# Related Rates in Calculus

I’m about to teach Related Rates in my Calculus class. And the book and the Internets aren’t helping me. Supposedly, related rates are so important because there are so many “real world” applications of it.

Like a  snowball melting, a ladder falling, a balloon being blown up, a stone creating a circular ripple in a lake, or two people/boats/planes/animals moving away from each other at a right angle.

Weird exemplars — I wonder where they got started and why they still hold so much water in every textbook? Because seriously?!, a ladder sliding down a wall — when is anyone truly going to need to know the rate of change of the angle over time? Same with the melting snowball.

I’m not someone who needs a real world application to justify everything I teach. In fact, I rarely do. But when we’re teaching something and hold it up as “calculus in the real world,” I refuse to believe that this is the best we can come up with.

I am searching high and low for one true real world problem. No contrivances, but something where I can point to and say: “this calculation needed to get done and because it was, we now have ____.”

I am thinking that maybe figuring out how a radar gun calculates the speed of a car, especially if it is being used from a moving car, might have something good there.

So far, though, the closest I can get is here:

Rockets: A camera is mounted at a point so many feet from a rocket launching pad. The rocket rises vertically and the elevation of the camera needs to change at just the right rate to keep it in sight. In addition, the camera-to-rocket distance is changing constantly, which means the focusing mechanism will also have to change at just the right rate to keep the picture sharp. Related rates applications can be used to answer the focusing problem as well as the elevation problem.

A number of AP Calculus classes have their students make videos with related rates problems. But those problems are just like the others: contrived. It’s like using integration to do simple addition. This video is the exception; I love it.

Anyway, holla below in the comments if you got anything.

1. I went through the same crisis this semester and I ended up choosing the exact same problem as the only one that seemed reasonable (I found it in Stewart 4ed., problem 31 in the section on related rates). I wish I had a really sage piece of advice for you, but all I can offer is empathy.

2. Matt E says:

Wow, my very first thought was a radar gun–and then you go and mention it! Well I’m fresh out of ideas. :-)

The Lengthening Shadow: The Story of Related Rates
by Bill Austin, Don Barry Phillips, David Berman
Mathematics Magazine, February 2000, Volume 73, Number 1, pp. 3–12

4. samjshah says:

Dave, thanks. This is exactly what I was looking to find out. I actually studied the history of science (and a bit of math) in grad school, so this has more interest to me than just of a math teacher. PERFECT-O!

I’ve always liked related rates myself, but having some realistic problems to mention would be lovely.

6. The link to video at the bottom of the post is apparently broken.

1. Argh! And I don’t even remember what it was (I wrote this so long ago!)

7. Farah says:

Hi. I totally share your frustrations, although this is an old post. I remember a keen student asking me why you might want the instantaneous rate of change and why average rate wouldn’t suffice for things like water filling a pool etc. I have questioned since when in. true real scenarios, an instantaneous rate of change is actually necessary, and why average rate of change over a small interval simply isn’t good enough…thoughts?

1. Nope… no thoughts… Sadly… I want someone to come up with a book on “calculus used in various disciplines” and be honest/real and useful for a high school teaccher. Maybe one is out there, but I don’t know. If it is, I haven’t found it.

1. I can’t help with related rates examples, but in electronics it is very common to look at derivatives of current or voltage (the current into a capacitor is proportional to the derivative of the voltage across the capacitor, the voltage across an inductor is proportional to the current through the inductor). It is common to look at sinusoidal inputs to electronic systems, and the resulting analysis cannot be done with averages—you need to look at the phase and amplitude change of the sinusoid. These examples may be beyond first calculus classes in high school, because the students haven’t had the physics of capacitors and inductors yet. (It is also much easier to do everything with complex numbers and exp(i w t).)

2. Ah yes I remember doing all these with complex numbers. Thank you. Even though these “may be beyond first calculus classes in high school” I think it would be awesome to have a book with tons of these things in it, that would make sense to a calculus teacher so: (a) they can go through some of the uses with their classes and show them actual uses and (b) if they want to make an interdisciplinary unit that teaches some of the background material, they could. But mainly for (a). SOMEONE MAKE ONE!

3. Sam, I have a similar goal. I teach the first couple semesters of calculus and it’s always a diverse set of students. Premed, bio, computer scientists, physics, engineering and a couple others. Ideally the materials would allow a number of different tracks that students could identify with. For any given unit there might be five homework sets and a student could choose a biology themed set and work those problems. The sets wouldn’t be identical but the each set ought to capture/cover the same ideas. Along with those would be some bank of context-less questions that are optional if students want to practice the mechanics.

There’s a lot of overhead in introducing real, even simple, models from the sciences. For the teacher resources there needs to be not only summaries and solutions, but extensive bibliographies. Different teachers will be attracted to different models and the resource should supply a lot of choices.

I think differential equations offer a lot of possibility. While they’re usually taught in a fourth semester (aside from possibly a chapter in calculus II) they are full of problems. A single realistic and accessible model can be used for lots of different topics. Developing the motivation for dP/dt=rP as a model for growth takes time. Are there any equilibria? Are they stable? Slope fields. These are all questions that don’t require any derivative rules and focus on graphing and thinking about the meaning and context of the model. What are it’s shortcomings? How can it be improved? What does it mean to say that this is a model of a population? Can we use this model to make predictions and check those predictions against existing census data? What are the units of dP/dt? What are the units of r? What is the affect of adding a constant, dP/dt=rP+H? What are the units of H? What are possible meanings, in the context of the model of H? What effect, if any, does H have on the equilibria? What other functions can we add/subtract from rP to create other models?

This is conceptually very heavy and students don’t have any safety lines to hold onto (algorithms for computing) and I’m still trying to figure out the pacing and support to give them to develop the skills to work on these questions. These questions are robust, they scoff at TI-89s and wolframalpha.com. Though those tools can be brought in later to help. Population dynamics and ecology have been the areas I’ve had the most success with so far. Students are usually not familiar with the results (so they can’t “remember” them from chemistry or physics etc) yet they have lots of content knowledge about animals, predation, basic ecosystems. There are plenty of decent textbooks as well. Math is increasingly found in biology and ecology but the students tend to shy away from it more so than traditional hard sciences like physics and chemistry, as such the text books can be more user friendly for teachers looking for content and examples. Also, textbooks in these areas, unlike calculus books, are full of useful references to primary sources.

it’s not hard, for example, to work your away from the simple exponential growth model to logistic growth and then logistic growth with an added functions to mimic hunting, harvesting (think fisheries) or competition between species. These models, while still very simplified, are taken seriously by the scientific community and have affect on legislation. How much can we fish or hunt, how do we come up with sustainable models of consumption? While those questions won’t be answered in a beginning calculus course, it should become clear that those questions could be answered by other people and those people will need calculus, but not calculus in the sense of the chain rule or integration by parts (which is often done much better with a computer) but calculus in the sense of the strong conceptual tools for modeling change and the qualitative reasoning skills to discern the strengths and weaknesses of the model.

In summary. My own goal (slowed recently by becoming a dad :) is to create an inventory of real published models and to figure out how to cover the topics required in my department by using these models over and over and over again. How many different questions can we come up with on a model and to organize these models into groups that students can choose from. It’s a massive reorganization to be sure but I finally feel like there’s a way to make this material meaningful in 2014 to such diverse audiences. I hope these comments aren’t too long!

2. Farah, interesting question. I think it’s mostly convenience. Averages of functions are messy to deal with while derivatives are easy to calculate (consider the derivative of s(t)=Aexp(-gt)sin(wt) for a oscillating system with dissipation, the average value of this function over a small arbitrary interval delta-t is a far messier expression than the derivative). Derivatives give us local expressions about a phenomena and then we use integration to provide global predictions. It certainly depends on the scenario, but the mathematical sciences rely on derivatives as much as they rely on real numbers instead of rational numbers, even though every scientific measurement is rational. So in a sense we could just toss out irrational numbers, but they turn out to be incredibly helpful to understanding the world even though we’re not sure if the world is actually a continuum.

As for related rates: I share the frustration. Though calculus books are rife with this problem. Even if a problem seems more realistic there is seldom any source for further investigation. My best ‘cop-out’ explanation to my students in such situations is that these are “modeling calisthenics.” In the sciences you often need to develop a great deal of content knowledge before you can begin creating mathematical models. Given the diverse audience it will generally be hard to find situations that are both meaningful and widely accessible. Calculus simply isn’t everyday mathematics, it was built for science. Melting snowballs and growing piles of sand are silly, but also simple enough to be captured by the shared experiences of the class and their developing mathematical skills.

However, it does pain me to cover them. I’m slowly sifting through texts in mathematical social sciences, mathematical biology/ecology etc looking for basic ideas that could potentially be driven by a supported classroom discussion. There are not many of these and once we find them we’ll have to really wring them dry and use them over and over at every opportunity, but it’s worth a shot.

3. @becauseofbeauty: This sentiment made me giddy.
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Ideally the materials would allow a number of different tracks that students could identify with. For any given unit there might be five homework sets and a student could choose a biology themed set and work those problems. The sets wouldn’t be identical but the each set ought to capture/cover the same ideas. Along with those would be some bank of context-less questions that are optional if students want to practice the mechanics.
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I want this to exist!

8. Kathleen W. says:

So I’m the only person who actually thinks that the ladder problem is pretty cool? To me, it’s a basic system of two variables that you don’t need a lot of real-world expertise to understand, just some very simple geometry, and it shows how the rates are “related” because the variables are. I don’t need to understand any biology or economics to get that.