Exercise 34

The temperature on an unevenly heated metal plate positioned in the first quadrant of the $xy$-plane is given by $$\begin{gathered} C(x,y) = \frac{25xy+25}{(x-1)^2 + (y-1)^2 + 1}. \end{gathered}$$

Assume that temperature is measured in degrees Celsius and that $x$ and $y$ are each measured in inches. (Note: At no point in the following questions should you expand the denominator of $C(x,y)$.)

1. Determine $\frac{\partial C}{\partial x}\restrict{(x,y)}{}$ and $\frac{\partial C}{\partial y}\restrict{(x,y)}{}$.

2. If an ant is on the metal plate, standing at the point $(2,3)$, and starts walking in the direction parallel to the positive $y$ axis, at what rate will the temperature the ant is experiencing change? Explain, and include appropriate units.

3. If an ant is walking along the line $y = 3$ in the positive $x$ direction, at what instantaneous rate will the temperature the ant is experiencing change when the ant passes the point $(1,3)$?

4. Now suppose the ant is stationed at the point $(6,3)$ and walks in a straight line towards the point $(2,0)$. Determine the average rate of change in temperature (per unit distance traveled) the ant encounters in moving between these two points. Explain your reasoning carefully. What are the units on your answer?