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I have a function, which is $f(x) = e^{1/x}$. I want to calculate the error bound for the Trapezoid Rule, which formula is:

$|E|\leq K\frac{(a-b)^3}{12\cdot n^2}$

where $|f''(x)|\leq K$. What's the value of $K$ for the above function? If I calculated $f''(x)$ correctly, it should be: $f"(x) = \frac{e^{1/x}}{x^3}\left(\frac 1x + 2\right).$

Please correct me if I'm wrong. The boundaries are $[1, 2]$.

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    Yep, it's a decreasing function (a fact that can be checked from the derivative). Just a minor point, you actually want a "min" value of $K$ since it sounds like you want to find the least upper bound.2012-09-11

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If you look at $f''$ you should be able to convince yourself that it is decreasing over $[1,2]$. In that case, the maximum value is $f''(1)$, which is then $K$