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School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 5 SOLUTIONS DANIEL COREY Section 3.3 10. The augmented matrix 1 2 1 4 7 7 1 1 5 1 0 0 10 0 0 row reduces to 0 1 1 00 0 1 0 0 28 13 0 1 3 9 7 4 3 1 1 So the inverse of the matrix is 28 13 3 7 3 1 9 4 1 18. x1 1 x2 = 4 x3 1 1 1 2 28 7 1
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 1: -algebras and Borel sets The due date for this assignment is Thursday 9/9/2010. 1. Let A be the set of numbers in [0; 1] which admit decimal expansions such that the digits 2; 4; 6; 8 all appear at lea
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 1: -algebras and Borel sets The due date for this assignment is Thursday 9/9/2010. 1. Let A be the set of numbers in [0; 1] which admit decimal expansions such that the digits 2; 4; 6; 8 all appear at lea
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 3: Measurable Functions and Their Unexpected Properties The due date for this assignment is 9/30 1. Consider the Cantor function : [0; 1] P [0; 1]. It is dened as follows: For each ! n x 2 C , the Cantor
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 2: Measures, Outer Measures, and Weird Sets The due date for this assignment is Thursday 9/16/2010. 1. Let (X; A; ) be a measure space and fAi g a sequence of A-measurable sets such that 1 X i=1 (Ai ) < 1
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 2: Measures, Outer Measures, and Weird Sets The due date for this assignment is Thursday 9/16/2010. 1. Let (X; A; ) be a measure space and fAi g a sequence of A-measurable sets such that 1 X i=1 (Ai ) < 1
School: Yale
Course: Integral Calculus
Yale University, Department of Mathematics Math 115 Calculus Fall 2013 Final Exam Review Guide When/Where. The Final Exam will take place during 7:00 10:30 pm on Sunday, December 15th, 2013 in Davies Auditorium, located underground between Dunham Labs and
School: Yale
Week 7 Important Discrete Distributions Expectation and Variance Lecture 17. Bernoulli and Binomial Distributions. Expectation Revisited. Discrete Uniform Distribution. We have seen some examples that all outcomes of an experiment are equally likely. Let
School: Yale
Week 5 Lecture 11. Conditional distribution and Conditional density. Review. Discrete random variables X1 ; X2 ; ; Xn are mutually independent if P (X1 = x1 ; X2 = x2 ; ; Xn = xn ) = P (X1 = x1 ) P (X2 = x2 ) : : : P (Xn = xn ) . Continuous random variabl
School: Yale
Week 4 Lecture 8. Discrete conditional distribution. Examples 1: A doctor gives a patient a test for a particular cancer. Before the results of the test, the only evidence the doctor has to go on is that 1 woman in 1000 has this cancer. Experience has sho
School: Yale
Week 6 Lecture 14 Random Walks Drunkard Walk. Imagine now a drunkard walking randomly in an ideals ized 1 dimensional city ( or 2 dimensional, or 3 and higher dimensional city). The city is eectively innite and arranged in a 1 dimensional equally-spaced g
School: Yale
Week 3 Lecture 5. Expectation Probability Density Function: Let f (x) 0 and P (E ) as following Z P (X 2 E ) = f (x) dx. R f (x) dx = 1. Dene E Are the probability axioms satised? It is important to observe that there a similar paradox in the calculus Za
School: Yale
Week 2 Lecture 3. Expectation and Probability axioms. Random variable. A random variable is a real-valued function dened on the sample space, i.e., X (! ) is a function from to R. For example, for = fBB; BG; GB; GGg, your X could be the number of boys, th
School: Yale
Course: CalcOfFunctionsOfOneVariableII
SOLUTIONS TO EXAM 2, MATH 115, FALL 2012 1. (65 points) The graphs of f (x), g(x), and h(x) are shown below. You may assume that as x , the graphs of f , g, and h continue in a fashion similar to the trends observed in the graphs. 3 gx 2 1 hx f x 0 1 2 3
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 1 Math 115 October 3rd, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 2 Math 115 April 10th, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor w
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 2 Math 115 November 13th, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructo
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Mathematics 115b Mid-term # 1 2/19/08 Name: Section: Please answer all SIX questions; each question is worth TEN points. You may use the table of integrals supplied with the test. However, books, notes, calculators, computers, cellphones may NOT be used.
School: Yale
Course: Linear Algebra
Math 54 worksheet, September 14, 2009 1. Let T be the linear transformation from R3 to R3 which consists of rst rotating 30 degrees about the z axis and then rotating 90 degrees about the x axis. (I havent specied the directions of the rotations. Use whic
School: Yale
Course: Calculus
Math 1B - Fall 2006 10/09/2006 Integrals sin2 x cos2 x dx Use 2 sin x cos x = sin 2x, then use Ex 1. Ex 2. 0 1 (1 cos 4x) = sin2 2x. 2 4 dx 1 1 dx Try x = tan , dx = sec2 d. 2+1 4x 2 2 Ex 3. Ex 4. 4 dx 4 + e2x x Use Weierstrass substitutions, let t = tan
School: Yale
Course: Linear Algebra
Math 54 worksheet, September 21, 2009 1. Find the inverse of 1 A = 3 2 2 4 , 4 0 1 3 using both row reduction and the adjugate. Check that you get the same answer. A1 8 = 10 3 4 3 2 7 2 1 1 1 2 2. Use the adjugate to nd a formula for the inverse of a 2 2
School: Yale
STAT 241/541, Probability Theory with Applications Fall 2013 Instructor: Harrison H. Zhou (huibin.zhou@yale.edu) O ce hours: Wednesday 4:00-6:00pm (tentative) or by appointments, Room 204, 24 Hillhouse Ave., James Dwight Dana House. T.A.: Corey Brier <cor
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 5 SOLUTIONS DANIEL COREY Section 3.3 10. The augmented matrix 1 2 1 4 7 7 1 1 5 1 0 0 10 0 0 row reduces to 0 1 1 00 0 1 0 0 28 13 0 1 3 9 7 4 3 1 1 So the inverse of the matrix is 28 13 3 7 3 1 9 4 1 18. x1 1 x2 = 4 x3 1 1 1 2 28 7 1
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 1: -algebras and Borel sets The due date for this assignment is Thursday 9/9/2010. 1. Let A be the set of numbers in [0; 1] which admit decimal expansions such that the digits 2; 4; 6; 8 all appear at lea
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 1: -algebras and Borel sets The due date for this assignment is Thursday 9/9/2010. 1. Let A be the set of numbers in [0; 1] which admit decimal expansions such that the digits 2; 4; 6; 8 all appear at lea
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 3: Measurable Functions and Their Unexpected Properties The due date for this assignment is 9/30 1. Consider the Cantor function : [0; 1] P [0; 1]. It is dened as follows: For each ! n x 2 C , the Cantor
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 2: Measures, Outer Measures, and Weird Sets The due date for this assignment is Thursday 9/16/2010. 1. Let (X; A; ) be a measure space and fAi g a sequence of A-measurable sets such that 1 X i=1 (Ai ) < 1
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 2: Measures, Outer Measures, and Weird Sets The due date for this assignment is Thursday 9/16/2010. 1. Let (X; A; ) be a measure space and fAi g a sequence of A-measurable sets such that 1 X i=1 (Ai ) < 1
School: Yale
Course: Optimization I
Answers to assignment due on October 14th. 2. Program 1D: z * = min y b, subject to yA c. To place this thing in the format of Program 12.1, we need to express it as a maximization problem with nonnegative variables and equality constraints. Lets replace
School: Yale
Course: Calogero-MoserSystemsRepTheory
Solutions for Homework 1 Problem 1Shifrin 2.3.7: Problem 2Shifrin 2.3.8.bcefi: Problem 3Shifrin 2.3.16: Problem 4Shifrin 2.3.18 (solution by Bonnie Yn): Problem 5: Problem 6:
School: Yale
Course: CANCELLED
#3: (1 + 1/n)2 Limit of sequence: 0 (1+1/n)^3 (1+1/n)^4 (1+1/n)^n Graphs a, b, and c all approach zero. The graphs reach the limit of zero much faster as the exponential power increases. Graph d achieves a limit different from the previous graphs
School: Yale
Course: CANCELLED
2 1.5 1 0.5 -1 -0.5 0.5 k=0 1 1 0.5 -1 -0.5 0.5 1 -0.5 -1 k=1 1 0.5 -1 -0.5 0.5 1 -0.5 -1 k=3 0.2 0.1 -1 -0.5 -0.1 0.5 1 -0.2 k=5 0.03 0.02 0.01 -1 -0.5 -0.01 -0.02 0.5 1 k=7 0.004 0.002 -1 -0.5 -0.002 0.5
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 12 Solution Triet M. Le Page 116: 2: Let f (x) = 0, x0 e-1/x , x > 0. To show f is infinitely differentiable at x = 0. Clearly, f is infinitely differentiable on (-, 0) (0, ), with 0, x<0 f (n) (x) = -1/x P2n (1/x)e , x
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 4 SOLUTIONS DANIEL COREY Section 3.1 18. T is not linear since T (0, 0, 0) = (0, 4, 0) = (0, 0, 0) (see the solution to exercise 55 below). 22. The columns are linear dependent, so T is not one-to-one. The span of the columns is not all
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 3 SOLUTIONS DANIEL COREY Section 2.2 16. 9 5 2 5 3 A= 1 x x= 1 x2 13 b = 9 2 30. The columns do not span R2 , for example e1 = (1, 0) is not in the span of the columns. 40. h can be any real number except h = 12/5 (this value of h is pre
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 2 SOLUTIONS DANIEL COREY Section 1.4 2. This amounts to solving the system corresponding to the augmented matrix 1 0 0 1 1 1 0 0 55 0 1 0 1 0 1 1 0 55 which row reduces to 0 0 1 1 1 0 0 1 30 0 0 1 1 30 000 0 30 25 30 0 So the possible v
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 1 SOLUTIONS DANIEL COREY Section 1.1 18. This system is not in echelon form because the leading variable in the third row is to the left of the leading variable in the second row. 20. This system is in echelon form. The leading variables
School: Yale
Course: Calculus
Solutions to some limit convergence denition problems Dustin Cartwright October 2, 2006 These are some possible questions which I came up with involving the definition of a convergent series, and solutions to these questions. The solutions are probably mo
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 1 (due Friday 27 Jan 2012) Solutions 1. Graphs of Multivariable Functions a) Let f : R2 R be a function on R2 . Give R3 the standard (x, y, z ) coord
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 2 (Fri 03 Feb 2012) Solutions 1. CM 14.2 Exercise 28 Solution. xy y 2 exy xy x ye ln(ye ) = = =y x yexy yexy Problem 40 1 Solution. Well, f (65, 16
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 3 (Fri 10 Feb 2012) Selected Solutions 1. CM 14.4 Exercise 14 Solution. z |(x,y) = 1 (yey (x+y )2 + ey (x2 + xy x) Exercise 24 Solution. f |(1,2)
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 4 (Fri 17 Feb 2012) Selected Solutions 1. Let P be a point in R3 and let v be a direction vector at P . Find a parameterization of the line through P
School: Yale
STAT 241/541 Homework 2 Solution Prepared by James Hu Problem 1: Ch2.2:2 (a) c = 1/48 . b 1 1 xdx = (b2 a2 ) . 96 a 48 10 1 75 (c) Pr(X > 5) = xdx = , Pr(X < 7) = 48 96 5 (b) Pr(E ) = 7 1 45 xdx = , 96 2 48 24 3 and Pr(X 2 12X + 35 > 0) = 1 Pr(5 < X < 7)
School: Yale
Homework 2 Due September 13. Chapter 2.2: 2, 4, 6, 8, 12. Problem 6: Let X U (0; 1). (i) Find the density of Y = 1=X and EY . X (ii) Find the density of Y = tan 2 and EY . 1
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 11 Solution Triet M. Le Page 107: 3(a): Let P be a polynomial of odd degree. One sees that limx- P (x) = - and limx P (x) = . Let a, b R such that P (a) < 0 and P (b) > 0. Clearly, P is continuous on [a, b]. By the Interm
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 10 Solution Triet M. Le Page 104: 2: f is real on R and |f (x)| < M for all x R. Then for any x = y, the mean value theorem implies f (x) - f (y) = f (c), for some c between x and y. x-y Hence, for any > 0, |f (x) - f (y)
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 2 Math 115 April 10th, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor w
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 2 Math 115 November 13th, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructo
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Mathematics 115b Mid-term # 1 2/19/08 Name: Section: Please answer all SIX questions; each question is worth TEN points. You may use the table of integrals supplied with the test. However, books, notes, calculators, computers, cellphones may NOT be used.
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Math 115 Exam 1 Practice This exam is meant to suggest the kind, diculty, and number of problems on the real exam. Note that topics covered in the course, but not included on this practice exam, still might appear on the real exam. Directions Set aside 90
School: Yale
Course: CalcOfFunctionsOfOneVariableII
1. Find the areas between these curves x = y 2 + 2y 4 and x = y + 2. 2. Find the volume of the solid formed by rotating the region between y = x2 and y = 2x about y = 2. 3. (a) Find the area enclosed by the curve r = 1 cos(), 0 2. (b) Find its arclength.
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Math 115 Final Practice This exam is meant to suggest the kind, diculty, and number of problems on the real exam. Note that topics covered in the course, but not included on this practice exam, still might appear on the real exam. Directions Set aside 3 1
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 1 Math 115 October 3rd, 2012 Name:_ Instructor:_Section:_ 1. D o not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instruct
School: Yale
Course: Linear Algebra With Applications
School: Yale
Course: Linear Algebra With Applications
School: Yale
Course: Calculus
2 x2 + 4 dx. 1. Find 0 e 2. Find (ln x)2 dx. 1 1 /2 arctan x dx. x 0 4. If the curve y = 2x x2 , 0 x 1 is rotated about the x-axis, nd the area of the resulting surface. 3. Find (1)n to within 0.01. (You do not n+1 n=1 need to carry out the computation, b
School: Yale
Course: Linear Algebra
Math 54 quiz solutions October 12, 2009 1 Dene the null space of a matrix (3 points). If A is an n m matrix, the null space of A is the set of vectors x in Rm such that Ax = 0. 2 Dene what it means for a set of vectors to be linearly dependent (3 points).
School: Yale
Course: Calculus Of Functions With One Variable
112b Midterm II Wed Apr 9, 2008 90 minutes Please answer all 7 parts below, in your blue books. Show all your work and justify clearly where appropriate. You may not use your books, notes or calculators on this exam. Good luck! 1. (16 pts.) Find the deriv
School: Yale
Course: Linear Algebra
Math 54 quiz solutions October 30, 2009 1 Dene eigenspace (2.5 points). For a matrix A and an eigenvalue of A, the -eigenspace is the set of all vectors x such that Ax = x. 2 Dene what it means for a matrix to be diagonalizable (2.5 points). A matrix A is
School: Yale
Course: Linear Algebra
Math 54 quiz solutions December 2, 2009 1 Find the Fourier cosine series expansion of f (x) = 3 + x dened on 0 < x < (7 points). For n > 0, we compute an using the formula: an = 2 (3 + x) cos nx dx 0 Use integration by parts with u = 3 + x, dv = cos nx,
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 1 Math 115 October 3rd, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor
School: Yale
Course: CalcOfFunctionsOfOneVariableII
SOLUTIONS TO EXAM 2, MATH 115, FALL 2012 1. (65 points) The graphs of f (x), g(x), and h(x) are shown below. You may assume that as x , the graphs of f , g, and h continue in a fashion similar to the trends observed in the graphs. 3 gx 2 1 hx f x 0 1 2 3
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 9 Solution Triet M. Le 1. Page 89: 8, 10, 11, 13. 15. 2. Page 91: 3,6. 3. Page 94: 2, 3, 6. Page 89: 8: B S being not closed implies B does not contain all its limit points. Let 1 a S be a limit point of B such that a B
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: HW 8 Solution Triet Le Remark 1. Let U be any collection of subsets of a generic set S. Then c U U U = U U U c. 1: If K = {K } is a collection of compact subsets of a metric space (S, d) such that the intersection of every fini
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: HW 7 Soln of Selected Problems Triet Le 1) Page 78: 5: Solution 1. Let A R C with the usual metric d(x, y) = |x - y|. Pick any x A, then for any > 0, x + i /2 N (x). Hence, N (x) is not a subset of A, for all > 0. And so x is n
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 6 Solution Triet Le 1. Page 66: 2. Determine the coefficients {an } of the power series whose sum is (1 - n=0 z)-2 . Solution 1. Let f (z) = (1 - z)-2 = n=0 an z n . Then an = f (n) (0) , n! where f (n) (0) = (n + 1)!
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 4 Solutions to Selected Problems Triet Le October 11, 2007 1. Page 44: 10. Prove that M [0, ) = 0, 1 , where M is the Mandelbrot set. 4 Solution 1. To show [0, 1/4] M [0, ): Suppose c [0, 1/4]. Claim the sequence {zn
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 5 Solution Triet Le October 14, 2007 1. Let K be a fixed natural number and let be a map from N to N such that is 1-to-1 from {(n - 1)K, (n - 1)K + 1, ., nK} onto itself, for all n = 1, 2, 3. Suppose n=0 an converges.
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 3 Solution Triet Le October 3, 2007 1. Let {zn } be a complex sequence. Show that n lim zn = z lim Re(zn ) = Re(z) and lim Im(zn ) = Im(z). n n Solution 1. Let {zn } be a complex sequence. () : Suppose limn zn = z. Then
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 2 Solution Triet Le September 21, 2007 1. Show that for any reals x and y with x < y, there are a rational r and an irrational t such that x < r < y and x < t < y. Solution 1. Suppose that x > 0. Let z = y - x > 0 and let
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513, Homework 1 Solution Triet Le September 17, 2007 1. Recall: Z = {m - n : m, n N}. Each n N is identified with the expression (n+m)- m for any m N. Addition on Z is defined as: (m-n)+(m -n ) = (m+m )-(n+n ). Let k, m, n N such th
School: Yale
Course: Vector Analysis
Mathematics 250 Fall 2006 Selected Solutions Assignment #8 Proof Practice # 8: Suppose that A is an invertible k k matrix, and that B is another k k matrix |A-1 | such that |B - A| < |A1 | . Show that B is invertible, and that |B -1 | 1-|B-A|A-1 |
School: Yale
Course: Vector Analysis
Mathematics 250 Fall 2006 Selected Solutions, Assignment #6 Hubbard and Hubbard, Problem 3.3.5: Find the cardinality of the set of multiexponents of degree m in with n entries. (Equivalently, find the number of monomials of degree m in n variables.
School: Yale
Course: Integral Calculus
Yale University, Department of Mathematics Math 115 Calculus Fall 2013 Final Exam Review Guide When/Where. The Final Exam will take place during 7:00 10:30 pm on Sunday, December 15th, 2013 in Davies Auditorium, located underground between Dunham Labs and
School: Yale
Course: Calculus
Math 1B Problems, volume 2 Dustin Cartwright1 1. Which of the following are p-series or can be treated as p-series? Which are geometric series? If it is a p-series or geometric series, say whether it converges. 3 n (a) n=0 (b) 1 + (c) n=1 1 1 1 + 3 + 4 +.
School: Yale
Course: Calculus
Math 1B Problems, volume 3 Dustin Cartwright1 1. Let P (x) = 2 3x2 + 5x3. What is P (n)(0) for n = 1, 2, 3, 4, 5? What is the Taylor series for P ? 2. Show that the function J0(x) = (1)n x2n 22n(n!)2 n=0 satises the dierential equation x2J0 (x) + xJ0(x) +
School: Yale
Course: Calculus
Math 1B Problems, volume 4 Dustin Cartwright1 1. Solve y 4y = ex cos x using the method of undetermined coecients with the initial conditions y(0) = 1 and y (0) = 2. 2. Solve y + 3y + 2y = sin(ex ). 3. Solve y 2y + y = ex . 1 + x2 4. Find the general solu
School: Yale
Course: Calculus
Math 1B Problems Dustin Cartwright1 1. If f is a continuous function on [a, b] and b g(x) = f (t) dt x What is g (x)? 2. Find a function f and a number a such that x 4+ a f (t) dt = 2 x t2 for all x > 0. 3. Find the value of n lim n i=1 i3 n4 (Hint: The s
School: Yale
Week 7 Important Discrete Distributions Expectation and Variance Lecture 17. Bernoulli and Binomial Distributions. Expectation Revisited. Discrete Uniform Distribution. We have seen some examples that all outcomes of an experiment are equally likely. Let
School: Yale
Week 5 Lecture 11. Conditional distribution and Conditional density. Review. Discrete random variables X1 ; X2 ; ; Xn are mutually independent if P (X1 = x1 ; X2 = x2 ; ; Xn = xn ) = P (X1 = x1 ) P (X2 = x2 ) : : : P (Xn = xn ) . Continuous random variabl
School: Yale
Week 4 Lecture 8. Discrete conditional distribution. Examples 1: A doctor gives a patient a test for a particular cancer. Before the results of the test, the only evidence the doctor has to go on is that 1 woman in 1000 has this cancer. Experience has sho
School: Yale
Week 6 Lecture 14 Random Walks Drunkard Walk. Imagine now a drunkard walking randomly in an ideals ized 1 dimensional city ( or 2 dimensional, or 3 and higher dimensional city). The city is eectively innite and arranged in a 1 dimensional equally-spaced g
School: Yale
Week 3 Lecture 5. Expectation Probability Density Function: Let f (x) 0 and P (E ) as following Z P (X 2 E ) = f (x) dx. R f (x) dx = 1. Dene E Are the probability axioms satised? It is important to observe that there a similar paradox in the calculus Za
School: Yale
Week 2 Lecture 3. Expectation and Probability axioms. Random variable. A random variable is a real-valued function dened on the sample space, i.e., X (! ) is a function from to R. For example, for = fBB; BG; GB; GGg, your X could be the number of boys, th
School: Yale
Course: IntroFunctionsSeveralVariables
Physics 181 To dos:! 1. Register online on classesv2! 2. If pretty sure of taking class, you need a clicker! !Get a clicker from Bass library !Register the clicker on classesv2 under ! !Tests & Quizzes by Wed. Jan 19 3.3. Read Ch. 22 for Fri. Jan. 14! 4.
School: Yale
Course: IntroFunctionsSeveralVariables
Physics 181 University Physics Natural continuation of Physics 180 Todays handout: syllabus Requirements: book, clickers, classesv2, online homeworks: due before lecture (classesv2) written homeworks: due before lecture each Wed. Grading, policies, etc.
School: Yale
Course: CalcOfFunctionsOfOneVariableII
SOLUTIONS TO EXAM 2, MATH 115, FALL 2012 1. (65 points) The graphs of f (x), g(x), and h(x) are shown below. You may assume that as x , the graphs of f , g, and h continue in a fashion similar to the trends observed in the graphs. 3 gx 2 1 hx f x 0 1 2 3
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 1 Math 115 October 3rd, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 2 Math 115 April 10th, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructor w
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 2 Math 115 November 13th, 2013 Name: Instructor: Section: 1. Do not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instructo
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Mathematics 115b Mid-term # 1 2/19/08 Name: Section: Please answer all SIX questions; each question is worth TEN points. You may use the table of integrals supplied with the test. However, books, notes, calculators, computers, cellphones may NOT be used.
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Math 115 Exam 1 Practice This exam is meant to suggest the kind, diculty, and number of problems on the real exam. Note that topics covered in the course, but not included on this practice exam, still might appear on the real exam. Directions Set aside 90
School: Yale
Course: CalcOfFunctionsOfOneVariableII
1. Find the areas between these curves x = y 2 + 2y 4 and x = y + 2. 2. Find the volume of the solid formed by rotating the region between y = x2 and y = 2x about y = 2. 3. (a) Find the area enclosed by the curve r = 1 cos(), 0 2. (b) Find its arclength.
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Math 115 Final Practice This exam is meant to suggest the kind, diculty, and number of problems on the real exam. Note that topics covered in the course, but not included on this practice exam, still might appear on the real exam. Directions Set aside 3 1
School: Yale
Course: CalcOfFunctionsOfOneVariableII
Exam 1 Math 115 October 3rd, 2012 Name:_ Instructor:_Section:_ 1. D o not open this exam until you are told to do so. 2. Do not separate the pages of this exam. If they do become separated, write your name on every page and point this out to your instruct
School: Yale
Course: Linear Algebra With Applications
School: Yale
Course: Linear Algebra With Applications
School: Yale
Course: Calculus
2 x2 + 4 dx. 1. Find 0 e 2. Find (ln x)2 dx. 1 1 /2 arctan x dx. x 0 4. If the curve y = 2x x2 , 0 x 1 is rotated about the x-axis, nd the area of the resulting surface. 3. Find (1)n to within 0.01. (You do not n+1 n=1 need to carry out the computation, b
School: Yale
Course: Linear Algebra
Math 54 quiz solutions October 12, 2009 1 Dene the null space of a matrix (3 points). If A is an n m matrix, the null space of A is the set of vectors x in Rm such that Ax = 0. 2 Dene what it means for a set of vectors to be linearly dependent (3 points).
School: Yale
Course: Calculus Of Functions With One Variable
112b Midterm II Wed Apr 9, 2008 90 minutes Please answer all 7 parts below, in your blue books. Show all your work and justify clearly where appropriate. You may not use your books, notes or calculators on this exam. Good luck! 1. (16 pts.) Find the deriv
School: Yale
Course: Linear Algebra
Math 54 quiz solutions October 30, 2009 1 Dene eigenspace (2.5 points). For a matrix A and an eigenvalue of A, the -eigenspace is the set of all vectors x such that Ax = x. 2 Dene what it means for a matrix to be diagonalizable (2.5 points). A matrix A is
School: Yale
Course: Linear Algebra
Math 54 quiz solutions December 2, 2009 1 Find the Fourier cosine series expansion of f (x) = 3 + x dened on 0 < x < (7 points). For n > 0, we compute an using the formula: an = 2 (3 + x) cos nx dx 0 Use integration by parts with u = 3 + x, dv = cos nx,
School: Yale
Course: Linear Algebra
Math 54 quiz solutions November 23, 2009 1 Find a general solution to the system: (5 points) x (t) = 1 3 2 1 x(t) + . 2 1 The characteristic equation of the matrix is: 1 3 2 = 2 3 + 2 6 = 2 3 4 = ( 4)( + 1), 2 so the eigenvalues are 4 and 1. For = 4, 3 3
School: Yale
Course: Linear Algebra
Math 54 quiz solutions November 18, 2009 1 Find a general solution (6 points) for the system x (t) = Ax(t) A= 13 . 12 1 Find the solution with initial conditions x(0) = 2 (4 points). 0 The characteristic polynomial of A is: 1 12 3 = 1 2 + 2 36 = 2 2 35 =
School: Yale
Course: Linear Algebra
Math 54 quiz solutions November 4, 2009 1 Find the general solution to the dierential equation (5 points) y + y 2y = e2t The auxillary equation of the homogeneous dierential equation is r2 + r 2 = (r + 2)(r 1) so the roots are r = 2 and r = 1. Thus, the g
School: Yale
Course: Linear Algebra
Math 54 quiz solutions October 28, 2009 1 Find the equation y = 0 + 1 x of the least-squares line that best ts the data points: (1, 0), (2, 1), (4, 2), (5, 3). (5 points) We have the matrices: 1 1 X= 1 1 0 1 . and y = 2 3 1 2 4 5 The normal equations ar
School: Yale
Course: Linear Algebra
Math 54 quiz solutions October 21, 2009 1 Find an invertible matrix P and a matrix C such that 5 1 5 = P CP 1 1 where C = a b b . a (4 points) The characteristic equation is 5 1 5 = 5 6 + 2 + 5 = 2 6 + 10. 1 By the quadratic formula, its roots are: 6 36 4
School: Yale
Course: Linear Algebra
Math 54 quiz solutions October 14, 2009 1 Find matrices P and D, with D diagonal, such that: (4 points) 2 4 3 = P DP 1 . 1 The characteristic equation of the matrix is 2 4 3 = 2 3 + 2 12 = 2 3 10 = ( 5)( + 2). 1 (The factorization was found by looking for
School: Yale
Course: Linear Algebra
Math 54 quiz solutions September 25, 2009 1 Use Cramers rule to nd the value of x3 in the solution to the equation: 3 2 5 x1 2 1 1 3 x2 = 1 6 1 6 x 3 3 Call the matrix on the left hand side A. We compute the determinant of A by expanding by minors along
School: Yale
Course: Linear Algebra
Math 54 quiz solutions September 23, 2009 1 Find the determinant of h A= 3 4 2 1 . 1 3 6 5 (4 points). For what values of h is A invertible (2 points)? We expand by minors along the rst row: det A = h 6 5 3 1 3 4 1 3 1 +2 4 1 6 5 = h(6 5) 3(3 + 4) + 2(15
School: Yale
Course: Linear Algebra
Math 54 quiz solutions 1 Here is a matrix and 1 1 A= 2 4 September 16, 2009 its echelon form: 2 1 0 1 1 954 6 5 3 0 6 1 2 0 0 9 1 9 2 1 0 0 9 3 0 0 54 0 7 1 2 00 Find bases for Col(A) and Nul(A) and state the dimensions of these subspaces. (6 points) A ba
School: Yale
Course: Linear Algebra
Math 54 quiz solutions September 9, 2009 1. Describe the solutions to the following in parametric vector form: (3 points) x 3 1 4 0 1 0 2 1 2 5 x2 = 0 x3 1113 0 x4 We use elementary row operations to row reduce the matrix: 3140 1113 1113 2 1 2 5 2 1 2 5
School: Yale
Course: Linear Algebra
Math 54 quiz solutions September 2, 2009 1 Find all solutions to the system of equations: x1 4x2 + 7x3 = 5 x2 4x3 = 3 2x1 6x2 + 6x3 = 4 We form the augmented matrix and use row operations to echelon form: 1 4 7 5 1 4 7 5 1 4 0 1 4 3 0 1 4 3 0 1 2 6 6 4 0
School: Yale
Course: Linear Algebra With Applications
Math 222a - Linear Algebra & Applications Fall 2013 (Prof. Folsom) Yale University Study Guide for Midterm 2 General information about Midterm 2. (Friday Nov. 8th 10:30am-11:20am) Our second exam will consist of 3-5 problems, some consisting of multiple
School: Yale
Course: Linear Algebra With Applications
Math 222a - Linear Algebra & Applications Fall 2013 (Prof. Folsom) Yale University Study Guide for Midterm I General information about Midterm I. (Friday Oct. 4th 10:30am-11:20am) Our rst exam will consist of 3-5 problems, some consisting of multiple par
School: Yale
Course: Multivariable Calculus
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School: Yale
Course: Multivariable Calculus
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School: Yale
Course: Multivariable Calculus
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School: Yale
Course: Multivariable Calculus
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School: Yale
Course: Multivariable Calculus
Math 120b Exam 2 1. Suppose is the portion of the rst quadrant bounded by y = Evaluate 1 dA x2 + y 2 2. Evaluate /2 /2 y 0 y2 4 x2 and y = 1 x2 . sin(x) dxdy x 3. Integrating the vector elds along the paths indicated, which of these lines integrals is 0.
School: Yale
Course: Multivariable Calculus
Math 120b Exam 2 Solutions 1. In the rst quadrant, the curves y = 4 x2 and y = 1 x2 are the portions of the circles of radius 2 and 1 in the rst quadrant. That is, the region has this form. The natural coordinate system for this region is polar, and the r
School: Yale
Course: Multivariable Calculus
Math 120b Exam 1 Solutions 1. (a) A line is parallel to a plane if the direction vector of the line is perpendicular to the normal vector of the plane. The direction vector of this line is 1, 0, 1 ; the normal vector of this plane is 1, 2, 1 . Because 1,
School: Yale
Course: Calculus
Math 1B quiz solutions October 20, 2006 1 Does 10n converge or diverge? (5 points) n 3n+1 n=1 All the terms are positive, so we can use a limit comparison test with bn = 1/n: an 10n/(n 3n+1 ) = lim n bn n 1/n lim 1 n 3 = lim 10 3 n which diverges to becau
School: Yale
Course: Calculus
Math 1B quiz solutions October 18, 2006 1 Find the radius of convergence and the interval of convergence for xn . (3n)! n=0 (5 points) Use the ratio test: lim n an+1 xn+1 /(3(n + 1)! = lim n an xn/(3n) |x|(3n)! = lim n (3n + 3)! |x| = lim n (3n + 1)(3n +
School: Yale
Course: Calculus
Math 1B quiz solutions October 25, 2006 1 around x = 0. (Hint: Use a 1 + x2 power series you already know) (5 points) 1 Find a power series representation for This is similar to the expression for the geometric series, so 1 (1)n x2n = 1 + x2 n=0 2 Show th
School: Yale
Course: Calculus
Math 1B quiz solutions October 25, 2006 cn(x 3)n is a power series which converges absolutely for x = 0, what 1 If n=0 can you say about its radius of convergence? (3 points) What if it converges conditionally for x = 0? (2 points) This is a power series
School: Yale
Course: Calculus
Math 1B quiz solutions November 1, 2006 1 Find the power series for terms (2 points). 3 1 x (3 points). Write out the rst 3 non-zero Using the binomial series, we get: 3 1 x = (1 x)1/3 = n=0 = n=0 1/3 (x)n n (1)n 1/3 n x n (1/3)(1/3 1) 2 1/3 x+ x + 1! 2!
School: Yale
Course: Calculus
Math 1B quiz solutions November 3, 2006 1 Find the power series for doesnt use k n 1 (x+2)2 (2 points) and simplify it to the point that it , factorials (!), or products of the form 1 2 n (3 points). We have to use a little bit of algebra to get this into
School: Yale
Course: Calculus
Math 1B quiz solutions November 8, 2006 1 If y = F (x, y) is a dierential equation, y0 = y(x0 ) is an initial condition, and the step size is h, what is the next point produced by Eulers method? (Hint: the x coordinate is x0 + h) (5 points) The point (x0
School: Yale
Course: Calculus
Math 1B quiz solutions November 8, 2006 1 Sketch a direction eld for the dierential equation y = x/y (3 points) and sketch a solution (2 points). (Note that the dierential equation is not dened for y = 0). The solution curve is a half-circle centered at t
School: Yale
Course: Calculus
Math 1B quiz solutions November 14, 2006 1 Find the general solution to the equation y + y/x + x2 1 = 0. (5 points) This is a linear dierential equation after rearranging to get the standard linear form: y + y/x = x2 + 1 First we nd the antiderivative of
School: Yale
Course: Calculus
Math 1B quiz solutions November 15, 2006 1 Find the general solution to y x 2 = y + ex . (5 points) First, we rearrange to get the standard form for a linear dierential equation: y yx = xex 2 Now we nd the integrating factor: R e x dx = ex 2 /2 Then we mu
School: Yale
Course: Calculus
Math 1B quiz solutions (Note: November 29, 2006 tan u du = ln | sec u| + C ) 1 Find the general solution to y + 4y + 3y = ex + sin x using the method of undetermined coecients (7 points). The auxillary equation is r 2 + 4r + 3 = 0. This has solutions r =
School: Yale
Course: Calculus
Math 1B quiz solutions November 29, 2006 1 Find the general solution to y + 2y + y = 5 sin 2x using undetermined coecients (7 points). The auxillary equation is r 2 + 2r + 1 = 0 which has a root only at r = 1. Thus the complementary solution is y = ex + x
School: Yale
Course: Calculus
Math 1B quiz solutions December 6, 2006 1 Find a general solution to y + 2xy + 2y = 0 as a power series (10 points). We write y = n=0 anxn and then take the derivatives: nanxn1 y= n=1 y= n=2 n(n 1)anxn2 Then we substitute these into the dierential equatio
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Final (Tue 08 May 2012, 08:3011:00 am, MSC 301) Study Guide and Practice Exam Section I 1. Determine whether the following statements are always true (T), sometime
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Final (Tue 08 May 2012, 08:3011:00 am, MSC W301) Study Guide and Practice Exam Directions: You will have 2 hour and 30 minutes for the nal exam. No electronic devi
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Midterm # 1 (Tue 21 Feb 2012) Practice Exam Solution Guide Practice problems: The following assortment of problems is inspired by what will appear on the midterm e
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Midterm # 1 (Tue 21 Feb 2012) Practice Exam Directions: You will have 1 hour and 10 minutes for the midterm exam. No electronic devices will be allowed. No notes w
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Midterm # 2 (Thu 05 Apr 2012) Practice Exam Solution Guide Practice problems: The following assortment of problems is inspired by what will appear on the midterm e
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Midterm # 2 (Tue 03 Apr 2012) Practice Exam Directions: You will have 1 hour and 10 minutes for the midterm exam. No electronic devices will be allowed. No notes w
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Quiz # 1 St. Valuentines Day 2012 Solutions Directions: You had 20 minutes. No electronic devices were allowed. No notes were allowed. 1. Let S be the plane in R3
School: Yale
Course: Linear Algebra
Math 54 worksheet, September 14, 2009 1. Let T be the linear transformation from R3 to R3 which consists of rst rotating 30 degrees about the z axis and then rotating 90 degrees about the x axis. (I havent specied the directions of the rotations. Use whic
School: Yale
Course: Calculus
Math 1B - Fall 2006 10/09/2006 Integrals sin2 x cos2 x dx Use 2 sin x cos x = sin 2x, then use Ex 1. Ex 2. 0 1 (1 cos 4x) = sin2 2x. 2 4 dx 1 1 dx Try x = tan , dx = sec2 d. 2+1 4x 2 2 Ex 3. Ex 4. 4 dx 4 + e2x x Use Weierstrass substitutions, let t = tan
School: Yale
Course: Linear Algebra
Math 54 worksheet, September 21, 2009 1. Find the inverse of 1 A = 3 2 2 4 , 4 0 1 3 using both row reduction and the adjugate. Check that you get the same answer. A1 8 = 10 3 4 3 2 7 2 1 1 1 2 2. Use the adjugate to nd a formula for the inverse of a 2 2
School: Yale
Course: Linear Algebra
Math 54 worksheet, September 30, 2009 1. Write the solutions to the following system of equations in parametric vector form: 3x1 + 2x2 + 2x3 = 7 2x1 2x2 + 8x3 = 8 x1 + 4x2 6x3 = 1 The solutions are: x1 3 2 x2 = 1 + t 2 x3 0 1 2. Find bases for the row
School: Yale
Course: CalcFunctionsSeveralVariables
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 11 Due date - Nov 30 1. Section 4.5: 4, 16, 28, 34, 46. 2. The length a of the horizontal axis of an ellipse is decreasing at a rate of 2 inches per second, while the length of its vertical axis b is increasing at a rate of 1 inch p
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 6 Due date - Oct 12 1. Section 3.3: 4, 6, 14, 22, 40 (consider multiplying and dividing by something), 44. x x 2. Section 3.4: 8, 12, 34, 40, 50 (note that ee means e(e ) , and not (ee )x ), 60, 72. Practice problems (do but do not
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 10 Due date - Nov 16 1. Section 4.3: 2, 8, 14, 18, 22, 30. 2. Section 4.4: 4 (note that the instructions appear before question 1) , 8, 18, 20, 28, 32, 38. Practice problems (do but do not submit): 1. Section 4.3: 1, 7, 13, 17, 19,
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 12 Due date - Dec 7 1. Section 4.9: 2, 14, 20, 26, 28, 46, 54. 2. Secion 5.1: 4, 20. 3. Section 5.2: 34, 36. 4. Section 5.3: 8, 14, 24, 28, 36, 42, 68. Practice problems (do but do not submit): 1. Section 4.9: 3, 13, 15, 25, 27, 45,
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 9 Due date - Nov 9 1. Section 4.1: 4, 14, 22, 32, 40, 44, 56, 78. 2. Section 4.2: 2, 8, 12, 16, 18, 32. Practice problems (do but do not submit): 1. Section 4.1: 3, 13, 21, 33, 39, 47, 55. 2. Section 4.2: 1, 7, 9, 15, 19. 1
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 8 Due date - Nov 2 1. Section 3.9: 8, 12 (notice that the instructions for this problem appear just before quesiton 11), 16, 22, 30, 40. 2. Let f (x) = cos( x). (a) Find the linear approximation of f near x = 1 . 3 (b) Sketch the gr
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 7 Due date - Oct 19 1. Section 3.5: 6, 10, 14, 50, 54. 2. Section 3.6: 2, 10, 20, 28, 44, 48. 3. Section 3.8: 8, 18(a). 4. During the 1940s, scrolls which are now known as the Dead Sea Scrolls, were found in a cave in Khirbet Qumran
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 3 Due date - Sep 21 1. Section 2.2: 2, 6, 8, 20, 26 2. Section 2.3: 2, 8, 10, 16, 18, 22, 28, 40, 42, 48 3. Section 2.5: 4, 6, 14, 40, 50, 52 4. Let f ( x) = ( 0 x0 1 x>0 g ( x) = x2 (a) Show that lim f x!0 g (x) 6= f ( lim (g (x).
School: Yale
Course: Calculus Of Functions With One Variable
Math 112 - Problem set 4 Due date - Sep 28 1. Section 2.6: 4, 6, 16, 18, 20, 24, 40(c), 44, 62 2. Section 2.7: 6, 10(a), 16, 22, 24 3. Section 2.8: 2, 4, 8, 12, 22, 24, 40, 46 1
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 7 SOLUTIONS DANIEL COREY Section 5.1 4. M23 0 = 0 3 2 2 6 0 0 M31 6 2 = 1 6 4 1 1 0 0 . 6 14. a) Expanding along the bottom row det A = (1)(4+2) 2 1 4 3 0 0 0 2 1 + (1)(4+3) (2) 1 1 4 1 2 2 0 1 = (12 3) 2(4 + 4 4 1) = 3 1 b) Expanding al
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 6 SOLUTIONS DANIEL COREY Section 4.2 4. These vectors cannot form a basis since they are linearly dependent. (It looks like u1 and u3 are dependent. In any event, three vectors in R2 must be dependent) 8. Let A be the matrix whose rows a
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 5 SOLUTIONS DANIEL COREY Section 3.3 10. The augmented matrix 1 2 1 4 7 7 1 1 5 1 0 0 10 0 0 row reduces to 0 1 1 00 0 1 0 0 28 13 0 1 3 9 7 4 3 1 1 So the inverse of the matrix is 28 13 3 7 3 1 9 4 1 18. x1 1 x2 = 4 x3 1 1 1 2 28 7 1
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 4 SOLUTIONS DANIEL COREY Section 3.1 18. T is not linear since T (0, 0, 0) = (0, 4, 0) = (0, 0, 0) (see the solution to exercise 55 below). 22. The columns are linear dependent, so T is not one-to-one. The span of the columns is not all
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 3 SOLUTIONS DANIEL COREY Section 2.2 16. 9 5 2 5 3 A= 1 x x= 1 x2 13 b = 9 2 30. The columns do not span R2 , for example e1 = (1, 0) is not in the span of the columns. 40. h can be any real number except h = 12/5 (this value of h is pre
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 2 SOLUTIONS DANIEL COREY Section 1.4 2. This amounts to solving the system corresponding to the augmented matrix 1 0 0 1 1 1 0 0 55 0 1 0 1 0 1 1 0 55 which row reduces to 0 0 1 1 1 0 0 1 30 0 0 1 1 30 000 0 30 25 30 0 So the possible v
School: Yale
Course: Linear Algebra With Applications
LINEAR ALGEBRA HW 1 SOLUTIONS DANIEL COREY Section 1.1 18. This system is not in echelon form because the leading variable in the third row is to the left of the leading variable in the second row. 20. This system is in echelon form. The leading variables
School: Yale
Course: Calculus
Solutions to some limit convergence denition problems Dustin Cartwright October 2, 2006 These are some possible questions which I came up with involving the definition of a convergent series, and solutions to these questions. The solutions are probably mo
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 1 (due Friday 27 Jan 2012) Solutions 1. Graphs of Multivariable Functions a) Let f : R2 R be a function on R2 . Give R3 the standard (x, y, z ) coord
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 2 (Fri 03 Feb 2012) Solutions 1. CM 14.2 Exercise 28 Solution. xy y 2 exy xy x ye ln(ye ) = = =y x yexy yexy Problem 40 1 Solution. Well, f (65, 16
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 3 (Fri 10 Feb 2012) Selected Solutions 1. CM 14.4 Exercise 14 Solution. z |(x,y) = 1 (yey (x+y )2 + ey (x2 + xy x) Exercise 24 Solution. f |(1,2)
School: Yale
Course: Multivariable Calculus
Emory University Department of Mathematics & CS Math 211 Multivariable Calculus Spring 2012 Problem Set # 4 (Fri 17 Feb 2012) Selected Solutions 1. Let P be a point in R3 and let v be a direction vector at P . Find a parameterization of the line through P
School: Yale
STAT 241/541 Homework 2 Solution Prepared by James Hu Problem 1: Ch2.2:2 (a) c = 1/48 . b 1 1 xdx = (b2 a2 ) . 96 a 48 10 1 75 (c) Pr(X > 5) = xdx = , Pr(X < 7) = 48 96 5 (b) Pr(E ) = 7 1 45 xdx = , 96 2 48 24 3 and Pr(X 2 12X + 35 > 0) = 1 Pr(5 < X < 7)
School: Yale
Homework 2 Due September 13. Chapter 2.2: 2, 4, 6, 8, 12. Problem 6: Let X U (0; 1). (i) Find the density of Y = 1=X and EY . X (ii) Find the density of Y = tan 2 and EY . 1
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 1: -algebras and Borel sets The due date for this assignment is Thursday 9/9/2010. 1. Let A be the set of numbers in [0; 1] which admit decimal expansions such that the digits 2; 4; 6; 8 all appear at lea
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 1: -algebras and Borel sets The due date for this assignment is Thursday 9/9/2010. 1. Let A be the set of numbers in [0; 1] which admit decimal expansions such that the digits 2; 4; 6; 8 all appear at lea
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 3: Measurable Functions and Their Unexpected Properties The due date for this assignment is 9/30 1. Consider the Cantor function : [0; 1] P [0; 1]. It is dened as follows: For each ! n x 2 C , the Cantor
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 2: Measures, Outer Measures, and Weird Sets The due date for this assignment is Thursday 9/16/2010. 1. Let (X; A; ) be a measure space and fAi g a sequence of A-measurable sets such that 1 X i=1 (Ai ) < 1
School: Yale
Course: Measure Theory And Integration
Math 320 Measure Theory and Integration Assignment 2: Measures, Outer Measures, and Weird Sets The due date for this assignment is Thursday 9/16/2010. 1. Let (X; A; ) be a measure space and fAi g a sequence of A-measurable sets such that 1 X i=1 (Ai ) < 1
School: Yale
Course: Optimization I
Answers to assignment due on October 14th. 2. Program 1D: z * = min y b, subject to yA c. To place this thing in the format of Program 12.1, we need to express it as a maximization problem with nonnegative variables and equality constraints. Lets replace
School: Yale
Course: Calogero-MoserSystemsRepTheory
Solutions for Homework 1 Problem 1Shifrin 2.3.7: Problem 2Shifrin 2.3.8.bcefi: Problem 3Shifrin 2.3.16: Problem 4Shifrin 2.3.18 (solution by Bonnie Yn): Problem 5: Problem 6:
School: Yale
Course: CANCELLED
#3: (1 + 1/n)2 Limit of sequence: 0 (1+1/n)^3 (1+1/n)^4 (1+1/n)^n Graphs a, b, and c all approach zero. The graphs reach the limit of zero much faster as the exponential power increases. Graph d achieves a limit different from the previous graphs
School: Yale
Course: CANCELLED
2 1.5 1 0.5 -1 -0.5 0.5 k=0 1 1 0.5 -1 -0.5 0.5 1 -0.5 -1 k=1 1 0.5 -1 -0.5 0.5 1 -0.5 -1 k=3 0.2 0.1 -1 -0.5 -0.1 0.5 1 -0.2 k=5 0.03 0.02 0.01 -1 -0.5 -0.01 -0.02 0.5 1 k=7 0.004 0.002 -1 -0.5 -0.002 0.5
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 12 Solution Triet M. Le Page 116: 2: Let f (x) = 0, x0 e-1/x , x > 0. To show f is infinitely differentiable at x = 0. Clearly, f is infinitely differentiable on (-, 0) (0, ), with 0, x<0 f (n) (x) = -1/x P2n (1/x)e , x
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 11 Solution Triet M. Le Page 107: 3(a): Let P be a polynomial of odd degree. One sees that limx- P (x) = - and limx P (x) = . Let a, b R such that P (a) < 0 and P (b) > 0. Clearly, P is continuous on [a, b]. By the Interm
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 10 Solution Triet M. Le Page 104: 2: f is real on R and |f (x)| < M for all x R. Then for any x = y, the mean value theorem implies f (x) - f (y) = f (c), for some c between x and y. x-y Hence, for any > 0, |f (x) - f (y)
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 9 Solution Triet M. Le 1. Page 89: 8, 10, 11, 13. 15. 2. Page 91: 3,6. 3. Page 94: 2, 3, 6. Page 89: 8: B S being not closed implies B does not contain all its limit points. Let 1 a S be a limit point of B such that a B
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: HW 8 Solution Triet Le Remark 1. Let U be any collection of subsets of a generic set S. Then c U U U = U U U c. 1: If K = {K } is a collection of compact subsets of a metric space (S, d) such that the intersection of every fini
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: HW 7 Soln of Selected Problems Triet Le 1) Page 78: 5: Solution 1. Let A R C with the usual metric d(x, y) = |x - y|. Pick any x A, then for any > 0, x + i /2 N (x). Hence, N (x) is not a subset of A, for all > 0. And so x is n
School: Yale
Course: INTRODUCTION TO ANALYSIS
Math 301/ENAS 513: Homework 6 Solution Triet Le 1. Page 66: 2. Determine the coefficients {an } of the power series whose sum is (1 - n=0 z)-2 . Solution 1. Let f (z) = (1 - z)-2 = n=0 an z n . Then an = f (n) (0) , n! where f (n) (0) = (n + 1)!
School: Yale
STAT 241/541, Probability Theory with Applications Fall 2013 Instructor: Harrison H. Zhou (huibin.zhou@yale.edu) O ce hours: Wednesday 4:00-6:00pm (tentative) or by appointments, Room 204, 24 Hillhouse Ave., James Dwight Dana House. T.A.: Corey Brier <cor