Mean Value Theorem

What if a car averages sixty miles per hour — must it have been going exactly sixty at some moment? Yes! The Mean Value Theorem guarantees it, and here is why. We have a smooth curve with two points, A on the left and B on the right.

• Smooth curve, points A and B

The blue line connecting A and B is called the secant line. The slope of the secant line is the average rate of change from A to B.

The Mean Value Theorem says: somewhere between A and B, the tangent is parallel to the secant. Parallel lines have equal slopes, so the instantaneous rate equals the average rate at point C. Let us state the theorem properly now.

• Tangent parallel to secant at C

Here, a and b are the two endpoints, and c is the mystery point in between. The theorem needs two conditions: the curve must be continuous and smooth — no breaks or sharp corners.

• a, b = endpoints; c = mystery point

Let us try a worked example with simple numbers. Take f of x equals x squared, from a equals one to b equals three.

First compute the average slope: f of three minus f of one, divided by three minus one.

That gives eight over two, which equals four.

Now the derivative of x squared is two x, so set two c equal to four.

Solving gives c equals two, which is indeed between one and three — perfect!

Real life: a speed camera at the start and end of a tunnel records your average speed. If your average speed exceeds the limit, MVT proves you were speeding at some exact moment. Now your turn: for f of x equals x squared, from a equals two to b equals six, find c.

• Average speed → exact speeding moment

The Mean Value Theorem guarantees a point where instantaneous rate equals average rate — every time.