Notes to a video lecture on http://www.unizor.com
Steepness and Constant e
Another important concept we will be dealing with is a concept of a steepness of a curve that represents the graph of a real function y=f(x).
The concept of steepness of a curve at some point was introduced first in the chapter of Algebra dedicated to exponential functions. We have defined it, approximately, as a ratio of the increment of the Y-coordinate along the curve to an increment of the X-coordinate as we step forward from the original point where we want to measure the steepness.
We have also indicated that the smaller increment of
Using the terminology we are now familiar with, the steepnessof a curve that represents the graph of a real function y=f(x)or, simply, the steepness of a real function y=f(x) at some point x=r is a limit of the ratio of f(r+d)−f(r) to d as d→0,
When we introduced the exponential functions y=ax in the chapter "Algebra - Exponential Function", we have proven that the steepness of function y=ax at point x=0depends on the value of its basea, and, for a=2 the steepness of function y=2x is less than 1, while for a=3 the steepness of function y=3x is greater than 1.
Assuming that the steepness is smoothly increasing, as we increase the base from a=2 toa=3, it is reasonably to assume the existence of such base between these two values, for which the steepness at pointx=0 is equal exactly to 1.
This value, obviously, is not an integer and, as can be proven, not even rational. Traditionally it is designated by the letter e(probably, in honor of a famous mathematician Euler who extensively researched its properties) and is considered as a fundamental mathematical constant, like π.
The approximate value of this constant is 2.71.
Now we can state that
limd→0[er+d−er] /d = 1
The above can be accepted as a definition of a number e. There are a few others, all equivalent to this one, that is each definition can be proven based on another definition.
Here are a few.
limn→∞(1+1/n)n = e
limd→0(1+d)1/d = e
limn→∞[n·(n!)-1/n] = e
limn→∞(1/0!+1/1!+...+1/n!) = e
The last one can be formulated using infinite summation sign as
Σ∞n=0(1/n!) = e
In this course we will assume the definition of number e as a base of an exponential functiony=ex that has a steepness of 1 at point x=0.
It means that we consider as given the following statement, constituting the defining property of number e.
For any, however small, positive ε exists δ such that
IF |d| ≤ δ
THEN |(ed−e0)/d − 1| ≤ ε
or, considering e0=1,
|(ed−1)/d − 1| ≤ ε
For those math purists, the existence and uniqueness of this number e was not rigorously addressed. We just relied on the monotonic smoothness ofsteepness of an exponential function y=ax at point x=0 as we increase the base from a=2, when this steepness is less than1, to a=3, when it is greater than 1.