Math 472/572 - Assignment 1
Math 472/572 - Assignment 8
Due: Tuesday, November 28. Nothing accepted after Thursday, November 30. 10% off for being late. Please work by yourself. See me if you need help.
1. (3 points) Use Simpson's rule with h = 1/4 to approximate ) dx). You may use mathematical software to evaluate the sums involved.
2. (4 points) The error in Simpson's rule is - f(4)(() (b-a) h4 where ( is some point between a and b. Use this formula to get an upper bound for the absolute value of the error in your approximation in part a. You may use mathematical software to find the fourth derivative.
3. (3 points) How small would h have to be so that when you repeat the calculations in part a with this value of h, you are guaranteed to get an approximation which is within 10-6 of the true value of the integral? (You don't actually have to repeat the calculations with this value of h. Just find how small it should be.)
4. (3 points) Use the Simpson's rule with h = 1/8 to approximate ) dx).
5. (3 points) Using your answers to parts 1 and 4, estimate the error in your answer to part 4. Use the fact that the error is Ch4 + O(h6), where C is a constant independent of h and O(h6) denotes something whose absolute value is bounded by a constant times h6.
6. (2 points) Using your answers to parts 4 and 5, find a better approximation to ) dx).
7. (2 point) Use mathematical software to estimate ) dx). In Mathematica you could enter Integrate[1.0/Sqrt[1-x^4], {x, 0, 1}]. In MATLAB one way is to first create an M-file named fun.m with function y = fun(x) on one line and y = 1 ./ sqrt(1+ x.^4); on the second. Then enter quad('fun', 0, 1).
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