Problem 1

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

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

ANSWER

x <- c(5.6, 6.3, 7, 7.7, 8.4)
y <- c(8.8, 12.4, 14.8, 18.2, 20.8)

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

As shown in the linear regression model, the regression line is \(y = -14.8 + 4.257x\).

Verify the plot.

plot(x,y, xlab="", ylab="")
abline(r)
lines(c(5,9), -14.8+4.257*c(5,9), col="green")

Problem 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.

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

ANSWER

Partial derivatives:

\(f_x(x, y) = 24 - 6y^2\)

\(f_y(x, y) = -12xy - 24y^2\)

\(f_x\) and \(f_y\) are never underfined.

If \(24-6y^2=0\), then \(y^2 = 4\) and \(y = \pm2\).

If \(y=2\) and \(-12xy - 24y^2=0\), then \(-24x = 24\times 4\) and \(x=-4\).

If \(y=-2\) and \(-12xy - 24y^2=0\), then \(24x = 24\times 4\) and \(x=4\).

Calculate \(f(x, y)\).

\(f(4,-2) = 24\times 4 - 6\times4\times (-2)^2 - 8 \times (-2)^3 = 64\)

\(f(-4,2) = 24\times (-4) - 6\times(-4)\times 2^2 - 8 \times 2^3 = -64\)

Two critical points: \((4,-2,64)\) and \((-4, 2, -64)\).

Use Second Derivative test to determine if points are minimum, maximum or saddle.

Second partial derivatives:

\(f_{xx}=0\)

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

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

Then \(D(x,y) = f_{xx} f_{yy}-f^2_{xy} = -(-12y)^2 = -144y^2\).

\(D(x,y)<0\) for all \((x, y)\), On the basis of the Second Derivative Test, any critical point is a saddle point.

Problem 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?

ANSWER

\[ \begin{split} R(x,y) &= (81 - 21x + 17y)x + (40 + 11x - 23y)y \\ &= 81x-21x^2+17xy+40y+11xy-23y^2\\ &=81x+40y+28xy-21x^2-23y^2 \end{split} \]

\(R(2.3, 4.1)=81\times 2.3+40\times 4.1+28\times 2.3\times 4.1-21\times (2.3)^2-23\times (4.1)^2 = 116.62\)

Problem 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?

ANSWER

Consider \(x+y=96\), then \(x=96-y\).

\[ \begin{split} C(x,y) = C(96-y,y) &= \frac{1}{6} x^2 + \frac{1}{6} y^2 + 7x + 25y + 700 \\ &=\frac{1}{6} (96-y)^2 + \frac{1}{6} y^2 + 7\times (96-y) + 25y + 700 \\ &=\frac{1}{6}(y^2 - 192 y + 9216) + \frac{1}{6}y^2+672-7y+25y+700\\ &= \frac{1}{6}y^2 - 32y+1536+\frac{1}{6}y^2+18y+1372\\ &= \frac{1}{3}y^2 - 14y + 2908\\ &=C_1(y) \end{split} \]

\(C_1'(y) = \frac{2}{3}y-14\)

ww will find minimal value by considering \(C_1'(y)=\frac{2}{3}y-14=0\), then \(y=21\). Then \(x=96-y=75\).

On the bais of the analysis there has to be production of 75 units in Los Angeles and 21 units in Denver.

Problem 5

Evaluate the double integral on the given region.

\[ \int\int_R (e^{8x+3y}) dA, R:2\le x\le4\ and\ 2 \le y \le 4 \]

ANSWER

\[ \begin{split} \int_2^4\int_2^4 (e^{8x+3y})\ dy\ dx &= \int_2^4 (\frac{1}{3}e^{8x+3y})|_2^4\ dx\\ &= \int_2^4 ((\frac{1}{3}e^{8x+12})-(\frac{1}{3}e^{8x+6}))\ dx\\ &= \int_2^4 \frac{1}{3}e^{8x+6}(e^6-1)\ dx\\ &= \frac{1}{24}e^{8x+6}(e^6-1) |_2^4\\ &= \frac{1}{24}e^{32+6}(e^6-1)-\frac{1}{24}e^{16+6}(e^6-1)\\ &= \frac{1}{24}(e^6-1)(e^{38}-e^{22})\\ &= \frac{1}{24}(e^{44} - e^{38} - e^{28} + e^{22}) \end{split} \]