# Showing applications of calculus to intro students

So I'm wrapping up my second year teaching high school calculus, and my first as an AP course. In addition to review, I'd like to end the year by giving the students a taste of the kinds of real-world applications that they can now start to appreciate now that they have a conceptual understanding of the fundamentals of calculus. Obviously, we've seen plenty of word problems that show how optimation, related rates, and integrals can lead to effective decision-making, but I'd like to go a bit deeper.

Here are a few ideas I have of topics to show them. Can you suggest other topics in the same vein? My list is currently heavy on modeling with differential equations that are easy to conceptualize even if they are hard to solve. Other topics across the calculus curriculum that might engage students would be very welcome.

In rough order of difficulty to comprehend:

• The equation for continuously compounded interest basically comes from realizing that interest is the derivative of a saving account's balance and is proportional to the size of the balance.
• Same thing for radioactive decay being an exponential function, since the change in the number of radioactive ions at any instant is proportional to the total number of radioactive ions.
• Without getting too deep into physics, knowing that the acceleration of a moving frictionless spring is proportional and opposite to its distance from the equilibrium point suggest a sinusoidal motion of the spring since sine and cosine are equal to the negative of their second derivatives.
• The now-familiar shape of time vs. infected people in an epidemic can be inferred from the S-I-R model, which is a nonlinear system of differential equations but still easy to understand where the equations come from.
• Similarly, it is relatively intuitive to understand the Lotka-Volterra equations to model population sizes in a predator-prey dynamic.
• Honestly, I teach nearly all of these examples in my calculus class. I am particularly fond of using spreadsheets to model Lotka-Volterra, springs, and disease spread. For context, I teach at a community college, but most of my calculus students are high school students taking the class for dual enrollment credit. – Xander Henderson Apr 26 at 15:19
• All your examples seem like differential equations, which adds lots more complication beyond what's the main focus of a first course for high school kids on differential and integral calculus. – Ben Crowell Apr 26 at 19:15
• If your students have a bit of background of physics deriving electrical and gravitational fields for objects like a cone, sphere, ring, etc is a nice way to apply calculus. – Aatmaj Apr 27 at 11:11
• I'm actually working on a book on this very topic. If you would like to see a current version and give comments, check out my profile page. – johnnyb Apr 28 at 21:33

I worry about throwing a bunch of application/modeling that relies on diffyQs beyond the AP curriculum. You are combining tougher aspect (applications are harder than straight math) along with new math content. This is sort of what I hate about how PDEs are taught (relatively unfamiliar application, like string tension, plus the new math needed to solve it). In contrast, calculus and ODEs (and lower) seem to take an approach of teaching the equation solving first, before practicing/complicating with applications (i.e. word problems).

I do really like predator-prey though from your list. I would take an approach that emphasizes graphing, observing (maybe a bit of Excel plugging) rather than the terminology of Lotke Volterra. Independent of the math, what I like about this application is the gutty vitality and intuitive understanding of it. Animals eating each other. Thing that's more easy to think about than mechanical or financial topics (and definitely than electrical or chemicals, which are always more "invisible").

An alternate suggestion might be to cover 2nd order constant coefficient ODEs. Used to be part of the standard Calc BC in the 80s. They slightly watered down the class a few years ago.

While you're talking about physics, you might want to mention generally the way physics is computed in video games. The positions and velocities of various objects are updated every fraction of a second, with the new velocity being the old velocity plus acceleration (due to forces) times time and the new position being the old position plus velocity times time. You'll have to gloss over the vector bits though.

You can also talk about Newton's Method for numerically solving equations. This is actually how many computer algebra systems handle numerically solving equations.

More of a comment than an answer but I don't have the rep to post comments. When my AP calc professor was introducing derivatives, we dropped Barbie dolls out of a window and timed their descent lol. We found average speeds that way, and then he asked us if we could determine the speed it was falling at exactly 1 second. There's not a good way to do this without derivatives, which was a fun segue into the concept.

Generalizing more, the relationship of acceleration to velocity to position is always good for explaining derivatives and integrals, and even an intuitive proof for the fundamental theorem of calculus (if you're driving from one location to another, by the time you reach your destination all the tiny velocity * delta time increments summed up end up equaling the distance you traveled, or final position minus initial position [ F(b) - F(a) ]. And the units work out too, since m/s * s = m).

Here are several real-world problems that are mostly accessible to first-year students. Some of them require a little bit of differential equation manipulation, but a minimum amount:

• Geometry (where do equations of volumes of cones / spheres / etc come from)
• Deriving the Interest Rate Equation
• Deriving the Logistic Curve
• Deriving the Kelly Criterion
• Lotka-Volterra (predator/prey) models
• Physics:
• Deriving Kinematic equations
• Deriving kinetic energy equation
• Deriving escape velocity equation
• Deriving the rocket equation
• Deriving Newton's Law of cooling
• Electronics
• Calculating RMS voltage
• Calculating capacitor filling times
• Chemistry
• Several rate equations are based on calculus
• I am working on a book on this topic. If you want to see the current book status and/or provide feedback, check my profile. – johnnyb Apr 28 at 21:41
• Also, I cover almost all of these (except capacitor filling and chemistry) in my first-year high-school homeschool course in which I only have a total of 30 hours all year. – johnnyb Apr 28 at 21:54