Examples of error in the following topics:
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- The letters $\varepsilon$ and $\delta$ can be understood as "error" and "distance," and in fact Cauchy used $\epsilon$ as an abbreviation for "error" in some of his work.
- In these terms, the error $(\varepsilon)$ in the measurement of the value at the limit can be made as small as desired by reducing the distance $(\delta)$ to the limit point.
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- An example of such an integrand is $f(x) = \exp(x^2)$, the antiderivative of which (the error function, times a constant) cannot be written in elementary form.
- An important part of the analysis of any numerical integration method is to study the behavior of the approximation error as a function of the number of integrand evaluations.
- A method which yields a small error for a small number of evaluations is usually considered superior.
- Reducing the number of evaluations of the integrand reduces the number of arithmetic operations involved, and therefore reduces the total round-off error.
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- Because of this, transcendental functions can be an easy-to-spot source of dimensional errors.
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- This method becomes very complicated and is particularly error prone when doing calculations by hand.
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- An example of such an integrand $f(x)=\exp(-x^2)$, the antiderivative of which (the error function, times a constant) cannot be written in elementary form.
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- In general, this curve does not diverge too far from the original unknown curve, and the error between the two curves can be made small if the step size is small enough and the interval of computation is finite.
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- Taylor's theorem gives quantitative estimates on the error in this approximation.