 lim g lim 7 + + Oe sie limits re ieret so lim g( ) oes t eist. I the two oe sie limits h lim g woul hve eiste ee equl the h the sme vlue. Some Cotiuous Futios Prtil list o otiuous utios the vlues o or whih they re otiuous.. Polyomils or ll. 7. os( ) si ( ) or ll.. Rtiol utio, eept or s tht give ivisio y zero. 8. t ( ) se( ) provie. ( o) or ll. p p p p L,,,,, L 4. ( eve) or ll 0. 9. ot. e or ll. ( ) s( ) provie 6. l or > 0. L, p, p,0, p, p, L Itermeite Vlue Theorem Suppose tht ( ) is otiuous o [, ] let M e y umer etwee ( ) The there eists umer suh tht < < ( ) M.. Visit http://tutoril.mth.lmr.eu or omplete set o Clulus otes. 00 Pul Dwkis
Clulus Chet Sheet Derivtives Deiitio Nottio I y ( ) the the erivtive is eie to e ( ) ( + ). h h 0 lim h I y ( ) the ll o the ollowig re equivlet ottios or the erivtive. y ( ) y ( ) D I y ( ) ll o the ollowig re equivlet ottios or erivtive evlute t. y y D I y ( ) the,. m is the slope o the tget lie to y ( ) t the equtio o the tget lie t is y +. give y I ( ). ( ) ( ) Iterprettio o the Derivtive is the istteous rte o. hge o ( ) t.. I ( ) is the positio o ojet t time the is the veloity o the ojet t. Bsi Properties Formuls g re ieretile utios (the erivtive eists), re y rel umers,. ( ± g) ( ) ± g ( ). g g+ g Prout Rule Ê ˆ g g 4. Á Quotiet Rule Ë g g Power Rule ( ( g )) g( ) g This is the Chi Rule. ( ) 0 6. 7. ( ) ( si) ( os) os si ( t) se se set Commo Derivtives ( s) sot ( ot) s ( si ) ( os ) ( t ) + ( ) l ( ) ( e ) e ( l ( ) ), > 0 ( l ), 0 ( log ( ) ), > 0 l Visit http://tutoril.mth.lmr.eu or omplete set o Clulus otes. 00 Pul Dwkis
Clulus Chet Sheet Chi Rule Vrits The hi rule pplie to some speii utios.. ( È ( ) ) È ( ) ( ) Î Î. ( osè ( ) ) ( ) si ( ) Î ÈÎ ( ) ( ). ( e ) ( ) e 6. ( tè ( ) ) ( ) se ( ) Î ÈÎ ( ). ( l È ( ) ) 7. se Î [ ] se t ( ) ( ) 4. ( siè ( ) ) ( ) os È ( ) 8. ( t È ( ) ) Î Î Î +ÈÎ ( ) ( ( ) ) ( ) [ ( ) ] [ ( ) ] Higher Orer Derivtives The Seo Derivtive is eote s The th Derivtive is eote s ( ) ( ( ) is eie s ) ( ) is eie s ( ) ( ( )), i.e. the erivtive o the ( ) ( ) ( ( ) ), i.e. the erivtive o irst erivtive, ( ). the () st erivtive,. Impliit Dieretitio + y si y + y y here, so prouts/quotiets o y 9y Fi y i e. Rememer will use the prout/quotiet rule erivtives o y will use the hi rule. The trik is to ieretite s orml every time you ieretite y you tk o y (rom the hi rule). Ater ieretitig solve or y. e ( y ) e y yy ( y) y e y e y 9y 9 + + os + y + y + yy y y + i y 9y 9y e 9 os y 9 os y y y 9 9 Critil Poits is ritil poit o. ( ) 0 or. ( ) Iresig/Deresig Cove Up/Cove Dow oes t eist. provie either Iresig/Deresig. I ( ) > 0 or ll i itervl I the ( ) is iresig o the itervl I. < or ll i itervl I the. I ( ) 0 ( ) is eresig o the itervl I. or ll i itervl I the. I ( ) 0 ( ) is ostt o the itervl I. 9y e y 9y y9e os ( y) Cove Up/Cove Dow. I ( ) > 0 or ll i itervl I the ( ) is ove up o the itervl I. < or ll i itervl I the. I ( ) 0 ( ) is ove ow o the itervl I. Iletio Poits is iletio poit o ovity hges t. i the Visit http://tutoril.mth.lmr.eu or omplete set o Clulus otes. 00 Pul Dwkis
Asolute Etrem. is solute mimum o ( ) i ( ) ( ) or ll i the omi. is solute miimum o ( ). i ( ) ( ) or ll i the omi. Fermt s Theorem hs reltive (or lol) etrem t I, the is ritil poit o Clulus Chet Sheet. Etreme Vlue Theorem is otiuous o the lose itervl I [, ] the there eist umers so tht,.,,. ( ) is the s. m. i [, ],. is the s. mi. i [, ]. Fiig Asolute Etrem To i the solute etrem o the otiuous, use the utio ( ) o the itervl [ ] ollowig proess.. Fi ll ritil poits o ( ) i [, ].. Evlute ( ) t ll poits ou i Step.. Evlute ( ) ( ). 4. Ietiy the s. m. (lrgest utio vlue) the s. mi.(smllest utio vlue) rom the evlutios i Steps &. Etrem Reltive (lol) Etrem. is reltive (or lol) mimum o or ll er. ( ) i. is reltive (or lol) miimum o or ll er. ( ) i st Derivtive Test I is ritil poit o. rel. m. o ( ) i ( ) 0 the is > to the let o ( ) < 0 to the right o.. rel. mi. o ( ) i ( ) 0 < to the let o ( ) > 0to the right o. is. ot reltive etrem o ( ) i ( ) the sme sig o oth sies o Derivtive Test I Me Vlue Theorem,. is ritil poit o ( ) suh tht ( ) 0 the. is reltive mimum o ( ) i ( ) 0. is reltive miimum o ( ) i ( ) 0. my e reltive mimum, reltive miimum, or either i ( ) 0. Fiig Reltive Etrem /or Clssiy Critil Poits.. Fi ll ritil poits o. Use the st erivtive test or the erivtive test o eh ritil poit. <. >. I ( ) is otiuous o the lose itervl [ ] ieretile o the ope itervl (, ) the there is umer < < suh tht ( ). Newto s Metho I is the th guess or the root/solutio o ( ) 0 the (+) st guess is provie ( ) eists. + ( ) ( ) Visit http://tutoril.mth.lmr.eu or omplete set o Clulus otes. 00 Pul Dwkis
Clulus Chet Sheet Relte Rtes Sketh piture ietiy kow/ukow qutities. Write ow equtio reltig qutities ieretite with respet to t usig impliit ieretitio (i.e. o erivtive every time you ieretite utio o t). Plug i kow qutities solve or the ukow qutity. E. A oot ler is restig gist wll. The ottom is iitilly 0 t wy is eig pushe towrs the wll t 4 t/se. How st is the top movig ter se? E. Two people re 0 t prt whe oe strts wlkig orth. The gleq hges t 0.0 r/mi. At wht rte is the iste etwee them hgig whe q 0. r? is egtive euse is eresig. Usig Pythgore Theorem ieretitig, + y i + yy 0 Ater se we hve 0 7 so y 7 76. Plug i solve or y. 7 7( 4 ) + 76 y 0 i y t/se 4 76 4 We hve q 0.0 r/mi. wt to i. We use vrious trig s ut esiest is, seq i seq tqq 0 0 We kowq 0. so plug i q solve. se( 0.) t( 0.)( 0.0) 0 0. t/se Rememer to hve lultor i ris! Optimiztio Sketh piture i eee, write ow equtio to e optimize ostrit. Solve ostrit or oe o the two vriles plug ito irst equtio. Fi ritil poits o equtio i rge o vriles veriy tht they re mi/m s eee. E. We re elosig retgulr iel with E. Determie poit(s) o y + tht re 00 t o ee mteril oe sie o the losest to (0,). iel is uilig. Determie imesios tht will mimize the elose re. Mimize A y sujet to ostrit o + y 00. Solve ostrit or plug ito re. A y( 00y) 00y i 00y y Dieretite i ritil poit(s). A 004y i y By eriv. test this is rel. m. so is the swer we re ter. Filly, i. 00 0 The imesios re the 0. Visit http://tutoril.mth.lmr.eu or omplete set o Clulus otes. Miimize ( 0) ( y ) ostrit is + the y +. Solve ostrit or plug ito the utio. y y i + y + y y y+ Dieretite i ritil poit(s). y i y By the erivtive test this is rel. mi. so ll we ee to o is i vlue(s). i ± The poits re the (, ) (, ). 00 Pul Dwkis
Deiite Itegrl: Suppose o [, ]. Divie [, ] with D hoose Clulus Chet Sheet Itegrls Deiitios is otiuous ito suitervls o rom eh itervl. * i * The limâ ( i ) D. i AtiDerivtive : A tierivtive o ( ) is utio, F( ), suh tht F ( ) ( ). Ieiite Itegrl : ( ) F( ) + where F( ) is tierivtive o ( ). Fumetl Theorem o Clulus is otiuous o [, ] the Vrits o Prt I : u g( ) () t t is lso otiuous o [, ] () t t u ( ) u( ) È Î g t t. () t t v ( ) v( ) È v Î is otiuous o[, ], F( ) is u () t t u u v v F ) Prt I : I () Prt II : tierivtive o ( ) (i.e. F F. the ± ± ± ± g g g g 0 + or y vlue o. I ( ) g( ) o the I ( ) 0 o the 0 Properties [ ] [ v ], is ostt ( ) ( ), is ostt g I m ( ) M o the m ( ) ( ) M ( ) + + k k+ + +, + l + + l luu ul ( u) u+ u u e u e + Commo Itegrls osuu si u+ Visit http://tutoril.mth.lmr.eu or omplete set o Clulus otes. siuu osu+ se uu t u+ seutuu seu+ suotuu su+ s uu ot u+ tuu l seu + seuu l seu+ t u + u u u + + t u u si u + 00 Pul Dwkis
Clulus Chet Sheet Str Itegrtio Tehiques Note tht t my shools ll ut the Sustitutio Rule te to e tught i Clulus II lss. u Sustitutio : The sustitutio u g( ) will overt u g. For ieiite itegrls rop the limits o itegrtio. E. os u i u i u :: 8 i u i u Itegrtio y Prts : uv uv vu g ( g ) g ( ) ( u) u usig g 8 si( u) ( si( 8) si() ) 8 itegrl ompute u y ieretitig u ompute v usig v E. u v e i u ve e e + e e e + os os u u uv uv vu. Choose u v rom E. l v. u l v i u v ( ) l l l l l Prouts (some) Quotiets o Trig Futios m m For si os we hve the ollowig : For t se we hve the ollowig :. o. Strip sie out overt rest to osies usig si os, the use the sustitutio u os.. m o. Strip osie out overt rest to sies usig os si, the use the sustitutio u si.. m oth o. Use either. or. 4. m oth eve. Use oule gle /or hl gle ormuls to reue the itegrl ito orm tht e itegrte.. o. Strip tget set out overt the rest to sets usig t se, the use the sustitutio u se.. m eve. Strip sets out overt rest to tgets usig se + t, the use the sustitutio u t.. o m eve. Use either. or. 4. eve m o. Eh itegrl will e elt with ieretly. si si os os os si os Trig Formuls :, +, E. t se 4 ( se se ) tse 4 ( u ) uu ( u se ) 4 t se t se t se se se + 7 7 E. si os 4 si si si (si ) si os os os (os ) si os (u) u u4 u + u u ( os ) u u se + l os os + Visit http://tutoril.mth.lmr.eu or omplete set o Clulus otes. 00 Pul Dwkis
Clulus Chet Sheet Trig Sustitutios : I the itegrl otis the ollowig root use the give sustitutio ormul to overt ito itegrl ivolvig trig utios. i siq os q si q i seq t q se q + i tq se q + t q 6 E. 49 si q i osq q 4 4si 4os 4 9 q q os q Rell. Beuse we hve ieiite itegrl we ll ssume positive rop solute vlue rs. I we h eiite itegrl we ee to ompute q s remove solute vlue rs se o tht, Ï i 0 Ì Ó i < 0 I this se we hve 4 9 osq. Û ı 6 4 si q q si q 9 ( os ) ( os ) q q q s ot q q + Use Right Trigle Trig to go k to s. From sustitutio we hve siq so, From this we see tht ot 49 q. So, 6 4 49 49 + Prtil Frtios : I itegrtig P where the egree o Q P is smller th the egree o Q( ). Ftor eomitor s ompletely s possile i the prtil rtio eompositio o the rtiol epressio. Itegrte the prtil rtio eompositio (P.F.D.). For eh tor i the eomitor we get term(s) i the eompositio orig to the ollowig tle. Ftor i Q( ) Term i P.F.D Ftor i Q( ) + A + A+ B + + ( + ) k ( + + ) + + k Term i P.F.D A A Ak + + L + + + + A + B A k + Bk + L + + + + + k k 7+ ( )( + 4) E. 7+ 4 + 6 ( )( 4) + + + 4 + + 4 6 + 4 + 4 ( ) 4l + l + 4 + 8t Here is prtil rtio orm reomie. A ( + 4 + 4 + 4 7+ A B+ C + 4) + ( B+ C)( ) + Set umertors equl ollet like terms. 7 + A+ B + C B + 4A C Set oeiiets equl to get system solve to get ostts. A+ B 7 C B 4A C 0 A 4 B C 6 A lterte metho tht sometimes works to i ostts. Strt with settig umertors equl i 7 + A + 4 + B+ C. Chose ie vlues o plug i. previous emple : For emple i we get 0 A whih gives A 4. This wo t lwys work esily. Visit http://tutoril.mth.lmr.eu or omplete set o Clulus otes. 00 Pul Dwkis
Clulus Chet Sheet Applitios o Itegrls Net Are : ( ) represets the et re etwee the is with re ove is positive re elow is egtive. Are Betwee Curves : The geerl ormuls or the two mi ses or eh re, Èupper utio È Îlower utio Î & i Îright utio Îlet utio y i A È È y A y I the urves iterset the the re o eh portio must e ou iiviully. Here re some skethes o ouple possile situtios ormuls or ouple o possile ses. A ( y) g( y) y + A g A g g Volumes o Revolutio : The two mi ormuls re V A( ) V A y y. Here is some geerl iormtio out eh metho o omputig some emples. Rigs Cyliers A p ( ( outer rius) ( ier rius) ) A p ( rius) ( with / height) Limits: /y o right/ot rig to /y o let/top rig Limits : /y o ier yl. to /y o outer yl., y, y,, Horz. Ais use g( ), A( ). Vert. Ais use g( y ), A( y ) y. Horz. Ais use g( y ), A( y ) y. Vert. Ais use g( ), A( ). E. Ais : y > 0 E. Ais : y 0 E. Ais : y > 0 E. Ais : y 0 outer rius : ( ) ier rius : g( ) outer rius: + g( ) ier rius: + ( ) rius : y with : ( y) g( y) rius : + y with : ( y) g( y) These re oly ew ses or horizotl is o rottio. I is o rottio is the is use the y 0 se with 0. For vertil is o rottio ( > 0 0 ) iterhge y to get pproprite ormuls. Visit http://tutoril.mth.lmr.eu or omplete set o Clulus otes. 00 Pul Dwkis
Work : I ore o F( ) moves ojet i, the work oe is W Clulus Chet Sheet F Averge Futio Vlue : The verge vlue o ( ) o is vg ( ) Ar Legth Sure Are : Note tht this is ote Cl II topi. The three si ormuls re, L s SA p ys (rotte out is) SA p s (rotte out yis) where s is epeet upo the orm o the utio eig worke with s ollows. y ( ) s + i y, s + y i y, y y y () () s + t i t, y g t, t t t r s r + q i r q, q With sure re you my hve to sustitute i or the or y epeig o your hoie o s to mth the ieretil i the s. With prmetri polr you will lwys ee to sustitute. Improper Itegrl A improper itegrl is itegrl with oe or more iiite limits /or isotiuous itegrs. Itegrl is lle overget i the limit eists hs iite vlue iverget i the limit oes t eist or hs iiite vlue. This is typilly Cl II topi. Iiite Limit. lim t ( ). ( ) lim ( ) t. + + t t q t provie BOTH itegrls re overget. Disotiuous Itegr t. Disot. t : ( ) lim ( ). Disot. t : ( ) lim ( ). Disotiuity t < < : + t t provie oth re overget. Compriso Test or Improper Itegrls : I ( ) g( ) 0 o [, ) the,. I ov. the ov.. I ivg. the Useul t : I > 0 the For give itegrl ( ) ivie [, ] g p g overges i p > iverges or p. ivg. Approimtig Deiite Itegrls (must e eve or Simpso s Rule) eie D ito suitervls [, ], [, ],, [ ] 0 with 0, * * * Mipoit Rule : ªD È ( ) + + + ( ) the, * Î L, i i, i D ª È 0 + ++ + + + Î L D ª È 0 + 4 + + + + 4 + Î L is mipoit [ ] Trpezoi Rule : Simpso s Rule : Visit http://tutoril.mth.lmr.eu or omplete set o Clulus otes. 00 Pul Dwkis