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It is necessary to approximate

$$I(f) = \int\limits_a^bf(x)\,\mathrm{d}x$$ where f is a function that has $2$ continuous derivatives. If $M(f)$ and $T(f)$ are the approximations obtained through the Midpoint Rule and the Trapezoid Rule, respectively, and $f''$ is constant, what is the relationship between $M(f),T(f)$ and $I(f)$?

I did this

$$|E_T|= \frac{k(b-a)^3}{12h^2} \\ |E_M| = \frac{k(b-a)^3}{24h^2} \\ \frac{|E_T|}{|E_M|} = 2 \\ |E_I| = 0$$

The relationship is $|E_I| < |E_M| < |E_T|$.

Is this correct?

vitamin d
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Segmentation fault
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1 Answers1

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You appear to be beginning to analyze estimates of the potential errors in the uses of $M(f)$ and $T(f)$ as opposed to the estimates themselves.

You may be expected to notice that "$f''(x)$ is constant" (quite a hypothesis!) means that $f(x)$ is a quadratic function, for which the "Simpsons Rule" approximation, although it is not expressly mentioned here but you may have heard of, will be exact (i.e. "$S(f)$," the Simpsons approximation, will be $I(f)$).

And there is a well known linear relationship between the midpoint, trapezoid, and Simpsons approximation methods that is described below:

Relation between Simpson's Rule, Trapezoid Rule and Midpoint Rule

Note that the problem does not provide what number of partitions (or which partitions) are being used to evaluate the rules. The answer linked above (and simplicity of $f$) may shed light on that.

leslie townes
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