Applications of MVT for Integrals, suitable for calculus 1

Question

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?

Answer 1

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.

Answer 2

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.

Answer 3

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.