Spreadsheet programs have many uses for organizing and displaying data, as well as for arranging information into various formats. The following is an exploration of the parabolic function f(x) = x^2 and it's derivative f'(x) = 2x using Microsoft Excel. It is not my purpose to bore you with the simplicities of a spreadsheet program, (which are very easy to use by the way) but to demonstrate briefly how they can be used for functional analysis.

The following tables reflect values generated by predefining a single column to perform the desired computation.

X-Value	f(x)=x^2	f(x) Value
-4		16
-3		9
-2		4
-1		1
0		0
1		1
2		4
3		9
4		16

This is as expected, because the function square's each value put into it. So as x gets larger and larger, the function tends to infinity. Now let us investigate values of -1< x < 1:

X-Value	f(x)=x^2	f(x) Value
-0.5		0.25
-0.4		0.16
-0.2		0.04
-0.1		0.01
0.0001		0.00000001
0.002		0.000004
0.04		0.0016
0.5		0.25
0.9		0.81

Interesting. What does this spreadsheet tell us? Well it seems that as our x-input is getting smaller and smaller the functional value is tending to zero. Why is this?

Well, basically the function is just squaring a fraction (on this interval). We are taking a piece of a piece, a fourth of a fourth etc...Now how would this relate to the derivative?

Let us go back to our spreadsheet program and define a new column, which automatically takes the derivative at a certain point (or x-value in this case).

Now what do we know about the derivative?

The derivative is the slope of the tangent line at a particular place in the function. This is because tangent is defined as tan(*) = [sin(*)/ cos(*)] for a particular measure *, which can parametrize any point in Euclidean two-space. Further, when we look at this location (particular x-value); what exactly are we looking at? The slope is also the (change in vertical distance)/(change in horizontal distance), so this tells us the change of the function at that x-value, ergo, the instantaneous rate of change.

X-Value	f(x)=x^2	f(x) Value	f'(x) = 2x	Derivative Value
-0.5		0.25		-1
-0.4		0.16		-0.8
-0.2		0.04		-0.4
-0.1		0.01		-0.2
0.0001		0.00000001		0.0002
0.002		0.000004		0.004
0.04		0.0016		0.08
0.5		0.25		1
0.9		0.81		1.8

So now when examining the spreadsheet we can see that x is increasing it's rate of change as we go from negative infinity to positive infinity, and vice versa.

This corresponds to our parabola having:

A negative slope on the interval from (-• , 0).

A zero slope at 0 (no kidding).

A positive slope on (0,• ).

f(x) = x²

We can visually see this in the above picture, which corresponds to f(x)=x². Now when overlapped with it's derivative graph we can see the direct relationship (if only reality were this direct).

Now this is a very simple use of a spreadsheet program, but there are so many other possible mathematical uses. One may set up entire systems of spreadsheets connected to satellites, which arrange incoming data of lunar modules, as they interact with the earth's magnetic field. One can use a spreadsheet to organize a budget. Assume we did not know what function modeled a particular phenomenon (very lifelike case), but we did have measurements of whatever effect this event was causing. We could analyze the data and possibly come up with some information about the phenomenon.

The following is a fictitious example:

Spread of religious Propaganda during times of Great Crisis: Athens GA Vs NY.C.

Time is given in months, after time 6 the crisis is over.
Initial Radius is 40/100 of total area for Athens, 12/100 for NYC
Time of Phenomenon		Athens GA	New York City
0:00:00		40/100	12/100
0:00:01		41/100	12/100
0:00:02		43/100	13/100
0:00:03		45/100	13/100
0:00:04		50/100	17/100
0:00:05		50/100	19/100
0:00:06		60/100	25/100
0:00:07		61/100	25/100
0:00:08		62/100	24/100
0:00:09		63/100	23/100
0:00:10		63/100	22/100
0:00:11		62/100	20/100
0:00:12		62/100	17/100
0:00:13		61/100	15/100
0:00:14		60/100	12/100
0:00:15		55/100	11/100

Possible explanation of data: (Being merely biased speculation, although it is indicative of how one might choose to interpret spreadsheet data)

Athens: Having more of an inclination to be a religious type in Athens, we see a gradual increase in propaganda up until time: 04, as whatever crisis reaches it's maximum. We then see a frightening increase from t:04 through t:06. This may have some indication of the local populations resistance to propaganda regimes or may be indicative of some divine inspiration in the peoples hearts (either that or fear). Now we see something interesting after:06 (after the crisis passes). We still see temporary increases in spatial area coverage! Why? What possible reasons could there be? Maybe, it spread a little too wide.

NYC: Now in NYC we see something more interesting, there is almost no increase up until the time of the crisis. Then we see an almost overnight change in the dogma's spread. Then a gradual decrease, until (after short while) an almost rapid decline. It is as if the populace decided to accept the propaganda just in case the crisis came to pass, and then hold onto it for a little bit more (just to be sure). They then rapidly gave up whatever beliefs they held, seeing that they were no longer of any use. Interesting strategy.