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### TEST 2_S2011

Course: PGE 310, Spring 2012
School: University of Texas
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Word Count: 403

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#2A TEST (April 12, 2011) PGE 310 Name: Signature: UT EID: NOTES: 1. You have 1 hour and 30 minutes. Points will be deducted for tests turned in after 8:30 2. Write your name on all pages of the test 3. SHOW ALL WORK. Partial (or any credit) cannot be given for problems without work shown 1. Qualitative Questions (20 points) (a) Using the Taylor series (provided on your equation sheet), derive Newtons method...

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#2A TEST (April 12, 2011) PGE 310 Name: Signature: UT EID: NOTES: 1. You have 1 hour and 30 minutes. Points will be deducted for tests turned in after 8:30 2. Write your name on all pages of the test 3. SHOW ALL WORK. Partial (or any credit) cannot be given for problems without work shown 1. Qualitative Questions (20 points) (a) Using the Taylor series (provided on your equation sheet), derive Newtons method for solving for the root of a nonlinear equation. Show all work. (b) Briefly explain how false position and secant method are different. Be specific. (c) Explain when I might choose to use LU decomposition instead of Gauss Elimination. Give an example if it helps explain your point. (d) Explain how Gauss Seidel and Jacobi methods for solving systems of equations differ. Why is Gauss Seidel often much faster? (d) Explain how I would derive equations to curve fit data to an equation of the form y=a0 +a1*x + a2*x2 + a3*x3. You may use simple equations to explain your answer, but you dont need to actually do any calculations, derivations, or calculus. 2. (a) Use the first three iterations of the bisection method to estimate the root of the following nonlinear equation. Use initial guesses of xL=0 and xU=1.0 and calculate approximate the relative error. Show all work and fill in the table at the bottom of the page (15 points) x3 3x + 1 = 0 iteration First Second Third xr App relative error ------------- 3. (a) Use the first iteration of the Netwon-Raphson method and an initial guess of x=1; y=0 to solve the following system of equations. SHOW ALL WORK (20 points) 12 1 x + xy = 2 2 14 1 x + 3 yx 3 = 2 2 (b) Is the solution after the first iteration better than the initial guess? Use numbers/calculations to back up your answer. Your answer can be yes, no, or inconclusive but you must EXPLAIN. 4. The relationship between viscosity, , and shear rate, , that describes a certain surfactant used in enhanced oil recovery is given by the equation below. Given the provided data, use curve fitting to estimate the unknown parameters k and n. (20 points) = k n 1 = [1 15 75 200] = [1000 600 400 350] 5. Write a MATLAB function file called backsubstitution.m that solves for the x, when sent an upper triangular matrix (U) and right-hand-side vector (b). Recall that the system of equations Ux=b looks something like this: u11 u12 u22 u13 u23 u33 ... u1n x1 b1 ... u2 n x2 b2 ... u3n x3 = b3 O M M unn xn bn
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