## Activity: Partial Derivatives from a Contour Map

Static Fields 2022 (3 years)
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?
• This activity is used in the following sequences
• Media

This is a graph of the function $f(x,y)$:

1. Find the derivative of this function.
2. Find the derivative of this function at the leftmost of the indicated points.
3. Find the partial derivative of this function with respect to $x$ at the leftmost of the indicated points.

## Instructor's Guide

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:

1. Find the derivative of this function.
2. Find the derivative of this function at the leftmost of the indicated points.
3. Find the partial derivative of this function at the leftmost of the indicated points with respect to $x$.

### Student Conversations

See the comments in the solution to the Student Handout.
• assignment Contours

assignment Homework

##### Contours

Static Fields 2022 (5 years)

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.

1. Determine $\frac{\partial\mu}{\partial x}$ and $\frac{\partial\mu}{\partial y}$ at each of the four points.
2. On a printout of the figure, draw a qualitatively accurate vector at each point corresponding to the gradient of $\mu(x,y)$ using your answers to part a above. How did you choose a scale for your vectors? Describe how the direction of the gradient vector is related to the contours on the plot and what property of the contour map is related to the magnitude of the gradient vector.
3. Evaluate the gradient of $h(x,y)=(x+1)^2\left(\frac{x}{2}-\frac{y}{3}\right)^3$ at the point $(x,y)=(3,-2)$.

• group Sequential Stern-Gerlach Experiments

group Small Group Activity

10 min.

##### Sequential Stern-Gerlach Experiments
Quantum Fundamentals 2022 (3 years)
• group Number of Paths

group Small Group Activity

30 min.

##### Number of Paths

Student discuss how many paths can be found on a map of the vector fields $\vec{F}$ for which the integral $\int \vec{F}\cdot d\vec{r}$ is positive, negative, or zero. $\vec{F}$ is conservative. They do a similar activity for the vector field $\vec{G}$ which is not conservative.
• group Quantifying Change

group Small Group Activity

30 min.

##### Quantifying Change

In this activity, students will explore how to calculate a derivative from measured data. Students should have prior exposure to differential calculus. At the start of the activity, orient the students to the contour plot - it's busy.
• group Projectile with Linear Drag

group Small Group Activity

120 min.

##### Projectile with Linear Drag
Theoretical Mechanics (4 years)

Students consider projectile motion of an object that experiences drag force that in linear with the velocity. Students consider the horizontal motion and the vertical motion separately. Students solve Newton's 2nd law as a differential equation.
• keyboard Kinetic energy

keyboard Computational Activity

120 min.

##### Kinetic energy
Computational Physics Lab II 2022

Students implement a finite-difference approximation for the kinetic energy operator as a matrix, and then use numpy to solve for eigenvalues and eigenstates, which they visualize.
• assignment Directional Derivative

assignment Homework

##### Directional Derivative

Static Fields 2022 (5 years)

You are on a hike. The altitude nearby is described by the function $f(x, y)= k x^{2}y$, where $k=20 \mathrm{\frac{m}{km^3}}$ is a constant, $x$ and $y$ are east and north coordinates, respectively, with units of kilometers. You're standing at the spot $(3~\mathrm{km},2~\mathrm{km})$ and there is a cottage located at $(1~\mathrm{km}, 2~\mathrm{km})$. You drop your water bottle and the water spills out.

1. Plot the function $f(x, y)$ and also its level curves in your favorite plotting software. Include images of these graphs. Special note: If you use a computer program written by someone else, you must reference that appropriately.
2. In which direction in space does the water flow?
3. At the spot you're standing, what is the slope of the ground in the direction of the cottage?
4. Does your result to part (c) make sense from the graph?

• group Covariation in Thermal Systems

group Small Group Activity

30 min.

##### Covariation in Thermal Systems

Students consider how changing the volume of a system changes the internal energy of the system. Students use plastic graph models to explore these functions.
• group Calculating Coefficients for a Power Series

group Small Group Activity

30 min.

##### Calculating Coefficients for a Power Series
Theoretical Mechanics (7 years)

Power Series Sequence (E&M)

This activity starts with a brief lecture introduction to power series and a short derivation of the formula for calculating the coefficients of a power series for a particular function:

$c_n={1\over n!}\, f^{(n)}(z_0)$

After a brief lecture deriving the formula for the coefficients of a power series, students compute the power series coefficients for a $\sin\theta$ (around both the origin and $\frac{\pi}{6}$). The meaning of these coefficients and the convergence behavior for each approximation is discussed in the whole-class wrap-up.

• group Gravitational Force

group Small Group Activity

30 min.

##### Gravitational Force

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.

Learning Outcomes