## Activity: 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 Small Group Activity schedule 30 min. build Contour graph with P,V,S,T,U contours in color , handout for each student, A personal or shared writing space for each student to write/draw/sketch. description Student handout (PDF)
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Thermo Derivatives
What students learn
• To calculate derivatives on a contour map.
• Many different derivatives can be determined at a point.
• Think about derivatives as ratios of small changes.
• Reinforces the idea that a derivative is a ratio of small changes along a path
• In thermo, derivatives are often invertible?
• The sign of a derivative does not depend on the direction you're traveling along the path
• Media

Orient Yourself to the Physical System & the Graph: The plot shows various thermodynamic quantities for water vapor in a piston (cylindrical thermos with a movable top) at different states. From state (point) $A$ to state (point) $B$, estimate the following quantities:

 Verbal Description Symbol Estimate (with Units) Change in volume: $\Delta V_{A\rightarrow B}$ Change in entropy: $\Delta S_{A\rightarrow B}$ Change in temperature: $\Delta T_{A\rightarrow B}$ Change in pressure: $\Delta P_{A\rightarrow B}$ Change in internal energy: $\Delta U_{A\rightarrow B}$

Under what circumstances would you be willing to label these quantities with d's instead of $\Delta$'s? For example, $dV$ instead of $\Delta V$.

Discussion: What does “small” mean? Some students say “small” but not what it is small compared to. The instructor can talk about linearity in the whole class discussion.

Discussion: Can you take a limit experimentally? There can be some good discussion about this (I'm not convinced there is a right answer, but is a good thing to think about.)

Discussion: Linearity Once the data is linear, the secant line becomes the tangent line and the slope is the derivative even if the measured changes are “large”.

Determine a Rate: Pick two of the variables in the table and determine the rate of change of one with respect to the other from $A$ to $B$. What experiment could you do to measure this rate?

Discussion: Holding Entropy Fixed Between points A & B, the entropy doesn't change. The experiment the students describe should hold entropy constant, for example, by insulating the piston.

Discussion: Derivatives: “Under what circumstances would you be willing to call this rate a derivative?” Have students plot graphs of their functions to see how linear they are. Many students will not be willing to call this a derivative because we aren't taking a limit - talk about linear approximation.

Discussion: Derivative of which point? If a student is willing to call this rate a derivative, at which point is it the derivative? It is the best approximation to the derivative somewhere between the two points.

Discussion: Slope Is their derivative a slope? Of what?

Follow-up: How would you represent this derivative symbolically? Some students write $\Delta U / \Delta T$ vs. $dU/dT$; some students write subscript $S$ vs. $A \rightarrow B$. A good opportunity to talk about how the derivative ought to depend on the value of $S$.

Inverting Your Rate: Determine the reciprocal of the rate you calculated. Brainstorm a meaningful name for this rate.

Discussion: Experiment What experiment would you do to measure this quantity?

Discussion: Inversion The derivative can be flipped as long as the path is unchanged.

Follow-up: How would you represent this rate it symbolically?

Reversing the Path: How does the rate you previously calculated change if instead you went from state $B$ to state $A$?

Answer: The rate (and therefore the derivative) is the same.

Discussion: Sign Many students will initially say that the derivative changes sign.

SUMMARY PAGE

Goals

• To calculate derivatives on a contour map.
• Many different derivatives can be determined at a point.
• Think about derivatives as ratios of small changes.
• Reinforces the idea that a derivative is a ratio of small changes along a path
• In thermo, derivatives are often invertible?
• The sign of a derivative does not depend on the direction you're traveling along the path

Time Estimate: 30 minutes

Tools/Equipment

• Contour graph with P,V,S,T,U contours
• Student handout for each student
• A personal or shared writing space for each student to write/draw/sketch.

Intro

• Orient the students to the contour plot - it's busy.
• Exposure to differential calculus

Whole Class Discussion / Wrap Up

• How students report out their answers to Explore.
• Many students will not be willing to call the ratio of changes a derivative until you've taken the limit that the changes become infinitesimal. Talk about linear approximation and/or tables of data.
• Most students will have heard in a math class that derivatives are not invertible. In thermo, they often are as long as you don't change the path.
• Many students will think that if you reverse the path, the derivative will change sign. Discuss how the signs of both the numerator and denominator of the ratio of small changes swaps, leaving the derivative unchanged.

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

group Small Group Activity

30 min.

##### “Squishability” of Water Vapor (Contour Map)

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

group Small Group Activity

30 min.

##### Heat and Temperature of Water Vapor

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.
• 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 Ideal Gas Model

group Small Group Activity

30 min.

##### Ideal Gas Model

Students consider whether the thermo surfaces reflect the properties of an ideal gas.
• 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.
• group Directional Derivatives

group Small Group Activity

30 min.

##### Directional Derivatives
Vector Calculus I 2022

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 The Hillside

group Small Group Activity

30 min.

##### The Hillside
Vector Calculus I 2022 (2 years)

Students work in groups to measure the steepest slope and direction on a plastic surface, and to compare their result with the gradient vector, obtained by measuring its components (the slopes in the coordinate directions).
• assignment Contours

assignment Homework

##### Contours

Static Fields 2023 (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 Vector Differential--Curvilinear

group Small Group Activity

30 min.

##### Vector Differential--Curvilinear
Vector Calculus II 23 (11 years)

Integration Sequence

In this small group activity, students are given a picture as a guide. They then write down an algebraic expression for the vector differential in different coordinate systems (cartesian, cylindrical, spherical).

Use Vector Differential--Rectangular as an introduction. This activity can be done simultaneously with Pineapples and Pumpkins where students or the instructor cut volume elements out of pineapples and/or pumpkins to show the geometry.

• assignment Coffees and Bagels and Net Worth

assignment Homework

##### Coffees and Bagels and Net Worth
Energy and Entropy 2021 (2 years)

In economics, the term utility is roughly related to overall happiness. Many things affect your happiness, including the amount of money you have and the amount of coffee you drink. We cannot directly measure your happiness, but we can measure how much money you are willing to give up in order to obtain coffee or bagels. If we assume you choose wisely, we can thus determine that your happiness increases when you decrease your amount of money by that amount in exchange for increasing your coffee consumption. Thus money is a (poor) measure of happiness or utility.

Money is also a nice quantity because it is conserved---just like energy! You may gain or lose money, but you always do so by a transaction. (There are some exceptions to the conservation of money, but they involve either the Fed, counterfeiters, or destruction of cash money, and we will ignore those issues.)

In this problem, we will assume that you have bought all the coffee and bagels you want (and no more), so that your happiness has been maximized. Thus you are in equilibrium with the coffee shop. We will assume further that you remain in equilibrium with the coffee shop at all times, and that you can sell coffee and bagels back to the coffee shop at cost.*

Thus your savings $S$ can be considered to be a function of your bagels $B$ and coffee $C$. In this problem we will also discuss the prices $P_B$ and $P_C$, which you may not assume are independent of $B$ and $C$. It may help to imagine that you could possibly buy out the local supply of coffee, and have to import it at higher costs.

1. The prices of bagels and coffee $P_B$ and $P_C$ have derivative relationships between your savings and the quantity of coffee and bagels that you have. What are the units of these prices? What is the mathematical definition of $P_C$ and $P_B$?

2. Write down the total differential of your savings, in terms of $B$, $C$, $P_B$ and $P_C$.

3. Solve for the total differential of your net worth. Your net worth $W$ is the sum of your total savings plus the value of the coffee and bagels that you own. From the total differential, relate your amount of coffee and bagels to partial derivatives of your net worth.

Author Information
Raising Physics to the Surface
Keywords
Thermo Derivatives
Learning Outcomes