Exercise 15

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 about kinetic energy.

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

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