Question 1

\[\frac{y^{1/2}}{8x^{3/2}}\]

Question 2

\[f(x)*g(x)=3x^{-1}(x+5)^{1/2}\]

Question 3

\[R(x)-C(X)=-0.2x^2+11.2x-147\]

library(rootSolve)
par(mfrow=c(1,1))
fx <- function(x){-0.2*x^2+11.2*x-147}
s <- seq(0,50,by=1)
a <- fx(s)
roots <- uniroot.all(fx, c(0,50))
plot(a~s,type='l')
points(roots, rep(0,length(roots)), col='red',pch=21)

roots
## [1] 21 35

\[x=21,35\]

Question 4

library(Deriv)
myf1 <- function(x){9*x^8.2}
Deriv(myf1)
## function (x) 
## 73.8 * x^7.2

\[y'=73.8x^{7.2}\]

Question 5

myf1 <- function(x){-4*x^5-3*x^4-4*x^3-3}
Deriv(myf1)
## function (x) 
## -(x^2 * (3 * (4 + x * (3 + 4 * x)) + x * (3 + 8 * x)))

\[f'(x)=-4x^2(5x^2+3x+3)\]

Question 6

\[h=(4264^2-(2t)^2)^{1/2}\]

myf1 <- function(t){(4264^2-(2*t)^2)^(1/2)}
dmyf1 <- Deriv(myf1)
dmyf1
## function (t) 
## -(4 * (t/sqrt(18181696 - (2 * t)^2)))

\[f'(h)=-\frac{2t}{\sqrt{2132^2-t^2}}\]

dmyf1(1232)
## [1] -1.416092

\[f'(1232)=-1.416ft/sec\]

Question 7

myf1 <- function(x){(3*x^8-x^5)/(2*x^8+2)}
Deriv(myf1)
## function (x) 
## {
##     .e1 <- x^8
##     .e3 <- 2 + 2 * .e1
##     .e4 <- x^3
##     x^4 * ((3 * .e4 - 1) * (5 - 16 * (.e1/.e3)) + 9 * .e4)/.e3
## }

\[f'(x)=\frac{3x^8-x^5(15x^7)-(2x^8)(24x^7-5x^4)}{(2x^8+2)^2}\]

Question 8

a)

par(mfrow=c(1,2))
fx <- function(x){4*x+25*x^-1}
s=seq(-5,5,by=.1)
a=fx(s)
plot(a~s,type='l')
dfx=Deriv(fx)
b=dfx(s)
plot(b~s,type='l')
roots = uniroot.all(dfx, c(-5,5))
points(roots, rep(0,length(roots)), col='red',pch=21)

roots
## [1] -2.5  2.5

\[f'(x)=4-25x^{-2}\] \[x=\pm\frac{5}{2},0\]

b)

paste(dfx(-1),dfx(0),dfx(1),sep=',')
## [1] "-21,-Inf,-21"
paste(dfx(-3), dfx(-5/2), dfx(-2),sep=',')
## [1] "1.22222222222222,0,-2.25"
paste(dfx(2), dfx(5/2), dfx(3),sep=',')
## [1] "-2.25,0,1.22222222222222"
fx(-5/2)
## [1] -20
fx(5/2)
## [1] 20

\(x=0\) Nothing

Local Minimum = \((\frac{5}{2},20)\)

Local Maximum = \((\frac{-5}{2},-20)\)

Question 9

myfx1 <- function(x){-9*(-3*x^3+3*x+6)^7+7*(-3*x^3+3*x+6)+4}
Deriv(myfx1)(1)
## [1] 17635926

\[y'=-63(-3x^3+3x+6)^6(-9x^2+3)+7(-9x^2+3)\]

Question 10

a)

fx1 <- function(x){(x+5)*(x+4)^2}
dfx1 <- Deriv(fx1)
ddfx1 <- Deriv(dfx1)
dfx1
## function (x) 
## (2 * (5 + x) + 4 + x) * (4 + x)
ddfx1
## function (x) 
## 2 * (5 + x) + 3 * (4 + x) + 4 + x

\[f'(x)=3x^2+26x+56\] \[f''(x)=6x+26\]

b)

dfx1(-5)
## [1] 1
dfx1(uniroot(dfx1,c(-5,-4.5))$root)
## [1] -1.629429e-05
dfx1(-4.5)
## [1] -0.25
dfx1(uniroot(dfx1,c(-4.5,0))$root)
## [1] 4.287782e-07
dfx1(0)
## [1] 56

Increasing: \((-\infty,\frac{-14}{3})\cup(-4,\infty)\) Decreasing: \((\frac{-14}{3},-4)\)

c)

ddfx1(-5)
## [1] -4
ddfx1(uniroot(ddfx1,c(-5,-4))$root)
## [1] 1.776357e-15
ddfx1(-4)
## [1] 2

DOWN: \((-\infty,\frac{-26}{6})\) UP: \((\frac{-26}{6},\infty)\)

d)

fx1(-14/3)
## [1] 0.1481481
fx1(-4)
## [1] 0

Local Max: \((\frac{-14}{3},\frac{4}{27})\) Local Min: \((-4,0)\)

Question 11

a)

fx <- function(x){-7*x^2-84*x-140}
dfx <- Deriv(fx)
ddfx <- Deriv(dfx)
ddfx
## function (x) 
## -14

\[f''(x)=-14\]

b)

ddfx(-6)
## [1] -14

Local Max: \((-6,112)\)

Question 12

Yearly Carrying Cost = \(0.6x\) Reordering Cost = \(\frac{1500}{x}\)

\[y=0.6x+\frac{1500}{x}\]

fx <- function(x){0.6*x+1500/x}
dfx <- Deriv(fx)
uniroot(dfx,c(0,100))$root
## [1] 50

\[x=50\]

ddfx <- Deriv(dfx)
ddfx
## function (x) 
## 3000/x^3

\(y''=\frac{3000}{x^3} > 0\) when \(x=50\)

Lot Size = \(50\) and Orders = \(\frac{100}{50}=2\)

Question 13

fx <- function(x){18*x/(log(x^4))}
dfx <- Deriv(fx)
dfx
## function (x) 
## {
##     .e1 <- 4 * log(x)
##     18 * 1/.e1 - 72/.e1^2
## }

\[y'=\frac{18(\ln(x^4)-4)}{\ln^2(x^4)}\]

Question 14

a)

fx <- function(x){-5*x^3*exp(x)}
dfx <- Deriv(fx)
dfx
## function (x) 
## -(5 * (x^2 * (3 + x) * exp(x)))

\[f'(x)=-5(x^3+3x^2)e^x\]

b)

ddfx <- Deriv(dfx)
ddfx
## function (x) 
## -(5 * (x * ((2 + x) * (3 + x) + x) * exp(x)))

\[f''(x)=-5e^x(x^2+6x+6)\]

Question 15

fx <- function(x){8*((8*x+5)^2)^(1/3)}
integrate(Vectorize(fx),-3,1)
## 124.2966 with absolute error < 0.002

\[\int_{-3}^1 8\sqrt[3]{(8x+5)^2} \approx 124.2966\]

Question 16

library(Ryacas)
x <- Sym('x')
Integrate(1/(12+4*x),x)
## expression(log(4 * (x + 3))/4)

\[y=Ce^{4x}-3\]

Question 17

fx <- function(x){298/(x^(1/2))}
integrate(Vectorize(fx),137,232)
## 2102 with absolute error < 2.3e-11

\[\int_{137}^{232} \frac{298}{\sqrt{x}} dx + 238 = \$2102\]

QUestion 18

fx <- function(x){(4*x-3)^(1/2)}
dfx <- Deriv(fx)
ddfx <- Deriv(dfx)
d3fx <- Deriv(ddfx)
d4fx <- Deriv(d3fx)
d5fx <- Deriv(d4fx)
dfx(1)
## [1] 2
ddfx(1)/prod(1:2)
## [1] -2
d3fx(1)/prod(1:3)
## [1] 4
d4fx(1)/prod(1:4)
## [1] -10
d5fx(1)/prod(1:5)
## [1] 28

\[f(y)=f(1)+f'(1)(x-1)+\frac{f''(1)(x-1)^2}{2!}+\frac{f'''(1)(x-1)^3}{3!}+\frac{f^4(1)(x-1)^4}{4!}+\frac{f^5(1)(x-1)^5)}{5!}\]

\[f(y)=1+2(x-1)-2(x-1)^2+4(x-1)^3-10(x-1)^4+28(x-1)^5\]