1 Question 1

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

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

1.1 Answer

The equation of the regression line for the given points is \(y = 4.26x -14.80\)

## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##     -14.800        4.257

2 Question 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.

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

2.1 Answer

Find partial derivative:

\(f_{x} = 24 -6y^{2}\)

\(f_{y} = -12xy -24y^{2}\)

\(f_{xx} = 0\)

\(f_{xy} = -12y\)

\(f_{yy} = -12x -48y\)

Find critical points:

\(f_{x} = 24 -6y^{2} = 0 \Rightarrow y = [2,-2]\)

\(f_{y} = -12x(\pm 2) -24(\pm 2)^{2} = 0 \Rightarrow (x,y) = \{(-4,2),(4,-2)\}\)

\(f(-4,2) = -4\cdot24 -6\cdot(-4)\cdot4 -8\cdot8 = -64\)

\(f(4,-2) = 24\cdot4 -6\cdot4\cdot4 -8\cdot(-8) = 64\)

Find discriminant:

\(D(x,y) = f_{xx}f_{yy} -f^{2}_{xy} = 0 -(-12y)^{2} = -576 < 0\)

Therefore, there are two saddle points \((4,-2,64)\) and \((-4,2,-64)\) with no maxima or minima.

3 Question 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 81-21x+17y units of the “house” brand and 40+11x-23y units of the “name” brand.

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

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

3.1 Step 1

\(R(x,y)\)

\(= (81-21x+17y) \cdot x +(40+11x-23y) \cdot y\)

\(= 81x -21x^{2} +17xy +40y +11xy -23y^{2}\)

\(= -21x^{2} +81x + 28xy + 40y - 23y^{2}\)

3.2 Step 2

The revenus is $116.62 if she sells the “house” brand for $2.30 and the “name” brand for $4.10.

## [1] 116.62

4 Question 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 minimize the total weekly cost?

4.1 Answer

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

\(C(x) = \frac{1}{6}x^{2} +\frac{1}{6}(96-x)^{2} +7x+25(96-x) +700\)

\(= \frac{1}{6}x^{2} +1536 -32x +\frac{1}{6}x^{2} +7x +2400 -25x +700\)

\(= \frac{1}{3}x^{2} -50x +4636\)

Set \(C' = \frac{dC}{dx} = \frac{2}{3}x -50 = 0\)

\(x = 75\)

\(y = 96 -75 = 21\)

\(C'' = \frac{d^{2}C}{dx^{2}} = \frac{2}{3} > 0\)

\(C(75,21) = \frac{1}{3}x^{2} -50x +4636 = 2761\)

\(\because\) \(C''>0\), (75,21) is a relative minima.

\(\therefore\) Los Angeles plant should produce 75 units and Denver plant should produce 21 units each week to minimize the total weekly cost at $2,761.

5 Quetsion 5

Evaluate the double integral on the given region.

\[\iint_{R}\left( e^{8x+3y} \right) dA; \; R:2\leq x\leq 4 \; and \; 2\leq y\leq 4\] Write your answer in exact form without decimals.

5.1 Answer

\(\iint_{R}\left( e^{8x+3y} \right) dA; \; R:2\leq x\leq 4 \; and \; 2\leq y\leq 4\)

\(= \int_{y=2}^{y=4}\int_{x=2}^{x=4} \left( e^{8x+3y} \right) dxdy\)

\(= \int_{2}^{4} e^{8x} dx \cdot \int_{2}^{4} e^{3y} dy\)

\(= \left[ \frac{e^{8x}}{8} \right]_{2}^{4} \cdot \left[ \frac{e^{3y}}{3} \right]_{2}^{4}\)

\(= \left( \frac{e^{32}-e^{16}}{8} \right) \left( \frac{e^{12}-e^{6}}{3} \right)\)

\(= \frac{e^{44}-e^{38}-e^{28}+e^{22}}{24}\)