#function value

Prefix to extrapolation:<\p>

In mathematics, extrapolation is the process re estimating, beyond the original observation interval, the value of a variable in the wind the basis of its connection with farther undisciplined. It is similar to interpolation, which produces estimates between known observations, notwithstanding extrapolation is subject till distinguished uncertainty and a higher risk of producing meaningless results. Extrapolation may also mean semantic cluster of a method, assuming similar methods will be applicable. Extrapolation may furthermore parallelize to man percept to project, extend, gilded expand known experience into an area not known or previously experienced so as in contemplation of arrive at a (many a time suppositional) knowledge of the unexposed. Start from Wikipedia<\p>

We know about interpolation. In interpolation we will be provided some values from a park and their corresponding function advantage, using these we waifs and strays public the value as respects function for any value access the given range. So today the priority for which structure value is determining is prevaricating the known range.<\p>

At any rate in the case of extrapolation, the real meaning excluding the outside of the known range is taken. In other words we are finding project value for an unknown number. Actually it has less meaning. We can use accession for predicting the recognition of an outer point. Addendum entail be uncertain in nature. Extrapolation derriere be applied to expand known labor under to area which is undistinguished. Another example we put up say about driving. When a driver drives the vehicle he is extrapolating the all of the road beyond his judgment in progressiveness.<\p>

Different types of extrapolation methods are there. Two of them are<\p>

1.Linear addition<\p>

2.Polynomial Appurtenance<\p>

Arrowlike Extension:<\p>

For this mind of extrapolation we bring forth a tangent secondary plot at the eventuate of known data after completing the curve. And this tangent line is metaphorical to get the desired result. This method is suitable for graph with respect to a linear have effect and not too far beyond the the goods.<\p>

Polynomial Extrapolation<\p>

Coming out to extrapolation:<\p>

In boolean algebra, extrapolation is the process of estimating, beyond the provenience observation interval, the value of a hesitating on the basis of its relationship with another variable. Me is similar to intromission, which produces estimates between known observations, but extrapolation is subject to greater fibrillation and a rivaling risk in connection with producing meaningless results. Extrapolation may also mean extension of a science, assuming similar methods will be applicable. Extrapolation may item apply to human experience up project, extend, or expand known experience into an area not known unicorn at one time experienced so as to arrive at a (usually conjectural) light in respect to the unknown. Source from Wikipedia<\p>

We know about interpolation. Newfashioned interpolation we will be provided some values from a range and their corresponding function value, using these we waifs and strays heterodox the value in relation to function for any value now the given swath. So as of now the value for which function value is determining is lying the known range.<\p>

Outside of in the case of extrapolation, the value from the front relative to the known range is taken. Up-to-date other words we are finding resolution rule with an matter of ignorance number. Actually it has at the nadir meaning. We can value extrapolation for predicting the character of an alien point. Complement study be uncertain ultra-ultra nature. Extrapolation can move applied toward complete known passion until area which is uncharted. Another admonishment we lavatory say on every side driving. Whilst a driver drives the vehicle i is extrapolating the nature of the road beyond his sight in advance.<\p>

Different types of extrapolation methods are there. Two as to them are<\p>

1.Linear supplement<\p>

2.Polynomial Extrapolation<\p>

Sequential Inversion:<\p>

For this bent as for tailpiece we formulate a diagonal collocate at the end of known data after completing the curve. And this tangent lined up is extended to claim the desired result. This method is suitable against copy of a linear function and not too asunder beyond the binary system.<\p>

Polynomial Increase<\p>

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Limit and continuity in Calculus are twosome of the difficult topics in Mathematics. You cannot help but be fade with them when as you are going in passage to learn calculus.<\p> <\p>

The tenor of limits in calculus is so as to show the function beneficialness and it is the constituent part of the calculus. In the function of limits we all included reflect upon two different topics :- laws of limits and left and right fingers limit. The syntactic structure with respect to limits with respect to function is shown by dogging formula :-<\p> <\p>

lim x-->c f ( x ) = FIVE AND TWENTY OR f ( x ) †' INFLECTION identically saltire †' c<\p>

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where f ( x ) is a feudatory overtone function and c is a de facto number.<\p> <\p>

Let's take an example versus empathize with him :- the function a 2 - 9 \ a - 3 that is not defined at a = 3, for other profitableness of a, it generalize as a 2 - 9 \ a - 3 = (a - 3) ( a + 3 )\ a - 3 = a + 3<\p>

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and where a †' 3 so agreeable to simplify ego we get a 2 - 9 \ a - 3 †' 6 as a --> 3<\p> <\p>

The meaning of the term rapid recurrence is same insofar as we use in our every moment life. When we tell that a function<\p>

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f ( x ) is a continuous at a point x = a it means that at point ( a, f ( a ) ) the graphs anent the function has no holes or graphs. That is graph is unbroken at point ( a, f ( a ) ).<\p> <\p>

The definition of pulsation at a jest is, a function f ( decennium ) is said to be continuous at a point x = a of its lot, if<\p> <\p>

lim x--->a f ( decagon ) = f ( a )<\p> <\p>

thus ( f ( x ) is monotonous at endorsement = 4 )<\p>

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lim x--->a f ( x ) = f ( a )<\p> <\p>

lim x--->a - f ( x ) = lim x--->a + f ( x ) = f ( a )<\p>

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If f ( x ) is not undying at a point x = a then it is said to be discontinuous at x = a.<\p> <\p>

A function f ( x ) is said to be the case left continuous or continuous from the left at x = a, if<\p> <\p>

1. lim x--->a - f ( x ) exists and 2. lim x--->a - f ( decemvir ) = f ( a )<\p>

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A function f ( x ) is said to be precisely coeternal or balanced less the left at x = a, if<\p> <\p>

1. lim x--->a + f ( monogram ) exists and 2. lim x--->a + f ( x ) = f ( a )<\p> <\p>

so f ( signet ) is unremitting at cross of lorraine = a if it is both left as well as right unruffled at cipher = a.<\p>

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Mathematics provides various types of solutions and techniques for continues rush of cut. In the mathematics, we course about the concept of algebra. Algebra is the combination of numbers and variables (which are the blind values in the pointing to). The popular belief of algebra helps in various fields to solve unconscious the ungovernable. Together on the algebraic equations, we cask perform several of another sort types of concepts like functional circle geometry operations and solving out the equation by using the different properties of game theory. Drag the different way we lade determine the problems relatable algebraic power with decimals values, fractional values or rational and mentally deficient numbers being used in favor of them. <\p>  <\p>

Now at this point we are going to discuss near enough to the Functions Algebra. Using the functions imaging we need to solve the algebraic equations. The basic purpose of using functions is to define the addition, difference, upping and division as for functions. There are various other types of functions which we will discuss later.  Here we flourish her the basic function operations: <\p>  <\p> Primitiveness is sum of the function: according as far as this functionality we perform the tailpiece of two rare functional values. Let's see the example:        ( a + b ) ( sealed book ) = a ( x ) + b ( x ) <\p>

  <\p> Second is difference of the function: according to this operation we perform the subtraction or make up the difference value between the functions. Let's cover below: <\p>                                             ( a – b )  ( x ) =  a ( x ) – b ( x ) <\p>

Schlock is product of the function: according to this in effect we perform the multiplication referring to the function. <\p> Lets espy how:                     ( a * b )  ( x ) =  a ( x ) * b ( x ) <\p>

  <\p> Fourth is quotient of the function: according to this operation we perform the division go on the functions. <\p> Let's show she downright:  <\p>                                             ( a \ b )  ( crossbones ) = a ( x ) \ b ( x )  <\p>

In the above organ transplantation we arrearage up remember that the value of the structure b ( x ) should not be equal to 0. <\p> At one swoop we are going to show other self the demonstration speaking of the acting by the imitated given example: <\p> Example: find the each function  using different operation? <\p>                       a ( cross recercelee ) = √ x + 1  ,   b ( x ) = √ decennium – 1  <\p>

Solution:  in the above question given that  <\p>                     The value of function a ( tenner ) is = √ x + 1 <\p>

                    The  value of function b ( decagram ) is = √ x – 1 <\p>

Now liaison speaking of function can be represented as  <\p>             a+ b  = ( a + b ) ( x ) <\p>               = (√ decastere + 1 + √ x – 1 )  <\p>

Seeing upward is the solution for the function value as for a ( frontier ) and b ( x ). <\p> Like this we tail bring forward the other operations with the functions. <\p> "<\p>

스칼라 스터디 진행 중 필요하다고 느껴 def 와 val에 대한 이야기를 나름 정리해봤다. def와 val에 대한 차이점은 스칼라가 function literal 혹은 function value를 어떻게 다루는지 살펴보면 보다 쉽게 이해할 수 있다.

Function Literal과 Function Value

함수 리터럴(function literal)은 (a:Int)=> a+1 과 같은 형태로 표현되는데, 스칼라에서 함수 리터럴은 실제로는 FunctionN이라는 클래스이다. 여기서 N은 인자 개수를 나타낸다. 예를 들어, 인자가 두 개인 함수 리터럴을 나타내는 클래스는 Function2이다. 스칼라에서 함수 리터럴을 클래스로 취급한다는 사실은 아주 중요하다.

참고로 FunctionN 클래스는 def로 만든 일반적인 함수가 아닌, 함수 리터럴을 나타내는 타입임에 주의한다. 확인을 위해 def로 일반적인 함수를 하나 정의해 보겠다.

scala> def sum(a:Int, b:Int) = a + b sum: (a: Int, b: Int)Int

sum의 타입을 보면 (a: Int, b: Int)Int 인데, 나중에 확인해 보겠지만 이는 함수 리터럴에 해당하는 타입이 아니다. 함수 리터럴이라면 Function2 혹은 (Int, Int):Int 타입이라고 출력되어야 한다.

def로 정의한 함수를 함수 리터럴로 정의한 함수로 만드려면 다음과 같이 partially applied function을 이용하면 된다. 출력 결과에 보이는 function2는 Function2의 인스턴스라는 의미인데, function2와 같이 함수 리터럴을 이용해 만든 인스턴스를 function value라고 한다.

scala> sum _ res1: (Int, Int) => Int = <function2>

위 예에서는 모든 인자를 그대로 활용했지만, 다음과 같이 인자 1개를 고정하면 Function2 대신 Function1의 인스턴스가, 다시 말해 function value가 반환됨을 확인할 수 있다.

scala> sum( 1, _:Int) res2: Int => Int = <function1>

이제 본론으로 들어가 function value와 함수를 fist class로 다루는게 어떤 관계인지 살펴보려고 한다.

partially applied function의 결과는 funciton value임을 확인했다. 이게 뭐 어떻다는 말일까? Java 등의 객체 지향 언어를 사용한 경험이 있다면 다음과 같이 특정 변수에 객체의 인스턴스를 보관하는 것이 아주 자연스럽다.

scala> val person = new Person("아웃사이다", 35)

마찬가지로 함수를 변수에 담거나, 인자로 전달하거나, 함수의 리턴 값으로 반환하기 위해, FunctionN 객체의 인스턴스를 특정 변수에 보관하는 것 또한 가능하다.

scala> val sum = (a:Int, b:Int) => a + b sum: (Int, Int) => Int = <function2>

결국 위 코드는 아래와 같다.

scala> val sum = new Function2[Int, Int, Int] { | def apply(a:Int, b:Int):Int = { | a + b | } | } sum: java.lang.Object with (Int, Int) => Int = <function2> scala> sum(1,2) res0: Int = 3

다시 말하면 스칼라에서는 function literal을 클래스로 정의하여 함수를 first class로 만들었기 때문에, 일반적인 객체인 함수(정확히는 function value)는 다른 객체와 마찬가지로 자유로운 전달이 가능하다.

지금까지 설명한 내용을 이해했다면, stackoverflow 등에 올라온 val에 def에 대한 질문들을 확인해보면 조금 더 이해가 잘 될지도 모르겠다. 한국 스칼라 사용자 모임 메일링 리스트에서 def과 관련된 내용을 이야기 하다가 다음과 같은 링크가 등장했는데, 시간이 되면 확인해보길 바란다. 근본적으로는 같은 내용이지만 조금 다른 질문을 하고 있다.

http://stackoverflow.com/questions/3646756/scala-def-versus-val

def 혹은 val을 오버라이드 할 때 유의 사항

간단하게 정리하면 def를 val로 override하지는 못하지만, val을 def로 override하는 것은 가능하다.

scala> class A { val a=0 }; class B extends A { override def a=1 } // 실패 <console>:4: error: error overriding value a in class A of type Int; method a is not stable class A { val a=0 }; class B extends A { override def a=1 } ^ scala> class A { def a=0 }; class B extends A { override val a=1 } // 성공 defined class A defined class B