I'm going to give a lecture in a public contest (like a civil service examination to be a university public teacher in my country) about the intermediate value theorem in 45 minutes. I'm going to give this lecture to three professors (the examiners) but it's like I'm going to give it to undergraduate students.Since the proof of the intermediate value theorem is too hard to show to someone who's never learned real analysis, what should I do in this class? I'm open to suggestions.

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I&#039;m going to give a lecture in a public contest (like a civil service examination to be a university public teacher in my country) about the intermediate value theorem in 45 minutes. I&#039;m going to give this lecture to three professors (the examiners) but it&#039;s like I&#039;m going to give it to undergraduate students.Since the proof of the intermediate value theorem is too hard to show to someone who&#039;s never learned real analysis, what should I do in this class? I&#039;m open to suggestions.

Boatein · Accepted Answer

You should first start with some explanation of the theorem via graphic illustration and then present some key applications. Apart from those mentioned in the other answer one of the basic applications of intermediate value theorem is the existence of n&#039;th roots of positive real numbers. Usually when roots are introduced in high school no one really discusses their existence.Maybe you can also add some historical remarks about the theorem and the fact that the theorem was used much before it was finally proved. And in fact people did not even think it needed a proof.And then you can provide proofs based on multiple approaches. It is also important to stress that intermediate value property is not the privilege of continuous functions only and there are discontinuous functions which have this property and one important class of such functions is the class of derivatives.Note: I am not an educator by profession and the above is not based on any teaching experience but rather based on what I would have liked in a class on this theorem if I were a student.

Jaycee Mathis · Accepted Answer

You should mention some of its consequences, such as:1. every polynomial function from       R   into itself with odd degree has at least one root;2. if   a  ,  b  ∈      R   are such that       a    2    +      b    2    =  1 then there is some   θ  ∈  [  0  ,  2  π  ) such that   a  =  cos  ⁡  θ and   b  =  sin  ⁡  θ.

I'm going to give a lecture in a public contest (like a civil service examination to be a university

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