A one-meter long bar is heated unevenly, with temperature in ${}^\circ$C at a distance $x$ meters from one end at time $t$ given by $$H(x,t) = 90e^{-0.15t}\sin\!\left(\pi x\right)$$ $0\le x\le 1$. On a sheet of paper, sketch a graph of $H$ against $x$ for $t=0$ and $t=1$. Think about why your graphs make sense.

(a) Calculate each of:
$H_x(0.1,t) =$
$H_x(0.9,t) =$ .
(Be sure that you can say in words what the practical interpretation (in terms of temperature) of these two partial derivatives is, and why each has the sign that it does.)

(b) Calculate:
$H_t(x,t) =$ .
(Again, be sure that you understand why it has the sign that it does, and what its interpretation in terms of temperature is.)