Problems: In Problems $1-6$ deal with chain rules. The remaining problems are concerned with

Section $17\mathrm{.}4$ on Optimization for functions of two variables.

Given is $w=3x^2+xy$ and $x=2r-3s$ and $y=6r+s.$ Find $\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{r}}$ and $\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{s}}.$

Given is $w=\frac{x+e^y}{3y}$ and $x=rs+s{e}^r$ and $y=9r^3\sqrt[\phantom{2}]{s}.$ Find $\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{r}}$ and $\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{s}}.$

Given is $w=x^2+\sqrt[\phantom{2}]{x+z{y}^2}$ and $x=r^2-s^2$ and $y=rs$ and $z=r^2+s^3.$ Find $\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{r}}$ and $\frac{\text{d}\text{e}\text{l}\text{w}}{\text{d}\text{e}\text{l}\text{s}}.$

Given is $z=\sqrt[\phantom{2}]{x^2+y^2}$ and $x=e^{2t}$ and $y=e^{-2t}.$ Use a chain rule to find $\frac{dz}{dt}$ evaluated at

$t=1.$

Assume $w=u(s,t)=f(s^2-t^2,t^2-s^2)$ for where $f$ is a MATA$33$S function defined for all

real $s$ and $t.$ Verify that $t{u}_s(s,t)+s{u}_t(s,t)=0.$

Let $z=x^2+xy+y^2,x=s+t,$ and $y=st.$ Find $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{s}}$ and $\frac{\text{d}\text{e}\text{l}\text{z}}{\text{d}\text{e}\text{l}\text{t}}$ two ways:

$($a$)$ By first substituting $x$ and $y$ as functions of $s$ and $t$;

$($b$)$ By using a chain rule.

Section $17\mathrm{.}4,$ Pages $752-754$#$1-3,7,9,17,18,21,22,23,24,27.$

A delivery company accepts only rectangular boxes who length plus "girth" do not sum over

$108cm($The "girth" of a rectangular box is defined as the perimeter of a cross$-$section of the

box$).$ Find the dimensions of an acceptable box of largest volume.

In this question let $f(x,y)=x^2-y^2-2x+4y+6$