Let $f$ be the $R^3-R$ function defined by

$f(x,y,z)=x^{yz}.$

$($a$)$ Determine the rate of increase of $f$ at the point $(e,1,1)$ in the direction of the vector $(3,-2,1).$

$($b$)$ Determine the maximum rate of increase of $f$ at $(e,1,1).$ In which direction is the maximum rate of increase?

$($c$)$ At the point $(e,1,1),$ in which direction is the function constant $($i$.$e$.$ its value doesn't change$)?$

$[12]$

$2.($Sections $1\mathrm{.}3,6\mathrm{.}3,7\mathrm{.}6)$

A particle travels along the curve $C$ defined by $(x,y)=r_t,$ where

$r_t=(tcost,tsint,t^2)$;$t0.$

$($a$)$ Show that $C$ lies on the paraboloid $x^2+y^2=z.$

$($b$)$ Find a vector tangent to $C$ at the point $r_1.$

$(2)$

$($c$)$ Calculate the speed of the particle at time $t=1.$

$(3)$

$($d$)$ Find a Cartesian equation for the line $l$ tangent to the curve $r_{?}$ at the point $(0,\frac{}{2},\frac{{}^2}{4}).(5)$

$($e$)$ Let $f$ be the $R^3-R$ function defined by $f(x,y,z)=x+z.$ Use the General Chain Rule to determine the derivative function $.$

$3.$ Let $f$ be the $R^2-R$ function defined by

$f(x,y)=\frac{1}{x^2+y^2}$

and let $r_{?}$ be the $R-R^2$ function defined by

$r_t=(tcost,tsint).$

$($a$)$ Determine the composite function $f$@$r_{?}.($Simplify your answer$).$

$($b$)$ Determine $gra{d}_f(x,y)$ and $r_{{?}^{?}}(\text{'})(t).$

$($c$)$ Determine the derivative function by

i$.$ differentiating the expression obtained in $($a$)$

ii$.$ using the chain rule $($Theorem $7\mathrm{.}6\mathrm{.}1).$

Compare your answers.