Data 605 HW #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 )
pt1 <- c(5.6, 6.3, 7, 7.7, 8.4)
pt2 <- c(8.8, 12.4, 14.8, 18.2, 20.8)

reg <- lm(pt2 ~ pt1)
summary(reg)
## 
## Call:
## lm(formula = pt2 ~ pt1)
## 
## 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 ***
## pt1           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 = -14.8 + 4.2571x

  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
knitr::include_graphics('gif.gif')

  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_func <- function(x, y){
  x*(81 - (21*x) + (17*y)) + y * (40 + (11 * x) - 23*y)
}

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

revenue_func(2.3, 4.1)
## [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) = 1/6x^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?
knitr::include_graphics('Q5.gif')

  1. Evaluate the double integral on the given region. Write your answer in exact form without decimals.
knitr::include_graphics('P5.gif')

double_integral <- function(x,y) exp(8*x + 3 * y)

format(round(quad2d(double_integral, 2, 4, 2, 4), 17), scientific =  FALSE)
## [1] "534155947497083904"