Visualizing the Mean Value Theorem
The Mean Value Theorem states that if f(x) is a continuous function on the closed interval [a,b] which is differentiable on the open interval (a,b), then there exists c such that a<c<b and
This statement can be a little hard to parse, so let's break it down. We'll focus on understanding the right and left sides of the equation in the theorem geometrically.
is the derivative of f(x) at a point c; geometrically this is the slope of the tangent line to the graph of f(x) at c.
is the rise over run of the graph of f(x) between a and b; you may think of this as an approximation of the slope of f(x) on the interval (a,b), but you can also think of this quantity as the slope of the line through the points and on the graph of f(x). This line is called the secant line of through A and B.
So what the Mean Value Theorem says geometrically is the following:
1. Take any two points A and B on the graph of f(x).
2. Draw the secant line between these points.
3. Then as long as f(x) is continuous and differentiable between these two points there must be a third point C on the graph of f(x) in between A and B such that the slope of the tangent line at C is equal to the slope of the secant line between A and B.
4. In other words, one can always find a point C in between A and B on the graph of f(x) whose tangent line is parallel to the secant line between A and B
Applet Instructions
Above you should see the graph of a function f(x).You can change which function to graph by typing in the input box at the top right.
Drag points A and B then hit the checkbox labeled "show/hide secant line" to show the secant line between A and B.
Next try to drag point C in between A and B so that the tangent line is parallel to the secant line you just picked. If f(x) is continuous and differentiable between A and B you are guaranteed to find such a C!