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School: George Mason
Course: Quantitative Reasoning
Syllabus,Spring2014 Math106006,QuantitativeReasoning Instructor: Dr.LynnellMatthews Office:ExploratoryHallRoom4407 OfficeHours: MW4:30PMTO5:30PM Phone: 7039931981 Email: lmatthe4@gmu.edu Text: MathematicalIdeas,byMiller,HereenandHornsby,CustomEdition,Pear
School: George Mason
Math 677. Fall 2009. Homework #7 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.13: # 1 The system x = y + ax2 + bxy + cy 2 y = dx2 + exy + f y 2 Let LJ [h(x)] = Jh(x) Dh(x)J(x) an
School: George Mason
Math 677. Fall 2009. Homework #5 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.9: # 3 For the system x2 x1 x2 + x2 x3 1 3 2 x1 + x2 x3 x= 2 3 x1 x3 x3 x2 x2 x2 x5 1 3 3 the Liapu
School: George Mason
Math 677. Fall 2009. Homework #4 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.5: # 5 Determine the ow of the nonlinear system x = f (x), f (x) = (x1 , 2x2 +x2 )T 1 1 2 and show t
School: George Mason
Math 677. Fall 2009. Homework #6 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.11: # 2 (a) x = x2 , y = y. Expansion of the function = p2 (x, ) is a nbhd of the origin has the for
School: George Mason
Course: Ordinary Differential Equations
Math 677. Fall 2009. Homework #2 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 6: # 4 The system x = Ax with 1 1 0 0 1 1 0 0 A= 0 0 0 2 0 0 1 2 has eigenvalues 1,2 = a1 ib1 = 1
School: George Mason
Course: Ordinary Differential Equations
Math 677. Material: Fall 2009. Final Exam Review. Perko, Chapters 1, 2, 3.1-3.3, 4.2-4.3; lectures 1-24. 1. Linear systems x = Ax: Diagonalization and Jordan representation Properties of operator exponentials Solution to IVP x = Ax, x(0) = x0 , stability
School: George Mason
Course: Ordinary Differential Equations
Math 677. Material: Fall 2009. Exam I Review. Perko, Chapters 1.1-2.7; lectures 1-12. 1. Linear systems x = Ax: Diagonalization and Jordan representation (a) Case of distinct roots (b) Case of complex roots (c) Case of repeated roots (d) Jordan canonical
School: George Mason
Course: Numerical Analysis
l^ lufafcfw_t 6Ef, (-; r,r*, s/"tn , tlx =( , Ex- f,t, +2xz:/ - V*r* xz-zz ,+=(!il s4!) 6qdsuffiitz;\ba Gat sian ;!,X7=O4 &n;aahbq Xyeq Xt*Zo= | =)xF I So2nfiort: F =/t\ la,e =&lAilb- ,;T Sp* A'-Q4tr) YieR^ 'Rn = tbcfw_a1 ?yff?*, Uxed?q Jl r,e/cu(1) YzcJf
School: George Mason
Course: Numerical Analysis
MaIn osg, f*r*nd"'lr'on i ftt, a f) '(0,t" po;ruh f*pocri*: f, s.b, f ftt) -/' t\ \, '4,*$ee w* h /t* ,b 0UtL U|z*t?,b+ g.+ ' dath- 2 -slcp D Aan &. ryeu Ctil" 4 tntayola-*) .-> nan4.ral, P4 +cfw_rA -) lxi, tattoaa2 e-t. =-r I 4 J) fuc -r t1'* 4ony )ea.t
School: George Mason
Course: Numerical Analysis
/'lacfw_A luliatuw-, . +t , ftrd'# = f 6s, '. w-*Jn.Hg!l'! t-X 'f tll llx =rt-xt [r - ll,e*(Vrl'lr iUtt = _lf ,-L.1 fuilt'f ir.-Dlcfw_. =L , ilg' Gu^uian 4)n, ptorJttrr,' k i.f'p.rrr =[ti |'t]"6J ilag 'daatinutcc ,. Ta 6h*r 'Jtrn-*nr. f'=Lu W kru Y r ofrn
School: George Mason
Course: Numerical Analysis
b,;ir*, S l,l "t/, 685. Pf&tu*, S tu/z 9,'nfr) t,'nk+L) . -uat ptatuzlr/ aaAtz .'? U,df = ,hf,&) wt. ddaet ia ,t = a/.c,/, tv fs;^ s vl LF %r* , lr/ ft+ti- r;il)l ls;n,(iU/4,kt K.')O ;rhr *Q Y rv llr< ,inx $ O F+o, x l, g?fX -v y' ("e) Silv*t urd+u .<' vu
School: George Mason
Course: Ordinary Differential Equations
a /W4577 - Lecnrc + ,hvdpl )rr, - 2.r?y2, I ennk^ t'ul a*alcx 2j =q * i4. Vr,.Vk)Wt'. lf4, s u,!iu' - =a | i.l W'Xry' ., VuVc*, 4,+, =[U, ihr/.tu$cfw_& ., V,UJ a*-/ t Bt. o I \r I I . 'B;l B; ry b a&* "cfw_ tr,*ft"n c^) B \/ = P-t*P=f '/t*atfe,nto =(^:-:.
School: George Mason
Course: Ordinary Differential Equations
,lt fhla'rt 677 , Leattuc ? , h/ha4r. C/4ea&,a cc fttou.r*o Lenr>1.1 -ftF:+ cfw_^ G,D < 6t cuufr:-? e. V cav,?aeb te* c'f D / ).ilrA.c fr 3u a.tfi*aa Lat, (r^, y*) e D , tL (c,r) ?, (t) - haoh*trdb c*. W I 4[isfir(q*) f ?@-t*.qu, .tcl, ol f* =';t;!r='^ I,
School: George Mason
Course: Ordinary Differential Equations
elA - rcat)hraL7OEs (f=(k, 1(a,t) icfw_ f b+s aat 4,P, o, t + dutou,t|a4 c6tfct' En tvnl)u,n' ce.tt 4) tf $ iJ Cnt*t'utau4, ra Ucitlt a, S&AOn ' -! 'A& eatt &, ,wn-u$yil =g*Z/" o :r:"=(;h). o) ,\ = f(.) r) o*oA G' hrt so sob,rt'*o x&)=;q t*r t' fr;i i,hbe
School: George Mason
Course: Ordinary Differential Equations
ls Expor*n-tt"ala a/ lflafr errz 2pznt*ns ie ctu'-" z /., lR' -' lR n fl'azat q,er*/-r G l tR^) t" ' rrTl = ra * I rft )l , rxl = (i n,)' - u ccfw_r,/za', \'-' ' Lrcfw_s i 4 /ir11 7-O, ltTr/=o e)T=o t' z r;l'zg" /nf= ^n*f Axf'. Zxia ^,".(!/;ff1)1" t = -'n
School: George Mason
Course: Ordinary Differential Equations
tl , We la.eAr^) . fl ?aqa'ru lrut h tlt'Jry Evl.i,* x4c*'*, Carr. + h" ?taZ &sfiaa/ 2tr- -Z + ght -, PeL P-' uh*. tL=(x', c ('1, lr^l P=iS r=l) o 'g'/ - tl Cr*.T-, cfw_ + \ f=tr aa*aa*) F-^ l;nu,* ahel*q: J| fteR*x2t, w;(A a7*abcs JLC 2J=4)t;3; , i't,"h
School: George Mason
Course: Ordinary Differential Equations
@ q 'it x* =) qyV,eta!4 spinaZQr>1l :) @-Dttr,ttdA cV,ra!- 0) l4 (t lif < *=)r* 4 2r=ilrlLrwlo c>lfre'r ,Y* lrl/e ) w;tt sfir,', E', gei - stak, u*a&4 euls*= b i =tlx rt= D(r") Je = ff+) ? SrU - s@4. ruubfu- ,naa1f>&/l , erist aa^d av- ha$at 6 Es. (r a* k
School: George Mason
Course: Analytic Geometry And Calculus I
'!Flt G,s / ,/ .:* il I .U. :!+-: +:iep- ffiLffiN&l t=-%qTzo&r$ +LQr-scfw_oX -E&>-+(m-cfw_F= " tfb +Zkx)-gdcfw_-=o :tK ( x=cx-tss 4r ( K -"1 ffim6rV*) =o \t^(\ -r -i '( O ?Frt =t-tu) ta.*-[- 11 6-X 1 uu I u,t- - a-F . X \d -It) +Irx\: t>xl+.ts f*Ll"e[na6J
School: George Mason
Course: Analytic Geometry And Calculus I
MfArH /t3 - vA M _1_ -VAM a -I U@StoN_1- a Ltt nclJj I r \ r-t"/) Li,uSfx)=3 K& G) Yes (t*cqu$e lgcfw_a): /- +r') I @) [t* Sfrl /xd-,rot cxisf (t"cq ue"- fi ' ou-sibb( tt,- i h /t+C @\ tiw -Prx) K-)s C") lt, x+ d- +o +04 ") -^ Yac C) po. Itw pe):S X-Jz Vd
School: George Mason
Course: Analytic Geometry And Calculus I
:Ft Nfft -+'@= lr -_= M-Ttffi.f: 4 trfrPMS # If [ittt- btt^)=VE.+a'f ilrF l0r Uw-T [Xq+rD+r-3 t-Tq*>Lr t04/ I i"-llo-1,t I^+d cfw_-?r>1' \n->o cfw_6 q=s=t'(- lr -,-trr) 64-f t6*)=-Bxs ;T cfw_-ga -3 tl tl .-'? xL+ te : lt:L)-= I 4 _ :cc)j'r4 ttzc-sf,) +>q"
School: George Mason
Course: Analytic Geometry And Calculus I
uw( MATH 113 - QUIZ 2 - 7r StrPTtrMBER 2012 Answer all of the following questions in the space provided. Show all work as partial credit rnay be given. Answers without justification) even if they are correct, will earn no credit. L. (b pts.) lrnd t2 lllq
School: George Mason
Course: Analytic Geometry And Calculus I
" t)EV_ t MATH 113 - QUIZ 4 _ 25 SEPTEMBER 2012 Answer all of the following questions in the space provided. Show all work as partial credit may be given. Answers without justification, if they are correct, will earn no ",r.n credit. 1. (5 pts.) Assume th
School: George Mason
Course: Analytic Geometry And Calculus I
MATH 113 - QUIZ 6 - 16 OCTOBER 2012 Answer all of the following questions in the space provided. Show all work as partial credit may be given. Answcrs witliout justification, even if they are corrcct, will car.n ncl credit. 1. (5 pts.) Find /'(r) r J\r):
School: George Mason
Math 677. Fall 2009. Homework #7 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.13: # 1 The system x = y + ax2 + bxy + cy 2 y = dx2 + exy + f y 2 Let LJ [h(x)] = Jh(x) Dh(x)J(x) an
School: George Mason
Math 677. Fall 2009. Homework #5 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.9: # 3 For the system x2 x1 x2 + x2 x3 1 3 2 x1 + x2 x3 x= 2 3 x1 x3 x3 x2 x2 x2 x5 1 3 3 the Liapu
School: George Mason
Math 677. Fall 2009. Homework #4 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.5: # 5 Determine the ow of the nonlinear system x = f (x), f (x) = (x1 , 2x2 +x2 )T 1 1 2 and show t
School: George Mason
Math 677. Fall 2009. Homework #6 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.11: # 2 (a) x = x2 , y = y. Expansion of the function = p2 (x, ) is a nbhd of the origin has the for
School: George Mason
Course: Ordinary Differential Equations
Math 677. Fall 2009. Homework #2 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 6: # 4 The system x = Ax with 1 1 0 0 1 1 0 0 A= 0 0 0 2 0 0 1 2 has eigenvalues 1,2 = a1 ib1 = 1
School: George Mason
Course: Ordinary Differential Equations
Math 677. Fall 2009. Homework #1 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2: # 3 Write the following linear DE with const coecients in the form of the linear system x = Ax and
School: George Mason
Course: Quantitative Reasoning
Syllabus,Spring2014 Math106006,QuantitativeReasoning Instructor: Dr.LynnellMatthews Office:ExploratoryHallRoom4407 OfficeHours: MW4:30PMTO5:30PM Phone: 7039931981 Email: lmatthe4@gmu.edu Text: MathematicalIdeas,byMiller,HereenandHornsby,CustomEdition,Pear
School: George Mason
Course: Numerical Analysis
GMU Department of Mathematical Sciences Math 685: Numerical Analysis Spring 2010 Syllabus Instructor: Prof. Maria Emelianenko Email: memelian@gmu.edu Phone: (703) 993-9688 Office: Room 226A, Science and Tech I Office Hours: Mon 1-3pm and by appt Time and
School: George Mason
MATH 1224; Summer II 2011; CRN 70873 Vector Geometry - Course Contract M-R 11:00AM-12:15PM, MCB 321 Instructor: Idir Mechai Room: MCB 321 Office: 461-Q McBryde Hall E-mail Address: mechaii@vt.edu Office Hours: M/W: 12:15-02:00 PM T / R: 02:00-03:00 PM Oth
School: George Mason
Course: Discrete Mathematics I
Syllabus: Math 125 001 Discrete Mathematics I 14321, Spring 2007 Date range Time & place Instructor Email Phone Office hours Course description Jan 22 - May 16, 2007 Lecture: MWF 10:30 am 11:20 am, Robinson Hall B122 Alexei V. Samsonovich asamsono
School: George Mason
Course: Intro Calcbusiness Applicatio
MATH 108 - Introductory Calculus with Business Applications Spring 2008 Coure Time and Place: Tuesday and Thursday 12:00-1:15 pm, 376 Bull Run Hall, Prince William Campus Instructor: Saleet Jafri Email: sjafri@gmu.edu Office: 328G Occoquan Building,
School: George Mason
Course: Quantitative Reasoning
Syllabus,Spring2014 Math106006,QuantitativeReasoning Instructor: Dr.LynnellMatthews Office:ExploratoryHallRoom4407 OfficeHours: MW4:30PMTO5:30PM Phone: 7039931981 Email: lmatthe4@gmu.edu Text: MathematicalIdeas,byMiller,HereenandHornsby,CustomEdition,Pear
School: George Mason
Math 677. Fall 2009. Homework #7 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.13: # 1 The system x = y + ax2 + bxy + cy 2 y = dx2 + exy + f y 2 Let LJ [h(x)] = Jh(x) Dh(x)J(x) an
School: George Mason
Math 677. Fall 2009. Homework #5 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.9: # 3 For the system x2 x1 x2 + x2 x3 1 3 2 x1 + x2 x3 x= 2 3 x1 x3 x3 x2 x2 x2 x5 1 3 3 the Liapu
School: George Mason
Math 677. Fall 2009. Homework #4 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.5: # 5 Determine the ow of the nonlinear system x = f (x), f (x) = (x1 , 2x2 +x2 )T 1 1 2 and show t
School: George Mason
Math 677. Fall 2009. Homework #6 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.11: # 2 (a) x = x2 , y = y. Expansion of the function = p2 (x, ) is a nbhd of the origin has the for
School: George Mason
Course: Ordinary Differential Equations
Math 677. Fall 2009. Homework #2 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 6: # 4 The system x = Ax with 1 1 0 0 1 1 0 0 A= 0 0 0 2 0 0 1 2 has eigenvalues 1,2 = a1 ib1 = 1
School: George Mason
Course: Ordinary Differential Equations
Math 677. Fall 2009. Homework #1 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2: # 3 Write the following linear DE with const coecients in the form of the linear system x = Ax and
School: George Mason
Course: Ordinary Differential Equations
a /W4577 - Lecnrc + ,hvdpl )rr, - 2.r?y2, I ennk^ t'ul a*alcx 2j =q * i4. Vr,.Vk)Wt'. lf4, s u,!iu' - =a | i.l W'Xry' ., VuVc*, 4,+, =[U, ihr/.tu$cfw_& ., V,UJ a*-/ t Bt. o I \r I I . 'B;l B; ry b a&* "cfw_ tr,*ft"n c^) B \/ = P-t*P=f '/t*atfe,nto =(^:-:.
School: George Mason
Course: Ordinary Differential Equations
,lt fhla'rt 677 , Leattuc ? , h/ha4r. C/4ea&,a cc fttou.r*o Lenr>1.1 -ftF:+ cfw_^ G,D < 6t cuufr:-? e. V cav,?aeb te* c'f D / ).ilrA.c fr 3u a.tfi*aa Lat, (r^, y*) e D , tL (c,r) ?, (t) - haoh*trdb c*. W I 4[isfir(q*) f ?@-t*.qu, .tcl, ol f* =';t;!r='^ I,
School: George Mason
Course: Ordinary Differential Equations
elA - rcat)hraL7OEs (f=(k, 1(a,t) icfw_ f b+s aat 4,P, o, t + dutou,t|a4 c6tfct' En tvnl)u,n' ce.tt 4) tf $ iJ Cnt*t'utau4, ra Ucitlt a, S&AOn ' -! 'A& eatt &, ,wn-u$yil =g*Z/" o :r:"=(;h). o) ,\ = f(.) r) o*oA G' hrt so sob,rt'*o x&)=;q t*r t' fr;i i,hbe
School: George Mason
Course: Ordinary Differential Equations
ls Expor*n-tt"ala a/ lflafr errz 2pznt*ns ie ctu'-" z /., lR' -' lR n fl'azat q,er*/-r G l tR^) t" ' rrTl = ra * I rft )l , rxl = (i n,)' - u ccfw_r,/za', \'-' ' Lrcfw_s i 4 /ir11 7-O, ltTr/=o e)T=o t' z r;l'zg" /nf= ^n*f Axf'. Zxia ^,".(!/;ff1)1" t = -'n
School: George Mason
Course: Ordinary Differential Equations
Math 677. Fall 2009. Homework #3 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 1: # 4 IVP x = x3 , x(0) = 2 has solution in the form x(t) = 1 all t (, 8 ) and lim x(t) = . 2 , 18t
School: George Mason
Course: Ordinary Differential Equations
tl , We la.eAr^) . fl ?aqa'ru lrut h tlt'Jry Evl.i,* x4c*'*, Carr. + h" ?taZ &sfiaa/ 2tr- -Z + ght -, PeL P-' uh*. tL=(x', c ('1, lr^l P=iS r=l) o 'g'/ - tl Cr*.T-, cfw_ + \ f=tr aa*aa*) F-^ l;nu,* ahel*q: J| fteR*x2t, w;(A a7*abcs JLC 2J=4)t;3; , i't,"h
School: George Mason
Course: Ordinary Differential Equations
@ q 'it x* =) qyV,eta!4 spinaZQr>1l :) @-Dttr,ttdA cV,ra!- 0) l4 (t lif < *=)r* 4 2r=ilrlLrwlo c>lfre'r ,Y* lrl/e ) w;tt sfir,', E', gei - stak, u*a&4 euls*= b i =tlx rt= D(r") Je = ff+) ? SrU - s@4. ruubfu- ,naa1f>&/l , erist aa^d av- ha$at 6 Es. (r a* k
School: George Mason
Course: Ordinary Differential Equations
prea: Un fi | Ax *.r,2rv, t7 = (0 I x0 =Yo Sln*: icfw_ Ko Colas to * ()ru t< /D+ P = )rht htlbrlsh &e Jua Q'at (r" otl* u^,#, t/<r\ tJ tb ltwa;e.,t fuent h il1 ,IgCurl'ott) Us, k=-|,pto. z \ At ,r W) A^.r-r: x&). =eMleb) xft) =e*rrb) W X, l: :71,ffr: a u,
School: George Mason
Course: Ordinary Differential Equations
lnrn 67r, ffi ,?t*'a'/g v ry 'r4, hr,n't ys/ut<t. * =Ax B *l'riur, -A#/Z"T_ff,0, ry, h (U, r' t'lg QE, ve+t, o, An ur) - ei;t r+ rP B= 9g,r/ elatpet F, cfw_,ge i - E" = fr l*eip*,-cfw_ni,ur'/ai'oJJ s*atr4' ftffi#) .= fucfw_?,rr,/4r,.oJ .1 ' cfw_Spi[r,',f;
School: George Mason
Course: Ordinary Differential Equations
n G17, /tt Lcc*+:*e lo, M. cRh -opan f eC'(e) Pt: Eq /cA*' a( - T/ua Sct ts eakJ f/*, (e tf )rw*ztaatt Cad4 f* ezet Wft cJ cfw_* ut<o t ryXfe h r'n i effz) vte*. r, ?r 6>cA , ,ardh&,Cufu mar'ft&fs -& t.At Aedt4 Tfu &- (t. rvl it? fctc*?n-( a fhl = l-I:*r,
School: George Mason
Course: Ordinary Differential Equations
Math 677. Material: Fall 2009. Final Exam Review. Perko, Chapters 1, 2, 3.1-3.3, 4.2-4.3; lectures 1-24. 1. Linear systems x = Ax: Diagonalization and Jordan representation Properties of operator exponentials Solution to IVP x = Ax, x(0) = x0 , stability
School: George Mason
Course: Ordinary Differential Equations
7j YJ /\!/, - 67r'. LectLre /cfw_ afr bbft y' ,2oo9 tLt-,'tazt (t "l ly'e-.ttt , A'i W x(t) = ("Ko .-)' ' t) r -/ ftr, r.rup(r,cfw_ ' (J ,$r ^t :) zlt)- aeat d.:cfw_ir. r., ux I s/ea-zo f T,:-X, Z,=Cr-t ?, = c, o!* 4 ( z, = !zrf (;4, /lr -) /+= ("' f)'n,*
School: George Mason
Course: Ordinary Differential Equations
Math 677. Material: Fall 2009. Exam I Review. Perko, Chapters 1.1-2.7; lectures 1-12. 1. Linear systems x = Ax: Diagonalization and Jordan representation (a) Case of distinct roots (b) Case of complex roots (c) Case of repeated roots (d) Jordan canonical
School: George Mason
Course: Ordinary Differential Equations
(t arl-, Leofu:c 7 t (/ACmn*ru it-tqrthi) Larn. A@VCQR) Pft)>o nh'nnesa +' ilh-cs' s,'& . cfw_il:2H +1!Vt'ton'nrae*c( A; cfw_*ry 6ig,ci- e i < I ftt rte d'u'/* 14 fer$'u'ta, &o*, o rf 2&1 e2 % $v' n) (arcl+ *b) exlb g=irffl:"^ A. - T'&) =a 2k) /t) 2 = Vra
School: George Mason
Course: Numerical Analysis
l^ lufafcfw_t 6Ef, (-; r,r*, s/"tn , tlx =( , Ex- f,t, +2xz:/ - V*r* xz-zz ,+=(!il s4!) 6qdsuffiitz;\ba Gat sian ;!,X7=O4 &n;aahbq Xyeq Xt*Zo= | =)xF I So2nfiort: F =/t\ la,e =&lAilb- ,;T Sp* A'-Q4tr) YieR^ 'Rn = tbcfw_a1 ?yff?*, Uxed?q Jl r,e/cu(1) YzcJf
School: George Mason
Course: Numerical Analysis
MaIn osg, f*r*nd"'lr'on i ftt, a f) '(0,t" po;ruh f*pocri*: f, s.b, f ftt) -/' t\ \, '4,*$ee w* h /t* ,b 0UtL U|z*t?,b+ g.+ ' dath- 2 -slcp D Aan &. ryeu Ctil" 4 tntayola-*) .-> nan4.ral, P4 +cfw_rA -) lxi, tattoaa2 e-t. =-r I 4 J) fuc -r t1'* 4ony )ea.t
School: George Mason
Course: Numerical Analysis
/'lacfw_A luliatuw-, . +t , ftrd'# = f 6s, '. w-*Jn.Hg!l'! t-X 'f tll llx =rt-xt [r - ll,e*(Vrl'lr iUtt = _lf ,-L.1 fuilt'f ir.-Dlcfw_. =L , ilg' Gu^uian 4)n, ptorJttrr,' k i.f'p.rrr =[ti |'t]"6J ilag 'daatinutcc ,. Ta 6h*r 'Jtrn-*nr. f'=Lu W kru Y r ofrn
School: George Mason
Course: Numerical Analysis
Math 685/CSI 700/OR 682 Homework 3 given 02/15/2010 Problem 1. Suppose that both sides of an arbitrary system of linear equations Ax = b is premultiplied by a nonsingular diagonal matrix. Does this change the true solution x? Does this aect the conditioni
School: George Mason
Course: Numerical Analysis
b,;ir*, S l,l "t/, 685. Pf&tu*, S tu/z 9,'nfr) t,'nk+L) . -uat ptatuzlr/ aaAtz .'? U,df = ,hf,&) wt. ddaet ia ,t = a/.c,/, tv fs;^ s vl LF %r* , lr/ ft+ti- r;il)l ls;n,(iU/4,kt K.')O ;rhr *Q Y rv llr< ,inx $ O F+o, x l, g?fX -v y' ("e) Silv*t urd+u .<' vu
School: George Mason
Course: Numerical Analysis
1 Simple Calculations with MATLAB 1.1 Introduction and a Word of Warning MATLAB is an incredibly powerful tool, but in order to use it safely you need to be able to understand how it works and to be very precise when you enter commands. Changing the way y
School: George Mason
Course: Numerical Analysis
Math 685/CSI 700/OR 682 Homework 2 given 02/01/10 Problem 1. It is good to know your limits. For your computer system nd: (a) the largest possible oating-point number; (b) the largest integer I s.t. (I + 1) 1 equals I; (c) the smallest possible positive o
School: George Mason
Course: Numerical Analysis
/(oa 68t, Lecftse 4, -'aT')"W -YT| SH O o epdit ir*l = + (l- zrrl-'u)0
School: George Mason
Course: Numerical Analysis
Math 685/CSI 700/OR 682 Homework 1 given 01/25/10 The following are some practice problems I would like you to do. Solutions will be discussed in class on 02/01/10. Problem 1. What do the following pieces of Octave/Matlab code accomplish? (a) x = (0 : 40)
School: George Mason
Course: Numerical Analysis
efl 6s5 bq+*e,5 tleftnds .fu &a2t . inailbe , D l/on"l g*s fr'TA lls go+Cfi F= A'8 e fastab *. . + d)c' Xlxr'.,Xn llt' sftr" - \tzco*ctxtt,"ix? 1l v' )t' JG +Ctx;1 *. *hx; ,!n= vA=[,xt., sfa . <l - \'-'. */=cfw_&)-(tart W:":') etlinri'*tt x,^1 l=[a,] I r,
School: George Mason
Course: Numerical Analysis
GMU Department of Mathematical Sciences Math 685: Numerical Analysis Spring 2010 Syllabus Instructor: Prof. Maria Emelianenko Email: memelian@gmu.edu Phone: (703) 993-9688 Office: Room 226A, Science and Tech I Office Hours: Mon 1-3pm and by appt Time and
School: George Mason
Course: Numerical Analysis
Il4nt/, 68g, Leefi,r*" g U*, *6. f- F s , Q -nft "rl * 1=)-,rlr)ft, (r - A(rr+ *rftr+lA : n -A(A?*+r)-hV ft+$lf , /sl< t ?*aq. tFls dh)' u.oL / e*-s)1l + = 7l <fu lF-, ll < Xuililej'l/:/:l'/xtt
School: George Mason
Course: Numerical Analysis
afure t , x=?t IVffiS; Ponn lly/.i,rlcl I XE*t ] .-r hct b U sahal , fitf b Xtr 'll ae atat)t flo"i/tprduth'l ryL-yfi:< 2t)2t1 ,', >2; ,(ah c-lifl Q^*, * Q,Qa .+ Xe ha b *, cl*aur ,L, rr) cut tUfur X. J r/rVr*, +qlaVa ,tl*ocryb. Wv Ce4, tD-t*w &S Xt*r Xr-
School: George Mason
Course: Analytic Geometry And Calculus I
a I l&o ocfw_ Li*,ft @prcfw_ cfw_*-.;,.^ I;^! fu,cfw_.7 vo-Q Co [.,^/,.1 a l. *,cfw_t" Trrro +^r^Jo'.^g-,n[*.,l o (il s cfw_*-t cfw_* ,noor,tS c*',qV[s r$ /rQcfw_ - roL, *fo.';[-+ (.," ,-ocfw_e a( cU*A @ +t^L [iwi"[. CI H*w k Ci*o( roto"r'[.a-? cfw_ -[":=
School: George Mason
Course: Analytic Geometry And Calculus I
'!Flt G,s / ,/ .:* il I .U. :!+-: +:iep- ffiLffiN&l t=-%qTzo&r$ +LQr-scfw_oX -E&>-+(m-cfw_F= " tfb +Zkx)-gdcfw_-=o :tK ( x=cx-tss 4r ( K -"1 ffim6rV*) =o \t^(\ -r -i '( O ?Frt =t-tu) ta.*-[- 11 6-X 1 uu I u,t- - a-F . X \d -It) +Irx\: t>xl+.ts f*Ll"e[na6J
School: George Mason
Course: Analytic Geometry And Calculus I
Q,ar> i ,',r K-> aI L TutsJy s. a,t, 2.4 -Ptx) = L ['rn hl0: M +-[ O li *.'l-t ?;*a v,uan'ttj cfw_. 'QxrystLol'( +-0"*+ Qv oJu cfw_ c cfw_. o/o, wL voc[ws ocfw_ $frl 0-'^ @ ttf cfw_C"l W K-)ot^ ,- , , ur*,| +. a, X naaf a \aut g1_, @ hrnicfw_s Ar,*-[ t.",
School: George Mason
Course: Analytic Geometry And Calculus I
MfArH /t3 - vA M _1_ -VAM a -I U@StoN_1- a Ltt nclJj I r \ r-t"/) Li,uSfx)=3 K& G) Yes (t*cqu$e lgcfw_a): /- +r') I @) [t* Sfrl /xd-,rot cxisf (t"cq ue"- fi ' ou-sibb( tt,- i h /t+C @\ tiw -Prx) K-)s C") lt, x+ d- +o +04 ") -^ Yac C) po. Itw pe):S X-Jz Vd
School: George Mason
Course: Analytic Geometry And Calculus I
A: ? ,3 t e,'t [i* -fcn: L tiar> K-) 61 e [ir cfw_fn = cfw_ fq) 4o,. ( e", ^a uu ptn) Xo. *roh ovol +n'*cho.lc ("W ;cfw_^q,u,cfw_, n* h w,i P'u^en\ cfw_ noh;"'J ,!eL^o'),. +"); , C*g! oir. f ,r,'l'P*.ho'r,tl Q1twhe ^ ^() ('rl,rr* W cfw_*ln.hous o'^l ["9 t
School: George Mason
Course: Analytic Geometry And Calculus I
:Ft Nfft -+'@= lr -_= M-Ttffi.f: 4 trfrPMS # If [ittt- btt^)=VE.+a'f ilrF l0r Uw-T [Xq+rD+r-3 t-Tq*>Lr t04/ I i"-llo-1,t I^+d cfw_-?r>1' \n->o cfw_6 q=s=t'(- lr -,-trr) 64-f t6*)=-Bxs ;T cfw_-ga -3 tl tl .-'? xL+ te : lt:L)-= I 4 _ :cc)j'r4 ttzc-sf,) +>q"
School: George Mason
Course: Analytic Geometry And Calculus I
q%Ff!a Q'ror* cfw_,nT-,un l'i Z 2"3, J?,+ rcfw_ e Li ,4^ , ft a',no( qcfw_rA-tod [),r^ik o\ trns "iQ- [ir,t Srx)- L xotp tiu, -[ ft) : t'ce P-fm x->a- tr r f*'1yt/L[n' o'lo -f fX): 6n^ a> o a (tqg -9 a=2, -+ (A- !s* / )' (VaSe d er-f",ut rLl-,.-/) t+ a> I
School: George Mason
Course: Analytic Geometry And Calculus I
i Qrl,* 3 ?.s, a,e, Tt"o D.o,ti"o\-l tlo PO CA LCU LATa PS $ct :cfw_tK) 3 \t(oFr=wL^ @- .t+h nn, tcl tt _ h^'r h+o t^ $ )(o & N*J q P",u,t r = 3-V ct:3 Ft* q l"ty^+ [t r.t ot q:3. I (3,9 cfw_a) : (ar *F-. TJ\ L4o'-A T)-ilffi \lk=*w q\btLA2- q cfw_:t\,r)L
School: George Mason
Course: Analytic Geometry And Calculus I
',- r- t. cfw_f Y'|-oa4ttc ^i- ^ cfw_'(x) - D,.ln +Orlo[*+'V\t'Lnlc-ttq I i1 [ry'n lr-)d @ T- \cqcfw_\4rth',n' ffi 'w' |wl-i ".^t "[ +-r,Wt- il;';'Yl'ttqrtr+, r *f* rlshs g- ca 1*'/rb TlrnUl d,J tA4a +-" o vlrtt,I) rr*, k'?rC cfw_'fA L^ 'r*)'= stqr" otult
School: George Mason
Course: Analytic Geometry And Calculus I
Q,] , 3.I Qutz 4 Erqw t tZuap 7-J6 (t* I Lt/eJ D'par.ro.hw ureto -) - +lCA : ti., w-d cfw_irxl t tT. ocfw_ cfw_=4 ^'/-!A ( ?o.jr) \^ I r,rt^a + c, j'n*f h + a1 x' / ttrl - rr\x^-l bn,.rq h* is /r ti*ontt *-+ 616, De v tvou* lroS! S u \^',t s ivcfw_s- otrs
School: George Mason
Course: Analytic Geometry And Calculus I
Qr,tt* 3 2,Sr& ,6 T,fl ?.rc\( D,rt'n "'+ Li^ucfw_ . t I I | ,Xq ,. \t/ Q a,\v.z^ocfw_ q l,tqv,q A, n _ J I ^ Jgp',i.'ho+ J e0;th* cfw_ ti,1^,+ (o, co"JJ (inr Srn: ) A t r,t +b' t] v va.<a'l* | My i^cfw_i,h-e [^ Y-)^ [*o h'^ ? cfw_r^) $*r. X Weva-\ t-/ E u
School: George Mason
Course: Analytic Geometry And Calculus I
d Doirvothnor rts , cfw_(x+ t')-+ fx) +'ro: liror h+.) I I afohqr, Ircfw_Ta^ptretqnqzt O S["f: cfw_ t qt xi F\ , I, Fq'[-r cfw_ C|"ryl l 1",^"7-ar^7 (t,u "l- yn*/ln"O @ [+ fr^e lai J = cfw_t) +t^-,. +'rA N W cl,*^?n ocfw_ y irzt sf.,unta,aouS d'tcfw_. cfw
School: George Mason
Course: Introduction To Advanced Mathematics
1.2 Conditionals and Biconditionals A. Conditional: 1. Read implies If then whenever only if is sufficient for is necessary for T T F F T F T F Why is 2. T F T T true when is equivalent to T T T F T F F F T F T T is false? 3. Definition 1. is t
School: George Mason
Course: Introduction To Advanced Mathematics
2.1 Basic Concepts of Set Theory. A. Belonging 1. A set is thought of as a collection of elements. 2. A fundamental property of a set is determining whether an element does or does not belong to . If it does, we write . So given , either or . 3. Typically
School: George Mason
Course: Introduction To Advanced Mathematics
1.3 Quantifiers A. Open Sentences 1. Recall that is not a proposition. On the other hand, if we write Then for each value of , is a proposition. is called an open sentence. 2. Definition 2. The truth set of an open sentence is the set of all such that i
School: George Mason
Course: Introduction To Advanced Mathematics
1.1 Propositions and Connectives 1. Deductive Reasoning a. Drawing correct conclusions from assumptions. b. Conclusions are logically necessary. c. Conclusions must be correct. 2. Inductive Reasoning a. Postulating general principles based on observations
School: George Mason
Course: Introduction To Advanced Mathematics
2.2 Set Operations. A. Intersections, Unions, Differences. 1. Definition 1. Let and be sets. i. If then and If then and If then and . ii. . iii. . B. Proving Theorems involving sets. Theorem 2.6. Let , , and i. ii. iii. iv. If Proof: then be sets. C. Russ
School: George Mason
Course: Introduction To Advanced Mathematics
1.4-1.7 Proofs and Proof Methods A. Basic assumptions about 1. 2. Each is closed under addition (+) and multiplication ( ) and the usual commutative, associative, and distributive laws hold. 3. For , a. there is a unique number such that for all , (additi
School: George Mason
Course: Introduction To Advanced Mathematics
3.3. Partitions. A. Definition and Examples 1. Definition 1. Let be a nonempty set. A partition of is a collection of subsets of satisfying a. if b. if c. , then and , and . then either or 2. Partitions and Equivalence Relations. a. Example. Consider the
School: George Mason
Course: Introduction To Advanced Mathematics
3.4. Ordering Relations. A. Partial Orderings 1. Idea: Partial orders stand in relation to in the same way that equivalence relations stand in relation to . 2. Properties of . a. Reflexive? b. Symmetric? c. Transitive? 3. Example. divides on . 4. Example
School: George Mason
Course: Introduction To Advanced Mathematics
3.2. Equivalence Relations. A. Definition and Examples 1. Definition 1. Let is reflexive if is symmetric if is transitive if be a relation on a set . is an equivalence relation if it is reflexive, symmetric and transitive. 2. Example. The identity relatio
School: George Mason
Course: Introduction To Advanced Mathematics
3.1. Cartesian Products and Relations A. Cartesian Products 1. Definition 1. Let and Cartesian product of is the set 2. be sets. The and , denoted , is an ordered pair, meaning that it is a two-element set in which the order matters. Given two ordered pai
School: George Mason
Course: Introduction To Advanced Mathematics
2.4/2.5. Mathematical Induction. A. Idea behind induction 1. Suppose you have an open sentence defined for all , and you want to prove . 2. Induction says you can do this as follows: a. Prove . This is called the base case. b. Prove that . This is called
School: George Mason
Course: Introduction To Advanced Mathematics
2.3 Extended Set Operations and Indexed Families of Sets. A. Families of Sets 1. Definition 1. A set of sets is called a family of sets. If is such a family then 2. Example. is a family of sets. Note that 3. Example. Given a set family of sets. is a Recal
School: George Mason
Course: Ordinary Differential Equations
Math 677. Material: Fall 2009. Final Exam Review. Perko, Chapters 1, 2, 3.1-3.3, 4.2-4.3; lectures 1-24. 1. Linear systems x = Ax: Diagonalization and Jordan representation Properties of operator exponentials Solution to IVP x = Ax, x(0) = x0 , stability
School: George Mason
Course: Ordinary Differential Equations
Math 677. Material: Fall 2009. Exam I Review. Perko, Chapters 1.1-2.7; lectures 1-12. 1. Linear systems x = Ax: Diagonalization and Jordan representation (a) Case of distinct roots (b) Case of complex roots (c) Case of repeated roots (d) Jordan canonical
School: George Mason
Course: Numerical Analysis
l^ lufafcfw_t 6Ef, (-; r,r*, s/"tn , tlx =( , Ex- f,t, +2xz:/ - V*r* xz-zz ,+=(!il s4!) 6qdsuffiitz;\ba Gat sian ;!,X7=O4 &n;aahbq Xyeq Xt*Zo= | =)xF I So2nfiort: F =/t\ la,e =&lAilb- ,;T Sp* A'-Q4tr) YieR^ 'Rn = tbcfw_a1 ?yff?*, Uxed?q Jl r,e/cu(1) YzcJf
School: George Mason
Course: Numerical Analysis
MaIn osg, f*r*nd"'lr'on i ftt, a f) '(0,t" po;ruh f*pocri*: f, s.b, f ftt) -/' t\ \, '4,*$ee w* h /t* ,b 0UtL U|z*t?,b+ g.+ ' dath- 2 -slcp D Aan &. ryeu Ctil" 4 tntayola-*) .-> nan4.ral, P4 +cfw_rA -) lxi, tattoaa2 e-t. =-r I 4 J) fuc -r t1'* 4ony )ea.t
School: George Mason
Course: Numerical Analysis
/'lacfw_A luliatuw-, . +t , ftrd'# = f 6s, '. w-*Jn.Hg!l'! t-X 'f tll llx =rt-xt [r - ll,e*(Vrl'lr iUtt = _lf ,-L.1 fuilt'f ir.-Dlcfw_. =L , ilg' Gu^uian 4)n, ptorJttrr,' k i.f'p.rrr =[ti |'t]"6J ilag 'daatinutcc ,. Ta 6h*r 'Jtrn-*nr. f'=Lu W kru Y r ofrn
School: George Mason
Course: Numerical Analysis
b,;ir*, S l,l "t/, 685. Pf&tu*, S tu/z 9,'nfr) t,'nk+L) . -uat ptatuzlr/ aaAtz .'? U,df = ,hf,&) wt. ddaet ia ,t = a/.c,/, tv fs;^ s vl LF %r* , lr/ ft+ti- r;il)l ls;n,(iU/4,kt K.')O ;rhr *Q Y rv llr< ,inx $ O F+o, x l, g?fX -v y' ("e) Silv*t urd+u .<' vu
School: George Mason
Course: Numerical Analysis
1 Simple Calculations with MATLAB 1.1 Introduction and a Word of Warning MATLAB is an incredibly powerful tool, but in order to use it safely you need to be able to understand how it works and to be very precise when you enter commands. Changing the way y
School: George Mason
Course: Numerical Analysis
/(oa 68t, Lecftse 4, -'aT')"W -YT| SH O o epdit ir*l = + (l- zrrl-'u)0
School: George Mason
Course: Numerical Analysis
efl 6s5 bq+*e,5 tleftnds .fu &a2t . inailbe , D l/on"l g*s fr'TA lls go+Cfi F= A'8 e fastab *. . + d)c' Xlxr'.,Xn llt' sftr" - \tzco*ctxtt,"ix? 1l v' )t' JG +Ctx;1 *. *hx; ,!n= vA=[,xt., sfa . <l - \'-'. */=cfw_&)-(tart W:":') etlinri'*tt x,^1 l=[a,] I r,
School: George Mason
Course: Numerical Analysis
Il4nt/, 68g, Leefi,r*" g U*, *6. f- F s , Q -nft "rl * 1=)-,rlr)ft, (r - A(rr+ *rftr+lA : n -A(A?*+r)-hV ft+$lf , /sl< t ?*aq. tFls dh)' u.oL / e*-s)1l + = 7l <fu lF-, ll < Xuililej'l/:/:l'/xtt
School: George Mason
Course: Numerical Analysis
afure t , x=?t IVffiS; Ponn lly/.i,rlcl I XE*t ] .-r hct b U sahal , fitf b Xtr 'll ae atat)t flo"i/tprduth'l ryL-yfi:< 2t)2t1 ,', >2; ,(ah c-lifl Q^*, * Q,Qa .+ Xe ha b *, cl*aur ,L, rr) cut tUfur X. J r/rVr*, +qlaVa ,tl*ocryb. Wv Ce4, tD-t*w &S Xt*r Xr-
School: George Mason
Course: Analytic Geometry And Calculus I
a I l&o ocfw_ Li*,ft @prcfw_ cfw_*-.;,.^ I;^! fu,cfw_.7 vo-Q Co [.,^/,.1 a l. *,cfw_t" Trrro +^r^Jo'.^g-,n[*.,l o (il s cfw_*-t cfw_* ,noor,tS c*',qV[s r$ /rQcfw_ - roL, *fo.';[-+ (.," ,-ocfw_e a( cU*A @ +t^L [iwi"[. CI H*w k Ci*o( roto"r'[.a-? cfw_ -[":=
School: George Mason
Course: Analytic Geometry And Calculus I
Q,ar> i ,',r K-> aI L TutsJy s. a,t, 2.4 -Ptx) = L ['rn hl0: M +-[ O li *.'l-t ?;*a v,uan'ttj cfw_. 'QxrystLol'( +-0"*+ Qv oJu cfw_ c cfw_. o/o, wL voc[ws ocfw_ $frl 0-'^ @ ttf cfw_C"l W K-)ot^ ,- , , ur*,| +. a, X naaf a \aut g1_, @ hrnicfw_s Ar,*-[ t.",
School: George Mason
Course: Analytic Geometry And Calculus I
A: ? ,3 t e,'t [i* -fcn: L tiar> K-) 61 e [ir cfw_fn = cfw_ fq) 4o,. ( e", ^a uu ptn) Xo. *roh ovol +n'*cho.lc ("W ;cfw_^q,u,cfw_, n* h w,i P'u^en\ cfw_ noh;"'J ,!eL^o'),. +"); , C*g! oir. f ,r,'l'P*.ho'r,tl Q1twhe ^ ^() ('rl,rr* W cfw_*ln.hous o'^l ["9 t
School: George Mason
Course: Analytic Geometry And Calculus I
q%Ff!a Q'ror* cfw_,nT-,un l'i Z 2"3, J?,+ rcfw_ e Li ,4^ , ft a',no( qcfw_rA-tod [),r^ik o\ trns "iQ- [ir,t Srx)- L xotp tiu, -[ ft) : t'ce P-fm x->a- tr r f*'1yt/L[n' o'lo -f fX): 6n^ a> o a (tqg -9 a=2, -+ (A- !s* / )' (VaSe d er-f",ut rLl-,.-/) t+ a> I
School: George Mason
Course: Analytic Geometry And Calculus I
i Qrl,* 3 ?.s, a,e, Tt"o D.o,ti"o\-l tlo PO CA LCU LATa PS $ct :cfw_tK) 3 \t(oFr=wL^ @- .t+h nn, tcl tt _ h^'r h+o t^ $ )(o & N*J q P",u,t r = 3-V ct:3 Ft* q l"ty^+ [t r.t ot q:3. I (3,9 cfw_a) : (ar *F-. TJ\ L4o'-A T)-ilffi \lk=*w q\btLA2- q cfw_:t\,r)L
School: George Mason
Course: Analytic Geometry And Calculus I
',- r- t. cfw_f Y'|-oa4ttc ^i- ^ cfw_'(x) - D,.ln +Orlo[*+'V\t'Lnlc-ttq I i1 [ry'n lr-)d @ T- \cqcfw_\4rth',n' ffi 'w' |wl-i ".^t "[ +-r,Wt- il;';'Yl'ttqrtr+, r *f* rlshs g- ca 1*'/rb TlrnUl d,J tA4a +-" o vlrtt,I) rr*, k'?rC cfw_'fA L^ 'r*)'= stqr" otult
School: George Mason
Course: Analytic Geometry And Calculus I
Q,] , 3.I Qutz 4 Erqw t tZuap 7-J6 (t* I Lt/eJ D'par.ro.hw ureto -) - +lCA : ti., w-d cfw_irxl t tT. ocfw_ cfw_=4 ^'/-!A ( ?o.jr) \^ I r,rt^a + c, j'n*f h + a1 x' / ttrl - rr\x^-l bn,.rq h* is /r ti*ontt *-+ 616, De v tvou* lroS! S u \^',t s ivcfw_s- otrs
School: George Mason
Course: Analytic Geometry And Calculus I
Qr,tt* 3 2,Sr& ,6 T,fl ?.rc\( D,rt'n "'+ Li^ucfw_ . t I I | ,Xq ,. \t/ Q a,\v.z^ocfw_ q l,tqv,q A, n _ J I ^ Jgp',i.'ho+ J e0;th* cfw_ ti,1^,+ (o, co"JJ (inr Srn: ) A t r,t +b' t] v va.<a'l* | My i^cfw_i,h-e [^ Y-)^ [*o h'^ ? cfw_r^) $*r. X Weva-\ t-/ E u
School: George Mason
Course: Analytic Geometry And Calculus I
d Doirvothnor rts , cfw_(x+ t')-+ fx) +'ro: liror h+.) I I afohqr, Ircfw_Ta^ptretqnqzt O S["f: cfw_ t qt xi F\ , I, Fq'[-r cfw_ C|"ryl l 1",^"7-ar^7 (t,u "l- yn*/ln"O @ [+ fr^e lai J = cfw_t) +t^-,. +'rA N W cl,*^?n ocfw_ y irzt sf.,unta,aouS d'tcfw_. cfw
School: George Mason
Course: Introduction To Advanced Mathematics
1.2 Conditionals and Biconditionals A. Conditional: 1. Read implies If then whenever only if is sufficient for is necessary for T T F F T F T F Why is 2. T F T T true when is equivalent to T T T F T F F F T F T T is false? 3. Definition 1. is t
School: George Mason
Course: Introduction To Advanced Mathematics
2.1 Basic Concepts of Set Theory. A. Belonging 1. A set is thought of as a collection of elements. 2. A fundamental property of a set is determining whether an element does or does not belong to . If it does, we write . So given , either or . 3. Typically
School: George Mason
Course: Introduction To Advanced Mathematics
1.3 Quantifiers A. Open Sentences 1. Recall that is not a proposition. On the other hand, if we write Then for each value of , is a proposition. is called an open sentence. 2. Definition 2. The truth set of an open sentence is the set of all such that i
School: George Mason
Course: Introduction To Advanced Mathematics
1.1 Propositions and Connectives 1. Deductive Reasoning a. Drawing correct conclusions from assumptions. b. Conclusions are logically necessary. c. Conclusions must be correct. 2. Inductive Reasoning a. Postulating general principles based on observations
School: George Mason
Course: Introduction To Advanced Mathematics
2.2 Set Operations. A. Intersections, Unions, Differences. 1. Definition 1. Let and be sets. i. If then and If then and If then and . ii. . iii. . B. Proving Theorems involving sets. Theorem 2.6. Let , , and i. ii. iii. iv. If Proof: then be sets. C. Russ
School: George Mason
Course: Introduction To Advanced Mathematics
1.4-1.7 Proofs and Proof Methods A. Basic assumptions about 1. 2. Each is closed under addition (+) and multiplication ( ) and the usual commutative, associative, and distributive laws hold. 3. For , a. there is a unique number such that for all , (additi
School: George Mason
Course: Introduction To Advanced Mathematics
3.3. Partitions. A. Definition and Examples 1. Definition 1. Let be a nonempty set. A partition of is a collection of subsets of satisfying a. if b. if c. , then and , and . then either or 2. Partitions and Equivalence Relations. a. Example. Consider the
School: George Mason
Course: Introduction To Advanced Mathematics
3.4. Ordering Relations. A. Partial Orderings 1. Idea: Partial orders stand in relation to in the same way that equivalence relations stand in relation to . 2. Properties of . a. Reflexive? b. Symmetric? c. Transitive? 3. Example. divides on . 4. Example
School: George Mason
Course: Introduction To Advanced Mathematics
3.2. Equivalence Relations. A. Definition and Examples 1. Definition 1. Let is reflexive if is symmetric if is transitive if be a relation on a set . is an equivalence relation if it is reflexive, symmetric and transitive. 2. Example. The identity relatio
School: George Mason
Course: Introduction To Advanced Mathematics
3.1. Cartesian Products and Relations A. Cartesian Products 1. Definition 1. Let and Cartesian product of is the set 2. be sets. The and , denoted , is an ordered pair, meaning that it is a two-element set in which the order matters. Given two ordered pai
School: George Mason
Course: Introduction To Advanced Mathematics
2.4/2.5. Mathematical Induction. A. Idea behind induction 1. Suppose you have an open sentence defined for all , and you want to prove . 2. Induction says you can do this as follows: a. Prove . This is called the base case. b. Prove that . This is called
School: George Mason
Course: Introduction To Advanced Mathematics
2.3 Extended Set Operations and Indexed Families of Sets. A. Families of Sets 1. Definition 1. A set of sets is called a family of sets. If is such a family then 2. Example. is a family of sets. Note that 3. Example. Given a set family of sets. is a Recal
School: George Mason
Course: Advanced Calculus I
I F:t l.t*ul @ B A.n-^sp r.^ LR, ir./cfw__ Vwa C/ Tc co\tuvcfw_ l= bta C [a'w r Ltcfw_ A s tR a'd, tocfw_ cfw_e^T tcfw_ nv of-nar * Tw* cfw_-hr,.t tt q @ cfw_-t"(. s ,a'L*wn fanQ,. Cevwt ^6 i*- A 1e";, [,./e . "fAl 8,Ory, !$, Cr.r u".i.^-'cfw_fu .*[(a^-0
School: George Mason
Course: Advanced Calculus I
Fra u^.y [e l. . hvrn out L s [. tut^ihfl" *0 r[*-l tL is (,uq V*+cfw_"*1- e : c\ nr-g nu*ko^, S" ; ea*r,r, U"-te T LL 4s q arcfw_-cfw_n t0* 'i [n-\.r" Pltt"/ r-r'1/ La a^rb Ca"rly "J! e conv(w^?A hA tlP q*/ e-. T.* s |.^o, ,8* R w-a rr,t*-s cfw_ \r cr.u&
School: George Mason
Course: Advanced Calculus I
1 .1 The Real Number System. A. What do we need to know about IR.? 1. R. is an ; Archimedean ordered field. 2. Q is also an Archimedean ordered field. 3. What are some differences between R. and e? a. Q is missing some important numbers like t/2, rc, e, e
School: George Mason
Course: Advanced Calculus I
h(L;)Ft;,! ^& fi)=(ilt tcfw_;) 'a$rutn4e w lr/ ' /r Yor f* nl-.l "Ap+l,ox D hfi1+rXu ' u fi =' (A-'YJ| 'nXo, ,X o* e\-yD ;-T - (A+rxJ i * - n\l(oT aarr Y'cfw_i+(A*', QA l4-n - lrO$ f ;g4 -ry'6un^Jh'y fYvb q tu re(Wtu ) v fi 'Vtd-ltn, y'X 'frvwp4oD ,A + tX
School: George Mason
Course: Advanced Calculus I
F. B. Density of Q in R. 1. Definition 1. A subset S c IR is called dense in R. if for all x IR, there is a sequence se in S such that sk + x. 2. Remark. Equivalently we can say that S is dense in IR if every open interval in R contains an element of S. P
School: George Mason
Course: Advanced Calculus I
4. Example. limsupn-6 )y1 - t^= cfw_t\= ;tP llf f*] 1 Iiminfr-* flt -L G2 u\ n r,r-t ( htl [. wt )n = C\ =o [t 1. [t t,rn 1 -o 7u li* t^: [t* o : Q ) I 5. Theorem 3. xn if and only if limsupft_-+axn ltminfn-*r, - Proof' - L. (+\ S,qlf,*t i^* L^we wut[ s c
School: George Mason
Course: Advanced Calculus I
,/ l,q I s_e tL +t,d S cc,a,t I* ,t,1?^ d4 0, uu trx ,f , Y r"cfw_p^ rr,L F tr"T td\S= [J]o Fact ! Car^,rvrcfw_e g-: t) Jfl SQ u/ ; . ltl: f Ulp 9qw vva UUY +=', ",(" +L* lo$uot. t L Sn @ :- cfw_6,6r,5E,-.J o^f &n @"[^ ff^,Ir)o s4' o*(c lL w*w'. (6^*nt A,
School: George Mason
Course: Advanced Calculus I
?n|i r,. t'l n I u,'l y ,^-(, , !* + Dqo\oou./ fRcfw_ *1urQ, P= |r,t,-/ : I L(cfw_ rP) Yvt L:/ t/ (cfw_rP)= xal L (x; * xi-,) L = ivcfw_ cfw_f ,cfw_," Fi,t X( fr; -r,XL f rVi (rr- rL-,) Mi -5np ;*ro L *-/ r vL^vl* s*.7 L(T,P) Pr i Vc-"'t73 + cfw_tcfw_ rlc
School: George Mason
Course: Advanced Calculus I
cfw_eClqttl cfw_rn )o cfw_31;'s xc ) F.b] o^/ C'r sq/nt t$&)"tx (-^ (r "lL Pe L'I b], -t tP)>o, t In* t^a "q t Let P = [rq/ - -,f ul Tt^o,^ 4 S" te* P tr, f wt h ? e [^i-,X1]l ['- '( la,t] Qn, q'[ [.'oos-f l qd L, rt -P; Ttr^ rr?)= !'[?'oAx; ?G,fd) )o $:"
School: George Mason
Course: Advanced Calculus I
i l,-ar .l n-\ J. lK'l : -r l,t r' , (-x\ kvtt[-.rt*l MWa^'L V,51r ["'D W, I Qzt, <l Vr.b]: V, 'ls] : P,ffiil1 [( t-xs) ^t = ,uP lrtil 'tp [.'D [o,il s Ict-rq-r\:;H t-,lrj :1P, .A I [ l^-P[tqF fxl,u\ : Yel Yol Fs, t :- lr^ -to 4s ft+do [, cfw_| &: WLy xoe
School: George Mason
Course: Advanced Calculus I
Erdvq fr2, R,-\^J l t+ Xur Is L.,w"W+cfw_r,.^Srl-Us gu,Jvttq \ K* f^$ a- co.yt @ Ccyr,"l-.o P-Jr[-,;^e t+ X^ Var : SwVt-1ry'14^or Fo* V-',^"h"( tL ( U@ C"?( ,o^y^* Xw l, ,rtrcfw_ Vt \/v^ou. lrt"s ^ocfw_ ru,cfw_ ,pl| [.x*L* oo , heF oua 2r-^( Xy.: + I Xn=
School: George Mason
Course: Advanced Calculus I
?l ( + hrn,+ ch4f vt'^sulL n (q, t) , o ( \*r L a'\ ^+) s '>o fs >o s,cfw_ U /,y< k,h, \x -JIz S e> \cfw_rt-cfw_r0l zt G) n*, a- x ll G) Y'.w / u ,t r$ c.n* o^ (arq] SWut ,-n,-C c*cfw_ cfw_v\ &.'b)= (q,4 u[ucfw_,t) S*), O t-e\ r>cv, Firt/ \, t" 6^ &1r) .
School: George Mason
Course: Numerical Analysis / Numerical Methods
MATH 446 / OR 481 SAUER SPRING 2014 Review Problems 24 April 2014 1. Describe the most efcient method you can nd (in the sense of the minimum number of multiplications needed) for evaluating the polynomial P (x) = a3 x3 + a7 x7 + a11 x11 + a15 x15 + a19 x
School: George Mason
Course: Ordinary Differential Equations
a /W4577 - Lecnrc + ,hvdpl )rr, - 2.r?y2, I ennk^ t'ul a*alcx 2j =q * i4. Vr,.Vk)Wt'. lf4, s u,!iu' - =a | i.l W'Xry' ., VuVc*, 4,+, =[U, ihr/.tu$cfw_& ., V,UJ a*-/ t Bt. o I \r I I . 'B;l B; ry b a&* "cfw_ tr,*ft"n c^) B \/ = P-t*P=f '/t*atfe,nto =(^:-:.
School: George Mason
Course: Ordinary Differential Equations
,lt fhla'rt 677 , Leattuc ? , h/ha4r. C/4ea&,a cc fttou.r*o Lenr>1.1 -ftF:+ cfw_^ G,D < 6t cuufr:-? e. V cav,?aeb te* c'f D / ).ilrA.c fr 3u a.tfi*aa Lat, (r^, y*) e D , tL (c,r) ?, (t) - haoh*trdb c*. W I 4[isfir(q*) f ?@-t*.qu, .tcl, ol f* =';t;!r='^ I,
School: George Mason
Course: Ordinary Differential Equations
elA - rcat)hraL7OEs (f=(k, 1(a,t) icfw_ f b+s aat 4,P, o, t + dutou,t|a4 c6tfct' En tvnl)u,n' ce.tt 4) tf $ iJ Cnt*t'utau4, ra Ucitlt a, S&AOn ' -! 'A& eatt &, ,wn-u$yil =g*Z/" o :r:"=(;h). o) ,\ = f(.) r) o*oA G' hrt so sob,rt'*o x&)=;q t*r t' fr;i i,hbe
School: George Mason
Course: Ordinary Differential Equations
ls Expor*n-tt"ala a/ lflafr errz 2pznt*ns ie ctu'-" z /., lR' -' lR n fl'azat q,er*/-r G l tR^) t" ' rrTl = ra * I rft )l , rxl = (i n,)' - u ccfw_r,/za', \'-' ' Lrcfw_s i 4 /ir11 7-O, ltTr/=o e)T=o t' z r;l'zg" /nf= ^n*f Axf'. Zxia ^,".(!/;ff1)1" t = -'n
School: George Mason
Course: Ordinary Differential Equations
tl , We la.eAr^) . fl ?aqa'ru lrut h tlt'Jry Evl.i,* x4c*'*, Carr. + h" ?taZ &sfiaa/ 2tr- -Z + ght -, PeL P-' uh*. tL=(x', c ('1, lr^l P=iS r=l) o 'g'/ - tl Cr*.T-, cfw_ + \ f=tr aa*aa*) F-^ l;nu,* ahel*q: J| fteR*x2t, w;(A a7*abcs JLC 2J=4)t;3; , i't,"h
School: George Mason
Course: Ordinary Differential Equations
@ q 'it x* =) qyV,eta!4 spinaZQr>1l :) @-Dttr,ttdA cV,ra!- 0) l4 (t lif < *=)r* 4 2r=ilrlLrwlo c>lfre'r ,Y* lrl/e ) w;tt sfir,', E', gei - stak, u*a&4 euls*= b i =tlx rt= D(r") Je = ff+) ? SrU - s@4. ruubfu- ,naa1f>&/l , erist aa^d av- ha$at 6 Es. (r a* k
School: George Mason
Course: Ordinary Differential Equations
prea: Un fi | Ax *.r,2rv, t7 = (0 I x0 =Yo Sln*: icfw_ Ko Colas to * ()ru t< /D+ P = )rht htlbrlsh &e Jua Q'at (r" otl* u^,#, t/<r\ tJ tb ltwa;e.,t fuent h il1 ,IgCurl'ott) Us, k=-|,pto. z \ At ,r W) A^.r-r: x&). =eMleb) xft) =e*rrb) W X, l: :71,ffr: a u,
School: George Mason
Course: Ordinary Differential Equations
lnrn 67r, ffi ,?t*'a'/g v ry 'r4, hr,n't ys/ut<t. * =Ax B *l'riur, -A#/Z"T_ff,0, ry, h (U, r' t'lg QE, ve+t, o, An ur) - ei;t r+ rP B= 9g,r/ elatpet F, cfw_,ge i - E" = fr l*eip*,-cfw_ni,ur'/ai'oJJ s*atr4' ftffi#) .= fucfw_?,rr,/4r,.oJ .1 ' cfw_Spi[r,',f;
School: George Mason
Course: Ordinary Differential Equations
n G17, /tt Lcc*+:*e lo, M. cRh -opan f eC'(e) Pt: Eq /cA*' a( - T/ua Sct ts eakJ f/*, (e tf )rw*ztaatt Cad4 f* ezet Wft cJ cfw_* ut<o t ryXfe h r'n i effz) vte*. r, ?r 6>cA , ,ardh&,Cufu mar'ft&fs -& t.At Aedt4 Tfu &- (t. rvl it? fctc*?n-( a fhl = l-I:*r,
School: George Mason
Course: Ordinary Differential Equations
7j YJ /\!/, - 67r'. LectLre /cfw_ afr bbft y' ,2oo9 tLt-,'tazt (t "l ly'e-.ttt , A'i W x(t) = ("Ko .-)' ' t) r -/ ftr, r.rup(r,cfw_ ' (J ,$r ^t :) zlt)- aeat d.:cfw_ir. r., ux I s/ea-zo f T,:-X, Z,=Cr-t ?, = c, o!* 4 ( z, = !zrf (;4, /lr -) /+= ("' f)'n,*
School: George Mason
Course: Ordinary Differential Equations
(t arl-, Leofu:c 7 t (/ACmn*ru it-tqrthi) Larn. A@VCQR) Pft)>o nh'nnesa +' ilh-cs' s,'& . cfw_il:2H +1!Vt'ton'nrae*c( A; cfw_*ry 6ig,ci- e i < I ftt rte d'u'/* 14 fer$'u'ta, &o*, o rf 2&1 e2 % $v' n) (arcl+ *b) exlb g=irffl:"^ A. - T'&) =a 2k) /t) 2 = Vra
School: George Mason
Course: Introductory Calculus With Business Applications
1.3. Linear Functions Denition A linear function is a function that changes at a constant rate with respect to its independent variable. The graph of a linear function is a straight line. The equation of a linear function can be written as y = mx + b wher
School: George Mason
Course: Introductory Calculus With Business Applications
2.6. Implicit Differentiation and Related Rates Example Find dy 1 if x + = 4. dx y Implicit Differentiation Suppose an equation denes y implicitly as a differentiable function of x. To nd the derivative of y, 1. Differentiate both sides of the equation wi
School: George Mason
Course: Introductory Calculus With Business Applications
2.4. The Chain Rule If y = f (u) is a differentiable function of u and u = g(x) is in turn a differentiable function of x, then the composite function f (g(x) is a differentiable function of x whose derivative is given by the product dy dy du = dx du dx o
School: George Mason
Course: Introductory Calculus With Business Applications
2.3. Product and Quotient Rules; Higher-Order Derivatives The Product Rule If f (x) and g(x) are differentiable at x, then so is their product and d d d [f (x)g(x)] = f (x) [g(x)] + g(x) [f (x)] dx dx dx or equivalently (fg) = fg + gf Example Differentia
School: George Mason
Course: Introductory Calculus With Business Applications
1.5. Limits Behavior of f (x) as x approaches c Consider the behavior of f (x) = as x approaches 1. x f (x) 0.8 -1.2 0.9 -1.1 0.99 -1.01 x 2 3x + 2 (x 1)(x 2) = x 1 x 1 1 undened 1.01 -0.99 1.1 -0.9 1.2 -0.8 As x approaches 1, f (x) approaches 1. Denition
School: George Mason
Course: Introductory Calculus With Business Applications
1.2. The Graph of a Function Denition The graph of a function f consists of all points (x, y) where x is in the domain of f and y = f (x); that is, all points of the form (x, f (x). Example Graph the function f (x) = x 2 . x and y intercepts Denition The
School: George Mason
Course: Introductory Calculus With Business Applications
1.1. Functions Loosely speaking, a function consists of two sets and a rule that associates elements in one set with elements in the other. Denition A function is a rule that assigns to each objects in a set A exactly one object in a set B. The set A is c
School: George Mason
Course: Introductory Calculus With Business Applications
1.4. Functional Models Basic Goal : Developing mathematical methods for dealing with practical problems. Mathematical Modeling Stage. 1 (Formulation) Identify key variables and establish equations relating those variables. Stage. 2 (Analysis of the Model)
School: George Mason
Course: Introductory Calculus With Business Applications
1.6. One-sided Limits and Continuity One-sided Limit If f (x) approaches L as x tends toward c from the left (x < c), we write lim f (x) = L. xc Likewise, if f (x) approaches M as x tends toward c from the right (x > c), then lim+ f (x) = M. xc Example F
School: George Mason
Course: Introductory Calculus With Business Applications
2.2. Techniques of Differentiation The Constant Rule For any constant c, d [c] = 0 dx The Power Rule For any real number n, d n [x ] = nx n1 dx Example Differentiate the function y = x 5. The Constant Multiple Rule If c is a constant and f (x) is diffenen
School: George Mason
Course: Introductory Calculus With Business Applications
2.1 The Derivative The derivative of a function The derivative of the function f (x) with respect to x is the function f (x) given by f (x + h) f (x) . h h0 f (x) = lim The process of computing the derivative is called differentiation, and we say that f (
School: George Mason
Course: Introductory Calculus With Business Applications
4.2. Logarithmic Functions If x is a positive number, then the logarithm of x to the base b(b > 0, b = 1), denoted logb x, is the number y such that by = x; that is, y = logb x if and only if Example Evaluate log10 1, 000. Example Solve the equation log4
School: George Mason
Course: Introductory Calculus With Business Applications
4.1 Exponential Functions Denition of bn for rational values of n (and b > 0) Integer Powers: If n is a positive integer, bn = b b b n factors Fractional Powers: If n and m are positive integers, m m bn/m = ( b)n = bn Negative Powers: bn = Zero Power: b0
School: George Mason
Course: Introductory Calculus With Business Applications
4.4. Additional Exponential Models General Procedure for Sketching the Graph Step 1. Find the domain of f (x). Step 2. Find and plot all intercepts. Step 3. Determine all vertical and horizontal asymptotes and draw them. Step 4. Find f (x) and determine t
School: George Mason
Course: Introductory Calculus With Business Applications
3.4. Optimization Absolute Maxima and Minima of a function Let f be a function dened on an interval I containing the number c. Then f (c) is the absolute maximum of f on I if f (c) f (x) for all x in I. f (c) is the absolute minimum of f on I if f (c) f (
School: George Mason
Course: Introductory Calculus With Business Applications
3.3. Curve Sketching Vertical Asymptotes The vertical line x = c is a vertical asymptote of the graph of f (x) if either lim f (x) = + (or ) xc or lim f (x) = + (or ) xc + Vertical Asymptotes Example Determine all vertical asymptotes of the graph of g(x)
School: George Mason
Course: Introductory Calculus With Business Applications
3.1. Increasing and Decreasing Functions; Relative Extrema Increasing and Decreasing Functions Let f (x) be a function dened on the interval a < x < b, and let x1 and x2 be two numbers in the interval. Then f (x) is increasing on the interval if f (x2 ) >
School: George Mason
Course: Introductory Calculus With Business Applications
3.2. Concavity and Points of Inection Denition If f (x) is differentiable on the interval a < x < b, then the graph of f is concave upward on a < x < b if f is increasing on the interval concave downward on a < x < b if f is decreasing on the interval Con
School: George Mason
Course: Introductory Calculus With Business Applications
4.3. Differentiation of Logarithmic and Exponential Functions Derivative of ln x d 1 (ln x) = dx x Example for x > 0 Differentiate the function f (x) = x ln x. Differentiation of Logarithmic Functions The Chain Rule for Logarithmic Functions If u(x) is a
School: George Mason
Course: Introductory Calculus With Business Applications
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School: George Mason
Course: Introductory Calculus With Business Applications
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School: George Mason
Course: Introductory Calculus With Business Applications
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School: George Mason
Course: Introductory Calculus With Business Applications
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School: George Mason
Course: Introductory Calculus With Business Applications
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School: George Mason
Course: Introductory Calculus With Business Applications
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School: George Mason
Course: Introductory Calculus With Business Applications
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School: George Mason
Course: Introductory Calculus With Business Applications
s^ l De[)'Vr4hres Denvrhves c4re J,rtetFlS re,\,abd Io li'a 'ls : @ Fn) : ,f ., L 3=T fie lrurt ilar'o"tfi tlq\ ot'ud (Yl tYt^f ) herr slo 0L C3d^f-q M: tA/t (3+h\ - 3 rrz 6t-h) -'1 V\nqtn h-o \trna h-l o I\iaa hro t\ y1 of 1 3tuul cfw_-14q-" slopq ' t] +
School: George Mason
Course: Introductory Calculus With Business Applications
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School: George Mason
Course: Introductory Calculus With Business Applications
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School: George Mason
Course: Introductory Calculus With Business Applications
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School: George Mason
Course: Introductory Calculus With Business Applications
)q NA 0r- S",L 4.-+- rj q (oll eth'on e \gu'l pF -obJ-4t t'tacfw_r A:f1,3,s,1,?1 Hgq A t> lh s-i o$ o Jcfw_ I^nuru/ w\Arnabtn fror vs O ,1" 40 4,3, .1-V\!. lnuu,tb fhq e\ Q,n r e.ul'r o llnt T\orF 3 \ I ts eu cfw_-q\^ebscfw_ V,1 ,q c^rQ- Srr\ A slpu,tartt
School: George Mason
Course: Analytic Geometry And Calculus I
'!Flt G,s / ,/ .:* il I .U. :!+-: +:iep- ffiLffiN&l t=-%qTzo&r$ +LQr-scfw_oX -E&>-+(m-cfw_F= " tfb +Zkx)-gdcfw_-=o :tK ( x=cx-tss 4r ( K -"1 ffim6rV*) =o \t^(\ -r -i '( O ?Frt =t-tu) ta.*-[- 11 6-X 1 uu I u,t- - a-F . X \d -It) +Irx\: t>xl+.ts f*Ll"e[na6J
School: George Mason
Course: Analytic Geometry And Calculus I
MfArH /t3 - vA M _1_ -VAM a -I U@StoN_1- a Ltt nclJj I r \ r-t"/) Li,uSfx)=3 K& G) Yes (t*cqu$e lgcfw_a): /- +r') I @) [t* Sfrl /xd-,rot cxisf (t"cq ue"- fi ' ou-sibb( tt,- i h /t+C @\ tiw -Prx) K-)s C") lt, x+ d- +o +04 ") -^ Yac C) po. Itw pe):S X-Jz Vd
School: George Mason
Course: Analytic Geometry And Calculus I
:Ft Nfft -+'@= lr -_= M-Ttffi.f: 4 trfrPMS # If [ittt- btt^)=VE.+a'f ilrF l0r Uw-T [Xq+rD+r-3 t-Tq*>Lr t04/ I i"-llo-1,t I^+d cfw_-?r>1' \n->o cfw_6 q=s=t'(- lr -,-trr) 64-f t6*)=-Bxs ;T cfw_-ga -3 tl tl .-'? xL+ te : lt:L)-= I 4 _ :cc)j'r4 ttzc-sf,) +>q"
School: George Mason
Course: Analytic Geometry And Calculus I
uw( MATH 113 - QUIZ 2 - 7r StrPTtrMBER 2012 Answer all of the following questions in the space provided. Show all work as partial credit rnay be given. Answers without justification) even if they are correct, will earn no credit. L. (b pts.) lrnd t2 lllq
School: George Mason
Course: Analytic Geometry And Calculus I
" t)EV_ t MATH 113 - QUIZ 4 _ 25 SEPTEMBER 2012 Answer all of the following questions in the space provided. Show all work as partial credit may be given. Answers without justification, if they are correct, will earn no ",r.n credit. 1. (5 pts.) Assume th
School: George Mason
Course: Analytic Geometry And Calculus I
MATH 113 - QUIZ 6 - 16 OCTOBER 2012 Answer all of the following questions in the space provided. Show all work as partial credit may be given. Answcrs witliout justification, even if they are corrcct, will car.n ncl credit. 1. (5 pts.) Find /'(r) r J\r):
School: George Mason
Course: Analytic Geometry And Calculus I
UEL MATH 113 - I I QUIZ 3 - 18 SEPTEMBER 2012 Answer all of the following questions in the space provided. Show all work as partial credit may be given. Answers without justification) even if they are correct, will earn no credit. 1'(5pts.)Letg(r):4.Evalu
School: George Mason
Course: Analytic Geometry And Calculus I
IvIATH 113 - QUIZ 5 - 2 OCTOBER 2012 Answer all of the following questions in the space provided. Show all work as partial crcdit uray bc givi,n. Arrsrvers without justificatioll, evell if they are cclrrect, will earn ntr credit. 1. (5pts.) Fincltheerluat
School: George Mason
Course: Analytic Geometry And Calculus I
V e12s (d AJ _j_ MATH 113 - QUrZ 1 - 4 SEPTEMBER 2012 Answer all of the following questions in the space provided. Show all work as partial credit may be given. Attsrvers without justification) even if they are correct, will earn no credit. Consider the p
School: George Mason
Course: Analytic Geometry And Calculus I
MATH 113 - QUIZ 7 - 23 OCTOBER 2012 Answer all of the following questions in the space provided. Show all work as partial credit may be given. Answers without justification, even if they are correct, will earn no credit. 1. (5 pts.) Find /'(r) and f"(r) i
School: George Mason
Course: Analytic Geometry And Calculus I
MATH 113 - QUIZ 8 6 NOVEN4BER 2012 Answer all of the following questions in the space provided. Show all work as partial credit may be given. Answers without justification) even if thcy are correct, will carn no credit. 1. (5 pts.) F ind all of the critic
School: George Mason
Course: Analytic Geometry And Calculus I
MATH 113 QUIZ 11 _4DECEMBER2012 Ans*-er all of the foltowing questions in the space provided. Show all work as partial credit Illay bc giveo. Ansrvers without justification, even il they are correctj will earn no credit. 1. (5 pts.) Approximate the area u
School: George Mason
Course: Analytic Geometry And Calculus I
\IATH Ll3 QUIZ 12 11DECE\IBER 2012 ,q.nswer ! l of ihe lirllorving questions in the sprce provicied. Show .rll work ru prrrtial credit n)ay be given. Answers wilhout juslifiraiion, oven if they al'e corrcct. rvill carll ncr 1. cfw_4 pts. cach) trso thc Rr
School: George Mason
Course: Analytic Geometry And Calculus I
MATH 113 - QUIZ r0 - 2l NOVEMBER 2012 Answer all of the following questions in the space provided. Show all work as partial credit may be given. Answers without justification, even if they are correct, will earn no credit. 1. Consider the function f (n) (
School: George Mason
Course: Analytic Geometry And Calculus I
MATH 113 QUIZ I- 13 NOVEMBER 2012 Answer all of the following questions in the space provided. Show all work as partial credit may be given. Answers without justification) even if they are correct, will earn no credit. *2 1. (5 pts.) For the function f (r
School: George Mason
Course: Introduction To Advanced Mathematics
MATH 290 EXAM 2 SOLUTIONS 1. (5 pts.) Let a1 = 3, a2 = 9, and for n 3, an = 5an1 6an2 . Use the Principle of Complete Induction to prove that for all n N, an = 3n . Solution: We will prove this using the Principle of Complete Induction. If n = 1 or n = 2
School: George Mason
Course: Introduction To Advanced Mathematics
MATH 290 EXAM 1 SOLUTIONS 1. (5 pts. each) (a) Is [Q (P = Q)] = P a tautology, contradiction, or neither? Justify your answer with a truth table or portion of a truth table. (b) Give a useful denial in propositional form of P Q = R. Solution: (a). The exp
School: George Mason
Course: Advanced Calculus I
MATH 315 EXAM 5 SOLUTIONS 1. (10 pts.) Suppose that f C[a, b]. Prove that there is a number xM [a, b] such that f achieves its maximum value at xM , that is, for all x [a, b], f (x) f (xM ). (You may assume without proof that f is bounded on [a, b].) Solu
School: George Mason
Course: Advanced Calculus I
MATH 315 EXAM 1 SOLUTIONS 1. (10 pts.) Prove that for all real numbers a and b, |ab| = |a|b|. (For real numbers x, |x| = x if x 0 and x if x < 0. Hint: You must consider several cases.) Solution: There are three cases to consider. First assume that both a
School: George Mason
Course: Advanced Calculus I
MATH 315 EXAM 3 SOLUTIONS 1. (10 pts.) Prove directly from the denition that a subset of R that satises the HeineBorel Property must be bounded. (A set A R satises the Heine-Borel property if every open cover of A admits a nite subcover.) Solution: Let A
School: George Mason
Course: Advanced Calculus I
MATH 315 EXAM 4 SOLUTIONS 1. (10 pts.) Let f (x) be a function with domain Df and suppose that a is a cluster point of Df . Prove that if for every sequence xn Df \ cfw_a such that xn a, f (xn ) L, then limxa f (x) = L. (limxa f (x) = L, a a cluster point
School: George Mason
Course: Advanced Calculus I
MATH 315 EXAM 2 SOLUTIONS 1. (10 pts.) Prove directly, using the denition of convergence, that if xn and yn both converge then xn yn converges. (Hint: You may assume without proof that a convergent sequence is bounded, but you may not assume any other res
School: George Mason
Course: PRECALCULUS
Math LL4 Lim Exam 3 April 28, 20L4 1. Show all work (an answer with no work shown may receive no point). No electronic devices are allowed. 2. Write your name on both this sheet and PARTS I and II of your answer sheets. Works on Part I and II must be on s
School: George Mason
Course: PRECALCULUS
/a* t /- z. r,q. t l.re l+xv f*3e* 4 J-r6 atu< : 4 _u+t d,.<= *l*o^=te Lt= l4,hr< du=$o dx rL ,- i,'i;/-r) J. =+tg. =+G z d* = u=.h/x'.tt) dn= ft, l" x /- (x'*') rl*At I *'^ = 'q=ttaz '/-u lu=, zPv:dx v=x - ! #, '" : x,/-,k?,) -?k + A rto,i'x + t 7/= x d*
School: George Mason
Course: PRECALCULUS
# *tu* L fwsilsvr / , I r?;n) wdr! /i/lzur'./r d , e'o /LrM, /y ig ) i &.ar?* ,l* /^L\ 5 l" t*;x e.'ex c(* r J ti- ;in-x ) L4r2 x *f x f 'Jii-h')*cfw_u' e dx*cfw_ iX 5, Sl< ,<' af tt = cfw_d-x tcfw_ x rL 7 j sifl tr * .s stttcfw_ * 2. i,; J /'^ r d,<" irc
School: George Mason
Course: PRECALCULUS
U*uti'mT =r9 t,rya* /.' L ,/ty,E,^,*, ,l4alt rlr cfw_q"A ilr- silm cfw_t rF* spr4zs aLiiF +6,f*' 4 F +: f* , Synnp ux3 S,Yl-, rtry t7 a. fr*tnz S^e"vt^Q.-r l-r F bi i c4= 3t J.A /- (tJ L r= -3 .L *?h t+ = 3*r _J^ -cfw_ t : 2. At+or*rro -'hef/p, t'/L f /a,
School: George Mason
Course: PRECALCULUS
/a^t r /, qhf wers vontirv' 7 ( :x-t J dx /xl-rcfw_ -t 6*'nt1-l - (Ex 'sr)(zx+t 'X* = .4* I ) + (3t-,)(zx*r) 3x-t -Y*t= n(zx+t)* /,;-U, +*t = O+ /t 2x+ t 8(S*-,) (-*-t) =8(-*) -'. 8= t -J L -f*t tt o=i E B k-,-.r = fr1|r,)= A'* ;) ?El =. -L f * * 4. J (*t
School: George Mason
Course: PRECALCULUS
Math 114 Final Exam May 8, 2008 1. The curve y = sin x, 0 x /2, has arc length given by the integral: /2 1 + cos2 x dx 0 2. The surface area of the surface generated by revolving the curve y = sin x, 0 x /2 about the x-axis is given by the integral: /2 2
School: George Mason
Course: PRECALCULUS
Math 114 Lim Exam 3 April 28,2OL4 f . Siro* all work (an answer with no work shorrn may receive no point). No electronic devices are allowed. 2. Write your name on both this sheet and PARjIS I and II of your answer sheets. Works on Part I and II must be o
School: George Mason
Course: PRECALCULUS
9r, 4wa,.e-tS /, j J. A lt(-t t?x-t z o+ . tt B -a- t3 -r = A'G*') +o 3 i: 3 fLrt* =L J J*-t -^ T G3 L /^ - f x+f -L *cfw_licie*+ ? ;o h,n-'+ = t=A X=0: 3. = n,+ t? cfw_z l- 3t(- t dx + J , B.(* ) :. fi=2 aA - xznT*'- ? fr ID ?- (-$-,) '" g* 3 /J' - L xt
School: George Mason
Course: Introductory Calculus With Business Applications
Math 108, Solution of Midterm Exam 1 1 Specify the domain of each of the following functions. x2 4x + 3 (a) f (x) = 2 x +x2 Solution. Since division by any nonzero number is possible, the domain of f is the set of all numbers satisfying x2 + x 2 = 0. Sinc
School: George Mason
Course: Introductory Calculus With Business Applications
Lecturer: Sangwook Kim Oce : Science & Tech I, 226D math.gmu.edu/ skim22 Solution of Quiz 1 1 Find all values of x such that f (g(x) = g(f (x), where f (x) = x2 + 2 and g(x) = x 1. Solution. Replace x by g(x) = x 1 in the formula for f (x) to get f (g(x)
School: George Mason
Course: Introductory Calculus With Business Applications
Lecturer: Sangwook Kim Oce : Science & Tech I, 226D math.gmu.edu/ skim22 Solution of Quiz 0 1 Find the interval or intervals consisting of all real numbers x that satisfy the given inequality. (a) |x 1| 3 Solution. Rewrite the given inequality as 3 x 1 3
School: George Mason
Course: Introductory Calculus With Business Applications
Math 108, Solution of Midterm Exam 3 1 Find an equation of the tangent line to the curve x3 + y 3 = 2xy at the point (1, 1). Solution. Dierentiating both sides of the given equation with respect to x, we get d x3 + y 3 dx dy 3x2 + 3y 2 dx dy 3x2 + 3y 2 dx
School: George Mason
Course: Introductory Calculus With Business Applications
Lecturer: Sangwook Kim Oce : Science & Tech I, 226D math.gmu.edu/ skim22 Solution of Quiz 5 1 Let f be a function dened by f (x) = x5 5x4 3x + 2. (a) Find intervals on which the graph of f is concave up and concave down. Solution. The rst derivative of f
School: George Mason
Course: Introductory Calculus With Business Applications
Lecturer: Sangwook Kim Oce : Science & Tech I, 226D math.gmu.edu/ skim22 Solution of Quiz 5 1 The rst derivative of a certain function () is given by () = 3 (2 5)2 . (a) Find intervals on which is increasing and decreasing. 5 The derivative of is continu
School: George Mason
Course: Introductory Calculus With Business Applications
Lecturer: Sangwook Kim Oce : Science & Tech I, 226D math.gmu.edu/ skim22 Solution of Quiz 4 1 Find if 3 + 3 = 2 + . Solution. Dierentiating both sides of the given equation with respect to , we get ] ] [ 2 [ 3 + 3 = + 32 + 3 2 = 2 + 1 2 = 2 32 (3 1) 2
School: George Mason
Course: Introductory Calculus With Business Applications
Math 108, Solution of Midterm Exam 2 1 List all values of x for which the following function f (x) is not continuous. Explain the reason. 2 x + 2x 3 2 if x < 1 x 4x + 3 f (x) = if 1 x < 3 2x 4 2 x 2x 3 if x 3 x2 + 2x 3 is a rational function dened
School: George Mason
Course: Introductory Calculus With Business Applications
Lecturer: Sangwook Kim Oce : Science & Tech I, 226D math.gmu.edu/ skim22 Solution of Quiz 2 1 Decide if the following function is continuous at x = 2. Explain the reason. 2 x x2 2 if x < 2 x 3x + 2 f (x) = 2 x x if x 2 Solution. We need to verify the
School: George Mason
Course: Introductory Calculus With Business Applications
Lecturer: Sangwook Kim Oce : Science & Tech I, 226D math.gmu.edu/ skim22 Solution of Quiz 3 1 Find the rate of change 3 + 1 for the function = 2 when = 1. +1 Solution. By the quotient rule, the derivative of = 3 + 1 with respect to is given by 2 + 1 [ ]
School: George Mason
Course: Introductory Calculus With Business Applications
Lecturer: Sangwook Kim Oce : Science & Tech I, 226D math.gmu.edu/ skim22 Solution of Quiz 6 1 Find all real numbers that satisfy 22x = 83 . Solution. The given equation is satised if and only if 22x = (23 )3 22x = 29 2x=9 x = 7 Thus, 22x = 83 if and only
School: George Mason
Course: Introductory Calculus With Business Applications
Lecturer: Sangwook Kim Oce : Science & Tech I, 226D math.gmu.edu/ skim22 Solution of Quiz 3 1 Find the rate of change dy x3 + 3 for the function y = 2 when x = 1. dx x +1 Solution. By the quotient rule, the derivative of y = x3 + 3 with respect to x is gi
School: George Mason
Course: Partial Differential Equations
Math 678. Fall 2011. Final Exam. Due Wednesday 12/14/11 by 5pm Solutions should represent individual work, with all necessary details. Only facts discussed in class or given in the main textbook can be used without proof (except the facts known from calcu
School: George Mason
Course: Quantitative Reasoning
Ill use ^ for exponents, so x^2 means you enter x2. Number 1: 6 6 Number 2: -117 Number 3: Number 4: Choice B Number 5: 6 Number 6: 2 ^ 5/54 The power is 5/54 Number 7: 3 Number 8: 20 Number 9: -12 + 92 Number 10: 20 Number 11: (1+16a) 7a Number 12: Numbe
School: George Mason
Course: Probability
December 2, 2010 Math 351 - Test 3 1. An urn has two black and one white ball. An experiment consists of taking two balls from the urn. Call it success if both of them are black and call it failure if they are not both black. After you have chosen the two
School: George Mason
Course: Probability
@ lfr =i,;. lftlffiArr'v /' 4 raccsf, : Y /t o-r ./or64",n!\ at ry o/ (ild177'q, 1 e?u' s fxta (/2 \ lr ft cfw_d, @ cfw_*<t, :.7 =J ).= n<B t9- l&) Jrf /.<r t4 r'cfw_ 47rp ;( +</z [gar /r o '/+Vrdf/ t? 6 s /r :'l ru @ 'ru /.2 @t0) = \2- /z<-tf ! r 4 a 6 -
School: George Mason
Course: Probability
October 31, 2010 Math 351 - Test 3 Some comments on the test. 1. Problem 1: (a) Most correctly deduced that p = 1/3. You should view it as an urn problem. You can then calculate p either by using combinations, or by conditional probability (it is the prob
School: George Mason
Course: Probability
October 28, 2010 Math 351 - Test 2 1. Suppose urn I has one red and three blue balls; suppose urn II has one red and ve blue balls. Pick one ball from each urn. You get $2 for each red one and $1 for each blue one. Let X be the total number of dollars you
School: George Mason
Course: Probability
September 28, 2010 Math 351 - Test 1 1. Suppose we have ve identical red books, six identical green books, and three dierent yellow books. (a) In how many dierent ways can they be arranged on a shelf? (b) In how many dierent ways can they be arranged on a
School: George Mason
Course: Introduction To Advanced Math
MATH 290 FINAL EXAM SOLUTIONS 1. (12 pts.) Prove that for all integers a, and b, and all nonzero integers c, ac|bc if and only if a|b. Which of the implications remains true if c = 0? Solution: Let a and b be integers and let c be a nonzero integer. (=) S
School: George Mason
Course: Introduction To Advanced Math
MATH 290 10 JUNE 2010 EXAM 4 Answer all of the following questions on the answer sheets provided. Show all work, as partial credit may be given. This test will be counted out of a total of 60 points. 1. Dene the relation S on R R by (x, y )S (z, w) if and
School: George Mason
Course: Introduction To Advanced Math
MATH 290 EXAM 4 SOLUTIONS 1. Dene the relation S on R R by (x, y )S (z, w) if and only if x z and y w. (a) (8 pts.) Prove that S is a partial order on R R. (b) (4 pts.) Is S a linear order? Why or why not? (c) (4 pts.) Find an upper bound for the rectangl
School: George Mason
Course: Introduction To Advanced Math
MATH 290 3 JUNE 2010 EXAM 3 Answer all of the following questions on the answer sheets provided. Show all work, as partial credit may be given. This test will be counted out of a total of 60 points. n 1. (16 pts.) Prove by induction that for all natural n
School: George Mason
Course: Introduction To Advanced Math
MATH 290 EXAM 3 SOLUTIONS n 1. (16 pts.) Prove by induction that for all natural numbers n, Solution: 1 (k k !) = (n + 1)! 1. k=1 Assume that n = 1. Then for n = 1. (k k !) = 1 1! = 1 = 2! 1. Hence the result holds k=1 n Let n be any natural number and as
School: George Mason
Course: Introduction To Advanced Math
MATH 290 27 MAY 2010 EXAM 2 Answer all of the following questions on the answer sheets provided. Show all work, as partial credit may be given. This exam will be counted out of a total of 60 points. 1. (10 pts.) Prove that for any integers a, b, and c, if
School: George Mason
Course: Introduction To Advanced Math
MATH 290 EXAM 2 SOLUTIONS 1. (10 pts.) Prove that for any integers a, b, and c, if a|b and a|c then a|(b c). Solution: Let a, b, and c be integers. Assume that a|b and a|c. We must show that a|(b c). Since a|b there is an integer n such that b = na, and s
School: George Mason
Course: Introduction To Advanced Math
MATH 290 20 MAY 2010 EXAM 1 Answer all of the following questions on the answer sheets provided. Show all work, as partial credit may be given. This test will be counted out of a total of 60 points. 1. (8 pts.) Prove that P = (Q R) is equivalent to (P Q)
School: George Mason
Math 677. Fall 2009. Homework #7 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.13: # 1 The system x = y + ax2 + bxy + cy 2 y = dx2 + exy + f y 2 Let LJ [h(x)] = Jh(x) Dh(x)J(x) an
School: George Mason
Math 677. Fall 2009. Homework #5 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.9: # 3 For the system x2 x1 x2 + x2 x3 1 3 2 x1 + x2 x3 x= 2 3 x1 x3 x3 x2 x2 x2 x5 1 3 3 the Liapu
School: George Mason
Math 677. Fall 2009. Homework #4 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.5: # 5 Determine the ow of the nonlinear system x = f (x), f (x) = (x1 , 2x2 +x2 )T 1 1 2 and show t
School: George Mason
Math 677. Fall 2009. Homework #6 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2.11: # 2 (a) x = x2 , y = y. Expansion of the function = p2 (x, ) is a nbhd of the origin has the for
School: George Mason
Course: Ordinary Differential Equations
Math 677. Fall 2009. Homework #2 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 6: # 4 The system x = Ax with 1 1 0 0 1 1 0 0 A= 0 0 0 2 0 0 1 2 has eigenvalues 1,2 = a1 ib1 = 1
School: George Mason
Course: Ordinary Differential Equations
Math 677. Fall 2009. Homework #1 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 2: # 3 Write the following linear DE with const coecients in the form of the linear system x = Ax and
School: George Mason
Course: Ordinary Differential Equations
Math 677. Fall 2009. Homework #3 Solutions. Part I. Exercises are taken from Diential Equations and Dynamical Systems by Perko, 3rd edition. Problem Set 1: # 4 IVP x = x3 , x(0) = 2 has solution in the form x(t) = 1 all t (, 8 ) and lim x(t) = . 2 , 18t
School: George Mason
Course: Numerical Analysis
Math 685/CSI 700/OR 682 Homework 3 given 02/15/2010 Problem 1. Suppose that both sides of an arbitrary system of linear equations Ax = b is premultiplied by a nonsingular diagonal matrix. Does this change the true solution x? Does this aect the conditioni
School: George Mason
Course: Numerical Analysis
Math 685/CSI 700/OR 682 Homework 2 given 02/01/10 Problem 1. It is good to know your limits. For your computer system nd: (a) the largest possible oating-point number; (b) the largest integer I s.t. (I + 1) 1 equals I; (c) the smallest possible positive o
School: George Mason
Course: Numerical Analysis
Math 685/CSI 700/OR 682 Homework 1 given 01/25/10 The following are some practice problems I would like you to do. Solutions will be discussed in class on 02/01/10. Problem 1. What do the following pieces of Octave/Matlab code accomplish? (a) x = (0 : 40)
School: George Mason
Course: Introduction To Advanced Mathematics
MATH 290 WRITING ASSIGNMENT 1 SOLUTIONS 1. Prove that for all real numbers a and b, |a| b if and only if b a b. Solution: Let a and b be real numbers. (=). Assume that |a| b. We will consider two cases. Assume rst that a 0. In this case, |a| = a. Since |a
School: George Mason
Course: Introduction To Advanced Mathematics
MATH 290 WRITING ASSIGNMENT 4 SOLUTIONS 1. Prove that for any sets A and B, (A B) (B A) = (A B) (A B). Solution: Let A and B be sets. Assume that (x, y) (A B) (B A). We must show that (x, y) (A B) (A B). Since (x, y) (A B) (B A), (x, y) (A B) and (x, y) (
School: George Mason
Course: Introduction To Advanced Mathematics
MATH 290 WRITING ASSIGNMENT 2 SOLUTIONS 1 1 1. Prove that for every real number x > 0, x + 2 and that x + = 2 if and only if x x x = 1. Solution: 1 Let x > 0. We must show that x + 2. Since the square of any real number x is nonnegative, (x 1)2 0. Multipl
School: George Mason
Course: Introduction To Advanced Mathematics
MATH 290 WRITING ASSIGNMENT 3 SOLUTIONS 1. Prove that for any sets A and B, (A B) (B A) = (A B) (A B). Solution: Let A and B be sets. Assume rst that x (A B) (B A). We will show that x (A B) (A B). Since x (A B) (B A), x A B or x B A. This means that ther
School: George Mason
Course: Introduction To Advanced Mathematics
MATH 290 WRITING ASSIGNMENT 5 SOLUTIONS 1. Use a form of induction to prove that for all natural numbers n 2, either n is prime or n has a prime factor p with p n. Solution: The proof will use the Principle of Complete Induction. Assume that n = 2. Since
School: George Mason
Course: Introduction To Advanced Mathematics
MATH 290 WRITING ASSIGNMENT 9 SOLUTIONS 1. Prove that if xn L and yn M , then xn yn LM . Solution: There is more than one way to prove this. The rst proof will use the fact that a convergent sequence is bounded. Specically we state: Lemma 1. If yn M then
School: George Mason
Course: Introduction To Advanced Mathematics
MATH 290 WRITING ASSIGNMENT 7 SOLUTIONS 1. Problem 19(a), (b), and (d), p. 173. Solution: Let A be a set with partial order R and for each a A, dene the set Sa = cfw_x A: xRa. (a). Let a, b A and assume that aRb. We must show that Sa Sb . Let x Sa . This
School: George Mason
Course: Introduction To Advanced Mathematics
MATH 290 WRITING ASSIGNMENT 8 SOLUTIONS 1. Problem 5 p. 171. Solution: In order to show that the relation R is a partial order, we must show that it is reexive, antisymmetric and transitive. Let x N. Then x = 20 x so that xRx since 0 Z and 0 0. Hence R is
School: George Mason
Course: Introduction To Advanced Mathematics
MATH 290 WRITING ASSIGNMENT 6 SOLUTIONS 1. Let a and b be natural numbers, and let d be the smallest natural number such that there exist integers x and y such that ax + by = d. Prove that d = GCD(a, b). (Hint: Letting c = GCD(a, b), rst prove that c d by
School: George Mason
Course: Advanced Calculus I
MATH 315 HOMEWORK 1 SOLUTIONS Exercise 1.3. Solution: By Axiom 5, 0 + 0 = 0. Therefore, by Axiom 4 (Distributivity), a0 = a(0 + 0) = a0+a0. Since a0 R, by Axiom 6 there is a number a0 such that a0+a0 = 0. Therefore, by Axiom 3, 0 = a0 + a0 = a0 + (a0 + a0
School: George Mason
Course: Advanced Calculus I
MATH 315 HOMEWORK 3 SOLUTIONS Exercise 1.51. Solution: Note rst that it follows from Theorem 1.4.1 (iii) and the fact that a constant sequence is convergent that if zn converges then c zn converges for any constant c. Also, by Theorem 1.4.1 (i) and (ii),
School: George Mason
Course: Advanced Calculus I
MATH 315 HOMEWORK 6 SOLUTIONS Exercise 4.6. Solution: (a) For x (0, 1], the function is clearly continuous since it is the product of a continuous function with the composition of two continuous functions. To see that it is continuous at x = 0, it is suci
School: George Mason
Course: Advanced Calculus I
MATH 315 HOMEWORK 6 SOLUTIONS Exercise 3.3. Solution: We will use the Darboux criterion to show that this function is Riemann integrable. Let > 0. We must nd a partition of [a, b] with the property that U (1(p) , P ) L(1(p) , P ) < . Choose for example th
School: George Mason
Course: Advanced Calculus I
MATH 315 HOMEWORK 4 SOLUTIONS Exercise 1.83. Solution: Let On = (1/n, 2) for all n N. Then I claim that O = cfw_On : n N is an open cover of E. To see this, let x E. Then x = 1/m for some m N. Then since 1/(m + 1) < 1/m, 1/m (1/(m + 1), 2) = Om+1 . Hence
School: George Mason
Course: Advanced Calculus I
MATH 315 HOMEWORK 2 SOLUTIONS Exercise 1.19. Solution: Let > 0. Then there there is an N1 N such that if n N1 then |sn L| < and an N2 N such that if n N2 then |un L| < . Letting N = max(N1 , N2 ), this means that if n N then L < sn tn un < L + . But this
School: George Mason
Course: Advanced Calculus I
MATH 315 HOMEWORK 5 SOLUTIONS Exercise 2.37. Solution: Consider the function g(x) = x f (x). Since f is continuous on [0, 1], then so is g. Since 0 f (x) 1, g(0) = 0 f (0) = f (0) 0, and g(1) = 1 f (1) 0. If either g(0) = 0 or g(1) = 0 then we are done si
School: George Mason
Course: Partial Differential Equations
Math 678. Homework 3 Solutions. #1 We need to derive the formula for the solution of the IVP ut u + cu = f. in Rn (0, ) u = g, on Rn cfw_t = 0 Some observations: (1) the solution to the non-homogeneous problem can be obtained from the solution to the homo
School: George Mason
Course: Partial Differential Equations
Math 678. Fall 2011. Homework #2. Due Wednesday 09/28/11 in class. Solutions should represent individual work, with all necessary details. Only facts discussed in class or given in the main textbook can be used without proof (except the facts known from c
School: George Mason
Course: Partial Differential Equations
Math 678. Homework 2 Solutions. #3, p.85 Following the proof of the mean value property, denote (r) = u(y )dS (y ) B (0,r ) u(y )dS = f (y )dS . and use the same argument to show that (r) = B (0,r ) r Then x some > 0 and notice that (r) ( ) = as 0. Sinc
School: George Mason
Course: Partial Differential Equations
Math 678. Homework 1 Solutions. #2, p.12 Think about circles denoting the partial derivative, with m consecutive circles representing an m-th order derivative in the corresponding variable. Dividers are placed between variables, and can be separated by m
School: George Mason
Course: Partial Differential Equations
Math 678. Fall 2011. Homework #1. Due Wednesday 09/14/11 in class. Solutions should represent individual work, with all necessary details. Only facts discussed in class or given in the main textbook can be used without proof. Only selected problems will b
School: George Mason
Course: Partial Differential Equations
Math 678. Fall 2011. Homework #3. Due Wednesday 10/12/11 in class. Solutions should represent individual work, with all necessary details. Only facts discussed in class or given in the main textbook can be used without proof (except the facts known from c
School: George Mason
Course: Partial Differential Equations
Math 678. Homework 4 Solutions. #1 (a) In order to get the general solution of uxy = 0, use integration in one of the variables, e.g. in y rst. This gives ux = f (x) and u(x, y ) = f (x)dx + G(y ). Denote F (x) = f (x)dx to get the conclusion: general sol
School: George Mason
Course: Partial Differential Equations
Math 678. Fall 2011. Homework #6. Due Wednesday 11/30/11 in class. Solutions should represent individual work, with all necessary details. Only facts discussed in class or given in the main textbook can be used without proof (except the facts known from c
School: George Mason
Course: Partial Differential Equations
Math 678. Homework 6 Solutions. #1 x1 ux1 + 2x2 ux2 + ux3 = 3u, u(x1 , x2 , 0) = g (x1 , x2 ) First we identify F (p, z, x) = x1 p1 + 2x2 p2 + p3 3z = 0. This gives Fp = (x1 , 2x2 , 1), Fx = (p1 , 2p2 , 0), Fz = 3. The CE system looks like x(s) = Fp = z (
School: George Mason
Course: Partial Differential Equations
Math 678. Fall 2011. Homework #5. Due Wednesday 11/16/11 in class. Solutions should represent individual work, with all necessary details. Only facts discussed in class or given in the main textbook can be used without proof (except the facts known from c
School: George Mason
Course: Partial Differential Equations
Math 678. Fall 2011. Homework #4. Due Wednesday 11/02/11 in class. Solutions should represent individual work, with all necessary details. Only facts discussed in class or given in the main textbook can be used without proof (except the facts known from c
School: George Mason
Course: Partial Differential Equations
Math 678. Homework 5 Solutions. #1 Consider a subsolution of the heat equation vt v 0 in UT . (a) The proof follows the argument given in Theorem 3, p.53-54, with the exception being that (r) 0, from where it follows that v (x, t) 1 4rn v (y, s) E (x,t;r
School: George Mason
School: George Mason
Course: Analytic Geom Calculus III
Math 213-002 Test 1 September 0,2011 2 Name: Show a ll w ork. ( No w ork, n o p oints.) T his t est i s c losed b ook a nd c losed n otes, n o calculators o r c ell p hones a re a llowed. U se o nly t he m ethods f rom c lass. G ood l uck! (1) ( 10 p oint
School: George Mason
MATH 621, Algebra I Assignment sheet 11 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): For a ring R, the least n N such that n = 1 + + 1 (n times) = 0 is called the characteristic of R a
School: George Mason
MATH 621, Algebra I Assignment sheet 10 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): Let R be an integral domain. If a R is a unit and b R, show there is a unique automorphism of R[X ]
School: George Mason
MATH 621, Algebra I Assignment sheet 9 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): Let R = cfw_m + n 2 : m, n Z show that R is a subring of R. Show further that R (Z2 , +, ) where Z2
School: George Mason
MATH 621, Algebra I Assignment sheet 8 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): Let G and H be nite groups of relatively prime orders. Show that Aut(G H ) Aut(G) Aut(H ). = Here Au
School: George Mason
MATH 621, Algebra I Assignment sheet 7 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): Let G be a group with |G| = pn m where p is a prime, gcd(p, m) = 1 and p > m 1. Show that G has a un
School: George Mason
MATH 621, Algebra I Assignment sheet 6 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): Let G be an abelian group with |G| = pn for some n N where p is a prime number. Show that G has a co
School: George Mason
MATH 621, Algebra I Assignment sheet 5 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): Show that a group G on four elements, |G| = 4 is abelian. Let N4 be the subset of the symmetric grou
School: George Mason
MATH 621, Algebra I Assignment sheet 4 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): A group G is called solvable if there is a normal chain, or sequence G = G0 G1 Gn = cfw_e, with each
School: George Mason
MATH 621, Algebra I Assignment sheet 3 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): Show that a subgroup of index two of a group is a normal subgroup. (2): Show that the automorphisms
School: George Mason
MATH 621, Algebra I Assignment sheet 2 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): Let G be a group and S G a subset. Show that C (S ) = cfw_g G : gs = sg for every s S is a subgroup
School: George Mason
MATH 621, Algebra I Assignment sheet 1 Spring 2012 - George Mason University. Professor: Geir Agnarsson (geir@math.gmu.edu) Homework to be handed in: (1): Let f : A B and g : B C be functions. Prove the following implications: f and g injective g f injec
School: George Mason
Course: Probability
~/ Math 3 51 - Homework 8 Due i n c lass o n T hursday. October 5, 2010 1. (This comes partially from problem 18 i n the text.) A t otal of 46% of the people eligible to vote in a certain city classify themselves as Independents, 30% as Liberals an
School: George Mason
Course: Probability
October 1, 2010 M ath 3 51 - Homework 7 1. Suppose we have t hree c ards. T he first is red on b oth sides, t he second is b lue o n b oth sides. a nd t he t hird is r ed Oil olle side a nd blue 011 t he o ther. Label t he sides of t he first c ard a s
School: George Mason
Course: Probability
il~? M ath 3 51 - Homework 6 D ue i n c lass o n T hursday. September 21, 2010 1. If y ou select eight cards from a s tandard deck of 52 cards, w hat is t he p robability you g et two of one denomination, two of a nother d enomination, t hree o f a t hi
School: George Mason
Course: Probability
S eptember 16, 2010 M ath 3 51 - Homework 5 Due i n c lass o n T uesday. L Say we have 20 c ards: 5 are blue (color 1) a nd n umbered 1 t o 5, 5 a re green (color 2) a nd l lumbered 1 t o 5, 5 a re red (color 3) a nd n umbered 1 t o 5, 5 are yellow (
School: George Mason
Course: Probability
September 14, 2010 M ath 3 51 - Homework 4 Due i n c lass o n T hursday. 1. Three sets A, B , C which lie in a sample space S satisfy t he following: P (A) .3 P (B) = .33 P (C) .17 P (AB) .08 P (AC) = .07 P (BC) = .1 P (A C BCC C ) .4 ~t : PCA~C.c/:.
School: George Mason
Course: Probability
S eptember 9, 2010 M ath 3 51 - Homework 3 Due i n c lass o n T hursday. In e ach o f t hese p roblems s how e xactly w hat i t i s y ou a re c alculating a nd t hen c omplete t he c alculation u sing a c alculator. I t's o k t o a nswer e ach q uesti
School: George Mason
Course: Probability
S eptember 7, 2010 M ath 3 51 - Homework 2 D ue i n c lass o n T hursday. I n e ach o f t hese p robJems s how e xactly w hat i t i s y ou a re c alcuJating a nd t hen c omplete t he c alculation u sing a c alculator. I t's o k t o a nswer e ach q uestion
School: George Mason
Course: Probability
S eptember 3, 2010 M ath 3 51 - Homework 1 Due i n c lass n ext T uesday. In e ach o f t hese p roblems s how e xactly w hat i t i s y ou a re c alculating a nd t hen c omplete t he c alculation u sing a c alculator, f or e xample ( ~ )( ~) 4 0. I t's
School: George Mason
Course: Introduction To Advanced Math
MATH 290 WRITING ASSIGNMENT 3 SOLUTIONS 1. Dene the Fibonacci sequence cfw_f1 , f2 , f3 , . . . by f1 = f2 = 1 and for n > 2, fn = n n fn1 + fn2 . Use induction to prove that for all natural numbers n, fn = , where 1+ 5 1 5 = and = . (Hint: and are the so
School: George Mason
Course: Quantitative Reasoning
Syllabus,Spring2014 Math106006,QuantitativeReasoning Instructor: Dr.LynnellMatthews Office:ExploratoryHallRoom4407 OfficeHours: MW4:30PMTO5:30PM Phone: 7039931981 Email: lmatthe4@gmu.edu Text: MathematicalIdeas,byMiller,HereenandHornsby,CustomEdition,Pear
School: George Mason
Course: Numerical Analysis
GMU Department of Mathematical Sciences Math 685: Numerical Analysis Spring 2010 Syllabus Instructor: Prof. Maria Emelianenko Email: memelian@gmu.edu Phone: (703) 993-9688 Office: Room 226A, Science and Tech I Office Hours: Mon 1-3pm and by appt Time and
School: George Mason
MATH 1224; Summer II 2011; CRN 70873 Vector Geometry - Course Contract M-R 11:00AM-12:15PM, MCB 321 Instructor: Idir Mechai Room: MCB 321 Office: 461-Q McBryde Hall E-mail Address: mechaii@vt.edu Office Hours: M/W: 12:15-02:00 PM T / R: 02:00-03:00 PM Oth
School: George Mason
Course: Discrete Mathematics I
Syllabus: Math 125 001 Discrete Mathematics I 14321, Spring 2007 Date range Time & place Instructor Email Phone Office hours Course description Jan 22 - May 16, 2007 Lecture: MWF 10:30 am 11:20 am, Robinson Hall B122 Alexei V. Samsonovich asamsono
School: George Mason
Course: Intro Calcbusiness Applicatio
MATH 108 - Introductory Calculus with Business Applications Spring 2008 Coure Time and Place: Tuesday and Thursday 12:00-1:15 pm, 376 Bull Run Hall, Prince William Campus Instructor: Saleet Jafri Email: sjafri@gmu.edu Office: 328G Occoquan Building,