library(Deriv)

Question 1

f1 = function(x) 1 - (exp(- lambda * x))

answer1 = Deriv(f1)
answer1
## function (x) 
## lambda * exp(-(lambda * x))

Question 2

f2 = function(x) (x - a) / (b - a)
answer2 = Deriv(f2)
answer2
## function (x) 
## 1/(b - a)

Question 3

f3 = function(x) (x - a) ^ 2 / ((b - a) (c - a))
answer3 = Deriv(f3)
answer3
## function (x) 
## 2 * ((x - a)/(b - a)(c - a))

Question 4

f4 = function(x) 1 - ((b - x) ^ 2 / ((b - a) (c - a)))
answer4 = Deriv(f4)
answer4
## function (x) 
## 2 * ((b - x)/(b - a)(c - a))

Question 5

f5 = function(x) (3 * (x ^ 3))
answer5 = integrate(Vectorize(f5), 0, 10)
answer5
## 7500 with absolute error < 8.3e-11
library(mosaicCalc)
## Loading required package: mosaicCore
## Registered S3 method overwritten by 'mosaic':
##   method                           from   
##   fortify.SpatialPolygonsDataFrame ggplot2
## 
## Attaching package: 'mosaicCalc'
## The following object is masked from 'package:stats':
## 
##     D

Question 6

f6 = antiD(x * lambda * exp(- lambda * x) ~ x)
f6
## function (x, lambda, C = 0) 
## {
##     numerical_integration(.newf, .wrt, as.list(match.call())[-1], 
##         formals(), from, ciName = intC, .tol)
## }
## <environment: 0x00000000164c3620>

Question 7

a <- 0
b <- -1
f7 = function(x) (1 / (b - a))
answer7 = integrate(Vectorize(f7), 0, .5)
answer7
## -0.5 with absolute error < 5.6e-15

Question 8

f8 = antiD(x ^ {alpha + 1} / gamma * alpha * beta ^ {alpha} * e ^ {beta} ~ x)
## Warning in if (regexpr(rhsVar, deparse(lform[[2]], width.cutoff = 500))
## == : the condition has length > 1 and only the first element will be used

## Warning in if (regexpr(rhsVar, deparse(lform[[2]], width.cutoff = 500))
## == : the condition has length > 1 and only the first element will be used

## Warning in if (regexpr(rhsVar, deparse(lform[[2]], width.cutoff = 500))
## == : the condition has length > 1 and only the first element will be used
f8
## function (x, alpha, gamma, beta, e, C = 0) 
## {
##     numerical_integration(.newf, .wrt, as.list(match.call())[-1], 
##         formals(), from, ciName = intC, .tol)
## }
## <environment: 0x0000000016f5ba38>
library(matlib)

Question 9

m <- rbind(c(1, 2, 3), c(3, 3, 1), c(4, 6, 8))
m
##      [,1] [,2] [,3]
## [1,]    1    2    3
## [2,]    3    3    1
## [3,]    4    6    8
gaussianElimination(m, diag(3))
##      [,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
answer9 = inv(m)
answer9
##      [,1] [,2]  [,3]
## [1,] -4.5 -0.5  1.75
## [2,]  5.0  1.0 -2.00
## [3,] -1.5 -0.5  0.75
answer9a = solve(m)
answer9a
##      [,1] [,2]  [,3]
## [1,] -4.5 -0.5  1.75
## [2,]  5.0  1.0 -2.00
## [3,] -1.5 -0.5  0.75

Question 10

answer10 = det(m)
answer10
## [1] -4
library(pracma)
## 
## Attaching package: 'pracma'
## The following objects are masked from 'package:matlib':
## 
##     angle, inv
## The following object is masked from 'package:mosaicCore':
## 
##     logit

Question 11

answer11 = lu(m)
answer11
## $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

Question 12

answer12 = answer9%*%answer9a
answer12
##         [,1]   [,2]    [,3]
## [1,]  15.125  0.875 -5.5625
## [2,] -14.500 -0.500  5.2500
## [3,]   3.125 -0.125 -1.0625