# Lesson 13

Exponential Functions with Base $e$

### Lesson Narrative

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.

### Learning Goals

Teacher Facing

- Comprehend that we can use $e^r$ as the growth factor of $f(t) = e^{rt}$ when we assume the rate $r$ is applied continuously.
- Interpret the parameters in the equations of exponential functions with the form $f(t)=P \boldcdot e^{rt}$.

### Student Facing

- Let’s look at situations that can be modeled using exponential functions with base \(e\).

### Required Materials

### Required Preparation

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.)

### Learning Targets

### Student Facing

- I understand that $e$ is used in exponential models when we assume the growth rate is applied at every moment.

### CCSS Standards

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