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midterm1_2008

# midterm1_2008 - COT 3502 Computer Model Formulation Midterm...

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COT 3502 Computer Model Formulation Midterm Exam 1 February, 14, 2007 Closed books, notes, and computers. Calculators should be used only for arithmetic calculations (not for plotting or analytical derivations). 1. (20 points) Consider the following function. -4 -3 -2 -1 0 1 2 -20 -15 -10 -5 0 5 10 15 20 x y (a) Using this graph, perform 2 iterations of the bisection method. The initial interval has the following lower and upper boundaries: x l = -1 and x u = 2. Make sure to clearly label the plot! (b) Estimate the number ( n ) of the bisection method iterations needed to obtain an approximation x ( n ) r to the root x 0 with precision epsilon1 n = | x ( n ) r - x 0 | < 10 - 6 . Assume that the initial interval is the same as in part (a). (c) Can a bracketing method converge to the root at x = - 2? Why or why not? (d) Consider the method of false position. If your initial interval is x l = - 4, x u = 2, to which of the two roots will the iterations converge? 1

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2. (20 points) Assume that you need to compute a derivative of some function f ( x ) at a point x = x 0 . Only two values of this function are available, f ( x 0 - h ) and f ( x 0 + 2 h ). Derive a finite-difference formula for
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