Second-Order Partial Derivatives

The temperature on a heated metal plate positioned in the first quadrant of the $xy$-plane is given by

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 m…

Let $f(x,y) = 8 - x^2 - y^2$ and $g(x,y) = 8 - x^2 + 4xy - y^2$.

1. Determine $f_x$, $f_y$, $f_{xx}$, $f_{yy}$, $f_{xy}$, and $f_{yx}$.

2. Evaluate each of the partial derivatives in (a) at the point $(0,0)$.

3. What do the values in (b) suggest about the behavior of $f$ near $(0,0)$? Plot a graph of $f$ and compare what you see visually to what the values suggest.

4. Determine $g_x$, $g_y$, $g_{xx}$, $g_{yy}$, $g_{xy}$, and $g_{yx}$.

5. Evaluate each of the partial derivatives in (d) at the point $(0,0)$.

6. What do the values in (e) suggest about the behavior of…

Let $f(x,y) = \frac{1}{2}xy^2$ represent the kinetic energy in Joules of an object of mass $x$ in kilograms with velocity $y$ in meters per second. Let $(a,b)$ be the point $(4,5)$ in the domain of $f$.

1. Calculate $\frac{ \partial^2 f}{\partial x^2}$ at the point $(a,b)$. Then explain as best you can what this second-order partial derivative tells us about kinetic energy.

2. Calculate $\frac{ \partial^2 f}{\partial y^2}$ at the point $(a,b)$. Then explain as best you can what this second-order partial derivative tells us abou…