Quick point about equations and graphs

Hi,
mliae asked me to make some simpler post so she understands this post. So here it comes, she said that it was long since she used equations:


Well equation is something like this:

x=1

That is quite clear. Of course you can have very complicated equation with many “unknowns” which are usually noted as letters, x for example. All thats easy and it says that something (x) is equal to one, it has the value of 1.

We can manipulate these equations if we abide one rule: both sides (from left and right of equal sign) have to be manipulated. If we add one we have to do it on both sides:

x+1=1+1     —>    x+1=2

Easy. We can substitute in equations if we work with more:

a=b+42
2a+3=x

2(b+42)+3=x    (“a” was substitued by b+42 because that is what it equals to.)


Now in the post that is our concern we used this equation:

To get here we have to use graph but I did not stop on that very much so I will go through it again.

Points on graph have two coordinates. This is because graph has 2 dimensions. These coordinates are usually called x and y and they are noted like this:

(x;y)

x says how much the point is to the left or right and y says how up or down (or closer/ further)

We had two points there on the graph:

Now we will take the blue point as stacionary of course but since we are working in general and not with specific numbers, it should not matter.

So black point has coordinates x and y (x;y)

Since we want it to be general we will left it like this except the y. This is because we are going to derive functions and in those y=f(x) which means that y coordinate of the point is f(x). That is the notation that is used. We are working with the function f that gave coordinate y to our black point.

black_point(x;f(x))
blue_point(x+h;f(x+h))

The h should be clear from last time. It is the distance from the black point. Then when you insert this into the “slope” equation which I talked about here, you will get what you want.

Dragallur

Feel free to ask for more clarification.

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).

Derivatives made easy 1) Slope on curve

Hi,
today I am going to explain one fundamental equation that is used to calculate derivatives.


Derivative of a function basically says what is the rate of change or the slope of that function. You can have both rate of change of whole function which gives you another function or just at one point which equals to some number. The fundamental thing is to find the slope. Last time I explained this only for a line, not curve. On line the slope is constant but not on curve.

On the gif above you can see a function. There is a black point that is on the same place. Blue point is getting closer and closer with each step. All the time the black line creates secant line to the curve, it cuts it on two points. When it is red it is tangent line which means that it does not cuts it at all. It is the only one step in which it is just touching. To get the rate of change of curve you need to find this tangent line. Lets call the distance between the two points “h”. The tangent line will exist when h approaches zero.

THE equation. Sometimes f'(x) is written on the left.

This thing above is the equation that we will get when we take coordinates of both points.

black point(  x;f(x)   )                    [1]
blue point  (  x+h;f(x+h)  )           [1]

Just to clear things out, the first thing in the brackets is the x coordinate and the second y coordinate. As I said blue point is moved to side by the distance h and as the distance approaches 0 we are getting tangent line of curve. In the last post I showed that you need to only divide the difference of y coordinates by difference of x coordinates. Check it out if you do not know what I mean.

We will do exactly it to get right part of the equation above without the limit. Only h will be left in the denominator because there is the difference in x coordinates which means:

x+h-x=h

Now to get the tangent line we only add the limit and we have got the fundamental equation of derivates. This we can use for any curve or line that we want to derive. The left part is just a notation for derivation.

Dragallur

PS: be sure to check out this page that lets you interactively create secant and tangent lines.

[1]Just in case you do not understand why there is f(), it is basicly the same as y though we are in functions so we use this kind of notation and yes, f(x)=y. We just add h to get the y coordinate of blue point.

 

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.

 

How to! 5) Multiply two digit numbers in your head

Hi,
today I was sitting outside, chilling out and just as that I started to multiply numbers always adding one: 1*1=1 2*2=4 3*3=9 and so on. And I wondered, how far can this go without paper or calculator?


Well I finished with 100*100 without leaving single number and after a while I started to use this kind of fast and intuitive technique which in this post will be used only to multiply two same numbers smaller than 100.

Lets say you need to multiply 47*47.   (40+7)*(40+7)

Fourty will be called “base number”. The first thing you need to do is to multiply base numbers to get your base. This is simple because 40*40 is the same as 4*4 plus two zeros which means 1600.

Base is quite easy to remember so you do not need to concentrate on that much. Now we need to combine base numbers with, lets call them “pariah numbers”.

In our case pariah number is 7. You multiply it with base number to get 40*7=280 or simply 4*7=28 (+ zero from base number). Because we have two same brackets we need to multiply this twice and add it which is the same as multiplying your result by 2.

280*2=560

Now just add this “double number” to your base which should be quite easy:

1600+560=2160

Repeat this one few times and then add the product of two pariah numbers to get the result:

2160+(7*7)=2160+49=2209


Try this few times, and if you are sure in this simple case try any two digit numbers. It is quite nice way to warm up your brain 😉

Dragallur

Limits made easy and continuity

Hi,
yesterday I wrote some basics about functions and today I will write about limits again, and when they are defined and when they are not.


Just to remind you, we will use limits on functions. Those functions will usually be plotted on graph so it is easier to work with them. Our first function we work with will be the one below:

If you have not seen that yet, filled dot means that the point is inside of the function and circle means that the point is not there. Now when we speak of limit and if it is defined we have to get the same results as we approach from both sides.

What do I mean by this? Limit is basicly just approaching some number very very close. Take for example x=1. At that point in the function above, y equals 5. If we wanted to take the limit it would go as follow:  

limx→1   f(x)                           [1]

Now if we think about it before we get the result, we know that you can approach one thing in 1st dimension in two directions. Those are in limits noted as from the negative side (-) and positive side (+).

For the limit I wrote, (limx→1   f(x)) from both sides we will get the normal result as if we wanted to know what the function outputs but sometimes we just can not use the limit because  it does not exist.

Such an example would be for the following limit (still working with the first function):

limx→2   f(x)

In this limit we are trying to approach the number two but from the graph we can see that it is strange at that point. The function jumps so we should check the limit from both sides.

limx→2+   f(x) = 1
limx→2   f(x) = 2

Since the limits do not equal each other, the one before does not exist! We can have so called “one-sided limits” but they are not of much use as far as I know. Now we can actually create amazing definition for continous function!

Continous function is such that does not have any jumps or spaces in it.

example of a discontinuous function with a hole

 

 

The function on the left is discontinous. This is because on one point there is space, which means that function is not defined there.. this is problem if you want to be continous.

 

 

example of a discontinuous function with limits from left and right not equal.

 

Another function is discontinous even that it is defined on all Xs. The problem lies in what I told you about, if we took the limitx→3 f(x) = undefined
This is because from positive and negative side we will get different result.

 

example of a discontinuous function where f(a) and lim f(x) as x approaches a not equal

Again the function is not continous. This time the  limit at 2 exists but it does not equal the value of the function.

limitx→2 f(x) ≠ f(2)

 

 

example of discontinuous function where the limit does not exist, vertical asymptote.

 

 

This one on is not continous either. This is because if we approach from negative values we will get -∞ but if we approach from positive values we will get +∞. The limits do not equal so the function can not be continous.

 

From these points we can summarize what function has to have to be continous at one point.

  1. Point there is defined, it is not empty circle!
  2. Limit on the point exists.
  3. The limit equals to the point on the function.

Or also:

  1. f(a) is defined
  2. limx→ a f(x) exists
  3. limx→ a f(x) = f(a)

This is all from me for now, thanks to “analyzemath.com“. I did not need to create all the graphs again thanks to them.

Dragallur

[1]Sadly wordpress is not able to create some normal indexing so I will write it like this. Normally limit is noted as this:

Proper notation for limit where x approaching something is in the bottom.

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

What does the 3rd Kepler’s law say?

Hi,
today I want to do a short post about the 3rd Kepler’s law. I kind of really like it because it has very simple explanation but lot of uses at the same time.


The law goes as follow:{\frac  {T_{1}^{2}}{T_{2}^{2}}}={\frac  {a_{1}^{3}}{a_{2}^{3}}}

T stands for time and for semi-major axis of ellipse, that is basicly radius for planets since

What is semi-latus rectum?

their orbit is highly circular. The index and 2 stands for first and second object, basicly you are comparing two objects with each other though they must orbit the same body. This is very useful since you can compare anything in Solar System orbiting Sun with Earth. Why is it useful? Because Earth’s semi-major axis is 1AU and orbit lasts for 1 year which means that this fraction will disappear and you are left only with the object you want to calculate with.

Where did this even came from? The prove for this equation is very simple and basicly stands on the fact that centripetal force equals gravitational force for our orbiting object.

Fg=Fc

We can find the equations for both of these forces and from them finally get to the Kepler’s law:KeplerLaw3

Ok, before you start to freak out, this is completely easy. First line is clear, I have accidentaly indexed Fd instead of Fc because in Czech the force is called “dostředivá”.

Second line shows the forces and their equations, third canceles the mass of the orbiting body and the radius of orbit. Since v=s/t we can write it down as is shown. Also watch out because s is whole orbit so s^2=4π^2

The equation that you have in fifth and six line is also usable equation! It is more general and does not need the second orbiting body but it needs the mass of object. From this equation you can also figure out the mass of Sun which is completely amazing! (You have to watch out for the right units!)

After the small space I have divided the equation by the same one except that it works with some other object orbiting the same star (or planet..), with this step I will get easily rid of all the π, gravitational constant and mass of the center object.

Now we have the original 3rd Kepler’s law!

Dragallur

PS: in the prove we also assumed that r=a which means that planets orbit on circles not ellipses but it is accurate enough

How to pile up stuff

Hi,
today I will write about block (in my case) or also book stacking problem. This is fascinating problem and I want you to try to take twenty cards or same blocks. Your quest is to stack them on top of each other but at the same time try to hang them over side of table as much as you can.


It should look something like this:

How much “overhang” can you theoreticly do? With twenty cards the overhand will be maximally 1.79886982857. You can only put one card on another of course.

What is this thing anyway?

Well there is nice physics and math involved behind!

Lets start with the proof why you can actually do this in the first place. Try to ask someone around “How much is it possible to overhang infinte amount of blocks?” most people will probably tell you that the answer is 1/2 which is kind of intuitive, but not true as you already know if you tried the experiment that I told you to do in the beginning.

It all has to do with the center of mass. This is the place that you are trying to balance at the edge of table. As soon as the center is moved behind the edge it will fall.

For first card it is easy, you can truly get overhang of only 1/2 of the width of your block.

For two objects you can get better, actually 3/4! The first block (highest) will stay the same, it will be exactly at half of the previous one to maximize the effect. For this the center of mass will be on the edge. What about the one under that? Well there comes the problem because it combines center of mass with the one above which means that it cannot stretch so far. It carries two times the mass, lets say 2M

First block goes like this: X*M where M is the mass and X is the distance it can stretch, it is 1/2. This must equal also for the second block which carries two so there are two “masses”.

X*2M=X*M (X is different value every time)

X*2M=1/2M (because as I said X for first block is 1/2)

2X=1/2 (divides by M)

X=1/4

the second block has to be one fourth of its width from the table.

For third one — X*3M=1/2M …. X=1/6

Fourth = 1/8

As you continue you will find that this goes in particular series called “harmonic series”.

1/2 + 1/4 + 1/6 + 1/8…[1]

This series sums up to infinity so you can theoreticly make any overhang you want!

For the overhang of 1 you need only four — 1/2+1/4+1/6+1/8=25/24=1.04166

For overhang two you need 31!
For three you need 227!
For four you need 1,674!
For five you need 12,367!
For six you need 91,381!
For ten you need 272,400,601!

As you can see it increases pretty rapidly. 

Now just enjoy me trying to get to the best possible overhang:

Dragallur

[1] Harmonic series are just 2x bigger: 1+ 1/2 + 1/4 + 1/6 + 1/8…

I used these pages as resources both for pictures and for information: 1) 2) (only first picture)

Easy way to get excentricity

Hi,
it has been few days since I looked on some problems from astronomy contest. One of the problem was about excentricity of Pluto.


This has to do with ellipses since according to Kepler’s laws, planets are orbiting on ellipses, shapes just a little different from circles, at least when you consider their equation.

For circle equation is x2+y2=1

For ellipse equation is x2/a+y2/b=1

Sun is always the focus of the elipse, above those two points are -c,0 and c,0. For planets those ellipses are much less excentric which means that in the equation above, “a” and “b” are fairly similar.

In the problem I knew only perihelium and afelium of Pluto.

e = \frac{\varepsilon}{a}=\frac{\sqrt{a^2-b^2}}{a}

“e” is excentricity. “a” is semi-major axis. “b” is semi-minor axis. “ε” is linear excentricity (not really important).

Since the equation goes as the one above you need both semi-major and semi-minor axis to get the result. From knowing afelium and perihelium I easily got semi-major axis. To get semi-minor you must know that the distance from focal point to the top or bottom of ellipse is equal to semi-major axis, from this you can use pythagorean theorem and then all this information insert into the equation. All went right and with perihelium of 29.66 AU and afelium of 49.32 AU the excentricity is 0.246 which is just right, if excentricity is equal to 1 than it is parabola and if greater it is hyperbola.

Dragallur