Course Hero - We put you ahead of the curve!
You have requested the below document.
Sign up now to view this document for free!
- Title: sol2
- Type: Notes
- School: CUNY Queens
- Course: MATH 122
- Term: Fall
Coursehero >> New York >> CUNY Queens >> MATH 122
Path: CUNY Queens >> MATH >> 141 Fall, 2008
Path: CUNY Queens >> MATH >> 141 Fall, 2008
Path: CUNY Queens >> MATH >> 141 Fall, 2008
Path: CUNY Queens >> MATH >> 141 Fall, 2008
Path: CUNY Queens >> MATH >> 141 Fall, 2008
Path: CUNY Queens >> MATH >> 141 Fall, 2008
Path: CUNY Queens >> MATH >> 141 Fall, 2008
Path: CUNY Queens >> MATH >> 207 Fall, 2008
Path: CUNY Queens >> MATH >> 207 Fall, 2008
Path: CUNY Queens >> MATH >> 207 Fall, 2008
Path: CUNY Queens >> MATH >> 207 Fall, 2008
Path: CUNY Queens >> MATH >> 207 Fall, 2008
Path: CUNY Queens >> MATH >> 207 Fall, 2008
Path: CUNY Queens >> MATH >> 207 Fall, 2008
Path: CUNY Queens >> MATH >> 207 Fall, 2008
Path: CUNY Queens >> MATH >> 207 Fall, 2008
Path: CUNY Queens >> MATH >> 207 Fall, 2008
Path: CUNY Queens >> MATH >> 207 Fall, 2008
Path: CUNY Queens >> MATH >> 208 Fall, 2008
Path: CUNY Queens >> MATH >> 208 Fall, 2008
Path: CUNY Queens >> MATH >> 208 Fall, 2008
Path: CUNY Queens >> MATH >> 208 Fall, 2008
Path: CUNY Queens >> MATH >> 208 Fall, 2008
Path: CUNY Queens >> MATH >> 223 Spring, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 223 Spring, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 223 Spring, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 223 Spring, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 223 Spring, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 223 Spring, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 231 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 328 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Path: CUNY Queens >> MATH >> 631 Fall, 2008
Course Hero has millions of student submitted documents similar to the one below including study guides, homework solutions, papers, and exam answer keys.
Find millions of documents here - Study Guides, Homework Solutions, Papers, Exam Answer Keys and more.
Course Hero has millions of course related materials that will enable you to learn better, faster and get an A in all your courses.
Below is a small sample set of documents:
sol5.pdf
Path: CUNY Queens >> MATH >> 122 Fall, 2008
Description: ...
rev1_08.pdfPath: CUNY Queens >> MATH >> 141 Fall, 2008
Description: Math 141 Midterm 1 Review Sheet 9/27/2008 Here is an itemized list of the material that the rst midterm is based upon. Make sure that you study them carefully. For each topic, review the worked out examples in the book, examples that I did in lecture...
s8.pdfPath: CUNY Queens >> MATH >> 141 Fall, 2008
Description: HW 8 solutions Page 1 HW 8 solutions Page 2 ...
s11.pdfPath: CUNY Queens >> MATH >> 141 Fall, 2008
Description: HW 11 solutiuons Page 1 HW 11 solutiuons Page 2 HW 11 solutiuons Page 3 HW 11 solutiuons Page 4 HW 11 solutiuons Page 5 HW 11 solutiuons Page 6 ...
141ex1sol_08.pdfPath: CUNY Queens >> MATH >> 141 Fall, 2008
Description: Math 141 First Midterm Solutions Problem 1. [12 points] In each case, nd the limit: (i) lim x2 2x 3 x+1 x1 Both the numerator and denominator tend to zero as x 1. The key is to factor the numerator: x1 lim x2 2x 3 (x + 1)(x 3) = lim = lim (x ...
rev2sol_08.pdfPath: CUNY Queens >> MATH >> 141 Fall, 2008
Description: Math 141 Practice Test II Solutions 1. 1 x+ x 2 f (x) = = f (x) = 2 x + 1 1 x+ x x 1 1 1 2 =2 x+ x x 1 =2 x 3 . x 1 5 3 1 t (t2 + t + 1) = t 2 (t2 + t + 1) = t 2 + t 2 + t 2 53 31 1 1 = g (t) = t 2 + t 2 + t 2 . 2 2 2 g(t) = 1 r) ( 2r )(r2 ...
s4.pdfPath: CUNY Queens >> MATH >> 141 Fall, 2008
Description: HW4 solutions Page 1 ...
s7.pdfPath: CUNY Queens >> MATH >> 141 Fall, 2008
Description: HW 7 solutions Page 1 HW 7 solutions Page 2 ...
note0.pdfPath: CUNY Queens >> MATH >> 207 Fall, 2008
Description: This note will give a proof for the higher dimensional version of the chain rule. It will make use of the following simple fact: Lemma. Suppose T : Rn Rm is a linear map. Then, there is a constant C 0 such that T(x) C x for every x Rn . Proof. Le...
hw7.pdfPath: CUNY Queens >> MATH >> 207 Fall, 2008
Description: Math 207 Homework 7 due on Thursday 12/4/08 Problem 1. Find the critical points of the function f (x, y) = (x2 + y2 ) ex and determine their type. Problem 2. Find the global maximum and minimum of the function f (x, y) = x2 2xy + 2y over the closed ...
hw3.pdfPath: CUNY Queens >> MATH >> 207 Fall, 2008
Description: Math 207 Homework 3 due on Thursday 9/25/08 Problem 1. Let f : R2 R2 be the counter-clockwise rotation by angle and g : R R2 be the reection through the y-axis. (i) Find the matrices associated with f and g. (ii) Use (i) to nd the matrix associa...
hw3sol.pdfPath: CUNY Queens >> MATH >> 207 Fall, 2008
Description: Here are the solutions to HW 3 last two problems: Problem 3. Suppose A is a 2 2 matrix and 1 , 2 are the roots of the quadratic equation det(xI A) = 0 (these numbers are often called the eigenvalues of A). Show that 1 + 2 = tr(A) Solution. Let A = ...
hw6sol.pdfPath: CUNY Queens >> MATH >> 207 Fall, 2008
Description: Here is the solution to HW6 Problem 1 which we didnt discuss in class: Problem 1. Consider the path c(t) = (a cos t, a sin t, bt) tracing out a helix in R3 . Here a, b are positive constants. (i) Verify that the velocity vector v(t) = c (t) makes a c...
hw5.pdfPath: CUNY Queens >> MATH >> 207 Fall, 2008
Description: Math 207 Homework 5 due on Tuesday 11/4/08 Problem 1. Write the Jacobian matrix of each of the following maps: f (x) = (cos x, sin x) f (x, y) = (x3 , x2 y, xy2 , y3 ) f (x, y, z) = e(x 2 +y2 +z2 ) Problem 2. Suppose f : R R is a given twice di...
rev2_08.pdfPath: CUNY Queens >> MATH >> 207 Fall, 2008
Description: Math 207 Workshop 2 Thursday 11/13/08 1. Suppose c : R R3 is a differentiable path and c(t) has a local maximum or minimum at t0 . Show that the tangent vector c (t0 ) is perpendicular to c(t0 ). Interpret this geometrically. 2. Find all differentia...
linear_eq.pdfPath: CUNY Queens >> MATH >> 207 Fall, 2008
Description: ...
hw2.pdfPath: CUNY Queens >> MATH >> 207 Fall, 2008
Description: Math 207 Homework 2 due on Thursday 9/18/08 Problem 1. Suppose f : R2 R2 is the linear map which satises f (i) = i + j f (j) = 2i j. Compute f (3i 4j) and f ( f (3i 4j) in terms of i and j. Problem 2. There is only one linear map f : R3 R3 which...
hw6.pdfPath: CUNY Queens >> MATH >> 207 Fall, 2008
Description: Math 207 Homework 6 due on Tuesday 11/11/08 Problem 1. Consider the path c(t) = (a cos t, a sin t, bt) tracing out a helix in R3 . Here a, b are positive constants. (i) Verify that the velocity vector v(t) = c (t) makes a constant angle with the z-a...
207ex2sol_08.pdfPath: CUNY Queens >> MATH >> 207 Fall, 2008
Description: Math 207 Second Midterm Solutions Part I. In each case, decide whether the statement is true or false. Each correct answer has 2 points. The set {x R2 : x 1} is open. False. The point x = (1, 0) is in this set, but no ball centered at x is totall...
note1.pdfPath: CUNY Queens >> MATH >> 208 Fall, 2008
Description: Here is a rigorous proof of Fubinis Theorem on the equality of double and iterated integrals. The present version is slightly more general than the one stated in the textbook. Fubinis Theorem. Let f : Q = [a, b] [c, d] R be integrable. Suppose that...
VC5IntSup.pdfPath: CUNY Queens >> MATH >> 208 Fall, 2008
Description: Page i Internet Supplement for Vector Calculus Fifth Edition Version: October, 2003 Jerrold E. Marsden California Institute of Technology Anthony Tromba University of California, Santa Cruz W.H. Freeman and Co. New York Page i Contents Preface...
h3.pdfPath: CUNY Queens >> MATH >> 208 Fall, 2008
Description: Math 208 Homework 3 due on Thursday 3/1/07 Problem 1. Evaluate the following triple integrals: (i) D zex+y dx dy dz, where D is the box [0, 1] [0, 1] [0, 1]. xy 2 z 3 dx dy dz, where D is the solid in R3 bounded by the surface z = xy and the plane...
h6.pdfPath: CUNY Queens >> MATH >> 208 Fall, 2008
Description: Math 208 Homework 6 due on Thursday 4/26/07 Problem 1. Use Greens Theorem to evaluate y 2 dx + x dy, when is (i) the positively oriented square with vertices (1, 1). (ii) the positively oriented circle of radius 2 centered at the origin. Problem ...
h1.pdfPath: CUNY Queens >> MATH >> 208 Fall, 2008
Description: Math 208 Homework 1 due on Thursday 2/8/07 Problem 1. Use Cavalieris principle to prove the well-known formula V = 4 r3 for 3 the volume of a solid sphere of radius r. Problem 2. Compute the iterated integrals 1 0 0 /2 /2 1 (y cos x + 2) dx dy and v...
hw4.pdfPath: CUNY Queens >> MATH >> 223 Spring, 2008
Description: Math 328 Homework 4 due on Thursday 3/3/05 Problem 1. Let f (x) = x for 0 < x < 2 and extend f as a 2-periodic function x to the real line. Sketch the graphs of f (x) and F (x) = 0 f (t) dt over [4, 4]. Without computing anything, determine whether...
hw4.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 4 due on Thursday 3/3/05 Problem 1. Let f (x) = x for 0 < x < 2 and extend f as a 2-periodic function x to the real line. Sketch the graphs of f (x) and F (x) = 0 f (t) dt over [4, 4]. Without computing anything, determine whether...
hw7.pdfPath: CUNY Queens >> MATH >> 223 Spring, 2008
Description: Math 328 Homework 7 due on Thursday 3/31/05 Problem 1. Consider the wave equation 0 < x < , t > 0 utt = uxx u(0, t) = u(, t) = 0 t>0 u(x, 0) = sin(3x), u (x, 0) = sin(2x) 0 < x < t Verify that the solution u(x, t) given by the separation of v...
hw7.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 7 due on Thursday 3/31/05 Problem 1. Consider the wave equation 0 < x < , t > 0 utt = uxx u(0, t) = u(, t) = 0 t>0 u(x, 0) = sin(3x), u (x, 0) = sin(2x) 0 < x < t Verify that the solution u(x, t) given by the separation of v...
hw6.pdfPath: CUNY Queens >> MATH >> 223 Spring, 2008
Description: Math 328 Homework 6 due on Thursday 3/17/05 Problem 1. Find the (formal) solution of the ut = 4uxx u(0, t) = 1, ux (, t) = 0 u(x, 0) = x heat equation 0 < x < , t > 0 t>0 0<x< Problem 2. Show that the (formal) solution of the heat equation 0...
hw6.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 6 due on Thursday 3/17/05 Problem 1. Find the (formal) solution of the ut = 4uxx u(0, t) = 1, ux (, t) = 0 u(x, 0) = x heat equation 0 < x < , t > 0 t>0 0<x< Problem 2. Show that the (formal) solution of the heat equation 0...
hw5.pdfPath: CUNY Queens >> MATH >> 223 Spring, 2008
Description: Math 328 Homework 5 due on Thursday 3/10/05 Problem 1. Consider the function f (x) = C 0 0<x< L 2 L 2 x < L, that the (formal) solution of the heat 0 < x < L, t > 0 t>0 0<x<L nx . L where C and L > 0 are constants. Show equation ut = k uxx u(0,...
hw5.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 5 due on Thursday 3/10/05 Problem 1. Consider the function f (x) = C 0 0<x< L 2 L 2 x < L, that the (formal) solution of the heat 0 < x < L, t > 0 t>0 0<x<L nx . L where C and L > 0 are constants. Show equation ut = k uxx u(0,...
hw10.pdfPath: CUNY Queens >> MATH >> 223 Spring, 2008
Description: Math 328 Homework 10 due on Thursday 5/12/05 Problem 1. Recall that the solution to the one-dimensional heat equation ut = kuxx < x < , t > 0 u(x, 0) = f (x) is given by (xy)2 1 u(x, t) = f (y)e 4kt dy. 4kt Compute u(x, t) when the initial condi...
hw10.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 10 due on Thursday 5/12/05 Problem 1. Recall that the solution to the one-dimensional heat equation ut = kuxx < x < , t > 0 u(x, 0) = f (x) is given by (xy)2 1 u(x, t) = f (y)e 4kt dy. 4kt Compute u(x, t) when the initial condi...
hw2.pdfPath: CUNY Queens >> MATH >> 223 Spring, 2008
Description: Math 328 Homework 2 due on Thursday 2/17/05 Problem 1. Find the Fourier series of the following functions: (i) f (x) = cos x, < x < +. (ii) f (x) = cos x, < x < , extended as a -periodic function on the real line. 2 2 (iii) f (x) = 0 1 < x < 0 , ...
hw2.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 2 due on Thursday 2/17/05 Problem 1. Find the Fourier series of the following functions: (i) f (x) = cos x, < x < +. (ii) f (x) = cos x, < x < , extended as a -periodic function on the real line. 2 2 (iii) f (x) = 0 1 < x < 0 , ...
rev1sol_05.pdfPath: CUNY Queens >> MATH >> 231 Fall, 2008
Description: Math 231 Practice Test 1 Solutions Problem 1. (i) Set 24 6 A = 4 5 6 3 1 2 x y x= z 18 24 . b= 4 . .9 . . . 12 . . . 23 . . . (ii) We form the augmented matrix [A . b] . 2 4 6 . 18 1 . 12 3 2 R1 . . . b] = 4 5 6 . [A . 4 5 6 . 24 ...
h3.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 3 due on Thursday 3/1/07 Problem 1. Let f (x) = sin x for 0 < x < . (i) Sketch the graph of the -periodic extension of f over a few periods. Is this a piecewise smooth function? (ii) Compute the Fourier series of this extension. Use...
h1.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 1 due on Thursday 2/8/07 Problem 1. (i) Find the general solution u = u(x) of the rst order linear ODE x2 u + x u = 1. Then write down a formula for the solution which satises u(1) = 2. (ii) Find the general solution u = u(x) of the...
final_06.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Final Exam May 23, 2006 The exam has 5 problems, each worth 20 points. Return your work to Kiely 421 on Friday 5/26 at 1:15 pm. Your solutions must be complete and clear, showing all the steps along the way. You can use any result that ha...
sol6.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: ...
h9.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 9 due on Thursday 4/26/07 Problem 1. Find the Fourier integral of the functions f (x) = |x| + 1 |x| 1 0 |x| > 1 and 1 0<x<1 g(x) = 1 1 < x < 0 0 |x| > 1 For what values of x does each of the equalities F I(f )(x) = f (x) hold? ...
h10.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 10 due on Tuesday 5/8/07 Problem 1. Suppose f (x) has the Fourier transform F (). If a = 0, show that 1 f (ax) has the Fourier transform |a| F ( ). a Problem 2. Consider the function f (x) = ex 0 x0 . x<0 (i) Find the Fourier tr...
h2.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 2 due on Tuesday 2/20/07 Problem 1. Find the Fourier series of the following functions: (i) f (x) = cos x, < x < +. (ii) f (x) = cos x, < x < , f (x + ) = f (x) for all x. 2 2 (iii) f (x) = 0 1 < x < 0 , f (x + 2) = f (x) for al...
mid_06.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Midterm Exam March 21, 2006 The exam has 5 problems, each worth 20 points. Return your work on Thursday 3/23 before the start of lecture. Solutions must be complete and clear, showing all the steps along the way. Neatness counts! You can...
sol9.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: ...
sol7.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: ...
sol8.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: ...
mid_06sol.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Midterm Exam Solutions Problem 1. Consider the 2-periodic function dened by f (x) = 1 x + 1 1 < x 0 0x<1 and f (x + 2) = f (x). (i) Sketch the graph of f and its Fourier series over [3, 3]. Since f is piecewise smooth, the Basic Convergenc...
h6.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 6 due on Thursday 3/29/07 Problem 1. Consider the wave equation 0 < x < , t > 0 utt = uxx u(0, t) = u(, t) = 0 t>0 u(x, 0) = sin(3x), u (x, 0) = sin(2x) 0 < x < t Verify directly that the solution u(x, t) given by the separa...
s4.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: ...
s9.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: ...
s11.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: ...
s8.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: ...
s3.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: ...
mid_07.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Midterm Exam March 20, 2007 The exam has 5 problems, each worth 20 points. Return your work on Thursday 5/22 at the start of lecture. Solutions must be complete and clear, showing all the steps along the way. Neatness counts! You can use...
hh1.pdfPath: CUNY Queens >> MATH >> 328 Fall, 2008
Description: Math 328 Homework 1 due on Thursday 2/8/07 Problem 1. (i) Find the general solution u = u(x) of the rst order linear ODE x2 u + x u = 1. Then write down a formula for the solution which satises u(1) = 2. (ii) Find the general solution u = u(x) of the...
hw10.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 320 Homework 10 Due Tuesday December 2, 2003 Problem 1. Let f : R R be the function dened by f (x) = x2 x + 2 if x 1 if x > 1 Prove that f is not dierentiable at x = 1. Problem 3. What is wrong with the following proof of the Cauchy Mean ...
hw1.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 319/320 Homework 1 Problem 1. Write (in words) the negation of each of the following statements: (i) Jack and Jill are good drivers. (ii) All roses are red. (iii) Some real numbers do not have a square root. (iv) If you are rich and famous, you ...
hw5.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 320 Homework 5 Due Thursday October 16, 2003 Recall the following denitions: A set S R has the Bolzano-Weierstrass (BW) property if every innite subset of S has an accumulation point in S. A collection {Ui } of open sets in R is an open cover...
hw7.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 320 Homework 7 Due Thursday October 30, 2003 Problem 1. True or false? Justify your answer. If xn yn for all n and limn yn = , then limn xn = . The sequence {2cos n } has a convergent subsequence. If xn = 0 for all n and limn xn = 0, then...
hw8.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 320 Homework 8 Due Thursday November 6, 2003 Problem 1. True or false? Justify your answer. If lim supn xn = 2, then xn > 1.999 for all large n. There exists a sequence {xn } such that inf{xn : n N} = 0 even though lim inf n xn = 1. If f an...
rev2.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 320 Midterm 2 Review Sheet November 13, 2003 The exam covers everything up to the end of Chapter 5. The emphasis will be on the material we have discussed since the rst midterm, namely sequences, limits, and continuity. Here is a sample syllabus...
rev3sol.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Solutions to the practice problems Solution. Since |x y| < |x|, we have |x| < x y < |x| or x |x| < y < x + |x|. This shows x cannot be zero (for otherwise 0 < y < 0!). If x > 0, we get 0 < y < 2x, so xy > 0. If x < 0, we get 2x < y < 0, so again x...
hw4.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 319/320 Homework 4 Due Thursday October 2, 2003 Recap: For a given set S R, we dene the interior of S as int(S) = {x : some neighborhood of x is contained in S} the accumulation set of S as S = {x : every neighborhood of x contains a point of...
proj.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 320 Short Project Information Sheet The paper should address a single topic that is related to but not covered in the course. For example, it could be about an important concept, a theorem, a famous construction, or a famous counterexample (see...
hw9.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 320 Homework 9 Due Thursday November 13, 2003 Problem 1. True or false? Justify your answer. If f : D R is continuous and D is a closed set, then f (D) is a closed set. If f : D R is continuous, then |f | : D R is continuous. (Here |f | is...
hw6.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 320 Homework 6 Due Thursday October 23, 2003 Problem 1. True or false? Give a brief proof or a counterexample. If limn |xn | = 5, then either limn xn = 5 or limn xn = 5. If 0.9999 < xn < 1.0001 for all n 500, then limn xn = 1. If limn xn =...
hw6sol.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 320 Homework 6 solutions Problem 1. True or false? Give a brief proof or a counterexample. If limn |xn | = 5, then either limn xn = 5 or limn xn = 5. FALSE: Consider xn = (1)n 5. If limn xn = 1, then xn < 2 for all but nitely many values of n....
exam2_sol.pdfPath: CUNY Queens >> MATH >> 631 Fall, 2008
Description: Math 320 Second Midterm Solutions Problem 1. (i) Dene what it means for a set S R to be compact. Then state the HeineBorel Theorem. S is compact if every open cover of S has a nite subcover. The Heine-Borel Theorem states that S is compact if and on...