ME218 F2010 2nd Midterm Exam

ME218 F2010 2nd Midterm Exam - Department of Mechanical...

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Unformatted text preview: Department of Mechanical Engineering The University of Texas at Austin ME 218 Engineering Computational Methods Second Midterm Examination Fall 2010 Instructor: Prof. Dragan Djurdjanovic November 4th, 2010 (8:00 am – 9:15 am) Student Name:_____________________ Student EID:___________________ Lab Section Number (or time):___________________ I have neither given nor received any unauthorized aid on this exam, nor have I concealed any violations of the honor code. Signature:____________________ Note: There are 4 problems in this exam. First problem carries 15 points, second problem carries 30 points, third problem carries 35 points and the fourth problem carries 20 points. Be careful and through in your work and GOOD LUCK!!! 1 Problem 1: (15 points) Write out the system of linear equations needed to fit a quadratic polynomial to the following set of five data points using the least squares method discussed in class (x is the independent variable, y is the dependent variable). Please just write out the system of equations (either one by one or using matrix notation) and do NOT try to solve the system. xi yi 1 4.1 2 6.9 3 11.8 4 20.1 5 27.8 2 Problem 2: (30 points) The following data gives the population of the world using approximate census data from 1850 thru 2000. Year Population (billions) 1850 1900 1950 2000 1.3 1.6 3 6 Our goal is to use this data to create an exponential model in the form p( t ) = p0e λt that we can use to approximate the population on other years. (a) Set up the system of linear equations that must be solved to find the exponential least‐squares curve fit for the given data. (10 points) € (b) Solve the system of equations you found in part (a) and use that solution to find parameters po and λ. (10 points) −1 a b 1 d −b Please note that = ad − bc −c a c d € 3 (c) Use the parameters found in part (b) to estimate the population in 2050. (4 points) (d) Use the parameters you found in part (b) to estimate when the population will exceed 7 billion. (6 points) 4 Problem 3: (35 points) Let us observe equations describing 2 ellipses. x2 y2 + = 1 4 9 2 € x y2 + = 1 9 4 These 2 ellipses will intersect in 4 different points and we will use the vectorial Newton‐ Raphson’s method to numerically search for one of them. € (a) Transform the two equations given above into the form f ( x, y ) = 0 g( x, y ) = 0 that we can solve using the vectorial Newton‐Raphson’s method. (5 points) € € (b) Express the Jacobian matrix for the system of non‐linear equations you found in part (a). Please use symbolic calculations (i.e. express the Jacobian using generic variables x and y). (10 points) 5 (c) Starting from the initial guess ( x 0 , y 0 ) = (1.50, 2.50) , please conduct 2 iterations of the vectorial Newton‐Raphson’s method to numerically solve the system of linear equations you found in part (a). (15 points) € (There is one more part of this question on the next page) 6 (d) Obviously, if you had enough time to perform the procedure you did in part (c) a bit longer, you would converge to a solution (x,y) representing one of the intersection points for the two ellipses considered in this problem. Discuss in words how you would pursue other intersection points for this problem (please stay very conceptual – no calculations here). (5 points) 7 Problem 4: (20 points) (a) Compute the following integral using trapezoidal rule assuming that the step size h = 0.1. Round up your results and represent all numbers with only 2 decimal points. (15 points) (b) Please suggest one way of improving the accuracy of numerical evaluation of this integral (we discussed 2 possible ways of improving the accuracy – please recall at least one). (5 points) 8 ...
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