Mean Value Theorem Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

The Mean Value Theorem is a theorem that in some ways seems trivial, but turns out to be important in connecting derivatives to the general behavior of functions, to antiderivatives, etc. Our textbook presents the Mean Value Theorem in section 4.4. This discussion helps you appreciate what the theorem says and some of its significance.

To start, the conclusion to the Mean Value Theorem is usually stated in the form of an equation, namely that if f is a function that is continuous on interval [a,b] and differentiable on (a,b), then there is a value c between a and b such that

Instead of an equation, can you paraphrase the conclusion to the Mean Value Theorem in English, and particularly, can you paraphrase it in English in a way that makes clear why the word “mean” (as in “average”) appears in the name of the theorem?