Super C, the human cannonball, is shot into the air at 35 mph, but his average vertical velocity is zero. In this lesson, you will use Rolle’s theorem to explain what this means about Super C’s flight.

## Vertical Velocity

jpg” alt=”Vertical Velocity Average Rate of Change Graph” /> |

Let’s go back and look at Super C, the human cannonball. Let’s again graph his height as a function of time. Can we use the **mean value theorem** to say anything about Super C’s flight path? Remember that the mean value theorem says that in a region where our cannonball has some rate of change, there is a point on his path where the instantaneous rate of change – his velocity at that second – is going to be equal to his average velocity. So if I’m looking at his height as a function of time, I’m looking at his vertical velocity. His average vertical velocity – his average rate of change – is actually zero. His vertical velocity is zero because his end point minus his start point is going to be zero; he’s landing on the ground, and his height is no higher than it was when it started.

What this means according to the mean value theorem, is that his instantaneous rate of change, vertically, must also be zero at some point in time on his path. If you look at his height as a function of time, this makes sense. Up at the apex here, he has no vertical velocity. His instantaneous rate of change equals his average rate of change, and they both equal zero.

## Understanding Rolle’s Theorem

This leads into **Rolle’s theorem**, which is really just a specialized case of the average value theorem. Rolle’s theorem says that if the average rate of change is zero, specifically because the start and end points are both at zero, then the instantaneous rate of change equals zero at somewhere along that path.

## Rolle’s Theorem as a Graph

Let’s look at what this means more mathematically. Say you have the function *y*=*f(x)*. Here, *x* is on the horizontal axis and *y* is on the vertical axis. You have some region, *a* to *b*, where both *f* at *a* and *f* at *b* equal zero. So your endpoints are at zero.

Rolle’s theorem says that somewhere between *a* and *b*, you’re going to have an instantaneous rate of change equal to zero. This means that somewhere between *a* and *b*, the tangent to your curve is going to be zero. Just looking at this graph, you can see that the tangent to the curve has zero slope at four points.For this particular graph, you could look at the region *a* to *b*, but you could also look at the region *a* to *c*. *a* and *c* are both at *y*=0, because *f* at *a* equals zero and *f* at *c* equals zero. And some point between *a* and *c*, the slope of the tangent equals zero by Rolle’s theorem. You could look at the region between *a* and *d* and say the same thing.

The region between *a* and *e* and say the same thing again. Between any of these points where *y*=0 at both the start and the end; there is a tangent equal to zero somewhere in between those two points.

## Rolle’s Theorem as an Equation

So if you have a function *f(x)* = (2 – *x*)(*x* – 1)(4 – *x*)(6 – *x*), you know that *f(x)* is going to equal zero at 1, 2, 4 and 6. Because *f(x)*=0 at those four points, you know that in between those four points, we have some place where the tangent to this line equals zero. Specifically, somewhere between *x*=1 and *x*=2, the tangent to this curve will equal zero. Somewhere between *x*=2 and *x*=4, there will be a tangent to the curve that is equal to zero.

Somewhere between *x*=4 and *x*=6, there will be a tangent to the curve that equals zero.

## Lesson Summary

So let’s recap. **Rolle’s theorem** says that for some function, *f(x)*, over the region *a* to *b*, where *f(a)* = *f(b)* = 0, there is some place between *a* and *b* where the instantaneous rate of change (the tangent to that curve) will equal zero. It will have a slope of zero.

This is true as long as you can always find the instantaneous rate of change over that region.