Derivatives made easy 2) Non-differentiable functions

Hi,
so I continue with this tutorial. Today I will cover when function is not differentiable. If you want to read the basics check out the last post, it links to limits and continuity which will be important today.


Just to remind ourselves here is the formula that I explained last time:

It uses limit so logically if the point we are trying to derive is not continous, which means that the limit does not exist there we can not derive. This is not the only thing that can happen to us. Sometimes function is completely continous but from negative side it grows exponentially and from positive side it is linear. Check out the nice picture below:

From the whole picture the most important point is the second one. It says: “continous (no gap) but not smooth, not differentiable”. To find this out algebraically you need to know the equations of both left and right part of the function.[1]

If you know them you will insert it into the derivation equation that we used last time as f(x). You have to add h of course as it is the difference between your two points. If both of your solutions equal (right and left side) you know that the function IS differentiable.

Dragallur

PS: ask me if you want to expand any part of the post. I have no problem whatsoever with it and it will make me even happy to write some extra parts.

[1]Yes function can be made up of different parts only on intervals. You can even write something like (odd Xs are equal to 2).

Limits made easy and slope

Hi,
today I am going to explain to you very basic concept called the slope. This is going to be important for the basic understanding of derivation.


Slope describes how much is line steep and if it is decreasing, horizontal or vertical. There is simple equation for it.

m=\frac{y_2-y_1}{x_2-x_1}.

m is the slope of a line

m = \tan (\theta)\!

This equation equals the first one because trigonometry

So to get the slope of line at given point you simply pick one point on it which has two coordinates: x1 y1. Then you pick another point to the right side and it is going to have coordinates x2 y2.

Here you can see why tan(theta) is equal to m

The subtraction creates this kind of triangle from which you can either calculate the angle or the m value which is the steepness.

Lets make an example (from wikipedia), lets say we haw line that runs through two points: A=(1,2) B=(13,8)

When we have such a information we can calculate the slope because:

x1=1
y1=2
x2=13
y2=8

All parts of equation are set:m = \frac{\Delta y}{\Delta x} = \frac{y_2 - y_1}{x_2 - x_1} = \frac{8 - 2}{13 - 1} = \frac{6}{12} = \frac{1}{2}

To show you in practice that such slope is not very big, watch the gif below.

Slope is used often in everyday life for the steepness of road. Often you will see sign as this one in mountains.

This type means that for 100 meters in horizontal direction you will go up by 10 meters.

Next time I will talk about steepness but this time not only on line where the steepness is always the same but also on curve where it changes every moment though derivatives are capable of solving such problems.

Dragallur

PS: all pictures were from wikipedia page so I will not link them one by one.

 

Limits made easy and heart equation

Hi,
I said that I started to learn limits. Since often I write about the things I just learn I will start this series today with the very basic. (The start is actually so easy you wont even think that it is precalculus or something)


Basicly the context for limits are functions. I have personally never learned functions on

Basic idea of function

their own and I do plan to do that but I do not think it is so important right now. You just
have to have this idea that when you have function you will give it some input and it will give out some output (kind of black box).
It is important to say that one input corresponds only to one output. You have already probably seen some graphs so it is good to say that you can plot a function.

 

The thing with outputs and inputs can be nicely illustrated on one thing: imagine a class of kids and that you measure them.

Carl – 157 cm
Ann – 152 cm
Caroline – 160 cm
… and so on

Now you will plot them on graph next to each other. In big enough class you would probably have more kids with same height.Limits made easy1

On the left we have three example kids from one class. As is written, Ann has 152 centimeters, Susan 152 too but Jacob has 149,155 and 161 centimeters. Such class would not make a great function! The interesting thing is of course that to one height (152 cm) you can match more people (Ann and Susan), this makes sense. But you can not match one person to more heights! This is the way function works. You could just change names and heights for x and y and you would be there[1]. So if you graph some function it will never be vertical and basicly no two values will ever be above each other, when you on the other hand plot equation you can easily get graph where there are two points above each other:

Heart

Equation stolen from twitter and modified for better look.

So… what role do limits play here? As you probably already heard, limit tolds us what value will function give us when we give it input really close to some number. If you have function like

f(x)=x^2                  (f(x) is the way functions are noted, you could use other symbols)

You can ask for limit of x approaching to any real number and you would get the same as if you would calculate x to the power of 2. The real usage of limits comes when there is point where you can not get some nice value and also I think that it is defined by limits if function is continous (without jumps) or not.

I will slowly continue in the next post 😉

Dragallur

PS: some of my readers mathematicians (or anybody), if there please point out any mistakes, I am just learning this so it would be great to know my mistakes!

[1] To make it even clearer I made this extra picture:

Limits made easy2

Turn your brain into supermassive black hole

Hi,
here it comes here it goes! Today I am writing probably the last post from Brazil since I am leaving in Tuesday. Maybe tommorow after seeing Christ in Rio, I will be able to write another one.
Anyway, here comes the promised post about Ackermann function.

Ackermann function is a function with two inputs and it is growing extremely fast. To see the results it is best to make a table. Ackermann
function is written like this: A(m,n).
There are few types of Ackermann function because some people made their modifications of it to fit their plans. Here I will mention the most famous one.
 A(m, n) = \begin{cases} n+1 & \mbox{if } m = 0 \\ A(m-1, 1) & \mbox{if } m > 0 \mbox{ and } n = 0 \\ A(m-1, A(m, n-1)) & \mbox{if } m > 0 \mbox{ and } n > 0. \end{cases} So lets see what this function means:
A(m,n) is the input of this. If m=0 then you will make n+1 and find it in the table which is below.
If m is bigger than 0 and n=0 then you will call Ackermann function again with m-1 and n=1.
Last one, when you have m>0 and n>0 then you call the function again with m lowered by one and n will be defined by another A function which obeys

m\n 0 1 2 3 4 n
0 1 2 3 4 5 n + 1

the rules again. Lets see the table. This is the first row which is extremely simple. M is the vertical axis. So when the arguments are A(0,0) then you go as follows: m=0 which means that you higher n by one which is 1 and that is the result.

If A(3,2) it gets very messy: here you go.
A(3,2)
A(2,A(3,1)  — because n and m are higher than 0 you lower m by one and then you call another function with n lowed by one.
A(2,A(2,A(3,0) — both are still higher, so you do the same thing again..
A(2,A(2,A(2,1) — change the function from A(3,0) to A(2,1) because n is 0 according to rule above
A(2,A(2,A(1,A(2,0)
A(2,A(2,A(1,A(1,1)
A(2,A(2,A(1,A(0,A(1,0)
A(2,A(2,A(1,A(0,A(0,1)
A(2,A(2,A(1,A(0,2)
A(2,A(2,A(1,3)
A(2,A(2,A(0,A(1,2)
A(2,A(2,A(0,A(0,A(1,1)
A(2,A(2,A(0,A(0,A(0,A(1,0)
A(2,A(2,A(0,A(0,A(0,A(0,1)
A(2,A(2,A(0,A(0,A(0,2)
A(2,A(2,A(0,A(0,3)
A(2,A(2,A(0,4)
A(2,13) — I skipped lot of the steps because it is such a mess when you have to do it whole again but it equals 29.
=29

Values of A(mn)
m\n 0 1 2 3 4 n
0 1 2 3 4 5 n + 1
1 2 3 4 5 6 n + 2 = 2 + (n + 3) - 3
2 3 5 7 9 11 2n + 3 = 2\cdot(n + 3) - 3
3 5 13 29 61 125 2^{(n + 3)} - 3
4 13

={2^{2^{2}}}-3

65533

={2^{2^{2^{2}}}}-3

265536 − 3

={2^{2^{2^{2^{2}}}}}-3

{2^{2^{65536}}} - 3

={2^{2^{2^{2^{2^{2}}}}}}-3

{2^{2^{2^{65536}}}} - 3

={2^{2^{2^{2^{2^{2^{2}}}}}}}-3

\begin{matrix}\underbrace{{2^2}^{{\cdot}^{{\cdot}^{{\cdot}^2}}}} - 3\\n+3\end{matrix}
5 65533

=2\uparrow\uparrow\uparrow 3 - 3

2\uparrow\uparrow\uparrow 4 - 3 2\uparrow\uparrow\uparrow 5 - 3 2\uparrow\uparrow\uparrow 6 - 3 2\uparrow\uparrow\uparrow 7 - 3 2\uparrow\uparrow\uparrow (n+3) - 3
6 2\uparrow\uparrow\uparrow\uparrow 3 - 3 2\uparrow\uparrow\uparrow\uparrow 4 - 3 2\uparrow\uparrow\uparrow\uparrow 5 - 3 2\uparrow\uparrow\uparrow\uparrow 6 - 3 2\uparrow\uparrow\uparrow\uparrow 7 - 3 2\uparrow\uparrow\uparrow\uparrow (n+3) - 3

Here is the rest of the start of the table. It increases rapidly since it repeats over and over again (this is called recursion). You see that you have to use Knuth`s up-arrow notation.
Now you see why A(G64,G64) is such spawn of hell.
The reason for Ackermann function to exist and to be so famous is that it is one of the first functions that are used in computability theory. It is the theory which asks what means that the function is not computable and how much not computable it is. Computable functions are those for which we can find some algorithms, and algorithms are very important. For example in computations.
It seems for example that there is no computable function for finding prime numbers or at least no efficient one.

Dragallur

1) Triangles: Prologue

Hi guys,
i am starting new row of posts from mathematics. (Of course I will also do fundamental forces.)
I would like to talk about triangles.

So they are geometrical objects with three edges and three vertices.
Triangle illustration.svgThere are many kinds of triangles because you can have different angles at vertices and your edges are different length. All of these properties fall under some rules. For example if you add up alpha, beta and gama angles you will always get 180 degrees.
Equilateral kind of triangle is special one because its sides are the same lenght with 60 degrees angles. Equilateral Triangle

Isosceles are those with same lenght of two sides and scalenes are those that are kind of chaotic because their sides or angles are not equal to others.
There are also kinds with one angle with 90 degrees. Then there is type of triangle with all angles smaller than 90 and, one with one angle bigger than 90.

There are similarities between triangles  if they meet some conditions.
First kind of condition is called SAS = side angle side
It means that if in one triangle has two sides and angle between them coresponding to second triangle´s sides and angle with same length (for sides) and measure (for angles) it is similar.
ASA = angle side angle
two angles “touching” side corresponding to second triangle
SSS = side side side
AAS = angle angle side
only one angle touching side

Sorry guys for this short and boring post, i just wanted to say this before i start to talk about trigonometric functions which are kind of my favourite.
Dragallur