I am looking for realistic applications of the average AND instantaneous rate of change, that can serve as an entry point to calculus for students. The main-idea is to show them a (simplified) problem of the real world that needs(!) to be solved by:

  1. Modeling the situation upfront from measurements (Turning measurement into a function and a graph)
  2. Calculating the average rate of change
  3. Calculating the instantaneous rate of change

So I'm looking for a big question to pursue/problem to solve while learning about those three aspects. Every time we move to the next aspect we learn something new to solve this big problem.

What do I mean by "realistic"?

For example, I could create a hypothetical situation including a car and a mountain. I could go on and tell the students that we could figure out the average steepness of the mountain's silhouette and even the precise steepness of a single point of that curve by using derivatives in order to find out whether or not the car will make it up the hill.

Now here's the big question: Is this really a realistic application? The main issues I'm having with this is, that there are too many open questions:

  • Is it really done this way? Why would anyone model a function for the mountain and calculate the rate of change instead of just measuring the angle of the steepest looking part using a spirit level? This hypothetic situation doesn't answer the question why it needs to be solved using calculus.
  • Why would we want to calculate the precise steepness of a single point? Just for the sake of being more precise? Why don't we just stick with an almost as precise approximation by using the average rate of change with a really small interval? Are there or have there ever been any real consequences because someone used such an "approximation"?
  • If the values on road gradient signs are really calculated this way, who is responsible for doing this calculations?

So what I mean by realistic is, that I'm looking for an example of an application, where real persons in the real world really need to mathematically model a situation before they need to calculate the rate of change.

  • $\begingroup$ How about population models where both growth over a period and instantaneous growth are of interest? $\endgroup$
    – Aeryk
    Jan 8, 2016 at 7:03
  • 1
    $\begingroup$ @Aeryk I also thought about using such an example. The problem is, that I actually wasn't able to find out why both growth over a period and instantaneous growth would be of interest to us. I know this sounds silly, but if my goal was to predict the future population size from existing data I would be done after modeling the function that describes the existing set of data, since all I had to do was to insert the year I want to predict for t. $\endgroup$ Jan 8, 2016 at 10:58
  • $\begingroup$ Example: The data for the population size of the US from 1900 to 2000 could be modeled by f(t)=t^2 (Let's just assume this would be realistic for simplicity's sake) whereas "t" stands for the years since 1900. All I had to do, to calculate the value for the year 2015 was to insert 115 for t and I'd be done, wouldn't I? If this is the case I'll have a hard time explaining why we need to learn about the rate of change. $\endgroup$ Jan 8, 2016 at 10:59
  • $\begingroup$ I think something involving climate change with historical data could be of interest. The are some data sets here, for example: sustainabilitymath.org/PfaffCalc.html $\endgroup$
    – ncr
    Jan 8, 2016 at 13:11
  • $\begingroup$ Why a mountain, and not just a trip by car from A to B? Mean speed is simple to compute, instantaneous speed is given by the speedometer. Can even mix in direction, if it is a curvy road... $\endgroup$
    – vonbrand
    Jan 8, 2016 at 14:08

3 Answers 3


I've always found mechanics to be my friend when it came to explaining the necessity of calculus. What about something like a rocket taking off into outer space?

  • The process needs to be modelled up front because of the cost and risk to life involved in a failed mission.
  • The average rate of change needs to be calculated in order to ensure that the rocket gains enough speed to reach escape velocity, otherwise the mission will fail.
  • The instantaneous rate(s) of change need to be calculated in order to ensure that the rocket materials and crew can cope with the stress of acceleration.

Pretty much everyone knows what a rocket is and most people find them at least vaguely exciting. Everyone has at least a passing experience with gravity and acceleration, so the real world necessity is relatively obvious.

  • $\begingroup$ Thanks for this suggestion! This really seems to be a plausible explanation for the necessity of calculating the rate(s) of change. I took this question to space.stackexchange.com (link) to see if this is really done this way. $\endgroup$ Jan 9, 2016 at 16:18

I would appreciate a really good answer to this. Somehow my brain is not up to it.

Basically, as far as I can tell, all real rates are average rates. Every observed change takes place in a time and the magnitude of the change divided by the magnitude of the time gives us the average rate. In an instant, there is no change and the magnitude of the time is zero (or possibly infinitesimal, if you're of that bent). Instantaneity is the result of our imaginations inventing models that are logically consistent. They are fictional, in a sense, but more like historical novels, they sometimes approximate reality.

The reason instantaneous rates are important is that, because of the Mean Value Theorem, derivatives are good approximations to average rates over small intervals of time. And derivatives are easier to think with than finite differences. I don't say they're easier to compute with. That may be true, sometimes. They usefulness is in how easy they are to think about, as well as in their effectiveness at approximating the changes we observe.

The real application to me is how we use them to think about problems. This has led to advances in mathematics. Try to think of all the times the MVT is used. I'm preparing to teach numerical analysis, and it is amazing to see how much of what we understand about effective numerical approximation is based on the MVT. This is real work, done by real people, to develop methods that help others solve problems within specified tolerances.

In other words, the actual real applications are theoretical. And to understand the world and set up its problems in a form to be solved, you need a workable mathematical model. When I was a young person, this was what attracted me into science and engineering, that the mathematics I was learning was giving me a grasp of the world that would allow me to solve some of its problems.

There are many applications of derivatives. In some, an equation with derivatives is set up to be integrated. In others, something, such as energy, is to be minimized. Sometimes, the derivative of the potential function indicates the force on the system. As alluded to before, the derivative may be used to help bound the error from discretization or truncation of continuous processes. In applications outside mathematics, the calculus problem has usually been solved in the form of an algebraic formula to be applied.

Persons who merely carry out the work of calculating the solution, filling out the reports and so forth, they just need good training in their specific tasks. Sometimes you get the training on the job; sometimes you enroll in a certificate program. The ones who need to solve problems need to understand the principles and to be able to set the problem up. One concern is that when you do anything, whether you have the first or second type of job, you're likely to run into problems now and then. If you've worked anywhere for a length of time, you've probably run into people who had trouble solving problems outside their ordinary tasks.

To circle back again to my point. The real application is to use derivatives to think about things. Many things are subject to mathematical formulations. They may be in mathematics. They may be in the sciences. Derivatives may be used to solve problems and advance our understanding in mathematics and the sciences. Or they be used to understand mathematical principles in the sciences, which are just problems that other people have solved.

An example, taken from the real world, but without formulas and so not quite like what the OP seeks: It should be clear to one who knows Newton's second law, force equals the mass times the acceleration $x''(t)$, that turning off the water is potentially dangerous. It never seems to be, but that's because of building codes; so one might be forgiven for never thinking so. A whole column of water, stretching back to who know where accelerates when the taps are opened. That momentum has to be stopped when the tap is turned off. Turn it off quickly and the water has to stop quickly. So the acceleration $x''(t)$ has be large, and thus the force has to be large. So the pipes have to be strong (enough).

I was talking with a civil engineer who had built dams in Iran in the 1950s. He said he never had used calculus on the job. But I was thinking that turning off a valve on a dam must be quite a dangerous thing. He told me about one young engineer who he had told to go shut off a valve. He soon saw the pressure gauge at the station shoot up and get pegged at max. Then it shot down and was pegged at zero. Then up. Finally it went down and stayed down. Then he knew. He went to the window and saw a geyser shooting up from the busted pipe.

Another example: When you're in a tight turn in a car and the driver suddenly straightens out the steering wheel, your body tends to be thrown toward the other side of the car. Why, when the acceleration in the car has changed to zero? It's because of the tension in the muscles in your torso that are exerting a force to supply the centripetal acceleration to your body to keep it in the same trajectory as the car. When the car straightens out, the tension in your body continues the centripetal acceleration, until you relax. So if the third derivative is too great, the acceleration changes faster than your body can react, and your torso tends to follow the extension of the curve that car used to be tracking.

It's much easier to think in these terms than in terms of finite differences (average rates).

  • $\begingroup$ I think this is a very good answer and helps dispel the mythology that the value of learning mathematics is immediate practical application. As you say, "the actual real applications of calculus are theoretical." Without calculus, you cannot understand the language of differential equations, and so you cannot really understand Newton's equations of motion, Maxwell's equations, wave equation, Schrodinger, Navier Stokes, etc. Without this theory, so much of our technology would have never emerged. $\endgroup$
    – user52817
    Jan 9, 2016 at 18:01
  • $\begingroup$ "Without this theory, so much of our technology would have never emerged." --> Then this should be explained to the student by giving examples, right? $\endgroup$ Jan 9, 2016 at 18:13
  • $\begingroup$ We could also rephrase that question: Why was calculus invented? What was to be gained by it's inventors? If it was out of sheer curiosity, what inspired that curiosity? What was the bigger picture behind it? $\endgroup$ Jan 9, 2016 at 18:48
  • $\begingroup$ Examples? Sure, for motivation perhaps, but not as a direct teaching tool. You would not teach young children to read by having them read from James Joyce's Ulysses, or by reading a technical article on NASA Deep Space Network. $\endgroup$
    – user52817
    Jan 9, 2016 at 18:50
  • $\begingroup$ "Sure, for motivation perhaps" --> You're right, that's the exact point of having such an example. It should serve as an entry point for each chapter of the curriculum. It's supposed to answer the probably most frequent question of students: "Why the heck am I doing this? Why should I care?" $\endgroup$ Jan 9, 2016 at 18:58

This follows up on I Stanley's mention of mechanics, and particularly kinematics.

I'm not sure if this works well for you, but, if you're willing to discuss two-dimensional motion (where the direction changes with time), the difference between the average and the instantaneous rates of change becomes obvious.

Basic examples could include circular motion (e.g. a planet's orbit) or a ballistic trajectory. The latter perhaps being a particularly good example, since in the x-axis you have a constant velocity, such that the average explains everything, while in the y-axis, the average velocity is zero, so you must examine the instantaneous change to make sense of the motion.

This also reminded me of Steven Strogatz's wonderful discussion of chase problems in the Pursuit chapter of "The Calculus of Friendship". It is very easy to formulate and explain such examples that arouse curiosity, but require a deep dive into calculus to solve.


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