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.