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9.6 - Staying cool / Matching
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<h2>9.6 - Staying Cool</h2>
<p>Let $T(x,y)$ be the temperature in a 10 ft by 10 ft room on a winter night.  One coreer of the room is at (0,0) and the opposite corner is at (10,10).  For each of the following functions, $T(x,y)$,
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<li>Draw, or describe in words, or plot with GeoGebra a graph of the temperature function.  Useful prototype commands in GeoGebra: (This should plot one quadrant of a sphere)
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function(sqrt(81-x^(2)-y^(2)), 0, 10, 0, 10)
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z=4
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<p>The last four numbers are the $x$-range to plot, $0\lt x\lt 10$, and then the $y$-range to plot, also 0 to 10. The command <tt>z=4</tt> will display the plane perpendicular to $\uv z$ at a height of $z=4$.  This lets you easily see what part of the surface is above 4 and what part is below 4.

<>Now, for each of the Temperature functions, $T(x,y)$ for this 10 by 10 foot room, plot $z=T(x,y)$.  Use your plot to answer these questions:
<li>Describe the likely floor locations of the heating vents.  Sketch the locations on a 10 by 10 square.
<li>Suppose you like to sleep with a temperature of 70${}^o$ or less.  Where would you put your bed? Again, sketch locations on a square.
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<ol>
<li> $T(x,y)=78-\frac{1}{10}\left[ x^2+(y-5)^2 \right]$
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<li> $T(x,y)=\frac12x-y+75$
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<h2>
9.6 - The matching game
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<p>Match each function with its graph.  Give reasons for your choices.
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<td>1)&nbsp;$f(x,y)=\frac{1}{x+1}+\sin y$. 
<p class="answer">the fraction blows up as $x\to-1$.  Otherwise, if $x=k$, then $z\propto C+\sin(y)$.
<br>$\Rightarrow$ III</td>
<td>2)&nbsp;$f(x,y)=\sqrt{4-x^2-y^2}$
<p class="answer">Horizontal traces ($z=k$) are circles.  Impossible for $z$ to be greater than $\sqrt 4=2$.<br>$\Rightarrow$ VI</td>
<td>3)&nbsp;$f(x,y)= \cos(x+y^2)$
<p class="answer">If $y^2=k$, then $z\propto \cos(x+k)$. Also, the $\cos(\text{ })$ function never gets above 1 or below -1, so $-1< z < +1$ as well.<br>
$\Rightarrow$ V
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<td>4)&nbsp;$f(x,y)=\ln(x^2+y^2+1)$
<p class="answer">Lowest possible value occurs when $x=y=0$:  $f(0,0)=\ln 1=0$.
Horizontal traces are circles.<br>
$\Rightarrow$ I</td>
<td>5) $f(x,y)=x^2\sqrt y$
<p class="answer">If $y=k$, $z\propto x^2$.  If $x=k$, $z\propto \sqrt y$.  Undefined for $y\lt 0$.<br>
$\Rightarrow$ IV</td>
<td>6) $f(x,y)=x^3y$
<p class="answer">When $y=k$, $z\propto x^3$.  When $x=k$, $z=\propto x$.<br>
$\Rightarrow$ II</td>
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<img src="0g/9.6.matching.jpg">



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