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School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 3 c c From the textbook solve the problems 29, 30, 31, 32 and 33 from the Chapter 4. Solve the problems 1, 2, 4, 5 and 6 from the Chapter 4 additional exercises at http:/www.athenasc.com
School: UCLA
Sule Ozler Economics 12 Fall 2011 READ THE EXPLANATIONS IN BOLD LETTERS CAREFULLY! MID-TERM EXAM - GOOD LUCK! WRITE YOUR NAME AND ID# You will not be allowed to ask questions during the exam. This exam has three sections. Each section has multiple questio
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi c c Homework 1 From the textbook solve the problems 1, 2, 3, 5, 6, 8, 11 and 14 at the end of the Chapter 4. And also the problems below: Problem 1. Give examples of (not independent) random variab
School: UCLA
Spring 2011- Professor Sule Ozler Name Economics 121 ID# MID-TERM EXAM-II WRITE YOUR NAME AND ID# This exam has TWO sections. Each section has multiple questions. Sections I is 40 points, section II is for a total of 60 points. I. MULTIPLE CHOICE: Choose
School: UCLA
Course: Mathemtcl Modeling
Math 142-2, Homework 1 Solutions April 7, 2014 Problem 34.4 Suppose the growth rate of a certain species is not constant, but depends in a known way on the temperature of its environment. If the temperature is known as a function of time, derive an expres
School: UCLA
Spring 2011- Professor Sule Ozler Name Economics 121 ID# FINAL EXAM WRITE YOUR NAME AND ID# No questions will be answered during the exam There will not be any bathroom breaks during the exam You are permitted to have only your writing equipment on your d
School: UCLA
Chapter Review Exercises SOLUTION 811 The function whose value we want to approximate is f (x, y, z) = x 2 + y2 + z We use the linear approximation at the point (7, 5, 70), hence h = 7.1 - 7 = 0.1, k = 4.9 - 5 = -0.1, and l = 70.1 - 70 = 0.1. We g
School: UCLA
MULTIVARIABLE CALCULUS (MATH 32A) FALL 2004 LECTURE NOTES 10/1/04 Office Hours: MS 7344, 11:00 12:40 Mon., Wed., Fri. E-mail: lchayes@math.ucla.edu Homework: 20% (each problem graded on a 2-pt. scale). PRINT OUT HOMEWORK FROM COURSE WEBPAGE Midterm
School: UCLA
Course: Linear Algebra
115A-6, Syllabus and General informations Fall 2012 Prerequisite: Math 33A Text: Linear Algebra, by Friedberg, Insel, and Spence, with a supplement on Languages and Proofs, and Induction (custom edition for UCLA). A link to the errata of the book is provi
School: UCLA
Course: Probability Theory
Probability 170A/2 Winter 2015 information sheet January 6, 2015 Instructor: Email: Huy Tran. tvhuy@math.ucla.edu. Website: https:/sites.google.com/site/tranvohuy/teaching/m170a2w2015 Textbook: Introduction to Probability by D. P. Bertsekas and John N. Ts
School: UCLA
Course: Multivariable Calculus
Theorem Quiz Info Math 32B, Week 10 There will be a theorem quiz in your discussion section in week 10. The theorem quiz will cover the three major theorems of multivariable calculus that we have discussed so far: the Fundamental Theorem of Line Integrals
School: UCLA
Course: MULTIVARIABLE CALCULUS
Mathematica for Rogawski's Calculus 2nd Edition 2010 Based on Mathematica Version 7 Abdul Hassen, Gary Itzkowitz, Hieu D. Nguyen, Jay Schiffman W. H. Freeman and Company New York 2 Mathematica for Rogawski's Calculus 2nd Editiion.nb Copyright 2010 Mathem
School: UCLA
School: UCLA
School: UCLA
Course: Linear Algebra
Let F be a eld, and S = cfw_s1 , s2 , s3 be a set with exactly three elements. Consider the vector space F(S, F ) all functions from S to F with the standard function addition and scalar multiplication. Find a basis for F(S, F ) (and prove thats indeed a
School: UCLA
Course: Linear Algebra
Find a basis for V = cfw_M M2 (R)/tr(M ) = 0, a subspace of 2 2 real valued matrices. 1
School: UCLA
Course: SYSTMS-DIFFNTL EQTN
Section 2.4, Problem 27 We are asked to show whether the transformation y1 x + x2 =1 y2 x1 x2 y1 for which the solution is not unique y2 is invertible. As we saw in section, there are y = 1 1+0 0+1 = = 0 10 01 and there are y for which no solution exists
School: UCLA
Course: SYSTMS-DIFFNTL EQTN
Section 1.2, Problem 42 The key to this problem is the assumption that the vehicles leaving the area during the hour were exactly the same as those entering it. At the end of the hour, there are no cars left over, in any of the streets or intersections. T
School: UCLA
Course: SYSTMS-DIFFNTL EQTN
REVIEW FOR QUIZ 2 Econ 11, Sections 2A/2B, Week 4 Matt Miller This week we will review a few items that are likely to come up on the quiz on Thursday. I include some of the answers here, but the remainder (along with graphs) will be drawn up in section. 1
School: UCLA
Course: ANALYSIS
Math 131a Lecture 2 Spring 2009 Midterm 1 Name: Instructions: There are 4 problems. Make sure you are not missing any pages. Unless stated otherwise (or unless it trivializes the problem), you may use without proof anything proven in the sections of the b
School: UCLA
Mathematics 131A Fall 2007 FINAL Your Name: Signature: INSTRUCTIONS: This is a closed-book test. Do all work on the sheets provided. If you need more space for your solution, use the back of the sheets and leave a pointer for the grader. Good luc
School: UCLA
Course: Probability Theory
Midterm 3, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Printed name: Signed name: Student ID number: Instructions: Read problems very carefully. Please raise your hand if you have questions at any time. The correct nal answer alone is n
School: UCLA
Course: Probability Theory
Midterm 2, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Printed name: Signed name: Student ID number: Instructions: Read problems very carefully. Please raise your hand if you have questions at any time. The correct nal answer alone is n
School: UCLA
Course: Probability Theory
Midterm 1 practice, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Name and student ID: Question Points 1 10 2 10 3 10 4 10 5 10 Total: 50 Score 1. (a) (2 points) If is a sample space, what is P(). Solution: P() = 1 (b) (2 points) If P(A) =
School: UCLA
Math 3C Midterm 1 Solutions Name: BruinID: Section: Read the following information before starting the exam. This test has six questions and seven pages. It is your responsibility to ensure that you have all of the pages! Show all work, clearly and in o
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 3 c c From the textbook solve the problems 29, 30, 31, 32 and 33 from the Chapter 4. Solve the problems 1, 2, 4, 5 and 6 from the Chapter 4 additional exercises at http:/www.athenasc.com
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi c c Homework 1 From the textbook solve the problems 1, 2, 3, 5, 6, 8, 11 and 14 at the end of the Chapter 4. And also the problems below: Problem 1. Give examples of (not independent) random variab
School: UCLA
Course: Mathemtcl Modeling
Math 142-2, Homework 1 Solutions April 7, 2014 Problem 34.4 Suppose the growth rate of a certain species is not constant, but depends in a known way on the temperature of its environment. If the temperature is known as a function of time, derive an expres
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 4 c c solutions From the textbook solve the problems 42, 43 b) and 44 b), c) from the Chapter 4. Solve the problems 26, 27 and 29 from the Chapter 4 and problem 1 from the Chapter 7 addi
School: UCLA
Course: Introduction Of Complex Analysis
EE101 Engineering Electromagnetics Winter 2013 2/14/13 Homework #5 Due: Thursday Feb 21 10:00 AM Hand in to the TA at beginning of class. No late homework is accepted (see grading policy posted on eeweb). If you cannot make it to class, you must slip the
School: UCLA
Math32a HW1 Due on Monday, Jan 14 All the numbered problems are from the textbook Multivariable Calculus by Rogawski (2nd edition). Note: if your textbook is Early Transcendentals by Rogawski (2nd edition), then all your chapters are shifted by 1 down. Th
School: UCLA
Course: Calculus
31B Notes Sudesh Kalyanswamy 1 Exponential Functions (7.1) The following theorem pretty much summarizes section 7.1. Theorem 1.1. Rules for exponentials: (1) Exponential Rules (Algebra): (a) ax ay = ax+y (b) ax ay = x y axy (c) (a ) = axy (2) Derivatives
School: UCLA
School: UCLA
School: UCLA
The Poisson distribution is a discrete distribution. It is often used as a model for the number of events (such as the number of telephone calls at a business, number of customers in waiting lines, number of defects in a given surface area, airplane arriv
School: UCLA
3C Notes Sudesh Kalyanswamy 1 Counting (12.1) 1.1 Guiding Questions for the Chapter Here are some questions for you to ask yourself as you read. If you can answer them all by the end, youre in pretty good shape as far as understanding the material. After
School: UCLA
Course: Linear Algebra
Math 33A, Sec. 2 Linear Algebra and Applications Spring 2012 Instructor: Professor Alan J. Laub Dept. of Mathematics / Electrical Engineering Math Sciences 7945 phone: 310-825-4245 e-mail: laub@ucla.edu Lecture: MWF 2:00 p.m. 2:50 p.m.; Rolfe 1200 Office
School: UCLA
Course: Calculus
Sample Syllabus Philosophical Analysis of Contemporary Moral Issues Winter 2015, MW 8-9:50 Instructor: Robert Hughes (hughes@humnet.ucla.edu) Instructor office hours: Dodd 378, M 11-12 and W 10-11 or by appointment TAs: Kelsey Merrill (kmerrill@humnet.ucl
School: UCLA
Course: Math 170
Probability Theory, Math 170A, Winter 2015 - Course Info Instructor: Tonci Antunovic, 6156 Math Sciences Building, tantunovic@math.ucla.edu, www.math.ucla.edu/~tantunovic Instructor Oce Hours: Monday 10-12am and Wednesday 9-10am in 6156 Math Sciences Buil
School: UCLA
Course: Math 1
Math1 Fall2012 Lecture1(8am),Lecture2(10am)andLecture3(1pm) Instructor Paige Greene E-Mail: paige@math.ucla.edu Office Number: MS 6617 B Office Hours: Mon and Wed (copy from me in class) Course Goals: The purpose of Math 1 is to give you a strong preparat
School: UCLA
Math33BDifferentialEquations Winter2015,lecture3 Time:MWF3:003:50p.m. Room:MS4000A Discussionsections: 3A T3:00P3:50PGEOLOGY6704 3B R3:00P3:50PGEOLOGY4645 Instructor:YuZhang Office:MS6617F Officehours:Monday2:003:00p.m. Wednesday4:006:00p.m. Email:yuzhang
School: UCLA
Math 3B Integral calculus and differential equations Prof. M. Roper Young Hall CS 76, MWF 2-2.50pm. All inquiries about this class should be posted to the class Piazza page. The only emails that I will respond to are questions about how to login to Piazza
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 3 c c From the textbook solve the problems 29, 30, 31, 32 and 33 from the Chapter 4. Solve the problems 1, 2, 4, 5 and 6 from the Chapter 4 additional exercises at http:/www.athenasc.com
School: UCLA
Sule Ozler Economics 12 Fall 2011 READ THE EXPLANATIONS IN BOLD LETTERS CAREFULLY! MID-TERM EXAM - GOOD LUCK! WRITE YOUR NAME AND ID# You will not be allowed to ask questions during the exam. This exam has three sections. Each section has multiple questio
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi c c Homework 1 From the textbook solve the problems 1, 2, 3, 5, 6, 8, 11 and 14 at the end of the Chapter 4. And also the problems below: Problem 1. Give examples of (not independent) random variab
School: UCLA
Spring 2011- Professor Sule Ozler Name Economics 121 ID# MID-TERM EXAM-II WRITE YOUR NAME AND ID# This exam has TWO sections. Each section has multiple questions. Sections I is 40 points, section II is for a total of 60 points. I. MULTIPLE CHOICE: Choose
School: UCLA
Course: Mathemtcl Modeling
Math 142-2, Homework 1 Solutions April 7, 2014 Problem 34.4 Suppose the growth rate of a certain species is not constant, but depends in a known way on the temperature of its environment. If the temperature is known as a function of time, derive an expres
School: UCLA
Spring 2011- Professor Sule Ozler Name Economics 121 ID# FINAL EXAM WRITE YOUR NAME AND ID# No questions will be answered during the exam There will not be any bathroom breaks during the exam You are permitted to have only your writing equipment on your d
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 4 c c solutions From the textbook solve the problems 42, 43 b) and 44 b), c) from the Chapter 4. Solve the problems 26, 27 and 29 from the Chapter 4 and problem 1 from the Chapter 7 addi
School: UCLA
Course: ANALYSIS
Math 131a Lecture 2 Spring 2009 Midterm 1 Name: Instructions: There are 4 problems. Make sure you are not missing any pages. Unless stated otherwise (or unless it trivializes the problem), you may use without proof anything proven in the sections of the b
School: UCLA
Mathematics 131A Fall 2007 FINAL Your Name: Signature: INSTRUCTIONS: This is a closed-book test. Do all work on the sheets provided. If you need more space for your solution, use the back of the sheets and leave a pointer for the grader. Good luc
School: UCLA
Course: Probability Theory
Midterm 3, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Printed name: Signed name: Student ID number: Instructions: Read problems very carefully. Please raise your hand if you have questions at any time. The correct nal answer alone is n
School: UCLA
Course: Probability Theory
Midterm 2, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Printed name: Signed name: Student ID number: Instructions: Read problems very carefully. Please raise your hand if you have questions at any time. The correct nal answer alone is n
School: UCLA
Course: Probability Theory
Midterm 1 practice, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Name and student ID: Question Points 1 10 2 10 3 10 4 10 5 10 Total: 50 Score 1. (a) (2 points) If is a sample space, what is P(). Solution: P() = 1 (b) (2 points) If P(A) =
School: UCLA
Course: MATH20F
Hector Ordorica Section: B51 / B02 TA: Matthew Cecil Exercise 3.1 Answer: D can be: 1. D = 0 0 00 2. D = 1 0 01 3. D = 0 1 10 Exercise 3.2 (a) Input: A = [1, 2, 0; 2, 1, 2; 0, 2, 1] B = [3, 0, 3; 1, 5, 1; 1, 1, 3] x = [1; 2; 3] C = (5*A^2*B - 3*A')^2 Outp
School: UCLA
Introduction to Probability 2nd Edition Problem Solutions (last updated: 11/25/11) c Dimitri P. Bertsekas and John N. Tsitsiklis Massachusetts Institute of Technology WWW site for book information and orders http:/www.athenasc.com Athena Scientific, Belmo
School: UCLA
Math 3C Midterm 1 Solutions Name: BruinID: Section: Read the following information before starting the exam. This test has six questions and seven pages. It is your responsibility to ensure that you have all of the pages! Show all work, clearly and in o
School: UCLA
Winter 2011- Professor Sule Ozler Name Economics 121 ID# FINAL EXAM WRITE YOUR NAME AND ID# This exam has three sections. Each section has multiple questions. Sections I and II are 20 points each, section III is for a total of 60 points. I. MULTIPLE CHOIC
School: UCLA
Course: MATH20F
Hector Ordorica Section: Tues 2-3 TA: Greene Exercise 2.1 (a) (b) (c) Input: C = [5 2 1; 6 0 1; 4 -6 2] d = [-1; 2; 3] x = C\d Output: x=0.9091 -1.0455 -3.4545 (d) Input: C*x-d Output: C*x-d ans = 1.0e-015 * -0.4441 -0.4441 0.8882 Exercise 2.2 Input: C =
School: UCLA
Course: Actuarial Math
MATH 115A - Practice Final Exam Paul Skoufranis November 19, 2011 Instructions: This is a practice exam for the nal examination of MATH 115A that would be similar to the nal examination I would give if I were teaching the course. This test may or may not
School: UCLA
School: UCLA
Course: Introduction Of Complex Analysis
EE101 Engineering Electromagnetics Winter 2013 2/14/13 Homework #5 Due: Thursday Feb 21 10:00 AM Hand in to the TA at beginning of class. No late homework is accepted (see grading policy posted on eeweb). If you cannot make it to class, you must slip the
School: UCLA
Math32a HW1 Due on Monday, Jan 14 All the numbered problems are from the textbook Multivariable Calculus by Rogawski (2nd edition). Note: if your textbook is Early Transcendentals by Rogawski (2nd edition), then all your chapters are shifted by 1 down. Th
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 3 c c From the textbook solve the problems 29, 30, 31, 32 and 33 from the Chapter 4. Solve the problems 1, 2, 4, 5 and 6 from the Chapter 4 additional exercises at http:/www.athenasc.com
School: UCLA
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 6 c c Solve the problems 18 a) c) d) e) and 19 from the Chapter 7 additional exercises at http:/www.athenasc.com/prob-supp.html And also the problems below: Problem 1. Denote points P0 ,
School: UCLA
Course: Mathemtcl Modeling
Math 142 Homework 1 Solution 34.5 (a) If the growth rate is 0, the discrete model becomes Nm+1 Nm = tfm , And the solution is Nm = N0 + t(f0 + f1 + fm1 ), which is analogous to integration, since let t = mt, then t t(f0 + f1 + fm1 ) f (s)ds 0 for t small
School: UCLA
Course: Mathemtcl Modeling
Math 142 Homework 2 Solution 34.16 We will rstly proceed similarly as 34.14(c). We assume the theoretical growth has the form N0 eR0 t , and we hope to gure out which N0 and R0 t the data best. 34.14(b) suggests that N0 should be equal to the initial popu
School: UCLA
MATH 33A/2 PRACTICE MIDTERM 2 Problem 1. (True/False) . 1 1 1 Problem 2. Let v1 = 1 , v2 = 0 , v3 = 1 . 0 1 1 3 (a) Show that B = 1 , v2 , v3 ) is a basis for R . (v 1 (b) Let u = 1 . Find the coordinates of u with respect to the basis B. 2 (c) Find the c
School: UCLA
MATH 32A FINAL EXAMINATION March 21th, 2012 Please show your work. You will receive little or no credit for a correct answer to a problem which is not accompanied by sucient explanations. If you have a question about any particular problem, please raise y
School: UCLA
Math 131A/2 Winter 2002 Handout #1a Instructor: E. Eros, MS 6931. Lecture Meeting Time: MWF 2:00PM-2:50PM Location: MS 5117 Recitation TA: to be announced, MS 5117 T 2:00P-2:50P Oce hours (tentative): TF 4-5. Text Fundamental Ideas of Analysis by Michael
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 5 c c From the textbook solve the problems 4 and 5 from the Chapter 5. Solve the problems 2, 4, 6, 7, 9 and 10 from the Chapter 7 additional exercises at http:/www.athenasc.com/prob-supp
School: UCLA
Course: 245abc
Math 202A Homework 13 Roman Vaisberg November 28, 2007 Problem 17. Suppose f is dened on R2 as follows: f (x, y ) = an if n x < n + 1 and n y < n + 1, (n 0); f (x, y ) = an if n x < n + 1 and n + 1 y < n + 2, (n 0); while f (x, y ) = 0 elsewhere. Here an
School: UCLA
INSTRUCTORS SOLUTIONS MANUAL A BRIEF COURSE IN MATHEMATICAL STATISTICS Elliot A. Tanis Hope College and Robert V. Hogg University of Iowa August 22, 2006 ii Contents Preface v 1 Probability 1.1 Basic Concepts . . . . . 1.2 Methods of Enumeration 1.3 Condi
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 2 c c From the textbook solve the problems 17, 18, 19, 22, 23 and 24 from the Chapter 4. Solve the problems 21, 22, 24, 30 from the Chapter 4 additional exercises at http:/www.athenasc.c
School: UCLA
Math 3C Midterm 2 Name: BruinID: Section: Read the following information before starting the exam. This test has six questions and fourteen pages. It is your responsibility to ensure that you have all of the pages! Show all work, clearly and in order, i
School: UCLA
Homework 4 Solutions Math 3C Due: Friday Nov 2, 9 a.m. #1(a) Let X be a random variable that equals the number of defective items in the sample. When the sampling is done with replacement, X is binomially distributed with parameters n = 5 and p = 5/20 = 1
School: UCLA
Course: Probability Theory
Mathematics Department, UCLA P. Caputo Fall 08, midterm 1 Oct 22, 2008 Midterm 1: Math 170A Probability, Sec. 2 Last name First and Middle Names Signature UCLA ID number (if you are an extension student, say so) Please note: 1. Provide the information ask
School: UCLA
Course: Introduction Of Complex Analysis
EE101 Engineering Electromagnetics Winter 2013 1/24/12 Homework #3 Due: Thursday Jan 31 10:00 AM Hand in to the TA at beginning of class. No late homework is accepted (see grading policy posted on eeweb). If you cannot make it to class, you must slip the
School: UCLA
Course: 245abc
Math 245A Homework 2 Brett Hemenway October 19, 2005 Real Analysis Gerald B. Folland Chapter 1. 13. Every -nite measure is seminite. Let be a -nite measure on X, M. Since is -nite, there exist a sequence of sets cfw_En in M, with (En ) < and En = X . n=1
School: UCLA
Course: Mathemtcl Modeling
Math 142 Homework 0 Solution 32.11 The easier way: (slightly unrigorous, but its ne for this class) Suppose the bank compounds the interest n times a year, and let t := 1/n. With the additional deposit (or withdrawl) D (t), the balance at time t + t is gi
School: UCLA
FINS3635 S2/2011 Put-Call Parity Matthias Thul Last Update: September 14, 2011 This documents shows you how a the general put-call relationship for European options can be obtained by simple no-arbitrage arguments and gives some examples of how it can be
School: UCLA
Course: 245abc
Math 245A Homework 1 Brett Hemenway October 19, 2005 3. a. Let M be an innite -algebra on a set X Let Bx be the intersection of all E M with x E . Suppose y Bx , then By Bx because Bx is a set in M which contains y . But Bx By because if y E for any E M,
School: UCLA
MATH 32A FIRST MIDTERM EXAMINATION October, 18th, 20101 Please show your work. You will receive little or no credit for a correct answer to a problem which is not accompanied by sufficient explanations. If you have a question about any particular problem,
School: UCLA
Course: MATH20F
Hector Ordorica 1/16/2007 Tues 2-3 Exercise 1.1 Input: H=8;e=5;c=3;t=20;o=15;r=18; O=15;r=18;d=4;o=15;r=18;i=9;c=3;a=1; HectorOrdorica = H + e + c + t + o + r + O + r + d + o + r + i + c + a Output: HectorOrdorica = 152 Exercise 1.2 Input: z = 25-(100-7ex
School: UCLA
Course: Discrete Math
Practice First Test Mathematics 61 Disclaimer: Listed here is a selection of the many possible sorts of problems. actual test is apt to make a different selection. 1. Let S5 be the set of all binary strings of length 5 (such as 11001). The Let E be
School: UCLA
Math 31A Midterm 1 Solutions 1. [3 points] The derivative of (a) 1 (b) (c) x2 x)2 x3 Brent Nelson is: 1 x2 1 2 x3 (3x 1 1 + x2 + 8x2 x + 3) (d) None of the above. Solution. We rst apply the quotient rule d dx x2 x)2 x3 = d d x3 dx [(x2 x)2 ] (x2 x)2 dx (
School: UCLA
Course: Mathemtcl Modeling
Math 142 Homework 1 Solution 32.3 (a) Use N (nt) to denote the population at time nt. Then for any n > 0, we have N (nt) = (1 + (b d)t)N (n 1)t) + 1000. (b) Given that N (0) = N0 , we hope to solve the equation above and get the general form for N (nt). T
School: UCLA
Management 1A Summer 2004 Danny S. Litt EXAM 2 SOLUTION Name: _ Student ID No. _ I agree to have my grade posted by Student ID Number. _ (Signature) PROBLEM 1 2 3 4 5 6 7 8 TOTAL POINTS 20 25 20 20 30 30 30 25 200 SCORE MANAGEMENT 1A Problem 1
School: UCLA
Name: Student ID: Section: 2 Prof. Alan J. Laub Math 33A MIDTERM EXAMINATION I Spring 2008 Instructions: Apr. 25, 2008 (a) The exam is closed-book (except for one page of notes) and will last 50 minutes. (b) Notation will conform as closely as possible to
School: UCLA
1008 C H A P T E R 16 M U LTI P L E I N T E G R AT I O N (ET CHAPTER 15) Therefore, D is defined by the inequalities 1 u uv Since x = v+1 and y = v+1 , we have y 2, x 3 y+x 6 y = v+1 = v u x v+1 uv and y+x = u u(v + 1) uv + = =u v+1 v+1 v
School: UCLA
Course: 31b
Math 31B, Lecture 4, Fall 2011 Exam #2, 9 November 2011 Name: ID Number: Section and TA: You have 50 minutes for the exam. No calculators, phones, notes, or books allowed. You must show your work for credit. Hint: n n=0 cr Question: 1 2 3 4 5 6 7 Tota
School: UCLA
S E C T I O N 17.1 Vector Fields (ET Section 16.1) 1061 19. F = x, 0, z SOLUTION This vector field is shown in (A) (by process of elimination). x x 2 + y2 + z2 , y x 2 + y2 + z2 , z x 2 + y2 + z2 20. F = SOLUTION The unit radial vector field i
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 4 c c From the textbook solve the problems 42, 43 b) and 44 b), c) from the Chapter 4. Solve the problems 26, 27 and 29 from the Chapter 4 and problem 1 from the Chapter 7 additional exe
School: UCLA
Course: Actuarial Math
MATH 115A - Assignment One - Solutions of Select Non-Graded Problems Paul Skoufranis October 3, 2011 1.2 Question 13) Let V denote the set of ordered pairs of real numbers. If (a1 , a2 ) and (b1 , b2 ) are elements of V and c R, dene (a1 , a2 ) + (b1 , b
School: UCLA
Course: Linear Algebra
Name: Student ID: Prof. Alan J. Laub June 9, 2011 Math 33A/1 FINAL EXAMINATION Spring 2011 Instructions: (a) The exam is closed-book (except for one two-sided page of notes) and will last two and a half hours. No calculators, cell phones, or other electro
School: UCLA
Course: Mathemtcl Modeling
Math 142-2, Homework 2 Solutions April 7, 2014 Problem 35.3 Consider a species in which both no individuals live to three years old and only one-year olds reproduce. (a) Show that b0 = 0, b2 = 0, d2 = 1 satisfy both conditions. All two year olds die befor
School: UCLA
Chapter Review Exercises SOLUTION 811 The function whose value we want to approximate is f (x, y, z) = x 2 + y2 + z We use the linear approximation at the point (7, 5, 70), hence h = 7.1 - 7 = 0.1, k = 4.9 - 5 = -0.1, and l = 70.1 - 70 = 0.1. We g
School: UCLA
MULTIVARIABLE CALCULUS (MATH 32A) FALL 2004 LECTURE NOTES 10/1/04 Office Hours: MS 7344, 11:00 12:40 Mon., Wed., Fri. E-mail: lchayes@math.ucla.edu Homework: 20% (each problem graded on a 2-pt. scale). PRINT OUT HOMEWORK FROM COURSE WEBPAGE Midterm
School: UCLA
Course: Linear Algebra
115A-6, Syllabus and General informations Fall 2012 Prerequisite: Math 33A Text: Linear Algebra, by Friedberg, Insel, and Spence, with a supplement on Languages and Proofs, and Induction (custom edition for UCLA). A link to the errata of the book is provi
School: UCLA
Course: Probability Theory
Probability 170A/2 Winter 2015 information sheet January 6, 2015 Instructor: Email: Huy Tran. tvhuy@math.ucla.edu. Website: https:/sites.google.com/site/tranvohuy/teaching/m170a2w2015 Textbook: Introduction to Probability by D. P. Bertsekas and John N. Ts
School: UCLA
Course: Multivariable Calculus
Theorem Quiz Info Math 32B, Week 10 There will be a theorem quiz in your discussion section in week 10. The theorem quiz will cover the three major theorems of multivariable calculus that we have discussed so far: the Fundamental Theorem of Line Integrals
School: UCLA
Course: Multivariable Calculus
Info and Study Suggestions for Final Exam Math 32B, Fall 2012 The nal exam will be cumulative. It will cover Chapters 15 and 16 of the textbook, except Section 15.5. There will be somewhat more emphasis on material that did not appear on the midterm exams
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Course: Multivariable Calculus
Sample Questions for Final Exam Math 32B, Fall 2012 These problems are intended to show you the sorts of problems that might appear on the exam and the approximate level of diculty you can expect. Some parts of the actual exam will probably be easier than
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Course: Introduction Of Complex Analysis
Homework I part 1 2 2 2 1.(10 pts) (a). Show that if z1 +z2 +z3 = z1 z2 +z1 z3 +z2 z3 and w1 = z2 z1 and w2 = z3 z1 , 2 2 then w1 + w2 = w1 w2 . 2 2 (b). Suppose w1 + w2 = w1 w2 and w1 = w2 , w1 = 0, w2 = 0. Show that 0, w1 , w2 form the vertices of an eq
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Course: DIFFERENTIAL EQUATIONS
Hour Exam I Mathematics 33B 1. Find the solution to dy dt 2 + 2ty = et with y(0) = 3. 2 Solution: This was a pretty easy one. The integrating factor is et note that 2 2 d et ( dy + 2ty) = dt (et y) so you get dt 2 2 d t2 (e y) = et et = 1. dt 2 Taking the
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Multivariable Calculus Oliver Knill Math 21a, Fall 2011 These notes contain condensed two pages per lecture notes with essential information only. Remaining space was lled with problems. Harvard Multivariable Calculus Math 21a, Fall 2011 Math 21a: Multiva
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Heraclitus the Paradoxographer: Peri Apiston, On Unbelieveable Tales Stern, Jacob, 1940- Transactions of the American Philological Association, Volume 133, Number 1, Spring 2003, pp. 51-97 (Article) Published by The Johns Hopkins University Press DOI: 10.
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Math32a Fall 2012 Midterm1 Summary R.Kozhan The material for the midterm includes (from Rogawskis Multivariable Calculus (2nd ed): Sections 13.113.5; Sections 12.112.2; Sections 14.114.4. Bring your ID card to the exam. No calculators, no books, no not
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Math32a R. Kozhan Midterm2 Summary Midterm2 will be focused on the sections listed below, and will not explicitly test the knowledge of the material included for Midterm1. However the student is assumed to know it and be able to use it whenever needed. Th
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4: EIGENVALUES, EIGENVECTORS, DIAGONALIZATION STEVEN HEILMAN Contents 1. 2. 3. 4. 5. 6. Review Diagonal Matrices Eigenvectors and Eigenvalues Characteristic Polynomial Diagonalizability Appendix: Notation 1 1 2 4 6 7 1. Review Lemma 1.1. Let V be a nite-d
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1: INTRODUCTION, FIELDS, VECTOR SPACES, BASES STEVEN HEILMAN Abstract. These notes are mostly copied from those of T. Tao from 2002, available here Contents 1. 3. 4. 5. 6. 7. 8. Introductory Remarks Fields and Vector Spaces Three Fundamental Motivations f
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3: ROW OPERATIONS, THE DETERMINANT STEVEN HEILMAN Contents 1. 2. 3. 4. 5. Review Row Operations Rank of a Matrix The Determinant Appendix: Notation 1 1 3 8 10 1. Review Theorem 1.1 (Dimension Theorem/ Rank-Nullity Theorem). Let V, W be vector spaces over
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2: LINEAR TRANSFORMATIONS AND MATRICES STEVEN HEILMAN Contents 1. 2. 3. 4. 5. 6. 7. 8. Review Linear Transformations Null spaces, range, coordinate bases Linear Transformations and Bases Matrix Representation, Matrix Multiplication Invertibility, Isomorph
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1 Overview Economics is the problem of the allocation of scarce resources. Microeconomics seeks to address the problem through the action of individuals. The basic assumption of Microeconomics is that individuals behave rationally. In this context, ration
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Homework 6 Partial Solutions Math 170A/1, Summer 2012 Other exy 0 1. Suppose that X, Y have joint PDF fX,Y (x, y) = x > 0, y > 0 . otherwise (a) Check that this is actually a joint PDF. (b) Find P(X > 1, Y < 1). (c) Find the marginal CDF and PDF of X. (d)
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Homework 6 Due: August 2 Math 170A/1, Summer 2012 Textbook Chapter 3: # 15, 18, 19, 21, 23, 25, 34 Other 1. Suppose that X, Y have joint PDF fX,Y (x, y) = exy 0 x > 0, y > 0 . otherwise (a) Check that this is actually a joint PDF. (b) Find P(X > 1, Y < 1)
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Homework 5 Partial Solutions Math 170A/1, Summer 2012 Other 1. Let X be a random variable with PDF given by fX (x) = c(1 x2 ) if 1 < x < 1 . 0 otherwise (a) What is the value of c? (b) Compute the CDF of X. (c) Compute E[X]. [c = 3/4; FX (x) = 1 2 3 + 4 x
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Homework 4 Partial Solutions Math 170A/1, Summer 2012 Other 1. Let X be a geometric random variable with parameter p. That is, P(X = k) = (1 p)k1 p, for k = 1, 2, . . . 1 1p and var(X) = . [Hint: treat x = 1 p as a variable, and p p2 interpret k(1 p)k1 =
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Homework 5 Due: July 26 Math 170A/1, Summer 2012 Textbook Chapter 3: # 1, 2, 5, 6, 7, 8, 11, 12, 13 Other 1. Let X be a random variable with PDF given by fX (x) = c(1 x2 ) if 1 < x < 1 . 0 otherwise (a) What is the value of c? (b) Compute the CDF of X. (c
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Homework 3 Partial Solutions Math 170A/1, Summer 2012 Other 1. An urn contains balls numbered 1 through N . Select n N of them, and let X be the largest number on any ball that was drawn. Find the probabilty mass function of X. [P(X = k) = k1 n1 / N n for
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Homework 2 Partial Solutions Math 170A/1, Summer 2012 Other 1. If two fair dice are rolled, what is the conditional probability the the rst one lands on 6, given that the sum of the dice is k? Compute the probability for k = 2, . . . , 12. [0, 0, 0, 0, 0,
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Homework 1 Partial Solutions Math 170A/1, Summer 2012 Here are solutions to the Other problems. Note that solutions to the Textbook problems may be found on the Textbooks website. See me or the TA for solutions to the Supplementary problems. Other 1. Befo
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Homework 1 Due: June 28, 2012 Math 170A/1, Summer 2012 Homework sets will generally consist of problems from the textbook, which appear at the end of each chapter. There are also supplementary problems that appear on the website for the textbook: http:/ww
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Homework 2 Due: July 5, 2012 Math 170A/1, Summer 2012 Textbook Chapter 1: # 14, 19, 24, 27, 30, 33, 36, 39 Supplementary Chapter 1: # 13, 18, 26, 28, 29 Other 1. If two fair dice are rolled, what is the conditional probability the the rst one lands on 6,
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Homework 3 Due: July 12 Math 170A/1, Summer 2012 Textbook Chapter 2: # 2, 4, 8, 9, 10, 14, 15, 17, 20, 22 Supplementary Chapter 2: # 1, 4, 5 Other 1. An urn contains balls numbered 1 through N . Select n N of them, and let X be the largest number on any b
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18 General Equilibrium We dene a perfectly competitive equilibrium to consist of: prices for all goods production choices for all rms consumption choices for all individuals such that: 1. rms/industries maximize prots 2. consumers maximize utility subj
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Course: INTEGRATION AND INFINITE SERIES
MATH 31B REVIEW SHEET FINAL EXAM Make sure to check CCLE for additional materials. Any handouts Prof. Aschenbrenner has given and any practice exercises will also be fair game for the exam. Combine this review sheet with the review sheets for Midterms 1 &
School: UCLA
Course: INTEGRATION AND INFINITE SERIES
MATH 31B REVIEW SHEET MIDTERM 2 Make sure to check CCLE for additional materials. Any handouts Prof. Aschenbrenner has given and any practice exercises will also be fair game for the exam. All page numbers etc. are from Rogawski. 8.2: HW: 2, 8, 10, 12, 30
School: UCLA
Course: Calculus
MATH 31B REVIEW SHEET MIDTERM 1 Make sure to check CCLE for additional materials. Any handouts Prof. Aschenbrenner has given and any practice exercises will also be fair game for the exam. All page numbers etc. are from Rogawski. 7.1: HW: 16, 20, 22, 21,
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Course: INTRODUCTION TO TOPOLOGY
Renzos Math 490 Introduction to Topology Tom Babinec Chris Best Michael Bliss Nikolai Brendler Eric Fu Adriane Fung Tyler Klein Alex Larson Topcue Lee John Madonna Joel Mousseau Nick Posavetz Matt Rosenberg Danielle Rogers Andrew Sardone Justin Shaler Smr
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Course: INTERMEDIATE CALCULUS
Answer Key for Review Packet for Exam #3 Math 12-D. Benedetto Interval of Convergence: Find the interval and radius of convergence for each of the following power series. Analyze convergence at the endpoints carefully, with full justication. 1. n=1 2. n=1
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Course: INTERMEDIATE CALCULUS
Review Packet for Exam #2-Second Part Math 12D. Benedetto Series: Find the sum for each of the following series (all of which converge): 2 Ccvrs 1j 96. -, - - 4 I - 97. = i; (7j 5- LJ , Ly 0O 1i / V 1 n1 2 )fl+1 1 3n+1 3 (-I /2 / / - /t 3 I / vi 100. _ fl
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Course: INTERMEDIATE CALCULUS
Review Packet for Exam #3 Math 121-D. Benedetto Interval of Convergence: Find the interval and radius of convergence for each of the following power series. Analyze convergence at the endpoints carefully, with full justication. 1. n=1 2. n=1 3. n=1 4. n=0
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Course: CALC III FOR ENGINEER STUDENTS
3.1 Dot Products The dot product of two vectors u and v is a number and it is denoted by u . v. First consider the case where the two vectors u and v are specified by a magnitude and direction. In that case the dot product is defined by u . v = (magnitude
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Course: CALC III FOR ENGINEER STUDENTS
3.2 Cross Products The cross product is only defined for vectors in three dimensions. For three dimensional vectors u and v that are specified by a magnitude and direction, the cross product uv is the vector defined as follows. 1. uv points in the directi
School: UCLA
Course: CALCULUS 4
Review Exam 3. Sections 6.1-6.6. 5 or 6 problems. 50 minutes. Laplace Transform table included. Exam: November 11, 2008. Problem 2 Example Use Laplace Transform to nd y solution of y 2y + 2y = (t 2), y (0) = 1, y (0) = 3. Solution: Compute the LT of the e
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Course: CALCULUS 4
Review of Linear Algebra (Sect. 7.2) The dot product of n-vectors. The matrix-vector product. A matrix is a function. The inverse of a square matrix. The determinant of a square matrix. n n systems of linear algebraic equations. The dot product of n-vecto
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Course: CALCULUS 4
Review for Exam 2. 5 or 6 problems. No multiple choice questions. No notes, no books, no calculators. Problems similar to homeworks. Exam covers: Regular-singular points (5.5). Euler dierential equation (5.4). Power series solutions (5.2). Variation of pa
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Course: MULTIVARIABLE CALCULUS
Mathematica for Rogawski's Calculus 2nd Edition 2010 Based on Mathematica Version 7 Abdul Hassen, Gary Itzkowitz, Hieu D. Nguyen, Jay Schiffman W. H. Freeman and Company New York 2 Mathematica for Rogawski's Calculus 2nd Editiion.nb Copyright 2010 Mathem
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School: UCLA
Course: Linear Algebra
Let F be a eld, and S = cfw_s1 , s2 , s3 be a set with exactly three elements. Consider the vector space F(S, F ) all functions from S to F with the standard function addition and scalar multiplication. Find a basis for F(S, F ) (and prove thats indeed a
School: UCLA
Course: Linear Algebra
Find a basis for V = cfw_M M2 (R)/tr(M ) = 0, a subspace of 2 2 real valued matrices. 1
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Course: SYSTMS-DIFFNTL EQTN
Section 2.4, Problem 27 We are asked to show whether the transformation y1 x + x2 =1 y2 x1 x2 y1 for which the solution is not unique y2 is invertible. As we saw in section, there are y = 1 1+0 0+1 = = 0 10 01 and there are y for which no solution exists
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Course: SYSTMS-DIFFNTL EQTN
Section 1.2, Problem 42 The key to this problem is the assumption that the vehicles leaving the area during the hour were exactly the same as those entering it. At the end of the hour, there are no cars left over, in any of the streets or intersections. T
School: UCLA
Course: SYSTMS-DIFFNTL EQTN
REVIEW FOR QUIZ 2 Econ 11, Sections 2A/2B, Week 4 Matt Miller This week we will review a few items that are likely to come up on the quiz on Thursday. I include some of the answers here, but the remainder (along with graphs) will be drawn up in section. 1
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Course: SYSTMS-DIFFNTL EQTN
Solutions to Linear Algebra, Fourth Edition, Stephen H. Friedberg, Arnold J. Insel, Lawrence E. Spence Jephian Lin, Shia Su, Zazastone Lai July 27, 2011 Copyright 2011 Chin-Hung Lin. Permission is granted to copy, distribute and/or modify this document un
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Course: SYSTMS-DIFFNTL EQTN
Linear Algebra Done Right, Second Edition Sheldon Axler Springer Contents Preface to the Instructor Preface to the Student Acknowledgments Chapter 1 ix xiii xv Vector Spaces Complex Numbers . . . . . Definition of Vector Space . Properties of Vector Space
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Course: SYSTMS-DIFFNTL EQTN
This page intentionally left blank CALCULUS SECOND EDITION Publisher: Ruth Baruth Senior Acquisitions Editor: Terri Ward Development Editor: Tony Palermino Development Editor: Julie Z. Lindstrom Associate Editor: Katrina Wilhelm Editorial Assistant: Tyler
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Course: SYSTMS-DIFFNTL EQTN
SUGGESTED REVIEW FOR QUIZ 2 Econ 11, Sections 2A/2B, Week 4 Material for Review Based on what we have covered in lecture, and what it is likely we will cover next week in lecture, the following set of readings/review should be very useful. Keep in mind th
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Course: SYSTMS-DIFFNTL EQTN
Solutions Chapter 3 October 23, 2013 3.3) M RS = M Ux M Uy = U/x U/y a) y M RS = x M Ux = y Marginal Utility of x is constant in x b) 2 2xy M RS = 2x2 y = M Ux = 2xy 2 y x Marginal Utility of x is increasing in x c) y M RS = 1/x = x 1/y M Ux = 1/x Margina
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Course: INTRODUCTION TO TOPOLOGY
LECTURE PLAN FOR MATH 472 INTRO TO TOPOLOGY Fall 2010 1 Aug 23 Announcements 1. Exams 2. Hwk 3. Oce times/talk to each other rst 4. Feedback Draw pictures about things that are equal or dierent in dierent geometries. Explore: euclidean 3d (LENGTH, ANGLES
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COUNTING RANDOM VARIABLES EX = n P(X = n) + . Var x = E(X2) (EX)2 DISTRIBUTIONS CONTINUOUS NORMAL UNIFORM EXPONENTIAL CLT Binomial for with replacement. Above is without replacement = hyper. Ex of bi and hyper is n x (k/n) which is np
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School: UCLA
School: UCLA
SECTION: Penultimate Math 3A, Section 4, Fall 2013 Not December 13, 2013 NAME/ID: I have read and understood the Student Honor Code, and this exam reects my unwavering commitment to the principles of academic integrity and honesty expressed therein. SIGNA
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SECTION: Not Midterm 1 Math 3A, Section 4, Spring 2013 NAME/ID: I have read and understood the Student Honor Code, and this exam reects my unwavering commitment to the principles of academic integrity and honesty expressed therein. SIGNATURE/ID: SCORES: 1
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SECTION: Not Midterm 2 Math 3A, Section 4, Not Spring 2013 Not November 20, 2013 NAME/ID: I have read and understood the Student Honor Code, and this exam reects my unwavering commitment to the principles of academic integrity and honesty expressed therei
School: UCLA
School: UCLA
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rS [0 .- ( I 2. i A)(; '" : O.D ~ -\: ./" ~ p>'- -"'(),c)H~t D" TUM ;:. W~ VCu' ('/.t'cfw_):;.- VWcfw_ [f)~ V(t.+l'C) 'f 2 C:r,; ['i,'-() :. 0- o~ VtM-iJ,) v'cu ~r) -(9< 12 ; (9 ,0 g ~ \T-l 1 (\0. "1 ) CO" ') Q,O'f yo V. "1 - ~"'f) 0, ~ J '" . 1,"'1,. 1,-
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School: UCLA
Course: Probability For Life Sciences Students
Discrete Random Variables and Joint Distribution Sections 12.4.12 Math 3C/3 UCLA George J. Schaeer May 2nd, 2014 Probability functions Remember that a probability function is a function P with the following properties: The domain (set of inputs) is the se
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Course: Probability For Life Sciences Students
The Binomial Distribution and Population Sampling Section 12.4.3 Math 3C/3 UCLA George J. Schaeer May 5th, 2014 Analyzing (discrete) random variables The following are important in the analysis of a DRV: The correct context or application for the random v
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Course: INTEGRATION AND INFINITE SERIES
Math 31B. Spring 14. WEEK 2 SUMMARY. Remark 1. The hyperbolic trigonometric functions are sinh() := 1 2 e e , cosh() = 1 2 e + e . They are to the hyperbola x2 y 2 = 1 what the usual trinogometric functions sin(), cos() are to the circle x2 + y 2 = 1, in
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Course: INTEGRATION AND INFINITE SERIES
Math 31B. Spring 14. WEEK 7. Part 1. Taylors theorem: It gives a way of approxmating a function by a polynomial 1 1 1 (N ) a) + f 00 (a)(x a)2 + . . . + f (k) (a)(x a)k + . . . + f (a)(x a)N + RN (f, x, a) 2 k! N! Here f (k) (a) represents the value at a
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Course: INTEGRATION AND INFINITE SERIES
Math 31B. Spring 14. WEEK 8. Convergence tests. We continue our study of methods to decide the convergence or divergence of a series 1 X an n=1 We already learned about the comparison and integral tests. Now we will discuss the divergence and Leibnitz tes
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Course: INTEGRATION AND INFINITE SERIES
Math 31B. Spring 14. WEEK 1 SUMMARY. Remark 1. We introduced the natural logarithm, arising as a denite integral of the function x1 , x 1 log(x) := dt. 1 t (by the way, the symbol := should be read as we are dening the thing on the left to represent the t
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33A PRACTICE EXAM SOLUTIONS SPRING 2011 A. ADAM AZZAM The amount of detail contained herein is meant to provide the diligent student with a means to check their answers. By no means do I guarantee the detail presented would merit a 100% on the nal exam, n
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Course: Linear Algebra
Linear Algebra and Its Applications Fourth Edition Gilbert Strang x y z Ax y Ay b b b 0 0 z Az 0 Contents Preface iv 1 . . . . . . . . 1 1 4 13 21 36 50 66 72 . . . . . . . 77 77 86 103 115 129 140 154 . . . . . . 159 159 171 180 195 211 221 2 3 Matrices
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Course: Linear Algebra
INTRODUCTION TO LINEAR ALGEBRA Fourth Edition MANUAL FOR INSTRUCTORS Gilbert Strang Massachusetts Institute of Technology math.mit.edu/linearalgebra web.mit.edu/18.06 video lectures: ocw.mit.edu math.mit.edu/gs www.wellesleycambridge.com email: gs@math.mi
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Math 104: Introduction to Analysis SOLUTIONS Alexander Givental HOMEWORK 1 1 1.1. Prove that 12 + 22 + + n2 = 6 n(n + 1)(2n + 1) for all n N. Put f (n) = n(n + 1)(2n + 1)/6. Then f (1) = 1, i.e the theorem holds true for n = 1. To prove the theorem, it su
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Math 312, Intro. to Real Analysis: Homework #6 Solutions Stephen G. Simpson Friday, April 10, 2009 The assignment consists of Exercises 17.3(a,b,c,f), 17.4, 17.9(c,d), 17.10(a,b), 17.14, 18.5, 18.7, 19.1, 19.2(b,c), 19.5 in the Ross textbook. Each exercis
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Ordered Fields and the Triangle Inequality Will Rosenbaum Updated: April 9, 2013 Department of Mathematics University of California, Los Angeles In this note, we prove the triangle inequality for ordered fields. Before going into the statement and proof o
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Math 312, Intro. to Real Analysis: Homework #5 Solutions Stephen G. Simpson Friday, March 20, 2009 The assignment consists of Exercises 14.3, 14.4, 14.6, 14.13, 15.3, 15.4, 15.7 in the Ross textbook. Each problem counts 10 points. In solving some of these
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1 |f | and f 2 are integrable when f is integrable Lemma 1.1. Let f : [a, b] R be a bounded function and let P = cfw_x0 , x1 , . . . , xn be a partition of [a, b]. Then for each i cfw_1, 2, . . . , n, Mi (f ) mi (f ) = supcfw_|f (x) f (y)| : x, y [xi1 ,
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Math 310 - hw 8 solutions Friday, 6. Nov 2009 18.4, 18.6, 18.10; 18.4 Let S R and suppose there is a sequence (xn )n in S that converges to a number x0 S . Show that there exists / an unbounded continuous function on S . 1 for x S. x x0 Then f is continuo
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CZ2105 Numerical Methods I, Tutorial 2 Department of Computational Science National University of Singapore TA: Dr. Zhao Yibao 1. For the given function f (x), let x0 = 0, x1 = 0.6, and x2 = 0.9. Construct the Lagrange interpolation polynomial of degree (
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Course: ANALYSIS
Math 131a Lecture 2 Spring 2009 Midterm 1 Name: Instructions: There are 4 problems. Make sure you are not missing any pages. Unless stated otherwise (or unless it trivializes the problem), you may use without proof anything proven in the sections of the b
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Mathematics 131A Fall 2007 FINAL Your Name: Signature: INSTRUCTIONS: This is a closed-book test. Do all work on the sheets provided. If you need more space for your solution, use the back of the sheets and leave a pointer for the grader. Good luc
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Course: Probability Theory
Midterm 3, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Printed name: Signed name: Student ID number: Instructions: Read problems very carefully. Please raise your hand if you have questions at any time. The correct nal answer alone is n
School: UCLA
Course: Probability Theory
Midterm 2, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Printed name: Signed name: Student ID number: Instructions: Read problems very carefully. Please raise your hand if you have questions at any time. The correct nal answer alone is n
School: UCLA
Course: Probability Theory
Midterm 1 practice, Math 170b - Lec 1, Winter 2013 Instructor: Toni Antunovi c c Name and student ID: Question Points 1 10 2 10 3 10 4 10 5 10 Total: 50 Score 1. (a) (2 points) If is a sample space, what is P(). Solution: P() = 1 (b) (2 points) If P(A) =
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Math 3C Midterm 1 Solutions Name: BruinID: Section: Read the following information before starting the exam. This test has six questions and seven pages. It is your responsibility to ensure that you have all of the pages! Show all work, clearly and in o
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Course: Actuarial Math
MATH 115A - Practice Final Exam Paul Skoufranis November 19, 2011 Instructions: This is a practice exam for the nal examination of MATH 115A that would be similar to the nal examination I would give if I were teaching the course. This test may or may not
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MATH 33A/2 PRACTICE MIDTERM 2 Problem 1. (True/False) . 1 1 1 Problem 2. Let v1 = 1 , v2 = 0 , v3 = 1 . 0 1 1 3 (a) Show that B = 1 , v2 , v3 ) is a basis for R . (v 1 (b) Let u = 1 . Find the coordinates of u with respect to the basis B. 2 (c) Find the c
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MATH 32A FINAL EXAMINATION March 21th, 2012 Please show your work. You will receive little or no credit for a correct answer to a problem which is not accompanied by sucient explanations. If you have a question about any particular problem, please raise y
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Math 3C Midterm 2 Name: BruinID: Section: Read the following information before starting the exam. This test has six questions and fourteen pages. It is your responsibility to ensure that you have all of the pages! Show all work, clearly and in order, i
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Course: Probability Theory
Mathematics Department, UCLA P. Caputo Fall 08, midterm 1 Oct 22, 2008 Midterm 1: Math 170A Probability, Sec. 2 Last name First and Middle Names Signature UCLA ID number (if you are an extension student, say so) Please note: 1. Provide the information ask
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MATH 32A FIRST MIDTERM EXAMINATION October, 18th, 20101 Please show your work. You will receive little or no credit for a correct answer to a problem which is not accompanied by sufficient explanations. If you have a question about any particular problem,
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Course: Discrete Math
Practice First Test Mathematics 61 Disclaimer: Listed here is a selection of the many possible sorts of problems. actual test is apt to make a different selection. 1. Let S5 be the set of all binary strings of length 5 (such as 11001). The Let E be
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Math 31A Midterm 1 Solutions 1. [3 points] The derivative of (a) 1 (b) (c) x2 x)2 x3 Brent Nelson is: 1 x2 1 2 x3 (3x 1 1 + x2 + 8x2 x + 3) (d) None of the above. Solution. We rst apply the quotient rule d dx x2 x)2 x3 = d d x3 dx [(x2 x)2 ] (x2 x)2 dx (
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Management 1A Summer 2004 Danny S. Litt EXAM 2 SOLUTION Name: _ Student ID No. _ I agree to have my grade posted by Student ID Number. _ (Signature) PROBLEM 1 2 3 4 5 6 7 8 TOTAL POINTS 20 25 20 20 30 30 30 25 200 SCORE MANAGEMENT 1A Problem 1
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Course: 31b
Math 31B, Lecture 4, Fall 2011 Exam #2, 9 November 2011 Name: ID Number: Section and TA: You have 50 minutes for the exam. No calculators, phones, notes, or books allowed. You must show your work for credit. Hint: n n=0 cr Question: 1 2 3 4 5 6 7 Tota
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Course: Linear Algebra
Name: Student ID: Prof. Alan J. Laub June 9, 2011 Math 33A/1 FINAL EXAMINATION Spring 2011 Instructions: (a) The exam is closed-book (except for one two-sided page of notes) and will last two and a half hours. No calculators, cell phones, or other electro
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Management 1A Summer 2004 Danny S. Litt EXAM 1 SOLUTION Name: _ Student ID No. _ I agree to have my grade posted by Student ID Number. _ (Signature) PROBLEM 1 2 3 4 5 6 7 8 9 10 TOTAL POINTS 20 20 20 20 20 20 20 20 20 20 200 SCORE MANAGEMENT 1A
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Name: Student ID: Section: 2 Prof. Alan J. Laub Math 33A FINAL EXAMINATION Spring 2008 Instructions: June 9, 2008 (a) The exam is closed-book (except for one page of notes) and will last two and a half (2.5) hours. (b) Notation will conform as closely as
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Course: Math 32A
MATH 32A: SECOND PRACTICE MIDTERM EXAMINATION Summer 2008 1 1. (20 points) The C be a curve dened by the position function r(t) =< sin 2t, t, cos 2t >. (a) Calculate the equation of the Normal plane at the point (0, , 1). Its easier than you think.
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Management 1A Fall 2007 Danny S. Litt EXAM 1 Solutions I agree to have my grade posted by Student ID Number _ _ (Signature) (Student ID Number) Name: _ PROBLEM 1 2 3 4 5 6 7 8 TOTAL POINTS 30 30 20 30 20 20 20 30 200 SCORE MANAGEMENT 1A NAME: _
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Course: Introduction Of Complex Analysis
Quiz 1 Solutions Problem 1: Show that zw = zw. Solution: Let z = x + iy, w = u + iv. Then zw = (x iy)(u iv) = xu yv i(yu + xv) zw = (x + iy)(u + iv) = xu yv + i(yu + xv) = xy yv i(yu + xv) = zw Problem 2: For which n is i an nth root of unity? Solution: n
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Course: Introduction Of Complex Analysis
Quiz 2 Solutions Problem 1: Find the coordinates of log i. Solution: log i = log(|i|) + iArg(i) = 0 + i( + 2k) 2 for k and integer. Problem 2: Show that |ez | e|z| for all z C, Solution: z = rei = r(cos + i sin ), |ez | = |er(cos +i sin ) | = |er cos |eir
School: UCLA
Course: Linear Algebra
Course 115A - Midterm2 Y. Tendero November 19th, 2012 Name: ID Number: Signature: Instructions: Show all work to receive full credit. Feel free to use the back of each paper but please indicate that you have done so. No calculator or any document allowed.
School: UCLA
Course: Linear Algebra
Course 115A - Midterm1 Y. Tendero October 19th, 2012 Name: ID Number: Signature: Instructions: Show all work to receive full credit. Feel free to use the back of each paper but please indicate that you have done so. No calculator or any document allowed.
School: UCLA
Course: Real Analysis
MATH 131B 2ND PRACTICE MIDTERM Problem 1. State the books denition of: (a) A complete metric space (b) and (c) Convergence of a series of real numbers (d) Normed vector space; Banach space Solution. See book. Problem 2. Let be a metric space with a metric
School: UCLA
Course: Math 31A
UCLA Math 31APractice Midterm #2 SolutionsSummer Session C, 2013 1. (20 points) For each of the parts, (a)(e), compute the derivative of the given function. (a) (4 points) (x + sin x)2 Answer: d d (x + sin x)2 = 2 (x + sin x) (x + sin x) dx dx = 2 (x + si
School: UCLA
Course: Math 31A
UCLA Math 31APractice FinalSummer Session C, 2014 1. (20 points) For each of the parts, (a)(e), compute the integral. (a) (4 points) 2 4 x2 dx 0 (b) (4 points) 3 |x2 1|dx 0 1 UCLA Math 31APractice FinalSummer Session C, 2014 (c) (4 points) sin2 (2x 1) cos
School: UCLA
Course: Math 31B
Math 31B, Practice Midterm 2 Solutions Ian Coley November 18, 2014 Solution to 1. (a) We are checking the series n=1 en 1 . 2n We dont need to nd the value of this sum, but we just need to check. Therefore we should use one of our tests. Lets try the com
School: UCLA
Course: Probability Theory
Math 394 B&C. Probability I. Summer 2014. Homework 3, due July 16 Homework 3, due July 16 Problems from old actuarial exams are marked by a star. Problem 1*. Upon arrival at a hospital emergency room, patients are categorized according to their condition
School: UCLA
Course: Probability Theory
Math 394 B&C. Probability I. Summer 2014. Homework 1, due July 2 Homework 1, due July 2 Combinatorics Problem 1. There are 8 apartments for 6 people. Each person chooses one apartment, and each apartment can host no more than one person. How many choices
School: UCLA
Course: Probability Theory
Solution of the Final Sangchul Lee December 18, 2014 1 Solution 1.1 Problem 1 For every claim write if it is true or false. Write a short explanation. (a) The sum of two continuous random variables is a continuous random variable. (b) Let X, Y be random v
School: UCLA
Course: CALCULUS FOR LIFE SCIENCE STUDENTS
Math 3A Lecture 1 Fall 2013 Exam 1 Name _ Section _ UCLA ID _ Please write neatly and show your work . NO calculators Cell phones OFF Problem 1 2 3 4 5 Points Points you earned 1. A lab has a culture of bacteria where each individual bacteria takes 3 hour
School: UCLA
Course: CALCULUS FOR LIFE SCIENCE STUDENTS
1. Outside temperature over the course of a day can be modeled as a sinusoidal function. Suppose you know the high temperature for the day is 63 degrees and the low temperature of 37 degrees occurs at 6 AM. Assuming t is the number of hours since midnight
School: UCLA
Course: CALCULUS FOR LIFE SCIENCE STUDENTS
Practice for Final Exam You will not be using a calculator on the final exam. Write your answer in exact form when you practice these problems. For curiosity, you can calculate the value using your calculator . 1. In Fahrenheit, 212 is the temperature for
School: UCLA
Course: CALCULUS FOR LIFE SCIENCE STUDENTS
Multiple Choice 1. The decay rate is equal to ln(2) divided by the half-life, so the answer is C. 2. The function f (x) has limit 2 as x and limit as x , the answer is C. 3. The function is dened when the part under the square root is nonnegative, so for
School: UCLA
Course: Introduction To Discrete Structures
MIDTERM 1 KEY Math 61 Spring 2013 Pietro Poggi-Corradini Monday Apr. 22. Justify your answers carefully Each problem is out of 10. 1. Prove by induction that the number of unordered pairs from [n](= cfw_1, 2, . . . , n) is equal to n(n 1) . 2 Base case n
School: UCLA
Course: Introduction To Discrete Structures
MIDTERM 2 KEY Math 61 Spring 2013 Pietro Poggi-Corradini Monday May 13. Justify your answers carefully Each problem is out of 10. 1. Terziglio is an Italian card game played with a 40-cards deck and 3 players, say Alice, Bob and Carol, plus a Deadman. At
School: UCLA
School: UCLA
Course: Introduction To Discrete Structures
MIDTERM 2 (MATH 61, FALL 2013) Your Name: (must be in ink) UCLA id: (must be in ink) Math 61 Section: Date: The rules: You MUST simplify completely and BOX all answers with an INK PEN. You are allowed to use only this paper and pen/pencil. No calculators.
School: UCLA
Course: Introduction To Discrete Structures
MATH 61: MIDTERM 2 1 a) Paths of type 1 a 2, where a cfw_1 , ., 5 : 5 possibilities. b) Paths of type 1 a b 2 , where a cfw_1 , 3 , 4 , 5 and b cfw_2, 3, 4 : 4 3 = 12 possibilities. c) There is no path P4 that starts and ends in the same vertex class, so
School: UCLA
Course: INTEGRATION AND INFINITE SERIES
Math 31B, Practice Midterm 1 Solutions Ian Coley October 23, 2014 Solution to 1. (a) We should use a logarithm to make things easier for us. Set lim xsin x = L. + x0 Then ln lim xsin x + x0 = lim ln(xsin x ) = ln L + x0 since we can move any continuous fu
School: UCLA
Course: INTEGRATION AND INFINITE SERIES
31B-2 Midterm 1 Name: , UID: , I read, understood, and will follow the instructions given below. Signature: August 20, 2014 Instructions This is a 90-minute closed-book exam. The textbook, lecture notes, and all personal scratch papers are not allowed dur
School: UCLA
Course: INTEGRATION AND INFINITE SERIES
Math 31B, Midterm 1 Solutions Ian Coley October 28, 2014 Problem 1. Find the following: (a) d dx (x2 + 1)(x2 + 2) (x3 + 1)(x3 + 2) . (b) d xsin x . dx Solution to 1. (a) We can use the quotient rule here, but we can also use logarithmic dierentiation, whi
School: UCLA
Course: INTEGRATION AND INFINITE SERIES
Practice Problems-Solutions December 12, 2014 Problem 1 Evaluate the following limit: lim x0 1 1 2 x tan2 x . Solution We combine the denominators into one: 1 1 tan2 x x2 = 2 x2 tan2 x x tan2 x Then using a2 b2 = (a b)(a + b), we have (tan x x)(tan x + x)
School: UCLA
Course: INTEGRATION AND INFINITE SERIES
Practice Problems December 5, 2014 Problem 1 Evaluate the following limit: 1 1 2 x tan2 x lim x0 . Problem 2 Evaluate the following limit: (ln x)100 . x x0.01 lim Problem 3 You put a cup of hot coee with temperature 100 C inside an ice box with temperatur
School: UCLA
Course: INTEGRATION AND INFINITE SERIES
Math 31B, Midterm 2 Solutions Ian Coley December 2, 2014 Solution to 1. (a) We need to nd /2 cos3 x dx. 0 We should rst use the substitution 1 sin2 x = cos2 x. This gives /2 (1 sin2 x) cos x dx. 0 Let u = sin x, so du = cos x dx. This also gives us the ne
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 3 c c From the textbook solve the problems 29, 30, 31, 32 and 33 from the Chapter 4. Solve the problems 1, 2, 4, 5 and 6 from the Chapter 4 additional exercises at http:/www.athenasc.com
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi c c Homework 1 From the textbook solve the problems 1, 2, 3, 5, 6, 8, 11 and 14 at the end of the Chapter 4. And also the problems below: Problem 1. Give examples of (not independent) random variab
School: UCLA
Course: Mathemtcl Modeling
Math 142-2, Homework 1 Solutions April 7, 2014 Problem 34.4 Suppose the growth rate of a certain species is not constant, but depends in a known way on the temperature of its environment. If the temperature is known as a function of time, derive an expres
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 4 c c solutions From the textbook solve the problems 42, 43 b) and 44 b), c) from the Chapter 4. Solve the problems 26, 27 and 29 from the Chapter 4 and problem 1 from the Chapter 7 addi
School: UCLA
Course: Introduction Of Complex Analysis
EE101 Engineering Electromagnetics Winter 2013 2/14/13 Homework #5 Due: Thursday Feb 21 10:00 AM Hand in to the TA at beginning of class. No late homework is accepted (see grading policy posted on eeweb). If you cannot make it to class, you must slip the
School: UCLA
Math32a HW1 Due on Monday, Jan 14 All the numbered problems are from the textbook Multivariable Calculus by Rogawski (2nd edition). Note: if your textbook is Early Transcendentals by Rogawski (2nd edition), then all your chapters are shifted by 1 down. Th
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 3 c c From the textbook solve the problems 29, 30, 31, 32 and 33 from the Chapter 4. Solve the problems 1, 2, 4, 5 and 6 from the Chapter 4 additional exercises at http:/www.athenasc.com
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 6 c c Solve the problems 18 a) c) d) e) and 19 from the Chapter 7 additional exercises at http:/www.athenasc.com/prob-supp.html And also the problems below: Problem 1. Denote points P0 ,
School: UCLA
Course: Mathemtcl Modeling
Math 142 Homework 1 Solution 34.5 (a) If the growth rate is 0, the discrete model becomes Nm+1 Nm = tfm , And the solution is Nm = N0 + t(f0 + f1 + fm1 ), which is analogous to integration, since let t = mt, then t t(f0 + f1 + fm1 ) f (s)ds 0 for t small
School: UCLA
Course: Mathemtcl Modeling
Math 142 Homework 2 Solution 34.16 We will rstly proceed similarly as 34.14(c). We assume the theoretical growth has the form N0 eR0 t , and we hope to gure out which N0 and R0 t the data best. 34.14(b) suggests that N0 should be equal to the initial popu
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 5 c c From the textbook solve the problems 4 and 5 from the Chapter 5. Solve the problems 2, 4, 6, 7, 9 and 10 from the Chapter 7 additional exercises at http:/www.athenasc.com/prob-supp
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 2 c c From the textbook solve the problems 17, 18, 19, 22, 23 and 24 from the Chapter 4. Solve the problems 21, 22, 24, 30 from the Chapter 4 additional exercises at http:/www.athenasc.c
School: UCLA
Homework 4 Solutions Math 3C Due: Friday Nov 2, 9 a.m. #1(a) Let X be a random variable that equals the number of defective items in the sample. When the sampling is done with replacement, X is binomially distributed with parameters n = 5 and p = 5/20 = 1
School: UCLA
Course: Introduction Of Complex Analysis
EE101 Engineering Electromagnetics Winter 2013 1/24/12 Homework #3 Due: Thursday Jan 31 10:00 AM Hand in to the TA at beginning of class. No late homework is accepted (see grading policy posted on eeweb). If you cannot make it to class, you must slip the
School: UCLA
Course: 245abc
Math 245A Homework 2 Brett Hemenway October 19, 2005 Real Analysis Gerald B. Folland Chapter 1. 13. Every -nite measure is seminite. Let be a -nite measure on X, M. Since is -nite, there exist a sequence of sets cfw_En in M, with (En ) < and En = X . n=1
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Course: Mathemtcl Modeling
Math 142 Homework 0 Solution 32.11 The easier way: (slightly unrigorous, but its ne for this class) Suppose the bank compounds the interest n times a year, and let t := 1/n. With the additional deposit (or withdrawl) D (t), the balance at time t + t is gi
School: UCLA
Course: 245abc
Math 245A Homework 1 Brett Hemenway October 19, 2005 3. a. Let M be an innite -algebra on a set X Let Bx be the intersection of all E M with x E . Suppose y Bx , then By Bx because Bx is a set in M which contains y . But Bx By because if y E for any E M,
School: UCLA
Course: Mathemtcl Modeling
Math 142 Homework 1 Solution 32.3 (a) Use N (nt) to denote the population at time nt. Then for any n > 0, we have N (nt) = (1 + (b d)t)N (n 1)t) + 1000. (b) Given that N (0) = N0 , we hope to solve the equation above and get the general form for N (nt). T
School: UCLA
Course: Probability Theory
Probability Theory, Math 170b, Winter 2013, Toni Antunovi - Homework 4 c c From the textbook solve the problems 42, 43 b) and 44 b), c) from the Chapter 4. Solve the problems 26, 27 and 29 from the Chapter 4 and problem 1 from the Chapter 7 additional exe
School: UCLA
Course: Actuarial Math
MATH 115A - Assignment One - Solutions of Select Non-Graded Problems Paul Skoufranis October 3, 2011 1.2 Question 13) Let V denote the set of ordered pairs of real numbers. If (a1 , a2 ) and (b1 , b2 ) are elements of V and c R, dene (a1 , a2 ) + (b1 , b
School: UCLA
Course: Mathemtcl Modeling
Math 142-2, Homework 2 Solutions April 7, 2014 Problem 35.3 Consider a species in which both no individuals live to three years old and only one-year olds reproduce. (a) Show that b0 = 0, b2 = 0, d2 = 1 satisfy both conditions. All two year olds die befor
School: UCLA
Course: Real Analysis
Homework 8 Solutions Math 131A-3 1. Problems from Ross. (29.5) Suppose |f (x) f (y )| (x y )2 for all x, y in R. Then for any given a R, we )f )f have | f (xxa (a) | |x a|, so by the squeeze theorem, as x a, | f (xxa (a) | 0. Therefore f (a) = 0. Since a
School: UCLA
Course: Algebra
UCLA Mathematics 110A: selected solutions from homework #1 David Wihr Taylor July 2, 2010 Introduction When reading these solutions always keep in mind the common techniques being used. The point of homework, and subsequently these solutions, is to give y
School: UCLA
Course: Real Analysis
Math 131A Analysis Summer Session A Homework 3 Solutions n 1. Let (an ) be a bounded sequence and let (bn ) be a sequence such that lim bn = 0. Prove that lim an bn = 0. n Proof. Since (an ) is bounded, there exists M > 0 such that |an | < M for a
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1 1. Prove Proposition 3.1. Proposition 3.1 Let be a signed measure on . If is an increasing sequence in , then lim . If is a decreasing sequence in and is finite, then lim . Answer) Let be a signed measure on . a. If is an increasing sequence in
School: UCLA
Course: Linear Algebra
Ch. 1 Dept. of ECE, Faculty of Engineering University of Tehran Linear Algebra Homework # I Chapter 1: Vector Spaces Exercises to be handed: marked by *. Due date: Sunday Esfand 9th 1383. 1. Let V = cfw_ (a1 , a2) : a1 , a2 R . Define addition of elements
School: UCLA
Course: MULTIVARIABLE CALCULUS
This page intentionally left blank Students Solutions Manual to accompany Jon Rogawskis Single Variable CALCULUS SECOND EDITION BRIAN BRADIE Christopher Newport University ROGER LIPSETT W. H. FREEMAN AND COMPANY NEW YORK 2012 by W. H. Freeman and Company
School: UCLA
Course: Introduction Of Complex Analysis
Homework 1 Due: January 13 Math 132/1, Winter 2015 Textbook, Part 1: I.1 # 1 (a, b, c, f, g, h), 2, 3, 4, 5 I.2 # 1, 2, 4, 5(a), 7, 8 I.4 # 1 (a, c, f), 2 (a, c, f), 3 (a, b) Other 1. Let M22 R denote the set of 2 2 matrices with real entries. Dene a func
School: UCLA
Course: Introduction Of Complex Analysis
Homework 5 Due: February 10 Math 132/1, Winter 2015 Textbook, Part 1: IV.1 # 1, 2, 3, 4, 5, 9 IV.2 # 1, 2, 3 IV.3 # 1, 4 IV.4 # 1 (a, c, e), 2, 3 Other: 1. Show that f (z) = z does not have a primitive. [There are many ways to show this.] 2. Use Problem I
School: UCLA
Course: Introduction Of Complex Analysis
Homework 2 Due: January 20 Math 132/1, Winter 2015 Textbook, Part 1: I.5 # 2 (b, c), 3, 4 I.6 # 1, 2 (a, b, c, e), 3 I.7 # 1, 2, 3, 5 I.8 # 1 (b, c), 2, 4, 7 II.1 # 10, 11, 15, 16, 17 Other 1. Show that for all z 2 C, |ez | 1 if and only if Re(z) 0. When
School: UCLA
Course: Introduction Of Complex Analysis
Homework 7 Due: February 24 Math 132/1, Winter 2015 Textbook, Part 1: V.4 # 1 (a, b, d), 2, 3, 6, 7, 8, 11, 12, 13 V.5 # 1, 2 V.6 # 2, 3, 4 Other: 1. Recall that the Fibonacci sequence is cfw_Fn such that F0 = 0, F1 = 1, and Fn+2 = Fn+1 + Fn for n 0. Den
School: UCLA
Course: Introduction Of Complex Analysis
Homework 4 Due: February 3 Math 132/1, Winter 2015 Textbook, Part 1: III.1 # 1, 2, 3, 4, 5, 6 III.2 # 1, 2, 3 III.3 # 1 Other: 1. Read the supplemental notes on dierential forms, available on CCLE. 2. For z C \ cfw_0, nd (or look up) an expression for the
School: UCLA
Course: Introduction Of Complex Analysis
Homework 3 Due: January 27 Math 132/1, Winter 2015 Textbook, Part 1: II.2 # 1, 3, 4, 6 (see also II.2 # 14) II.3 # 1, 2, 3, 8 II.4 # 4, 5, 6 II.5 # 1 (a, c, e), 2, 4, 6 (use # 5), 7 II.7 # 1 (a, b, c), 3, 5, 6 Other 1. Redo problem II.2 #4 using the Cauch
School: UCLA
Course: Introduction Of Complex Analysis
Homework 8 Due: March 3 Math 132/1, Winter 2015 Textbook: V.7 # 1 (a, c, g, i), 2 (a, c, g, i), 3, 6, 7, 8 VI.1 # 1, 2, 5 Other: 1. Let f : D C be a function. The pre-image of a point c Range(f ) is the set of all points in the domain that f takes to c. M
School: UCLA
Course: Introduction Of Complex Analysis
Notes on Dierential Forms Math 132/4, Fall 2014 Denition of Dierential Forms Let U R2 be open. A 0-form on U is just a function f : U ! R. A 1-form on U is an expression P (x, y) dx + Q(x, y) dy, where P, Q : U ! R are functions. A 2-form on U is an ex
School: UCLA
Course: Introduction Of Complex Analysis
Homework 6 Due: February 17 Math 132/1, Winter 2015 Textbook, Part 1: IV.5 # 1, 2, 3, 4 V.1 # 1, 2, 6 V.3 # 1 (b, f, i), 2 (b, c, d), 3, 6 Other: 1. Suppose that f is an entire function whose real part is positive. Show that f is constant. [Hint: consider
School: UCLA
Course: Linear Algebra
WHAT ARE MATHEMATICAL PROOFS AND WHY THEY ARE IMPORTANT? introduction Many students seem to have trouble with the notion of a mathematical proof. People that come to a course like Math 216, who certainly know a great deal of mathematics - Calculus, Trigon
School: UCLA
Course: Linear Algebra
1 Homework 9 (Due Fri, Nov 30th) Ex1: Let A be a n n matrix with n distinct eigenvalues 1 , , n . Show that det(A) = 1 n and tr(A) = 1 + + n . 1 1 Ex2: Find a 2 2 matrix A which has v1 = and v2 = has eigenvectors, is 1 0 not equal to the identity matrix o
School: UCLA
Course: Linear Algebra
1 Homework 5 (Due Nov 2nd) Ex1: Do Ex5 section 2.2 Ex2: Do Ex1 section 2.3 a) and e) to j) Ex3: Do Ex 4c) section 2.3 Ex4: Do Ex 10 of section 2.3 Ex5: Do Ex 1 of section 2.4 Ex6: Do Ex 2 of section 2.4 Ex7: Do Ex 4 of section 2.4 Ex8: Do Ex 9 of section
School: UCLA
Course: Linear Algebra
1 Homework 1 (Due Oct 5th) Proof is the heart of mathematics. It distinguishes mathematics from the sciences and other disciplines. Courts of law deal with the burden of proof, juries having to decide whether the case against a defendant has been proven b
School: UCLA
Course: Linear Algebra
1 Homework 3 (Due Oct 19th) Ex1: Prove: Lemma 1.1. (span(S) is bigger than S) Let S be a subset of a vector space V then S span(S). Ex2: Prove the second part: (Any subspace containing S also contains span(S). ): Theorem 1.2. (span(S) is a subspace, you c
School: UCLA
Course: Linear Algebra
1 Homework 10 (Due Fri, Dec 7th) Ex1: Do Ex 2 section 6.1. Ex2: Do Ex 3 section 6.1. Ex3: Do Ex 4 section 6.1. Ex4: Do Ex 5 section 6.1. Ex5: Do Ex 8 section 6.1. Ex6: Do Ex 9 section 6.1. Ex7: Do Ex 17 section 6.1. Ex8: Do Ex 2 d,h,i of section 6.2. Ex9:
School: UCLA
Course: Linear Algebra
1 Homework 6 (Due Nov 9th) Ex1: Let = (1, 0), (0, 1) be the standard basis for R2 and let = (3, 4), (4, 3) be another basis for R2 . Let l be the line connecting the origin to (4, 3) and T : R2 R2 be the operation of reection through l (so if v R2 then T
School: UCLA
Course: Linear Algebra
1 Homework 8 (Due Nov 26th) Ex1: Let A and B be similar n n matrices. Show that A and B have the same set of eigenvalues (every eigenvalue of A is an eigenvalue of B and vice versa). Ex2: For this question, the eld of scalars will be complex numbers C ins
School: UCLA
Course: Linear Algebra
1 Homework 7 (Due Nov 16th) Ex1: Let U, V, W be nite-dimensional vector space, and let S : V W and T : U V be linear transformations. 1. Show that rank(ST ) rank(S) 2. Show that rank(ST ) rank(T ) 3. Show that nullity(ST ) nullity(T ) 4. Give an example w
School: UCLA
Course: Linear Algebra
1 Homework 2 (Due Oct 12th) For exercises, keep in mind that weve a double goal: linear algebra and proofs. Each answer must be carefully justied, step by step. No argument can be skipped. When one check a proof he/she does not think: just check that each
School: UCLA
Course: Linear Algebra
1 Homework 4 (Due Oct 26th) Ex1: Lemma 1.1. Let T : V W be a linear transformation. The null space N (T ) is a subspace. Proof. Write the proof. Ex2: Lemma 1.2. (R(T ) is a subspace) Let T : V W be linear, R(T ) is a subspace. Proof. Write the proof. Ex3:
School: UCLA
Course: Math 31B
Homework 7 Name: , UID: , Discussion session: November 14, 2014 Please use this as a cover sheet. 9.4: 3, 4, 18, 23 11.1: 41, 57, 68, 86 Problem 1 Find the limit: lim n2 1 cos n 1 n . Problem 2 (True/False) Determine that the following statements are true
School: UCLA
Course: Math 31B
Homework 8 Name: , UID: , Discussion session: December 1, 2014 Please use this as a cover sheet. 11.2: 17, 27, 38, 53 11.3: 34, 59, 79, 58 Problem 1 Show that the following series converges absolutely: (1)n1 n=2 cos n . n(ln n)2 Problem 2 Show that the fo
School: UCLA
Course: Math 31B
Homework 4 Name: , UID: , Discussion session: October 25, 2014 Please use this as a cover sheet. 8.1: 17, 20, 40, 76 8.2: Formula (14), (15), (19), (20) in p. 423. Problem 1 Find the following limit: 1 lim t0+ t ln xdx Problem 2 Find the following denite
School: UCLA
Course: Math 31B
Homework 6 Name: , UID: , Discussion session: November 7, 2014 Please use this as a cover sheet. 8.6: 60, 65, 73, 74 9.1: 23, 27, 35, 46 Problem 1 Find the integral: 0 x3 1 dx. +1 Problem 2 Determine whether the integral converges, nd the value if it conv
School: UCLA
Course: Math 31B
Homework 2 Name: , UID: , Discussion session: October 10, 2014 Please use this as a cover sheet. 7.3: 22, 71, 74, 84 7.4: 15, 17, 18, 28 Problem 1 A tank contains 100 gal of salt-water which contains 1 lb of salt initially. Assume that the salt-water is w
School: UCLA
Course: Math 31B
Homework 5 Name: , UID: , Discussion session: October 31, 2014 Please use this as a cover sheet. 8.3: 2, 6, 8, 9 8.5: 14, 19, 33, 35 Problem 1 Find the integral: 1 x2 + 2x + 2 dx. 0 Problem 2 Evaluate the improper integral: 0 1 dx. x2 + 1
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Course: Math 31B
Homework 3 Name: , UID: , Discussion session: October 17, 2014 Please use this as a cover sheet. 7.6: 4, 11 (Find the temperatures of A and B at t seconds, do not plot) 7.7: 9, 30, 34, 61, 63, 66 Problem 1 Find the following limit: sin x x . x0 tan3 x lim
School: UCLA
Course: Probability Theory
Probability Theory, Math 170A/2, Winter 2015 - Homework 2 From the textbook solve the problems 14, 16, and 19 at the end of the Chapter 1. Solve the problems 6, 8, 9, 15, 16 from the Chapter 1 additional exercises at http:/www.athenasc.com/prob-supp.html
School: UCLA
Course: Probability Theory
Probability Theory, Math 170A, Fall 2014, Homework 5 From the textbook solve the problems: 2, 3, 4 and 17 at the end of Chapter 2. From the supplementary problems http:/athenasc.com/CH2-prob-supp.pdf, solve problems 2-8 of Chapter 2. And also the problems
School: UCLA
Course: Probability Theory
Probability Theory, Math 170A/2, Winter 2015, Homework 3 Problems from the textbook: 22, 30, 34, 35 at the end of the Chapter 1. Problems from http:/www.athenasc.com/prob-supp.html: 19, 23, 24, 25, 28, 29, 30. And also the problems below: Problem 1. Show
School: UCLA
Course: Calculus
31B Notes Sudesh Kalyanswamy 1 Exponential Functions (7.1) The following theorem pretty much summarizes section 7.1. Theorem 1.1. Rules for exponentials: (1) Exponential Rules (Algebra): (a) ax ay = ax+y (b) ax ay = x y axy (c) (a ) = axy (2) Derivatives
School: UCLA
School: UCLA
School: UCLA
The Poisson distribution is a discrete distribution. It is often used as a model for the number of events (such as the number of telephone calls at a business, number of customers in waiting lines, number of defects in a given surface area, airplane arriv
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3C Notes Sudesh Kalyanswamy 1 Counting (12.1) 1.1 Guiding Questions for the Chapter Here are some questions for you to ask yourself as you read. If you can answer them all by the end, youre in pretty good shape as far as understanding the material. After
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3C Practice Problems for Midterm 1 Sudesh Kalyanswamy (1) Bills school schedule has four timeslots. Bill has 30 choices of classes for the rst timeslot, 20 for the second, 23 for the third, and 17 for the last. How many possible schedules are there? (2) C
School: UCLA
School: UCLA
Math 171 Spring 2013 Homework 8 (due Friday, June 7th at lecture time) (1) Read Sections 5.3 and 5.4. (2) Solve problems 2, 3, 4 and 6 from the end of chapter 5.
School: UCLA
School: UCLA
Course: INTEGRATION AND INFINITE SERIES
A dierential equation is an equation that relates the derivative of a function to that function (and possibly other functions). For example a dierential equation involving the function y (t) could look like dy (t) = ky (t) (1) dt for some constant k . Whe
School: UCLA
Course: Calculus
Answer Key B. Johnson Calculus AB Memory Quiz Important Relationships, Theorems, Rules, and Formulas Important Relationships 1. State the formal definition of derivative, include diagram. If y = f ( x ) then f ( x ) = lim h0 f ( x + h) f ( x) h 2. Where a
School: UCLA
Course: Linear Algebra
Math 33A, Sec. 2 Linear Algebra and Applications Spring 2012 Instructor: Professor Alan J. Laub Dept. of Mathematics / Electrical Engineering Math Sciences 7945 phone: 310-825-4245 e-mail: laub@ucla.edu Lecture: MWF 2:00 p.m. 2:50 p.m.; Rolfe 1200 Office
School: UCLA
Course: Calculus
Sample Syllabus Philosophical Analysis of Contemporary Moral Issues Winter 2015, MW 8-9:50 Instructor: Robert Hughes (hughes@humnet.ucla.edu) Instructor office hours: Dodd 378, M 11-12 and W 10-11 or by appointment TAs: Kelsey Merrill (kmerrill@humnet.ucl
School: UCLA
Course: Math 170
Probability Theory, Math 170A, Winter 2015 - Course Info Instructor: Tonci Antunovic, 6156 Math Sciences Building, tantunovic@math.ucla.edu, www.math.ucla.edu/~tantunovic Instructor Oce Hours: Monday 10-12am and Wednesday 9-10am in 6156 Math Sciences Buil
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Course: Math 1
Math1 Fall2012 Lecture1(8am),Lecture2(10am)andLecture3(1pm) Instructor Paige Greene E-Mail: paige@math.ucla.edu Office Number: MS 6617 B Office Hours: Mon and Wed (copy from me in class) Course Goals: The purpose of Math 1 is to give you a strong preparat
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Math33BDifferentialEquations Winter2015,lecture3 Time:MWF3:003:50p.m. Room:MS4000A Discussionsections: 3A T3:00P3:50PGEOLOGY6704 3B R3:00P3:50PGEOLOGY4645 Instructor:YuZhang Office:MS6617F Officehours:Monday2:003:00p.m. Wednesday4:006:00p.m. Email:yuzhang
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Math 3B Integral calculus and differential equations Prof. M. Roper Young Hall CS 76, MWF 2-2.50pm. All inquiries about this class should be posted to the class Piazza page. The only emails that I will respond to are questions about how to login to Piazza
School: UCLA
Course: Actuarial Math
Math 172A: Syllabus, Structure, Details and Policies (Fall 2014) Introduction to Financial Mathematics This course is designed to provide an understanding of basic concepts of actuarial/financial mathematics. The course touches most (but not all) syllabus
School: UCLA
Course: Abstract Algebra
Math 32B (Calculus of several variables), Section 2 Fall 2013 MWF 2:00-2:50 pm, 1178 Franz Hall UCLA Instructor: William Simmons, MS 6617A, simmons@math.ucla.edu. Oce Hours (held in MS 6617A): M: 8:00-8:55 am T: 10:30-11:30 am W: 3:30-4:30 pm Teaching Ass
School: UCLA
Math 115A, Linear Algebra, Lecture 6, Fall 2014 Exterior Course Website: http:/www.math.ucla.edu/heilman/115af14.html Prerequisite: MATH 33A, Linear algebra and applications. Course Content: Linear independence, bases, orthogonality, the Gram-Schmidt proc
School: UCLA
Math 32A, Calculus of Several Variables Lecture 1, Winter 2014 Instructor: Will Conley Email: wconley@ucla.edu Office: MS 6322 Office Hours: Tentatively Wednesdays 11:3012:30, Thursdays 1:302:30, and any other time by appointment. Website: http:/www.math.
School: UCLA
Math 33A, Section 2 Linear Algebra and Applications Winter 2014 Instructor: Professor Alan J. Laub Dept. of Mathematics / Electrical Engineering Math Sciences 7945 phone: 310-825-4245 e-mail: laub@ucla.edu Lecture: MWF 1:00 p.m. 1:50 p.m. Math Sciences 40
School: UCLA
Math1 Fall2012 Lecture1(8am),Lecture2(10am)andLecture3(1pm) Instructor Paige Greene E-Mail: paige@math.ucla.edu Office Number: MS 6617 B Office Hours: Mon and Wed (copy from me in class) Course Goals: The purpose of Math 1 is to give you a strong preparat
School: UCLA
Differential Equations MATH 33B-1 Instructor Details: Lecture 1: Instructor: Office: Office hours: MWF 12PM-12:50PM in MOORE 100 Prof. Zachary Maddock MS 6236 (MS := Mathematical Sciences) Mondays 1:30PM-2:30PM Wednesdays 9:50AM-10:50AM, 1:30PM-2:30PM Tea
School: UCLA
Course: Linear Algebra
Course information and Policy, v. of Oct.1st, 2013 Marc-Hubert NICOLE Linear Algebra with Applications Math 33A-1, Lecture 1, Fall 2013. U.C.L.A. -Instructor Marc-Hubert NICOLE Lectures: MWF 8:00-8:50 AM in Rolfe 1200. Ofce: MS 5230 (see Ofce Hours belo
School: UCLA
S YLLABUS : P HYSICS 1B, L ECTURE 1, W INTER 2012 I NSTRUCTOR : Prof. Troy Carter 4-909 PAB (310) 825-4770 tcarter@physics.ucla.edu O FFICE H OURS : M 10-11AM, T 1PM-2PM, F 1-2PM, other times by appointment C OURSE A DMINISTRATOR : Elaine Dolalas (handles
School: UCLA
Course: Introduction Of Complex Analysis
Math 132: Complex Analysis for Applications INSTRUCTOR: Kefeng Liu TIME and LOCATION: MWF 9:00-9:50AM, MS5117. OFFICE HOURS: TT 9:00-10:00AM in MS 6330 (Or by appointment) SYLLABUS: Official Syllabus Text: Complex Analysis by Tom Gamelin TA: Wenjian Liu,
School: UCLA
Math 3C, Probability for Life Sciences Students, Lecture 2, Spring 2012 Instructor: Thomas Sinclair E-mail: thomas.sinclair@math.ucla.edu Oce: Math Sciences (MS) 7304 Oce Hours: M 3:004:30, W 11:0012:30 TAs: 2a,b Theodore Gast Oce: MS 2961 Oce Hours: TBA
School: UCLA
Mathematics 31B: Integration and Innite Series. Fall 2012 Instructor: David Weisbart Email: dweisbar@math.ucla.edu Textbook: J. Rogawski, Single Variable Calculus, 2nd Ed., W.H. Freeman & Co. 31B Teaching Statement: Along with 31A, this course is the foun
School: UCLA
Course: Linear Algebra
Physiological Science 5 Issues in Human Physiology: Diet and Exercise Winter 2012 Instructor: Joseph Esdin, Ph.D. Office Hours: Mon 12:30-1:20 pm & Wed 10:30-11:20 am Office: 3326 Life Sciences Building Phone: (310) 825-4118 Email: yezzeddi@ucla.edu TA: D
School: UCLA
Course: Linear Algebra
Mathematics 33B: Dierential Equations. Winter 2012 Instructor: David Weisbart Email: dweisbar@math.ucla.edu Textbook: Polking, Boggess, Arnold, Dierential Equations, 2nd Ed., Pearson. 33B Teaching Statement: Since the time of Newton, the language of diere
School: UCLA
ULE ZLER, UCLA ECONOMICS DEPARTMENT BUCHE HALL 9361 OZLER@ECON.UCLA.EDU OFFICE HOURS: TUE & THR 1:00-1:45 AND BY APPOINTMENT ONLY FALL 2011 ECONOMICS 121- INTERNATIONAL TRADE THEORY Course Description In this course we will study alternative models of int
School: UCLA
Course: Math 26
Math 26B Section 2 Calculus II for the Social and Life Sciences Fall 2012 Instructor: Jill Macari Office: Brighton Hall 121 Phone: 278-7074 Email: jmacari@csus.edu Office Hours: Monday, Tuesday, and Wednesday 10:30 am 11:30 am; Thursday 12:30 pm 1:30 pm a
School: UCLA
Math 131A Course Outline Spring 2011 Text: Apostol, Calculus, Volume I, 2nd ed. Instructor: Betsy Stovall 1. Introduction. Crash review of basic propositional logic and set notation. (Chapter I.2) On your own: Read I.12 with an emphasis on I.2. You wil
School: UCLA
Course: 245abc
1. AssignmentdueSeptember7:ChapterI,4(ignorethe hypothesisthatKnormalizesH),5,6,7,8,9.In Problem8,thereisapairofmisprints:astheproblemis written,therearethreeunionsigns,wheretheindices arerespectivelyc,x_candx_c.Thefirstunionshould beoverelementsx_c;thes
School: UCLA
Course: Mathematics-finance
Course Syllabus Math 181: THE MATHEMATICS OF FINANCE Fall 2001 1 Background in Finance and Probability 1. Introduction and Course Description 2. Review of probability 3. Discrete Random Walks 4. Random walks with Gaussian increments 5. Equity model
School: UCLA
Course: Prob Life Sci Stdt
Math 3C (Spring 2008) Probability for Life Sciences Students Instructor: Roberto Schonmann www.math.ucla.edu/rhs Time/Place: Lecture 1: MWF 9:00-9:50 in MS 4000A. Lecture 2: MWF 10:00-10:50 in MS 4000A. Text: Calculus for Biology and Medicine (second