Lesson 13

This is the second of two lessons that introduce students to the constant \(e\). Here they encounter exponential functions of the form \(f(t) = P \boldcdot e^{rt}\).

Students first explore the behavior of functions written in this form, comparing it to that of functions written in the familiar form of \(f(t) = a \boldcdot b^t\) or \(f(t) = a \boldcdot (1+r)^t\), using tables and graphs to do so. Students learn that using the same small value of \(r\) in \(P \boldcdot e^{rt}\) and in \(P \cdot (1+r)^t\) leads to functions that are close but not exactly the same. Although it is beyond the scope of this course to go into the difference, they learn that the form with base \(e\) is often used to model situations where the growth rate \(r\) happens not just for each unit of time \(t\) but at every moment; that is, \(r\) is the continuous growth rate.

Next, after experimenting with some concrete representations of functions written with base \(e\), students interpret and analyze the parameters of such equations more formally. They pay close attention to the meaning of various quantities in context (MP2). Along the way, they notice similarities in the structure of the two forms of exponential equations.

The lesson includes an optional activity for students to practice graphing functions expressed with base \(e\) and adjusting the graphing window so that the graphs are useful for analysis and problem solving. 

For the first activity, only a calculator with an \(e\) button is needed, but students need graphing technology for the optional activity. Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if each student has their own device. (Desmos is available under Math Tools.)