Exercise 1

Find the equation of the regression line for the given points. Round any final values to the nearest hundreth, if necessary.

(5.6,8.8),(6.3,12.4),(7,14.8),(7.7,18.2),(8.4,20.8)

#assign values to the variable x
cat("Values on the X-Axis","\n")
## Values on the X-Axis
(x <- c(5.6,6.3,7,7.7,8.4))
## [1] 5.6 6.3 7.0 7.7 8.4
#assign values to the variable y
cat("\n","Values on the Y-Axis","\n")
## 
##  Values on the Y-Axis
(y <- c(8.8,12.4,14.8,18.2,20.8))
## [1]  8.8 12.4 14.8 18.2 20.8
#using the lm function to obtain linear regression model
(r <- lm(y~x))
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##     -14.800        4.257

The regression line is \(y= -14.8 + 4.257x\)

Next, we will plot our findings:

plot(y~x,
     xlab="X-Axis Data",
     ylab="Y-Axis Data",
     col="red",
     main="X and Y Scatterplot")
abline(r)

Exercise 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.

\[f(x,y)=24x-6xy^2-8y^3\] \[fx(x,y) = 24 - 6y^2\] \[fx(x,y) = -12xy - 24y^2\]

Exercise 3

A grocery store sells two brands of a product, the “house” brand and a “name” brand. The manager estimates that if she sells the “house” brand for x dollars and the “name” brand for y dollars, she will be able to sell 81 - 21 x + 17y units of the “house” brand and 40 + 11x - 23y units of the “name” brand.

Step 1

Find the reverse function R(x,y)

\[R = x(81-21x+17y) + y(40+11x+23y)\] \[ = 81x-21x^2+17xy+40y+11xy+23y^2\] \[ = -21x^2-23y^2+28xy+81x+40y\]

Step 2

What is the revenue if she sells the “house” brand for $2.30 and the “name” brand for $4.10?

#assign the value of 2.3 to variable x3
cat("\n","The x3 Variable","\n")
## 
##  The x3 Variable
(x3 <- 2.3)
## [1] 2.3
#assign the value of 4.1 to variable y3
cat("\n","The y3 Variable","\n")
## 
##  The y3 Variable
(y3 <- 4.1)
## [1] 4.1
#apply the revenue function to variable z3
cat("\n","The Revenue is:","\n")
## 
##  The Revenue is:
(z3 <- -21*(x3^2)-23*(y3^2)+28*x3*y3+81*x3+40*y3)
## [1] 116.62

Exercise 4

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

\[C(x,y)=\frac{1}{6}x^2+\frac{1}{6}y^2+7x+25y+700\]

where 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 minimze the total weekly cost?

\[x + y = 96\] \[C(x,y)=\frac{1}{6}x^2+\frac{1}{6}y^2+7x+25y+700\] \[C`(X)=\frac{1}{3}x+7\] \[0 = \frac{1}{3}x+7\] \[-7 = \frac{1}{3}x\] \[-21 = x\] \[C`(Y)=\frac{1}{3}y+25\] \[0 = \frac{1}{3}y+25\] \[-25 = \frac{1}{3}y\] \[-75 = y\]

Exercise 5

Evaluate the double integral on the given region

\[\int\int(e^{8x+3y})dA;R:2\leq\,x\leq4\; and \;2\leq\;y\leq4\]

Write your answer in exact form without decimals

\[e(8x+3y) = e(8 * x) + e(3 * y)\]