Let z=e^(x^2+3y)+x^3y^2, where x=tcos r and y = rt^4. Use the chain rule fot multivariable functions (Cals III Chain Rule) to find (delz)/(delr) and (delz)/delt). Give your answers in terms of r and t only. Be sure to show all of your work.

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.] ${\displaystyle \underset{(x,y)\to (0,0)}{lim}\frac{x^2-y^2}{\sqrt{x^2+y^2}}}$

for the function f(x,y)= $\sum _{i=1}^n[(x_i{)}^{4\ast y_i}+(1-y_i)ln{x}_i]$, y are binary variables, x nonnegative real values with fixed upper bound C. Does the minimization min f(x,y) have a unique solution or not?

Let ${\displaystyle F\overrightarrow{\to }r=<x,y,z>}$ and use the Divergence Theorem to calculate the (nonzero) volume of some solid in IR3 by calculating a surface integral. (You can pick the solid).

Solve the system of equation by the method of your choice.
${\displaystyle x^2+{(y-9)}^2=49}$
${\displaystyle x^2-7y=-14}$

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

The definition of a linear program is following:

Find a vector $x$ such that: $min{c}^{T}x$, subject to $Ax=b$ and $x\ge 0$.

Generally, $b$ is assumed to be a fixed constant. However is it possible to construct a program where values of $b$ are part of the optimization? Could I included b in the optimization by changing $Ax=b$ to $Ax-b=0$. If so, would I also be able to place constraints upon $b$ like $\sum b=1$ and $1>b>0$? Finally, would such a program be possible to solve efficiently?

I am trying to solve the linear program for Wasserstein Distance between two discrete distributions. In the standard case, b represents the marginals for each datapoint. I know the marginals for the target distribution but the marginals from my source distribution are unknown. I am wondering if there is an efficient way to optimize the marginals for my source distribution such that the Wasserstein distance is minimized.