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<title>
10.6 - Heat seeking kitten
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<h1>Heat seeking kitten</h1>

<p><img src="KittenContour.jpg" style="width:300px" class="rightalign">Let $T(x,y)=x^2-2xy$ be the temperature at a point $(x,y)$ in the region bounded by the curves $y=x$ and $y=x^2$. (Blue lines on the accompanying contour plot.)  Suppose that a kitten is crawling around the region.

<ol type="a">
<li> At $P=(\frac12,\frac13)$
, in which direction should the cat go to cool down as quickly as possible?: Draw a vector on the contour plot indicating the direction.  And find the components of the gradient and use it to justify your answer.

<li> At $P$, in what direction(s) could the kitten go to
maintain its current temperature?

<li> Where is the hottest point <i>inside  the region</i>? Mark it on the  contour plot. Explain your answer.

<li> If,  at $P$, the cat moves in such a way that for each change in its $x$ direction of 2
units, the change in the $y$ direction is –1 unit, find a value for $\frac{dT}{dt}$, the rate of change of the temperature, from the
kitten’s point of view.  (Give your answer some plausible  units).
</ul>

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