\(~\)
\(~\)
(5.6, 8.8), (6.3, 12.4), (7, 14.8), (7.7, 18.2), (8.4, 20.8)
# creating the points into x and y variables
= c(5.6, 6.3, 7, 7.7, 8.4)
x = c(8.8, 12.4, 14.8, 18.2, 20.8)
y
# regression model
= lm(y~x)
reg_lm
# summary of regression model
summary(reg_lm)
##
## Call:
## lm(formula = y ~ x)
##
## 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
Regression line is y = -14.80 + 4.26x
\(~\)
\(f(x, y) = 24x - 6xy^2 - 8y^3\)
#install.packages('emdbook')
library(emdbook)
curve3d(24*x - 6*x*y^2 - 8*y^3, xlim = c(-5,5), ylim = c(-5,5))
\(~\)
f(x) = \(24 - 6y^2\)
f(y) = \(-12xy - 24y^2\)
Min and Max:
\(24 - 6y^2 = 0\)
\(y^2 = 4\)
y = ± 2
\(-12xy - 24^2 = 0\)
xy = 8
x = ± 4
The critical points would be (-4, 2)(4, -2)
\(~\)
h = number of “house” brand units sold n = number of “name” brand units sold
The revenue function is:
R = h(81 - 21x + 17y) + n(40 + 11x - 23y)
\(~\)
= 2.3
x = 4.1
y <- -21*x^2 - 23*y^2 + 28*x*y + 81*x + 40*y
total
print(paste0("The revenue is $", total))
## [1] "The revenue is $116.62"
\(~\)
x = number of units to be produced in LA
y = number of units to be produced in Denver
x + y = 96 = total products to be produced by both cities
y = 96 - x
Apply this to the equation : \(C(x, y) = \frac{1}{6} x^2 + \frac{1}{6} y^2 + 7x + 25y + 700\)
\(C(x, y) = \frac{1}{6} x^2 + \frac{1}{6} (96-x^2) + 7x + 25(96 - x) + 700\)
\(C(x, y) = \frac{1}{6} x^2 + 1536 - 32x + 7x + 2400 - 25x + 700\)
\(C(x, y) = \frac{1}{3} x^2 - 50x + 4636\)
\(\frac{dc}{dx} = \frac{2}{3}x - 50 = 0\)
x = \(\frac{3}{2} * 50\) = 75 units when \(\frac{dc}{dx}\) is 0
y = 96 - 75 = 21
Solution: There should be 75 units produced in LA and 21 units produced in Denver
\(~\)
\(\int\int (e^{8x + 3y}) dA;R: 2\leq x \leq 4\) and \(2 \leq y \leq 4\)
<- function(y) {(1/8) * (exp(1)^16 - 1) * exp(1)^(3 * y + 16)}
integrand = integrate(integrand, lower = 2, upper = 4)
volume print(volume, digits = 17)
## 534155947497083840 with absolute error < 5930
# set e value
<- exp(1)
e # compute
<- ((e^44 - e^38)/24) - ((e^28 - e^22)/24)
answer # print without scientific notation
format(answer, scientific = FALSE)
## [1] "534155947497083840"