The Mean Value Theorem

Another application of the derivative is the Mean Value Theorem (MVT). This theorem is very important. One of its most important uses is in proving the Fundamental Theorem of Calculus (FTC), which comes a little later in the year. See last Fridays post Foreshadowing the MVT  for an  a series of problems that will get…

The Mean Value Theorem

Another application of the derivative is the Mean Value Theorem (MVT). This theorem is very important. One of its most important uses is in proving the Fundamental Theorem of Calculus (FTC), which comes a little later in the year. Here are some previous post on the MVT: Fermat’s Penultimate Theorem   A lemma for Rolle’s Theorem:…

Mean Tables

The AP calculus exams always seem to have a multiple-choice table question in which the stem describes function in words and students are asked which of 5 tables could be a table of values for the function.  Could be because you can never be sure without other information what happens between values in the table.…

The Mean Value Theorem II

The Rule of Four suggests that mathematics be studied from the analytical, graphical, numerical and verbal points of view. Proof can only be done analytically – using symbols and equations. Graphs, numbers and words aid in that, but do not by themselves prove anything. On the other hand numbers and especially graphs can make many…

The Mean Value Theorem I

The Mean Value Theorem says that if a function, f , is continuous on a closed interval [a, b] and differentiable on the open interval (a, b) then there is a number c in the open interval (a, b) such that . Actually, it says a lot more than that which we will consider in…

Rolle’s Theorem

Rolle’s theorem say that if a function is continuous on a closed interval [a, b], differentiable on the open interval (a, b) and if f (a) = f (b), then there exists a number c in the open interval (a, b) such that .  (“There exists a number” means that there is at least one such…

Fermat’s Penultimate Theorem

I have mixed feelings about proof in high school math and high school calculus. I am not one for proving everything. For one thing, it cannot be done and, if it could be done, proof would become the whole focus of high school math. Proofs are not the focus of first-year calculus or AP calculus.…