( ) ( ) 1 ( ) Algebra Cheat Sheet ( ) (, ) = ( ) + ( ) ( )( ) Basic Properties & Facts Arithmetic Operations. Properties of Inequalities. b a.

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1 Alger Chet Sheet Bsi Proerties & Fts Arithmeti Oertios Proerties of Iequlities If < the+ < + < + ( + ) If < > the < < If < < the > > Proerties of Asolute Vlue + if + if < , + + Trigle Iequlit Diste Formul Eoet Proerties If P (, ) P (, ) re two m + m m oits the iste etwee them is m m m m, m m m Proerties of Rils m m,if is o,if is eve (, ) ( ) + ( ) P P Comle Numers i i i ( + i) + ( + i) + + ( + ) i ( + i) ( + i) + ( ) i ( + )( + ) + ( + ) ( + i)( i) +, i i i + i + + i i + i + i + i Comle Moulus Comle Cojugte For omlete set of olie Alger otes visit htt://tutoril.mth.lmr.eu. 5 Pul Dwkis

2 Logrithms Log Proerties Defiitio log is equivlet to Emle log 5 euse Seil Logrithms l log turl log e log log ommo log where e.78888k Ftorig Formuls If is o the, + + L+ + ( )( L ) Solve 6 Logrithm Proerties log log log log log r rlog Ftorig Solvig log log + log log log log The omi of log is > Qurti Formul Solve + +, If If If ± 4 4 > - Two rel uequl sols. 4 - Reete rel solutio. 4 < - Two omle solutios. Squre Root Proert If the ± Asolute Vlue Equtios/Iequlities If is ositive umer or < < < > < or > Comletig the Squre (4) Ftor the left sie () Divie the oeffiiet of the 5 () Move the ostt to the other sie. 5 () Tke hlf the oeffiiet of, squre it it to oth sies (5) Use Squre Root Proert 9 9 ± ± 4 (6) Solve for 9 ± For omlete set of olie Alger otes visit htt://tutoril.mth.lmr.eu. 5 Pul Dwkis

3 Costt Futio or f Grh is horizotl lie ssig through the oit (, ). Lie/Lier Futio m+ or f m+ Grh is lie with oit (, ) sloe m. Sloe Sloe of the lie otiig the two,, is oits ( ) rise m ru Sloe iteret form The equtio of the lie with sloe m, is -iteret m+ Poit Sloe form The equtio of the lie with sloe m, is ssig through the oit ( ) + m( ) Prol/Qurti Futio h + k f h + k The grh is rol tht oes u if > or ow if < hs verte hk,. t Prol/Qurti Futio + + f + + The grh is rol tht oes u if > or ow if < hs verte t, f. Futios Grhs Prol/Qurti Futio + + g + + The grh is rol tht oes right if > or left if < hs verte t g,. Cirle h + k r Grh is irle with rius r eter hk,. Ellise ( h) ( k) + Grh is ellise with eter ( hk, ) with verties uits right/left from the eter verties uits u/ow from the eter. Herol ( h) ( k) Grh is herol tht oes left hk,, verties right, hs eter t uits left/right of eter smtotes tht ss through eter with sloe ±. Herol ( k) ( h) Grh is herol tht oes u hk,, verties ow, hs eter t uits u/ow from the eter smtotes tht ss through eter with sloe ±. For omlete set of olie Alger otes visit htt://tutoril.mth.lmr.eu. 5 Pul Dwkis

4 Commo Algeri Errors Error Reso/Corret/Justifitio/Emle Divisio zero is uefie! 9 ( ) , ( ) 9 Wth rethesis! ( ) A more omle versio of the revious error Bewre of iorret elig! + Mke sure ou istriute the -! See revious error ( + ) ( + ) ( + ) ( + ) More geerl versios of revious three errors Squre first the istriute! See the revious emle. You ot ftor out ostt if there is ower o the rethesis! Now see the revious error. For omlete set of olie Alger otes visit htt://tutoril.mth.lmr.eu. 5 Pul Dwkis

5 Right trigle efiitio For this efiitio we ssume tht < q < or < q < 9. oosite oosite siq hoteuse jet osq hoteuse oosite tq jet Trig Chet Sheet Defiitio of the Trig Futios hoteuse jet hoteuse sq oosite hoteuse seq jet jet otq oosite Uit irle efiitio For this efiitio q is gle. siq sq osq seq tq otq Fts Proerties Domi The omi is ll the vlues of q tht Perio e lugge ito the futio. The erio of futio is the umer, T, suh tht f ( q + T) f ( q). So, if w siq, q e gle is fie umer q is gle we osq, q e gle hve the followig erios. Ê tq, q Á+,, ±, ±, K Ë sq, q,, ±, ±, K Ê seq, q Á+,, ±, ±, K Ë otq, q,, ±, ±, K Rge The rge is ll ossile vlues to get out of the futio. - siq sq sq - - osq seq seq - - < tq < - < otq < q (, ) si ( wq) os( wq) Æ Æ t ( wq) Æ T s( wq) se( wq) Æ Æ q ot ( wq) Æ T T w T w w T w T w w 5 Pul Dwkis

6 Tget Cotget Ietities siq osq tq otq osq siq Reirol Ietities sq siq siq sq seq osq osq seq otq tq tq otq Pthgore Ietities si q + os q t q + se q + ot q s q Eve/O Formuls si - q -siq s - q -sq os - q osq se - q seq t - q -tq ot - q -otq Perioi Formuls If is iteger. si q + siq s q + sq os q + osq se q + seq t q + tq ot q + otq Doule Agle Formuls si q siqosq os os -si t q q q ( q ) q - os -si q tq - t q Degrees to Ris Formuls If is gle i egrees t is gle i ris the t 8 fi t t 8 8 Formuls Ietities Hlf Agle Formuls si q -os( q) os q + os( q) -os( q ) t q + os q Sum Differee Formuls si ± sios ± ossi os ± osos msisi t ± t t ( ± ) m t t Prout to Sum Formuls sisi Èos os Î osos Èos os Î sios Èsi( ) si ( ) Î ossi Èsi( + ) -si ( -) Î Sum to Prout Formuls Ê + Ê - si + si siá osá Ë Ë Ê + Ê - si - si osá si Á Ë Ë Ê + Ê - os + os osá osá Ë Ë Ê + Ê - os - os -siá si Á Ë Ë Cofutio Formuls Ê Ê siá - q osq osá - q siq Ë Ë Ê Ê sá - q seq seá - q sq Ë Ë Ê Ê tá - q otq otá - q tq Ë Ë 5 Pul Dwkis

7 Uit Cirle Ê Á- Ë Ê Á- Ë,, 5 6 Ê Á-, Ë (, ) Ê Á, Ë 4 Ê Á Ë 6, Ê Á Ë, (-,) 8 6 (, ) Ê Á-,- Ë 7 6 Ê Á-,- Ë 5 4 Ê Á-,- Ë (, - ) Ê Á,- Ë 6 Ê Á,- Ë Ê Á,- Ë For orere ir o the uit irle (, ) : osq siq Emle Ê5 Ê5 osá si Á - Ë Ë 5 Pul Dwkis

8 Defiitio - si is equivlet to si - os is equivlet to os - t is equivlet to t Domi Rge Futio Domi Rge - si os - - t - < < - < < Iverse Trig Futios Iverse Proerties - - os os os os q q ( -( ) ) -( ( q) ) -( ) -( q ) si si si si q t t t t q Alterte Nottio - si rsi os t - - ros rt Lw of Sies, Cosies Tgets g Lw of Sies si si sig Lw of Cosies + - os os osg Mollweie s Formul + os ( - ) si g Lw of Tgets - t - + t + ( -g) ( g) ( -g) ( g) - t + t + - t + t + 5 Pul Dwkis

9 Clulus Chet Sheet Limits Defiitios Preise Defiitio : We s lim f ( ) L if Æ for ever e > there is > suh tht wheever f - L < e. < - < the Workig Defiitio : We s lim f L if we mke f ( ) s lose to L s we wt tkig suffiietl lose to (o either sie of ) without lettig. Right h limit : lim + Æ Æ f L. This hs the sme efiitio s the limit eet it requires >. Left h limit : lim - Æ f L. This hs the Limit t Ifiit : We s lim sme efiitio s the limit eet it requires egtive. <. Reltioshi etwee the limit oe-sie limits lim f ( ) L fi lim f ( ) lim f ( ) L lim f ( ) lim f ( ) L Æ Æ Æ lim lim fi lim f f + - Æ Æ Assume lim lim f Æ Æ g. limèf ( ) lim f ( ) Î Æ Æ. limèf ( ) ± g( ) lim f ( ) ± lim g( ) Î Æ Æ Æ. limèf ( ) g( ) lim f ( ) lim g( ) Î Æ Æ Æ Note : sg( ) if >. lim e & lim e Æ Æ-. liml ( ) & lim l ( ) Æ Æ + Æ f L if we mke f ( ) s lose to L s we wt tkig lrge eough ositive. There is similr efiitio for lim Æ- eet we require lrge egtive. Ifiite Limit : We s lim f Æ f L if we mke f ( ) ritrril lrge ( ositive) tkig suffiietl lose to (o either sie of ) without lettig. There is similr efiitio for lim f Æ eet we mke f ( ) ritrril lrge Æ Æ f Æ fi lim Does Not Eist Proerties oth eist is umer the, 4. ( ) È f lim f Æ lim Í Æ Îg lim g Æ 5. limèf ( ) lim f ( ) È Î Î Æ Æ 6. limè f ( ) lim f ( ) Î Æ Æ Bsi Limit Evlutios t ± sg - if <. -. If r > the lim r Æ r 4. If r > is rel for egtive the lim r Æ- 5. eve : lim Ʊ 6. o : lim & lim Æ - f L Æ ( ) rovie g Æ Æ- lim - 7. eve : lim L sg Ʊ o : lim L sg Æ o : lim L sg Æ Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus otes. 5 Pul Dwkis

10 Clulus Chet Sheet Cotiuous Futios If f ( ) is otiuous t the Cotiuous Futios Comositio f ( ) is otiuous t lim g ( ) the Æ ( ) lim lim f g f g f Æ Æ Ftor Cel lim lim Æ - Æ lim 4 Æ Rtiolize Numertor/Deomitor lim lim Æ9 9-8 Æ lim lim Æ9 Æ Comie Rtiol Eressios Ê Ê - ( + h) lim Á - lim hæ h h hæ Ë + há ( h) Ë + Ê -h - lim lim hæ há ( h) - hæ Ë + ( + h) Evlutio Tehiques L Hositl s Rule lim f f f Æ ( ) f If lim or lim Æ g Æ g f f lim lim Æ g Æ g ± ± the, is umer, or - Polomils t Ifiit q re olomils. To omute ( ) ( ) lim Ʊ q( ) of oth ( ) ftor lrgest ower of i q( ) out q the omute limit lim lim lim 5 - Æ- 5- Æ- Æ- - 5 ( -) Pieewise Futio Ï + 5 if <- lim g( ) where g( ) Ì Æ- Ó - if - Comute two oe sie limits, lim g lim Æ- Æ- lim g lim Æ- Æ- Oe sie limits re ifferet so lim g( ) Æ- oes t eist. If the two oe sie limits h lim g woul hve eiste ee equl the Æ- h the sme vlue. Some Cotiuous Futios Prtil list of otiuous futios the vlues of for whih the re otiuous.. Polomils for ll. 7. os( ) si ( ) for ll.. Rtiol futio, eet for s tht give ivisio zero. 8. t ( ) se( ) rovie. ( o) for ll. L, -,-,,, L 4. ( eve) for ll. 9. ot 5. e for ll. ( ) s( ) rovie 6. l for >. L, -, -,,,, L Itermeite Vlue Theorem Suose tht f ( ) is otiuous o [, ] let M e umer etwee f ( ) The there eists umer suh tht < < f ( ) M. f. Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus otes. 5 Pul Dwkis

11 Clulus Chet Sheet Derivtives Defiitio Nottio If f ( ) the the erivtive is efie to e f ( ) ( + ) -. h h Æ f lim h f If f ( ) the ll of the followig re equivlet ottios for the erivtive. f f ( ) f ( ) Df If f ( ) ll of the followig re equivlet ottios for erivtive evlute t. f f Df If f ( ) the,. m f is the sloe of the tget lie to f ( ) t the equtio of the tget lie t is f + f -. give If f ( ). ( f ) f ( ) Iterrettio of the Derivtive f is the istteous rte of. hge of f ( ) t.. If f ( ) is the ositio of ojet t time the f is the veloit of the ojet t. Bsi Proerties Formuls g re ifferetile futios (the erivtive eists), re rel umers,. ( f ± g) f ( ) ± g ( ). fg f g+ fg Prout Rule Ê f f g- fg 4. Á Quotiet Rule Ë g g - Power Rule ( f ( g )) f g( ) g This is the Chi Rule 5. ( ) ( ) ( si) ( os) os - si ( t) se se set Commo Derivtives ( s) - sot ( ot) - s - ( si ) - - ( os ) ( t ) + ( ) l ( ) ( e ) e ( l ( ) ), > ( l ), ( log ( ) ), > l Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus otes. 5 Pul Dwkis

12 Clulus Chet Sheet Chi Rule Vrits The hi rule lie to some seifi futios. -. ( Èf ( ) ) Èf ( ) f ( ) Î Î 5. ( osèf ( ) ) f ( ) si f ( ) Î - ÈÎ f ( ) f ( ). ( e ) f ( ) e 6. ( tèf ( ) ) f ( ) se f ( ) Î ÈÎ f ( ). ( l Èf ( ) ) 7. se Î [ ] se t f ( ) f ( ) 4. ( sièf ( ) ) f ( ) os Èf ( ) 8. ( t - Èf ( ) ) Î Î Î +ÈÎf ( ) ( f ( ) ) f ( ) [ f ( ) ] [ f ( ) ] Higher Orer Derivtives The Seo Derivtive is eote s The th Derivtive is eote s ( ) f ( f f ( ) is efie s ) f f ( ) is efie s f ( ) f ( ) -, i.e. the erivtive of the f f, i.e. the erivtive of. first erivtive, f ( ) ( ( ) ) the (-) st erivtive, f ( - ) ( ) Imliit Differetitio + si + here, so routs/quotiets of -9 Fi if e. Rememer will use the rout/quotiet rule erivtives of will use the hi rule. The trik is to ifferetite s orml ever time ou ifferetite ou tk o (from the hi rule). After ifferetitig solve for. e ( ) e ( ) - - e e os fi -9-9 e 9 os 9 os 9 9 Critil Poits is ritil oit of f. f ( ) or. f ( ) Iresig/Deresig Cove U/Cove Dow oes t eist. rovie either Iresig/Deresig. If f ( ) > for ll i itervl I the f ( ) is iresig o the itervl I. < for ll i itervl I the. If f ( ) f ( ) is eresig o the itervl I. for ll i itervl I the. If f ( ) f ( ) is ostt o the itervl I e e -os ( ) Cove U/Cove Dow. If f ( ) > for ll i itervl I the f ( ) is ove u o the itervl I. < for ll i itervl I the. If f ( ) f ( ) is ove ow o the itervl I. Ifletio Poits is ifletio oit of f ovit hges t. if the Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus otes. 5 Pul Dwkis

13 Asolute Etrem. is solute mimum of f ( ) if f ( ) f ( ) for ll i the omi. is solute miimum of f ( ). if f ( ) f ( ) for ll i the omi. Fermt s Theorem f hs reltive (or lol) etrem t If, the is ritil oit of Clulus Chet Sheet f. Etreme Vlue Theorem f is otiuous o the lose itervl If [, ] the there eist umers so tht,.,,. f ( ) is the s. m. i [, ],. f is the s. mi. i [, ]. Fiig Asolute Etrem To fi the solute etrem of the otiuous, use the futio f ( ) o the itervl [ ] followig roess.. Fi ll ritil oits of f ( ) i [, ].. Evlute f ( ) t ll oits fou i Ste.. Evlute f ( ) f ( ). 4. Ietif the s. m. (lrgest futio vlue) the s. mi.(smllest futio vlue) from the evlutios i Stes &. Etrem Reltive (lol) Etrem. is reltive (or lol) mimum of f f for ll er. f ( ) if. is reltive (or lol) miimum of f f for ll er. f ( ) if st Derivtive Test If is ritil oit of. rel. m. of f ( ) if f ( ) f the is > to the left of f ( ) < to the right of.. rel. mi. of f ( ) if f ( ) < to the left of f ( ) > to the right of. is. ot reltive etrem of f ( ) if f ( ) the sme sig o oth sies of Derivtive Test If Me Vlue Theorem,. is ritil oit of f ( ) suh tht f ( ) the. is reltive mimum of f ( ) if f ( ). is reltive miimum of f ( ) if f ( ). m e reltive mimum, reltive miimum, or either if f ( ). Fiig Reltive Etrem /or Clssif Critil Poits. Fi ll ritil oits of f ( ).. Use the st erivtive test or the erivtive test o eh ritil oit. <. >. If f ( ) is otiuous o the lose itervl [ ] ifferetile o the oe itervl (, ) f - f the there is umer < < suh tht f ( ). - Newto s Metho If is the th guess for the root/solutio of f ( ) the (+) st guess is rovie f ( ) eists. f - + f ( ) ( ) Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus otes. 5 Pul Dwkis

14 Clulus Chet Sheet Relte Rtes Sketh iture ietif kow/ukow qutities. Write ow equtio reltig qutities ifferetite with reset to t usig imliit ifferetitio (i.e. o erivtive ever time ou ifferetite futio of t). Plug i kow qutities solve for the ukow qutit. E. A 5 foot ler is restig gist wll. The ottom is iitill ft w is eig ushe towrs the wll t 4 ft/se. How fst is the to movig fter se? E. Two eole re 5 ft rt whe oe strts wlkig orth. The gleq hges t. r/mi. At wht rte is the iste etwee them hgig whe q.5 r? is egtive euse is eresig. Usig Pthgore Theorem ifferetitig, + 5 fi + After se we hve - 7 so Plug i solve for. 7 7( - 4 ) + 76 fi ft/se We hve q. r/mi. wt to fi. We use vrious trig fs ut esiest is, seq fi seq tqq 5 5 We kowq.5 so lug i q solve. se(.5) t(.5)(.) 5. ft/se Rememer to hve lultor i ris! Otimiztio Sketh iture if eee, write ow equtio to e otimize ostrit. Solve ostrit for oe of the two vriles lug ito first equtio. Fi ritil oits of equtio i rge of vriles verif tht the re mi/m s eee. E. We re elosig retgulr fiel with E. Determie oit(s) o + tht re 5 ft of fee mteril oe sie of the losest to (,). fiel is uilig. Determie imesios tht will mimize the elose re. Mimize A sujet to ostrit of + 5. Solve ostrit for lug ito re. A ( 5-) 5- fi 5 - Differetite fi ritil oit(s). A 5-4 fi 5 B eriv. test this is rel. m. so is the swer we re fter. Fill, fi The imesios re the 5 5. Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus otes. Miimize f ( ) ( ) ostrit is the +. Solve ostrit for lug ito the futio. f - fi Differetite fi ritil oit(s). f - fi B the erivtive test this is rel. mi. so ll we ee to o is fi vlue(s). - fi ± The oits re the (, ) (, ) -. 5 Pul Dwkis

15 Defiite Itegrl: Suose f o [, ]. Divie [, ] with D hoose Clulus Chet Sheet Itegrls Defiitios is otiuous ito suitervls of from eh itervl. * i * The limâ ( i ) f f D. Æ i Ati-Derivtive : A ti-erivtive of f ( ) is futio, F( ), suh tht F ( ) f ( ). Iefiite Itegrl : f ( ) F( ) + where F( ) is ti-erivtive of f ( ). Fumetl Theorem of Clulus f is otiuous o [, ] the Vrits of Prt I : u g( ) f () t t is lso otiuous o [, ] f () t t u ( ) f u( ) È Î g f t t f. f () t t v ( ) f v( ) - È v Î f is otiuous o[, ], F( ) is u f () t t u f u v f - v F f ) Prt I : If () Prt II : ti-erivtive of f ( ) (i.e. f F F. the - ± ± ± ± f g f g f g f g f - f f If f ( ) g( ) o the If f ( ) o the Proerties f f [ ] [ v ], is ostt f ( ) f ( ), is ostt () f g f f f t t f f If m f ( ) M o the m ( - ) f ( ) M ( -) k k+ + +, - + l + + l luu ul ( u) - u+ u u e u e + Commo Itegrls osuu si u+ siuu - osu+ se uu t u+ seutuu seu+ suotuu - su+ s uu - ot u+ tuu l seu + seuu l seu+ t u + u u - t u + + u u - si u + - Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus otes. 5 Pul Dwkis

16 Clulus Chet Sheet Str Itegrtio Tehiques Note tht t m 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 iefiite itegrls ro the limits of itegrtio. E. 5 os u fi u fi u :: 8 fi u fi u Itegrtio Prts : uv uv- vu g f ( g ) g ( ) f ( u) u usig g si( u) ( si( 8) -si() ) 8 5 itegrl omute u ifferetitig u omute v usig v E u v e fi u v-e e - e + e -e - e + 5 os os u u uv uv - vu. Choose u v from E. 5 l v. u l v fi u v ( ) l l - l - 5l 5 -l - Prouts (some) Quotiets of Trig Futios m m For si os we hve the followig : For t se we hve the followig :. o. Stri sie out overt rest to osies usig si - os, the use the sustitutio u os.. m o. Stri 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 hlf gle formuls to reue the itegrl ito form tht e itegrte.. o. Stri tget set out overt the rest to sets usig t se -, the use the sustitutio u se.. m eve. Stri 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 ifferetl. si si os os os si - os Trig Formuls :, +, E. t se 5 4 ( se se ) tse 4 ( u ) uu ( u se ) 5 4 t se t se t se - - se - se 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 se + l os - os + Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus otes. 5 Pul Dwkis

17 Clulus Chet Sheet Trig Sustitutios : If the itegrl otis the followig root use the give sustitutio formul to overt ito itegrl ivolvig trig futios. - fi siq os q - si q - fi seq t q se q - + fi tq se q + t q 6 E. 4-9 si q fi osq q 4 4si 4os q q os q - Rell. Beuse we hve iefiite itegrl we ll ssume ositive ro solute vlue rs. If we h efiite itegrl we ee to omute q s remove solute vlue rs se o tht, Ï if Ì Ó - if < 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 4-9 q. So, Prtil Frtios : If itegrtig P where the egree of Q P is smller th the egree of Q( ). Ftor eomitor s omletel s ossile fi the rtil frtio eomositio of the rtiol eressio. Itegrte the rtil frtio eomositio (P.F.D.). For eh ftor i the eomitor we get term(s) i the eomositio orig to the followig 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 ( )( 4) ( ) 4l - + l t Here is rtil frtio form reomie. A ( A B+ C + 4) + ( B+ C)( -) -+ Set umertors equl ollet like terms. 7 + A+ B + C- B + 4A- C Set oeffiiets equl to get sstem solve to get ostts. A+ B 7 C- B 4A- C A 4 B C 6 A lterte metho tht sometimes works to fi ostts. Strt with settig umertors equl i 7 + A B+ C -. Chose ie vlues of lug i. revious emle : For emle if we get 5A whih gives A 4. This wo t lws work esil. Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus otes. 5 Pul Dwkis

18 f Clulus Chet Sheet Alitios of Itegrls Net Are : ( ) reresets the et re etwee f the -is with re ove -is ositive re elow -is egtive. Are Betwee Curves : The geerl formuls for the two mi ses for eh re, Èuer futio È Îlower futio Î & fi Îright futio - Îleft futio f fi A - È È f A If the urves iterset the the re of eh ortio must e fou iiviull. Here re some skethes of oule ossile situtios formuls for oule of ossile ses. A f ( ) -g( ) A f -g Volumes of Revolutio : The two mi formuls re V A f g g f A V A( ). Here is some geerl iformtio out eh metho of omutig some emles. Rigs Cliers A (( outer rius) - ( ier rius) ) A ( rius)( with / height) Limits: / of right/ot rig to / of left/to rig Limits : / of ier l. to / of outer l. f, f, f, f, Horz. Ais use g( ), A( ). Vert. Ais use g( ), A( ). Horz. Ais use g( ), A( ). Vert. Ais use g( ), A( ). E. Ais : > E. Ais : E. Ais : > E. Ais : outer rius : - f ( ) ier rius : - g( ) outer rius: + g( ) ier rius: + f ( ) rius : - with : f ( ) - g( ) rius : + with : f ( ) - g( ) These re ol few ses for horizotl is of rottio. If is of rottio is the -is use the se with. For vertil is of rottio ( > ) iterhge to get rorite formuls. Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus otes. 5 Pul Dwkis

19 Work : If fore of F( ) moves ojet i, the work oe is W Clulus Chet Sheet F Averge Futio Vlue : The verge vlue of f ( ) o is fvg f ( ) - Ar Legth Surfe Are : Note tht this is ofte Cl II toi. The three si formuls re, L s SA s (rotte out -is) SA s (rotte out -is) where s is eeet uo the form of the futio eig worke with s follows. ( ) s + if f, s + if f, () () s + t if f t, g t, t t t r s r + q if r f q, q With surfe re ou m hve to sustitute i for the or eeig o our hoie of s to mth the ifferetil i the s. With rmetri olr ou will lws ee to sustitute. Imroer Itegrl A imroer itegrl is itegrl with oe or more ifiite limits /or isotiuous itegrs. Itegrl is lle overget if the limit eists hs fiite vlue iverget if the limit oes t eist or hs ifiite vlue. This is till Cl II toi. Ifiite Limit. lim t f f ( ). f ( ) lim f ( ) tæ. + + tæ t q - tæ- f f - - f rovie BOTH itegrls re overget. Disotiuous Itegr t. Disot. t : f ( ) lim f ( ). Disot. t : f ( ) lim f ( ) -. Disotiuit t < < : + t tæ f f f rovie oth re overget. Comriso Test for Imroer Itegrls : If f ( ) g( ) o [, ) f the,. If ov. the ov.. If ivg. the Useful ft : If > the For give itegrl f ( ) ivie [, ] g g overges if > iverges for. f ivg. Aroimtig Defiite Itegrls - (must e eve for Simso s Rule) efie D ito suitervls [, ], [, ],, [ ] - with, * * * Mioit Rule : ªD È ( ) ( ) the, f f f f * Î L, i i-, i D f ª Èf + f ++ f + + f f - + Î L D f ª Èf + 4f + f + + f 4f f Î L is mioit [ ] Trezoi Rule : Simso s Rule : Visit htt://tutoril.mth.lmr.eu for omlete set of Clulus otes. 5 Pul Dwkis