In this sequence of small whiteboard questions, students are shown the contour graph of a function of two variables and asked to find the derivative. They discover that, without a function to differentiate, they must instead think of the derivative as a ratio of small changes. This requires them to pick two nearby points. Which two?
1. Gradient Sequence | Contours >>
This is a graph of the function \(f(x,y)\):
- Find the derivative of this function.
- Find the derivative of this function at the leftmost of the indicated points.
- Find the partial derivative of this function with respect to \(x\) at the leftmost of the indicated points.
This activity is a set of SWBQ questions about finding a partial derivative from a contour graph. Show the contour graph and then ask, in an appropriate order depending on student responses:
assignment Homework
Shown below is a contour plot of a scalar field, \(\mu(x,y)\). Assume that \(x\)
and \(y\) are measured in meters and that \(\mu\) is measured in kilograms.
Four points are indicated on the plot.
group Small Group Activity
10 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
Mechanics Gravitational Force Gravitational Potential Energy Derivatives Introductory Physics
Students examine a plastic "surface" graph of the gravitational potential energy of a Earth-satellite system to make connections between gravitational force and gravitational potential energy.group Small Group Activity
120 min.
group Small Group Activity
30 min.
group Small Group Activity
30 min.
assignment Homework
Learn more about the geometry of \(\vert \vec{r}-\vec{r'}\vert\) in two dimensions.
Make a sketch of the graph \begin{equation} \vert \vec{r} - \vec{a} \vert = 2 \end{equation}
for each of the following values of \(\vec a\): \begin{align} \vec a &= \vec 0\\ \vec a &= 2 \hat x- 3 \hat y\\ \vec a &= \text{points due east and is 2 units long} \end{align}
group Small Group Activity
30 min.
group Small Group Activity
60 min.
Mechanics Gravitational Potential Energy Zero of Potential Introductory Physics
Students examine a plastic “surface” graph of the gravitational potential energy of an Earth-satellite system to explore the properties of gravitational potential energy for a spherically symmetric system.