The temperature $T$ (in degrees Celsius) of a metal plate, located in the xy-plane, at any point $(x,y)$ is given by $T=T(x,y) =24(x^{2}+y^{2})^{2}$.

Solution  (a) The rate of change of temperature in the direction of the positive $x$-axis is given by $$\begin{equation*} T_{x}(x, y)=2(24)(x^{2}+y^{2})(2x)=96\;x(x^2+y^2) \end{equation*}$$

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At the point $(1,-2)$, $T_{x}(1,-2)=96(1)(5) = 480$. This means that as one moves in a horizontal direction to the right away from the point $(1,-2)$, the temperature of the plate increases at the rate of $480 ^\circ{\rm C}$ per unit of distance.

(b) The rate of change of temperature in the direction of the positive $y$-axis is given by $$\begin{equation*} T_{y}(x, y)=2(24)(x^{2}+y^{2})(2y)=96\;y(x^2+y^2) \end{equation*}$$

At the point $(1,-2)$, $T_{y}(1,-2)=96(-2) (5) =-960$. This means that as one moves in a vertical direction up from the point $(1,-2)$, the temperature of the plate decreases at the rate of $960 {^\circ{\rm C}}$ per unit of distance.