The forum discussion focuses on calculating the error involved in the trapezoidal and midpoint methods for the integral ∫cos(x^2) from 0 to 1 with n=8. The error bounds are defined by the equations |Et|<= k(b-a)^3/(12n^2) and |Em|<=k(b-a)^3/(24n^2). The second derivative of the function, f''(x) = -4x^2cos(x^2), is evaluated at x=1, yielding f''(1) = -3.844, confirming that the second derivative is negative, which is relevant for error estimation. The maximum value of |f''(x)| over the interval [0, 1] is crucial for determining k in the error formulas.
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sugarxsweet said:Homework Statement
I am having issues figuring out the error involved in the trapezoidal and midpoint methods for ∫cos(x^2) from 0 to 1 with n=8
Homework Equations
|Et|<= k(b-a)^3/(12n^2)
|Em|<=k(b-a)^3/(24n^2)
The Attempt at a Solution
f(x)=cos(x^2)
f'(x)=-2xsin(x^2)
f''(x)=-4x^2cos(x^2)
f''(1) = -3.844
Is it right given that f''(1) is negative? Just wanted to make sure! Thanks