In this page we are going to study exponential functions.
You can move the red dots to see the graph of different exponential functions.
Exponential functions are related to geometric progressions:
Exponential functions can be seen as extensions of geometric succesions to the Real Numbers. But it is not so easy to make this extension.
You can think that it is easy to understand the meaning of a function like 2x
(or 3x or 10x) because we can calculate thinks like
But, the meaning of
There is no simple way to defining 2x for irrational x.
In general, this kind of functions are called Exponential Functions
b is called the base of the exponential function.
We expect that this function satisfy the fundamental equation
Some of these functions are increasing (when b < 1)
And some are decreasing (when b > 1)
When an exponential function is increasing, 'at the end' it increases very rapidly even if the base is only a little bigger than 1.
We can see that all exponential functions have a very similar shape. But there is a exponential function very special. Its base is a number but it seems odd that we call number e (Euler was the first to use this notation for this number).
Tangents at 0. Existence of number e
If we look at the tangent at x=0 of such functions there is a base for which the slope of the tangent of the function at x=0 is 1. We are going to call such number as e
"The existence of a number e having the above property can be motivated as follows. Let a, b be numbers > 1 and suppose that a < b. Then for all numbers x, we have
If b is very large, then the curve y=bx will have a very steep slope at x=0.
It is plausible that as a increases from numbers close to 1 (and > 1) to very large numbers, the slope of ax at x=0 increases continuously from values close to 0 to large values, and therefore for some value of a, which we call e, this slope is precisely equal to 1. Thus in this naive approach, e is the number such that the slope of ex at x=0 is equal to 1. " (Serge Lang)
We shall find out eventually how to compute e. Its value is
Using limits, the slope of the tangent at x= 0 is (if this limit exits)
Notice that, if this limit exists, only depends on b.
Then we are saying that there is a number e such as
Thinking about the derivative of an exponential function, if that limit exits and using the property of the exponentials, we can write
This is to say, the derivative of an exponential function is a multiple of the function (we do not know yet the value of this factor)
And the number e has a very nice property
Moving the red point that it is not over the vertical axis you can see different exponential functions. You can see the relation between the derivative of a function in a point and the tangent line. You can see a right triangle and that the hypotenuse is parallel to the tangent.
What happens when the base is e?.
If you move the two red points you can see the graph the more general exponential functions:
An exponential function is fixed when we know two points.
In this page we played with exponential functions but we have problems with the definition of such functions. Remember that we do not know the meaning of
The idea is to introduce logarithms first and then use logarithms to define exponential functions.
A. I. Markushevich, Areas and Logarithms, D.C. Heath and Company, 1963.
Serge Lang, A First Course in Calculus, Third Edition, Addison-Wesley Publishing Company.
Tom M. Apostol, Calculus, Second Edition, John Willey and Sons, Inc.
Michael Spivak, Calculus, Third Edition, Publish-or-Perish, Inc.
Otto Toeplitz, The Calculus, a genetic approach, The University of Chicago Press, 1963.
Kenneth A. Ross, Elementary Analysis: The Theory of Calculus, Springer-Verlag New York Inc., 1980.