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I'm about to give a first-semester calculus lecture covering the mean value theorem for integrals:

If $f$ is continuous on $[a,b]$, then there is some $c\in(a,b)$ such that $(b-a)f(c)=\int_a^b f(x)\,dx$.

In past semesters, I've shown examples in which I confirm that this theorem holds for some specific $f(x)$ and $[a,b]$, by solving for $c\in(a,b)$. But this is just checking the theorem -- not actually applying it. A "real" application occurs, for example, in the proof of Taylor's remainder formula, but my students aren't ready for that example.

What is a good "real" application of this theorem, suitable for students in first-semester calculus?

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    $\begingroup$ Problem 3 on math.uci.edu/sites/math.uci.edu/files/2A_final_samp1.pdf and Problem 7 on math.uci.edu/sites/math.uci.edu/files/2A_final_samp2.pdf may be the type of question you're looking for. $\endgroup$ – user2139 Apr 29 '16 at 4:07
  • $\begingroup$ fmlin, yes, these are the types of things I'm looking for. But will you think I'm changing the rules if I say I'd prefer an example where the connection to integration is clearer? $\endgroup$ – swensonj Apr 29 '16 at 13:58
  • $\begingroup$ I think it's interesting that you are using the word "application" to mean "using the MVT to prove something else". In most contexts with which I am familiar, "application" means "real-world application", e.g something about how a car driving with a non-constant acceleration must at some moment be traveling at an instantaneous velocity that is equal to its average velocity over the entire trip. $\endgroup$ – mweiss May 3 '16 at 17:21
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I think the best "real" application of the mean value theorem for integrals is to make a rigorous proof of the fundamental theorem of calculus.

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    $\begingroup$ You took the words right off of my keyboard. This was my first year teaching calculus also, and the lesson in which I did this was definitely a highlight of our year. $\endgroup$ – Andrew May 4 '16 at 2:59
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You can show that a function $f(x)$ that is continuous and differentiable everywhere with two roots has at least one value such that $f'(c)=0$. Follows from the mean value theorem since $(b-a)f'(c)=f(a)-f(b)$, or $(b-a)f'(c)=0$ since $x=a,b$ are roots.

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    $\begingroup$ This feels backwards, morally speaking: Rolle's theorem is regarded as scaffolding to prove the Mean-value theorem as opposed to as an (easy, since it's a fortiori) corollary of the latter, as far as I have seen. $\endgroup$ – Vandermonde May 17 '16 at 18:30
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The construction of the delta distribution as a limit of box functions. Consider $$h_{c}(t) = \begin{cases} 0 & t > |c|,\\ \frac{1}{2c} & t \leq |c|.\end{cases}$$ One wants to motivate defining the delta distribuion as the limit (of the linear functionals given by integrating against) $h_{c}(t)$ when $c \to 0$. For any function $f$ continuous in a neighborhood of the origin, by the mean value theorem for integrals and the continuity of $f$, the limit of $$\int_{-\infty}^{\infty}f(t)h_{c}(t)\,dt = \frac{1}{2c}\int_{-c}^{c}f(t)\,dt$$ when $c \to 0$ exists and equals $f(0)$. This shows how to define $\delta(t)$ as the nonexistent limit $\lim_{c\to 0}h_{c}(t)$, by interpreting functions as functionals via integration. It indicates the possibilities of extending the function concept in a useful way. Students that study physics or engineering will later see the delta function handled formally in those contexts, and it might help them to understand its formal properties to have been explained its relation to the usual function concept.

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