Objectives: In this tutorial, we discuss Rolle's Theorem and the Mean Value Theorem. We look at some applications of the Mean Value Theorem that include the relationship of the derivative of a function with whether the function is increasing or decreasing. We develop the First Derivative Test and look at some examples where the First Derivative Test is applied. After working through these materials, the student should be able

• to understand Rolle's Theorem and the Mean Value Theorem;
• to verify the Mean Value Theorem;
• to use the First Derivative Test to find intervals where a function is increasing or decreasing and to use this information to find local maxima and local minima.

Modules:

 Rolle's Theorem. Let f be a function which is differentiable on the closed interval [a, b]. If f(a) = f(b) then there exists a point c in (a, b) such that f '(c) = 0.

• Discussion [Using Flash]

 Mean Value Theorem. Let f be a function which is differentiable on the closed interval [a, b]. Then there exists a point c in (a, b) such that • Discussion [Using Flash]

• Drill problems on using the Mean Value Theorem.

 Corollary. Let f be a differentiable function such that the derivative f ' is positive on the closed interval [a, b]. Then f is increasing on [a, b]. Let f be a differentiable function such that the derivative f ' is negative on the closed interval [a, b]. Then f is decreasing on [a, b].

• Discussion [Using Flash]

 First Derivative Test. Suppose that c is a critical point of the function f and suppose that there is an interval (a, b) containing c. If f '(x) > 0 for all x in (a, c) and f '(x) < 0 for all x in (c, b), then c is a local maximum of f. If f '(x) < 0 for all x in (a, c) and f '(x) > 0 for all x in (c, b), then c is a local minimum of f.

• Drill problems on using the First Derivative Test.