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# Why is this of interest???? watch

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1. Given a function f[x], and a positive integer k, tell how to set a number c such that if p[x] is another function with
l p[x]-f[x]l<c for 0<=x<=0.5, then
[int. from 0 to 0.5]f[x]dx= [int. from 0 to 0.5]p[x]dx
with an error of less than 10^-k. Why is this of interest?
2. Let c = 2*10^(-k).

Then for all x such that 0 <= x <= 0.5,

p(x) - 2*10^(-k) < f(x) < p(x) + 2*10^(-k)

So

(int from 0 to 0.5) [p(x) - 2*10^(-k)] dx
< (int from 0 to 0.5) f(x) dx
< (int from 0 to 0.5) [p(x) + 2*10^(-k)] dx

[(int from 0 to 0.5) p(x) dx] - 10^(-k)
< (int from 0 to 0.5) f(x) dx
< [(int from 0 to 0.5) p(x) dx] + 10^(-k)

(int from 0 to 0.5) f(x) dx . and . (int from 0 to 0.5) p(x) dx . are hence less than 10^(-k) apart.

This is of interest because p(x) might be easier to integrate than f(x). [Eg, f(x) = 1/sqrt(1 - x^10) is hard to integrate, but p(x) = 1 + (1/2) x^10 + (3/8) x^20 is easy.]

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Updated: February 26, 2006
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