Solution-HW4

# Solution-HW4 - Homework#4 Solution 13.2 Given x x x x f 12...

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Unformatted text preview: Homework #4 Solution 13.2 Given ( ) x x x x f 12 2 5 . 1 4 6 + − − = (a) Plot the function. (b) Use analytical methods to prove that the function is concave for all values of x . (c) Differentiate the function and then use a root-location method to solve for the maximum f(x) and the corresponding value of x . SOLUTION: (a) Plot (b) ( ) 24 45 2 4 < − − = ′ ′ x x x f for all x , so the function is concave for all values of x . (c) ( ) 12 8 9 3 5 = + − − = ′ x x x f Using bisection method, we obtain: Root = 0.916915 So ( ) 69729 . 8 916915 . = f 13.3 Solve for the value of x that maximizes f(x) in Prob. 13.2 using the golden- section search. Employ initial guesses of x l = 0 and x u = 2 and perform three iterations. SOLUTION: Follow Example 13.1 For x l = 0, x u = 2, d = 0.618 × 2 = 1.236 X 1 = 0 + 1.236 = 1.236, x 2 = 2 – 1.236 = 0.764 The follow table can be generated: Interation x l x 2 x 1 x u 1 0 0.764 1.236 2 2 0 0.472 0.764 1.236 3 0.472 0.764 0.944 1.236 4 0.764 0.994 1.056 1.236 5 0.764 0.875 0.944 1.056 . . . . . . . . . . . . . . . 11 0.9117 0.9180 0.9218 0.9280 With ε a = 0.677% and x opt = 0.9179 and f(x opt ) = 8.6979 13.4 Repeat Prob. 13.3, except use quadratic interpolation. Employ initial guesses of x = 0 , x 1 = 1...
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## This note was uploaded on 01/20/2011 for the course ENG 115 taught by Professor Rocke during the Spring '10 term at UC Davis.

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Solution-HW4 - Homework#4 Solution 13.2 Given x x x x f 12...

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