12 Chapter 12 Exercises

Exercise 12. 1 “The global rise in CO_2_ has nearly leveled off.” — BBC News, Nov. 6, 2025

The meaning of the above sentence seems plain. Or is it?

Which of these do you think is closest to the meaning of the BBC quote?

ml1

The BBC news item is based on measurements of CO_2_ at the Mauna Loa observatory in Hawaii from March 1958 through October 2025. The data are available here. The graphic below shows the average measurement each year. The data have been annotated with “slope roses,” a visual key for measuring a rate of change. (The units are ppm-per-year.)

What is the rate of change of CO_2_ concentration with respect to time in 1970? Pick the closest answer.

If the rate of change in (2) had held from 1970 through all the intervening time up to 2025, what would the current CO_2_ concentration be? Pick the closest answer.

In year 2000, what was the rate of change in CO_2_ concentration with respect to time?

#| label: ml4 #| inline: true 1. 0.5 ppm/year 2. 1.0 ppm/year 3. 1.5 ppm/year 4. 2.0 ppm/year [correct] 5. 2.5 ppm/year


5. Estimate the rate of change in CO_2_ in 2020-2025.


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:::


6. Explain, as best you can, how the BBC quote is reflected in the actual data.

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:::
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File ID: rabbit-fight-ring

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---
id: "bear-pay-fork"
created: "Tue Oct 14 10:55:26 2025"
attribution: TBA
---





::: {#exr-bear-pay-fork}
To illustrate, consider @fig-nhanes-height which shows height versus age for a sample of about 5,000 people in the US. To guide the eye, a smooth curve has been drawn through the data to show the average height at each age.

::: {#fig-nhanes-height}


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![](Exer-12-rates-of-change_files/figure-html/unnamed-chunk-5-1.png){width=672}
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Height versus age for a sample of about 5,000 people in the US. The smooth blue curve has been added to guide the eye to the average height at each age.
:::

[[MAYBE PLOT OUT THE GROWTH **rate** for each person, to contrast it with the rate of change, which instead changes with time.]]

It pays to look carefully at @fig-nhanes-height to understand the different aspects of the graphic. First, each small, black dot gives the height and age of a single individual. There are 5000 dots corresponding to the 5000 different people in the sample. In contrast, the blue curve shows *aggregate* features across all 5000 people. Only a handful of individual dots fall on the curve. The curve indicates the average height for each year. 

Each individual person appears only once @fig-nhanes-height. It is impossible, therefore, to look at the change in height with respect to age for an individual. 

HAVE THEM CALCULATE THE RATE OF CHANGE AT DIFFERENT AGES.
:::
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File ID: bear-pay-fork

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---
id: "bear-sing-closet"
created: "Wed Nov 12 13:02:44 2025"
attribution: TBA
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Turn this material into exercises 

::: {#exr-bear-sing-closet}

## Quantities and functions (snapshots and movies) {#sec-snapshot-movie}

We start by turning the tables to show how a *rate* is a special case of a *rate of change*. The context to illustrate our discussion will be *prices*: why they are what they are.

In many cultures, such as those in rich countries, consumers almost always deal with prices as they are. We are used to looking at a price tag (or its electronic equivalent) then judging whether the price suits the value of the thing being considered and making a decision to buy or pass. But there are exceptions, for example coupons. Most of us have encountered another exception: haggling over a price. (Most professionals prefer a more dignified term than "haggling": *negotiating*.)

Two common situations where there is not a price tag are air travel and university education. When shopping for an airplane ticket you are likely to encounter many different prices for similar or identical services. We'll use that situation as our first example of the application of rates of change.

First, it's important to identify an intuition held by many or most people about *price* and *value*. People tend to think that a given object or service has a corresponding value. Philosophers of economics such as Adam Smith or Karl Marx offered a "theory of value" and attempted to calculate values by looking at the inputs required to produce the object.^[Although Smith and Marx are regarded as occupying opposing extremes of politics, their theories of value are very similar and reduce the value of *all* inputs into the labor required to generate those inputs: the "labor theory of value."] Let's use such a theory to try to understand airline ticket prices.

The service provided is, more or less, transporting people from location A to B. To provide that service, the airline needs to purchase or lease an airplane, pilots, stewards, support personnel, and, importantly, fuel. Imagine that the inputs needed to provide the service of a single flight from A to B cost $100,000. Naturally, this isn't the service price because there are many passengers. According to a simple theory of value, the ticket price should be a rate: the cost of providing the single flight divided by the number of passengers. If there are 100 passengers, the ticket price should be around $1000 per passenger.

Notice the units of ticket price: dollars per passenger. That's the clue that we are dealing with a rate. By convention, ticket prices are quoted in terms of money, say 1000 dollars. But that convention obscures where the price is coming from. 

We can draw a picture of the situation, as in @fig-ticket-price-1.


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![](Exer-12-rates-of-change_files/figure-html/fig-ticket-price-1-1.png){#fig-ticket-price-1 width=672}
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To calculate the ticket price, we divide total cost by the number of passengers. On the graph of cost versus passengers, the ticket price is the *slope* connecting *two* points. For the right-most point, we need data: the cost of providing the service and the number of passengers. But for the left-most point, we do not need any data: the position will always be (0,0). In terms of the photography analogy, we take a snapshot of the situation when there are 100 passengers. But we can equally well see the situation as comparing two frames of a movie: a frame taken at zero passengers and a frame taken at 100 passengers. The ticket price, which we described as a *rate*, is also a *rate of change*. Indeed, every rate can be seen as a rate of change, comparing one frame at the actual coordinate (cost = $100000, passengers = 100) configuration and a second frame at the fictitious (cost = 0, passengers = 0) value.

@fig-ticket-price-1 looks like the graph of a function: cost as a function of the number of passengers. At first, this may seem unrealistic. The cost of providing the service is fixed, it doesn't vary (much) with the number of passengers.

There are a few reasons why it's worthwhile to conceptualize cost as a function of passengers. Here are three, but there are more:

- The airline could replace the aircraft with a smaller one, for instance a 40-passenger "commuter" plane. This would considerably lower the cost.

- The airline still has to pay for the plane, even if its not being flown. So the additional cost of actually flying is only a fraction of the total cost. 

- The plane is needed for additional services. In the simplest case, the plane is needed for the return flight from B to A. Even if there are few passengers flying on the A to B leg, there might be many more people who want to fly from B to A on the particular day of the week. 

Since this isn't the place to get into details of the airline business, let's focus on just one issue: the passengers.

Some passengers are willing to pay quite a lot for the flight. This includes some business passengers who don't pay directly for the flight and also those people who have so much money that they aren't concerned with the cost. These are the people who you walk past in first- and business-classes when you board the plane to get to your seat far in the back near the bathrooms. For the sake of discussion, let's imagine that there of 10 passengers who are willing to pay $5000 for the flight A -> B and another 15 who are willing to pay $2000. Such passengers are rewarded by more space and comfort as well as other status symbols. (Airport gate agent: "First we open boarding to the Emerald Class passengers.")

A similar story goes for another class of passengers: the people whose schedules are uncertain but pressing. Such passengers want a refundable ticket. This allows them to reserve several flights just in case, but pay for only one flight. For our example, let's put this at 10 passengers who payed $1500 for the refundable tickets, but ended up on our A -> B flight.

These three hypothetical types of passengers let us fix points on the graph of cost versus passenger numbers. We'll arrange them in the order they are likely to board the plane: First class, business class, coach but refundable.

Class | Number | Ticket price | Revenue from class | Cumulative passengers |  Cumulative revenue
------|--------|--------------|---------------|---------| -------
First | 10     | $5000 | $50,000 | 10 | $50,000
Business | 15  | $2000 | $30,000 | 25 | $80,000
Refundable | 10 | $1500 | $15,000 | 35 | $95,000


::: {.cell}
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![](Exer-12-rates-of-change_files/figure-html/fig-airline-curve-1.png){#fig-airline-curve width=672}
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@fig-airline-curve shows the total revenue from tickets versus the number of tickets. There is one segment for each class of passenger. The vertical change of each segment is the revenue generated by the corresponding class. The horizontal change for each segment is the number of passengers in each class. The *slope*---that is, the *rate of change*---is the ticket price for each class. 

The blue dashed line shows the situation for the remaining 65 seats. To reach the $100,000 in revenue, only an additional $5000 is needed from those seats. The fare for those seats is the slope of the blue segment: $77 per passenger. Naturally, the airline prefers to make a profit. The $77 per passenger is the minimum fare in order to break even. Suppose they charge $250 per passsenger for the blue seats, a quarter of the price compared to @fig-ticket-price-1. The total revenue raised will be $111,000, providing room for $11,000 in profit while still giving passengers a low fare.


[[In-class discussion or example: College tuition?? pony-draw-laundry]]

:::
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File ID: bear-sing-closet

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id: "ash-ride-window"
created: "Wed Nov 12 13:04:12 2025"
attribution: TBA
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::: {#exr-ash-ride-window}
Another simple value statement compares flights. Suppose location C is twice as far as B from A. The cost of providing the service A -> C will be higher. Common intuition often suggests that the ticket price for the A -> C flight will be about twice that of B -> A.

:::
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File ID: as-ride-window

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id: "rhinosaurus-buy-screen"
created: "Wed Nov 12 13:05:08 2025"
attribution: TBA
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::: {#exr-rhinosaurus-buy-screen}

In [Chapter 11](Chap-11-dynamics.html#sec-dynamics) we referred to the *state* of a dynamical system. It would be more accurate to use the term  "**instantaneous state**" for the purpose; the state at each particular instant of time. 
It's intuitive to think about the instantaneous state as the system as you would see it in a photograph, frozen in time. The inventors of the mathematical methods behind the modeling of dynamical systems were [Isaac Newton and Gottfried Willhelm Leibnitz](https://en.wikipedia.org/wiki/Leibniz%E2%80%93Newton_calculus_controversy) in the decades around 1700. But photography was not invented until around 1850 so they could not draw on this intuition. They developed their methods using the tools of geometry and algebra.

Although it's intuitive to think of the instantaneous state by analogy to a photograph, this analogy is critically flawed. To see why, consider the photo of a pendulum clock in @fig-pendulum-photos(a). It's easy to see the *position* of the pendulum, but you can't tell from the photo which way the pendulum is swinging. The instantaneous state of the pendulum involves *both* the position *and* velocity. The photograph doesn't capture the *velocity*. 

::: {#fig-pendulum-photos}
#| layout-ncol: 2
#| fig-cap: "Photographs of a pendulum clock. Source: (a) shutterstock.com#193799621; (b) stock.adobe.com/search?k=pendulum+clock "
#| fig-subcap:
#| - Still photo
#| - Time exposure

![](www/clock-pendulum-still.png)

![](www/clock-photograph-time-exposure.png)
:::

@fig-pendulum-photos(b), a time exposure, gives a better sense of the velocity, but it would be very difficult to measure the velocity at any instant from the blurry view of the pendulum.

About 200 years after Newton and Leibniz, another technology was invented: the *motion picture*. This technology involves taking a *sequence* of still photographs separated by small intervals of time. Each of the photographs is called a **frame**. The frames are typically separated by 33 milliseconds. 

::: {#fig-clock-video}



Video of a pendulum clock. Set the playback speed to the slowest possible to be able to discern the individual frames. 

:::

You can measure the pendulum's position from a single frame, but to find the velocity you need *two* frames. Measure the position in each of two successive frames. You *calculate* the velocity by taking the change in position between the frames and dividing by the change in time (e.g. 33 milliseconds). Picking up on the language in [Section 3.5](Chap-03-rates.html#sec-rates-of-change-brief), velocity is the *rate of change* in position with respect to time.

Because a *calculation* is required to measure the *rate of change*, the mathematical methods came to be called **calculus**. Courses in calculus are usually presented in terms of algebra, which makes them inaccessible to many students. But the methods can also be represented arithmetically. We will emphasize the arithmetic approach.

@sec-rates-of-change-brief introduced the notion of *rate of change*. The situation addressed by rates of change involves two quantities. One is the *primary quantity*, typically a quantity that is of particular interest to us. The other is the *context-setting* quantity. We take two snapshots of the system that relates these quantities. We'll call these snapshots **from** and **to**.

If we label the primary quantity as ${\cal P}$ the two snapshots give us two values for $\cal P$, which we'll write as ${\cal P}_\text{from}$ and ${\cal P}_\text{to}$. The *change in $\cal P$* between the two snapshots is 
$${\cal P}_\text{to} - {\cal P}_\text{from}\ .$$

The context-setting quantity---we'll call it $\cal C$---similarly has a value in each of the snapshots. The change in $\cal C$ is $${\cal C}_\text{to} - {\cal C}_\text{from} .$$

The *rate of change of ${\cal P}$ with respect to ${\cal C}$* is a conversion factor that tells us how to convert a change in ${\cal C}$ into the corresponding change in ${\cal P}$. But a rate of change is a different kind of beast than the conversion factors we examined in @sec-quantity-framework when looking how to change one unit into another unit---say feet to meters, or Euros to USD. Those @sec-quantity-framework were *dimensionless* because the *to* and *from* quantities always have the same dimensions and are "flavors of one." But a rate of change is different: the *primary* and *context setting* quantities need not (and generally don't) have the same dimension. A *rate of change* is a conversion *between* dimensions. We go from the dimension of the *from* quantity into the dimension of the *to* quantity.



For the sake of discussion, let's image a system that uses several different factors to account for a person's *mood*. 

```{mermaid}
graph LR
  A(Day of year)
  A --> C[Sun's brightness]
  A --> D[Length of daylight]
  D --> J(Change in length of daylight)
  E[Work-related stress]
  F[Romantic situation]
  L[Family situation]
  G[Concern about global issues]
  K[Personal stress]
  H[Mood]
  D --> H
  C --> H
  E --> H
  G --> H
  J --> H
  F --> K
  L --> K
  A --> L
  K --> H
  H --> F
  
classDef exogenous fill:#fff,stroke-width:1px



class A exogenous
class E exogenous
class F exogenous
class G exogenous

Note the four endogenous quantities: Day of year, romantic situation, work-related stress, global issues. None of these have any input arrows. Then there are the links between quantities. We won’t be concerned about most of these, and some are tongue-in-cheek such as the holiday-induced family stress.

Focus on the connections between sunlight and mood. The model has a direct link between brightness, which varies over the year, and mood. There’s also a link between length of daylight—winter short, summer long—and mood. And still another link of sunlight to mood: the change in the length of daylight. At the time of the Spring equinox (in March), the days are getting longer fast. At the time of the Autumnal equinox (In September) the days are getting shorter fast. (In mid-summer and mid-winter, day length changes only slowly.)

Suppose you are building a model for the purpose of studying SAD (Seasonal Affective Disorder), a form of depression associated with the changing seasons. Many SAD sufferers feel the onset of symptoms in the Fall and recover toward Spring.

IN BOTH FALL AND SPRING the sun’s brightness and the length of day are the same, yet SAD onset is associated with Fall, not Spring. WHATS GOING ON? It’s the DERIVATIVE of length of day that distinguishes FALL FROM SPRING.

SHOW GRAPHS OF BRIGHTNESS AND LENGTH OF DAY.

File ID: rhinosaurus-buy-screen

Exercise 12. 2 MAKE THIS ABOUT HOW OUR UNDERSTANDING OF GROWTH concerns the instantaneous rate of change rather than the ratio of height to age.

[[Example: a person’s height divided by their age versus the change of height per year. Maybe use the NHANES height data, fitting a 3rd or 4th order natural spline. Show three graphs: height versus age, height/age versus age, change in height over one year/one year versus age.]]

File ID: sister-make-scarf

Exercise 12. 3 Spotting derivatives in the wild

Stock and flow. START WITH A SIMPLE, physical example: A bathtub filling up with water. Where does the water come from: the faucet. There’s also a loss of water through the drain. Amount of water is a function of time. The rate of change in that function is the flow rate. FOCUS ON THE UNITS.

12.1 File ID: hamster-say-blouse

Exercise 12. 4 Partial derivatives

File ID: owl-take-bulb

Exercise 12. 5 Tragedy of the commons problem.

In the Medieval period, an area of land was held for the common use of peasants or tenants. Typically this was used for grazing cattle.

SKETCH An upside-down U function for productivity versus number of cattle. Find the point at which productivity is maximal. Let’s assume there are 50 tenants sharing the land. Anachronistically, a management consultant advised the community that the productivity would be near optimal if each tenant had one or two grazing animals. (This puts the peak of the U at about 50-100 animals.) One tenant looking at the situation suspects he will do better by adding another animal. This might push the animal population beyond the optimal production point, but only a little. And all of the marginal benefits of the new animal accrue to that one tenant, so from his point of view, the addition is certainly worthwhile. Other tenants see this, and each adds another animal for his or her own benefit. What happens to the production now?

Suppose one tenant is inspired by the motto, “Think globally. Act locally.” This tenant resolves not to contribute to the problem and distains to add an extra animal. Explain whether this eco-minded tenant is rewarded or punished in terms of production for their resolution.

Suppose there were a tax imposed on each tenant for each animal they graze. Tax would be zero for the tenant’s first animal, then go up (per animal) for each additional animal. The tax should be set at that amount that overall productivity is diminshed if every tenant had that number of grazing animals. What should be the tax on the second and on the third animal per tenant?

This setting (without the tax) is called the “Tragedy of the Commons.” With the tax, it is called variously “sensible resource management” or “an infringement on individual liberty by an intrusive government,” depending on the person’s point of view. Think of an example of “tragedy of the commons” that exists in the contemporary world.

12 Chapter 12 Exercises

12.1 File ID: hamster-say-blouse

12.2 File ID: big-sit-room

12.3 In-class project?

AI generated