First Exam

1. (20 points) Find the limit, if it exists, or show that the limit does not exist.

   1. $$\displaystyle{\lim_{(x,y) \rightarrow (0,0)} \frac{x^{2} - y^{2}}{x^{2} + y^{2}}}$$

   2. $$\displaystyle{\lim_{(x,y) \rightarrow (6,3)} x y \cos(x-2 y)}$$

2. (20 points) Show that the function $f$ does not have a limit at $(0,0)$ by examining the limits of $f$ as $(x,y) \rightarrow (0,0)$ along the line $y=x$ and along the parabola $y=x^{2}$ :
   $$f(x,y) = \frac{x^{2} y}{x^{4} + y^{2}}$$

3. (10 points) Find the partial derivatives.
   1. $f_{x}$ if $f(x,y)= 5x^{2} y^{3} + 8 x y^{2} - 3 x^{2}$
   2. $${\frac{\partial}{\partial x} \ln (y e^{xy})}$$
4. (20 points) Find the quadratic Taylor polynomials about $(0,0)$ for $e^{x} \cos y$ .
5. (40 points) Find the critical points and classify them as local maxima, local minima, saddle points, or none of these.
   1. $f(x,y)=x^{3} - 3 x + y^{3} - 3y$
   2. $f(x,y)=\frac{1}{2} x^{2} + 3 y^{3} + 9 y^{2} - 3 x y + 9 y - 9 x$