MATH105 April 2017
• Q1 (a) • Q1 (b) • Q1 (c) • Q1 (d) • Q1 (e) • Q1 (f) • Q1 (g) • Q1 (h) • Q1 (i) • Q1 (j) • Q1 (k) • Q1 (l) • Q1 (m) • Q1 (n) • Q2 (a) • Q2 (b) • Q3 • Q4 (a) • Q4 (b) • Q5 (a) • Q5 (b) • Q5 (c) • Q5 (d) • Q5 (e) • Q6 (a) • Q6 (b) •
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For any function with continuous fourth derivative, the error in the Simpson's rule approximation is bounded above by
where K is a constant such that
on the interval
We need to determine an appropriate constant K. To do this we need to compute the fourth derivative of the integrand
Letting , we have
On the interval [1,2], this fourth derivative has absolute value bounded above by 2. So we can select K = 2. Since b = 2 and a = 1, we get the error estimate
We would like this error to be bounded above by This will happen if
We take the fourth root of both sides:
So the size of our partitions must be smaller than 1/10. Because the interval over which we are integrating has length 1, a value of corresponds to N = 10, so the error will be bounded above by provided that .