2016년 3월 25일 금요일

Differential Function : ln x




Before you see this post,

see

A Property of Limit of A Function : f(x)g(x)


and












Limit of sin(x)/x and tan(x)/x



Before you see this post,

see

A Property of Limit of A Function : f(x)/g(x),


Trigonometric Function : Sine, Cosine and Tangent,


Graph of Trigonometric Function : Sine, Cosine and Tangent


and












First, Let's divide the formula by sinθ.



There is a result.




Second, Let's divide the formula by tanθ. 



There is a result.





A Property of Limit of A Function : log_r [f(x)]





f(x) is a continuous function.


δ=δ0


e ( mathematical constant )




Before you see this post,

see


A Property of Limit of A Function : f(x)g(x)


and












A Property of Limit of A Function : f(x)/g(x)



Before you see this post,

see


A Property of Limit of A Function : f(x)g(x)






f(x) is a continuous function.

let me show you that a property of limit of a function : 1/f(x).


δ=δ0



Use property of limit of a function : f(x)g(x) like the picture below.


Result will be showed like the picture above.






2016년 3월 23일 수요일

Differential of A Function : f(x) / g(x)




Before you see this post,

see

A Property of Limit of A Function : f(x)g(x)




f(x) and g(x) are continuous functions.






Differential of A Function : f(x)g(x)



Before you see this post,

see

A Property of Limit of A Function : f(x)g(x)




f(x) and g(x) are a continuous functions




Differential of A Function : f(x)+g(x)


Before you see this post,

see


A Property of Limit of A Function : f(x)+g(x)




f(x) and g(x) are a continuous functions.






Hyperoperation






If you want to get more information about this topic,

Search Knuth's up-arrow notation or Hyperoperation

or

click

Knuth's up-arrow notation


or




1^3+2^3+3^3+...+n^3




Before you see this post,

see

(a+b)^n











(a+b)^n


Before you see this post,

see

Factorial, Permutation and Combination










2016년 3월 22일 화요일

Factorial, Permutation and Combination







n! -> n factorial
0! = 1 



   nPr -> n permutation r 
nPn = n!
n ≥ r

nPr means the number of methods 
to choose r different elements among n different elements 
with order.






nCr -> n combination r

      n ≥ r     

nCr means the number of methods
to choose r different elements among n different elements
without order.








if you want to more about factorial, permutation and combination

click

Factorial


Permutation

and


Combination








Power series : 1^2+2^2+3^2+...+n^2


Before you see this post,

see

Power series : 1+2+3+...+n =?


and 

(a+b)^n













Power series : 1+2+3+...+n =?

Continuous Function by ε,δ







Definition of continuous function by epsilon-delta (ε,δ) and its conditions


correction : lim [x->p] f(p) -> lim [x->p] f(x) at no. 3 on a picture above.





2016년 3월 21일 월요일

Radian

Graph of Trigonometric Function : Sine, Cosine and Tangent




Before you see this post,

see

Trigonometric Function : Sine, Cosine and Tangent









Strawberries

Trigonometric Function : Sine, Cosine and Tangent




               In Right-angled triangle, they are defined like the picture below.


sine -> sin
cosine -> cos
tangent -> tan






In Cartesian coordinate system, 
we can notate trigonometric functions, sine, cosine and tangent, 
like the picture above.







Sum of Angles of An n-polygon




Before you see this post,

see

Sum of Angles of Quadrangle, Pentagon, Hexagon






Let's make an n-polygon.

To make a figure that is an angle more than a former one.
put a triangle to a former one.


we can make an n-polygon by process like the picture above.



Sum of Angles of Quadrangle, Pentagon, Hexagon




Before you see this post,

see



Proof of Sum of Angles of A Triangle



Before you see this post,

see




[Pic. 1]

Use The parallel postulate, we can find those facts in [Pic. 1]




[Pic. 2]

There is a triangle.

0°<{α,β,γ}<180°




[Pic. 3]

We can draw like [Pic. 3] by




[Pic. 4]

let's draw l4 is parallel with l2.




[Pic. 5]


We can find sum of angles of the triangle by

facts in [Pic. 1].

α+β+γ = 180°

So, sum of angles of the triangle is 180°.





5 Postulates of Euclidean Geometry



1. "To draw a straight line from any point to any point."






2. To produce [extend] a finite straight line continuously in a straight line."







3. "To describe a circle with any centre and distance [radius]."








4. "That all right angles are equal to one another."

( right angle = 90˚ )










5. The parallel postulate
"That, if a straight line falling on two straight lines make the interior angles on the same side less than two right angles, the two straight lines, 
if produced indefinitely, meet on that side on which are the angles less than the two right angles."





If you want to know more about Euclidean Geometry,

click

or search Euclidean Geometry on SEARCH TAP.