P3 Siplifyig Jie Esclte, the ost fous high school clculus techer of ll tie, hd ber i his clssroo tht red Clculus does t hve to be de esy, it lredy is How true tht essge is As I lredy etioed, AP Clculus is siply two ides, istteous rte of chge, which is divisio process, d res of irregulr regios, ultiplictive process To ke it eve sipler, the, clculus is bout dividig d ultiplyig The thig tht kes clculus such chllegig course is iroiclly ot the clculus itself, but rther ll the tedious lgebr tht is eeded to ipleet the clculus I this sectio, we will exie soe of the lgebric gystics we ll eed A theticl expressio is oe of two types, either phrse like whe i doubt, ultiply by the pge uber or collectio of ubers, vribles, d/or opertiol sigs Siplifyig vrieties of the secod type llow us to work with sller, ore geble thigs without loss of geerlity Workig with expressios is very differet tht workig with coditiol equtios There re oly hdful of thigs you c do with expressio without chgig its vlue These iclude fctorig, ddig clever fors of zero (copletig the squre), d ultiplyig by clever for of oe (gettig coo deoitor) Wht is et by siplifyig is to ke expressio sller, ore codesed, coti less stuff, with fewer ters or fewer fctors, etc There s o defiitive siplest for i geerl, but I thik you will begi to otice whe expressio is stripped dow s fr it possibly c O ultiple-choice questios, of course, you ight hve to siplify to poit tht resebles (exctly) the correct swer choice We ll strt with siplifyig by fctorig, of which there re severl thigs to look for, surized i the chrt below Look for coo fctor Look for specil product like o Differece of two squres: b b b o Perfect squre trioil: b b b o Su/Differece of Cubes 3 b 3 b b b 3 b 3 b b b 3 Fctorble trioil (Trget Su/Trget Product) Guess d Check 5 Groupig 6 Usig sythetic divisio Let s work few Pge of 9
Exple : 3 x 9x Siplify x 7x Tht ws too esy Exple : 3 x 8x 0x 0 Siplify x Pge of 9
Alredy the level of sophistictio is icresig, but hopefully you re still followig log oky Hopefully you re ot too bored either A coplex (or copoud) frctio is expressio tht hs frctios withi frctios It ight look itiidtig, but I ssure you it s ll brk d o bite Ulike it s close reltive, the copoud frcture, delig with these frctios is either piful or excrucitigly piful There is tidy lgebric euver tht will efficietly erdicte these coplex or copoud frctios It ivolves ultiplyig by clever for of oe tht ivolves the lest coo ultiple (LCM) of ll the iiture deoitors Here s oe Exple 3: Siplify x 5 x 5 Soeties these coplex frctios c be i disguise It s your job to recogize the This will require reider bout egtive d rtiol expoets Here it is: d b This es tht ll rdicl expressios c be writte s rtiol expoets, with the root off the rdicl beig the deoitor of the expoet Fctors with egtive expoets c becoe fctor o the other side of frctio siply by chgig the sig of its expoet like c be equivletly writte s 3 x x Whe siplifyig expressios ivolvig rdicls or vribles i the deoitor, it is esier to write the i expoetil for; however, whe evlutig these expressios for prticulr x-vlues, it is esier to do so i their rdicl or deoitor for 3 b Pge 3 of 9
Exple : Siplify 36 x 6 x x Soeties frctiol expressios coti rdicls or rtiol expoets (hidde rdicls, reeber?) Siplifyig these type of expressios usully ivolves process clled rtioliztio, which is fcy e for lgebriclly ovig rdicls fro either uertor to deoitor or vice-vers You did this quite bit i Preclculus, especilly i your Trig uit Rtios like bece the equivlet The purpose of this ws to ke the Uit Circle ore uifor I geerl, both expressios re cosidered eqully siplified I fct, oe could ke rguet tht the first oe is ore siplified There will be ties, though, whe workig with vrible expressios where we will eed to pull out the se trick to circuvet lgebric tight spot Exple 5: Rtiolize x 3 3 x Soeties the process ivolves oly ubers Pge of 9
Exple 6: Siplify by rtiolizig 5 Soeties siplifyig expressios es cretig the first, kig outi out of ole hill before kig the outi bck ito ole hill Before doig tht, it s iportt to uderstd fuctio ottio d how to evlute fuctio Fuctio ottio y f x is so useful becuse it provides efficiet wy to see both iput d output, idepedet d depedet vrible, x- d y- vlue Your experiece i evlutig fuctios is probbly liited to pluggig i specific vlues i for x If you ve studied copositio of fuctios, they you ve bee lucky eough to plug vrible expressio cotiig x i for x Whe this hppes, it s very useful to revert bck to your dys before lgebr, whe you used spces, rectgulr boxes, or prethesis to represet ukow, s i the follow coditiol equtio fro those glorious dys of your theticl yesteryer 5 8 Bck the you got gold str for writig 3 iside the box You perhps the got wr, fuzzy feelig o the iside whe your th techer told you tht you just solved d lgebric equtio (or ot) We ll do the se thig evlutig the fuctio Rewrite it s f f x x x x Now whtever ppers i the prethesis o the left side of the equtio will pper i ech d every prethesis i the expressio o the right-hd side of the equtio We ll evlute this fuctio ow for selected iputs Pge 5 of 9
Exple 7: If f, evlute the followig () f (b) f (c) f x f e (e) f si x (f) f pik elepht x (d) You get the ide While this teplte writig is ot ecessry step for success i AP Clculus, it provides systetic wy to void creless errors d perhps world of cofusio Reeber tht it s the lgebr tht kes clculus so difficult If there is prove ethod for kig it less difficult, it s worth ipleetig Believe e, oe issed egtive sig c rui your whole dy It s very frustrtig to hve to retrce your steps to fid subtle error It s ofte better to just rework the proble secod tie workig ore crefully, ore slowly, d of course, ore correctly Wheever we plug i vrible expressio for x ito fuctio, we re cretig ew fuctio, uch like we did with trsfortios This process is type of copositio Uderstdig copositio of fuctios will be criticl to your success lter o whe you ler to tidifferetite d itegrte Exple 8: If f ( x) 3x 5x, siplify the expressio f x h f x h Pge 6 of 9
Alost ore iportt th copositio of fuctios is the decopositio of fuctios This ivolves idetifyig which fuctio is o the iside d which fuctio cotiig it is o the outside of give fuctio Exple 9: h x Decopose 3 x ito two fuctios f d g such tht hx f gx f g x Very ofte you will hve to siplify expressios or equtios ivolvig expoetil expressios d/or logrithic equtios The rules for siplifyig either type of expressios re very siilr, sice d expoet is othig ore th log, d log is othig ore th expoet Here re the bsic properties you ll be workig with Expoets Logriths 0, 0 l 0 l e l l l l l l l l, 0 l x e x l e log b l x x l x Logs were iveted by Scottish thetici Joh Npier Origilly clled Artificil Nubers, logs re very rel ideed, d Npier s logs gve us the rules for workig strictly with the expoets of ubers writte s powers of coo bse This ws useful for workig with very sll, icroscopic ubers s well s very lrge, strooicl ubers Scietific ottio is bsed o this usig bse te, the coo bse Ay log expressio c be equivletly be writte s d expoetil expressio The coversio forul is log b x y b y x I clculus, we will pririly use bse e, rther th bse 0, the coo bse The uber e, ot to be cofused with the letter e or televisio etwork E!, is pproxitely 78 It ws discovered d ed by Swiss thetici Leohrd Euler (proouced oiler ) It is rgubly the ost fous d iportt irrtiol uber i ll of thetics Clled the turl bse, this uber is ot oly helthier d lower i low-desity lipoproteis th bse 0, but occurs ofte i ture If you hve versio to cllig it the turl bse, feel free to cll it by soe of its coo (er, turl) oikers: Euler s uber, the Bker s uber, or li Pge 7 of 9
Exple 0: Fid the exct solutio to the equtio x x 6 3 solvig the equtio grphiclly o your clcultor, the give 3-decil pproxitio Verify by Logs ke evlutig d siplifyig expressios so uch fu d soeties frustrtig Becuse there re so y differet fors of the se correct swer, you eed to be dept d gile eough to esily ipulte betwee differet equivlet fors of the se expressio The AP ex does t require you to siplify your swers o the free-respose portio, but o the ultiple-choice sectio, you ight hve to siplify your swer beyod poit you would orlly stop Here s d exple tht de it o the Silver Scree i the 988 ovie Std d Deliver strrig Edwrd Jes Olos s the lte Clculus guru Jie Esclte It ws ctul questio o the 985 AP Clculus ex (o-clcultor portio) Which of the followig is equl to l? (A) l3 l (B) l 8 l (C) t e dt (D) l xdx (E) dt t You do t kow how to trslte swer choices (C), (D), d (E) yet, but by the ed of the yer, oce you ve stered how to do tht, this questio will red ore like the followig: Which of the followig is equl to l? (A) l3 l (B) l 8 l (C) e e (D) l 3 (E) l l The first two swer choices re bit for the esy prey who hve t eorized or sufficietly prcticed their log properties Choices (C) d (E) should be oticebly wrog for yoe with log or e experiece, icludig strtig cpfires d wiig spellig bees By process of eliitio, the correct swer choice ust be (E) This c be rrived t by two differet ethods l l l 0 or l l l l Perhps better questio would be the followig: Pge 8 of 9
Exple : Which of the followig is NOT equl to l? (A) l l (B) l (C) l6 (D) l8 l (E) ll e (F) l l e (G) l l Note: O the ctul AP ex, the lst swer choice is NOT lwys the correct swer choice Be creful of overgeerliztios d uprove theories bsed o liited epiricl evidece Pge 9 of 9