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

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

and let $C$ be the contour curve of $f$ at level $e^2.$

$($a$)$ Find a Cartesian equation for the tangent line $l$ to $C$ at the point $(1,1).$

$($b$)$ Sketch the contour curve $C$ together with the line $l$ in $R^2.$ Show all intersections with the axes.

$($c$)$ Find an equation for the tangent plane $V$ to the graph of $f$ in $R^3$ at the point $(1,1,e^2).$

$($d$)$ At which point on the graph of $f$ is 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$].$

$5.($Section $7\mathrm{.}9$ and Chapter $8)$

Consider the $3-$dimensional vector field $E_{?}$ defined by

$F_{x,y,z}=(2xz+\frac{1}{2}y{z}^2+x+y,\frac{1}{2}x{z}^2+x+y{z}^2,x^2+xyz+y^{2}z).$

$($a$)$ Write down the Jacobian matrix $J_E(x,y,z).$

$($b$)$ Determine $div{F}_{x,y,z}.$

$(2)$

$($c$)$ Determine $cur{l}_F(x,y,z).$

$(2)$

$($d$)$ Why does $E_{?}$ have a potential function? Give reasons for your answer, referring to the relevant definitions and theorems in the study guide.

$($e$)$ Find a potential function of $E_{?}.$

$(6)$

$[15]$

$6.($Chapter $9)$

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

$f(x,y)=\frac{sin(x+y)}{x+y}$

Find the third order Taylor polynomial about the point $(0,0).$

$[10]$