Step 1 Apply Multiply rule \[\left(\frac{64x^3}{y^3}\right)^{-1/2}\] Step2 Apply exponent rule \[\left(\frac{y^3}{64x^3}\right)^{1/2}\] \[\left(\frac{y^{3/2}}{64^{1/2} x^{3/2}}\right)\] \[\left(\frac{y^{3/2}}{8 x^{3/2}}\right)\]
#install.packages('Deriv')
library(Deriv)
exp01 <- expression(((64*x*y^-2)/(x^-2*y))^(-1/2))
Simplify(exp01)## expression(1/sqrt(64 * (x^3/y^3)))2 . If \(f(x) =\sqrt{(x+5)} \space and \space {g(x) = {3/x}}\) Find \(f(x) g(x)\)
Step1 : \[(f.g)(x) =f(x).g(x) \] \[(x+5)^{1/2} * {3/x} \]
f <- function(x) sqrt(x + 5)
g <- function(x) 3/x
fxgx <- function(x) 3*sqrt(x+5)/x
fxgx## function(x) 3*sqrt(x+5)/x3 . Find the break-even point(s) for the revenue and cost functions below.
\[R(x) = 24.2x-0.2x^2 \] \[C(x) = 13x+147 \]
library(mosaic)## Loading required package: dplyr##
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## New to ggformula? Try the tutorials:
## learnr::run_tutorial("introduction", package = "ggformula")
## learnr::run_tutorial("refining", package = "ggformula")## Loading required package: mosaicData## Loading required package: Matrix##
## The 'mosaic' package masks several functions from core packages in order to add
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## max, mean, min, prod, range, sample, sumrevenueFn <- function(x) { 24.2 * x - .2 * x ^ 2}
costFn <- function(x) { 13*x + 147}
breakEven = function(x) (24.2 * x - .2 * x ^ 2 - 13*x - 147)
mosaic::findZeros(breakEven(x)~x)## x
## 1 21
## 2 354 . Find the derivative for the following function \(y = 9x^{8.2}\)
function04 = function(x) 9*x^8.2
Deriv::Deriv(function04)## function (x)
## 73.8 * x^7.25 . Find the derivative for \(f(x) = -4x^5-3x^4-4x^3-3\)
function05 = function(x) -4*x^5-3*x^4-4*x^3-3
Deriv::Deriv(function05)## function (x)
## -(x^2 * (3 * (4 + x * (3 + 4 * x)) + x * (3 + 8 * x)))6 . A pole that is 4264 feet long is leaning against a building. The bottom of the pole is moving away from the wall at a rate of 2 ft./sec. How fast is the top of the pole moving down the wall when the top is 3480 feet off the ground? \[x^2+y^2 =( 4264)^2 \] \[x^2+y^2 = (4264)^2 \] \[2x\frac{dx}{dt}+2y\frac{dy}{dt} = 0\] \[\frac{dy}{dt} = -\frac{x}{y} \frac{dx}{dt}\] \[x = \sqrt{(4264^2-3480^2)} \]
x = sqrt(4264^2-3480^2)
x## [1] 2464\[\frac{dy}{dt} = - {(2464/3480)} * 2 \]
-(2464/3480)*2## [1] -1.4160927 . Use the Product Rule or Quotient Rule to find the derivative. \[f(x) = \frac{3x^8 - x^5}{2x^8+2}\]
function07 = function(x) ((3*x^8-x^5)/(2*x^8+2))
Deriv::Deriv(function07)## function (x)
## {
## .e1 <- x^8
## .e3 <- 2 + 2 * .e1
## .e4 <- x^3
## x^4 * ((3 * .e4 - 1) * (5 - 16 * (.e1/.e3)) + 9 * .e4)/.e3
## }function08 = function(x) 4*x+25*x^-1
derFn08 = Deriv::Deriv(function08)
derFn08## function (x)
## 4 - 25/x^2mosaic::findZeros(derFn08(x)~x)## x
## 1 -2.5
## 2 2.5derFn08(-3)## [1] 1.222222derFn08(-1)## [1] -21derFn08(3)## [1] 1.222222localMinFn8 = derFn08(-2.5)
localMinFn8## [1] 0localMaxFn8 = derFn08(2.5)
localMaxFn8## [1] 09 . Evaluate \(\frac{dy}{dx}\) at x= 1 for the below function \[{y = -9u^2+7u+4} , {where} \space {u = -3x^3+3x+6}\]
function09 = function(x) {-9*(-3*x^3 + 3*x + 6)^2 + 7 * (-3*x^3 + 3*x + 6) + 4}
dervFn09 = Deriv::Deriv(function09)
dervFn09## function (x)
## {
## .e1 <- x^2
## (3 - 9 * .e1) * (7 - 18 * (6 + x * (3 - 3 * .e1)))
## }dervFn09(1)## [1] 60610 . Consider the following function: \(f(x) = (x+5) (x+4)^2\)
function10 = function(x) (x+5)*(x+4)^2
firstDerFn10 = Deriv::Deriv(function10)
firstDerFn10## function (x)
## (2 * (5 + x) + 4 + x) * (4 + x)secondDerFn10 = Deriv::Deriv(firstDerFn10)
secondDerFn10## function (x)
## 2 * (5 + x) + 3 * (4 + x) + 4 + xmosaic::findZeros(firstDerFn10(x) ~ x)## x
## 1 -4.6666
## 2 -4.0000firstDerFn10(0)## [1] 56firstDerFn10(-4.5)## [1] -0.25firstDerFn10(-5)## [1] 1#decreasing : (-4.6666, -4)
# increasing : (-infinity, -4.6666), (-4, infinity)mosaic::findZeros(secondDerFn10(x)~x)## x
## 1 -4.3333secondDerFn10(0)## [1] 26secondDerFn10(-5)## [1] -4#concave up: (-4.3333, infinity)
#concave down: (-infinity, -4.3333)mosaic::findZeros(firstDerFn10(x)~x)## x
## 1 -4.6666
## 2 -4.0000localMin = function10(-4)
localMin## [1] 0localMax = function10(-4.6666)
localMax## [1] 0.1481481inflectionPoint = mosaic::findZeros(secondDerFn10(x)~x)
inflectionPoint## x
## 1 -4.333311 . Consider the function: \(f(x) = -7x^2-84x-140\)
function11 = function(x) -7*x^2 - 84*x - 140
firstDerFn11 = Deriv::Deriv(function11)
firstDerFn11## function (x)
## -(14 * x + 84)secondDerFn11 = Deriv::Deriv(firstDerFn11)
secondDerFn11## function (x)
## -14mosaic::findZeros(firstDerFn11(x)~x)## x
## 1 -6mosaic::findZeros(secondDerFn11(x)~x)## Warning in mosaic::findZeros(secondDerFn11(x) ~ x): No zeros found. You
## might try modifying your search window or increasing npts.## numeric(0)function11(-6)## [1] 11212 . A beauty supply store expects to sell 100 flat irons during the next year. It costs $1.20 to store one flat iron for one year. There is a fixed cost of $15 for each order. Find the lot size and the number of orders per year that will minimize inventory costs.
costFunction12 = function(x) (100/x * 1.2) + (x * (15 + 100/x))
dervCost = Deriv::Deriv(costFunction12)
mosaic::findZeros(dervCost(x)~x)## x
## 1 -2.8284
## 2 2.8284costFunction12(2.8284)## [1] 184.8528costFunction12(3)## [1] 185lotSize <- 100/2.8 # OR lotSize <- 100/313 . Find the derivative of the given expression: \(y= \frac{18x}{ln(x)^4}\)
#log is default natural loarithm in R
function13Log = function(x) {(18*x)/(log(x^4))}
Deriv::Deriv(function13Log)## function (x)
## {
## .e1 <- 4 * log(x)
## 18 * 1/.e1 - 72/.e1^2
## }14 . Consider the following function: \(f(x)= -5*x^3*e^x\)
myfunction14 = function(x) { -5*x^3*e^x}
firstDerFn14 = Deriv::Deriv(myfunction14)
firstDerFn14## function (x)
## {
## .e1 <- e^x
## -(5 * (x^2 * (3 * .e1 + .e1 * x * log(e))))
## }secondDerFn14 = Deriv::Deriv(firstDerFn14)
secondDerFn14## function (x)
## {
## .e1 <- e^x
## .e2 <- log(e)
## .e4 <- .e1 * x * .e2
## -(5 * (x * ((2 * .e1 + .e4) * (3 + x * .e2) + .e4)))
## }15 . Evaluate the definite integral below. Write your answer in exact form or rounded to two decimal places. \[\int_{-3}^1 (8^3{\sqrt{(8x+5)^2}}) dx\]
intFn15 = function(x) sqrt((8*x + 5)^2)
integrate(Vectorize(intFn15),-3,1)## 33.125 with absolute error < 3.7e-1316 . Solve the differential equation given below \[\frac {dy}{dx} = -4 (-3-y) \]
dydx = function(y) 12 + 4*y
y = function(x) e^(4*x) - 317 . The marginal cost of a product is given by \({238}+\frac{298}{\sqrt{x}}\) dollars per unit, where x is the number of units produced. The current level of production is 137 units weekly. If the level of production is increased to 232 units weekly, find the increase in the total costs. Round your answer to the nearest cent
marginalCost = function(x) 238 + (298/sqrt(x))
totalCost = function(x) 238*x + 596*sqrt(x)
totalCost(232) - totalCost(137)## [1] 2471218 . Find the Taylor polynomial of degree 5 near x = 1 for the following function: \(y=\sqrt{4x-3}\)
f = function(x) sqrt(4*x - 3)# Vector of length n+1 represents a polynomial degree n
#taylor(f,x0, n=4)
pracma::taylor(f, 1, n = 4)## [1] -10.00254 44.01025 -74.01549 58.01040 -17.00262