Class Participation $2$

Pret:$1*$ July$\_2025$ on CAMYAS

Example: $2$ Consider $f(x,y)=\sqrt[\phantom{2}]{x^2{y}^2}$

$($a$)$ Find the equation of a tangent plane at point $(3,4,5).$

$($b$)$ Find the normal vector at $(3,4,5).$

$($c$)$ Find the parametric equations of normal lines at $(3,4,5).$

$($d$)($i$)$ Find the angle between the tangent plane at $(3,4,5)$ and the $xy-$plane.

$($ii$)$ Find the angle between the tangent plane at $(3,4,5)$ and the plane $z=-x2y.$

Solution: Given $f(x,y)=\sqrt[\phantom{2}]{x^2{y}^2}.$

Set $z=f(x,y).$ Then $z=\sqrt[\phantom{2}]{x^2{y}^2},$ i$.$e$.,z-\sqrt[\phantom{2}]{x^2{y}^2}=0$

Then $z-f(x,y)=0=>z-\sqrt[\phantom{2}]{x^2{y}^2}=0.$

Set $F(x,y,z)=z-\sqrt[\phantom{2}]{x^2{y}^2}.$ The level surface is $S$:$F(x,y,z)=0.$

Now, $F(x,y,z)=z-\sqrt[\phantom{2}]{x^2{y}^2}.$ Please complete!