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Math 181 Honors Calculus
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Hello, this is gary from ma181. let's solve the extra credit problem. Here is the problem in italics:
"Extra Credit Problem
Suppose that f(x) is continuously differentiable on the interval [a,b]. Let N be a positive integer and let M = Max { |f'(x)| : a <= x <= b }. Let h = (b-a)/N and let R_N denote the "right endpoint" Riemann Sum for the integral
I = int( f(x), x=a..b).
In other words,
R_N = sum( f(a + n*h)*h , n=1..N ).
Explain why the error, E = | R_N - I |, satisfies
E < M(b-a)^2/N."
So what does this equation "E < M(b-a)^2/N" mean. This reads that the error is less than the Maximum value of the derivative of the function of x multiplied by the interval squared from x=a to x=b all divided by the total number of subintervals N.
I don't understand why this must be true. Maybe I'm wrong, but if f(x) were a horizontal line, wouldn't E=0 and M(b-a)^2/N also be =0. That would mean it is a false statement that E < M(b-a)^2/N. Are we to assume that E <= M(b-a)^2/N?
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