Let z(x,y)=e^(3xy), x(p,q)=p/q and y(p,q)=q/p are functions. Use multivariable chain rule of partial derivatives to find (i) (delz)/(delp) (ii) (delz)/(delq).

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

Requires Uploaded Supporting Analysis: Differentiate. ${\displaystyle y={(3x^2+5x+1)}^{\frac{3}{2}}}$

Using the Divergence Theorem, evaluate ${\displaystyle \int {\int }_{S}F.NdS}$, where ${\displaystyle F(x,y,z)=(z^{3}i-x^{3}j+y^{3}k)}$ and S is the sphere ${\displaystyle x^2+y^2+z^2=a^2}$, with outward unit normal vector N.

What is the Proximal Operator ($\mathrm{Prox}$) of the Pseudo $L_0$ Norm?
Namely:
${\mathrm{Prox}}_{\lambda {\Vert \cdot \Vert }_0}\left(\mathit{y}\right)=\arg \underset{\mathit{x}}{min}\frac{1}{2}{\Vert \mathit{x}-\mathit{y}\Vert }_2^2+\lambda {\Vert \mathit{x}\Vert }_0$
Where ${\Vert \mathit{x}\Vert }_0=\mathrm{n}\mathrm{n}\mathrm{z}(x)$, namely teh number of non zeros elements in the vector $\mathit{x}$.

The analysis of tooth shrinkage by
Loring Brace and colleagues at the University of Michigan’s Museum of Anthropology indicates that human tooth size is continuing to decrease and that the evolutionary process has not yet come to a halt. In northern Europeans, for example, tooth size reduction now has a rate of 1% per 1000 years. In about how many years will human teeth be 90% of their present size?

Suppose S is a region in the xy-plane with a boundary oriented counterclockwise. What is the normal to S? Explain why Stokes’ Theorem becomes the circulation form of Green’s Theorem.