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 your students ready for the MVT.

Here are some previous post on the MVT:

Fermat’s Penultimate Theorem   A lemma for Rolle’s Theorem: Any function extreme value(s) on an open interval must occur where the derivative is zero or undefined.

Rolle’s Theorem   A lemma for the MVT: On an interval if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b) and f(a) = f(b), there must exist a number in the open interval (a, b) where ‘(c) = 0.

Mean Value Theorem I   Proof

Mean Value Theorem II   Graphical Considerations

Darboux’s Theorem   The Intermediate Value Theorem for derivatives.

Mean Tables




Revised from a post of October 31, 2017


Graphing – an Application of the Derivative.

Graphing and the analysis of graphs given (1) the equation, (2) a graph, or (3) a table of values of a function and its derivative(s) makes up the largest group of questions on the AP exams. Most of the other applications of the derivative depend on understanding the relationship between a function and its derivatives.

Here is a list of posts on these topics. Since this list is rather long and the topic takes more than a week to (un)cover,

Tangents and Slopes

Concepts Related to Graphs

The Shapes of a Graph 

Open or Closed?  Concerning intervals on which a function increases or decreases.

Extreme Values


Joining the Pieces of a Graph

Using the Derivative to Graph the Function

Real “Real life” Graph Reading

Comparing the Graph of a Function and its Derivative  Activities on comparing the graphs using Desmos.

Writing on the AP Calculus Exams   Justifying features of the graph of a function is a major point-earner on the AP Exams.

Reading the Derivative’s Graph Summary and my most read post!




Revised from a post of October 10, 2017