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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: 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
School: UCSB
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
School: UCSB
Math4B SystemsofDifferentialEquations PhasePlaneAnalysis PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB A2x2systemof1storderlineardifferentialequationswillhavetheform x t ) Ax ( u1 ( t ) x( t ) u2 ( t ) a11 a12 A a21 a22 Thesolut
School: UCSB
Math4B SystemsofDifferentialEquations AutonomousExamples PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB Anautonomous2x2systemofDEshastheform dx 1 f ( x1 , x 2 ) dt dx 2 g( x 1 , x 2 ) dt Nullclinesarecurveswheref=0(vertical)org=0(hori
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: CACL WITH APPLI2
F f 4: \ *l \3 : c ?n Btn u c h 4 p s\b A ec l , n p. ) y p . , d K Xf . . n t c c un f , O B\ t i dc cfw_ Yv l i v B e ; = o - ; c R r] a m m T J h b: J ;* , : : vn T l . k h w dc B cfw_ (v \ A pi c k : . 5 5 (13 cfw_ l , [ l: -
School: UCSB
Course: CACL WITH APPLI2
Ch e n 0 ( c 1 : 30 - :w b s So u 0p 3 : m 67 2 3 +Ba l l f l r \ ec t wA L : Se c ( n : L S 1 06 1 g o w n T b pm , M o s (x t do 50 / . V a ; p . _ 10 1 n oe l HJ ;j l n / o on < , F Hw i n a l Ot ( . e l SA . 10 / J bK l low e " F u ow n 0 - c Yu M
School: UCSB
Course: CACL WITH APPLI2
Vo l u n w . c+ u p D r la : " t\ s o lld ( u c t (o n > : l r [ Ol v i d( , d , lu v p ih,s J i j e b )c n l o Acfw_ cn , cc 3. h p et w e 1 x tj c . 5t n n li UN ab , l It + f A > dj - n ; t a j v a v ;y . - o V lu m e dh . c x I dv A(x ) b ( E J T j
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
Course: MATH 3A
Math 34A Practice Final 1) Use the graph of y=10x to find approximate values: a) 500.3 b) y(0.65) 2) Integrate: a) (x b b) x 5e 3 + 6 dx x ) 6kx dx a ln( 2) c) (e e )dt t t ln(0.5) 3) The population of a certain town is increasing at a rate of 5+2x pe
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
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
School: UCSB
School: UCSB
School: UCSB
School: UCSB
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
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
Homework 12 MATH 244 Linear Analysis I (1) Do the following problems from 4.6: #18, 23, 24, 25. (See note below.) (2) Read 5.1 and 5.2. There are no problems assigned from these sections, and you do not need to worry about how to construct the matrix of a
School: UCSB
MATH 244 Homework 3 Linear Analysis I (1) Read 1.3 (at least p.2027) and do the following problems: #12, 13, 15. I recommend using technology to plot the slope eld and solution curves in 15, but you can also do it by hand. (2) For each of the following, u
School: UCSB
School: UCSB
Homework 2 MATH 244 Linear Analysis I (1) Read 1.2 and do the following problems: #9, 12, 19, 26, 32. (2) For each equation below, decide if it is linear or nonlinear. If it is linear, write it in the form of Denition 1.2.4 (p.11) and identify the coecien
School: UCSB
MATH 122A HW 10 SOLUTIONS RAHUL SHAH Problem 1. [4.49.1(a)(b)(c)(f)] Apply the Cauchy-Goursat theorem to show that f (z) dz = 0 C when the contour C is the unit circle |z| = 1 in either direction, and when a. f (z) = b. z2 ; z3 f (z) = zez ; c. f (z) = 1
School: UCSB
School: UCSB
MATH 122A HW 1 SOLUTIONS RAHUL SHAH Problem 1. [1.2.1a] Verify that ( 2 ) (1 2) = 2. Solution. ( 2 ) (1 2) = = = 2 + 2 2 2 2 2 2. Problem 2. [1.2.2] Show that a. (z) = (z); b. (z) = (z). Solution. a. Let z = x + y. Then (z) = (x y) = y. Also, (z) = y. Th
School: UCSB
MATH 122A HW 2 SOLUTIONS RAHUL SHAH Problem 1. [1.5.1(b)(d)] Solution. b. Notice that z z = = z z z z = = 1. = 4 + 4 1 = 3 4. Thus z = z. d. (2 + )2 Problem 2. [1.5.2] Solution. a. Let S = cfw_z C | ( ) = 2. Let z S. Then ( ) = 2 and thus () = 2. Hence (z
School: UCSB
School: UCSB
School: UCSB
MATH 122A HW 4 SOLUTIONS RAHUL SHAH Problem 1. [1.14.1] Find a domain in the z plane whose image under the transformation w = z 2 is the square domain bounded by u = 1, u = 2, v = 1 and v = 2. Solution. Notice that the preimage of the four given lines are
School: UCSB
MATH 122A HW 3 SOLUTIONS RAHUL SHAH Problem 1. [1.10.1(a)] Find the square roots of 2. Solution. Since + 2k 2 2 exp for k Z. Thus 1 (2) 2 = 2 exp + k 4 . Thus the two roots are c0 = 1 + and c1 = (1 + ). 1 Problem 2. [1.10.2(b)] Find all the roots of (8
School: UCSB
School: UCSB
MATH 122A HW 5 SOLUTIONS RAHUL SHAH Problem 1. [2.20.2] Show that a. a polynomial P (z) = a0 + a1 z + . . . + an z n an = 0 of degree n (n 1) is dierentiable everywhere, with derivative P (z) = a1 + 2a2 z + . . . + nan z n1 ; b. the coecients in the polyn
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
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
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 3A
Math 34A Practice Final 1) Use the graph of y=10x to find approximate values: a) 500.3 b) y(0.65) 2) Integrate: a) (x b b) x 5e 3 + 6 dx x ) 6kx dx a ln( 2) c) (e e )dt t t ln(0.5) 3) The population of a certain town is increasing at a rate of 5+2x pe
School: UCSB
Course: MATH 3A
Math 34A Practice Midterm #2 1. Using the graph of 10x provided (in your book), find the following: a) log(320) b) antilog(-3.7) c) 2750/320 2. a) Find the equation of the line that contains the point (2,5) and has slope 2. b) Find the point of intersecti
School: UCSB
Course: MATH 3A
Math 34A Derivative Practice Find the derivative with respect to x: 1) F(x) 5x 3 2x 1 5 2) G( x ) 3 2 x x 1 3) H( x ) x x n 4) p( x ) ax 5c 5) q( x ) 5 x 6) m( x ) 4e 2 x 7) f ( x ) 2 3 x Find the derivative of the following functions: t 8) y 3 t 100 2 5
School: UCSB
Course: MATH 3A
U Math34APracticeFinalFall2010 1)Usethegraphofy=10xtofindapproximatevaluesof a)500.3 b)y(0.65) 2)Findvaluesofxwherethefunctionf(x)=x 44x3+4 a)isincreasing b)isconcavedown c)hasaminimumpoint 3)Afunctionf(x)isgraphedbelow. a)Forwhatvaluesofxisf(x)<0? b)Forw
School: UCSB
Course: MATH 3A
CarProblems 1.Ittook8hourstodriveyourcar400miles.Onaverage,howfastwereyoudriving? 2.Youneedtodrivetoajobinterview250milesaway.Itiscurrently8amandtheinterviewisat1pm.If youcandriveatanaveragespeedof55mi/hr,willyougetthereintime? Ifnot,howmanyminuteslateare
School: UCSB
Course: MATH 3A
Math34A FinalReview PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB 1)Usethegraphofy=10xtofindapproximatevaluesof a)500.3 b)y(0.65) solutionforparta) firstwriteanequation:x=500.3 dothelogarithmofbothsides:log(x)=log(50 0.3) expandtherigh
School: UCSB
Course: MATH 3A
Math 34A Practice Midterm #2 1) Compute the following: a) The average rate of change of f(x) = 2x3 5x from x=1 to x=3. b) The average rate of change of f(x) = x2-1 from x=2 to x=2+h. c) The intersection point of the curves f(x) = 3x+1 and g(x) = -2x+6. 2)
School: UCSB
y Practice Final Exam 1 m=1 m=0 m=1 dy y = + 3 x: dx x a) Sketch the direction eld for this DE, using (light or dotted) isoclines for the slopes -1 and 0. 1. For the DE See picture at right y Isoclines: y = x + 3x = m y = 3x2 m x. (Note problem at (0, 0).
School: UCSB
18.03SC Final Exam Solutions 1. (a) The isocline for slope 0 is the pair of straight lines y = x. The direction eld along these lines is at. The isocline for slope 2 is the hyperbola on the left and right of the straight lines. The direction eld along thi
School: UCSB
18.03SC Final Exam 1. This problem concerns the differential equation dy = x 2 y2 dx () Let y = f ( x ) be the solution with f (2) = 0. (a) Sketch the isoclines for slopes 2, 0, and 2, and sketch the direction eld along them. (c) On the same diagram, sket
School: UCSB
Math 331.5: Homework 17 Solutions For the following problems, we use the dierential equation my + y + ky = g (t) where y is the displacement of the mass from its equilibrium position m is the mass is the damping coecient k is the spring constant g (t) is
School: UCSB
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
School: UCSB
Math4B SystemsofDifferentialEquations PhasePlaneAnalysis PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB A2x2systemof1storderlineardifferentialequationswillhavetheform x t ) Ax ( u1 ( t ) x( t ) u2 ( t ) a11 a12 A a21 a22 Thesolut
School: UCSB
Math4B SystemsofDifferentialEquations AutonomousExamples PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB Anautonomous2x2systemofDEshastheform dx 1 f ( x1 , x 2 ) dt dx 2 g( x 1 , x 2 ) dt Nullclinesarecurveswheref=0(vertical)org=0(hori
School: UCSB
Math4B SystemsofDifferentialEquations NonhomogeneousSystems PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB Asystemof1storderlineardifferentialequationswillhavetheform x t ) Ax g( t ) ( Wehavealreadyseentohowtosolvethehomogeneouscaseg(
School: UCSB
Math 4B Eulers Method Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Eulers Method will find an approximate solution to an initial value problem. Lets work through a simple example: Suppose we want to solve the following initial
School: UCSB
DifferentialEquations BasicConcepts PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB WhatisaDifferentialEquation? PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB WhatisaDifferentialEquation? ShortAnswer:Anequationinvolvi
School: UCSB
DifferentialEquations SecondOrderLinearDEs MechanicalOscillations PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB Acoupleofphysicalsituationscanbemodeledby2ndorderD.E.swithconstant coefficients.Mechanicaloscillations,suchasamassspringset
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: 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
School: UCSB
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
School: UCSB
Math4B SystemsofDifferentialEquations PhasePlaneAnalysis PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB A2x2systemof1storderlineardifferentialequationswillhavetheform x t ) Ax ( u1 ( t ) x( t ) u2 ( t ) a11 a12 A a21 a22 Thesolut
School: UCSB
Math4B SystemsofDifferentialEquations AutonomousExamples PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB Anautonomous2x2systemofDEshastheform dx 1 f ( x1 , x 2 ) dt dx 2 g( x 1 , x 2 ) dt Nullclinesarecurveswheref=0(vertical)org=0(hori
School: UCSB
Math4B SystemsofDifferentialEquations NonhomogeneousSystems PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB Asystemof1storderlineardifferentialequationswillhavetheform x t ) Ax g( t ) ( Wehavealreadyseentohowtosolvethehomogeneouscaseg(
School: UCSB
Math 4B Eulers Method Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Eulers Method will find an approximate solution to an initial value problem. Lets work through a simple example: Suppose we want to solve the following initial
School: UCSB
DifferentialEquations BasicConcepts PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB WhatisaDifferentialEquation? PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB WhatisaDifferentialEquation? ShortAnswer:Anequationinvolvi
School: UCSB
DifferentialEquations SecondOrderLinearDEs MechanicalOscillations PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB Acoupleofphysicalsituationscanbemodeledby2ndorderD.E.swithconstant coefficients.Mechanicaloscillations,suchasamassspringset
School: UCSB
DifferentialEquations SecondOrderLinearDEs PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB ASecondOrderLinearDifferentialEquationcan alwaysbeputintotheform: y p( t ) y q( t ) y g( t ) Thegeneralsolutionwillalwayshavetheform: y ( t ) y h
School: UCSB
DifferentialEquations SolvingFirstOrderLinearDEs PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB AFirstOrderLinearDifferentialEquationcan alwaysbeputintotheform: y p( t ) y f ( t ) Thegeneralsolutionwillalwayshavetheform: y ( t ) y h y p
School: UCSB
Math4B SystemsofDifferentialEquations MatrixSolutions Prepared by Vince Zaccone For Campus Learning Assistance Services at UCSB Asystemof1storderlineardifferentialequationswillhavetheform x t ) Ax ( HerewearedealingwithnvectorsandannxnmatrixA. x1( t ) a
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School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
School: UCSB
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School: UCSB
Course: Modern Algebra
Math 220A: Modern Algebra Simon RubinsteinSalzedo January 31, 2004 0.1 Introduction Professor: Adebisi Agboola. Oce Hours: Tuesday: 11:15-12:30, Wednesday: 11:15-12:30, Thursday: 11:15-12:30. Textbooks: Abstract Algebra by Dummit and Foote and Algebra by
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Course: Differential Geometry
Dierential Geometry Simon Rubinstein-Salzedo December 13, 2005 0.1 Introduction These notes are based on an undergraduate course in dierential geometry I took from Dr. Michael Crandall in the winter and spring of 2004. The primary textbook for that class
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Course: Modern Algebra
Math 220B: Modern Algebra Simon RubinsteinSalzedo March 11, 2004 0.1 Introduction Professor: Adebisi Agboola Oce Hours: Tuesday: 11:15-12:30, Wednesday: 11:15-12:30, Thursday: 11:15-12:30. 1 Chapter 1 Rings Denition 1.1 A ring (R, +, ) is an abelian group
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Course: Calculus On Manifolds
CCS Math 120: Calculus on Manifolds Simon RubinsteinSalzedo Spring 2004 0.1 Introduction These notes are based on a course on calculus on manifolds I took from Professor Martin Scharlemann in the Spring of 2004. The course was designed for rst-year CCS ma
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Course: Modern Algebra
Math 220C: Modern Algebra Simon RubinsteinSalzedo June 16, 2004 0.1 Introduction Professor: Adebisi Agboola Oce Hours: Tuesday: 11:15-12:30, Wednesday: 11:15-12:30, Thursday: 11:15-12:30. 1 Chapter 1 Preliminaries Question How can we tell whether a polyno
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Course: Modern Algebra
Group Theory Simon Rubinstein-Salzedo November 19, 2005 0.1 Introduction These notes are based on a graduate course in group theory I took from Dr. Ken Goodearl in the fall of 2004. The primary textbook for that class was Algebra by Larry Grove. Some mate
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Course: Game Theory
Econ 210B: Game Theory Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on game theory I took from Professor Rod Garratt in the Fall of 2005. The primary textbooks were Game Theory by Drew Fudenberg and Jean Ti
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Course: Differential Geometry
Math 240A: Dierentiable Manifolds and Riemannian Geometry Simon RubinsteinSalzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on dierentiable manifolds and Riemannian geometry I took from Professor Doug Moore in the Fall of 2005.
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Course: Noncumulative Noetherian Rings
Math 260Q: Noetherian Rings Simon Rubinstein-Salzedo Fall 2005 0.1 Introduction These notes are based on a graduate course on noetherian rings I took from Professor Ken Goodearl in the Fall of 2005. The textbook was An Introduction to Noncommutative Noeth
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Course: Algebraic Number Theory
Math 225AB: Algebraic Number Theory Simon RubinsteinSalzedo Winter and Spring 2006 0.1 Introduction Professor: Adebisi Agboola. Oce Hours: Tuesday: 11:15-12:30, Thursday: 11:15-12:30 (225A), Wednesday 10:0012:00 (225B). Textbooks: Algebraic Number Theory
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Course: Differential Geometry
Math 240B: Dierentiable Manifolds and Riemannian Geometry Simon RubinsteinSalzedo Winter 2006 0.1 Introduction These notes are based on a graduate course on dierentiable manifolds and Riemannian geometry I took from Professor Doug Moore in the Winter of 2
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Course: Differential Geometry
Math 240C: Dierentiable Manifolds and Riemannian Geometry Simon RubinsteinSalzedo Spring 2006 0.1 Introduction These notes are based on a graduate course on dierentiable manifolds and Riemannian geometry I took from Professor Doug Moore in the Spring of 2
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Course: Algebraic Number Theory
Math 225AB: Elliptic Curves Simon Rubinstein-Salzedo Winter and Spring 2007 0.1 Introduction These notes are based on a graduate course on elliptic curves I took from Professor Adebisi Agboola in the Winter and Spring of 2007. The textbooks were The Arith
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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
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Course: CACL WITH APPLI2
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Course: CACL WITH APPLI2
Ch e n 0 ( c 1 : 30 - :w b s So u 0p 3 : m 67 2 3 +Ba l l f l r \ ec t wA L : Se c ( n : L S 1 06 1 g o w n T b pm , M o s (x t do 50 / . V a ; p . _ 10 1 n oe l HJ ;j l n / o on < , F Hw i n a l Ot ( . e l SA . 10 / J bK l low e " F u ow n 0 - c Yu M
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Course: CACL WITH APPLI2
Vo l u n w . c+ u p D r la : " t\ s o lld ( u c t (o n > : l r [ Ol v i d( , d , lu v p ih,s J i j e b )c n l o Acfw_ cn , cc 3. h p et w e 1 x tj c . 5t n n li UN ab , l It + f A > dj - n ; t a j v a v ;y . - o V lu m e dh . c x I dv A(x ) b ( E J T j
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Course: Math 4A
Wolves & Rabbits Eigenvectors and Eigenvalues Finding Eigenvalues Finding Eigenvectors Eigenspaces Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 20, Feb. 28 2014 Based on the 2013 Millett and Scharlemann Lectures 1/21 Wolves & Rabbits
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Course: Math 4A
Last Time: Dot Product Distance Orthogonality Orthogonal sets Weights(coordinates) from orthogonal basis Orthonormal sets Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 23, March 7 2014 Based on the 2013 Millett and Scharlemann Lectures
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Course: Math 4A
Last Time: Eigenvalues and Eigenvectors Diagonalization Powers To the Matrix Diagonalization Again Triangular example Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 21, March 3 2014 Based on the 2013 Millett and Scharlemann Lectures 1/2
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Course: Math 4A
Last Time: Distance and Orthogonality Least Square Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 24, March 10 2014 Based on the 2013 Millett and Scharlemann Lectures 1/12 Last Time: Distance and Orthogonality Least Square Last Time: Di
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Course: Math 4A
Last Time: Diagonalization Thinking Complex Repeat! Dot product dened Dot product properties Vector length Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 22, March 5 2014 Based on the 2013 Millett and Scharlemann Lectures 1/14 Last Time
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Course: Math 4A
Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 19, Feb. 24 2014 Based on the 2013 Millett and Scharlemann Lectures 1/1 Last Time: Coordinates w.r.t. a Basis Suppose v1 , . . . vk is a basis for V. For any v V there is exactly one set c1
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Course: Math 4A
Last Time: Column Space, Image, Finding Basis Basis Again Coordinates Changing basis and coordinates in Rn Midterm Discuss Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 18, Feb. 21 2014 Based on the 2013 Millett and Scharlemann Lecture
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Course: Math 4A
Last Time: Null, Kernel, Span, Basis Column space of a matrix Image of linear transformation Back to Basis Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 17, Feb. 19 2014 Based on the 2013 Millett and Scharlemann Lectures 1/18 Last Time
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Course: Math 4A
Last Time: Vector Spaces And Subspaces Null-space of a matrix Kernel of linear transformation Linear Independence And Basis C Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 16, Feb. 14 2014 Based on the 2013 Millett and Scharlemann Lect
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Course: Math 4A
Last Time: Determinant via EROs More About Determinant The Geometry Going Abstract Dene & describe vector spaces Sub Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 14, Feb. 10 2014 Based on the 2013 Millett and Scharlemann Lectures 1/24
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Course: Math 4A
Last Time: Linear Independence Solutions of Homogeneous Systems Solutions of Non-homogeneous Systems Linear Transformatio Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 8, Jan. 24 2014 Based on the 2013 Millett and Scharlemann Lectures
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Course: Math 4A
Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 9, Jan. 27 2014 Based on the 2013 Millett and Scharlemann Lectures 1/1 Last Time: Linear Independence, Solution Sets In general, to see if a set of vectors cfw_a1 , . . . , an is linearly
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Course: Math 4A
Last Time: Vector Spaces Subspaces Examples and non-examples in Rn Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 15, Feb. 12 2014 Based on the 2013 Millett and Scharlemann Lectures 1/14 Last Time: Vector Spaces Subspaces Examples and n
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Course: Math 4A
Last Time: Invertible Matrix, Determinant Triangular And Almost Triangular Determinant And EROs Determinants via EROs Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 13, Feb. 7 2014 Based on the 2013 Millett and Scharlemann Lectures 1/22
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Course: Math 4A
Last Time: How to Invert Matrices The Core Theorem Tying Together Equivalent triple All tied together Determinants Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 12, Feb. 5 2014 Based on the 2013 Millett and Scharlemann Lectures 1/21 La
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Course: Math 4A
Last Time: Matrix of Linear Transformation One-to-One, Onto Matrix Algebra Invertible Matrices Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 10, Jan. 31 2014 Based on the 2013 Millett and Scharlemann Lectures 1/18 Last Time: Matrix of
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Course: Math 4A
Last Time: Matrix Inverse How to See Inverse The Core Theorem Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 11, Feb. 2014 Based on the 2013 Millett and Scharlemann Lectures 1/11 Last Time: Matrix Inverse How to See Inverse The Core The
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Course: Math 4A
Last Time: Span and Matrix-vector Multiplication Matrix Multiplication Back to SpanLinear Independence How to Determine (I Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 7, Jan. 22 2014 Based on the 2013 Millett and Scharlemann Lectures
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Course: Math 4A
Last Time: Echelon is Cool Vectors Picturing Vectors Properties of vectors Linear combination Span Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 5, Jan. 15 2014 Based on the 2013 Millett and Scharlemann Lectures 1/26 Last Time: Echelon
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Course: Math 4A
Last Time: Linear Combination And Span Matrix-vector multiplication Matrix equation Spanning the Whole Space Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 6, Jan. 17 2014 Based on the 2013 Millett and Scharlemann Lectures 1/20 Last Tim
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Course: Math 4A
Last Time: Solving System of Linear Equations Getting to Echelon Echelon examples (Reduced) echelon is cool Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 3, Jan. 10 2014 Based on the 2013 Millett and Scharlemann Lectures 1/16 Last Time
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Course: Math 4A
Last Time: Getting to Echelon (Reduced) echelon is cool Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 4, Jan. 13 2014 Based on the 2013 Millett and Scharlemann Lectures 1/7 Last Time: Getting to Echelon (Reduced) echelon is cool Gettin
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Course: Math 4A
Last Time: System of Linear Equations Solving a linear system Some examples Getting to echelon Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 2, Jan. 8 2014 Based on the 2013 Millett and Scharlemann Lectures 1/9 Last Time: System of Lin
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Course: Math 4A
Game Plan Whats Linear Algebra About? Linear Equations Systems of linear equations Linear Algebra with Applications Math 4A Xianzhe Dai UCSB Lecture 1, Jan. 6 2014 Based on the 2013 Millett and Scharlemann Lectures 1/14 Game Plan Whats Linear Algebra Abou
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School: UCSB
School: UCSB
School: UCSB
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School: UCSB
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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
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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: MATH 3A
Math 34A Practice Final 1) Use the graph of y=10x to find approximate values: a) 500.3 b) y(0.65) 2) Integrate: a) (x b b) x 5e 3 + 6 dx x ) 6kx dx a ln( 2) c) (e e )dt t t ln(0.5) 3) The population of a certain town is increasing at a rate of 5+2x pe
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Course: MATH 3A
U Math34APracticeFinalFall2010 1)Usethegraphofy=10xtofindapproximatevaluesof a)500.3 b)y(0.65) 2)Findvaluesofxwherethefunctionf(x)=x 44x3+4 a)isincreasing b)isconcavedown c)hasaminimumpoint 3)Afunctionf(x)isgraphedbelow. a)Forwhatvaluesofxisf(x)<0? b)Forw
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Course: MATH 3A
Math34A FinalReview PreparedbyVinceZaccone ForCampusLearningAssistance ServicesatUCSB 1)Usethegraphofy=10xtofindapproximatevaluesof a)500.3 b)y(0.65) solutionforparta) firstwriteanequation:x=500.3 dothelogarithmofbothsides:log(x)=log(50 0.3) expandtherigh
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Course: MATH 3A
Math 34A Practice Midterm #2 1) Compute the following: a) The average rate of change of f(x) = 2x3 5x from x=1 to x=3. b) The average rate of change of f(x) = x2-1 from x=2 to x=2+h. c) The intersection point of the curves f(x) = 3x+1 and g(x) = -2x+6. 2)
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y Practice Final Exam 1 m=1 m=0 m=1 dy y = + 3 x: dx x a) Sketch the direction eld for this DE, using (light or dotted) isoclines for the slopes -1 and 0. 1. For the DE See picture at right y Isoclines: y = x + 3x = m y = 3x2 m x. (Note problem at (0, 0).
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18.03SC Final Exam Solutions 1. (a) The isocline for slope 0 is the pair of straight lines y = x. The direction eld along these lines is at. The isocline for slope 2 is the hyperbola on the left and right of the straight lines. The direction eld along thi
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18.03SC Final Exam 1. This problem concerns the differential equation dy = x 2 y2 dx () Let y = f ( x ) be the solution with f (2) = 0. (a) Sketch the isoclines for slopes 2, 0, and 2, and sketch the direction eld along them. (c) On the same diagram, sket
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M A T H 3 4A Sp r i n g 1 4 Yu a n q i A pr i . Yo u r D ay P erm 2 . 4 35 \ / D Co p y i n NO On ly H o r t isc u s sio n ex p l ai n y ou r g so m eo n e 8 an s w e r a n d c h ec k y o u r s o l u t i o n s else s t e s t , o r an d d e l i b e r a t
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M A T H 3 4A Sp r i n g 1 4 Yu a n q i A pr i . Yo u r D ay P erm 2 . 4 35 \ / D C lea r l y Co p y i n NO On ly H o r t isc u s sio n ex p l ai n y ou r g so m eo n e 8 an s w e r a n d c h ec k y o u r s o l u t i o n s s t e s t , o r an d d e l i b e
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Course: Math 4A
Activity 6 Try to do each of the following problems with just your note card or no notes, as if it was the midterm. Question 1. Consider the following system of equations: xyz = 1 2x + 4y + z = a x 4y + bz = 3 1. Find all the values of a and b for which t
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Course: Math 4A
Mathematics 6A Winter 2013: Review for Midterm 2 February 20, 2013 Professor J. Douglas Moore Part I. Multiple Choice. There will be several multiple choice questions in which you need to circle the correct answer. These will be similar to i-Clicker quest
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Course: Math 4A
Mathematics 6A Winter 2013: Review for Final Exam March 15, 2013 Professor J. Douglas Moore Part I. Multiple Choice. There will be several multiple choice questions in which you need to circle the correct answer. Some examples can be found as i-Clicker qu
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MATH. 117 ANSWER KEY FOR SECOND MIDTERM FEB. 2014 1. [18 pts] Fill in complete and accurate denitions (not just notation) for each term. (Do not give the conditions from some theorem either, only the basic denition, as stated when the term was rst introdu
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MATH. 117 ANSWER KEY FOR FIRST MIDTERM FEB. 2014 1. [18 pts] Fill in complete and accurate denitions (not just notation) for each term. (Do not give the conditions from some theorem either, only the basic denition, as stated when the term was rst introduc
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Mathematics*108A*Midterm* * February*13,*2014* * Place*all*answers*in*your*BlueBook.*In*order*to*receive*complete*credit,*provide* complete*clearly*and*correctly*written*answers.*T/F*and*short*answers*must*be* include*short*explanations.* Section*I*T/R*an
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Math 108A Winter Quarter 2014 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 B
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Math 108A Winter Quarter 2014 Practice Problems for Midterm No notes or calculators are permitted on this exam. To obtain full credit you must show all work, provide complete proofs, and provide short but complete explanations for all questions in the Tru
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Math 8 Fall Quarter 2013 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 questio
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Course: Calculus With Applications I
MATH 3A - PRACTICE FIRST MIDTERM EXAM Answers Version A 1. Version B 4 5 1. 1 8 a) 1, 2 2. a = 1, b = 4 b) 0, 0 2. 3. 1 cos() 1, use squeeze thm. c) discontinuous at x = 0 3. lim f (x) = 0 = lim f (x) x0 4. x0+ 1 2 b) a) 5. f (0) = 2 and f (3) = 7, use I
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Course: Calculus With Applications I
MATH 3A - PRACTICE FIRST MIDTERM EXAM Spring 2009, Version B Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no exp
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Course: Calculus With Applications I
MATH 3A - PRACTICE SECOND MIDTERM EXAM Answers Version A 1. 2x 2x4 +5 y2 y 2. a) 21 m/s Version B 1. tan x tan y 2. b) 2 b) 5 seconds c) 4 3. 0.36 cos x a) | cos x| 3. 3x ln 3 3x+1 b) 4 1 x3 x 2 c) (3x + 2) ln x + x2 + 2 4. 4. 5. 1.25 m3 a) -1 3 m/min
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Course: Calculus With Applications I
MATH 3A - PRACTICE FIRST MIDTERM EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no exp
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Course: Calculus With Applications I
MATH 3A - PRACTICE FINAL EXAM Answers Version A 1. 2. ey + 3y cos x 3 sin x xey a) D = cfw_x = 0, x = 1 y = 0 hor. asympt, x = 1 vert. asympt. Version B 1. 2. 3. 2:12 pm 4. a) D = (, 1) (0, ) 1 y = 2 hor. asympt, no vert. asympt. b) x > 0 increasing, x <
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Course: Calculus With Applications I
MATH 3A - PRACTICE SECOND MIDTERM EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no ex
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Course: Calculus With Applications I
MATH 3A - PRACTICE FINAL EXAM Spring 2009, Version B Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no explanation
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Course: Calculus With Applications I
MATH 3A - PRACTICE FINAL EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no explanation
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Course: Calculus With Applications I
MATH 3A - PRACTICE SECOND MIDTERM EXAM Spring 2009, Version B Discussion time: Perm #: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no ex
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FINAL EXAM Spring 2009, Version B Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no explanatio
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FIRST MIDTERM EXAM Spring 2009, Version B Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no ex
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE SECOND MIDTERM EXAM Answers Version A 1. 2. Version B 5 ln(5/2) ln(3/2) 7 1. 200e 5 811 2 2 a) 2 cos x sin x 2. (x 1)2 ex 3 x +C c) cos 3 b) 3. t3 +C 3 1 t3 b) 3e 3 t + + 4 3 1 a) 3e 3 t + 4. x = 1, min; x = 1, max 5. ln 6 hours ln 81
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FIRST MIDTERM EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no ex
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FINAL EXAM Answers Version A 1. Version B a) 1.5, 1.8 1. z = 6x + 2y 1 b) 8.1 2. 3 a) x2 + e2x ln(2)x + C 2 b) 6x ln(x) + 5x + 9e3x+7 2. c) 2x3 + 14y c) 2(s + t) 6ts 3. a) y = ln b) 400 9 4 9 4 y 5. a) t2 + 3t 4 b) 10.5 m 3. 10 4. x =
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FIRST MIDTERM EXAM Answers Version A 1. 37 1 3 2. 4 3 b) 12 a) 102 t% b) after 100 days 16 4. 3 5. 1. 595 m 6 a) 21 b) 4 c) 1 + 6 x + 8x a) 2 c) 2x + 3 x 3. Version B 2. 1 hour 40 minutes 3. 25 2 4. between 62.6 and 69.2 liters 5. 3 ye
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE FINAL EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no explanatio
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Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE SECOND MIDTERM EXAM Spring 2009, Version B Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no e
School: UCSB
Course: Calculus For Social And Life Sciences
MATH 34B - PRACTICE SECOND MIDTERM EXAM Spring 2009, Version A Perm #: Discussion time: NAME: No Calculators. Solve each problem in the blue book. Number your solutions according to the corresponding problems. No points will be given for answers with no e
School: UCSB
Course: Vector Calculus I
Math 5B, Midterm 1 Review Problems - Solutions Fall 2006 1. Consider the two planes P1 , dened by the equation 2x y + z = 1, and P2 , given by the equation x + y z = 3. (a) Find parametric equations for the line that is the intersection P1 P2 of the plane
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Course: Vector Calculus I
Math 5B, Solutions to Final Review Problems Fall 2006 1 1 y 1. Integrate 0 sin x dx dy. x Solution. Since the integral sin x dx is too hard, we change the order of integration x so that we integrate with respect to y rst. This double integral is taken ove
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Course: Vector Calculus I
Math 5B, Midterm 2 Review Problems Fall 2006 1. (a) Convert the point (1, 1, 1) from rectangular to cylindrical coordinates. Solution. (It helps to draw pictures!) To convert to cylindrical coordinates, all 2 2 we need to do is change the x, y-coordinates
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Course: Vector Calculus I
Math 5B Midterm Exam Spring 2006 Your name: Your perm.: Your signature: Scores: 1. 2. 3. 4. 5. Total: (out of 100) Note: 12 extra credit points are included. 1 1. (22 points) 1)(15 points) Find the dierential dw of the function w = f (x, y, z) = x2 yz + x
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Course: Vector Calculus I
Math 5B - Midterm 1 Solutions 1. (a) Find parametric equations for the line that passes through the point (2, 0, 1) and is perpendicular to the plane with equation 4x y 2z = 1. Solution. The direction vector for this line is v = (4, 1, 2) and it must pass
School: UCSB
Course: Vector Calculus I
Math 5B, Midterm 2 Solutions Fall 2006 1. Suppose u = cos x + y and v = sin y x. Find x u v and y . v u (x,y) Solution. The Jacobian matrix (u,v) is simply the inverse of the Jacobian matrix sin x 1 (u,v) = . The determinant of this matrix is sin x cos y
School: UCSB
Course: Vector Calculus I
Math 5B, Midterm 2 Review Problems Fall 2006 1. (a) Convert the point (1, 1, 1) from rectangular to cylindrical coordinates. (b) Convert (2, /2, 2/3) from spherical to rectangular coordinates. 2. Suppose z and w are functions of x and y given by the equat
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Course: Vector Calculus II
Math 5C, Final Exam Review Problems Winter 2007 The Final Exam will cover material from the entire course, including several problems on vector calculus, and several on sequences and series. You should thus also review old homework problems from Chapters
School: UCSB
Course: Vector Calculus II
Name: , Perm No.2 Section Time : Math SC - Final Exam March 21, 2007 Instructions: 0 This exam consists of 9 problems, totaling 110 points, although it is out of 100 points, a You must show all your work and fully justify your answers in order to r
School: UCSB
Course: Vector Calculus II
Math 5C, Midterm 1 Review Problems Winter 2007 1. Compute C yz dx + 2x dy y dz where C is the straight line path from (1, 2, 1) to (1, 3, 0). 2. Find the surface area of the surface S, which is parametrized by x(u, v) = u v y(u, v) = u + v (u, v) = z(u, v
School: UCSB
Course: Vector Calculus II
Midterm 1 Solutions 1. [10 Points] Evaluate C ydx + ydy + xdz where C is the straight line path from (3, 1, 2) to (2, 1, 1). Does this integral depend on the path from (3, 1, 2) to (2, 1, 1)? Explain. Solution: Parametrize the curve by x = 3 t, y = 1 + 2t
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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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School: UCSB
MATH 122A HW 10 SOLUTIONS RAHUL SHAH Problem 1. [4.49.1(a)(b)(c)(f)] Apply the Cauchy-Goursat theorem to show that f (z) dz = 0 C when the contour C is the unit circle |z| = 1 in either direction, and when a. f (z) = b. z2 ; z3 f (z) = zez ; c. f (z) = 1
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School: UCSB
MATH 122A HW 1 SOLUTIONS RAHUL SHAH Problem 1. [1.2.1a] Verify that ( 2 ) (1 2) = 2. Solution. ( 2 ) (1 2) = = = 2 + 2 2 2 2 2 2. Problem 2. [1.2.2] Show that a. (z) = (z); b. (z) = (z). Solution. a. Let z = x + y. Then (z) = (x y) = y. Also, (z) = y. Th
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MATH 122A HW 2 SOLUTIONS RAHUL SHAH Problem 1. [1.5.1(b)(d)] Solution. b. Notice that z z = = z z z z = = 1. = 4 + 4 1 = 3 4. Thus z = z. d. (2 + )2 Problem 2. [1.5.2] Solution. a. Let S = cfw_z C | ( ) = 2. Let z S. Then ( ) = 2 and thus () = 2. Hence (z
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School: UCSB
MATH 122A HW 4 SOLUTIONS RAHUL SHAH Problem 1. [1.14.1] Find a domain in the z plane whose image under the transformation w = z 2 is the square domain bounded by u = 1, u = 2, v = 1 and v = 2. Solution. Notice that the preimage of the four given lines are
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MATH 122A HW 3 SOLUTIONS RAHUL SHAH Problem 1. [1.10.1(a)] Find the square roots of 2. Solution. Since + 2k 2 2 exp for k Z. Thus 1 (2) 2 = 2 exp + k 4 . Thus the two roots are c0 = 1 + and c1 = (1 + ). 1 Problem 2. [1.10.2(b)] Find all the roots of (8
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School: UCSB
MATH 122A HW 5 SOLUTIONS RAHUL SHAH Problem 1. [2.20.2] Show that a. a polynomial P (z) = a0 + a1 z + . . . + an z n an = 0 of degree n (n 1) is dierentiable everywhere, with derivative P (z) = a1 + 2a2 z + . . . + nan z n1 ; b. the coecients in the polyn
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Math 331.5: Homework 17 Solutions For the following problems, we use the dierential equation my + y + ky = g (t) where y is the displacement of the mass from its equilibrium position m is the mass is the damping coecient k is the spring constant g (t) is
School: UCSB
Course: Math 4A
Christian Searing Assignment hw5 due 11/10/2014 at 12:01pm PST Math4A-01-F14-BIGELOW 0 1 5. (1 pt) Let A = 0 2 . 0 4 Find dimensions of the kernel and image of A (or the linear transformation T (x) = Ax). , dim(Ker(A) = dim(Im(A) = . 1. (1 pt) Consider th
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Course: Math 4A
Christian Searing Assignment hw3 due 10/27/2014 at 10:22am PDT Math4A-01-F14-BIGELOW T (5u) = ( , T (5u + 6v) = ( ), T (6v) = ( , , T (v) = Correct Answers: 24 28 -7 (a*-8) (7*b) (-7*c) ) ) Correct Answers: T (u) = 1. (1 pt) Hello Let T : R2 R2 be a
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Course: Math 4A
Christian Searing Assignment hw7 due 12/01/2014 at 12:01pm PST Math4A-01-F14-BIGELOW 1. (1 pt) A square matrix is called a permutation matrix if it contains the entry 1 exactly once in each row and in each column, with all other entries being 0. All permu
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Course: Math 4A
Christian Searing Assignment hw7 due 12/01/2014 at 12:01pm PST Math4A-01-F14-BIGELOW 4. (1 pt) If 1. (1 pt) A square matrix is called a permutation matrix if it contains the entry 1 exactly once in each row and in each column, with all other entries being
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Course: Math 4A
Christian Searing Assignment hw5 due 11/10/2014 at 12:01pm PST Math4A-01-F14-BIGELOW \(\displaystyle\left.\begincfw_arraycfw_c \mboxcfw_1 \cr \mboxcfw_0 \cr \mboxcfw_2 \cr \endcfw_array\right.\) ,\(\displaystyle\left.\begincfw_arraycfw_c \mboxcfw_0 \cr \
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Course: Math 4A
Christian Searing Assignment hw3 due 10/27/2014 at 10:22am PDT Math4A-01-F14-BIGELOW 1. (1 pt) Hello Let T : R2 R2 be a linear transformation that sends the vector u = (5, 2) into (2, 1) and maps v = (1, 3) into (1, 3). Use properties of a linear transfor
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Course: Math 4A
Christian Searing Assignment hw2 due 10/20/2014 at 10:15pm PDT Math4A-01-F14-BIGELOW -4 -16 -23 1. (1 pt) Let v1 = 1 , v2 = 5 , v3 = 7 and 3 12 18 6 w = 0 . 8 1. Is w in cfw_v1 , v2 , v3 ? Type yes or no. 4. (1 pt) Let A be a 3x2 matrix. Suppose we know
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Course: Math 4A
Christian Searing Assignment hw8 due 12/08/2014 at 12:01pm PST Math4A-01-F14-BIGELOW YES NO 1. (1 pt) A is an n n matrix. -1 -2 4 4. (1 pt) The matrix A = -1 0 2 -1 -1 3 has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find th
School: UCSB
Course: Math 4A
Christian Searing Assignment hw8 due 12/08/2014 at 12:01pm PST Math4A-01-F14-BIGELOW -1 -2 4 4. (1 pt) The matrix A = -1 0 2 -1 -1 3 has two real eigenvalues, one of multiplicity 1 and one of multiplicity 2. Find the eigenvalues and a basis of each eigen
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Course: Math 4A
Christian Searing Assignment hw1 due 10/13/2014 at 07:25pm PDT Math4A-01-F14-BIGELOW 4. 1. (1 pt) Solve the system using matrices (row operations) 1 0 0 6x 9y= 39 9x 9y= 18 x= y= Correct Answers: 7 9 A. Unique solution: x = 0, y = 0, z = 0 B. Innitely
School: UCSB
Course: Math 4A
Christian Searing Assignment hw2 due 10/20/2014 at 10:15pm PDT Math4A-01-F14-BIGELOW one can solve the matrix equation Ax = c where the columns of A are -4 -16 -23 1. (1 pt) Let v1 = 1 , v2 = 5 , v3 = 7 and 3 12 18 6 w = 0 . 8 1. Is w in cfw_v1 , v2 , v
School: UCSB
Course: Math 4A
Christian Searing Assignment hw1 due 10/13/2014 at 07:25pm PDT Math4A-01-F14-BIGELOW 1. (1 pt) Solve the system using matrices (row operations) 6x 9y= 39 9x 9y= 18 x= B. No solutions C. Unique solution: x = 3, y = 4 D. Innitely many solutions E. Unique so
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Matthew Choi Assignment HW4 due 02/05/2015 at 11:59pm PST Math6B-01-W15-CHEN Determine whether the series converges, and if it converges, determine its value. 1. (1 pt) Use the limit comparison test to determine whether 6n3 6n2 + 16 an = 6 + 3n4 converge
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Matthew Choi Assignment HW2 due 01/22/2015 at 11:59pm PST Math6B-01-W15-CHEN 5. (1 pt) 1. (1 pt) n2 + sin(5n + 6) conn5 + 6 verges or diverges. If it converges, nd the limit. Converges (y/n): Limit (if it exists, blank otherwise): n1 11 21 Determine wheth
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Matthew Choi Assignment HW3 due 01/29/2015 at 11:59pm PST Math6B-01-W15-CHEN 4. (1 pt) The following series are geometric series or a sum of two geometric series. Determine whether each series converges or not. For the series which converge, enter the sum
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Matthew Choi Assignment HW2 due 01/22/2015 at 11:59pm PST Math6B-01-W15-CHEN n= (Note that because the validity of either of your answers depends on the other, if you enter only one, both will be marked wrong.) 1. (1 pt) n1 11 21 Determine whether the seq
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Matthew Choi Assignment HW3 due 01/29/2015 at 11:59pm PST Math6B-01-W15-CHEN 1. (1 pt) For each of the following, determine if it converges, and if so, determine what it converges to if you are able to do that using methods that we have learned in this co
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Matthew Choi Assignment HW4 due 02/05/2015 at 11:59pm PST Math6B-01-W15-CHEN 1. (1 pt) Use the limit comparison test to determine whether 6n3 6n2 + 16 an = 6 + 3n4 converges or diverges. n=16 n=16 1 (a) Choose a series bn with terms of the form bn = p n
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Matthew Choi Assignment HW1 due 01/15/2015 at 11:59pm PST Math6B-01-W15-CHEN C 1. (1 pt) Let C be the positively oriented circle x2 + y2 = 1. Use Greens Theorem to evaluate the line integral C 9y dx + 17x dy. x9 dx + 7x dy = 4. (1 pt) If curl F = (x2 + z2
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Matthew Choi Assignment HW1 due 01/15/2015 at 11:59pm PST Math6B-01-W15-CHEN 1. (1 pt) Let C be the positively oriented circle x2 + y2 = 1. Use Greens Theorem to evaluate the line integral C 9y dx + 17x dy. 4. (1 pt) If curl F = (x2 + z2 ) j + 2k, nd C F
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Matthew Choi Assignment HW5 due 02/12/2015 at 11:59pm PST Math6B-01-W15-CHEN 4. (1 pt) Determine whether the series is absolutely convergent, conditionally convergent, or divergent: 1. (1 pt) Determine whether the following series converges or diverges. c
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Matthew Choi Assignment HW5 due 02/12/2015 at 11:59pm PST Math6B-01-W15-CHEN 5. (1 pt) Match each of the following with the correct statement. A. The series is absolutely convergent. C. The series converges, but is not absolutely convergent. D. The series
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Matthew Choi Assignment HW6 due 02/19/2015 at 11:59pm PST Math6B-01-W15-CHEN 4. (1 pt) Find all the values of x such that the given series would converge. 1. (1 pt) Suppose that you are told that the Taylor series of 2 f (x) = x5 ex about x = 0 is x9 x11
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Matthew Choi Assignment HW6 due 02/19/2015 at 11:59pm PST Math6B-01-W15-CHEN c2 = 1. (1 pt) Suppose that you are told that the Taylor series of 2 f (x) = x5 ex about x = 0 is c3 = x9 x11 x13 + + + . 2! 3! 4! Find each of the following: c4 = x5 + x7 + x2 d
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Matthew Choi Assignment HW7 due 02/26/2015 at 11:59pm PST Math6B-01-W15-CHEN (c) Which function given below could the Fourier approxi1. (1 pt) (a) Suppose youre given the following Fourier mation F5 (x) model? ? 3 6 6 6 coefcients: a0 = , a1 = , a3 = , a5
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Matthew Choi Assignment HW7 due 02/26/2015 at 11:59pm PST Math6B-01-W15-CHEN 1. (1 pt) (a) Suppose youre given the following Fourier 3 6 6 6 coefcients: a0 = , a1 = , a3 = , a5 = , and 2 3 5 a2 , a4 , b1 , b2 , b3 , b4 , b5 are all zero. Find the followin
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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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School: UCSB
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School: UCSB
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School: UCSB
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School: UCSB
School: UCSB
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: 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
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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
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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
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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
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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
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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