Exercise 35

Consider the function $f$ defined by $f(x,y) = 8 - x^2 - 3y^2$.

1. Determine $f_x(x,y)$ and $f_y(x,y)$.

2. Find parametric equations in $\R^3$ for the tangent line to the trace $f(x,1)$ at $x=2$.

3. Find parametric equations in $\R^3$ for the tangent line to the trace $f(2,y)$ at $y=1$.

4. State respective direction vectors for the two lines determined in (b) and (c).

5. Determine the equation of the plane that passes through the point $(2,1,f(2,1))$ whose normal vector is orthogonal to the direction vectors of the two lines found in (b) and (c).

6. Use a graphing utility to plot both the surface $z = 8 - x^2 - 3y^2$ and the plane from (e) near the point $(2,1)$. What is the relationship between the surface and the plane?