CHAPTER 21
### Chapter 21 - The Mean Value Theorem

The Mean Value Theorem is a rather simple and obvious
theorem yet the same can not be said
about its implications in Calculus. Its proof will offer an important review of
the definition of the derivative and the integral. Let us begin with the graph
of
and its derivative

Before
continuing it is important to keep in mind that from the definition of the
derivative, the derivative of
,
gives us the slope
at any point on the graph of

The
area under the graph of the derivative from x=4 to x=6 is given by:

, of the graph of the anti-derivative from x=4 to x=6 is
exactly the area under its derivative bounded by the same interval. Remember
from the definition of the integral:

we calculated to be
10. If we draw a line through (4,8) and (6,18), and draw an imaginary base of
length
, then we get
somewhat of a triangle. The slope of this line is
. Therefore the area under the graph also equals the slope of
this line multiplied by the change in x, 2 or 5*2=10.

Now
we know that there is a point on the graph of
between x=4 and x=6
that has the same slope as the line connecting the endpoints of the interval
(4,8), (6,18). The reason we know that this is true is because the line
connecting the two endpoints is the direct way to link
the two points. Since the graph of
takes a curved way to
connect the points then somewhere along its path its direction must be parallel
to the connecting line if it is to ever reach the endpoint.

Now
that we know such a point exists and that the derivative of the graph is
defined at every point, then the change in y, 10, can be found by finding the
slope of the line that is parallel to the line through (4,8) and (6,18). This
slope is
. This tells us that there exists a value c between 4 and 6,
such that the derivative evaluated at this point c, gives the slope of the line parallel to the line connecting the
two endpoints on the anti-derivative. Since this line must lie between x=4
and x=6, then c must also lie in this
interval, and equals the value at which this line exists.

We can express this as:

Our results can be generalized:

This is the mean value theorem;
there exists a number c between a and b on the graph of f.(x) such that f.(c)
gives the slope of a tangent through f(c) that is parallel to the line through
(a, f(a)) and (b, f(b)). Conceptually speaking it is important to remember this
as:

The
area under the curve of a graph can be expressed either by evaluating the
integral at the endpoints **or** by
finding a number c, between a and b, and multiplying it by the change in x,
(b-a). In other words the net change in a function.s anti-derivative can be
found by just using the function itself. As we shall see, finding that
constant, c, is another problem itself.