pracfinal - Math 104A Practice Final Problems 1 Find the...

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Math 104A Practice Final Problems 1. Find the natural (free) cubic spline that interpolates the function values f (0) = 1, f (1) = 2, and f (2) = 9 in the interval [0 , 2]. 2. Let f (0 . 1) = 1 . 10517, f (0 . 2) = 1 . 22140, and f (0 . 3) = 1 . 34986. a) Find a first order approxi- mation to f 0 (0 . 2). b) Find a second order approximation to f 0 (0 . 2). 3. Suppose that N ( h ) is an approximation to M for every h > 0 and that M = N ( h ) + K 1 h + K 2 h 2 + K 3 h 3 + · · · for some constants K 1 , K 2 , K 3 , . . . . Use the values N ( h ), N ( h 3 ), and N ( h 9 ) to produce and O ( h 3 ) approximation to M . 4. The (simple) trapezoidal rule applied toR40f(x)dxgives the value 2 and Simpson’s rule givesthe value 1. What isf(2)? 5. Derive a quadrature formula for approximating Z 3 0 f ( x ) dx in terms of f (0), f (1), f (2) and f (3). 6. a) Define the degree of accuracy or precision of a quadrature formula. b) What’s the degreeof accuracy for the Composite Trapezoidal rule?c) What’s the degree of accuracy for theComposite Simpson’s rule?
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