# Differentials

## Prerequisites

Students should be able to:

- Determine an ordinary derivative from a symbolic expression.
- Hold variables constant when finding a derivative.
- Sketch graphs and identify points and changes between points on those graphs.

## In-class Content

- QUIZ/Derivatives pretest (10 min)

### Lecture: Differentials of 1D Functions (Lec - 20 min)

Main ideas

- Previewing the Math Bits theme of Variables and Representations
- Teaching differentials, small changes, and zapping with d

##### Lecture (30 minutes)

Ask students to make a sketch of f = 7x^{2} on the big white board.

What are the variables of interest? One idea here is to note that x is not the only variable here: f is also a variable from a physics perspective.

What representations do we have for this relationship? The symbolic equation is one, and the graph is another.

Introduce the differential quantities df and dx as small changes in f and x, respectively. Ask students to add df and dx to their graphs.

Then, ask students how df and dx are related to each other. Students should be able to articulate that this is a derivative.

Is the relationship the same if we choose a different point on the graph (a different initial x)?

Then relate df and dx using the symbolic representation: df = 14xdx.

- The following two steps can be replaced by other activities on the relevant Hour page.
- Give the students f = 5x
^{2}y^{3}. What are the variables? What representations do we have (if time, hand out surfaces and have students make sketches similar to the above)? What does a differential relationship look like? For this last one, use the generic form df = A dx + B dy, and talk about the fact that this is another representation. - Do some specific examples of using the zapping with d strategy (see Zapping with d).

### Activity: Total Differentials on a Surface

Link to Total Differentials on a Surface Activity

**Activity Highlights**

- This small group activity is designed to help students understand differentials in multivariable functions.
- Students work in small groups to generalize symbolic and graphical representations of differentials for a single variable function to a multivariable case using a plastic surface as a test case.
- The whole class wrap-up discussion emphasizes that differentials are a way of linearizing the relationship between variables by expressing how small changes in those variables are related.

### Activity: Evaluating Total Differentials

Link to Evaluating Total Differentials Activity

**Activity Highlights**

- This small group activity is designed to help students understand differentials in multivariable functions.
- Students work in small groups to generalize symbolic and graphical representations of differentials for a single variable function to a multivariable case using a plastic surface as a test case.
- The whole class wrap-up discussion emphasizes that differentials are a way of linearizing the relationship between variables by expressing how small changes in those variables are related.

### Activity: Covariation in Thermal Systems

Link to Covariation in Thermal Systems Activity

**Activity Highlights**

- This small group activity is designed to help students understand relationships between thermodynamic state variables.
- Students work in small groups to investigate states on a raising physics surface, and consider cycles.
- The whole class wrap-up discussion emphasizes the meaning of a “state” variable.

## Homework for Energy and Entropy

- (mbZapWithD)
*Find the differential of special functions.* - (mbDifferentials)
*Finding the differential abstractly.* - (mbDerivVSDiff)
*Find the differential of special functions.* - (mbMathematicaContours)
*Plot contours, Change of Variable.* - (CoffeeAndBagels)
*A thermo-like system. Requires Clairaut's theorem.*