Answer
a) $\frac{\partial{T}}{\partial{x}}$- Rate of change in temperature $T$ with respect to longitude $x$
$\frac{\partial{T}}{\partial{y}}$- Rate of change in temperature $T$ with respect to latitude $y$
$\frac{\partial{T}}{\partial{t}}$- Rate of change in temperature with time $t$
b) Positive, Negative, Positive
Work Step by Step
a) $\frac{\partial{T}}{\partial{x}}$ refers to the rate of change in temperature $T$ with respect to longitude $x$, $\frac{\partial{T}}{\partial{y}}$ refers to the rate of change in temperature $T$ with respect to latitude $y$, and $\frac{\partial{T}}{\partial{t}}$ refers to the rate of change in temperature with time $t$.
b) $f_{x}(158,21,9)$ would be positive as the West is defined as the positive $x$-direction (longitude), so the rate of change in temperature $T$ with respect to longitude $x$, $\frac{\partial{t}}{\partial{x}}$, would be positive as the temperature increases the further West one travels.
$f_{y}(158,21,9)$ would be negative as the North is defined as the positve $y$-direction (latitude), so the rate of change in temperature $T$ with respect to latitude $y$, $\frac{\partial{T}}{\partial{y}}$ would be negative as the temperature decreases the further North one travels.
$f_{t}(158,21,9)$ would be positive as the scenario takes place in the morning. Since the Sun is still rising and more heat energy dispensed by the Sun reaches the location the further overhead its position, the location will receive more heat energy as the day progresses and hence temperature $T$ increases with time $t$. Thus the rate of change in temperature with respect to time $\frac{\partial{T}}{\partial{t}}$ would be positive.
