In the series: Note 7.

Version: 0.3

By: Albert van der Sel

Doc. Number: Note 7.

For who: for beginners.

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Status: Ready.

Maybe you need to pick up "some" basic "mathematics" rather

So really..., my emphasis is on "rather

So, I am really not sure of it, but I hope that this note can be of use.

Ofcourse, I hope you like my "style" and try the note anyway.

Preceding notes:

Note 1: Basic Arithmetic.

Note 2: Linear Equations.

Note 3: Quadratic Equations and polynomials.

Note 4: The sine/cosine functions.

Note 5: How to differentiate and obtain the derivative function .

Note 6: Analyzing functions.

This note: Note 7: The e

Each note in this series, is build "on top" of the preceding ones.

Please be sure that you are on a "level" at least equivalent to the contents up to, and including, note 6.

This note will be a very small note on the subjects of the e

It is important for one of the next notes (note 10).

Note 8 will have as it's main topic of finding "primitive functions". Then, note 9, is a very light touch

on "complex numbers". Next, note 10 will focus on "differential equations", and indeed,

especially the function e

One thing seems to be sure when you see such a function: when "x" is positive, those functions will rise steeply.

There are some numbers with a "sort of" special status, like '0', or '1', or π (as we have seen in note 4),

or, which we will also see in this note, the number "e".

The number "e" is close to about '2.7', but the fraction behind the dot (or comma), never ends.

As a "reasonable" aproximation, the number 2.71828 is often used in many calculations.

When you can keep it simple, then simply just 2.7 is used as an approximation.

The number "e" is important in Physics and many other sciences.

Many sorts of "growths", or even decays (like in nuclear decays), follows a formula where "e" is an

important element in. But let's first start with functions in the form "f(x)=2

the sin(x) and cos(x) functions etc..

Here we will meet a new class of functions, namely f(x)=b

These functions are easy to plot. In figure 1 below, I have plotted three examples, namely:

f(x)=2

f(x)=5

f(x)=0.3

Figure 1. f(x)=2

Le's concentrate for a moment on f(x)=2

just a few points, already gives a reasonable picture on how the function "looks like".

Let's take a look at f(x)=2

value of x: | value of y=f(x) |

x= -2 | y= 1/4, since 2^{-2}=1/4 |

x= -1 | y= 1/2, since 2^{-1}=1/2 |

x= 0 | y= 1, since 2^{0}=1 |

x= 1 | y= 2, since 2^{1}=2 |

x= 2 | y= 4, since 2^{2}=4 |

x= 3 | y= 8, since 2^{3}=8 |

Question:

Can you make a similar table for f(x)=5

Here are a few observations:

- If b > 1, then a fuction as f(x)=b

the faster the function "climbs". Suppose b=10, then when x=3, then f(x)=10

and when x=6, then f(x)=10

- If 0< b < 1, then a function as f(x)=b

When x is positive (like 2, 3 etc..), the function decreases fast. For example, for f(x)=0.3

that if x=2, then f(x)=0.3

- Note that those functions exhibit aymptotic behaviour. They never reach the x-axis completely, but they get "nearer and nearer".

For a example, for an exponential function with b > 1, then when x approaches "-infinity", then f(x) approaches '0'.

Or, lim

So, you might say that we have not much "news" here.

Figure 2. f(x)=e

Ofcourse, when x=1, we have f(x) = e

Note in figure 2, that e is about 2.7.

Around the year 1683, Jacob Bernoulli found that as "n" goes to infinity in the following expression,

the outcome "converges" to a very specific number. Today we call that specific number 'e'.

lim

So, if you would say: let's take n=100, then the expression would be (1 + 1/100)

Let's do some calculations. If you would calculate (1 + 1/10)

if you would calculate (1 + 1/100)

You see, if we only take n=100, then we are already pretty close to the "true" 'e'.

You get more and more closer, the larger 'n' is.

People with a background in economics or accounting would see a rather familiar formula.

It resembles the formula when you set away an amount of money, to a certain rate of interest, for period of "t" (n).

The formula "(1 + 1/n)

Yes, but what about physics? Just too much examples are available. Here is one that that also resembles the above.

Suppose you have N

which is a very high number of such nuclei, each have a (general) expectation value to decay.

Say that this expectation value is 6 hours, and you single out one of such nuclei, then there still is a chance

that you stare to it for several years. But on average, each nuclei has a good chance to decay in about 6h.

The "half-life" is defined to be that time, that halve of the material has decayed (with a very high probability).

The amount of original (not decayed) nuclei, can be described by:

N(t)= N

In this case the e

Many

all sorts of events where an "ensemble" growths or diminishes.

What is that with "e"? I don't think it's only math or physics. I bet that it's a matter for philosophy too.

Well, it's not uncommon to have such functions. Suppose we have these two:

u(x)=√ x

v(x)=x

Then u(v(x))= √ (x

So, if we let "v" operate on x first, en then let "u" operate in "v(x)", we have x again. So, if both functions

are applied, then nothing happens to "x". The funcion "u(v(x)" maps "x" onto itself.

If we look at these two again:

u(x)=√ x

v(x)=x

Then people also often say, that "v" is the

When two functions are related

but the use the

Thus: f(f

The functions "f" and " f

Note: (optional reading)

In other mathematical disciplines, like that of studying vectors, matrices etc.., when a "mapping" or "operator"

has an inverse operator,

where I is the Identity operator, or

Let's plot both √ x and it's inverse x

See figure 3. Note that if you would also plot the line "y=x", then it becomes visible that both functions

are "mirrored" through "y=x".

Ofcourse, when we say "x is mapped onto itself", it means the line y=x, since that function is f(x)=x,

thus for example f(f

Figure 3. x

the logarithmic function using the number '10' as it's "base", is very helpful.

A couple of examples may illustrate that.

The "claim" here, is that if g(x)=e

But, if that is true, then h(x)=ln(x), must be the inverse function of g(x)=e

We know how the the curve of e

Indeed, ofcourse it's possible to draw such an inverse curve, since you only need to mirror e

Sure, it would be great to

Figure 4. g(x)=e

Thus, if they are each others inverse, then it should be true that:

ln(e

lim _{h -> 0} |
f(x + h) - f(x) ---------------- h |

It's important to know, that the upper equation really is the "heart" of finding derivatives.

If needed, please check note 5 again, where the relation is fully explainend.

Now, using "e

lim _{h -> 0} |
e^{(x + h)} - e^{x}---------------- h |

Remember from note 1, that a

Thus:

lim _{h -> 0} |
e^{x} e^{h} - e^{x}---------------- h |

Now, we can "factor out" e

lim _{h -> 0 } |
e^{x} (e^{h} - 1)---------------- h |

So, we may write that as:

e^{x} lim _{h -> 0 } |
(e^{h} - 1)------------ h |

Note: Do you see that we have managed to pull out e

While the Limit itself converges to "1" (I should have proved that too), we have found the very remarkable fact, that:

f '(x)= e

d e^{x}------ dx |
= e^{x} |

This fact, that the derivative of a function

Indeed. It's a very remarkable fact. But we also know that e

So, it might be argued, that it's very remarkable status, should not surprise us at all.

for "quick" introductions in math. But I really like you to remember the following:

f '(x)= 1/x

d ln(x) ------ dx |
= 1/x |

Note that the function 1/x, shows asymptotic behaviour if "x" approaches "0".

We have seen 1/x before, so we were already aware of that fact.

The next note is a super quick intro on the "primitive integral" and "primitive functions".