Techniques on Partial Fractions

Transcription

1 Proeedigs o the MTYC rd l Coeree Miepolis Miesot 007 eri Mthetil ssoitio o Two Yer Colleges Tehiqes o Prtil Frtios Tigi Wg PhD Proessor o Mthetis Okto Coity College 600 Est Gol Rod Des Plies IL 6006 Phoe: Eil: tigi@oktoed strt: Sie grphig lltors sh s TI-89 d opter lger systes do Prtil Frtio Deopositio PFD soe tehers py less ttetio o PFD lthogh PFD is qite tehil we eliee PFD lso ele stdets to ler thetil ides d ethods Reetly there he ee soe ppers disssig ot PFD [ ] The stdrd ethod o PFD is the Method o Udeteried Coeiiets MUC with whih the lltio is ote tedios Howeer we orlize the proess o prtil rtios d the ll lltios o prtil rtios siply re relted to three orls I dditio PFD otied y MUC is ote ot redy or itegrtio We proposed dieret ort o PFD tht e or iedite itegrtio The tehiqes disssed here he ot ee od i tetooks d these ltertie tehiqes ispire stdets to eplore d derstd prtil rtios Tehiqes o Prtil Frtio Deopositio Tehiqe : ll deopositios o prtil rtio e doe y pplyig the ollowig three orls: Forl I: [ p ][ p ] p p Forl II: where d 0 Forl III: where d 0 Tehiqe : Whe we itegrte rtiol tio we rek the rtiol tio ito prtil rtios i ters o derities o deoitors: I the deoitor hs tor where is ot Tehiqes o Prtil Frtios Tigi Wg

2 Proeedigs o the MTYC rd l Coeree Miepolis Miesot 007 eri Mthetil ssoitio o Two Yer Colleges Tehiqes o Prtil Frtios Tigi Wg torle i rel ers d 0 the we he the ollowig prtil rtios: B B B Proo o Forls Forl I p p p p ] ][ [ Forl I ws disssed i [] y Hg d eples e od there The et two orls he ot ee see i y plitios Forl II where d 0 Proo o Forl II y deritio: Let γ The the ertors stisy γ Tke the or Tke 0 the γ γ so γ Tke the γ or Sole or we get Thereore Forl III where d 0

3 Proeedigs o the MTYC rd l Coeree Miepolis Miesot 007 eri Mthetil ssoitio o Two Yer Colleges Tehiqes o Prtil Frtios Tigi Wg Rerk Swithig the positios etwee d i the oe orl iplies tht 0 We swith the positios o d i eltig is esier I people pply Forl I Proo o Forl III y deritio Let the rtio he the ollowig deopositio: where re deteried oeiiets By opletig the strtio o the right hd side we get ] [ Thereore 0 0 Eqtio idites Mltiplied y Eqtio eoes 4 0 Eqtio 4 is Eqtio we get 0 or d Thereore

4 Proeedigs o the MTYC rd l Coeree Miepolis Miesot 007 eri Mthetil ssoitio o Two Yer Colleges Tehiqes o Prtil Frtios Tigi Wg This opletes the proo o Forl III Proo o Forl III y Veriitio Eples Eple 4 4 y Forl III let 4 7 d

5 Proeedigs o the MTYC rd l Coeree Miepolis Miesot 007 eri Mthetil ssoitio o Two Yer Colleges Rerk Prtil rtio deopositio is osidered oplete i it oly otis B C epressios like d where d re positie itegers d is prie i rel ers Eple torig the deoitor pls d is deopositio y Forl III 4 4 Eple pls d is seprtig the rtios y Forl II y Forl II Itegrtio y Prtil Frtios i Ters o the Deritie o Ftor The oo ort o prtil rtios with qdrti tor whih is ot torle i rel ers is B B B Bt itegrtig rtiol tio with this ort stdets still e rstrted ese it is ot redy or itegrtio For eple y deoposig the rtiol tio d d Tehiqes o Prtil Frtios Tigi Wg

6 Proeedigs o the MTYC rd l Coeree Miepolis Miesot 007 eri Mthetil ssoitio o Two Yer Colleges My stdets still he diilties to iish d lost ll lls tetooks se this ort o prtil rtios To ke the deopositio redy or itegrtio y sstittio I tght stdets with ltertie ort: I the deoitor hs tor where is ot torle i rel ers d 0 the we he the ollowig prtil rtios: B B B Forts d re eqilet Bt hs the ollowig dtges: It is redy or itegrtio y sstittio Fidig deteried oeiiets o is esier th ese we se the zero o ' Let s reisit Eqtio : id d Soltio: B C [ B] C Let C C Let the zero o ' B C or B d B Let 0 B C 4 Ths d d 4 d the itegrl e iished ore eetiely th Reeree: Brzier Rihrd d Bo Egee 007 How to Copte the Prtil Frtio Deopositio Withot Relly Tryig The MTYC Reiew Vol 9 No 0-9 Cheri Chokri 007 ltertie Method to the Clssil Prtil Frtio Deopositio The MTYC Reiew Vol 9 No -7 Hg XCheg 99 Shortt i Prtil Frtios College Mthetis Jorl Vol No 4 4 Tehiqes o Prtil Frtios Tigi Wg