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…

Foreshadowing the MVT

The Mean Value Theorem (MVT) is proved by writing the equation of a function giving the (directed) length of a segment from the given function to the line between the endpoints as you can see here. Since the function and the line intersect at the endpoints of the interval this function satisfies the hypotheses of Rolle’s…

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.…

Darboux’s Theorem

Jean Gaston Darboux was a French mathematician who lived from 1842 to 1917. Of his several important theorems the one we will consider says that the derivative of a function has the Intermediate Value Theorem property – that is, the derivative takes on all the values between the values of the derivative at the endpoints…

Mean Numbers

Here is a problem for you and your students. The numbers are mean until you get to the end when they all become very nice and well-behaved.   You could give this to your students individually or as a group exploration. Give each person or group a different function and/or different intervals. Choose a function…