Prof. Ming Gu, 861 Evans, tel: 2-3145Email: [email protected]://www.math.berkeley.edu/∼mgu/MA128A2008S/Math128A: Numerical Analysis SampleMidterm IIThis is a closed book, closed notes exam. You need to justify every one of youranswers. Completely correct answers given without justification will receive littlecredit. Do as much as you can. Partial solutions will get partial credit. Look overthe whole exam to find problems that you can do quickly. You need not simplify youranswers unless you are specifically asked to do so.Problem Maximum Score Your Score1 42 243 244 245 24Total 1001. (10 Points)Your Name:Your SID:Your GSI:Math128A: Numerical Analysis Sample Midterm II 22. Boole’s Rule for numerical integration on the interval [a, b] is given byI4(x) =2h45(7f(a) + 32f(a + h) + 12f(a + 2h) + 32f(a + 3h) + 7f(b)) .(a) Show that the degree of precision of this formula is 5.(b) Develop the Composite Boole’s Rule for integration on [a, b].Math128A: Numerical Analysis Sample Midterm II 33. Construct the natural cubic spline that approximatesf(x) =sin xxat the nodes −1, 0, 1.Math128A: Numerical Analysis Sample Midterm II 44. Suppose thatL = limh→0f(h) and L − f(h) = c6h6+ c9h9+ · · · .Find a combination of f(h) and f(h/2) that is a much better estimate of L.Math128A: Numerical Analysis Sample Midterm II 55. Given h = 0.1, f(0) = 0, f(0.1) = 0.01 and f(0.2) = 0.04, find a second order approximationto
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