*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

*steepness*) - the better our approximation would be.

Using the terminology we are now familiar with, the

*steepness*of a curve that represents the graph of a real function

**or, simply, the**

*y=f(x)**steepness*of a real function

**at some point**

*y=f(x)***is a limit of the ratio of**

*x=r***to**

*f(r+d)−f(r)***as**

*d***,**

*d→0*that is,

*lim*[

_{d→0}*]*

**f(r+d)−f(r)**

**/d**When we introduced the exponential functions

**in the chapter "Algebra - Exponential Function", we have proven that the**

*y=a*^{x}*steepness*of function

**at point**

*y=a*^{x}**depends on the value of its base**

*x=0***, and, for**

*a***the**

*a=2**steepness*of function

**is less than**

*y=2*^{x}**, while for**

*1***the**

*a=3**steepness*of function

**is greater than**

*y=3*^{x}**.**

*1*Assuming that the

*steepness*is smoothly increasing, as we increase the base from

**to**

*a=2***, it is reasonably to assume the existence of such base between these two values, for which the**

*a=3**steepness*at point

**is equal exactly to**

*x=0***.**

*1*This value, obviously, is not an integer and, as can be proven, not even rational. Traditionally it is designated by the letter

**(probably, in honor of a famous mathematician Euler who extensively researched its properties) and is considered as a fundamental mathematical constant, like**

*e***.**

*π*The approximate value of this constant is

**.**

*2.71*Now we can state that

*lim*[

_{d→0}*]*

**e**^{r+d}−e^{r}

**/d = 1**The above can be accepted as a definition of a number

**. There are a few others, all equivalent to this one, that is each definition can be proven based on another definition.**

*e*Here are a few.

*lim*(

_{n→∞}*)*

**1+1/n**

^{n}= e*lim*(

_{d→0}*)*

**1+d**

^{1/d}= e*lim*[

_{n→∞}*]*

**n·(n!)**^{-1/n}

**= e***lim*(

_{n→∞}*)*

**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

**as a base of an exponential function**

*e***that has a**

*y=e*^{x}*steepness*of

**at point**

*1***.**

*x=0*It means that we consider as given the following statement, constituting the defining property of number

**.**

*e**For any, however small, positive*

IF|

**ε**exists**δ**such thatIF

*| ≤*

**d**

*δ*THEN

**|**(

**)/**

*e*^{d}−e^{0}

*d − 1*| ≤*ε*or, considering

**,**

*e*^{0}=1**|**(

**)/**

*e*^{d}−1

*d − 1*| ≤*ε*For those math purists, the existence and uniqueness of this number

**was not rigorously addressed. We just relied on the monotonic smoothness of**

*e**steepness*of an exponential function

**at point**

*y=a*^{x}**as we increase the base from**

*x=0***, when this**

*a=2**steepness*is less than

**, to**

*1***, when it is greater than**

*a=3***.**

*1*
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