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5 Pages

### polarGraphs

Course: M 162, Spring 2007
School: Calvin
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Word Count: 1185

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162 MATH Calculus II Computer Laboratory Topic: Intro to Mathematica, Part II, &amp; Intersections of Polar Functions Goals of the lab: To re-familiarize students with some basic operations in Mathematica, such as how to define a function, and the basic structure to plotting commands. To understand polar coordinates better and how polar functions work. 1 1.1 Review and Extensions A Summary of Previous...

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Calvin - M - 162
MATH 162: Calculus II, Sections B/CInformation for Exam 3 to take place on Apr. 5, 2007General Information. The exam focuses on material covered in frameworks for the dates Feb. 27 Mar. 30. Problems during this time have been assigned from section
Calvin - M - 162
MATH 162: Calculus II, Sections B/CInformation for Exam 2 to take place on Mar. 7, 2007General Information. 1. The exam focuses on material from the frameworks for the dates Feb. 13Mar. 1, 2007. This roughly corresponds to material found in Section
Calvin - M - 162
MATH 162: Calculus IIFramework for Tues., Feb. 27 Functions of Multiple VariablesDenition: A function (or function of n variables) f is a rule that assigns to each ordered n-tuple of real numbers (x1 , x2 , . . . , xn ) in a certain set D a real nu
Calvin - M - 162
Mathematics 162: Final Exam InformationTime and Location: 9:00 am-noon pm on Thursday, December 16, NH 251 Exam Syllabus: The final exam will cover the following sections: Chapter 7: Sections 1-4, 6, 7 Chapter 8: Material from `substitute chapter',
Calvin - M - 231
1Gaussian Elimination: A New Spin on the Solution of n Linear Equations in n UnknownsBy now systems of two equations in two variables are quite familiar to us. We have studied them in practically every mathematics course prior to the Calculus. We
Calvin - M - 243
MATH 143 Inference practice problems 1. An automobile manufacturer wants to know what percentage of its customers are dissatisfiedwith the performance of their newly-purchased vehicle. (a) Name the procedure we have learned best suited for determini
Calvin - M - 231
MATH 231A Runge-Kutta Notes: for a 2-equation, 1st-order systemFor our (general) problem from class dx/dt = f (t, x, y), dy/dt = g(t, x, y), x(t0 ) = x0 y(t0 ) = y0we get our approximate solution (xn , yn ) at time tn , n = 1, 2, . . . via the ite
Calvin - M - 232
Current Errata1 p. 118: About a third of the way up from the bottom there is a line containing &quot;A1 , A2 , dots, An &quot;. It should read &quot;A1 , A2 , . . . , An &quot;. [D. Moelker] p. 119: In about the middle of the page there appears the matrix B1 B2
Calvin - M - 231
MATH 231, Worksheet Finding inverse Laplace transforms Solutions1. Using partial fraction expansion, we have 1 s2 (s2 + 4) = A B Cs + D + 2+ 2 . s s s +4Multiplying through by the lowest commond denominator s 2 (s2 + 4), we get 1 = As(s2 + 4) + B(
Calvin - M - 232
Linear Algebra FormulasAngle between vectors u, v satisfies: Normal equations for Ax = b: cos =u,v |u|v|(AT A)x = AT bStatistics FormulasMeans and Variances Mean 1 xi x= niSample (of size n) Random variable X discrete X GeneralVariance
Calvin - M - 256
1Eigenvalues and EigenvectorsThe product Ax of a matrix A Mnn (R) and an n-vector x is itself an n-vector. Of particular interest in many settings (of which differential equations is one) is the following question: For a given matrix A, what are
Calvin - M - 333
MATH 333: Partial Differential EquationsProblem Set 1, Final version Due Date: Wed., Sept. 13, 2006Note: Certain of these &quot;problems&quot; do not require write-ups. 1. Look over the course webpage. In particular, read through the course syllabus, get acc
Calvin - M - 333
MATH 333: Partial Differential EquationsProject 7, Due Date: Mon., Oct. 30, 2006Let m be the mass of the electron in orbit around the proton of a hydrogen atom, e its charge, and h Planck's constant divided by 2. If (0, 0, 0) represents the positio
Calvin - M - 333
MATH 333: Partial Dierential EquationsProblem Set 9, Final version Due Date: Wed., Nov. 8, 20061. If we combine our assertion (made without proof) in class that the collection S = {1, cos(x), sin(x), cos(2x), sin(2x), . . .} is a complete orthogon
Calvin - M - 333
MATH 333: Partial Differential EquationsProject 9, Due Date: Mon., Nov. 13, 2006Because we have placed greater emphasis than in past semesters upon the numerical solution of PDEs, we have correspondingly placed less emphasis upon the eigenfunctions
Calvin - M - 333
MATH 333: Partial Dierential EquationsProblem Set 11, Final version Due Date: Wed., Nov. 22, 20061. We have, up until now, been dealing almost exclusively with what are called real-valued functions, functions whose outputs are real numbers (usuall
Calvin - M - 333
MATH 333: Partial Dierential EquationsProject 10, Due Date: Mon., Nov. 17, 2006(a) Under the usual inner product of L2 (a, b), the family S = {xn | n = 0, 1, 2, . . .} of monomials is not mutually orthogonal. Show that this is so in the particular
Calvin - M - 333
MATH 333: Partial Dierential EquationsProject 3, Due Date: Fri., Sept. 29, 2006Solve the following problem using Laplace transforms. utt = c2 uxx g, u(0, t) = 0, u(x, 0) = 0, ut (x, 0) = 0, x &gt; 0, t &gt; 0, t &gt; 0, x &gt; 0, x &gt; 0.According to my sourc
Calvin - M - 333
MATH 333: Partial Differential EquationsProject 1, Due Date: Fri., Oct. 20, 2006In this project you will study in greater depth some of the phenomena which can arise in first-order PDEs. In particular, you will study quasilinear PDEs (i.e., the coe
Calvin - M - 333
MATH 333: Partial Dierential EquationsProblem Set 12, Final version Due Date: Wed., Nov. 29, 20061. Do Exercise 4.22. For part (a), the explicit nite dierence scheme is the one from Exercise 4.20, part (c). 2. Do Exercise 6.2. 3. Compare and contr
Calvin - M - 333
MATH 333: Partial Differential EquationsProblem Set 13, Final version Due Date: Wed., Dec. 6, 20061. Chapter 7 of our book is devoted to the Dirichlet problem for Poisson's equation in two spatial dimensions -u = f, in R2 , u = g, in . Section 7.
Calvin - M - 333
MATH 333: Partial Differential EquationsProblem Set 2, Final version Due Date: Wed., Sept. 20, 20061. Do as much of Exercise 1.5 as you have time for, but hand in parts (b) and (c). 2. Do Exercise 1.6. 3. Consider the Cauchy problem for the wave eq
Calvin - M - 333
Table of Laplace Transforms function f (t) tn eat sin at cos at sinh at cosh at u(t)eat H(t - a)u(t - a) 1 - erf a/ 4t e-a /(4t) t ae-a /(4t) 4t3 u(n) (t) (u v)(t) tu(t) u(t)/ts2 2transform F (s) n!/sn+1 1/(s - a) a 2 + s2 a s 2 + s2 a a 2 -
Calvin - M - 333
MATH 333: Partial Dierential EquationsProblem Set 3, Final version Due Date: Wed., Sept. 27, 20061. Do Exercise 1.10. Here are some suggestions. A similar heat problem, one with = 1, is stated in equations (1.48), (1.49) of your text, and its solut
Calvin - M - 333
MATH 333: Partial Differential EquationsProblem Set 10, Final version Due Date: Wed., Nov. 15, 20061. Note: This is a scaled-back version of parts (a)(c) from Project 10. I feel secure including this problem here since no one has expressed an inte
Calvin - M - 333
MATH 333: Partial Dierential EquationsProject 5, Due Date: Fri., Oct. 13, 2006Consider the ordinary dierential equation d2 y xy = 0, dx2 &lt; x &lt; .Use Fourier transforms to show solutions of this ODE are scalar multiples of the Airy function Ai(x)
Calvin - M - 333
MATH 333: Partial Differential EquationsProblem Set 6, Final version Due Date: Wed., Oct. 18, 20061. Consider the derivative operator A = d4 /dx4 acting on some subset of functions in L2 (0, 1) which satisfy (0) = (0) = 0 = (1) = (1). This oper
Calvin - M - 333
MATH 333: Partial Dierential EquationsProject 4, Due Date: Fri., Oct. 13, 2006Find a resource that denes the two-dimensional Fourier transform (in an analogous way to how the 1-dimensional Fourier transform is dened in our text). There are certain
Calvin - M - 333
MATH 333: Partial Dierential EquationsProject 2, Due Date: Fri., Sept. 29, 2006Consider the vibrations of an innitely long thin string (one extending from position x = 0 up to x = +). The PDE model for this problem is utt = c2 uxx , u(0, t) = 0, u(
Calvin - M - 333
MATH 333: Partial Differential EquationsProblem Set 5, Final version Due Date: Wed., Oct. 11, 20061. Let , denote the usual inner product on Rn . Prove the following statement: An n-by-n matrix A is symmetric if and only if Ax, y = x, Ay for all
Calvin - M - 333
MATH 333: Partial Differential EquationsProblem Set 4, Final version Due Date: Wed., Oct. 4, 20061. Do Exercise 12.2(a). 2. Do Exercise 12.7. 3. Solve the boundary value problem uxx + uyy = 0, u(x, 0) = 1, where &gt; 0 is a constant. x R, y &gt; 0 u(x,