Logarithm

a>0

a>0 and l>0

Differential Function : [f(x)]^r = r[f(x)]^(r-1)

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see

Differential Function : ln x

and

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

q isn't zero.

Limit of A Function : a^[f(x)]

f(x) is a continuous function.

Differential Function : ln x

Before you see this post,

see

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

and

e ( mathematical constant )

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

Radian

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 : log(r) [f(x)]

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.

Lunch on March 25th, 2016

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

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Knuth's up-arrow notation

or

Hyperoperation

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

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see

(a+b)^n

(a+b)^n

Before you see this post,

see

Factorial, Permutation and Combination

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

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Factorial

Permutation

and

Combination

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

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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.

Radian

Graph of Trigonometric Function : Sine, Cosine and Tangent

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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

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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

Proof of Sum of Angles of A Triangle

Before you see this post,

see

5 Postulates of Euclidean Geometry 

[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

 2nd of  5 Postulates of Euclidean Geometry.

[Pic. 4]

let's draw l₄ is parallel with l_2.

[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,

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Here

or search Euclidean Geometry on SEARCH TAP.

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

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

Sandwich Theorem

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