Mean Value Theorem

The Mean Value Theorem connects the average rate of change with an instantaneous rate of change. It states that if a function is continuous AND differentiable on an interval then there is at least one point in the function where an instantaneous rate of change is equal to the average rate of change.

f '(c) = f(b)-f(a)/(b-a)

1. Explain what this means graphically by showing an example.


The graph shown here is f(x)= x^2+1. First of all the graph is both continuous and differentiable so it is a possible candidate for the Mean Value Theorem. Now let's apply a secant line (or an average rate of change)

The green line [y=2x+1] represents f(b)-f(a)/(b-a). The two points where the green line intersects our parabola are the points "a" and "b".

The tangent line is parallel to the secant line, meaning that instantaneous rate of change is equal to the average rate of change.

2. Explain why this only works for continuous and differentiable functions.
The Mean Value Theorem only applies to continuous and differentiable functions because if a graph has a discontinuity of any kind, then it disrupts the normal flow of the graph and it would mean that the interval has stops. You cannot find instantaneous or average rates of change when there is a chunk of the graph missing. The function also has to be differentiable because there needs to be a slope in order for any graph to have a rate of change.

This graph, y= |x|, is not differentiable at x=o because it has a corner. A "corner" has an infinite amount of tangent lines.