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School: UCSB
Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
School: UCSB
Course: Calc With Appli 2
MATH 3B EXAM I PRACTICE January 13, 2011 JEFFREY STOPPLE In these notes you will come up with your own practice exam questions. This will better help internalize the material. You should solve your own exam, or if you have a study partner you should switc
School: UCSB
Course: 117
Homework 2 Hctor Guillermo Cullar R e e os February 2, 2006 12.12 Let D be a nonempty set and suppose that f : D R and g : D R. Dene the function f + g : D R by (f + g)(x) = f (x) + g(x). (a) If f (D) and g(D) are bounded above, then prove that (f
School: UCSB
Homework 5 Math 104A, Fall 2010 Due on Tuesday, November 9th, 2010 1. Given xi , i = 0, 1, . . . , n, consider the Lagrange polynomials Ln,j for j = 0, 1, . . . , n. Prove that n Ln,j (x) = 1 for all x R. j =0 2. The following data is taken from a polynom
School: UCSB
Course: Vector Calculus 2
Math 5C: Exam #2 Solutions Date: July 16th , 2010 Score: out of 60 1. (10) Match each Maclaurin series to the function from the following list it represents by lling in the blank space below the series. (Note: All listed function are C at x = 0 under the
School: UCSB
Course: Probability Theory And Stochastic Processes
PStat 213A: Probability Theory and Stochastic Processes Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on probability theory and stochastic processes I took from Professor Raya Feldman in the Fall of 2005. Th
School: UCSB
Course: Linear Algebra
(\b \xcwew fusion WWĀ»- LEPLHHMHCSMfig} : .564. (4 645.? 1mm a?!" covxnblvmeUEIS . , 1. J _ a? J, ,3, "a WK firi SUI I UL] V3Ā§ 2 (Yimrm Div/V2.1 2/3 L? ifCHSUTLfIIH ;U{$ 5'9 mcpmtmnt 7:10 w ) w)- NOTC 5 !s b m Ā«.vtc: (mt. CH) puma A: [UUULNB] wm'm- g MU]
School: UCSB
Course: Linear Algebra
10/26/14 f. CLAS MxĆ©term Reviw WE ation 6M x + (23 +32 2 \ Q 3 Lk 3x 23 +2 r\ 3 2 Ox_lj+ll':/ O 1 VSTSTEWU - f. (cemgwmt) I'm.) CM ~um1q,U|e &o\m-om a pivot 'm exit/13 comm?! Umem sr\c\c'7(43md-ema WWW ho gammy) ,9/ H 0 451mm wpm m 6% [0001'] minnow * i
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Midterm 3 Review Copyright Daryl Cooper D.A.R.Y.L. March 2, 2015 How many practice Midterms have you done ? A=3 B=2 C=1 D = What midterm ? If you want to do well on the midterm: do not study later than 7pm the night before Then get a lot of rest.
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 25 Copyright Daryl Cooper D.A.R.Y.L. March 13, 2015 Substitute y = x n into the equation x 2 y 4xy + 4y = 0 and use this to nd the two values of n which give a solution. A = I have an answer B = working C = dont know what to do y = xn y =
School: UCSB
Math 34A Lecture 17 Copyright Daryl Cooper D.A.R.Y.L. Please do NOT come on stage November 9, 2012 Homework 7.13.43 Some biologists at UCSB have carefully recorded the number of elephant seal births in the Channel Islands from aerial photographs since the
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 23 Copyright Daryl Cooper D.A.R.Y.L. March 9, 2015 The graph of a function f (x) of one variable is the curve y = f (x) in the xy -plane. The tangent line to the graph at the point x = u is the straight line which goes through the point o
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 22 Copyright Daryl Cooper D.A.R.Y.L. March 6, 2015 f (x, y ) = 2x + 3y + xy + x 2 y 3 A = 6x 2 y B = 2y 3 C = 2 + y + 2xy 3 fx = f x = 2 + y + 2xy 3 fy = f y = 3 + x + 3x 2 y 2 fxx = x f x fyy = y f y fyx = y f x fxy = x f y = = 2y 3 6x 2
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 10 copyright Daryl Cooper D.A.R.Y.L. February 8, 2014 Webwork last night ? (A) was extremely slow/crashed (B) very slow (C) slightly slow (D) OK product rule d dx d dx (f (x) g (x) = f (x)g (x) + f (x)g (x) (sin(Kx) = K cos(Kx) Find deriv
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 8 D.A.R.Y.L. January 26, 2015 Trig ? (A) Done it, remember it (B) Done it, forgot it (C) Never done it Section (12.1) Sine Waves. Sine waves arise in many situations: geometry circles, triangles, navigation periodic phenomena child on a s
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 9 copyright: Daryl Cooper D.A.R.Y.L. February 8, 2014 Webwork last night ? (A) was extremely slow/crashed (B) very slow (C) slightly slow (D) OK sin() cos() 0 0 1 /2 1 0 0 1 3/2 1 0 (cos(/2) , sin(/2) = (0,1) You will nd it easier to reme
School: UCSB
Course: Calc With Appli 2
MATH 3B EXAM I PRACTICE January 13, 2011 JEFFREY STOPPLE In these notes you will come up with your own practice exam questions. This will better help internalize the material. You should solve your own exam, or if you have a study partner you should switc
School: UCSB
Course: Vector Calculus 2
Math 5C: Exam #2 Solutions Date: July 16th , 2010 Score: out of 60 1. (10) Match each Maclaurin series to the function from the following list it represents by lling in the blank space below the series. (Note: All listed function are C at x = 0 under the
School: UCSB
Course: Calculus
Being the final examination for Math 3B NO notes or calculators. READ all questions carefully. Make sure your answers are clearly marked and it is clear what work is relevant and should be graded. Each problem is worth 20 points. Note there is a blank pag
School: UCSB
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera March 17th, 2009 Answer the following 8 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for presen
School: UCSB
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera December 8th, 2010 Answer the following 7 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for pres
School: UCSB
Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
School: UCSB
Course: 117
Homework 2 Hctor Guillermo Cullar R e e os February 2, 2006 12.12 Let D be a nonempty set and suppose that f : D R and g : D R. Dene the function f + g : D R by (f + g)(x) = f (x) + g(x). (a) If f (D) and g(D) are bounded above, then prove that (f
School: UCSB
Homework 5 Math 104A, Fall 2010 Due on Tuesday, November 9th, 2010 1. Given xi , i = 0, 1, . . . , n, consider the Lagrange polynomials Ln,j for j = 0, 1, . . . , n. Prove that n Ln,j (x) = 1 for all x R. j =0 2. The following data is taken from a polynom
School: UCSB
Course: Graph Theory
HOMEWORK 3 SOLUTIONS (1) Show that for each n N the complete graph Kn is a contraction of Kn,n . Solution: We describe the process for several small values of n. In this way, we can discern the inductive step. Clearly, K1 , which is just one vertex, is a
School: UCSB
October 15, 2011 13:47 ast Sheet number 3 Page number xx cyan magenta yellow black October 10, 2011 15:10 frs Sheet number 3 Page number iii cyan magenta yellow black 10 th EDITION David Henderson/Getty Images CALCULUS EARLY TRANSCENDENTALS HOWARD ANTON I
School: UCSB
Course: Math 4A
Math 4A Midterm Review Prolbems Note: These problems are provided as review of the key ideas that well be tested on the midterm, but this is not a practice midterm. Because we are testing your understanding of concepts and not simply computational uency,
School: UCSB
Mathematics 4B Winter 2015: Review for Final March 12, 2015 Professor J. Douglas Moore YOU ARE ALLOWED ONE 3 x 5 CARD FOR THE FINAL EXAM. Recall that there is assigned seating for the nal exam. Please write your seat number on your card. You will need to
School: UCSB
School: UCSB
School: UCSB
School: UCSB
Math 3C Practice Word Problems Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1) A dangerous substance known as Chemical X is lethal if its concentration in the air is 100 parts per million by volume (ppmv). The half-life of che
School: UCSB
Course: Homological Algebra
Math 236B, Spring 2015, MWF 9-9:50, HSSB 1223 Homological Algebra Instructor: Birge Huisgen-Zimmermann, SH 6518, Oce hours M, F 11 - 12, W 12:30 1:30. Accompanying texts, as for the winter quarter: The manuscript I will put on the board will again serve
School: UCSB
Course: Abstact Algebra
Math 111C, Spring 2015, MWF 10 10:50, Building 387, Room 103 Introduction to Abstract Algebra Instructor: Birge Huisgen-Zimmermann, SH 6518 Oce Hours: Mon, Fri 11-12, Wed 12:30 - 1:30 Teaching Assistant: Nathan Schley, SH 6431P, Oce hours Tue, Thur 4-5. E
School: UCSB
Math 8: Transition to higher mathematics Syllabus Text: How to prove it (A structured approach) by D. J. Velleman, Second Edition Lectures: TR 12:30-1:45 ARTS 1353 Discussions : MW 5:00-5:50 HSSB 1223 MW 6:00-6:50 HSSB 1207 Instructor: Eleni Panagiotou Of
School: UCSB
Course: Math 4A
Math 4A - Jacob Course Information and Syllabus Spring Quarter 2015 Monday, Wednesday, Friday 1:00 - 1:50 Instructor: Bill Jacob Oce Hours: Monday and Wednesday 2:15 - 3:15, Friday 9:45 -10:45, and by appointment. Oce: South Hall 6719 Email: jacob@math.uc
School: UCSB
SYLLABUS FOR MATHEMATICS 4B CALCULUS WINTER 2015 Professor John Douglas Moore Office: South Hall 6714 Office hours: TuTh 3:30, W 1 Telephone: 893-3688 email: moore@math.ucsb.edu Lectures: Lotte Lehman Concert Hall TuTh 9:30-10:45 Text: Boyce, Diprima, Ele
School: UCSB
Course: Differential Equations
Math 4B UCSB, Spring 2015 Lecture: MWF 11:00-11:50, MUSICLLCH Textbook: The textbook for this course is optional. For those who would like a reference and extra practice problems the book Elementary Dierential Equations by Boyce and DiPrima will work. Ins
School: UCSB
Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
School: UCSB
Course: Calc With Appli 2
MATH 3B EXAM I PRACTICE January 13, 2011 JEFFREY STOPPLE In these notes you will come up with your own practice exam questions. This will better help internalize the material. You should solve your own exam, or if you have a study partner you should switc
School: UCSB
Course: 117
Homework 2 Hctor Guillermo Cullar R e e os February 2, 2006 12.12 Let D be a nonempty set and suppose that f : D R and g : D R. Dene the function f + g : D R by (f + g)(x) = f (x) + g(x). (a) If f (D) and g(D) are bounded above, then prove that (f
School: UCSB
Homework 5 Math 104A, Fall 2010 Due on Tuesday, November 9th, 2010 1. Given xi , i = 0, 1, . . . , n, consider the Lagrange polynomials Ln,j for j = 0, 1, . . . , n. Prove that n Ln,j (x) = 1 for all x R. j =0 2. The following data is taken from a polynom
School: UCSB
Course: Vector Calculus 2
Math 5C: Exam #2 Solutions Date: July 16th , 2010 Score: out of 60 1. (10) Match each Maclaurin series to the function from the following list it represents by lling in the blank space below the series. (Note: All listed function are C at x = 0 under the
School: UCSB
Course: Probability Theory And Stochastic Processes
PStat 213A: Probability Theory and Stochastic Processes Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on probability theory and stochastic processes I took from Professor Raya Feldman in the Fall of 2005. Th
School: UCSB
Course: Calculus
Being the final examination for Math 3B NO notes or calculators. READ all questions carefully. Make sure your answers are clearly marked and it is clear what work is relevant and should be graded. Each problem is worth 20 points. Note there is a blank pag
School: UCSB
Course: Differential Equations
-7 -3 -1 1 -2 26.(1 pt) 0 -2 -1 -2 -28 -12 -6 3 . 32.(1 pt) Let M = -2 3 Compute the rank of the above matrix -2 2 1 Find c1 , c2 , and c3 such that M 3 + c1 M 2 + c2 M + c3 I3 = 0, where 7 4 7 I3 is the identity 3 3 matrix. 27.(1 pt) 7 4 3 , c1 = 21 12
School: UCSB
Course: Graph Theory
HOMEWORK 3 SOLUTIONS (1) Show that for each n N the complete graph Kn is a contraction of Kn,n . Solution: We describe the process for several small values of n. In this way, we can discern the inductive step. Clearly, K1 , which is just one vertex, is a
School: UCSB
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera March 17th, 2009 Answer the following 8 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for presen
School: UCSB
Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera December 8th, 2010 Answer the following 7 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for pres
School: UCSB
Math 34A Lecture 17 Copyright Daryl Cooper D.A.R.Y.L. Please do NOT come on stage November 9, 2012 Homework 7.13.43 Some biologists at UCSB have carefully recorded the number of elephant seal births in the Channel Islands from aerial photographs since the
School: UCSB
October 15, 2011 13:47 ast Sheet number 3 Page number xx cyan magenta yellow black October 10, 2011 15:10 frs Sheet number 3 Page number iii cyan magenta yellow black 10 th EDITION David Henderson/Getty Images CALCULUS EARLY TRANSCENDENTALS HOWARD ANTON I
School: UCSB
Course: Differential Equations
Sample WeBWorK problems. 1.(1 pt) Which of the following are vectors in R2 ? A. (1,0,0) B. (1,0) C. x2 D. x E. 1 F. (0,1) G. None of the above Which of the following is the zero vector in R2 ? A. 0 B. (0,0,0) C. (0,0) D. None of the above Which pairs belo
School: UCSB
Course: Linear Algebra
(\b \xcwew fusion WWĀ»- LEPLHHMHCSMfig} : .564. (4 645.? 1mm a?!" covxnblvmeUEIS . , 1. J _ a? J, ,3, "a WK firi SUI I UL] V3Ā§ 2 (Yimrm Div/V2.1 2/3 L? ifCHSUTLfIIH ;U{$ 5'9 mcpmtmnt 7:10 w ) w)- NOTC 5 !s b m Ā«.vtc: (mt. CH) puma A: [UUULNB] wm'm- g MU]
School: UCSB
Course: Linear Algebra
10/26/14 f. CLAS MxĆ©term Reviw WE ation 6M x + (23 +32 2 \ Q 3 Lk 3x 23 +2 r\ 3 2 Ox_lj+ll':/ O 1 VSTSTEWU - f. (cemgwmt) I'm.) CM ~um1q,U|e &o\m-om a pivot 'm exit/13 comm?! Umem sr\c\c'7(43md-ema WWW ho gammy) ,9/ H 0 451mm wpm m 6% [0001'] minnow * i
School: UCSB
Course: Linear Algebra
:a ,I I' i. HW r} _ A $me mum :5 (Met! (kW H- If canmms The. _,Ā» - exam mm m cam vow am a Column, WM an other emch bctnq O. A vermuwhum matrices art ihVeVfM- F W MW Mw rm {WWW pe-miahom mmmx 'ooux uooi ivUO'] iron-c Jame] = v' \ U ()0 {g I m} ii. g; 5 cl
School: UCSB
Course: Linear Algebra
11/ 25M ame WOYk % % 1.A19 an wxm WWHYIX A c A mmm A w, W} mveere w and WWW 0 of A. Him/w, (took 0% i'w, prom/g a); JAWMO'M B a fa fund we cigewalues off A z , (Ā£01156, Ljou 3L;ch ma: +1: Take Tm irt 03"? )3 0 ' LE Ax =x for Sam; vemr x, Ā«W?! 7 is W Cien
School: UCSB
Course: Linear Algebra
Assignment hw2 due 10/20/2014 at 10:15am PDT _ah_#_#_ 3 12 40 1- (1 pt) Let V] = 2 , v2 = 9 , V3 = 29 and l 4 13 4 MI= 0 8 1. Is win {V1,V2,V3}? Type yes or no. 2. How many vectors are in {V1,V2,V3}? Enter int if the answer is innitely many. 3. How ma
School: UCSB
Course: Linear Algebra
mun. Iv Assignmm hwl due 10/13/2014 at 07:2spm PDT 1- (1 Pl) Soive the system using matrices (row Operations) rlr + 3y: 29 3x 7): -51 x 2 y = - Anni-arts) submitted: I -10 I 3 (correct) ms_y_ 2. (1 pt) The reduced row echelon form of a system of linear
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 23 Copyright Daryl Cooper D.A.R.Y.L. March 9, 2015 The graph of a function f (x) of one variable is the curve y = f (x) in the xy -plane. The tangent line to the graph at the point x = u is the straight line which goes through the point o
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Midterm 3 Review Copyright Daryl Cooper D.A.R.Y.L. March 2, 2015 How many practice Midterms have you done ? A=3 B=2 C=1 D = What midterm ? If you want to do well on the midterm: do not study later than 7pm the night before Then get a lot of rest.
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 22 Copyright Daryl Cooper D.A.R.Y.L. March 6, 2015 f (x, y ) = 2x + 3y + xy + x 2 y 3 A = 6x 2 y B = 2y 3 C = 2 + y + 2xy 3 fx = f x = 2 + y + 2xy 3 fy = f y = 3 + x + 3x 2 y 2 fxx = x f x fyy = y f y fyx = y f x fxy = x f y = = 2y 3 6x 2
School: UCSB
Course: Calculus For Social And Life Sciences
version Tardis Practice PRINT NAME Quality Bonus Math 34B Winter 2010 Midterm3 2 SCORE 25 Signature Prof D.A.R.Y.L. Put final answers in boxes on this page. SHOW WORK in the blue book. If the work in the blue book does not match the answer you give on thi
School: UCSB
Course: Probability Theory And Stochastic Processes
PStat 213A: Probability Theory and Stochastic Processes Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on probability theory and stochastic processes I took from Professor Raya Feldman in the Fall of 2005. Th
School: UCSB
Course: Linear Algebra
(\b \xcwew fusion WWĀ»- LEPLHHMHCSMfig} : .564. (4 645.? 1mm a?!" covxnblvmeUEIS . , 1. J _ a? J, ,3, "a WK firi SUI I UL] V3Ā§ 2 (Yimrm Div/V2.1 2/3 L? ifCHSUTLfIIH ;U{$ 5'9 mcpmtmnt 7:10 w ) w)- NOTC 5 !s b m Ā«.vtc: (mt. CH) puma A: [UUULNB] wm'm- g MU]
School: UCSB
Course: Linear Algebra
10/26/14 f. CLAS MxĆ©term Reviw WE ation 6M x + (23 +32 2 \ Q 3 Lk 3x 23 +2 r\ 3 2 Ox_lj+ll':/ O 1 VSTSTEWU - f. (cemgwmt) I'm.) CM ~um1q,U|e &o\m-om a pivot 'm exit/13 comm?! Umem sr\c\c'7(43md-ema WWW ho gammy) ,9/ H 0 451mm wpm m 6% [0001'] minnow * i
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Midterm 3 Review Copyright Daryl Cooper D.A.R.Y.L. March 2, 2015 How many practice Midterms have you done ? A=3 B=2 C=1 D = What midterm ? If you want to do well on the midterm: do not study later than 7pm the night before Then get a lot of rest.
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 25 Copyright Daryl Cooper D.A.R.Y.L. March 13, 2015 Substitute y = x n into the equation x 2 y 4xy + 4y = 0 and use this to nd the two values of n which give a solution. A = I have an answer B = working C = dont know what to do y = xn y =
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 24 Copyright Daryl Cooper D.A.R.Y.L. March 11, 2015 HW 15.4.3 A rectangular box must have a volume of 2 cubic meters. The material for the base and top costs $2 per square meter. The material for the vertical sides costs $8 per square met
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 1 Copyright Daryl Cooper D.A.R.Y.L. January 5, 2015 Use your iclicker to respond. When did you take 34A ? A = Last quarter with me B = Last year with someone else C = A long time ago in a galaxy far, far away D = Never took it Lectures on
School: UCSB
Math 6A - Vector Calculus I MWF, 8 - 8:50 AM, 1940 Buchanan Hall Instructor: Matt Porter Email: mattporter@math.ucsb.edu Oce Hours: MWF, 11 AM - 12 PM, in 6432U South Hall (or by appointment) TA: Ebrahim Ebrahim Email: ebrahim@math.ucsb.edu Oce hours: Fri
School: UCSB
Course: Homological Algebra
MATH 236B, S 2015. Further remarks on direct limits (generally useful) Let Ai , fij i,jI, ij be a directed system of left R-modules, with lim Ai = iI Ai /U , where U = ij (inj fij ini )(Ai ) iI Ai (here inj : Aj iI Ai is the canonical injection). Moreover
School: UCSB
Course: Homological Algebra
MATH 236 A, W 2015, Projective Modules over Local Rings Second installment: Showing freeness of projectives over local rings. Ill start by reminding you of the background weve assembled earlier. Kaplanskys Theorem (Theorem 11). Let R be any ring and M an
School: UCSB
Course: Homological Algebra
MATH 236B, S 2015. A brief introduction to limits of functors Let I be a small category, I its set of objects. Moreover, let F : I C be a contravariant functor. A limit of F is an object C of C, together with a family i HomC (C, F (i) for i I, with F (f )
School: UCSB
Course: Homological Algebra
MATH 236 B, S 2015 An example of a ring with diering right and left global dimensions Let R := ZQ 0 Q , short for the subring a b 0 c a Z, b, c Q . We claim M2 (Q) that r.gl.dim R = 1 while l.gl.dim R 2. Obviously R is not semisimple (it has a nonzero ni
School: UCSB
Course: Homological Algebra
MATH 236B, S 2015. Supplement re colimits: Pushouts and direct limits These notes pick up where our nal 236A class, afternoon of March 13, left o. A. Pushouts I was already rushed when I sketched the fact that pushouts are colimits. So Ill back up a littl
School: UCSB
Course: Abstact Algebra
A REMARK ON FINITE ABELIAN GROUPS THAT FEEDS INTO THE PROOF OF THEOREM 22 Auxiliary Proposition. Let A be a nite abelian group. Then there exists an element x A such that |y| divides |x| for all y A. Consequence: If A fails to be cyclic, there exists a na
School: UCSB
Course: Abstact Algebra
SUGGESTIONS AND SOME MOTIVATION FOR REVIEW OF 111B MATERIAL Prelude, partly doubling up on what I said in class: Whenever we analyze prototypes of algebraic structures, such as vector spaces, groups, and rings (now elds), we use homomorphisms to compare s
School: UCSB
Course: Abstact Algebra
KRONECKERS THEOREM, FIRST VERSION We know: Every embedding : F K of elds induces an embedding of polynomial rings : F [X] K[X] (i.e., an injective ring homomorphism); it sends any polynomial d d i i i=0 ai X F [X] to the polynomial i=0 (ai )X in K[X]. The
School: UCSB
Course: Abstact Algebra
MATH 111C, S 2015, EXISTENCE OF SPLITTING FIELDS, GENERAL VERSION Ill start with a reminder. In class, we proved: Theorem 14 [nite version]. Let F be a eld and P a nite set of nonconstant polynomials in F [X]. Then there exists a splitting eld for P over
School: UCSB
Math Methods 1 Lia Vas Line and Surface Integrals. Stokes and Divergence Theorems Review of Curves. Intuitively, we think of a curve as a path traced by a moving particle in space. Thus, a curve is a function of a parameter, say t. Using the standard vect
School: UCSB
Quadric Surfaces In 3-dimensional space, we may consider quadratic equations in three variables x, y, and z: ax2 + by 2 + cz 2 + dxy + exz + f yz + gx + hy + iz + j = 0 Such an equation denes a surface in 3D. Quadric surfaces are the surfaces whose equati
School: UCSB
School: UCSB
Vector Review Notation: Vectors in R3 can be written as an ordered triple: a (a x , a y , a z ) They can be writes in terms of the unit vectors in the x, y, and z directions: a axi a y az k j Or: a ax x a y y az z They can also be written as a column vect
School: UCSB
Calculus 3 Lia Vas Greens Theorem. Curl and Divergence Greens Theorem. Let C be a smooth curve r(t) = (x(t), y(t) with endpoints r(a) = (x(a), y(a) and r(b) = (x(b), y(b). A curve is called closed if r(a) = r(b). In this case, we say that C is positive or
School: UCSB
o9 _ Swces . _ . . . A . .s whee: .i5Ā¢_9._. . . . 2:. . .Wm.s. Lima} be FammgiiaeĆ©racing. . u.<w=. 147,.011' are UV) 6.1.7 a.5.b$.Ā¢i?9 1R3. The. laces-an. (24:31.2): (icuav>ay<%v>azmv>) - ' - R is a t43Ā¢Ā„Ā£.Ā£9$4"V<+5 5 ? f5?Ā„4Ā¢Ā¢. _ .Q.Ā¢.r.w.hĆ©<-k U :
School: UCSB
Some Common Curves and Their Parameterizations 1. The straight line between the points: P0 ( x0 , y0 , z 0 ) and P ( x1 , y1 , z1 ) 1 x ( x1 x0 )t x0 y ( y1 y 0 )t y 0 z ( z1 z 0 )t z 0 Or, as a position vector: c (t ) ( x1 x0 )t x0 , ( y1 y0 )t y0 , ( z1
School: UCSB
Some Common Surfaces and their Parameterizations 1. Any surface of the form z f ( x, y) xx yy z f ( x, y ) Or, as a position vector: ( x, y ) x, y, f ( x, y) 2. Any surface expressed in cylindrical coordinates as z f (r , ) x r cos( ) y r sin( ) z f (r ,
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Course: Differential Equations
Math 4B Midterm Review These are practice problems to help you prepare for your midterm you do not need to turn in solutions. You should think of this as a starting point for organizing your study plan. You should also review homework problems, lecture no
School: UCSB
Course: Differential Equations
Math 4B Midterm Review These are practice problems to help you prepare for your midterm you do not need to turn in solutions. You should think of this as a starting point for organizing your study plan. You should also review homework problems, lecture no
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Math 217: Dierential Equations Final Review Mark Pedigo 1 First-Order DEs Background Info Denition 1.1 (Dierential Equation). A dierential equation (abbreviated DE) is an equation relating an unknown function and its derivatives. Note: The solution is not
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Math 34A Lecture 17 Copyright Daryl Cooper D.A.R.Y.L. Please do NOT come on stage November 9, 2012 Homework 7.13.43 Some biologists at UCSB have carefully recorded the number of elephant seal births in the Channel Islands from aerial photographs since the
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 23 Copyright Daryl Cooper D.A.R.Y.L. March 9, 2015 The graph of a function f (x) of one variable is the curve y = f (x) in the xy -plane. The tangent line to the graph at the point x = u is the straight line which goes through the point o
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 22 Copyright Daryl Cooper D.A.R.Y.L. March 6, 2015 f (x, y ) = 2x + 3y + xy + x 2 y 3 A = 6x 2 y B = 2y 3 C = 2 + y + 2xy 3 fx = f x = 2 + y + 2xy 3 fy = f y = 3 + x + 3x 2 y 2 fxx = x f x fyy = y f y fyx = y f x fxy = x f y = = 2y 3 6x 2
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 10 copyright Daryl Cooper D.A.R.Y.L. February 8, 2014 Webwork last night ? (A) was extremely slow/crashed (B) very slow (C) slightly slow (D) OK product rule d dx d dx (f (x) g (x) = f (x)g (x) + f (x)g (x) (sin(Kx) = K cos(Kx) Find deriv
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 8 D.A.R.Y.L. January 26, 2015 Trig ? (A) Done it, remember it (B) Done it, forgot it (C) Never done it Section (12.1) Sine Waves. Sine waves arise in many situations: geometry circles, triangles, navigation periodic phenomena child on a s
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 9 copyright: Daryl Cooper D.A.R.Y.L. February 8, 2014 Webwork last night ? (A) was extremely slow/crashed (B) very slow (C) slightly slow (D) OK sin() cos() 0 0 1 /2 1 0 0 1 3/2 1 0 (cos(/2) , sin(/2) = (0,1) You will nd it easier to reme
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 7 Copyright Daryl Cooper D.A.R.Y.L. January 16, 2015 Click as you do d A dx 2e 3x d B dx (2x ) C (4e 2x + 6x 2 + a) dx 2 D 1 x 2 dx E What value of x minimizes f (x) = x 2 + 2bx + c A = 6e 3x B = (ln 2)2x C = 2e 2x + 2x 3 + ax + C D = 1/2
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 6 Copyright Daryl Cooper D.A.R.Y.L. January 23, 2015 After t seconds the speed of a rocket is v (t) = 6 + 20t meters/second. What is the average speed during the rst 10 seconds ? A = 1060 B = 1006 C = 106 D = 103 E = 160 C average speed =
School: UCSB
Course: Calculus For Social And Life Sciences
Math 34B Lecture 2 Copyright Daryl Cooper D.A.R.Y.L. January 7, 2015 From reading section (9.1) you should know Finding the area under a graph is useful because A = It is important to nd area B = The area under a graph represents something else we care ab
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 3 Copyright Daryl Cooper D.A.R.Y.L. January 10, 2014 Which of these is an antiderivative of 3x 2 A = 3x 2 + C B = 6x + C C = x 3 + 17 D = x 3 /3 + C E = none of the above Today Fundamental theorem and antiderivatives C because d x 3 + 17
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 4 Copyright Daryl Cooper D.A.R.Y.L. February 8, 2014 t Can use antiderivative to nd 0 f (x) dx How can we use antiderivative to nd answer b a f (x) dx = b 0 b a f (x) dx ? f (x) dx a 0 f (x) dx reason (area between a and b) = (area betwe
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 5 Copyright Daryl Cooper D.A.R.Y.L. January 16, 2013 velocity is the rate of change of distance. velocity v (t) = 2t + 5 cm/sec t = time in seconds How far does object move between t = 0 and t = 3 ? A = 20 B = 22 C = 24 D = 26 E = need hi
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 11 copyright Daryl Cooper D.A.R.Y.L. February 2, 2015 Todays theme: approximating functions. 12.6 Power Series Next Midterm: Monday Feb 10 Last lecture: 1/(1 + x) 1 x Increasing x by r% decreases 1/x by about r% A Euro is $1.10 To conver
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 12 copyright Daryl Cooper D.A.R.Y.L. February 6, 2015 At the start of 2010 there were 100,000 rats on an island. The number of rats was growing at an annual rate of 20,000 r.p.y. (rats per year). A disease was spreading through the popula
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 17 Copyright Daryl Cooper D.A.R.Y.L. February 20, 2015 Summary A dierential equation describes how a quantity y (t) is changing Exponential growth/decay equation dy dt y says how quickly the quantity changes is proportional to the quantit
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 18 Copyright Daryl Cooper D.A.R.Y.L. February 23, 2015 How to sketch the slope eld for the equation dy /dt = t 2 y 2 2 2 At the point (t0 , y0 ) draw a short line with slope t0 y0 . Do this for lots of dierent points (t0 , y0 ) to ll out
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 19 Copyright Daryl Cooper D.A.R.Y.L. February 25, 2015 Do you remember the tangent line approximation ? A = very well B = a bit but would like a review C = I forgot D = never heard of it The tangent line approximation (section 8.6) says f
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 20 Copyright Daryl Cooper D.A.R.Y.L. February 26, 2015 Chap. 15 Multi-Variable Calculus f (x, y ) = 3x 4 y 2 + y 3 + xy + 1 is a function of two variables. The partial derivative with respect to x is fx (x, y ) = x 3x 4 y 2 + y 3 + xy + 1
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 16 Copyright Daryl Cooper D.A.R.Y.L. February 18, 2015 How many Pink Panther movies have you seen ? A = Many B = one or two C = Pink what ? (13.7) Decay to a limiting value. Solve problems by playing a detective game with the dierential e
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 15 copyright Daryl Cooper D.A.R.Y.L. February 15, 2015 (13.7) Exponential Decay towards a limiting value dy dt = k(M y (t) with k > 0 The General Solution is y (t) = M + Ae kt A is an arbitrary constant Check this is a solution: d LHS = d
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 13 copyright Daryl Cooper D.A.R.Y.L. February 4, 2015 (13.1) & (13.2) Dierential Equations dy = 2t. dt Unknown is a function y (t). Solve by integrating y = 2t dt = t2 + C General solution is t 2 + C There are innitely many solutions one
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Course: Calculus For Social And Life Sciences
Math 34B Lecture 14 copyright Daryl Cooper D.A.R.Y.L. February 10, 2015 Review of Natural logs e = 2.7182818284590 n(x) = loge (x) = natural logarithm of x = how many es you multiply to get x Thus n(e x ) = x and e n(x) bit less than 3 =x In other word n
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Course: Math 4A
1 March 30, 2015 Lecture Synopsis and Further Background 1. In 8th grade mathematics you studied y = mx + b and f (x) = kx and solved kx = c. What do they mean? In this class we have to understand these expressions as meaning more than graphs of lines. Th
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Course: Math 4A
April 13, 2015 Lecture Synopsis and Further Background 1. Example. Consider the two sets of vectors 80 1 0 1 0 1 0 19 2 1 0 > > 1 > <B 5 C B 2 C B 1 C B 1 C> = B C,B C , B C , B C 2 R4 >@ 1 A @ 2 A @ 1 A @ 0 A> > > : ; 5 and 2 1 1 80 1 0 19 1 > > 2 > <B 2
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Course: Math 4A
1 April 3, 2015 Lecture Synopsis and Further Background 18. Denition. Two systems of m equations in n variables are called row equivalent (or simply equivalent) if one system can be obtained from the other system by performing a (nite) sequence of element
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Course: Math 4A
1 April 1, 2015 Lecture Synopsis and Further Background 8. Combination Charts. These ideas can be applied to thinking about linear transformations mapping R2 ! R2 where the output values are placed in a combination chart. (a) Use linearity for the linear
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Course: Math 4A
May 15, 2015 Lecture Synopsis and Further Background 13. Coordinates with Respect to a Basis We recall that if cfw_1 , 2 , . . . , n is a basis for Rn , then for every vector v v v n there is a unique sequence of real numbers a , a , . . . , a such that
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Course: Math 4A
May 13, 2015 Lecture Synopsis and Further Background 7. Calculating Determinants Using Gaussian Elimination Unless there are many zeros in the matrix, cofactor expansions can be a very tiresome method for computing determinants. We next show how a determi
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Course: Math 4A
May 11, 2015 Lecture Synopsis and Further Background 1. The 2 2 Determinant You may be acquainted with the denition of the 2 2 and 3 3 determinants from other courses. This is ne. We will extend this theory over the next two classes. Usually these small d
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Course: Math 4A
May 8, 2015 Lecture Synopsis and Further Background 11. Elementary matrices. We conclude this section by analyzing elementary row operations in terms of matrix multiplication. The key concept is that of an elementary matrix. Each of the three basic row op
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Course: Calc With Appli 2
MATH 3B EXAM I PRACTICE January 13, 2011 JEFFREY STOPPLE In these notes you will come up with your own practice exam questions. This will better help internalize the material. You should solve your own exam, or if you have a study partner you should switc
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Course: Vector Calculus 2
Math 5C: Exam #2 Solutions Date: July 16th , 2010 Score: out of 60 1. (10) Match each Maclaurin series to the function from the following list it represents by lling in the blank space below the series. (Note: All listed function are C at x = 0 under the
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Course: Calculus
Being the final examination for Math 3B NO notes or calculators. READ all questions carefully. Make sure your answers are clearly marked and it is clear what work is relevant and should be graded. Each problem is worth 20 points. Note there is a blank pag
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Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera March 17th, 2009 Answer the following 8 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for presen
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Introduction to Numerical Analysis Math 104A, Winter 2009 Instructor: Carlos J. Garc a-Cervera December 8th, 2010 Answer the following 7 questions. Show all your work for full credit. You must include your computer programs. Follow the guidelines for pres
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Course: Calculus For Social And Life Sciences
version Tardis Practice PRINT NAME Quality Bonus Math 34B Winter 2010 Midterm3 2 SCORE 25 Signature Prof D.A.R.Y.L. Put final answers in boxes on this page. SHOW WORK in the blue book. If the work in the blue book does not match the answer you give on thi
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Course: Calculus For Social And Life Sciences
version Tardis A PRINT NAME Quality Bonus Math 34B Winter 2010 Midterm3 2 SCORE 30 Signature Prof D.A.R.Y.L. Put final answers in boxes on this page. SHOW WORK in the blue book. If the work in the blue book does not match the answer you give on this test
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Course: Calculus For Social And Life Sciences
version PRU Tardis ' Math 3413 Fall 2010 Final Quality SCORE Prof D.A.R.Y.L. no calculators Bonus 2 71 Put nal answers in boxes on this page. Show high quality work in the blue book for all answers. Points may be awarded for this. Number your solutions
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Course: Calculus For Social And Life Sciences
version Tardis Red PRINT NAME Quality Bonus Math 34B Winter 2011 Midterm3 SCORE 2 25 Signature Prof D.A.R.Y.L. Put final answers in boxes on this page. SHOW WORK in the blue book. If the work in the blue book does not match the answer you give on this tes
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Course: Calculus For Social And Life Sciences
version Tardis Quality Bonus A PRINT NAME Math 34B Winter 2010 practice Midterm1 2 SCORE 30 Signature Prof D.A.R.Y.L. Put final answers in boxes on this page. SHOW WORK in the blue book. If the work in the blue book does not match the answer you give on t
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Course: Calculus For Social And Life Sciences
version Red Quality Bonus PRINT NAME Math 34B Winter 2011 Midterm 2 2 Signature Prof D.A.R.Y.L. Tardis SCORE 30 Put final answers in boxes on this page. SHOW WORK in the blue book. If the work in the blue book does not match the answer you give on this te
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Course: Calculus For Social And Life Sciences
w (I at local min x: at local max x: to. if version I PRINT NAB/[E Math 34B W 2015 . Midterm 2 Quality SCORE 29 - Bonus 2 Prof D.A.R.Y.L. Signature Put nal answers in boxes on this page. SHOW WORK in the blue book. If the work in the blue book does not
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EXAM 2 (VERSION 1) . - W3 Math 6A November 8, 2013 Name: By writing my name I swear to abide by the UCSB honor code. Indicate which section you attend: ( ) Thursday, 4-4550, Changliang Wang . ' - ' ' ' ( ) Thursday, 55:50, Changliang Wang ( ) Thursd
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EXAM 1 (VERSION 1) Math 6A October 18, 2013 Name: _ \-W_._J By writing my name I swear to abide by the UCSB honor code. When you turn in your test, indicate one: ( ) My note card is stapled to the back of this exam. ( ) I did not use a note card. ' Read a
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FINAL EXAM (VERSION 1) SO him} m- Math 6A December 13, 2013 Name: ~ _V_y By writing my name I swear to abide by the, UCSB honor code. Read all of the following information before starting the exam: You may nd a sheet of scratch taper at the back of the ex
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Course: CACL WITH APPLI2
3H Midterm # 1 v.3 W 201D an. 29, 21i] Instructor: X+ Dai Print Your Perm Number; NEEDE Circle your TAa name and Discussion time: RJ'ELHBIHJIHH Rani; 5pm' 5pm; TPm Qhtiih Sultan}? T8arn;Ā®; pm; Tpm; -' - {\{trll 3. _._,_, 13.11 tit-Jun.- hL-M'k high-rt U
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Course: MATH 3A
Calculus with applicatinna 1 Math 3A. Fall 2009 Instructur: Suukyung Jun Gambler 29111: M Answer the flibwing E questinn. Shaw all your war]: far full credit- Garrett answers with inmnsistent; war}: may not be given credit. Perm nu mh-er: M H II .5 TA
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Course: Abstact Algebra
MATH 111C, S 2015, OVERVIEW Basics Denitions. Field, subeld, eld homomorphism, prime eld, characteristic. Theorem 1. For any eld F , either char(F ) = 0 or char(F ) = p, where p is a positive prime in Z. Corollary 2. Let F0 be the prime eld of F . Then:
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Course: Abstact Algebra
MATH 111C, S 2015, OVERVIEW Basics Denitions. Field, subeld, extension eld, eld homomorphism, prime eld, characteristic. Theorem 1. For any eld F , either char(F ) = 0 or char(F ) = p, where p is a positive prime in Z. Corollary 2. Let F0 be the prime eld
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Course: Abstact Algebra
MATH 1110 NAME SECOND MIDTERM 15 May 2015, 10:0010:50 PERM NO. PLEASE WRITE NEATLY. When giying proofs, be sure to sort out tentative ideas on scratch paper, and to put down your argument in a logical sequence on the test paper. The scratch work will not
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Course: Abstact Algebra
MATH 1 1 1 C NAME FIRST MIDTERM 24 April 2015, 10:00-10:50 PERM NO. PLEASE WRITE NEATLY. When giving proofs, be sure to sort out tentative ideas on scratch paper, and to put down your argument in a logical sequence on the test paper. The scratch work Will
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Course: Abstact Algebra
MATH 111C, S 2015, OVERVIEW Basics Denitions. Field, subeld, eld homomorphism, prime eld, characteristic. Theorem 1. For any eld F , either char(F ) = 0 or char(F ) = p, where p is a positive prime in Z. Corollary 2. Let F0 be the prime eld of F . Then:
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Math 8 Spring Quarter 2015 Practice Problems for Midterm Solutions No notes or calculators are permitted on this exam. To obtain full midterm credit you must show all work, provide complete proofs, and provide short by complete explanations for all questi
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Math 8 Spring Quarter 2015 Practice Problems for Final No notes or calculators are permitted on this exam. You must show all work, provide complete proofs, and provide short by complete explanations for all questions in the True/False section in your Blue
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Course: CACL WITH APPLI2
Math 3B Final Practice Written by Victoria Kala victoriakala@umail.ucsb.edu SH 6432u Oce Hours: R 11:00 am - 12:00 pm Last updated 3/11/2015 I have chosen a problem from each section you have covered this quarter. 1. Find the average value of the function
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Course: CACL WITH APPLI2
Math 3B Final Practice Written by Victoria Kala victoriakala@umail.ucsb.edu SH 6432u Oce Hours: R 11:00 am - 12:00 pm Last updated 3/11/2015 I have chosen a problem from each section you have covered this quarter. 1. Find the average value of the function
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Course: CACL WITH APPLI2
Math 3B Final Practice Written by Victoria Kala victoriakala@umail.ucsb.edu Last updated 3/16/2015 There were some errors in the solutions. Below are the corrections. If you believe you nd an error please email me. 1. On Number 10, y should be multiplied
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Homework 1 Solutions 1. (#1.1.2 in Strauss) Which of the following operators are linear? (a) Lu = ux + xuy (b) Lu = ux + uuy (c) Lu = ux + u2 y (d) Lu = ux + uy + 1 (e) Lu = 1 + x2 (cos y )ux + uyxy [arctan(x/y )]u Solution: (a) Linear. (b) Nonlinear the
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Course: 117
Homework 2 Hctor Guillermo Cullar R e e os February 2, 2006 12.12 Let D be a nonempty set and suppose that f : D R and g : D R. Dene the function f + g : D R by (f + g)(x) = f (x) + g(x). (a) If f (D) and g(D) are bounded above, then prove that (f
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Homework 5 Math 104A, Fall 2010 Due on Tuesday, November 9th, 2010 1. Given xi , i = 0, 1, . . . , n, consider the Lagrange polynomials Ln,j for j = 0, 1, . . . , n. Prove that n Ln,j (x) = 1 for all x R. j =0 2. The following data is taken from a polynom
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Course: Graph Theory
HOMEWORK 3 SOLUTIONS (1) Show that for each n N the complete graph Kn is a contraction of Kn,n . Solution: We describe the process for several small values of n. In this way, we can discern the inductive step. Clearly, K1 , which is just one vertex, is a
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October 15, 2011 13:47 ast Sheet number 3 Page number xx cyan magenta yellow black October 10, 2011 15:10 frs Sheet number 3 Page number iii cyan magenta yellow black 10 th EDITION David Henderson/Getty Images CALCULUS EARLY TRANSCENDENTALS HOWARD ANTON I
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Course: Linear Algebra
:a ,I I' i. HW r} _ A $me mum :5 (Met! (kW H- If canmms The. _,Ā» - exam mm m cam vow am a Column, WM an other emch bctnq O. A vermuwhum matrices art ihVeVfM- F W MW Mw rm {WWW pe-miahom mmmx 'ooux uooi ivUO'] iron-c Jame] = v' \ U ()0 {g I m} ii. g; 5 cl
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Course: Linear Algebra
11/ 25M ame WOYk % % 1.A19 an wxm WWHYIX A c A mmm A w, W} mveere w and WWW 0 of A. Him/w, (took 0% i'w, prom/g a); JAWMO'M B a fa fund we cigewalues off A z , (Ā£01156, Ljou 3L;ch ma: +1: Take Tm irt 03"? )3 0 ' LE Ax =x for Sam; vemr x, Ā«W?! 7 is W Cien
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Course: Linear Algebra
Assignment hw2 due 10/20/2014 at 10:15am PDT _ah_#_#_ 3 12 40 1- (1 pt) Let V] = 2 , v2 = 9 , V3 = 29 and l 4 13 4 MI= 0 8 1. Is win {V1,V2,V3}? Type yes or no. 2. How many vectors are in {V1,V2,V3}? Enter int if the answer is innitely many. 3. How ma
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Course: Linear Algebra
mun. Iv Assignmm hwl due 10/13/2014 at 07:2spm PDT 1- (1 Pl) Soive the system using matrices (row Operations) rlr + 3y: 29 3x 7): -51 x 2 y = - Anni-arts) submitted: I -10 I 3 (correct) ms_y_ 2. (1 pt) The reduced row echelon form of a system of linear
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Course: Abstact Algebra
Math 111C, Spring 2015, Problem (9) (9) Show that the polynomials X 3 2 and X 3 3 in Q[X] do not have isomorphic splitting elds over Q. Proof. First note the following: Every polynomial f Q[X] \ Q has a unique splitting eld (over Q) which is contained in
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Course: Abstact Algebra
Math 111C, Spring 2015, Homework Assignments All assignments refer to our text, Abstract Algebra, third edition, by Dummit & Foote. March 30: p.519, # 5 April 1: p.519, # 7, more cleanly stated as follows: Consider the polynomials fn = x3 nx + 2 Z[x], for
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Math 8 Spring Quarter 2015 Homework 2 Due: Tuesday 14 April 1. Let A be the set cfw_a, cfw_1, a, cfw_4, cfw_1, 4, 4. Which of the following statements are true and which are false? (a) a A. (b)cfw_a A. (c) cfw_1, a A. (d) cfw_4, cfw_4 A (e) cfw_1, 4 A (f)
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Challenge Problem 1: Assigned: 29 June, 2015 Due By: 6 July, 2015 15 points Finding the Volume of an object as a limit of Riemann Sums In class, we have spent a bunch of time talking about how to nd the area under a curve as a limit of Riemann Sums. In pa
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Course: Advanced Linear Algebra
MATH 10813 Midterm # 2 F 2012 Nov. 19, 2012 Instructor: X. Dad Name _ All questions have equal points. Show complete work. Continue on the back of the page if you need more space. Good luck? 1. Let V = R2 be given the inner product (3:, y) = 4531311 +
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Math 108A - Home Work # 7 Solutions 1. For the following matrices: (a) Find the eigenvalues (over F = C). (b) Describe the eigenspace for each eigenvalue. (i.e., describe all the eigenvectors for each eigenvalue). (c) Determine whether F 2 or F 3 has a ba
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Math 108A - Home Work # 5 Solutions LADR Problems, p. 59-60: 4. We must show that V = null(T ) + F u and also that null(T ) F u = cfw_0. First suppose, v null(T ) F u. This means that v = au for some a F and T v = 0. Thus T (au) = T (v) = 0, which implies
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Math 108A - Home Work # 8 Solutions Spring 2009 1. LADR Problems, p. 94-95: 5. If (w, z) F 2 is an eigenvector for T with eigenvalue F , we have T (w, z) = (z, w) = (w, z). This implies that z = w and w = z. Hence z = 2 z and w = 2 w, and since z and w ca
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Math 108A - Home Work # 4 Solutions 1. Exercises 13, 14 on p. 36 in LADR. 13. Suppose U and W are subspaces of R8 with dim U = 3 and dim W = 5. If U + W = R8 , then dim U + W = dim R8 = 8. Thus dim(U W ) = dim U + dim W dim(U + W ) = 3 + 5 8 = 0. Since U
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Math 108A - Home Work # 3 Solutions Spring, 2009 1. LADR Problems, p. 35: 1. Solution. Let v V , and assume V = span(v1 , . . . , vn ). Then we can write v = c1 v1 + + cn vn = c1 (v1 v2 ) + (c1 + c2 )(v2 v3 ) + (c1 + c2 + c3 )(v3 v4 ) + + (c1 + + cn )vn .
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Math 108A - Home Work # 2 Solutions Spring 2009 1. From LADR: 5: Soluton. (a) Subspace. Closed under addition: If x1 +2x2 +3x3 = 0 and y1 +2y2 +3y3 = 0 then (x1 + y1 ) + 2(x2 + y2 ) + 3(x3 + y3 ) = 0 + 0 = 0. Closed under scalar multiplication: If x1 + 2x
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Math 108A - Home Work # 1 Solutions 1. For any z C, prove that z R if and only if z = z. Solution. Let z = a + bi C for a, b R. If z R, then b = 0 and z = a. Then z = a + 0i = a 0i = a = z. Conversely, if z = z, we have a + bi = a bi, which implies 2bi =
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Course: Math 4A
Math 4A Midterm Review Prolbems Note: These problems are provided as review of the key ideas that well be tested on the midterm, but this is not a practice midterm. Because we are testing your understanding of concepts and not simply computational uency,
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Mathematics 4B Winter 2015: Review for Final March 12, 2015 Professor J. Douglas Moore YOU ARE ALLOWED ONE 3 x 5 CARD FOR THE FINAL EXAM. Recall that there is assigned seating for the nal exam. Please write your seat number on your card. You will need to
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Math 3C Practice Word Problems Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB 1) A dangerous substance known as Chemical X is lethal if its concentration in the air is 100 parts per million by volume (ppmv). The half-life of che
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Math 6B Midterm Review #1 1 Greens Theorem (y + e 1. Compute the line integral x ) dx + (2x + cos(y 2 ) dy where C is the positively oriented boundary C of the region D enclosed by the parabolas y = x2 and x = y 2 . Solution: Here we use (curl form) Q P x
School: UCSB
Course: Calc With Appli 2
This is a study guide for the second Math 3B midterm. It indicates which types of problems you may be expected to answer on the midterm, with instructions on where to nd these topics in the Stewart calculus book. Plenty of examples can be found in the e
School: UCSB
Course: Homological Algebra
Math 236B, Spring 2015, MWF 9-9:50, HSSB 1223 Homological Algebra Instructor: Birge Huisgen-Zimmermann, SH 6518, Oce hours M, F 11 - 12, W 12:30 1:30. Accompanying texts, as for the winter quarter: The manuscript I will put on the board will again serve
School: UCSB
Course: Abstact Algebra
Math 111C, Spring 2015, MWF 10 10:50, Building 387, Room 103 Introduction to Abstract Algebra Instructor: Birge Huisgen-Zimmermann, SH 6518 Oce Hours: Mon, Fri 11-12, Wed 12:30 - 1:30 Teaching Assistant: Nathan Schley, SH 6431P, Oce hours Tue, Thur 4-5. E
School: UCSB
Math 8: Transition to higher mathematics Syllabus Text: How to prove it (A structured approach) by D. J. Velleman, Second Edition Lectures: TR 12:30-1:45 ARTS 1353 Discussions : MW 5:00-5:50 HSSB 1223 MW 6:00-6:50 HSSB 1207 Instructor: Eleni Panagiotou Of
School: UCSB
Course: Math 4A
Math 4A - Jacob Course Information and Syllabus Spring Quarter 2015 Monday, Wednesday, Friday 1:00 - 1:50 Instructor: Bill Jacob Oce Hours: Monday and Wednesday 2:15 - 3:15, Friday 9:45 -10:45, and by appointment. Oce: South Hall 6719 Email: jacob@math.uc
School: UCSB
SYLLABUS FOR MATHEMATICS 4B CALCULUS WINTER 2015 Professor John Douglas Moore Office: South Hall 6714 Office hours: TuTh 3:30, W 1 Telephone: 893-3688 email: moore@math.ucsb.edu Lectures: Lotte Lehman Concert Hall TuTh 9:30-10:45 Text: Boyce, Diprima, Ele
School: UCSB
Course: Differential Equations
Math 4B UCSB, Spring 2015 Lecture: MWF 11:00-11:50, MUSICLLCH Textbook: The textbook for this course is optional. For those who would like a reference and extra practice problems the book Elementary Dierential Equations by Boyce and DiPrima will work. Ins
School: UCSB
Course: Math 4A
Math 4A Syllabus Winter 2014 Lecture: MWF 10:00 10:50am, MUSIC LLCH Text: Linear Algebra with applications by David C. Lay, Addison-Wesley, 4th Edition. Material to be covered: Chapters 1-6 in the text book. iClicker: You should purchase an iClicker and b
School: UCSB
Math 108a Professor: Padraic Bartlett Syllabus for Math 108a Weeks 1-10 UCSB 2013 Basic Course Information Professor: Padraic Bartlett. Class time/location: MWF 9-9:50, Phelps 3505. Oce hours/location: TTh 2-3pm, South Hall 6516. Additionally, I am tea
School: UCSB
Analysis with an introduction to proof - Math 117 Spring2009 Monday, Wednesday, & Friday, 12:00-12:50pm, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.e
School: UCSB
Analysis with an introduction to proof - Math 117 Spring2009 Monday, Wednesday, & Friday, 12:00-12:50pm, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.e
School: UCSB
Intro. to Numerical Analysis - Math 104B Winter 2011 Tuesday & Thursday, 8:00-9:15am, South Hall 6635 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.edu/~cgarcia/Courses
School: UCSB
Math 104A, Fall 2010 Intro. to Numerical Analysis Tuesday & Thursday, 9:30-10:45am, 387 101 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: (805) 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.edu/~cgarcia/Courses/Mat
School: UCSB
Math 104A, Winter 2009 Intro. to Numerical Analysis Monday, Wednesday, & Friday, 9:00-9:50am, Arts 1426 Instructor: Carlos J. Garc a-Cervera. Oce: South Hall, 6707. Phone: (805) 893-3681. E-mail: cgarcia@math.ucsb.edu URL: http:/www.math.ucsb.edu/~cgarcia
School: UCSB
Math 108B Intro to Linear Algebra Winter 2010 Professor: Kenneth C. Millett Office: 6512 South Hall Office Hours: R 8:30 11:00 Email: millet@math.ucsb.edu Graduate Assistant: Tomas Kabbabe Office: 6432K South Hall Office Hour: W 10:00 11:00 Email: tomas@m
School: UCSB
Course: Numerical Analysis
University of California, Santa Barbara Department of Statistics & Applied Probability PSTAT 120B, Probability & Statistics, Spring 2010 Instructor: Jarad Niemi Email: niemi@pstat.ucsb.edu Course hours: MWF 10:00-10:50am in HFH 1104 TAs: Varvara Kulikova