What's magic about e = 2.718 ?
The exponential function is frequently confusing. It is useful to construct some examples. Start with a simple x-y set of coordinates:
Then add points for y = 2x for x = 0, 1, 2, 3 as well as -1,-2, -3 to get a rough idea of the exponential growth:
Now fill in the half and quarter integers with the square roots and quarter roots. An easy way to do this is to use your square root key to take the square root of 2 and the square root of that number to get 1.189207, which is the fourth root of 2. Now press the multiply X button twice to multiply the fourth root of 2 by anything. Successive presses of the = button will generate the following in-between points:
Now we are ready to explore the slope of the curve made by connecting the dots:
Draw lines tangent to the curve at x = 1, x = 2, and x = 3
We can see that the slope of the line at x = 1, x = 2, and x = 3 becomes greater as x and y increase, but it is always less than y, the dependent variable. For example, the slope of the line at x = 1 can be estimated by taking the two points before and after that point. It can also be estimated from the point the slope meets the y axis.
If you repeat this exercise for y = 3x, you will find that the slope of y vs. x will then be closer to but slightly greater than y.
But if you were to repeat the exercise for y = 2.718x, then you would find that the slope of y vs. x would then be equal to y. That is, the slope of y = ex is the value of the function itself, where e = 2.718 by our graphical experiment.
Incidentally, do the above experiments on a larger scale, where x and y can be large numbers, and the tangent line will cross the x axis at x = 2.718, and that will be independent of which base we choose, whether it be 2x, 3x, ex, or 10x. Put a car on an exponential curve, any exponential curve, and back it up to infinity and its headlights will shine at e = 2.718 on the horizontal axis.
The Banker's Value
Suppose your banker offered you a doubling bond, a bond which, in time, would double in value.
