[9.6] - Reading Assignment

Read section 9.6 in the textbook.

1. If $f$ is a function of two variables and $f (3, 4) = −1$, give the coordinates of one point on the graph of $f$.

   $$(x, y, z)=(3,4,-1)$$

2. This equation defines a particular quadric surface: $x^2 +\frac19y^2 +\frac14z^2 =1$. Why is this surface *not* the graph of a function $z=f(x,y)$?

   Let's say that $(x,y,z)$ are the coordinates of a point that satisfies the equation. Because of the squares in the equation, the point $(x,y,-z)$ would also solve the equation. But for $z(x,y)$ to be a function, there must be a unique value of $z$ for every pair $(x,y)$ in the domain of the function. [This equation corresponds to a "football"-like elliptical surface in 3-d.]

3. What do the vertical traces of the surface $z = 4x^2 + y^2$ look like? Sketch a representative one. What are the horizontal traces (contour lines) for $z > 0$? For $z < 0$?

   One vertical trace occurs if $x=k$. Then we'd have $z(y)=4k^2+y^2=C+y^2$: A set of parabolas.
   
   [Parabolas also for $y=k$.]
   
   Horizontal traces would correspond to $z=k$, so $k=4x^2+y^2$. If $k>0$, this is the equation of an ellipse in the $x-y$ plane. If $k<0$, there is no solution (no trace).

4. Muddy questions? Questions you wonder about?