Derivatives

1. $F(x|x\ge <!--\; \ge \; -->0)=1-<!--\; -\; -->e^{-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}x}$

library(Deriv)
myf1<-function(x){
   (1-exp(-lambda*x))
}
hw1<-Deriv(myf1)
hw1

## function (x) 
## lambda * exp(-(lambda * x))

2. $F(x|b>a)=\frac{x-<!--\; -\; -->a}{b-<!--\; -\; -->a}$

myf2 <- function(x){
   (x-a)/(b-a)
}
hw2<-Deriv(myf2)
hw2

## function (x) 
## 1/(b - a)

3. $F(x|a<x\le <!--\; \le \; -->c\le <!--\; \le \; -->b)=\frac{(x-<!--\; -\; -->a{)}^2}{(b-<!--\; -\; -->a)(c-<!--\; -\; -->a)}$

myf3 <- function(x){
   (x-a)^2/((b-a)*(c-a))
}
hw3<-Deriv(myf3)
hw3

## function (x) 
## 2 * ((x - a)/((b - a) * (c - a)))

4. $F(x|a\le <!--\; \le \; -->c<x<b)=1-<!--\; -\; -->\frac{(b-<!--\; -\; -->x{)}^2}{(b-<!--\; -\; -->a)(c-<!--\; -\; -->a)}$

myf4 <- function(x){
   1 - (b-x)^2/((b-a)*(c-a))
}
hw4<-Deriv(myf4)
hw4

## function (x) 
## 2 * ((b - x)/((b - a) * (c - a)))

5. ${\int <!--\; \int \; -->}_0^{10}3x^{3}dx$

library(mosaicCalc)

## Loading required package: mosaicCore

## 
## Attaching package: 'mosaicCalc'

## The following object is masked from 'package:stats':
## 
##     D

hw5<- antiD(3*x^3~x,x=10)
hw5

## function (x = 10, C = 0) 
## 3/4 * x^4 + C

hw5()

## [1] 7500

6. ${\int <!--\; \int \; -->}_0^{x}x\mathrm{\lambda <!--\; \lambda \; -->}{e}^{-<!--\; -\; -->\mathrm{\lambda <!--\; \lambda \; -->}x}dx$

library(mosaicCalc)
hw6<-antiD(x*lambda*(exp(-lambda*x))~x)
hw6

## function (x, lambda, C = 0) 
## {
##     numerical_integration(.newf, .wrt, as.list(match.call())[-1], 
##         formals(), from, ciName = intC, .tol)
## }
## <environment: 0x000000001b2018a0>

7. ${\int <!--\; \int \; -->}_0^{.5}\frac{1}{b-<!--\; -\; -->a}dx$

library(mosaicCalc)
hw7 <- antiD((1/(b-a))~x, x=.5)
hw7

## function (x = 0.5, C = 0, b, a) 
## (1/(b - a)) * x + C

With the following matrix,

$$\mathbf{X}=\left[\begin{array}{rrr}1& 2& 3\\ 3& 3& 1\\ 4& 6& 8\end{array}\right]$$

9. Invert it using Gaussian row reduction.

A=matrix(c(1,3,4,2,3,6,3,1,8),3,3)
AB <-  cbind(A, diag(3))
AB[2,] <- AB[2,] - 3*AB[1,]     
AB[3,] <- AB[3,] - 4 * AB[1,]     
AB[2,] <-  -1 * AB[2,] + AB[3,]         
AB[3,] <-  2 * AB[2,] + AB[3,]  
AB[3, ] <- AB[3,]/4 
AB[1,] <- AB[1, ] - 2 * AB[2,] 
AB[2, ] <- AB[2, ] - 4 * AB[3,]
AB[1, ] <- AB[1, ] + 5 * AB[3,]
AB

##      [,1] [,2] [,3] [,4] [,5]  [,6]
## [1,]    1    0    0 -4.5 -0.5  1.75
## [2,]    0    1    0  5.0  1.0 -2.00
## [3,]    0    0    1 -1.5 -0.5  0.75

10. Find the determinant.

det(A)

## [1] -4

11. Conduct LU decomposition

library(matrixcalc)
A=matrix(c(1,3,4,2,3,6,3,1,8),3,3)
lu.decomposition(A)

## $L
##      [,1]      [,2] [,3]
## [1,]    1 0.0000000    0
## [2,]    3 1.0000000    0
## [3,]    4 0.6666667    1
## 
## $U
##      [,1] [,2]      [,3]
## [1,]    1    2  3.000000
## [2,]    0   -3 -8.000000
## [3,]    0    0  1.333333

12. Multiply the matrix by it’s inverse.

hw12<-A %*% solve(A)
hw12

##               [,1]          [,2]         [,3]
## [1,]  1.000000e+00 -2.220446e-16 4.440892e-16
## [2,] -4.440892e-16  1.000000e+00 2.220446e-16
## [3,]  0.000000e+00  0.000000e+00 1.000000e+00