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When interpreting the meaning of the derivative in real phenomena, it may seem that the interpretation is in conflict with the definition of the derivative itself. The confusion is caused by the units of measurement. We measure continues quantities using discrete units. For example, every unit of measurement of time (e.g., hour, minute, second, etc.) is discrete. To see the confusion that may arise, suppose the distance at time $t$, of a small object moving along a straight line, measured from a conveniently chosen origin, is given by $f(t)=t^2+2t+1$, where distance is measured in meter and time is measured in second. Then $f^\prime(3)=8$ is interpreted by saying that at $3$ seconds after the object started to move its instantaneous velocity is $8$ meters per second. This may become confusing, because one may think that one second (as in per second) does not correctly capture the nature of the definition of $f'(3)$ that is defined as: $$\lim_{t\rightarrow3}\frac{f(t)-f(3)}{t-3}.$$
Additionally, since the instantaneous velocity of this object at $t=3$ seconds is also $0.008$ meters per millisecond, a student may ask why isn't $f^\prime(3)=0.008$?

So, what is an appropriate way of explaining the meaning of $f^\prime(3)=8$ in this context that avoids this confusion? Is it better to say $f^\prime(3)=8$ means: if we assume that at $t=3$ seconds the velocity of the object suddenly remains constant, then it will move 8 meters every second?

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    $\begingroup$ I think the issue has to do with distinguishing between some "real" application and a mathematical model for that application. Indeed, if one had a bag that was originally known to contain a total of 6 marbles and you decide to count the marbles (as a check on the number of marbles), do you worry about all the marble molecules that rub off between the original assertion that there are 6 marbles and the later count that there are 6 marbles? (I'm trying to give a really silly example for emphasis.) $\endgroup$ Mar 31, 2023 at 20:30
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    $\begingroup$ @SueVanHattum I have known many students who have difficulty with interpreting the meaning of derivative in a specific context like the example above. They have not asked me this question, but they have also not asked me any other question that could possibly tell me what the difficulty is. So I am wondering if the difficulty they have with interpretation may be because of this. $\endgroup$ Apr 1, 2023 at 20:44
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    $\begingroup$ @SueVanHattum The difficulty actually is better seen when we have a discrete quantity. For example, suppose the total cost (in dollars) of producing $q$ soccer balls per week by a company is given by a function $f(q)$. What is the meaning of $f^\prime(100)=50$? This may become confusing because the company can't make any fraction of one soccer ball (e.g., $100.1$ soccer balls) so how can one talk about $f^\prime(100)$? Examples like this are in Calculus textbooks. The confusion about the unit meter per second can arise, if a student thinks one second is like one soccer ball. $\endgroup$ Apr 1, 2023 at 21:30
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    $\begingroup$ Mahdi: I do lots of work in economic analysis. For the soccer ball production, we would just treat it as a continuous variable. Since the solution we get will be good enough. For one thing, we are probably interested in modes of operation, where the fraction diminishes in importance (e.g. over several weeks) or for that matter even the rate is probably a statistical abstraction and actual production has some random variation (probably higher than one ball/week). But the answer we get for pricing is still a good one to go with. $\endgroup$ Apr 1, 2023 at 21:43
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    $\begingroup$ You could also think of material (sand draining a glass, or even water draining a tank). Technically this is noncontinuous since individual atoms exist. But at 6.022 times 10^23 molecules/mole (e.g. 18 g of water), I think we can agree the abstraction is irrelevant and the measurement error is much higher than a single molecule of water. I mean...for that matter do you worry about diffraction of golf balls in flight or treat them under Newtonian mechanics? $\endgroup$ Apr 1, 2023 at 21:47

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Shades of Zeno:

https://en.wikipedia.org/wiki/Zeno%27s_paradoxes

And I don't think your confusion really has to do with calculus per se, but with the definition of speed (or velocity if we are using vectors). Even in a non calculus based physics, we can look at time/distance/speed/acceleration with various algebraic formulas. Calculus enables us to understand the relation of speed and acceleration and the like, better. But it's not required to understand speed itself.

Constant speed at less than a full time interval:

Per [thingie], does not mean you have to have at least one of them. Like if I do a ten minute trip (to keep it simple, assume constant speed, 10 miles covered in ten minutes), does that mean I didn't have a speed expressible in miles per hour? Can I get out of the a ticket? Or is my speed 60 miles per hour, even though I went less than one?

Speed as a function of time (non-constant):

Like if I drop a rock (in a vacuum), the velocity is -gt.

https://en.wikipedia.org/wiki/Free_fall#Uniform_gravitational_field_without_air_resistance

In this case, the speed is not constant, is changing. But that doesn't mean it doesn't have an actual value at certain points.

Or consider driving. Unless you are doing a long trip on the highway, there's a good chance your speed is non-constant the whole time. But that doesn't mean there wasn't an instantaneous speed. Just look at the dashboard.

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  • $\begingroup$ Thanks for your response. You are correct, the confusion is not with calculus, it is with how to interpret a calculus concept in real phenomena. In particular, what is the meaning of $f^\prime(3)=8$ in the above example? $\endgroup$ Mar 31, 2023 at 22:14
  • $\begingroup$ Consider the dropped ball example. Acceleration is constant versus time. Velocity is linear versus time. And position is quadratic versus time. Yes, a couple microseconds after 3 seconds, the speed will be a little different. But that's how continuous (ly varying) functions work. Even just algebraically. Again, I think you are just getting trapped up in the Zeno paradox of motion, maybe a slightly different version of it. $\endgroup$ Mar 31, 2023 at 22:18
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    $\begingroup$ Um...in your example, if speed is 8 m/s at time 3, then it is EXACTLY .008m/millisecond. Not approximately. Exactly. That's how unit conversions work. Regardless if it's an instantaneous speed (i.e. a speed that is varying over time) or constant. 8 meters/second times (second/1000 milliseconds) =.008 meters per millisecond. $\endgroup$ Mar 31, 2023 at 22:35
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    $\begingroup$ You are correct. I see where the confusion was. I have edited the question. But you have answered my question. $\endgroup$ Mar 31, 2023 at 23:23
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Chunky vs. smooth

In mathematics education literature, the concept of an instantaneous rate of change has been described in terms of "chunky images of change" vs. "smooth images of change," borrowing the language of Castillo-Garsow et al.

Chunky images of change involve imagining change as occurring in completed chunks. Two features characterize chunky images of change: a unit chunk whose repetition makes up the variation, and the lack of an image of variation within the unit chunk. Therefore, ongoing change is generated by a sequence of equal-sized chunks, and this makes measuring change essentially about counting how many chunks have occurred.

As you state, students may interpret "8 meters per second" to mean that the smallest unit of variation is one second. This is a chunky image of change. In contrast,

Smooth images of change involve imagining a change in progress. Ongoing change is generated by conceptualizing a variable as always taking on values in the continuous, experiential flow of time.... Smooth images of change are not the same as chunky images of change cut up really small. Smooth images of change involve an entirely different conceptualization of variation.

In this paradigm, there is no unit of variation at all. An object moves continuously, and at each instant, it has a speed. One way to help students envision this is, as @guest philsopher also suggested, to think about the speedometer of a car. As the car accelerates, the speedometer measures the instantaneous speed of the car as it varies over time. "The instantaneous speed of the car is 8 meters per second" means that the speedometer is pointing at 8 m/s (or 28.8 km/h or 17.9 mph.)

Real phenomena and calculus

Here's a metaphor that the authors use to explain how images of change relate to "reality."

A child throwing a ball through a paper wall sees a continuous motion and knows that the ball collides with the wall. A traditional animator animating a ball being thrown through a paper wall draws frames, and it's possible that the ball never overlaps with the wall in any frame.

Now, imagine the child watching the animated movie. Drawing from her own physical experience, the child might imagine the ball colliding with the wall. However in the reality of the animator, the ball never collided with the wall, because none of the frames he drew show the ball in contact with the wall. This difference in the child’s reality (relating the ball to physical experience and perceiving the movie as continuous motion in progress) and the animator’s reality (individual frames in which there is no motion) is analogous to the difference between smooth and chunky images of change.

We also note that the “true” nature of the reality itself does not matter. The child uses smooth thinking in both the real ball and animated ball cases by imagining the ball in continuous motion. Similarly, the animator, who imagines the animated ball as individual frames, could also imagine the real ball in frames by imagining change at the molecular, atomic, or quantum level. Images of change are conceptualizations and are thus in the mind of the beholder.

The physical world is full of quantities which students perceive as varying smoothly. This experiential knowledge is a basis for the mathematical concept of instantaneous rate of change. Accordingly, Castillo-Garsow et al. give this suggestion for teaching calculus.

It was only with the advent of modern analysis that a fully chunky (epsilon-delta) calculus was possible—a calculus in which limits are characterized in measurable intervals (chunks) without necessitating an appeal to the non-chunky idea of “infinitesimal” or ideas of smooth motion such as “approaching” in proof. Still, during the development of this calculus, mathematicians relied on smooth thinking to develop intuitions and conjectures before writing the formal, chunky proofs. Because we ask students to reason about change and rate long before they study analysis, it seems reasonable that beginning with smooth images of change could create the foundations for students’ consideration of the difficult-to-learn concepts of limit, rate of change/differentiation, and accumulation/integration.

Units and functions

What to do if a student asks "Why isn't $f'\!(3)=0.008?$" is an entirely different question that could merit an entire article. My answer is that when we write

$$f(t)=t^2+2t+1,$$

we've omitted the units of the coefficients, obscuring the unit-dependence of the equation. To make the unit-dependence explicit, we can write

$$x=(1\,\text{m}/\text{s}^2)\cdot t^2 + (2\,\text{m}/\text{s})\cdot t+1\,\text{m}.$$

By changing the units of the coefficients, we can see that this is equivalent to

$$x=(10^{-6}\,\text{m}/\text{ms}^2)\cdot t^2+(2\cdot10^{-3}\,\text{m}/\text{ms})\cdot t+1\,\text{m}.$$

Now, we can calculate

\begin{align*} \left.\frac{dx}{dt}\right|_{t=3\,\text{s}} &= 2\cdot(10^{-6}\,\text{m}/\text{ms}^2)\cdot (3\,\text{s})+(2\cdot10^{-3}\,\text{m}/\text{ms})\\ &= 2\cdot(10^{-6}\,\text{m}/\text{ms}^2)\cdot (3\cdot10^3\,\text{ms})+(2\cdot10^{-3}\,\text{m}/\text{ms})\\ &= 0.008\,\text{m/ms}. \end{align*}

The velocity of the object at $t=3\,\text{s}$ is indeed $0.008\, \text{m/ms}$. Unfortunately, this does not mean that $f'\!(3)=0.008$.

Note that, interpreted as $\mathbb{R}\to\mathbb{R}$ functions, the two expressions for $x$ are not equal. We have \begin{align*} f(t)&=t^2+2t+1,\\ g(t)&=0.000001t^2+0.002t+1, \end{align*} and using function notation, the student's idea would have to be written as $g'\!(3000)=0.008$.

Thus we can see that $\mathbb{R}\to\mathbb{R}$ functions and modern function notation fail to capture how functions are often treated in the sciences: quantities that vary in response to other quantities, which is the original meaning of function. Using Leibniz notation, "the velocity at time $t=t_0\!$" can be represented independent of units as $$\left.\frac{dx}{dt}\right|_{t=t_0}$$ However, if we use our two functions $f$ and $g$,

  • $f'\!(3)$ is the velocity in meters per second at $t=3$ seconds.
  • $g'\!(3)$ is the velocity in meters per millisecond at $t=3$ milliseconds.
  • $g'\!(3000)$ is the velocity in meters per millisecond at $t=3$ seconds.

This is why many equations in physics are written with variables and parameters instead, i.e. $$x=at^2+v_0t+x_0.$$ See also this post and discussion over on MathOverflow.

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    $\begingroup$ I love the speedometer. I tell my students that I think it probably used to be harder to imagine speed this way, before people used speedometers. $\endgroup$
    – Sue VanHattum
    Apr 1, 2023 at 19:14
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    $\begingroup$ @SueVanHattum Me, too. Then I ask, how do they think the speedometer measures the instantaneous speed of the car? $\endgroup$
    – user1815
    Apr 1, 2023 at 22:32
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    $\begingroup$ Hee hee. Me too. I thought it was a non-instantaneous measurement, like how many revolutions of something per 1/10th of a second, maybe. But it's actually analog (and instantaneous). Something to do with faster pushes something magnetic (?) further away or closer. (I clearly don't know these things from personal experience.) $\endgroup$
    – Sue VanHattum
    Apr 1, 2023 at 22:36
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    $\begingroup$ "the non-chunky idea of 'infinitesimal'" -- somehow, I always thought of infinitesimals as chunks. In fact, that seems their attraction. Different people can imagine things differently, I suppose. $\endgroup$
    – user1815
    Apr 2, 2023 at 1:53
  • $\begingroup$ @Raciquel I understand what you mean! I'm not sure in what sense the authors were using "non-chunky," but my idea is that infinitesimals allow us to combine a smooth image of change with chunky mathematical operations. For example, there is the metaphor of "zooming in" on a smooth curve until it becomes chunky at an infinitesimal scale to see the slope $dy/dx$. $\endgroup$ Apr 2, 2023 at 2:35
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So, what is an appropriate way of explaining the meaning of $f′(3)=8$ in this context that avoids this confusion?

I think this confusion can't be and should not be avoided, because it's at the very base of the concept of the derivative.

Feynman elaborates on this point in his "Lectures on Physics" in the chapter about motion, which is freely available here

https://www.feynmanlectures.caltech.edu/I_08.html#Ch8-S2 .

Main point:

Calculus was invented in order to describe motion, and its first application was to the problem of defining what is meant by going “60 miles an hour.”

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