|UBC Calculus Online Course Notes|
On this page we will calculate the slope of the exponential functions that we described earlier. This produces a startling result about the rate at which this function increases.
To calculate the slope, we will consider an interval between and , compute the ratio of average rate of change of the exponential function over this interval and then see what happens as the interval becomes very small.
The calculation begins like this:
This last line is especially convenient because it separates the part of the calculation which depends on and the part which depends on . Notice that the part which depends on is just the original value of the function . The other part which depends on is less clear cut---we have not seen a limit like this before. Let's begin to investigate it by setting in the expression:
This means that the limit is simply measuring the derivative of the function at . In other words,
Let's think about this for just a minute because it is telling us something very important. The derivative at is just a number which we could denote by . Doing so allows us to write the equation as
To express this in words, we could say that the rate of growth of the function is proportional to the function itself.
For your consideration:
Choosing a convenient base
We are now in position to see why a particular choice of base seems most convenient in calculus. For the function , you will have found the derivative
where . This means that taking the derivative of results in the original function multiplied by a funny constant. (Similarly, the derivative of is , but with the constant C=2.303).
We will now define a number, which we call , by requiring that . This gives the tidy result that
But how can we understand the value of ? Well, it is defined as the number which has the property that has slope 1 at . Notice in the demonstration below that as we change the base a , the steepness of the graph changes.
Now we are asking if we can choose the base so that the slope at x = 0 is 1. Since the tangent line at must also pass through , the tangent line should have the equation . In the demonstration is shown the graph of an exponential and the line . We would like to vary the exponential so that the graph has the line as its tangent at .
After playing around with this demonstration, you may believe that . The number is similar, in some ways, to : it is given to us by nature, we have no choice in the matter, and it is irrational (that is, it is not the ratio of two integers).
We have found some rather interesting facts here, and it is best to summarize them: