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I would like some advises which contents to teach in a 45 minutes class to first year undergraduates about the mean value theorem. I'm thinking to do the following:

  1. Motivation
  2. The theorem and the demonstration
  3. applications and examples

So my questions are:

  1. Do you think this is the best approach?

  2. If you taught this class, what would you teach in the items 1 and 3 above?

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  • $\begingroup$ Mean value theorem for derivatives or integrals? Or something else? $\endgroup$ – Steven Gubkin Dec 10 '17 at 14:37
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I think your proposed outline is fine. For these theorems in Calc I (also, e.g., the Intermediate Value Theorem), I try to start with a step 0 in which you pose a question to the students to get them to start thinking about the theorem before you state it formally (maybe this is your "Motivation" step but I interpret that more as why you're taking the time to state and prove this theorem in class.

The Cornell Good Questions Project has questions of this type. See page 36 of this pdf for example. Question 5, in particular, is how I might get the students thinking before getting all formal. It might also be useful to get them to try Question 6 before stating the theorem so that they can really see why the hypothesis is needed.

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I would assume Rolle's Theorem had been proved in a previous class. Then,

  • motivate from physics of motion: the mean value theorem simply says the average speed must match the instantaneous speed at some time during the motion.

I like to say what it doesn't say as well, for instance, if you tell the police that your average speed was 30mph since it takes 10 minutes to travel 5 miles to your house, they probably don't care to hear that when they stop you going 50mph in a particular 30mph zone. Silly motivations aside, then I would

  • prove the mean value theorem

To do this carefully, I think you can do all of the above in about 15-20 minutes if you prepare. Then in the remaining time:

  • give an example where you actually calculate $c$ in $(a,b)$ for which $f'(c) = \frac{f(b)-f(a)}{b-a}$. I usually try to make it clear that the MVT itself does not find $c$ it just tells us that our search is not futile. It is there somewhere to find.

Then, to my taste, one important application is to graphing. To prove $f$ is increasing on $(a,b)$ we need $a<x_1 <x_2<b$ implies $f(x_1) \leq f(x_2)$. Notice, $$ \frac{f(x_2)-f(x_1)}{x_2-x_1} = f'(c) \ \ \Rightarrow \ \ \boxed{f(x_2)-f(x_1) = f'(c)(x_2-x_1)} $$ thus if $f'(c) >0$ we find $$ f(x_2)-f(x_1) >0 $$ which shows $f(x_1)< f(x_2)$. In other words, if $f'(x) > 0$ for $x \in (a,b)$ then we can show $f$ is strictly increasing on $(a,b)$. Naturally, this discussion is limited to differentiable function. If we extend the analysis to piecewise differentiable, or just continuous functions then the derivative need not be defined and we may not be able to use calculus to analyze the graph of such a function. For those examples we have to rely on the definition of increasing function which is based on inequalities. In any event, the $\boxed{ equation}$ or it's close cousin $f(x_2) = f(x_1)+f'(c)(x_2-x_1)$ is how we often use the mean value theorem to linearize the function with it's derivative. Even when we can't find $c$ explicitly, the existence alone suffices to prove many theorems.

Alternatively, just "prove" what I say above with some appropriate pictures paired with the MVT. Pictures are obviously important in this lecture.

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The main uses of the mean value theorem are the proofs of many of the other theorems about derivatives. Presumably those will come some time after this 45-minute class. In the "motivation" include my first sentence above. In "examples" draw some graphs, and observe that there are (one or more) tangent lines parallel to the secant line.

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