WebTest Name: (Test)
x = c(5.6, 6.3, 7, 7.7, 8.4)
y = c(8.8, 12.4, 14.8, 18.2, 20.8)
lm(y~x)
##
## Call:
## lm(formula = y ~ x)
##
## Coefficients:
## (Intercept) x
## -14.800 4.257
The equation for the regression line for the given points is \[ y=4.257x\ -\ 14.800 \] 2. Find all local maxima, local minima, and saddle points for the function given below. Write your answer(s) in the form ( x, y, z ). Separate multiple points with a comma. Using the saddle points calculator: https://calculator-online.net/saddle-point-calculator/
\[ f(x,y)=24x-6xy^{2}-8y^{3} \] 1st Derivative Steps of: \[ \frac{d}{dx} \left( -6xy^{2}+24x-8y^{3}\right) =24-6y^{2} \] 2nd Derivative Steps of: \[ \frac{d}{dx} (24-6y^{2})=0 \] 1st Derivative Steps of: \[ \frac{d}{dx} \left( -6xy^{2}+24x-8y^{3}\right) =-12y(x+2y) \] 2nd Derivative Steps of: \[ \frac{d}{dx} (-12y(x+2y))=-12x-48y \] thus \[ thus\ if\ 24-6y^{2}=0\ then\ y=[2,-2] \] Now solve for x if y=[2,-2]: \[ f_{y}=-12xy-24y^{2}=0\ thus\ (-4,2)\ and\ (4,-2) \]
Now solve the equation: \[ -12x-48y=-12(8)-24(-4)^{2}=0 \]
f_eq = function(x,y){
z = 24*x - 6*x*y^2 - 8*y^3
return(c(x,y,z))
}
print(rbind(f_eq(-4,2),f_eq(4,-2)))
## [,1] [,2] [,3]
## [1,] -4 2 -64
## [2,] 4 -2 64
Answer: (-4,2,-64) and (4,2,64)
groc_rev = function(x,y){
g_rev = 28*x*y-23*y^2-21*x^2+40*y+81*x
return(g_rev)
}
(revenue = groc_rev(2.3,4.1))
## [1] 116.62
A company has a plant in Los Angeles and a plant in Denver. The firm is committed to produce a total of 96 units of a product each week. The total weekly cost is given by \[ C(x,y)=\frac{1}{6} x^{2}+\frac{1}{6} y^{2}+7x+25y+700 \], x is the number of units produced in Los Angeles and y is the number of units produced in Denver. How many units should be produced in each plant to minimize the total weekly cost? \[ x+y=96 \] \[ y=96-x \] \[ x=96-y \] Substitute the values into the equation: \[ C(x,96-x)=x^{2}+(96x)^{2}+7x+25(96-x)+700 \] \[ C(x,96-x)=x^{2}-50x+4636 \] Calculate the minimum value: \[ C\prime (x)=\frac{2}{3} x-50=0 \] \[ x=75 \] \[ 21=96-75 \] thus Denver should produce 21 units and Los Angeles 75 to minimize total weekly costs.
Evaluate the double integral on the given region. \[ \iint \left( {}e^{8x+3y}\right) dA;R:2\leq x\leq 4\ and\ 2\leq y\leq 4 \]
Write your answer in exact form without decimals. Using the Symbolad integral calculator https://www.symbolab.com/solver/double-integrals-calculator
\[ \int^{4}_{2} \int^{4}_{4} (e^{8x+3y})dxdy=\] \[ \int^{4}_{2} (\frac{1}{8} )(e^{3y+32}-e^{3y+16})dy= \] \[ \frac{e^{44}-e^{38}-e^{28}+e^{22}}{24}= \] \[ \frac{{}e^{0}}{24} = 0 \]