library(mosaicCalc)## Loading required package: mosaic## Registered S3 method overwritten by 'mosaic':
## method from
## fortify.SpatialPolygonsDataFrame ggplot2##
## The 'mosaic' package masks several functions from core packages in order to add
## additional features. The original behavior of these functions should not be affected by this.##
## Attaching package: 'mosaic'## The following objects are masked from 'package:dplyr':
##
## count, do, tally## The following object is masked from 'package:Matrix':
##
## mean## The following object is masked from 'package:ggplot2':
##
## stat## The following objects are masked from 'package:stats':
##
## binom.test, cor, cor.test, cov, fivenum, IQR, median, prop.test,
## quantile, sd, t.test, var## The following objects are masked from 'package:base':
##
## max, mean, min, prod, range, sample, sum## Loading required package: mosaicCore##
## Attaching package: 'mosaicCore'## The following objects are masked from 'package:dplyr':
##
## count, tally## The legacy packages maptools, rgdal, and rgeos, underpinning the sp package,
## which was just loaded, will retire in October 2023.
## Please refer to R-spatial evolution reports for details, especially
## https://r-spatial.org/r/2023/05/15/evolution4.html.
## It may be desirable to make the sf package available;
## package maintainers should consider adding sf to Suggests:.
## The sp package is now running under evolution status 2
## (status 2 uses the sf package in place of rgdal)##
## Attaching package: 'mosaicCalc'## The following object is masked from 'package:stats':
##
## DTurunan fungsi (diferensial) ialah fungsi lain dari suatu fungsi sebelumnya, misalnya fungsi f menjadi f’ yang memiliki nilai tak beraturan. Turunan dapat diketahui melalui sifat-sifatnya. Berikut sifat-sifat dari diferensiasi, Jika k suatu konstanta, f dan g fungsi-fungsi yang terdiferensialkan, u dan v fungsi – fungsi dalam x sehingga u =f(x) dan v =g(x) maka berlaku :
Untuk dapat melakukan sebuah program menghitung turunan pertama sebuah fungsi pada RStudio aktifkan operasi turunan terlebih dahulu melalui sintaks berikut :
findiff <- function(f, x, h, method=NULL){
if(is.null(method)){
warning("please select a method")
}else{
if(method == "forward"){
return((f(x+h)-f(x))/h)
}else if(method=="backward"){
return((f(x)-f(x-h))/h)
}else if(method=="central"){
return((f(x+h)-f(x-h))/(2*h))
}else{
warning("you can use method: forward, bacward, or central")
}
}
}Contoh Pengeoperasian Diferensiasi dalam sifatnya : Jika y = u*v dengan u = 3x4 dan v = 2x2 , Tentukan turunan pertama y !
y’ = u’v + u v′
u = 3x4 maka u’ = 12x3
v = 2x2 maka v’ = 4x
y’ = ( 12x3 * 2x2 ) + ( 3x4 * 4x)
y’ = 24x5 + 12x5
y’ = 36x5
Berikut merupakan pengoperasian Diferensiasi Fungsi dalam sifatnya melalui persoalan.
Hitunglah turunan pertama f(x) = 2x - 1 / x2 -1 Penyelesaian Secara Manual :
f(x) = 2x - 1 / x2 -1 f’(x) = 2 * ( x2 -1 ) - ( 2x - 1 ) * 2x / ( x2 -1 )2 f’(x) = -2x2 + 2x -2 / ( x2 -1 )2
Penyelesaian Menggunakan RStudio :
Hitunglah turunan pertama f(x) = 2x - 1 / x2 -1 dengan x = 2, dan h = 0.05
findiff(function(x)
((2*2)-1)/((2^2)-1), x=2, h=0.05,
method="central")## [1] 0## [1] 0Hitunglah turunan pertama y = ( 3x4 + 2x2 + x ) ( x2 + 7 ) Penyelesaian Secara Manual :
y = ( 3x4 + 2x2 + x ) ( x2 + 7 ) u = 3x4 + 2x2 + x maka u’ = 12x3+ 4x + 1 v = x2 + 7 maka v’ = 2x y’ = ( 12x3+ 4x + 1 ) * ( x2 + 7 ) + ( 3x4 + 2x2 + x ) * ( 2x ) y’ = 12x5 + 88x3 + x2 + 28x + 7 + 6x5 + 4x3 + 2x2 y’ = 18x5 + 92x3 + 3x2 + 28x + 7
Penyelesaian Menggunakan RStudio :
Hitunglah turunan pertama y = ( 3x4 + 2x2 + x ) ( x2 + 7 ) dengan x = 1 dan h = 0.05.
findiff(function(x)
(3*x^4+2*x^2+x)*(x^2+7) , x=1, h=0.05,
method="central")## [1] 148.3826## [1] 148.3826