Making Connections in Calculus
One of my favorite things about Calculus is that it extends and deepens much of the content from prior math courses: rate of change from Algebra, area and volume from Geometry, function behavior from Precalculus, optimization from Algebra 2, I could go on and on. I’ll never forget the day I was driving home from school and it dawned on me that point-slope form of a line WAS the Fundamental Theorem of Calculus! I KNEW there was a reason I had such strong affection for point-slope form–and it wasn't just for the ease of not having to solve for the y-intercept.
Although I could talk about this at length (future blog post??), here we will discuss a different connection we can make with previous content. This one has to do with solving a differential equation and the formula for continuously compounding interest.
In Algebra 2, students first learn the well-known formula for continuously compounding interest, and it’s usually written as A=Pe^(r*t), which we often just refer to as “Pert”. The formula is derived from the regular compound interest formula, A=P(1+r/n)^(n*t), when we take the limit as n (the number of compoundings) goes to infinity. The constant “e” ends up being the maximum growth factor when we have 100% interest. What?? While an investment earning 100% interest will simply double if interest is paid out annually, it will grow by more if interest is compounded more frequently. The absolute maximum amount in the account by the end of one year is 2.718 or e times the original investment. We explore this phenomenon in “Who Wants to be a Millionaire?” in Unit 3 of our Precalculus course. This idea of a limiting factor is tough for students, and many still walk away with only a partial understanding of this “Pert” equation, as in, they know when to use it and how, but still not necessarily why it works.
Cue AP Calculus. In Unit 7, students learn how to solve differential equations given an initial condition. In the final lesson of the unit, students look at differential equations where the rate of change is directly proportional to the current quantity, and they see that this is a model for exponential growth and decay. It is here where we can make valuable connections to the compound interest formula they learned in previous years. But of course we don’t want to spoil the fun by just telling them how they’re related. Let’s let them discover it for themselves!
For the past couple of years, I’ve used this extended warm-up on the day before or after Lesson 7.8 (depending on if you want students to jump straight to the exponential growth and decay formula or go through all the steps of SISIS and derive it from scratch. I prefer the long way.)
This set of questions was carefully designed to get students to derive the formula for continuously compounding interest: A=Pe^(r*t) where A is the value of the account, P is the principal, r is the interest rate, and t is the time.
One of my favorite moments of the entire school year was when students truly looked like they were having an epiphany on the day they worked through this warm-up. “My life is complete!” one student exclaimed. (I made the bold choice to interpret that comment as 100% authentic.)
Through this warm-up students walked away with three important understandings:
Continuously compounding interest represents a scenario where the rate of change of a quantity (the dollar amount in the account) is directly proportional to the current amount.
The constant of proportionality is the interest rate! (k=r)
The principal P is the initial condition y0 (when t=0)
Helping students see the connections between what they learned in the past and what they're learning now is one of the best ways to motivate and engage students. When students see math as a unified whole instead of as separate topics and ideas, their appreciation for the discipline grows!
In case you're curious, here are my solutions to the warm-up problem.