Week $4$

Consider the surface $S$ defined $by$ the implicit equation

$x^2+y^2+z^3=29$

$(a)$ Say briefly why the point vec$(a)=(1,1,3)$ lies $on$ the surface $S.$

$(b)$ Find a formula for the tangent plane $to$ the level$-$set $Sat$ the point vec$(a),$ expressing your answer

$in$ the form $px+qy+rz=c.$ State carefully any formula you use $to$ arrive $at$ your answer.

Suppose that $f(x,y)is$ a function $of$ two variables for which $we$ know that

$\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{x}}=x{y}^2+2y,$ and $,\frac{\text{d}\text{e}\text{l}\text{f}}{\text{d}\text{e}\text{l}\text{y}}=x^{2}y+2x$

Suppose now that $x$ and $y$ themselves vary $as$ functions $of$ variables $s$ and $t$ :

$x(s,t)=s^{2}t-2t^3,$ and $,y(s,t)=2st-5$

Let $F$ denote $f$ seen $as$ a function $ofs$ and $t$ :

$F(s,t)=f(x(s,t),y(s,t)).$

Use the chain rule $to$ calculate del$\frac{F}{d}$els $at(s,t)=(3,2).$