WebTest Name: (Test)

  1. Find the equation of the regression line for the given points. Round any final values to the nearest hundredth, if necessary. ( 5.6, 8.8 ), ( 6.3, 12.4 ), ( 7, 14.8 ), ( 7.7, 18.2 ), ( 8.4, 20.8 )
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

\[y=−14.800 + 4.257x\]

plot(y,x,col = "red",main = "points",
abline(lm(x~y)),cex = 1.3,pch = 16,xlab = "X-axis",ylab = "Y-axis")

  1. 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.

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

First partial derivative:

\[f_x=24−6y^2\]

\[f_y=−12xy−24y^2\]

\[f_{xy}=−12y \]

\[f_{xx}=0 \]

\[f_{yy}=−12x−48y\]

Critical points:

\[f_x=24−6y^2=0 ==> y = [2,-2]\]

Plug into equation:

\[f_y = −12xy − 24y^2 = 0 => (-4,2) & (4,-2)\]

Add Z coordinates:

\[f(−4,2)=−12(8)−24(−4)2\]

equation = function(x,y){
  z = 24*x - 6*x*y^2 - 8*y^3
  return(c(x,y,z))
}
print(rbind(equation(-4,2),equation(4,-2)))
##      [,1] [,2] [,3]
## [1,]   -4    2  -64
## [2,]    4   -2   64

Second partial derivative:

\[D=D(x,y)=f_{xx}f_{yy}−f^2_{xy} \] \[−(−12y)^2=−144y^2 ==> D(X,Y) < −576 \]

  1. 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  21x + 17y units of the “house” brand and 40 + 11x  23y units of the “name” brand.

Step 1. Find the revenue function R ( x, y ).

\[R(x)=(81−21x+17y)∗x \] \[R(y)=(40+11x−23y)∗y \]

Join:

\[R(x,y)=R(x)+R(y) \] \[R(x,y)=28xy−23y^2−21x^2+40y+81x \]

Step 2. What is the revenue if she sells the “house” brand for $2.30 and the “name” brand for $4.10?

revenue = function(x,y){
  rev = 28*x*y-23*y^2-21*x^2+40*y+81*x
  return(rev)
}
rev = revenue(2.3,4.1)
rev
## [1] 116.62

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:

\[C(x, y) = \frac{1}{6} x^2 + 16y^2 + 7x + 25y + 700\]

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 minimize the total weekly cost?

\[x+y=96 \] \[y=96−x \] \[x=96−y \]

Input values into the original equation:

\[C(x,96−x)=x^2+(96−x)^2+7x+25(96−x)+700 \]

\[C(x,96−x)=x^2−50x+4636 \]

Find the Minimum value that satisfies the equation:

\[ C′(x)=\frac{2}{3}x−50 = 0 \]

\[ x = 75 \]

\[ 21 = 96 − 75 \]

Denver = 21 adn Los Angeles = 75.

5.Evaluate the double integral on the given region.   \[∫∫_R(e^{8x+3y}) dA ; R:2<= x <= 4 and 2 <= y <= 4 \]

Write your answer in exact form without decimals.

\[∫^4_2∫^4_2(e^{8x+3y})dxdy\]

\[∫^4_2(\frac{1}{8})(e^{16}−1)e^{3y+16}dy\]

\[(e^{16}−1)/24e^{3y+16}+C\]

\[(e^{44}−e^{28})/24−(e^{38}−e^{22})/24\]

\[(1/24)(e^{22}−e^{28}−e^{38}+e^{44})\]

\[ = \frac{1}{24}.1\]