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I've to give a model lecture on Mean Value theorems(Rolle's theorem,Lagrange's Mean value theorem,Cauchy's Mean value theorem).But i do not know what content i should include in the lesson(I do not have any prior experience of teaching).I think of the following questions which i should present-

  • What is the motivation for Mean value theorems?
  • What is the need for Mean value theorems?
  • Why Rolle's theorem,Lagrange mean value theorem,Cauch mean value theorem are different?
  • What are the applications of mean value theorems in mathematics?

  • Mean value theorems are helpful in proving some advanced result in maths?

    I need suggestions/advice/critics about above questions?Are they enough,or i'm forgetting some other points?

I need help from the members of MSE in improving the above questions and adding some interesting problems(if there is any scope).

thanks and regards

paf
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Styles
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    Can you tell us who your audience is? A survey of MVTs for senior academics will be quite different to one for students in their first year of analysis :) – postmortes Aug 09 '18 at 10:30
  • @postmortes:It is for the job teaching first year analysis student,but the audience will be experienced teachers and professors. – Styles Aug 09 '18 at 10:35
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    @postmortes:http://www.fiitjeehr.com/RecruitmentPolicy/0069_SelectProcedure.pdf – Styles Aug 09 '18 at 10:37
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    They allow you to pass information from the function to their derivatives, and from the derivatives to the function. You can present an example of each direction. Typical examples of the first direction are proving existence (or counting) of solutions of equations in which the equation is the derivative. Typical examples of the second direction is proving inequalitites. –  Aug 09 '18 at 11:03
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    You can mention that all those theorems can be proven from each other, and are all equivalent to the continuity axiom of the real numbers. As a consequence you only really need one of them. Of course, depending on the problem, one of the versions can look ready to apply, while others would require more massaging. –  Aug 09 '18 at 11:06
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    You should definitely provide examples of the theorems failing when each of its hypotheses get weakened. At least do it for Lagrange's, or Rolle's. –  Aug 09 '18 at 11:08
  • @floodbaharak:Thanks you,its a great help you just did – Styles Aug 09 '18 at 11:58
  • @user582578: If we remove the condition of differentiability in Lagrange's theorem, then $\vert x \vert$ can be used to show the failure of the theorem. If the condition of continuity is neglected then the result will not hold trivially as derivative of function does not exist. – Styles Aug 11 '18 at 18:22

3 Answers3

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Remember that your primary objective in this model lecture is to communicate clearly and effectively to your scenario audience (first year analysis students), not to impress you actual (experienced) audience with the breadth and depth of your knowledge.

I imagine your biggest challenge when delivering your model lecture will be time management. The various versions of the MVT, with motivation, background, generalisations, connections, applications etc. is an enormous topic. You could easily fill a two or three hour lecture. But I expect you only have 30 or 45 minutes - and you will need to start from basics.

So focus on getting the essentials in first. Motivation is key - why do we want to prove the MVT ? Why is it not obvious ? Can't we just draw a graph ? Remember that results like the MVT look obvious at first glance to students who are used to working with polynomials and other well behaved, smooth functions. A few well chosen examples where the MVT property does not hold will help with motivation.

Next go through a simple and clear proof. Outline the overall proof strategy. Then walk through the proof step by step. Show how each of the conditions in the theorem statement is used in the proof. Don't forget that your scenario audience may not be completely comfortable with formal proofs, so take time over this.

You can then mention generalisations, applications etc. briefly at the end if you have time. And provide some further reading references/links so that the more able/interested students have something to take away.

gandalf61
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You need MVT to prove just about anything about derivatives. For example, $f'=0$ implies $f$ is constant. (Of course explaining what the need for MVT is depends on the audience. In a calculus class the students probably think it's obvious that $f'=0$ implies that $f$ is constant. In a context like that where they don't see the need to prove such things there really is no valid reason to prove MVT... Edit: I just saw the description of the context in that pdf. In a situation like that you might comment on how in math we do feel the need to prove things. Heh mention how the logical foundations of calculus were all fuzzy until we were saved by Weierstrass and Cauchy...)

Or a dramatic generalization: MVT plus the definition of the Riemann integral make it trivial to show that if $f$ is differentiable on $[a,b]$ and $f'$ is Riemann integrable then $f(b)-f(a)=\int_a^b f'$. Given $a=t_0<\dots<t_n=b$, write $$f(b)-f(a)=\sum_{j=1}^n(f(t_j)-f(t_{j-1})).$$ If you apply MVT to each term $f(t_j)-f(t_{j-1})$ what you get is precisely a Riemann sum for $\int_a^b f'$.

David C. Ullrich
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It depends on the audience. I assume you talk to first year undergraduate math students.

Of course, you must include the precise statement of the theorems (in particular, insist on the condition $f(a)=f(b)$ in Rolle's theorem) and geometric interpretation (except maybe for Cauchy's MVT), maybe with their proofs (depending on the audience: it's necessary for math students, probably not for some other students) and you must give examples so that students can understand how to apply these results in practice. Maybe you can give counter-examples.

One of the main consequences of MVT is the theorem linking variations of a differentiable function and sign of its derivative. If you have time, you may state (and prove) it.

Finally, beginning by including some motivation can't be bad, to say the least.

paf
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  • :thanks a lot for this helpful answer!Will you please explain "MVT is the theorem linking variations of a differentiable function and sign of its derivative."?or provide some URL for the same? – Styles Aug 09 '18 at 10:45
  • See page 3 of http://www2.clarku.edu/~djoyce/ma120/meanvalue.pdf – paf Aug 09 '18 at 13:40
  • :"You can explain Rolle's theorem by saying that if your average speed during a journey from A to B said 50kms/hour then there had to be a time when your instantaneous speed was 50kms/hour as well. The significance of Rolle's theorem" can be used as motivation? This is from the answer of Kaushik:https://math.stackexchange.com/questions/468838/importance-if-rolles-and-lagranges-theorem-in-daily-life-for-school-children?rq=1 – Styles Aug 12 '18 at 19:11