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Im supposed to explain the reasoning behind retaining a negative error from a trapezoidal approximation of the function: f[x] = e^4x.

I have graphed the function and it looks like this: Graph of Function

I think the main reason I received a negative error is because the function is concave up and the Trapezoidal Rule overestimates the approximation. Im not exactly sure if this is correct, but if it were to be and someone understands it better than I do, I would really appreciate an explanation.

Thank you for your time, and efforts.

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    $\begingroup$ That doesn't look like the graph of the function $-x^3+3$. $\endgroup$ – PJK Jul 6 '17 at 1:05
  • $\begingroup$ Im sorry you it was supposed to be e^4x -.- $\endgroup$ – Mike Jul 6 '17 at 1:07
  • $\begingroup$ I convinced myself that my reasoning was right though, because if you overestimate something, then subtract the actual value by your estimation you will get a negative answer $\endgroup$ – Mike Jul 6 '17 at 1:08
  • $\begingroup$ Yes, for concave functions $f''(x)>0$ so the error by trapezoidal approximation is negative. $\endgroup$ – PJK Jul 6 '17 at 1:13
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The trapezoidal rule is estimating your function as sequence of lines over a set of intervals.

i.e. at each point in the interval $[a,b], \lambda f(a) + (1-\lambda) f(b) $ with $\lambda \in [0,1]$ is being used as a stand in for $f(\lambda a + (1-\lambda) b)$

Since $f(x)$ is convex, $\lambda f(a) + (1-\lambda) f(b)\ge f(\lambda a + (1-\lambda) b)$

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