Exercise 13

The temperature on a heated metal plate positioned in the first quadrant of the $xy$-plane is given by $$\begin{gathered} C(x,y) = 25e^{-(x-1)^2 - (y-1)^3}. \end{gathered}$$

Assume that temperature is measured in degrees Celsius and that $x$ and $y$ are each measured in inches.

1. Determine $C_{xx}(x,y)$ and $C_{yy}(x,y)$. Do not do any additional work to algebraically simplify your results.

2. Calculate $C_{xx}(1.1, 1.2)$. Suppose that an ant is walking past the point $(1.1, 1.2)$ along the line $y = 1.2$. Write a sentence to explain the meaning of the value of $C_{xx}(1.1, 1.2)$, including units.

3. Calculate $C_{yy}(1.1, 1.2)$. Suppose instead that an ant is walking past the point $(1.1, 1.2)$ along the line $x = 1.1$. Write a sentence to explain the meaning of the value of $C_{yy}(1.1, 1.2)$, including units.

4. Determine $C_{xy}(x,y)$ and hence compute $C_{xy}(1.1, 1.2)$. What is the meaning of this value? Explain, in terms of an ant walking on the heated metal plate.