A metal plate has the shape of the region $x^2 + y^2 \leq 1$. The plate is heated so that the temperature at any point $(x,y)$ on it is indicated by
$$T(x,y) = 2x^2 + y^2 – y + 3$$
Find the hottest and coldest points on the plate and the temperature at each of these points. (Hint: Parametrize the boundary of the plate in order to find any critical points there.)
I know how to do the actual optimization part of this problem. I already found a critical point at (0,0.5) by setting the first partial derivatives equal to 0. My problem is, how do I parametrize the boundary to find the other ones? I've seen solutions where they used cos(t) and sin(t) – where and how did they know how to do that?
Best Answer
We parametrize $x=\sin t, y=\cos t$ (although it works if we switch $x$ and $y$ as well). We obtain: $$2\sin^2 t+\cos^2 t-\cos t+3.$$Take the derivative and solve: $$\sin t(2\cos t+1)=0.$$
Edit An important part I left out is that $0\leq t \leq 2\pi$. We choose this time interval because it traverses the circle exactly once.