midterm1

midterm1 - f ( x i +1 ) = f ( x i ) + f ( x i ) h + f 00 (...

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Numerical Analysis in Engineering ME 140A, Fall 2008 Midterm #1 October 30, 2008 1. (10 points) Imagine a water reservoir of circular shape, with radius R = 1 km and depth h ( r ). There is a pollutant in the water with concentration c ( r ). a) Which integral do you have to evaluate to find the volume of the reservoir? b) How much pollutant is in the reservoir? 2. (15 points) Consider the function y ( x ) = cos(2 πx ). Evaluate: Z 1 0 y ( x ) dx. a) analytically b) with the trapezoidal rule for n = 1. c) with the trapezoidal rule for n = 2. Explain your results. 1
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3. (25 points) Develop an optimal Gauss quadrature formula: Z 1 0 f ( x ) dx = c o f ( x o ) Determine the coefficient c o and the location x o such that you get exact results for: Z 1 0 f ( x ) dx if f ( x ) = x 2 and f ( x ) = x 3 . 4. (25 points) Given the Taylor Series
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Unformatted text preview: f ( x i +1 ) = f ( x i ) + f ( x i ) h + f 00 ( x i ) 2! h 2 + f (3) ( x i ) 3! h 3 + ... + f ( n ) ( x i ) n ! h n + R n demonstrate that f 00 ( x i ) =-f ( x i +2 ) + 16 f ( x i +1 )-30 f ( x i ) + 16 f ( x i-1 )-f ( x i-2 ) 12 h 2 is fourth order accurate. 5. (25 points) If you evaluate f 00 1 ( x i ) = f ( x i +2 )-2 f ( x i ) + f ( x i-2 ) 4 h 2 + O ± h 2 ² and f 00 2 ( x i ) = f ( x i +1 )-2 f ( x i ) + f ( x i-1 ) h 2 + O ± h 2 ² Show that the Richardson extrapolation f 00 R ( x i ) = 4 3 f 00 2 ( x i )-1 3 f 00 1 ( x i ) is fourth order accurate. Hint: use the Taylor series. 2...
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This note was uploaded on 01/04/2011 for the course ME 140A taught by Professor Meiburg during the Spring '08 term at UCSB.

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midterm1 - f ( x i +1 ) = f ( x i ) + f ( x i ) h + f 00 (...

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