Semester 2 Warmup
MultiV 2021-22 / Dr. Kessner

No calculator! Have fun!

1. Consider the function \(f(x,y) = 2x^2 -4x +3y^2 +12y +20\).

1. Find the equation of the tangent plane to the surface \(z = f(x,y)\) at \((x,y) = (0,0)\).

2. A critical point of \(f\) is a point \((x,y)\) where both \(f_x\) and \(f_y\) are either zero or undefined. Find all critical points of \(f\) (there is only one for this example).

3. At the critical point, find the linear approximation of \(f\).

4. Let \(d^2 \! f\) be the matrix of 2nd partial derivatives: \[ d^2 \! f = \begin{pmatrix} f_{xx} & f_{xy} \\ f_{yx} & f_{yy} \\ \end{pmatrix} \] Find \(d^2 \! f\) and \(\det d^2 \! f\) (at the critical point). What does \(d^2 f\) tell you about the shape of the surface at the critical point?

5. Complete the square to write the function in the form \(f(x,y) = a(x-h)^2 + b(y-k)^2 + c\). What does this tell you about the surface?

2. Do the same calculations for the function \(g(x,y) = -4x^2 +16x -5y^2 -11\).

3. Do the same calculations for the function \(h(x,y) = x^2 -2x -2y^2 +8y -9\).