Activity: Partial Derivatives from a Contour Map

Static Fields 2022 (4 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?
  • Media
    • activity_media/contoursfig1.png

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

    Gradient Sequence

    Static Fields 2022 (6 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

    E&M Conservative Fields Surfaces

    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 Gravitational Force

    group Small Group Activity

    30 min.

    Gravitational Force

    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 Equipotential Surfaces

    group Small Group Activity

    120 min.

    Equipotential Surfaces

    E&M Quadrupole Scalar Fields

    Students are prompted to consider the scalar superposition of the electric potential due to multiple point charges. First a single point charge is discussed, then four positive charges, then an electric quadrupole. Students draw the equipotential curves in the plane of the charges, while also considering the 3D nature of equipotentials.
  • group Quantifying Change

    group Small Group Activity

    30 min.

    Quantifying Change

    Thermo Derivatives

    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 Covariation in Thermal Systems

    group Small Group Activity

    30 min.

    Covariation in Thermal Systems

    Thermo Multivariable Functions

    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.
  • assignment Circle Vector, Version 2

    assignment Homework

    Circle Vector, Version 2
    Static Fields 2022 (6 years)

    Learn more about the geometry of \(\vert \vec{r}-\vec{r'}\vert\) in two dimensions.

    1. Make sketches of the following functions, by hand, on the same axes: \begin{align} y &= \sin x\\ y &= \sin(2+x) \end{align} Briefly describe the role that the number 2 plays in the shape of the second graph
    2. 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}

    3. Derive a more familiar equation equivalent to \begin{equation} \vert \vec r - \vec a \vert = 2 \end{equation} for arbitrary \(\vec a\), by expanding \(\vec r\) and \(\vec a\) in rectangular coordinates. Simplify as much as possible. (Ok, ok, I know this is a terribly worded question. What do I mean by “more familiar"? What do I mean by “simplify as much as possible"? Why am I making you read my mind? Try it anyway. Real life is not full of carefully worded problems. Bonus points to anyone who can figure out a better way of wording the question that doesn't give the point away.)
    4. Write a brief description of the geometric meaning of the equation \begin{equation} \vert \vec r - \vec a \vert = 2 \end{equation}

  • group Directional Derivatives

    group Small Group Activity

    30 min.

    Directional Derivatives
    Vector Calculus I 2022

    Directional derivatives

    Gradient Sequence

    This small group activity using surfaces relates the geometric definition of directional derivatives to the components of the gradient vector. Students work in small groups to measure a directional derivative directly, then compare its components with measured partial derivatives in rectangular coordinates. The whole class wrap-up discussion emphasizes the relationship between the geometric gradient vector and directional derivatives.
  • group Gravitational Potential Energy

    group Small Group Activity

    60 min.

    Gravitational Potential Energy

    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.

Learning Outcomes