R Markdown

This is an R Markdown document. Markdown is a simple formatting syntax for authoring HTML, PDF, and MS Word documents. For more details on using R Markdown see http://rmarkdown.rstudio.com.

When you click the Knit button a document will be generated that includes both content as well as the output of any embedded R code chunks within the document. You can embed an R code chunk like this:

1

  • \(y = -14.8+ 4.257x\)
x <- c(5.6, 6.3, 7, 7.7, 8.4)
y <- c(8.8, 12.4, 14.8, 18.2, 20.8)

lm(y~x)
## 
## Call:
## lm(formula = y ~ x)
## 
## Coefficients:
## (Intercept)            x  
##     -14.800        4.257

2

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

\[ f_x(x, y) = 24 - 6y^2....y^2 = +-4\]

\[fy(x,y)= -12xy - 24y^2 \]

plugging in y

  • at y=2 \[ -24x -96=0 ....x= -4 \]

  • at y= -2

\[ 24x-96=0....x =4 \]

pluggin in to f(x,y)

  • \[ 24(4) - 6(4)(-2^2)-8(-2)^3...= 64 \]
  • \[24(-4) - 6(-4)(2^2)-8(2)^3...=-64 \]

max min saddle test

\[ H= F_{xx}(x_o , y_0) f_{yy} (x_0, y_0) - f_{xy}(x_0, y_0))^2) \]

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

\(D=-144y^2\)

  • D is always negative and therefore our critical points are saddle points
    • (4,-2,64) and (-4,2,-64).

3

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

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

rev_fnc <- function(x,y){
    paste("revenue is, ", 81*x+40*y+28*x*y-21*x**2-23*y**2)
}

rev_fnc(2.3,4.1)
## [1] "revenue is,  116.62"

4

\[x+ y=96...x= 96-y\]

\[\frac{1}{6}x^2 +\frac{1}{6}y^2+ 7x + 25y +700 \]

\[\frac{1}{6}(96-y)^2 +\frac{1}{6}y^2+ 7(96-y) + 25y +700 \]

\[\frac{1}{6}(y^2 - 192 y + 9216) + \frac{1}{6}y^2+672-7y+25y+700 \]

96**2
## [1] 9216
96*7
## [1] 672
9216/6
## [1] 1536
192/6
## [1] 32

\[\frac{1}{6}y^2 - 32y+1536+\frac{1}{6}y^2+18y+672+700 \]

\[\frac{1}{6}y^2 - 32y+1536+\frac{1}{6}y^2+18y+672+700 \]

\[\frac{1}{3}y^2 - 14y + 2908 \]

  • Now take derivative and set = to 0 to find critical points

\[ \frac{2}{3} y-14=0\]

\[y=21 & x =75 \]

5

\[ A = \int_2^4\int_2^4 e^{(8x + 3y)} dA \\\] \[\int_2^4 e^{(8x)} \int_2^4e^{(3y)} \]

  • integrate

\[ \frac{1}{8} e^{8x}\Big|_2^4 * \frac{1}{3} e^{3y}\Big|_2^4 \]

  • evaluate at boundaries

\[ \frac{1}{8} (e^{32}-e^{16}) * \frac{1}{3}(e^{12}-e^{6}) \]

(1/8)*(exp(32)-exp(16))*(1/3)*(exp(12)-exp(6))
## [1] 5.341559e+17