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328 Math 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 n x . L where C and L > 0 are constants. Show equation ut = k uxx u(0, t) = u(L, t) = 0 u(x, 0) = f (x) is given by the series u(x, t) = 2C n=1 1 cos( n ) kn2 2 2 exp 2 t n L sin Problem 2. Two iron rods, each 20 cm long, are such that one is at temperature 100 C and the other at 0 throughout. C They are put in perfect contact at one end and their other ends are kept at 0 C. Use the result of problem 1 to show that after 600 seconds (= 10 minutes) the temperature at the point of contact is approximately 36 C. Assume the thermal di usivity k of iron is 0.15 cm2 /sec. Problem 3. Find the (formal) solution of the heat equation 0 < x < 1, t > 0 ut = uxx u (0, t) = ux (1, t) = 0 t > 0 x u(x, 0) = x 0 < x < 1. Identify the steady-state and transient temperatures.
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CUNY Queens >> MATH >> 328 (Fall, 2008)
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,...
CUNY Queens >> MATH >> 223 (Spring, 2008)
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...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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...
CUNY Queens >> MATH >> 223 (Spring, 2008)
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 , ...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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 , ...
CUNY Queens >> MATH >> 231 (Fall, 2008)
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 ...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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CUNY Queens >> MATH >> 328 (Fall, 2008)
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? ...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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CUNY Queens >> MATH >> 328 (Fall, 2008)
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CUNY Queens >> MATH >> 328 (Fall, 2008)
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CUNY Queens >> MATH >> 328 (Fall, 2008)
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...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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CUNY Queens >> MATH >> 328 (Fall, 2008)
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CUNY Queens >> MATH >> 328 (Fall, 2008)
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CUNY Queens >> MATH >> 328 (Fall, 2008)
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CUNY Queens >> MATH >> 328 (Fall, 2008)
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CUNY Queens >> MATH >> 328 (Fall, 2008)
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...
CUNY Queens >> MATH >> 328 (Fall, 2008)
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...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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 ...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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 ...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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 =...
CUNY Queens >> MATH >> 631 (Fall, 2008)
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....
CUNY Queens >> MATH >> 631 (Fall, 2008)
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...
CUNY Queens >> MATH >> 631 (Fall, 2008)
Math 319/320 Homework 3 solutions Problem 1. Show that for all real numbers x and y, | |x| |y| | |x y|. By the triangle inequality, we have |x| = |x y + y| |x y| + |y| = |x| |y| |x y| and |y| = |y x + x| |y x| + |x| = |x y| + |x| = |x y...
CUNY Queens >> MATH >> 631 (Fall, 2008)
Math 319/320 Worksheet 2 Problem 1. Fill in the blanks in the following proof of A (B C) = (A B) (A C). If x A (B C), then either x A or x B C. If x A, then x A B and x , so x . On the other hand, if x B C, then x and x , so again...
CUNY Queens >> MATH >> 631 (Fall, 2008)
Math 319/320 Review Sheet, Sept 26 2003 Here is a list of the topics that may be on the exam. Learn the important denitions; we will ask you to state some. 1. Logic and techniques of proof Quantiers and negating quantied statements; proofs by contrad...
CUNY Queens >> MATH >> 631 (Fall, 2008)
Math 319/320 Homework 2 Due Thursday September 18, 2003 Problem 1. Prove the identity: A (B C) = (A B) (A C). Problem 2. Let f : A B be a function. Suppose that C and {Cj , j N} are subsets of A and D is a subset of B. Are the following stat...
CUNY Queens >> MATH >> 631 (Fall, 2008)
Math 320 Homework 7 solutions Problem 1. True or false? Justify your answer. If xn yn for all n and limn yn = , then limn xn = . If xn = 0 for all n and limn xn = 0, then limn 1/xn = +. TRUE: Given M > 0, nd n0 such that yn < M whenever n n0 . ...
CUNY Queens >> CSCI >> 345 (Fall, 2008)
0 R Virtex-II Platform FPGAs: Complete Data Sheet 0 0 DS031 August 1, 2003 Product Specification This document includes all four modules of the Virtex-II Platform FPGA data sheet. Module 1: Introduction and Overview DS031-1 (v2.0) August 1, 200...
CUNY Queens >> CSCI >> 345 (Fall, 2008)
DK2 Handel-C advanced optimization Handel-C advanced optimization Celoxica, the Celoxica logo and Handel-C are trademarks of Celoxica Limited. All other products or services mentioned herein may be trademarks of their respective owners. Neither the...
CUNY Queens >> CSCI >> 345 (Fall, 2008)
Laboratory I September 10, 2003 Objective The objective of this session is to become familiar with the Celoxica DK development platform, using sample Handle-C projects provided with the kit. When you finish this lab, you should be able to configure ...
CUNY Queens >> CSCI >> 345 (Fall, 2008)
Hardware Laboratory Dr. Vickery CSCI 381.3/780 Section E6MDA Reg. Codes 3113/3114 Class Meetings Four 50-minute hours per week. Approximately Half lectures, Half closed labs Lectures: Remsen 105 Mondays Labs: SB A-227 Wednesdays Open Labs SB A-20...
CUNY Queens >> CSCI >> 345 (Fall, 2008)
DK2 Using the logic estimator Using the logic estimator Celoxica, the Celoxica logo and Handel-C are trademarks of Celoxica Limited. All other products or services mentioned herein may be trademarks of their respective owners. Neither the whole nor...
CUNY Queens >> CSCI >> 345 (Fall, 2008)
October 20, 2003 Laboratory IV Topics Laboratory V Topics Laboratory IV Topics Video Generation On-the-fly Using a framebuffer PAL provides three Laboratory V Topics Servomotor Operation (Futaba S3003) 50 Hz pulse stream Short, medium, long pulses ...
CUNY Queens >> CSCI >> 345 (Fall, 2008)
Platform Developer\'s Kit DSM FIR Filter Tutorial Manual For PDK v2.1 Celoxica, the Celoxica logo and Handel-C are trademarks of Celoxica Limited. All other products or services mentioned herein may be trademarks of their respective owners. Neither ...
CUNY Queens >> CSCI >> 345 (Fall, 2008)
DK2 Handel-C VGA graphics output Handel-C VGA graphics output Celoxica, the Celoxica logo and Handel-C are trademarks of Celoxica Limited. All other products or services mentioned herein may be trademarks of their respective owners. Neither the who...
CUNY Queens >> CSCI >> 345 (Fall, 2008)
September 15, 2003 Lab Access Key to A-205 available at main gate and department office. One Design Kit available in department office. Problems with first lab assignment? This weeks laboratory: Write a HandelC program from scratch. C/C+ Compilatio...
CUNY Queens >> CSCI >> 345 (Fall, 2008)
October 7, 2003 Three Ways of Doing I/O Tradeoffs Keyboard to Hex Display Keyboard to LCD Simulated and Real I/O Three Ways of Doing I/O Directly control FPGA Pins Handle-C Interfaces (bus_in, etc.) Use RC200E-specific macros Non-portable Cannot us...
Rio Hondo College >> MATH >> 140 (Fall, 2008)
Item Number:_ Degree Applicable _ Non-Degree Applicable _X_ Transferable _ RIO HONDO COMMUNITY COLLEGE CURRICULUM COMMITTEE COURSE REVISION FORM A Major Revision Form is completed for the following purposes: 1. Reinstating a course that has been del...
Rio Hondo College >> MATH >> 191 (Fall, 2008)
Item Number:_ Degree Applicable _ Non-Degree Applicable _X_ Transferable _ RIO HONDO COMMUNITY COLLEGE CURRICULUM COMMITTEE COURSE REVISION FORM A Major Revision Form is completed for the following purposes: 1. Reinstating a course that has been del...
Rio Hondo College >> MATH >> 250 (Fall, 2008)
Item Number:_ Degree Applicable _ Non-Degree Applicable _X_ Transferable _ RIO HONDO COMMUNITY COLLEGE CURRICULUM COMMITTEE COURSE REVISION FORM A Major Revision Form is completed for the following purposes: 1. Reinstating a course that has been del...
Rio Hondo College >> MATH >> 260 (Fall, 2008)
Item Number:_ Degree Applicable _ Non-Degree Applicable _X_ Transferable _ RIO HONDO COMMUNITY COLLEGE CURRICULUM COMMITTEE COURSE REVISION FORM A Major Revision Form is completed for the following purposes: 1. Reinstating a course that has been del...
Rio Hondo College >> MATH >> 140 (Fall, 2008)
Rubric for Evaluation of a Student Learning Outcome Course: Math 160 SLO: Given a quadratic equation in any form, the student will be able to simplify the equation, recognize the technique required to solve it, and correctly solve it. Faculty part...
Rio Hondo College >> MATH >> 191 (Fall, 2008)
Rubric for Evaluation of a Student Learning Outcome Course: Math 160 SLO: Given a quadratic equation in any form, the student will be able to simplify the equation, recognize the technique required to solve it, and correctly solve it. Faculty part...
Rio Hondo College >> MATH >> 250 (Fall, 2008)
Rubric for Evaluation of a Student Learning Outcome Course: Math 160 SLO: Given a quadratic equation in any form, the student will be able to simplify the equation, recognize the technique required to solve it, and correctly solve it. Faculty part...
Rio Hondo College >> MATH >> 260 (Fall, 2008)
Rubric for Evaluation of a Student Learning Outcome Course: Math 160 SLO: Given a quadratic equation in any form, the student will be able to simplify the equation, recognize the technique required to solve it, and correctly solve it. Faculty part...
Rio Hondo College >> MATH >> 140 (Fall, 2008)
Students Learning Outcome and Assessment Plan Rio Hondo College Department/program: Mathematics Date: 9/10/07 Course title/number: Math 30: Pre-Algebra Participating Faculty: Gisela Spieler-Persad, Joyce Low, Faye Daryoush, Jan La Turno, George Wan...
Rio Hondo College >> MATH >> 191 (Fall, 2008)
Students Learning Outcome and Assessment Plan Rio Hondo College Department/program: Mathematics Date: 9/10/07 Course title/number: Math 30: Pre-Algebra Participating Faculty: Gisela Spieler-Persad, Joyce Low, Faye Daryoush, Jan La Turno, George Wan...
Rio Hondo College >> MATH >> 250 (Fall, 2008)
Students Learning Outcome and Assessment Plan Rio Hondo College Department/program: Mathematics Date: 9/10/07 Course title/number: Math 30: Pre-Algebra Participating Faculty: Gisela Spieler-Persad, Joyce Low, Faye Daryoush, Jan La Turno, George Wan...
Rio Hondo College >> MATH >> 260 (Fall, 2008)
Students Learning Outcome and Assessment Plan Rio Hondo College Department/program: Mathematics Date: 9/10/07 Course title/number: Math 30: Pre-Algebra Participating Faculty: Gisela Spieler-Persad, Joyce Low, Faye Daryoush, Jan La Turno, George Wan...
Rio Hondo College >> MATH >> 140 (Fall, 2008)
1 Item Number:_ Degree Applicable _X_ Non-Degree Applicable _ Transferable _X_ RIO HONDO COMMUNITY COLLEGE CURRICULUM COMMITTEE COURSE REVISION FORM A Course Revision Form is completed for the following purposes: 1. Reinstating a course that has bee...
Rio Hondo College >> MATH >> 191 (Fall, 2008)
1 Item Number:_ Degree Applicable _X_ Non-Degree Applicable _ Transferable _X_ RIO HONDO COMMUNITY COLLEGE CURRICULUM COMMITTEE COURSE REVISION FORM A Course Revision Form is completed for the following purposes: 1. Reinstating a course that has bee...
Rio Hondo College >> MATH >> 250 (Fall, 2008)
1 Item Number:_ Degree Applicable _X_ Non-Degree Applicable _ Transferable _X_ RIO HONDO COMMUNITY COLLEGE CURRICULUM COMMITTEE COURSE REVISION FORM A Course Revision Form is completed for the following purposes: 1. Reinstating a course that has bee...
Rio Hondo College >> MATH >> 260 (Fall, 2008)
1 Item Number:_ Degree Applicable _X_ Non-Degree Applicable _ Transferable _X_ RIO HONDO COMMUNITY COLLEGE CURRICULUM COMMITTEE COURSE REVISION FORM A Course Revision Form is completed for the following purposes: 1. Reinstating a course that has bee...
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