Practice Final Exam. 3.) What is the 61st term of the sequence 7, 11, 15, 19,...?

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1 Discrt mth Prctic Fl Em.) Fd 4 (i ) i=.) Fd i= 6 i.) Wht is th 6st trm th squnc 7,, 5, 9,...? 4.) Wht s th 57th trm, 6,, 4,...? 5.) Wht s th sum th first 60 trms th squnc, 5, 7, 9,...?

2 6.) Suppos st A conts 4 objcts. How mny 9 objct substs A r thr? 7.) How mny wys r thr to choos nd ordr 49 objcts from collction 04 objcts? 8.) How mny diffrnt wys r thr to ordr 9 diffrnt objcts? 9.) You r dcortg room by choosg color to pt th wlls with nd color crpt to us th floor. You hv 6 diffrnt colors pt to choos from th wlls, nd diffrnt colors crpt to choos from th floor. How mny diffrnt wll nd floor color combtions could you crt? 0.) Writ ( 9 ) s n tgr stndrd m.

3 ************************************************ Algbr.) Writ ( 7 8 ) s rtionl numbr stndrd m..) Writ (7 ) s rtionl numbr stndrd m..) f = 0,thnwhtis 0 s rtionl numbr stndrd m? (Notic tht this is skg th y-trcpt th grph.) 4.) f >0, wht is () s rtionl numbr stndrd m? (Th nswr hs somthg to do with th -trcpt th grph ().) 5 5.) Writ 8 s rtionl numbr stndrd m.

4 6.) Fd whr ( ) = 8. 7.) Fd whr ( ) 5 = 7. 8.) Fd whr 4 () ( ) 8 =. 9.) Fd g f() if f() = nd g() =. 0.) Fd th vrs g() = 7 ( ). 4

5 .) Wht is th implid dom f() = ( 7)?.) Wht is th implid dom g() = 7 4?.) Fd

6 4.) Complt th squr: Writ 4 5thmα( β) γ whr α, β, γ R. 5.) How mny roots dos 4hv? 6.) Fd root 6. 7.) ( pts.) Compltly fctor. (Ht: is root.) 6

7 8.) y is th distnc btwn which two numbrs? 9.) Solv if < 4. ************************************************ Lr lgbr 0.) Wht s th dtrmnt th mtri blow? ( ) 5.) Fd th product ( ) ( ) 0 0.) Wht s th vrs th mtri blow? ( ) 4 7

8 .) Writ th followg systm thr lr qutions thr vribls s mtri qution y z = y z = y z = 0 4.) Solv, y, nd z if y = z nd =

9 ************************************************ Grphs 5.) Grph th followg functions:,,,,,,,,, (). 6.) Grph f :(, 0] R whr f() =. 7.) Grph g : {4,, } R whr g() =. 8.) Grph nd lbl its - ndy-trcpts (if thr r ny). 9.) Grph 4( ) nd lbl its vrt. 40.) Grph nd lbl its - ndy-trcpts (if thr r ny). 9

10 4.) Grph p(). (Lbl ll -trcpts.) p() = ( )( )( )( ) 4.) Grph r() (Lbl ll -trcpts nd ll vrticl symptots.) r() = ( )( ) 4( )( ) 4.) Grph h() = { if ; if =. 44.) Grph if (, ); m() = if = ; if (, ]. 0

11 Lst Nm: First Nm:.) 6.).) 7.).) 8.) 4.) 9.) 5.) 0.) 6.).) 7.).) 8.).) 9.) 4.) 0.) 5.).) 6.).) 7.).) 8.) 4.) 9.) 5.) 0.)

12 .).).) 4.)

13 S ************************************************ Grphs 5 C 5 C 7 8 t t 5 C 7 8 t 7.) Grph th lfollowg functions:,,,,, l, (). l,,, 5.) 7.) 7.)5.) S 7.) Grph 8.) 7.) f : (, ] R whr f () =. 7.) 7.) Grph g : {,, 5} R whr g() = 0. 9.) g() 0.) ( ) g() 0.) ( ) 0.).) Grph ( ) nd lbl its -trcpt. 7 8 t 4( t ) 5 C Grph nd its vrt. 5 lbl l its y-trcpt. Grph nd lbl.) -trcpts.) Grph p(). (Lbl ll.) C 7 ) p() = (0)( )( )( )( 0 8 t () () FLLNG BOUNDARES FLLNG BOUNDARES SEMSMPLE AND SOLV SEMSMPLE AND SOLV () () FLLNG BOUNDARES BOUNDARES OF COARSE MANFO () FLLNG OF MLADEN COARSE MANF BESTVNA, A.) 5.) 9.) g() 0 6.) ( ) MLADEN BESTVNA, A 4.) Grph r() (Lbl ll -trcpts nd ll vrticl symptots.) SEMSMPLE AND SOLVABLE ARTHMETC SEMSMPLE AND SOLVABLE ARTHMETC ().) FLLNG 5.) ) MANFOLDS BOUNDARES6.) OF ( COARSE N 9.) f ( 6.) f () ( ) Abstrct. p SEMSMPLE AND SOLVABLE ARTHMETC GROUPS () () BOUNDARES FLLNG OF MANFOLDS N Abstrct. W W provid provid 8 p 8.) f () 6.) 8.) fcoarse () r() = MLADEN BESTVNA, ALEX ALEX ESKN, & KEVN WORTM rliztion thorm L MLADEN BESTVNA, ESKN, & KEVN WORTM rliztion thorm L SEMSMPLE SOLVABLE GROUPS 8.) f OF () ARTHMETC 4( AND )(MANFOLDS ) FLLNG BOUNDARES COARSE MANFOLDS N FLLNG BOUNDARES OF COARSE N mtic (ovr numbr () 8.) f () MLADEN BESTVNA, ALEX ESKN, & KEVN WORTMAN mtic (ovr numbr SEMSMPLE AND SOLVABLE ARTHMETC GROUPS low both polyn AND SOLVABLE ARTHMETC GROUPS 8.) f () SEMSMPLE () Abstrct. towrds low dimnsions, dimnsions, both polyn Abstrct. W provid prtil rsults towrds conjctur conjctur MLADEN BESTVNA, ALEX ESKN, & KEVN WORTMAN FLLNG BOUNDARES OF COARSE MANFOLDS N 4.) 5.) () qulitis p 5.) OF Abstrct. 6.) qulitis nd nd fitnss fitnss rliztion thorm Lubotzky-Mozs-Rghunthn rliztion p () Lubotzky-Mozs-Rghunthn FLLNG BOUNDARES COARSE MANFOLDS N towrds conjcturl W provid GROUPS prtil rsults gnsemsmple AND SOLVABLE ARTHMETC As tool our pro, w As tool our pro, w mtic (ovr numbr filds or function filds) tht impl filds) tht impl MLADEN BESTVNA, ALEX mtic ESKN, & KEVN WORTMAN MLADEN BESTVNA, ALEX ESKN, & KEVN WORTMAN rliztion thorm GROUPS Lubotzky-Mozs-Rghunthn gnrith- SEMSMPLE AND SOLVABLE ARTHMETC isoprimtric quliti Abstrct. W provid prtil rsults conjcturl isoprimtric quliti lowtowrds dimnsions, uppr isoprim low dimnsions, both polynomil isoprim 5.) Grph mtic (ovr numbr filds or function filds) tht implis, bl s ss rliztion thorm Lubotzky-Mozs-Rghunthn rithbl tht tht ppr ppr s qulitis nd fitnss proprtis. qulitis MLADEN BESTVNA, ALEX ESKN, & KEVN WORTMAN Abstrct. W provid prtil rsults towrds conjcturl gnlow dimnsions, both polynomil uppr isoprimtric Abstrct. W provid prtil rsults towrds conjcturl gnpl, thus gnrliz mtic (ovr numbr filds or function filds) tht implis, if = ; As w lso provid polynomil polynomil uppr uppr bb As tool tool our pro, MLADEN BESTVNA, ALEX ESKN, & KEVN WORTMAN rliztion both fitnss thorm Lubotzky-Mozs-Rghunthn rithqulitis nd proprtis. rliztion thormh() Lubotzky-Mozs-Rghunthn rith= low dimnsions, polynomil uppr isoprimtric isoprimtric qulitis nd fitnss rsults rsults crt crt isoprimtric mtic (ovr numbr filds filds) uppr tht implis, As our pro, w lso provid polynomil mtic (ovr numbr filds or function tht implis, or function tool iffilds) =. Abstrct. W provid prtil rsults towrds conjcturl gnqulitis nd fitnss proprtis. bl tht ppr s sub prbolic prbolic s s bl dimnsions, both polynomil uppr crt isoprimtric isoprimtric qulitis nd fitnss rsults uppr solvabstrct. provid rsults conjcturl gnlow dimnsions, both polynomil uppr isoprimtric rliztion W thorm prtil Lubotzky-Mozs-Rghunthn rithaslow towrds tool our pro, w lso provid polynomil 0 pl, thus gnrlizg thorm Bu. Our m rsult is Thorm pl, thus gnrlizg thorm Bu. ppr 4.) () qulitis nd fitnss 4.) s5.) 5.) bl tht sub prbolic smisimrliztion thorm Lubotzky-Mozs-Rghunthn rithqulitis nd fitnss 5.) (ovr 6.) () 5.) tht 6.) () () mtic numbr fildsproprtis. or function filds) implis, proprtis. isoprimtric qulitis nd rsults crt solvfitnss som bckground. As tool our pro, w lso provid polynomil uppr pl, thus gnrlizg thorm Bu. mtic (ovr numbr filds or function filds) tht implis, As tool our pro, w lso provid polynomil uppr bl dimnsions, both polynomil uppr isoprimtric tht ppr s sub prbolic smisim6.) low Grph isoprimtric qulitis nd fitnss rsults ) crt low dimnsions, both polynomil uppr isoprimtric isoprimtric qulitis nd fitnss rsults crt solv5.).) 5.) Thorm 7.) g() 8.) 9.) g() 0.) ( 9.) g() 0 6.) ( ) 9.) f () 6.) ( solv5 blow. B sttg pl, thus gnrlizg thorm Bu. nd fitnss proprtis. 7.) qulitis g() 8.) 4.) ) rsult 9.) g() 0.) ( ) 5.) 6.) Our m is it bl tht ppr s sub prbolic smisimour m rsult is Thorm 5 blow. B sttg it, qulitis nd fitnss proprtis. bl tht ppr s sub prbolic smisim if [, ); g() 0.) 7.) ( p() ) As 9.) tool our pro, w lso provid polynomil uppr 4 som pl polynomil, thus gnrlizg bckground. thorm ( Bu. 4.) fitnss )it,w provid Aspl tool our pro, w lso provid, thus gnrlizg thorm Bu. Our m rsult is uppr Thorm 50.) blow. B sttg 9.) g() 6.) ( ) som bckground. isoprimtric qulitis nd rsults crt solv m() = rsults if =crt ; solv ppr isoprimtric qulitis nd fitnss 9.) g() 0.) ( (B ) ) som bl tht prbolic smisim7.) p() 40.) g( Our bckground. m rsult is Thorm 56.) blow. sttg it, w provid 4.) sssub bl tht ppr sub prbolic smisimpl, thus gnrlizg thorm Bu.if (, ). som bckground. pl, gnrlizg Bu. m rsult Thorm 5 blow. sttg it, w provid 5 blow. Our m thus rsult is Thorm Bissttg it, w providb thorm 4.) Our 5.) 5.) 6.) () () som bckground. som bckground..) 4( ).) Our 5.) 8 sttg blow. B ( ) it, it,w f B ) Our m m rsult rsult isis Thorm Thorm blow. sttg wprovid provid 8.)7.) ( 7.)6.) 556.) 9.) ( )() som som bckground. bckground. 7.)6.) 7.) ( p()4.) 6.) ) ) ( ( ) ) 7.)6.) 8.) 0 ( 8.)7.) ( ) ) 6.) 40.) g() 7.) p() 6.) ) 7.) ( 8.) q() 4 8.)

14 5.) 6.) 5.)4.) 6.)5.) () () 6.) 7.) 6.) 7.) 7.) 6.) 7.) 5.) () 5.)4.) 6.) 6.)5.) () () 5.)4.) () 7.) 8.) ) ).) 4(.).) 4(.) 9.) 4( ) 40.).) 4( ).) ) 5.).) 5.) ( 6.) ( ) 9.) f () 6.) 4.) 5.) 7.) 8.) 8.) 7.) 8.) 8.) 8.) 9.) 8.) 4.) 5.) 7.)6.) 9.) 9.) 9.) 5.) () 6.) () 7.) ( ) 8.) ( ) 9.) 9.) 0.) 0.) 0.) 6.) 7.) 0.).) 0.) 9.) 5.) 6.)9.) fl(( 0.) 0.).).) 7.) ( 8.) ( ) ).).).) 6.) 7.) ( 7.) 8.) 7.) 8.) ( ) ) () ) *.) 7.)6.).) ( * * ** ** * ***** * * *** *** * *** *** ** ** * ***** ** ** ** ** ** ******.).).).) p() 4.) r().) p() 4.) r() 4.) p() 4.).) p() 4.) r() p() 6.) r() 7.) l 7.) p() 40.) g() 7.) 40.) 7.) ** g( 6.) ( ) 4.) 6.) ( )****** ** ******.) p() 4.) *rqc) ( * *** **** **** * ***** * * * ******* *** ** ** ** ***** * * * * * 8.) * ** ***** *** * *** ************************************************************************ ** ** * ***** ** ** ****** * * ****** * * ** *****.) p()4.) 4.) r() 4.) rqc).)rqc) p() 6.) 7.) 8.) f () 9.) f () -~ - -l -, 7.) () ) 0.)9.) g()g()8.) ( - i ~, -~ - -l -, -l -~ i -, ~ i, 9.)8.) f ()f () 0.)9.) g()g() f ()9.) g() 9.)8.) f ()0.) 0.)9.) g()g() 9.)8.) f ()f () g() 5.) h() 6.) m() 4.) h() 44.) 5.) h() 6.) m() 5.) h() 6.) m() 8.) fm() ().) 5.) ) 8.) 8.) q() 9.) f( () 5.) q() ) 6.) 6.) ( 9.) f () 9.)8.) f ()f () 7.) ( p() 4.) 6.) ) 9.) 0.) g() 9.) f (g 0.)9.) g()g().) 5.)4( ) ) 6.) (.) 9.) f () 4 7.) p() 40.)( g() ) 6.) 40.) g(

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