ma174_hw7 - that interpolates the function f x at the...

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Assigned: Monday, November 14, 2011 Due Date: Monday November 21, 2011 MATH 174/274: Homework VII “Splines and Numerical Integration” Fall 2011 NOTE: For each homework assignment observe the following guidelines: 1. [Taken from Burden & Faires, 5th edition, 1993] A natural cubic spline S on [0 , 2] is defined by S ( x ) = ( S 0 ( x ) = 1 + 2 x - x 3 if 0 x < 1 S 1 ( x ) = 2 + b ( x - 1) + c ( x - 1) 2 + d ( x - 1) 3 if 1 x 2 . Find b , c , and d . 2. [Taken from Burden & Faires, 5th edition, 1993] A clamped cubic spline S for a function f ( x ) on [1 , 3] is defined by S ( x ) = ( S 0 ( x ) = 3( x - 1) + 2( x - 1) 2 - ( x - 1) 3 if 1 x < 2 S 1 ( x ) = a + b ( x - 2) + c ( x - 2) 2 + d ( x - 2) 3 if 2 x 3 . Given that f 0 (1) = f 0 (3), find a , b , c , and d . 3. Consider the following 4 equally spaced points on the interval [ x 0 , x 3 ]: x j = x 0 + jh [ j = 0 , 1 , 2 , 3] , where h = ( x 3 - x 0 ) / 3. (a) Construct all the Lagrange polynomials L 3 ,j ( x ) that correspond to the points x 0 , x 1 , x 2 , and x 3 . (b) Use these Lagrange polynomials to construct the interpolating polynomial
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Unformatted text preview: ) that interpolates the function f ( x ) at the points x , x 1 , x 2 , and x 3 . (c) Integrate the interpolating polynomial P 3 ( x ) to derive the following Newton-Cotes formula often referred to as Simpson’s three-eigths rule : Z x 3 x f ( x ) dx ≈ 3 h 8 h f ( x ) + 3 f ( x 1 ) + 3 f ( x 2 ) + f ( x 3 ) i . 4. Consider the following integral: Z 4 3 x √ x 2-4 dx. Compute this integral: (a) Exactly by hand. (b) Approximately with the Trapezoidal rule and compute the error (error = | approx-exact | ). (c) Approximately with Simpson’s rule and compute the error (error = | approx-exact | ). (d) Approximately with Simpson’s three-eigths rule and compute the error (error = | approx-exact | ). 1...
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