HW15

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

df_lm <- lm(data = df , y~x)
summary(df_lm)

## 
## Call:
## lm(formula = y ~ x, data = df)
## 
## Residuals:
##     1     2     3     4     5 
## -0.24  0.38 -0.20  0.22 -0.16 
## 
## Coefficients:
##             Estimate Std. Error t value Pr(>|t|)    
## (Intercept) -14.8000     1.0365  -14.28 0.000744 ***
## x             4.2571     0.1466   29.04 8.97e-05 ***
## ---
## Signif. codes:  0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
## 
## Residual standard error: 0.3246 on 3 degrees of freedom
## Multiple R-squared:  0.9965, Adjusted R-squared:  0.9953 
## F-statistic: 843.1 on 1 and 3 DF,  p-value: 8.971e-05

y = 4.26x - 14.8

Find all local maxima, local minima, and saddle points for the function given below.

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

First Derivitive:

\[ fx = 24−6y^2 \] \[ fy =−12xy−24y^2 \]

Critical Points / Saddle Points / Min/Max:

24−6y^2 = 0 y^2 = 4 y = + / - 2

-12xy - 24*4 = 0 xy = 8

x = - / + 4

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

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

Find the revenue function R ( x, y )

R(x,y) = x(81 - 21x + 17y) + y(40 + 11x - 23y)
81x - 21x^2 + 17yx + 40y + 11xy - 23y^2
-21x^2 + 28xy + 81x + 40y - 23y^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 

(-21)*x^2 + 28*x*y + 81*x + 40*y - 23*y^2

## [1] 116.62

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) = (1/6)x^2 + (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?

We know the total cost can be found by:

c(x,y) = (1/6)x^2 + (1/6)*y^2 + 7x + 25y + 700

and that

x + y = 96
x = 96 - y

c(y) = (1/6)(96-y)^2 + (1/6)*y^2 + 7(96 - y) + 25y + 700
c(y) = (1/3)y^2−14y+2908
c’(y) = (2/3)y-14

c’(y) = 0 y = 21 (units produced in denver)
x = 75 (units produced in LA)

Evaluate the double integral on the given region.