Asked by David
How large does N have to be in order to approximate the integral of cos(x^2) from 0 to 1, using the Midpoint Rule, with error at most 10^(-6)?
Answers
Answered by
MathMate
Use the value of the error term:
max. error ≤M2(b-a)^sup3;/(24N²)
where
M2 is the maximum absolute value of f"(x) on the interval [a,b].
Here f(x)=cos(x²)
f'(x)=-2xsin(x²), and
f"(x)=-2sin(x²)-4x²cos(x²)
(check my differentiation.)
For other cases other than middle-sum, see:
http://en.wikipedia.org/wiki/Riemann_sum
max. error ≤M2(b-a)^sup3;/(24N²)
where
M2 is the maximum absolute value of f"(x) on the interval [a,b].
Here f(x)=cos(x²)
f'(x)=-2xsin(x²), and
f"(x)=-2sin(x²)-4x²cos(x²)
(check my differentiation.)
For other cases other than middle-sum, see:
http://en.wikipedia.org/wiki/Riemann_sum
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