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9.6 - Reading questions
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<h2>[9.6] - Reading Assignment
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<p class="show"><b>Read</b> section 9.6 in the textbook.
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<li>If $f$ is a function of two variables and $f (3, 4) = −1$, give the coordinates of one point on the graph of $f$.

<p class="justme answer">$$(x, y, z)=(3,4,-1)$$
<li>This equation defines a particular quadric surface: $x^2 +\frac19y^2 +\frac14z^2 =1$.  Why is this surface *not* the graph of a function $z=f(x,y)$?
<p class="justme answer">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 <b>for $z(x,y)$ to be a function, there must be a <u>unique</u> value of $z$ for every pair  $(x,y)$ in the domain</b> of the function. [This equation corresponds to a "football"-like elliptical surface in 3-d.]
<li>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 $z < 0$?
<p class="justme answer">One vertical trace occurs if $x=k$.  Then we'd have $z(y)=4k^2+y^2=C+y^2$: A set of parabolas.
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[Parabolas also for $y=k$.]<br><br>

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).
<li>Muddy questions? Questions you wonder about?
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