The temperature $T$ (in ${}^{\circ}$C) at any point in the region $-10 \leq x \leq 10$, $-10 \leq y \leq 10$ is given by the function $$T(x,y) = 100 - x^2 - y^2.$$

(a) On a sheet of paper, sketch isothermal curves (curves of constant temperature) for $T=100^{\circ}$ C, $T=75^{\circ}$ C, $T=50^{\circ}$ C, $T=25^{\circ}$ C, and $T=0^{\circ}$ C.

(b) Suppose a heat-seeking bug is put down at each of the following points on the $xy$-plane. Thinking of the positive $y$ direction as north and the positive $x$ direction as east, pick the direction in which it should move to increase its temperature fastest.
(i) $(-7.5, 7.5)$ the bug should go ? N NE E SE S SW W NW
(ii) $(7.5, 0)$ the bug should go ? N NE E SE S SW W NW
(iii) $(0, -7.5)$ the bug should go ? N NE E SE S SW W NW

(Think about how your answers to these questions are related to the level curves through that point!)