**IV.H APPENDIX: The Fundamental Theorem of Calculus: Proof for
Positive Continuous Functions.**
© 2000 M. Flashman

Here is a proof of the first draft version of the Fundamental Theorem
announced in Section IV.H. It follows the same argument presented in Example
IV.G.3 without much substantial change in the organization or the key concepts.
If you feel ambitious you might try to create this proof on your own with
Example IV.G.3 as a template.
**Theorem IV.4.** **[The Fundamental Theorem Of Calculus For Differential
Equations]**

Suppose
that **P is a positive continuous function on [A,B].**

Then **there is a function F so that F'(***t*) = P(*t*) for
all *t* where A < *t *< B.

In fact, **F can be defined at x = ***t* to be the area of
the region enclosed by the X-axis, the graph of the function P (i.e., Y
= P(X) ), the line X = A, and the line X = *t*. See **Figure
IV.HA.i.**

**Figure IV.HA.i**

**Proof:** For *t >* A let **F(***t*) denote the
area of the region enclosed by the X- axis, X = A, X =* t*, and Y
= P(*x*).

Notice that F(A) = 0 because the "region" in this case is a line and
its area is 0.

**We must show that F'(***a*) = P(*a*) for *a* > A,
so that F is a solution for the differential equation for the interval
[A,B]

We interpret the expression **F(***a*+*h*) - F(*a*) when
*h *> 0

Consider the graph in **Figure IV.HA.ii.**
On this graph the region enclosed by the X-axis, X=*a*, X = *a*+*h*,
and the graph of Y = P(X) has area F(*a *+ *h*) - F(*a*).

**Figure IV.HA.ii**

Examining this region more carefully in **Figure
IV.HA.iii**, we notice that since P is continuous on [*a*, *a
*+ *h*], P will have extreme values.

**Figure IV.HA.iii**

We let ** c**_{*} and c^{*} denote the points
in [*a*, *a *+ *h*] where P has its** minimum and maximum
values.**

Furthermore the region we are considering lies inside the rectangle
with base the segment of the X-axis between X = *a* and X = *a *+
*h *and height** P(c**^{* }).

The region also contains the rectangle with base the segment of the
X-axis between X = *a* and X = *a *+ *h* and height** P(c**_{*}).
This geometric situation means that ** **

**P(c**_{*})*h* < F(*a* + *h*)
- F(*a*) < P(c^{*})h,

**thus** **P(c**_{*}) < [F(*a*+h)
- F(*a*)]/h < P(c^{*}).
But because of the continuity of the function P, as $h \to
0$, $c* \to a$ and $c^* \to
a$, so $P(c*)\to P(a)$
and $P(c^*)\to P(a)$. Thus,
by the Sandwich Lemma of Chapter I, we have
**[Remember, we assumed ***h *> 0.]
It can be shown similarly that to

**Thus**
**E.O.P. **

**Notes**: 1. Another proof of this result in a slightly more general
setting is given in the next chapter.
2. **Geometric Interpretation of F'(***a*) = P(*a*):
This result has the same informational content as Barrow's Theorem discussed
in Chapter 0. At this stage you might review the statement of Barrow's
Theorem in terms of how to draw the tangent line at a point on the area
curve determined by a given curve and the X-axis. Under an appropriate
interpretation of the derivative as the slope of the tangent line, the
proof of Barrow's Theorem given in the appendix to Chapter 0 gives a geometric
argument that can be modified to justify Theorem IV.4 as well.
3.** Velocity Interpretations of F'(***a*) = P(*a*):
Consider P as a velocity function for some moving object, X, and F as a
position function for another moving object,Y, with its position at time
*t* being equal to the area of the region described in the theorem.
Under this interpretation, the inequality
**P(c**_{*}) h < F(a+h) - F(a) < P(c^{*})h
expresses a statement that the actual change in position of object Y for
the time interval [a,a+h] will be between the distance travelled by an
object always moving at object X's smallest velocity and the distance travelled
by an object always moving at object X's greatest velocity.
Under the same interpretation, the inequality ** **
**P(c**_{*}) < [F(a+h) - F(a)]/h < P(c^{*})
expresses the statement that the average velocity for object Y on [a,a+h]
is between the minimum and maximum instantaneous velocities of object X
for that interval. Thus when *h* is close to zero, **the instantaneous
velocity of Y will be close to P(***a*)! |