$(Sections2\mathrm{.}12,3\mathrm{.}2,7\mathrm{.}5,7\mathrm{.}8)$

Consider the $R^2-R$ function defined $by$

$f(x,y)=e^{x^2+y^2}$

and let $Cbe$ the contour curve $offat$ level $e^2.$

$(a)$ Find a Cartesian equation for the tangent line $ltoCat$ the point $(1,1).$

$(b)$ Sketch the contour curve $C$ together with the line $lin{R}^2.$ Show all intersections with the

axes.

$(c)$ Find $an$ equation for the tangent plane $Vto$ the graph $offin{R}^{3}at$ the point $(1,1,e^2).(4)$

$(d)At$ which point $on$ the graph $offis$ the tangent plane parallel $to$ the plane $2e^{2}x-2e^{2}y-z=$

$4?$

$[Hint$: two planes are parallel $if$ and only $if$ their normal vectors are parallel$].$