Functions Of Several Variables

1. Find and sketch the domain of the function

   $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=\sqrt{1+x-y^2}$.

   Find the range of $f$.

2. Make a rough sketch of a contour map for the function whose graph is shown.

   

3. Draw a contour map of the function $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=y-\ln <!--FUNCTION\; APPLICATION-->\left(x\right)$ showing several level curves.
4. Sketch the contour map and the graph of the function $f<!--FUNCTION\; APPLICATION-->\left(x,y\right)=x^2+9y^2$.

5. Match each given function with its graph and its contour map. Explain.

   1. $z=\sin <!--FUNCTION\; APPLICATION-->\left(xy\right)$
   2. $z=e^x\cos <!--FUNCTION\; APPLICATION-->\left(y\right)$
   3. $z=\sin <!--FUNCTION\; APPLICATION-->\left(x-y\right)$
   4. $z=\sin <!--FUNCTION\; APPLICATION-->\left(x\right)-\sin <!--FUNCTION\; APPLICATION-->\left(y\right)$
   5. $z=\left(1-x^2\right)\left(1-y^2\right)$
   6. ${\displaystyle z=\frac{x-y}{1+x^2+y^2}}$

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6. Describe the level surfaces of the function $f<!--FUNCTION\; APPLICATION-->\left(x,y,z\right)=x^2-y^2+z^2$.
   Family of hyperboloids of one or two sheets with axis the $y\text{-axis}$.