Homework 15

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)
df <- data.frame (x,y)

#round the coefficients of the regression line to the nearest hundredth
round(coef(lm(y~x, data=df)),2)
## (Intercept)           x 
##      -14.80        4.26

Regression Line equation: \(y = -14.80 + 4.26x\)

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.

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

Partial derivative : \[f_x = 24 - 6y^2\] \[f_y = -12xy - 24y^2\] Critical points Set \(f_x = 0\)

\[24 - 6y^2 = 0\]

\[6y^2 = 24\] \[y^2 = 4\] \[y = +/- 2\]

Substituting the y values \(f_y = 0\) \[-12xy -24y^2 = 0\] \[xy + 2y^2 = 0\]

Substitute y=2
\[2x + 2*2^2 = 0\]

\[2x + 8 = 0\] \[2x = -8 \] \[x = -4 \]

Substitute y=-2

\[-2x + 8 = 0\] \[2x = 8\] \[x = 4\]

Substitute x & y values (-4,2)

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

\[z= 24(-4) - 6(-4)(2^2) - 8(2^3)\] \[z = -96 + 96 - 64\] \[x = -64\]

Substitute x & y values (4,-2)

\[f(x,y) = 24x - 6xy^2 - 8y^3\] \[z= 24(4) - 6(4)(-2^2) - 8(-2^3)\]

\[z = 96 - 96 + 64\] \[z = 64\]

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 ).

Revenue = units_sold x price \[R(x,y) = x(81 - 21x + 17y) + y(40 + 11x -23y)\]

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

\[ = -21x^2 - 23y^2 + 28xy + 81x + 40y\]

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

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?

Given: x + y = 96 y = 96 - x

\[C(x, y) = \frac{1}{6} x^2 + \frac{1}{6} y^2 + 7x + 25y + 700\] \[ = \frac{1}{6} x^2 + \frac{1}{6} (96 - x)^{2} + 7x + 25(96 - x) + 700\] \[ = \frac{1}{6} x^2 + \frac{1}{6} (9216 - 192x + x^2) + 7x + 2400 - 25x + 700\]

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

\[ = \frac{1}{3} x^2 - 50x + 4636\]

Setting the derivative to zero

\[C' = \frac{2}{3}x - 50\] \[0 = \frac{2}{3}x - 50\] \[\frac{2}{3}x = 50\] \[x = (50 * 3)/2 = 75\]

The number of units produced in LA to minimize the weekly cost is 75 units.

y = 96 - 75 = 21

5. Evaluate the double integral on the given region.

\[\int \int (e^{8x + 3y})dA; R:2 \le x \le 4, 2 \le y \le 4 \] Write your answer in exact form without decimals.

\(u = 8x + 3y, du = 8\)

\[\frac{1}{8} \int^4_2 e^u du\] \[= \frac {e^{8x + 3y}}{8}^4_2 \]

\[= \frac {e^{8*4 + 3y}}{8} - \frac{e^8(2) + 3y}{8} \] \[= \frac {e^{32 + 3y}}{8} - \frac{e^{16} + 3y}{8} \] \[ = \int^4_2 \frac{e^{32 + 3y} - e^{16 + 3y} }{8} dy\] \[ = \frac{1}{8} \int^4_2 e^{32 + 3y} dy - \frac{1}{8} \int^4_2 e^{16 + 3y} dy\] \[ = (\frac{1}{8}) \frac{e^{32 + 3y}}{3}^4_2 - (\frac{1}{8}) \frac{e^{16 + 3y}}{3}^4_2 \]

\[ = (\frac{1}{8}) (\frac{e^{32 + 3(4)}}{3} - \frac{e^{32 + 3(2)}}{3}) - (\frac{1}{8}) (\frac{e^{16 + 3(4)}}{3} - \frac{e^{16 + 3(2)}}{3})\]

\[ = (\frac{1}{8}) (\frac{e^{32 + 12}}{3} - \frac{e^{32 + 6}}{3}) - (\frac{1}{8}) (\frac{e^{16 + 12}}{3} - \frac{e^{16 + 6}}{3})\] \[ = \frac{e^{44}}{24} - \frac{e^{38}}{24} - \frac{e^{28}}{24} - \frac{e^{22}}{24} \]
\[ = \frac{e^{44} - e^{38} - e^{28} + e^{22}}{24}\]

(1/24)* (exp(1)^44 - exp(1)^38 - exp(1)^28 - exp(1)^22)
## [1] 5.341559e+17