The Mean Value Theorem and Its Consequences (Part I)

The Mean Value Theorem & Its Consequences (Part I)

Rolle's Theorem is a theorem in calculus that states that if f(x) is a differentiable function defined on an interval [a, b] and if f'(x) = 0 at some point c in [a, b], then there exists at least one number d in (a, b) such that f(d) = 0. The theorem is named after Pierre de Fermat and Nicole Rolle.

We will study the Mean Value Theorem, which is important in its own right and has many applications.

We will then look at Rolle's theorem and use it to prove the Mean Value Theorem.

After that, we'll look at two applications of Mean Value Theorem, first of all in finding anti-derivatives and also there is L'Hopital's Rule which helps us to find limits of certain types of functions. Let's begin with Rolle's Theorem. Before we read the statement of the theorem, look at such an example:

The graph of this function is given by the blue line.

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You see that there is a point at which the tangent is horizontal, and there is a point c such that f(c) = 0. Rolle's Theorem says the following: Let f be a function that is continuous in this closed interval and differentiable at every point in the open interval.

Now, suppose at two endpoints a and b, the value of the function are equal. That means if you look at these two points, they are at the same height.

If at the two endpoints of an interval, a function is equal to the same height, then there must exist at least one point c within that interval such that the derivative is zero.

That means the tangent line is horizontal.

Now, this is obvious if we look at the graph. The proof is not difficult. Recall the Extreme Value Theorem; f is continuous on the closed interval a b, and since it is monotonic and defined on this interval, it has an absolute maximum and an absolute minimum as well. Now, where are these absolute maximums and minimums located?

First, if the absolute maximum occurs at an interior point, this point must be a critical point. In other words, the derivative must equal zero or not exist at this point. But we already assumed that the function is differentiable at every point in the interval a b.

The derivative at the interior point must be equal to zero, so the case b is out. If both extreme values, the maximum and minimum occur at the endpoints, a and b then that means that the maximum must equal the minimum because f(a) equals f(b).

When the maximum and minimum are equal to each other, then all other values must be the same. In other words, the graph of the function must be horizontal (and therefore a constant function); f(x) = c for every point c in the interval.

In both cases one and two, we see that there is always a c whose tangent is horizontal. So we can now come to the important Mean Value Theorem. Before we look at this theorem's statement, let's look at its graph.

In the graph above, f(x) is shown on the interval a b. From this graph one could imagine that there should be another point on the graph where the tangent is parallel to the red line joining the endpoints. This is exactly what the Mean Value Theorem says.

The Mean Value Theorem states that for a continuous function in the closed interval a b and differentiable at every point in the open interval a b, there exists a certain point c such that derivative at c equals this ratio: f(b) minus f(a) over b minus a. Well, what is this ratio here?

The slope of the red line joining the two endpoints of this interval is exactly equal to f(b) minus f(a). So b minus a is the difference between the values of the interval and so the height over the horizontal distance is equal to the slope.

The Mean Value Theorem states that there is always a point c between a and b whose tangent is equal to the slope of this line. This is very much like the Rolle's Theorem--if we turn this graph by a certain angle so that the red line becomes horizontal and then make the green line horizontal as well, we are back in the situation described by Rolle's Theorem. So therefore we can prove the Mean Value Theorem by applying Rolle's Theorem.

We define a new function h(x), which we call `the left-hand side' by the right-hand side. So from f(x) we make up the new function h(x) in the following way: f(a) minus this one here. It is actually not that complicated. f(a) is just one number, okay?

The ratio that appears in the Mean Value Theorem is just a number, and then so a number plus another number times x minus a. It's a very simple function, and it is actually the equation of the line joining two points: point A and point B. We just try to subtract this straight line from the function f.

Notice that the derivative of h equals the derivative of f minus this, because this simple function when differentiated gives you this.

Now we check that h(x) satisfies the condition of Rolle's Theorem, namely that h at a equals h at b. If we put x equal to a, then h is zero. If we put x equal to b, then h is also zero.

When h(a) is equal to h(b) and therefore we can apply Rolle's Theorem, there must exist a point c in the interval a b such that the derivative of h equals zero. If you put in x equals zero, you deduce that f dash c equals this which is exactly what the Mean Value Theorem states.