Consider this multivariable function. f(x,y)=xy+2x+y−36a) What is the value of f(2,−3)?b) Find all x-values such that f (x,x) = 0

Lynbrook West , an apartment complex , has 100 two-bedroom units. The montly profit (in dollars) realized from renting out x apartments is given by the following function.
${\displaystyle P\left(x\right)=-12x^2+2136x-41000}$
To maximize the monthly rental profit , how many units should be rented out?
What is the maximum monthly profit realizable?

Use polar coordinates to find the limit. [Hint: Let ${\displaystyle x=r\cos \text{and}y=r\sin }$ , and note that (x, y) (0, 0) implies r 0.] ${\displaystyle \underset{(x,y)\to (0,0)}{lim}\frac{x^2-y^2}{\sqrt{x^2+y^2}}}$

(a) In a regression analysis, the sum of squares for the predicted scores is 100 and the sum of squares error is 200, what is ${\displaystyle R^2}$?
(b) In a different regression analysis, 40% of the variance was explained. The sum of squares total is 1000. What is the sum of squares of the predicted values?

Solve the given differential equation by separation of variables.
${\displaystyle {(e^y+1)}^2{e}^{-y}dx+{(e^x+1)}^6{e}^{-x}dy=0}$

Find the complex zeros of the following polynomial function. Write f in factored form.
${\displaystyle x^4+2x^3+22x^2+50x-75}$
The complex zeros of f are ?
Use the complex zeros to factor f.

Analysis of daily output of a factory during a 8-hour shift shows that the hourly number of units y produced after t hours of production is
${\displaystyle y=140t+\frac{1}{2}{t}^2-t^3,0\le t\le 8}$
a) After how many hours will the hourly number of units be maximized? hr
b) What is the maximum hourly output? ${\displaystyle \frac{units}{hr}}$

Write down the definition of the complex conjugate, ${\displaystyle \bar{z}\text{ ,if }z=x+iy}$ where z,y are real numbers. Hence prove that, for any complex numbers w and z,
${\displaystyle \overline{w\stackrel{―}{z}}=\bar{w}z}$