Discovering the MVT

Today’s Blog is an exploration that will lead up to the Mean Value Theorem (MVT) and, I hope, help your students better understand the MVT and why it is true. While you may do this by hand, using a graphing calculator will make things way easier. This is good calculator practice and can be done…

Riemann Sum & Table Problems (Type 5)

For updated information for the special 2020 exams click here. AP  Questions Type 5: Riemann Sum & Table Problems Tables may be used to test a variety of ideas in calculus including analysis of functions, accumulation, theory and theorems, and position-velocity-acceleration, among others. Numbers and working with numbers is part of the Rule of Four and table problems…

Then there is this – Existence Theorems

Existence Theorems An existence theorem is a theorem that says, if the hypotheses are met, that something, usually a number, must exist. For example, the Mean Value Theorem is an existence theorem: If a function f is defined on the closed interval [a, b] and differentiable on the open interval (a, b), then there exists…

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…