Chapter Three: Section Two

Rolle’s Theorem and the Mean Value Theorem

Chapter Three: Section Two Look at the image below of a continuous,

differentiable function on a closed interval. This interval is chosen so that the functional output is equal at the two endpoints. What this diagram shows us is that there is a critical value in this region. Must this be the case?

Chapter Three: Section Two

Try to draw a continuous, differentiable function on a closed interval where the endpoints have equal functional outputs, but there is not a critical value in the region…

Chapter Three: Section Two Pretty tough, isn’t it?

In fact, according to Rolle’s Theorem it is impossible. Rolle’s Theorem states that, for a function f that is continuous on a closed interval [a,b] and differentiable on the open interval (a,b), if f(a)=f(b), then there must exist some value c, where a < c < b such that c is a critical value of the function f.

Chapter Three: Section Two This result, while powerful, is pretty limited in its

usefulness. How often will we have the good fortune to be

considering a piece of a function where the output values at the two endpoints are identical?

Not often, I would suspect.

Chapter Three: Section Two What we need to consider is the larger implication

here. What Rolle’s Theorem says is that the slope of the tangent line at some point in this closed interval is equal to the slope of the secant line joining the endpoints. Here, the slope happens to be 0. Can we generalize this and say that the slope of the tangent must always have a point where it is equal to the slope of the secant line joining the endpoints of such a region?

Chapter Three: Section Two I want you to take a few minutes here and do

some drawing. Sketch a few examples of continuous,

differentiable functions. Imagine cutting this function on an interval and joining the endpoints of this region. Does it look like there must be a tangent in the region that is parallel to this secant line?

Chapter Three: Section Two Do any of your sketches look like this;

Chapter Three: Section Two This image, copied below again, suggests a very

important result in Calculus. That result is called the Mean Value Theorem.

Chapter Three: Section Two The Mean Value Theorem tells us that, for a

function f that is continuous on the closed interval [a,b] and differentiable on the open interval (a,b) there must exist some value c such that a < c < b that satisfies the following equation;

f b f af c

b a

Chapter Three: Section Two What this formula says to us is this;

In a nice, smooth, continuous function on any interval there is a point where the instantaneous rate of change (the derivative) is equal to the average rate of change (the slope of the secant line).

In less formal terms, an example of what the Mean Value Theorem guarantees is that if I travel 200 miles in four hours there must be some instant in that time interval when I am traveling at exactly 50 miles per hour. Can you convince yourself that this is true?