Online study resources available anywhere, at any time
High-quality Study Documents, expert Tutors and Flashcards
Everything you need to learn more effectively and succeed
We are not endorsed by this school |
- Course Hero has verified this tag with the official school catalog
We are sorry, there are no listings for the current search parameters.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
We don't have any study resources available yet.
School: George Mason
Course: Number Theory
MATH 301 Homework 2 1. 3.2 no.3 Show that there are no prime triplets, that is, primes p, p+2, and p + 4, other than 3, 5, and 7. 2. 3.2 no. 6 Find the smallest prime between n and 2n for these values of n. (a) 3 (b) 5 (c) 19 (d) 31 3. 3.2 no. 8 Find the
School: George Mason
Course: Elementary Differential Equations
MATH 214 EXAM 1 VERSION 1 SOLUTIONS 1. (5 pts. each) Consider the initial value problem y = y (y 2)2 , y (0) = y0 . (a) Find all equilibrium solutions to this equation. Do not attempt to solve the IVP. (b) Describe the long term behavior (that is, the beh
School: George Mason
Course: Math For Elem School I
First, draw a triangle on a piece of paper. Then cut out the triangle. Label the angles of your triangle. Fold the corner, that is on the opposite side of the longest side (hypotenuse), to meet the longest side. Lastly, fold the other two corners to meet
School: George Mason
Course: Math For Elem School I
After you get the measurements asked for, convert them to units based on inches. 1. 2. a. b. i. 3. a. b. 4. a. b. i. 5. a. b. 6. a. Area: The area of the top half circle is : ( 3.1416) A = ( r2 ) (3.1416x7x7) 76.97 cm2 The area of the rectangle (2cm by 10
School: George Mason
Course: Quantitative Reasoning
Math 106 add to, unknown result Terry has 20 grapes. He got 15 more from Abby. How many grapes does he have now? add to, unknown change Terry had 20 grapes. After getting some from Abby, he has 35 grapes. How many grapes did he get from Abbby (draw 35, co
School: George Mason
Course: Quantitative Reasoning
Math 106 Final exam Set Collection of objects Cardinality The # of elements in a set. Repeated # do not count. Equality of Set Every element in A is in B. Infinite Not whole # Finite Whole # Subset Set A is a subset of B if every element of A is also an e
School: George Mason
Course: Elementary Differential Equations
MATH 214 EXAM 1 VERSION 1 SOLUTIONS 1. (5 pts. each) Consider the initial value problem y = y (y 2)2 , y (0) = y0 . (a) Find all equilibrium solutions to this equation. Do not attempt to solve the IVP. (b) Describe the long term behavior (that is, the beh
School: George Mason
Course: Discrete Mathematics I
9.11. E wmhme Relaw DaM-(w Ah Wax/«lame make on A 5:)" A is 4. MY? «Mm 6Q. On A W 85 (Molve) mmmocc MA HMRVC. E1A221¢AJ>M A (e. 41¢ 50 04 A11 W14 in. 48 word Ml 3R =f(&,4)'AXAl 5 am) 5 L4v¢ «Kc same pawsa. 7.) Ld A (4 41¢. 50(- 94 «11 rovioews o4 -K¢ 111;
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
Course: Number Theory
MATH 301 Homework 2 1. 3.2 no.3 Show that there are no prime triplets, that is, primes p, p+2, and p + 4, other than 3, 5, and 7. 2. 3.2 no. 6 Find the smallest prime between n and 2n for these values of n. (a) 3 (b) 5 (c) 19 (d) 31 3. 3.2 no. 8 Find the
School: George Mason
Course: Math For Elem School I
First, draw a triangle on a piece of paper. Then cut out the triangle. Label the angles of your triangle. Fold the corner, that is on the opposite side of the longest side (hypotenuse), to meet the longest side. Lastly, fold the other two corners to meet
School: George Mason
Course: Math For Elem School I
After you get the measurements asked for, convert them to units based on inches. 1. 2. a. b. i. 3. a. b. 4. a. b. i. 5. a. b. 6. a. Area: The area of the top half circle is : ( 3.1416) A = ( r2 ) (3.1416x7x7) 76.97 cm2 The area of the rectangle (2cm by 10
School: George Mason
Course: Quantitative Reasoning
Math 106 add to, unknown result Terry has 20 grapes. He got 15 more from Abby. How many grapes does he have now? add to, unknown change Terry had 20 grapes. After getting some from Abby, he has 35 grapes. How many grapes did he get from Abbby (draw 35, co
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: Number Theory
MATH 301 Homework 2 1. 3.2 no.3 Show that there are no prime triplets, that is, primes p, p+2, and p + 4, other than 3, 5, and 7. 2. 3.2 no. 6 Find the smallest prime between n and 2n for these values of n. (a) 3 (b) 5 (c) 19 (d) 31 3. 3.2 no. 8 Find the
School: George Mason
Course: Elementary Differential Equations
MATH 214 EXAM 1 VERSION 1 SOLUTIONS 1. (5 pts. each) Consider the initial value problem y = y (y 2)2 , y (0) = y0 . (a) Find all equilibrium solutions to this equation. Do not attempt to solve the IVP. (b) Describe the long term behavior (that is, the beh
School: George Mason
Course: Math For Elem School I
First, draw a triangle on a piece of paper. Then cut out the triangle. Label the angles of your triangle. Fold the corner, that is on the opposite side of the longest side (hypotenuse), to meet the longest side. Lastly, fold the other two corners to meet
School: George Mason
Course: Math For Elem School I
After you get the measurements asked for, convert them to units based on inches. 1. 2. a. b. i. 3. a. b. 4. a. b. i. 5. a. b. 6. a. Area: The area of the top half circle is : ( 3.1416) A = ( r2 ) (3.1416x7x7) 76.97 cm2 The area of the rectangle (2cm by 10
School: George Mason
Course: Quantitative Reasoning
Math 106 add to, unknown result Terry has 20 grapes. He got 15 more from Abby. How many grapes does he have now? add to, unknown change Terry had 20 grapes. After getting some from Abby, he has 35 grapes. How many grapes did he get from Abbby (draw 35, co
School: George Mason
Course: Quantitative Reasoning
Math 106 Final exam Set Collection of objects Cardinality The # of elements in a set. Repeated # do not count. Equality of Set Every element in A is in B. Infinite Not whole # Finite Whole # Subset Set A is a subset of B if every element of A is also an e
School: George Mason
Course: Quantitative Reasoning
Critical thinking ASSUME/DOES NOT ASSURE that one will reach the truth. does not assure Critical thinking MAY/MAY NOT be distinguished from feelings. may What are the two important mathematical properties of a compass? . complementary angle Two angles who
School: George Mason
Course: Quantitative Reasoning
Polya's Problem 1. Understand 2.Devise a plan 3.Work the plan 4. CHECK Devise a plan means to. draw a picture, guess and check, equation, table/chart, etc Arithmetic Sequence(def) a fixed difference is added or subtracted to the previous term to get the n
School: George Mason
Course: Discrete Mathematics I
9.11. E wmhme Relaw DaM-(w Ah Wax/«lame make on A 5:)" A is 4. MY? «Mm 6Q. On A W 85 (Molve) mmmocc MA HMRVC. E1A221¢AJ>M A (e. 41¢ 50 04 A11 W14 in. 48 word Ml 3R =f(&,4)'AXAl 5 am) 5 L4v¢ «Kc same pawsa. 7.) Ld A (4 41¢. 50(- 94 «11 rovioews o4 -K¢ 111;
School: George Mason
Course: Discrete Mathematics I
CD 63. Po eon-vhf: Prom; 14 Pawééh, n. 0111945 a. VJ? Cato m [was . in. M4( r\¥>m+u&u of (9+ 9n¢ 4x caved» "mm W 2: 9 Q <2? <2 1: E] Ws] Wag/4m: Gévem {hm quQ a. 51w: W¢ Xmas Mn (41% 7., Prove at +wo m Man 5. .94 Wk 0K. l6 Iv/tg} «3. $4; Prolazeu; SLa
School: George Mason
Course: Discrete Mathematics I
NOMM. V. =Y\(n~\3(hZ)'.3'Z-\, 0'. = V(n,0)=l I P(v\,r\=h.(n-\)-.-(nr+l> ) OéVéh. End» Pam.) = e. g =39, PR3} =¥.6.S= 2m, Deé'znx'km. A pormme 04 A at}! 04 8:, m ¢vrmzambd a! an in, a and «In some 9rd. Envy dc, ac!) {angé'oal'cué 041 are 4/1 pam- h-Bm 04
School: George Mason
Course: Discrete Mathematics I
em. m+kawsukca kaukm Tke Who;th 01H¢m~kml Inauokm Gris/ea a. s4¢mev SP mucumima 4m E, upped-e 44 1. ? 65 +ru¢ {0" some. puke»er no) 2. I4 9) as +me 4- some. par-(1'.wa L991 KznhHw, H- as +rue «(or "sz new", what/r kid. T1 {P a; {-rue 4191- we! JJda/O'b na
School: George Mason
Course: Number Theory
MATH 301 Homework 11 1. 9.1 no.2 Determine the following orders a) ord11 3 b) ord17 2 c) ord21 10 d) ord25 9 2. 9.1 no.8 Show that the integer 20 has no primitive roots. 3. 9.1 no.10 How many primitive roots does 13 have? Find a set of this many incongrue
School: George Mason
Course: Number Theory
MATH 301 Homework 10 1. 8.3 no.4 What is the plaintext message that corresponds tothe ciphertext 1213 0902 0539 1208 1234 1103 1374 that is produced using modular exponentiation with modulus p = 2591 and encryption key e = 13? 2. 8.3 no.6 With modulus p =
School: George Mason
Course: Number Theory
# 7 ersqrfra. )-sl-46oo) = Qb')S(s')-_= (u z;(as:r) l/onurr.o.l. ,",., -, i_rz] c.) -Qefg-=D = Xn.B,f) -yo , QAse)=Cr"*; = to[-tzt(a-) *tlp_ = 4:5_!,ao.6:_8zq ! Yq, - P) 0 6ao\.-, 3. -z = at-27= is-6 -tzt .j's- 4-rr-t3 O( it.3-15?) : (r'-iTfrt.-)rt:')(a)
School: George Mason
Course: Number Theory
MATH 301 Homework 9 1. 8.1 no.6 Decrypt the message RTOLK TOIK, which was encrypted using the ane transformation C 3P + 24(mod 26). 2. 8.1 no.8 The message KYVMR CLVFW KYVBV PZJJV MVEKV VE was encrypted using a shift transformation C P + k(mod 26). Use fr
School: George Mason
Course: Number Theory
MATH 301 Homework 6 1. 7.3 no.1 Find the six smallest even perfect numbers. (Please justify that the sixth even perfect number comes from a Mersenne prime.) 2. 7.3 no.4 Find a factor of each of the following integers: a) 2111 1 b) 2289 1 c) 246189 1 3. 7.
School: George Mason
Course: Number Theory
MATH 301 Homework 7 1. 7.1 no.2 Find the value of the Euler -function for the following integers. a) 100 b) 256 c) 1001 d) 2 3 5 7 11 13 e) 10! f) 20! 2. 7.1 no.4 Find all positive integers n such that (n) has each of the following values. Be sure to prov
School: George Mason
Course: Number Theory
MATH 301 Homework 5, due TUESDAY, Oct. 15 1. 4.3 no. 4: Find all the solutions of each of the following systems of linear congruences. (a) x 4 (mod 11), x 3 (mod 17) (b) x 1 (mod 2), x 2 (mod 3), x 3 (mod 5) (c) x 0 (mod 2), x 0 (mod 3), x 1 (mod 5), x 6
School: George Mason
Course: Elementary Differential Equations
MATH 214 EXAM 1 VERSION 1 SOLUTIONS 1. (5 pts. each) Consider the initial value problem y = y (y 2)2 , y (0) = y0 . (a) Find all equilibrium solutions to this equation. Do not attempt to solve the IVP. (b) Describe the long term behavior (that is, the beh
School: George Mason
Course: Discrete Mathematics I
9.11. E wmhme Relaw DaM-(w Ah Wax/«lame make on A 5:)" A is 4. MY? «Mm 6Q. On A W 85 (Molve) mmmocc MA HMRVC. E1A221¢AJ>M A (e. 41¢ 50 04 A11 W14 in. 48 word Ml 3R =f(&,4)'AXAl 5 am) 5 L4v¢ «Kc same pawsa. 7.) Ld A (4 41¢. 50(- 94 «11 rovioews o4 -K¢ 111;
School: George Mason
Course: Discrete Mathematics I
CD 63. Po eon-vhf: Prom; 14 Pawééh, n. 0111945 a. VJ? Cato m [was . in. M4( r\¥>m+u&u of (9+ 9n¢ 4x caved» "mm W 2: 9 Q <2? <2 1: E] Ws] Wag/4m: Gévem {hm quQ a. 51w: W¢ Xmas Mn (41% 7., Prove at +wo m Man 5. .94 Wk 0K. l6 Iv/tg} «3. $4; Prolazeu; SLa
School: George Mason
Course: Discrete Mathematics I
NOMM. V. =Y\(n~\3(hZ)'.3'Z-\, 0'. = V(n,0)=l I P(v\,r\=h.(n-\)-.-(nr+l> ) OéVéh. End» Pam.) = e. g =39, PR3} =¥.6.S= 2m, Deé'znx'km. A pormme 04 A at}! 04 8:, m ¢vrmzambd a! an in, a and «In some 9rd. Envy dc, ac!) {angé'oal'cué 041 are 4/1 pam- h-Bm 04
School: George Mason
Course: Discrete Mathematics I
em. m+kawsukca kaukm Tke Who;th 01H¢m~kml Inauokm Gris/ea a. s4¢mev SP mucumima 4m E, upped-e 44 1. ? 65 +ru¢ {0" some. puke»er no) 2. I4 9) as +me 4- some. par-(1'.wa L991 KznhHw, H- as +rue «(or "sz new", what/r kid. T1 {P a; {-rue 4191- we! JJda/O'b na
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: 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: 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: Number Theory
MATH 301 Homework 2 1. 3.2 no.3 Show that there are no prime triplets, that is, primes p, p+2, and p + 4, other than 3, 5, and 7. 2. 3.2 no. 6 Find the smallest prime between n and 2n for these values of n. (a) 3 (b) 5 (c) 19 (d) 31 3. 3.2 no. 8 Find the
School: George Mason
Course: Math For Elem School I
First, draw a triangle on a piece of paper. Then cut out the triangle. Label the angles of your triangle. Fold the corner, that is on the opposite side of the longest side (hypotenuse), to meet the longest side. Lastly, fold the other two corners to meet
School: George Mason
Course: Math For Elem School I
After you get the measurements asked for, convert them to units based on inches. 1. 2. a. b. i. 3. a. b. 4. a. b. i. 5. a. b. 6. a. Area: The area of the top half circle is : ( 3.1416) A = ( r2 ) (3.1416x7x7) 76.97 cm2 The area of the rectangle (2cm by 10
School: George Mason
Course: Quantitative Reasoning
Math 106 add to, unknown result Terry has 20 grapes. He got 15 more from Abby. How many grapes does he have now? add to, unknown change Terry had 20 grapes. After getting some from Abby, he has 35 grapes. How many grapes did he get from Abbby (draw 35, co
School: George Mason
Course: Number Theory
MATH 301 Homework 11 1. 9.1 no.2 Determine the following orders a) ord11 3 b) ord17 2 c) ord21 10 d) ord25 9 2. 9.1 no.8 Show that the integer 20 has no primitive roots. 3. 9.1 no.10 How many primitive roots does 13 have? Find a set of this many incongrue
School: George Mason
Course: Number Theory
MATH 301 Homework 10 1. 8.3 no.4 What is the plaintext message that corresponds tothe ciphertext 1213 0902 0539 1208 1234 1103 1374 that is produced using modular exponentiation with modulus p = 2591 and encryption key e = 13? 2. 8.3 no.6 With modulus p =
School: George Mason
Course: Number Theory
# 7 ersqrfra. )-sl-46oo) = Qb')S(s')-_= (u z;(as:r) l/onurr.o.l. ,",., -, i_rz] c.) -Qefg-=D = Xn.B,f) -yo , QAse)=Cr"*; = to[-tzt(a-) *tlp_ = 4:5_!,ao.6:_8zq ! Yq, - P) 0 6ao\.-, 3. -z = at-27= is-6 -tzt .j's- 4-rr-t3 O( it.3-15?) : (r'-iTfrt.-)rt:')(a)
School: George Mason
Course: Number Theory
MATH 301 Homework 9 1. 8.1 no.6 Decrypt the message RTOLK TOIK, which was encrypted using the ane transformation C 3P + 24(mod 26). 2. 8.1 no.8 The message KYVMR CLVFW KYVBV PZJJV MVEKV VE was encrypted using a shift transformation C P + k(mod 26). Use fr
School: George Mason
Course: Number Theory
MATH 301 Homework 6 1. 7.3 no.1 Find the six smallest even perfect numbers. (Please justify that the sixth even perfect number comes from a Mersenne prime.) 2. 7.3 no.4 Find a factor of each of the following integers: a) 2111 1 b) 2289 1 c) 246189 1 3. 7.
School: George Mason
Course: Number Theory
MATH 301 Homework 7 1. 7.1 no.2 Find the value of the Euler -function for the following integers. a) 100 b) 256 c) 1001 d) 2 3 5 7 11 13 e) 10! f) 20! 2. 7.1 no.4 Find all positive integers n such that (n) has each of the following values. Be sure to prov
School: George Mason
Course: Number Theory
MATH 301 Homework 5, due TUESDAY, Oct. 15 1. 4.3 no. 4: Find all the solutions of each of the following systems of linear congruences. (a) x 4 (mod 11), x 3 (mod 17) (b) x 1 (mod 2), x 2 (mod 3), x 3 (mod 5) (c) x 0 (mod 2), x 0 (mod 3), x 1 (mod 5), x 6
School: George Mason
Course: Number Theory
Homeutz;n* _ : t- (9 _S*go":<- +tor-(q'4+L)a+t)-er (S Sr+) or 3*.' 1, *\u^ . q-1-.3X+l ,*sa-a*A = 3t+3:3(1+r) - a"/- a.'z -3(X+a1 a+\=-3t+6 - "-^- 'eo4 JO -kr'&v,r . t's -qnA atY -.-s- t o* ?d*-.ls -cfw_q s^tj %rqY f;>'n 4, a+ a^Aat-'cfw_-+o o.\l U. p-t'a
School: George Mason
Course: Number Theory
/-'7fl$b tl I \J 77rzo 'JS: : = ?L1 -e1 cfw_o Ho,ut*ro rlc-f, u grCs) +t e.-! n;o 1," B (o A. S-* =l C-r.ra of \_JT a pzc<&pn;r* 4o tl*-b"4_IQ r= 1,:, * g r,a,.o+o,or^"\ 54 ti no* pel al,^r;4f ,-t^ psr*hp,.r* tM @c@Joffi*, f_r-* : (a'-':d'_l-_to*,11f,'L -
School: George Mason
Course: Number Theory
MATH 301 Homework 6 1. 6.2 no.1 Show that 91 is a pseudoprime to the base 3. 2. 6.2 no.2 Show that 45 is a pseudoprime to the bases 17 and 19. 3. 6.2 no. 6 Show that if n = (a2p 1)/(a2 1), where a is an integer, a > 1, and p is an odd prime not dividing a
School: George Mason
Course: Number Theory
+S $ns1enS a)K aY 6^rcfw_ ,t), xt3 (^"/- t+)_ tl- It +t, it: f,+f l-t" r*o*o t . *- t (r+) ( a) + 3(t)(-l) I ^a c) - rCD r<; B6'@tq= 34 -x1^.:i>*nrfue yz"l ,)3" I Cby o,'peohr^) t(rsXD+a(roXt) + a(e)C.>: r5*2o+r8.53 ^eoC".dz) tcfw_&=ac,'"*D 12"6 + ?l9: .,
School: George Mason
Course: Number Theory
MATH 301 Homework 4, due MONDAY, Oct. 7 1. 4.1 no. 4: Show that if a is an even integer, then a2 0 (mod 4), and if a is an odd integer, then a2 1 (mod 4). 2. 4.1 no. 5: Show that if a is an odd integer, then a2 1 (mod 8). 3. 4.1 no. 8: Find the least posi
School: George Mason
Course: Number Theory
HAU Etr)o ek+ q SO lutrops ,^ l- +1 ?f"- x ) l't't Tt_c,="lkrl (t\./' ty- Sg. Supp.s*- 0. = Ee _,L t's t/e^r _Say tF Sc,^e t"t k 3e" , +l,a^ n.-qL.7 tJ' = TK *_X,9 kl+ 4k+l oi k=l q $o- s>y'e"*y^t, +ao. ?r> r So nQ. " : sC".odrl)" -tC6) =-6 Ca): z( a): r
School: George Mason
Course: Number Theory
'@r ftoqe u)orL .-. tr 3 S-l*h:, ? (a+b, a b -ta-+b) -7 (arb,-Zb>=@!b,ea), a ri coruatottl f p 2.3,t|, /o 5*=_/,S (arb, 4 -b ) 7A"s /.u 7D , 2+ 2r > .(z= z -3' 7, 4o= 2,.7, /6f,7 +!'rz r(a.5 c4 lo-d:r(z) rzo- & =9(,g)t t _, <o_- (6617o, re r)= r rf- 76()+
School: George Mason
Course: Number Theory
MATH 301 Homework 3 1. 3.3 no.10 Show that if a and b are integers with (a, b) = 1, then (a + b, a b) = 1 or 2. 2. 3.3 no. 20 Find three mutually relatively prime integers from among the integers 66, 105, 42, 70, and 165. 3. 3.3 no. 24 Show that if k is a
School: George Mason
Course: Ordinary Differential Equation
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
Course: Ordinary Differential Equation
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
Course: Ordinary Differential Equation
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
Course: Ordinary Differential Equation
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: 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