1. Find the equation of the regression line for the given points. Round any final values to the nearest hundredth, if necessary.
x <- c(5.6, 6.3, 7, 7.7, 8.4)
y <- c(8.8, 12.4, 14.8, 18.2, 20.8)
regressionline <- lm(y ~ x)
regressionline
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##     -14.800        4.257

\(equation: y= −14.8 + 4.257x\)

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

\(\frac{df}{dx} = \frac{d}{dx}(24x-6xy^2-8y^3)\) \(0=24-6y^2\) \(24=6y^2\) \(4=y^2\) \(y=2,-2\)

\(0=-12xy-24y^2\) y = 2, x = -4 y = -2, x = 4

critical points are (-4,2)(4,2)

\(f_{xx} = 0\) \(f_{yy} = -12 - 48y\) \(f_{xy} = -12y\)

\(D(-4,2) = 0 * 84 - 24^2 = -576\) \(D(4,-2) = 0 * -108 + 24^2 = -576\)

Both critical points are saddle points

x <- c(4,-4)
y <- c(-2,2)
df <- data.frame(x,y)
df$z <- 24 * df$x - 6* df$y * (df$x^2) - 8 * df$y^3
print(df$z)
## [1]  352 -352

(4, -2, 352), (-4, 2, -352)

  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 ).

Revenue = \(x(81−21x+17y) + y(40+11x−23y)\) R(x, y) = \(81x−21x^2+28xy+40y−23y^2\)

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

x <- 2.3
y <- 4.1

z <- 81*x - 21*(x^2) + 28*x*y  + 40*y - 23*(y^2) 
z
## [1] 116.62
  1. 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?

x+y=96 y = 96 - x

Put Y into the equation = \(x^2 + (96 - x)^2 + 7x + 25(96-x) + 700\) $ = x^2 + 1536 - 32x + + 7x + 2400 - 25x + 700$ $ = x^2 - 50x + 4636$

Minima critical point : C′(x)=0

\(C′(x)=\frac{2}{3}x−50=0\) \(x = 75\)

75 + y = 96 \(y = 21\)

  1. Evaluate the double integral on the given region.