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.
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 PAGEGoals
- 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 Small Group Activity
30 min.
group Small Group Activity
30 min.
group Small Group Activity
30 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
30 min.
group Small Group Activity
30 min.
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
30 min.
vector calculus coordinate systems curvilinear coordinates
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 Homework
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.
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 \(P_B\)?
Write down the total differential of your savings, in terms of \(B\), \(C\), \(P_B\) and \(P_C\).