Suppose you are a professional mountain biker who is going to do a downhill race. The slope of the mountain will smoothly increase your speed, or accelerate you, until you reach the bottom, where your speed is at its highest. This is a real-world application of one of the most useful formulas in calculus: the mean value theorem. This calculus tutorial will guide you in fully understanding what the theorem is and does.

What is the mean value theorem?
The textbook demonstration of the mean value theorem is this:

However, this is an absolutely confusing and, well, meaningless equation, unless you define the circumstances surrounding it. In order for the mean value theorem to be true, three criteria must be met:

1. The function must be continuous on the closed interval [a,c].
2. The function must be differentiable on the open interval (a,c).
3. (b, f(b)) must be a point that lies on the interval (a,c).
With those three criteria met, the equation for the mean value theorem makes more sense. The derivative of the function at some x-coordinate b along the function must be equal to the average rate of change (slope) from a to c. Thus, in a problem involving the mean value theorem, you are looking for when (at what b) the derivative is equal to the average rate of change.

Comparing it to the biker scenario
Earlier, we talked about a biker's travel down a mountainous slope. Let us check each of the criteria to see if this qualifies for evaluation under the mean value theorem:
1. The mountain slope is continuous. There are no deadly chasms along the way.
2. The mountain slope does not have any magical irregularities that interrupt the flow of time and position.
3. There are an infinite number of points available to choose from along the slope.
Looks like a match to me!

Example problem
Let's say the biker's position with respect to time is modeled by...

where f(t) is measured in feet and t is measured in seconds. The biker starts at the peak of the mountain, 3500 ft., and goes to sea level (0 ft.). At what time does his velocity equal the average velocity of the ride?

To begin, let's find the average velocity from the beginning (a, f(a)) to the end (c, f(c)). Finding f(a) is easy enough: it's (0, 3500). Finding f(c) isn't much harder, just replace f(t) with 0 and solve for t. You should get:

Therefore, the average velocity of this problem is found by finding the slope between (0, 3500) and (59.1608, 0). We find his average velocity is -59.1608.

:

Now all that's left is finding where the derivative of f(t) is -59.1608. Through the power rule and basic algebra,

Thus, we can conclude that at 29.5804 seconds, the biker's velocity is equal to his average velocity for the entire ride.

Conclusion
With these skills and the example provided, I hope you now have a better understanding of the mean value theorem, its meaning, and its context in the real world.