Activity: Partial Derivatives from a Contour Map

Static Fields 2022 (3 years)
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
    • assignment Contours

      assignment Homework

      Contours
      Static Fields 2022 (4 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 Quantifying Change (Remote)

      group Small Group Activity

      30 min.

      Quantifying Change (Remote)

      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.
    • group ``Squishability'' of Water Vapor (Contour Map)

      group Small Group Activity

      30 min.

      “Squishability” of Water Vapor (Contour Map)

      Thermo Partial Derivatives

      Students determine the “squishibility” (an extensive compressibility) by taking \(-\partial V/\partial P\) holding either temperature or entropy fixed.
    • 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 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 Heat and Temperature of Water Vapor (Remote)

      group Small Group Activity

      5 min.

      Heat and Temperature of Water Vapor (Remote)

      Thermo Heat Capacity Partial Derivatives

      In this introduction to heat capacity, students determine a derivative that indicates how much the internal energy changes as the temperature changes when volume is held constant.
    • assignment Directional Derivative

      assignment Homework

      Directional Derivative
      Static Fields 2022 (4 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 Charged Sphere

      group Small Group Activity

      30 min.

      Charged Sphere

      E&M Introductory Physics Electric Potential Electric Field

      Students use a plastic surface representing the potential due to a charged sphere to explore the electrostatic potential, equipotential lines, and the relationship between potential and electric field.
    • group Electric Potential of Two Charged Plates

      group Small Group Activity

      30 min.

      Electric Potential of Two Charged Plates
      Students examine a plastic "surface" graph of the electric potential due to two changes plates (near the center of the plates) and explore the properties of the electric potential.

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 at the leftmost of the indicated points with respect to \(x\).

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.

Learning Outcomes