library(Deriv)
f<-function(x){1-exp(-lx)}
Deriv(f)

## function (x) 
## 0

library(Deriv)
f<-function(x){(x-a)/(b-a)}
Deriv(f)

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

library(Deriv)
f<-function(x){(x-a)^2 / (b-a)(c-a)}
Deriv(f)

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

library(Deriv)
f<-function(x){1-((b-x)^2 / (b-a)(c-a))}
Deriv(f)

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

f<-function(x){3*x^3}
integrate(f,0,10)  

## 7500 with absolute error < 8.3e-11

library(mosaicCalc)

## Loading required package: mosaicCore

## 
## Attaching package: 'mosaicCalc'

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

f<-function(x)x*antiD(exp(-x)~x)-(1-exp(-x))
antiD(f~x)

## function (x, C = 0, f) 
## f * x + C

a<-0
b<-.5
f<-function(x){1/(b-a)}
integrate(Vectorize(f),a,b)

## 1 with absolute error < 1.1e-14

8. unknown

A<-matrix(c(1,3,4,2,3,6,3,1,8),ncol=3)
solve(A)

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

A<-matrix(c(1,3,4,2,3,6,3,1,8),ncol=3)
determinant(A)

## $modulus
## [1] 1.386294
## attr(,"logarithm")
## [1] TRUE
## 
## $sign
## [1] -1
## 
## attr(,"class")
## [1] "det"

library(Matrix)
A<-matrix(c(1,3,4,2,3,6,3,1,8),ncol=3)
lu(A)

## 'MatrixFactorization' of Formal class 'denseLU' [package "Matrix"] with 4 slots
##   ..@ x       : num [1:9] 4 0.75 0.25 6 -1.5 ...
##   ..@ perm    : int [1:3] 3 2 3
##   ..@ Dimnames:List of 2
##   .. ..$ : NULL
##   .. ..$ : NULL
##   ..@ Dim     : int [1:2] 3 3

expand(lu(A))

## $L
## 3 x 3 Matrix of class "dtrMatrix" (unitriangular)
##      [,1]       [,2]       [,3]      
## [1,]  1.0000000          .          .
## [2,]  0.7500000  1.0000000          .
## [3,]  0.2500000 -0.3333333  1.0000000
## 
## $U
## 3 x 3 Matrix of class "dtrMatrix"
##      [,1]       [,2]       [,3]      
## [1,]  4.0000000  6.0000000  8.0000000
## [2,]          . -1.5000000 -5.0000000
## [3,]          .          . -0.6666667
## 
## $P
## 3 x 3 sparse Matrix of class "pMatrix"
##           
## [1,] . . |
## [2,] . | .
## [3,] | . .

A<-matrix(c(1,3,4,2,3,6,3,1,8),ncol=3)
A%*%solve(A)

##               [,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