Mth 15 Iteredite Alger Stud Guide for E 3 (Chpters 7, 8, d 9) You use 3 5 ote crd (oth sides) d scietific clcultor. You re epected to kow (or hve writte o our ote crd) foruls ou eed. Thik out rules d procedures ou eeded to kow for hoework. For eple: epoet rules, etc For E 3 ou will eed to e le to: 1. Evlute rdicl epressios. Negtive uers uder rdicls with eve idices re ot rel, ut egtive uers uder rdicls with odd idices re egtive. We use solute vlues to sigif tht vriles re ot egtive uder eve roots. 7.1. Chge rdicl epressios to epoetil epressios, d vice-vers, to evlute or siplif. 7. 1 * * * 3. Appl the rules of epoets to rtiol d egtive epoets. 7. * product rule: * power rule: ( ) or * zero epoet rule: 0 1 * dd/sutrct with se se: o rule! 1 * quotiet rule: * egtive epoet rule: d 1 * frctio rised to egtive epoet: * epded power rule: 4. Siplif squre root epressios usig the product rule d the quotiet rule for rdicls. 7.3 * product rule for rdicls: * quotiet rule for rdicls: 5. Add or sutrct squre root epressios. You ight eed to siplif squre roots or fid coo deoitors efore the c e coied. Reeer ol like rdicls e coied (vriles d their epoets hve to e the se oth iside d outside the root to e like). 7.4 6. Multipl two squre roots usig the distriutive propert or the FOIL ethod. If possile, siplif squre roots tht pper i the product. 7.5 7. Siplif quotiet ivolvig squre roots. Rtiolize deoitors. 7.5 * For deoitors with 1 ter: get rid of the root i the otto ultiplig top d otto wht the deoitor eeds to coe out of the rdicl. * For deoitors with ters: get rid of the root i the otto ultiplig top d otto the deoitor s cojugte (the se ioil ut with opposite sigs i the iddle). 8. Solvig rdicl equtios : 7.6 * For oe rdicl: get the rdicl loe o oe side of the equl sig, rise oth sides to the power of the ide, d solve the reiig equtio.
* For two rdicls: get ech rdicl to ech side of the equl sig, rise oth sides to the power of the ide, d solve the reiig equtio. * For two rdicls d o-rdicl: get oe rdicl o oe side d the other rdicl d ordicl to the other side of the equl sig, rise oth sides to the power of the ide. You will still hve rdicl d eed to repet the process. 9. Solve for specific vriles i foruls ivolvig rdicls. 7.6 10. Perfor opertios ivolvig cople uers. This icludes covertig rdicls to cople uers, ddig, sutrctig, ultiplig, d dividig cople uers. 7.7 11. Siplif powers of i 7.7 1. Solve qudrtic equtios usig the squre root propert. Solve qudrtic equtios copletig the squre. 8.1 * The Squre Root Propert: If, the * To solve + + c = 0 copletig the squre: 1) set up the equtio so tht the vrile ters re o the left of the equl sig, i stdrd for, d the costt ter is o the right. Bsicll, get it ito the for c. ) divide, so the coefficiet of is 1. 3) coplete the squre tkig oe-hlf the coefficiet of the -ter, squrig it, d ddig this qutit to oth sides of the equtio. Bsicll, dd to oth sides. 4) fctor the Perfect Squre Trioil o the left side of the equtio d siplif the right side. Reeer, it lws fctors ito 5) use the priciple of squre roots 6) solve the reiig equtio 13. Solve qudrtic equtios usig the qudrtic forul. 8. * For 4c + + c = 0, ou c solve for usig The Qudrtic Forul. 14. Grph qudrtic equtios usig the is of setr, the verte, d the -itercepts. 8.3 * The is of setr is the verticl lie tht goes dow the iddle of the prol, through the verte. Sice the is of setr is lie the equtio of this lie is. * The verte is the iiu poit for, or the iu poit for. verte:, f ( ). * The verte is used to solve pplictio proles tht require ou to solve for the iiu or iu vlues, i.e.: highest height of projectile, iu re of rectgle, iiu cost, etc...
* The -itercepts re the poits where the grph crosses the -is (where =0). Ech tie ou set equtio equl to zero d solve for ou re fidig the -itercepts. To deterie the uer of -itercepts qudrtic equtio hs, clculte the discriit. Discriit 4c If 4c 0 Solutios to c 0 Grph of f ( ) c The equtio hs two uequl REAL solutios. Two -itercepts. If 4c is perfect squre, the solutios re RATIONAL uers.) If 4c 0 If 4c is NOT perfect squre, the solutios re IRRATIONAL CONJUGATES. The equtio hs ol oe REAL solutio. It would e doule root d if,, d c re rtiol, it would e RATIONAL uer. Oe -itercept. If 4c 0 The equtio hs NO REAL solutios. It hs two IMAGINARY solutios. The would e COMPLEX CONJUGATES. No -itercepts. * Usig trsltios (shiftig the grph) is es w grph qudrtic equtio puttig it ito the for f ( ) ( h) k, with verte: ( h, k), d deteries whether the grph will e rrower or wider th the origil grph. *The greter the solute vlue of, the rrower the grph* 15. Solve pplictio proles usig vrious techiques for solvig qudrtic equtios. 8.3 * There re 4 ws tht we kow how to solve qudrtic equtio. 1.) Fctorig,.) Squre Root Propert, 3.) Copletig the Squre, 4.) Usig the Qudrtic forul. You use of these 4 ethods to solve qudrtic equtio otied fro pplictio prole. i.e.: fidig the tie it tkes, t, for projectile to e certi height, h (t).
16. Solve equtios tht c e de ito qudrtic equtios usig u-sustitutios. 8.4 * With the equtio writte i descedig order, let u the iddle vrile or ioil, the u the first vrile or ioil. Rewrite the equtio usig u to solve, the replce u with wht u=, so ou c solve wht ou were origill sked to solve. 17. Solve qudrtic, poloil, d rtiol iequlities i oe vrile. 8.5 * Fid the zeros of the iequlit d the vlues tht ke the fuctio udefied d set the s oudr regios o uer lie. Test ech regio. If the stteet is true, shde tht regio. Write the shded regio i itervl ottio. 18. Grph epoetil fuctios. 9.1 19. Solve pplictio proles ivolvig epoetil fuctios. 9.1 r t * Be prepred to clculte popultio growth, dec, Copoud iterest forul: A P( 1 ), d so o. 0. Fid the coposite of two fuctios. 9. ( f g)( ) f ( g( )) es first write the f-fuctio, ut replce ll s with ig lk ( ). Iside the ig lk ( ) write the g-fuctio d siplif. 1. Deterie whether fuctio is oe-to-oe. 9. * A fuctio is 1-to-1 if it psses the horizotl d the verticl lie test. (Drw horizotl or verticl lie through the grph. If it itersects the lie ore th oce, it fils the test.) 1-to-1 NOT 1-to-1 * Fuctios with ledig odd epoets re 1-to-1. Eple: f ( ), 3 d cue root fuctios re 1-to-1. Eple: f ( ), f ( ) 3 f ( ), squre root * Fuctios with eve ledig epoets re 1-to-1 whe the doi is restricted. Eple: f ( ), 0, f ( ), 0
. Fid the iverse of 1-to-1 fuctio. 9. Steps to fid the iverse fuctio: ) Replce f () with. ) Iterchge the vriles d. c) Solve the equtio for. d) Replce with f 1 ( ). This is the iverse of the fuctio, f (). e) Verif f () d f 1 ( 1 1 ) re iverses ( f f )( ) f ( f ( )) d ( f 1 f f 1 )( ) ( f ( )) 3. Covert fro epoetil for to logrithic for. 9.3 Defiitio: If log, the 4. Grph logrithic fuctios. 9.3 * log is rell, so we teporril look t grph the poits (, ). This is the grph of log., crete the poits (, ) d the 5. Approite coo logriths d powers of 10. 9.3 * log log * log es 10 6. Use the properties of turl logriths to siplif. 9.3 10 * Nturl epoetil fuctio: f ( ) e * Nturl logriths: log l If l, the e. e * Properties: * le 1 * l e * l l * l l l * l l l * e
7. Use the properties of logriths to epd or siplif logrithic epressios. 9.4 * product rule: log log log * idetit: log 1 * quotiet rule: log log log * zero power: log 1 0 * power rule: log log * idetit d power: log 8. Use the chge of se forul to clculte logriths. 9.4 * Chge of se forul: log log log 9. Solve epoetil d logrithic equtios. 9.5 * If, the. * If log log, the. 30. Solve pplictio proles ivolvig logriths. 9.6 * Use the defiitio of logriths to solve for the vrile s the epoet. 31. Solve equtios d pplictio proles ivolvig turl logriths. 9.6 * Be prepred to clculte popultio growth, Cotiuousl copouded iterest forul: rt kt A Pe, Rdioctive dec: A A e o, d so o.