Derivatives

Find the derivatives with the respect to x of the following.

\[F(x|x >= 0) = 1 - e^{-\lambda x}\]

library(mosaicCalc)
library(mosaic)
library(Deriv)
funcA <- function(x){ (1-exp(-lambda*x))}
solutionA <- Deriv(funcA)
solutionA
## function (x) 
## lambda * exp(-(lambda * x))

\[F(x|b > a) = \frac{x-a}{b-a}\]

funcB <- function(x){(x-a)/(b-a)}
solutionB <- Deriv(funcB)
solutionB
## function (x) 
## 1/(b - a)

\[F(x|a < x <= c <= b) = \frac{(x - a)^2}{(b-a)(c-a)}\]

funcC <- function(x){ (x-a)^2/((b-a)*(c-a)) }
solutionC <- Deriv(funcC)
solutionC
## function (x) 
## 2 * ((x - a)/((b - a) * (c - a)))

\[F(x|a <= c < x < b)=1-\frac{(b-x)^2}{(b-a)(c-a)}\]

funcD <- function(x){ 1 - (b-x)^2/((b-a)*(c-a)) }
solutionD <- Deriv(funcD)
solutionD
## function (x) 
## 2 * ((b - x)/((b - a) * (c - a)))

Integrals

Solve the following definite and indefinite integrals

\[\int_0^{10} 3x^3 dx\]

funcIA <- function(x){ 3*x^3 }
integrate(funcIA, 0, 10)
## 7500 with absolute error < 8.3e-11

\[\int_0^{x} x \lambda e^{-\lambda x} dx\]

funcIB <- antiD(x*lambda*(exp(-lambda*x))~x)
funcIB(x = 10,lambda = 8)
## [1] 0.125

\[\int_0^{.5} \frac{1}{b-a} dx\]

funcIC <- function (x, C = 0, a, b) { 1/(a - b) * x }
integrate(funcIC, a = 10, b  = 8, lower = 0, upper = 0.5)
## 0.0625 with absolute error < 6.9e-16

\[\int_0^{10} x \frac{1}{r( \alpha )\beta \alpha} x^ \alpha -1 e^ {-\beta x} dx\]

funcID <- antiD(( 1/(gamma(alpha)*beta^alpha))*(x^alpha)*exp(-(beta*x))~x )
funcID(x = 10, alpha = 8, beta = 7)
## [1] 3.43893e-14

Linear Algebra

With the following matrix,

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

Invert it using Gaussian row reduction.

\[\mathbf{X} = \left[\begin{array} {rrr} 1 & 2 & 3 \\ 3 & 3 & 1 \\ 4 & 6 & 8 \end{array}\right] \] \[\mathbf{I} = \left[\begin{array} {rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \]

-3R1+R2->R2
-4R1+R3->R3

\[\mathbf{X} = \left[\begin{array} {rrr} 1 & 2 & 3 \\ 0 & -3 & -8 \\ 0 & -2 & -4 \end{array}\right] \]

\[\mathbf{I} = \left[\begin{array} {rrr} 1 & 0 & 0 \\ -3 & 1 & 0 \\ -4 & 0 & 1 \end{array}\right] \]

1/-3R2->R2 \[\mathbf{X} = \left[\begin{array} {rrr} 1 & 2 & 3 \\ 0 & 1 & 2.7 \\ 0 & -2 & -4 \end{array}\right] \]

\[\mathbf{I} = \left[\begin{array} {rrr} 1 & 0 & 0 \\ 1 & 0 & 0 \\ -4 & 0 & 1 \end{array}\right] \]

-2R2+R1->R1 2R2+R3->R3 \[\mathbf{X} = \left[\begin{array} {rrr} 1 & 0 & -2.3 \\ 0 & 1 & 2.7 \\ 0 & 0 & 1.3 \end{array}\right] \]

\[\mathbf{I} = \left[\begin{array} {rrr} -1 & 0.7 & 0 \\ 1 & 0 & 0 \\ -2 & -1 & 1 \end{array}\right] \]

0.769026 x R3->R3 \[\mathbf{X} = \left[\begin{array} {rrr} 1 & 0 & -2.3 \\ 0 & 1 & 2.7 \\ 0 & 0 & 1 \end{array}\right] \]

\[\mathbf{I} = \left[\begin{array} {rrr} -1 & 0.7 & 0 \\ 1 & 0 & 0 \\ -2 & -1 & 0.8 \end{array}\right] \]

2.3 x R3 + R1->R1 -2.6 x R3 + R2->R2 \[\mathbf{X} = \left[\begin{array} {rrr} 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{array}\right] \]

\[\mathbf{I} = \left[\begin{array} {rrr} -5 & -1 & 1.8 \\ 5 & 1 & -2 \\ -2 & -1 & 0.8 \end{array}\right] \]

Find the determinant.

mat001<-matrix(c(1,3,4,2,3,6,3,1,8),3,3)
print(mat001)
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    3    3    1
## [3,]    4    6    8
det001 <- det(mat001)
print(det001)
## [1] -4

Conduct LU decomposition

U

-3R1 + R2

\[\mathbf{X} = \left[\begin{array} {rrr} 1 & 2 & 3 \\ 0 & -3 & -8 \\ 4 & 6 & 8 \end{array}\right] \]

-4R1 + R3

\[\mathbf{X} = \left[\begin{array} {rrr} 1 & 2 & 3 \\ 0 & -3 & -8 \\ 0 & -2 & -4 \end{array}\right] \]

-0.666666666667R2 + R3

\[\mathbf{X} = \left[\begin{array} {rrr} 1 & 2 & 3 \\ 0 & -3 & -8 \\ 0 & 0 & 1.3 \end{array}\right] \]

L

\[\mathbf{X} = \left[\begin{array} {rrr} 1 & 0 & 0 \\ 3 & 1 & 0 \\ 4 & 0.666666666667 & 1 \end{array}\right] \]

Multiply the matrix by it’s inverse.

mat001<-matrix(c(1,3,4,2,3,6,3,1,8),3,3)
print(mat001)
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    3    3    1
## [3,]    4    6    8
#Inverse
invMatrix001 <- solve(mat001)
print(invMatrix001)
##      [,1] [,2]  [,3]
## [1,] -4.5 -0.5  1.75
## [2,]  5.0  1.0 -2.00
## [3,] -1.5 -0.5  0.75
#Multiply Matrix by its Incerse (Correct Answer)
print(mat001 %*% invMatrix001)
##               [,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
#Casewise Multiplication/ Vectorization
print(mat001 * invMatrix001)
##      [,1] [,2]  [,3]
## [1,] -4.5   -1  5.25
## [2,] 15.0    3 -2.00
## [3,] -6.0   -3  6.00