x = c(5.6, 6.3, 7, 7.7, 8.4)
y = c(8.8, 12.4 ,14.8,18.2 ,20.8)

fit = lm(y~x)
summary(fit)
## 
## 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
#Beta is 4.2571 and intercept is -14.8
  1. partial derivate w.r.t X \[f_x(x,y) = 24-6y^2 = 0 \] \[y = -2,2 \] partial derivate w.r.t Y \[f_y(x,y) = -12xy - 24y^2 = 0 \] \[ x = -2y \] \[ x = -4, 4 \] We need to compute Z for each case: \[ z = f(x,y) = 24x - 6xy^2 - 8y^3 \] \[ (4, -2, z), (-4, 2, z) \]
z_1 = (24*4)-(6*4*4)-(8*-8) # with (4,-2)
z_2 = (24*-4)-(6*-4*4)-(8*8) # with (-4,2)
z_1
## [1] 64
z_2
## [1] -64
#With formula D(x,y)=f(xx)(x,y)fyy(x,y)-[fxy(x,y)]^2, we find saddle points.

#If D>0 and fxx>0, local min
#If D>0 and fxx<0, local max
#If D<0, saddle point

\[D = -144y^2 \] We know that D<0, for any y. Thus, all critical points are saddle points.

  1. step 1.

\[R(x,y) = price_x * x + price_y * y \] \[ = (81-21x+17y)x + (40 + 11x -23y)y \] \[ = 81x - 21x^2 + 28xy + 40y - 23y^2 \] Step 2.

calculation <- function(x,y)
  {
   r = 81 * x - 21* x^2 + 28 *y*x + 40*y  - 23*y^2
   return(c(r))
  }

print(calculation(2.3,4.1))
## [1] 116.62
#The revenue is $116.62.
  1. \[ c(x,y) = \frac{1}{6}x^2 + \frac{1}{6}y^2 + 7x + 25y + 700 \]

We know x + y = total number of units: \[ x + y = 96 \] \[ x = 96 - y \]

Plug x into cost function: \[ c(x,y) = \frac{1}{6}(96-y)^2 + \frac{1}{6}y^2 + 7(96-y) + 25y + 700 \] \[ c(y) = \frac{1}{3}y^2 - 14y + 2908 \]

First derivative for y: \[ c'(y) = \frac{2}{3}y-14 = 0 \] \[ y = 21 \] \[ x = 75 \] 75 units in L.A and 21 units in Denver are required to minimize the cost.

  1. \[\int\limits_{2}^{4}e^{3y}\int\limits_{2}^{4}e^{8x}dxdy \] \[\frac{e^{32} - e^{16}}{8}\int\limits_{2}^{4}e^{3y}dy \] \[(\frac{e^{32} - e^{16}}{8}) (\frac{e^{12} - e^{6}}{3}) \] \[e^{16}e^{6}(\frac{e^{16} - 1}{8})(\frac{e^{2}-1}{3}) \] \[\frac{e^{16}e^{6}}{24}(e^{16} - 1)(e^{2} - 1)\]