# How does the formula #1/90((b-a)/2)^5(f^(4)(zeta))# work for calculating error?

When dealing with approximation of integrals, one is not interested in finding exact values. So, we don't either require to know the error value exactly. In fact, if we compute the exact error, then we can retrieve the exact value of the integral by simply adding (subtracting) the error to (from) the approximated value. In a few words, searching for the exact error is the same thing as searching for the exact value.

The interesting fact about error (actually the reason why approximation is so widely used and useful, especially in applications) is that it can be estimated, often considering the worst case. In this manner we don't have exact information, but we are sure that the error on the approximated value is bounded. Once this bound is known, we could decide if the precision accomplished by the approximation is enough or not, even in the worst case (and maybe search for a way to improve the approximation...).

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The formula ( \frac{1}{90} \left( \frac{b-a}{2} \right)^5 f^{(4)}(\zeta) ) is derived from Simpson's rule for numerical integration. It estimates the error between the true value of an integral and the approximation obtained through Simpson's rule. In the formula:

- (a) and (b) represent the endpoints of the interval of integration.
- (f^{(4)}(\zeta)) denotes the fourth derivative of the integrand (f(x)) evaluated at some point (\zeta) within the interval.
- ( \left( \frac{b-a}{2} \right)^5 ) scales the error estimate based on the width of the interval raised to the fifth power.
- ( \frac{1}{90} ) is a constant factor specific to Simpson's rule.

The formula provides an estimate of the maximum error between the actual integral value and the value obtained using Simpson's rule.

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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