Lesson 12

The Number $e$

Let’s learn about the number $e$.

Match each equation to a situation it represents. Be prepared to explain how you know. Not all equations have a match.

$f(t) = 400 \boldcdot (0.5)^{0.1t}$

$g(t) = 400 \boldcdot (1.25)^{0.1t}$

$h(t) = 400 \boldcdot (0.75)^{0.1t}$

$j(t)=400 \boldcdot (2)^{10t}$

$k(t) = 400 \boldcdot (2)^{0.1t}$

A scientist begins an experiment with 400 bacteria in a petri dish. The population doubles every 10 hours. The function gives the number of bacteria $t$ hours since the experiment began.
A patient takes 400 mg of a medicine. The amount of medicine in her bloodstream decreases by 25% every 10 hours. The function gives the amount of medicine left in her bloodstream after $t$ hours of taking the medicine.
The half-life of a radioactive element is 10 years. There are 400 g of the element in a sample when it is first studied. The function gives the amount of the element remaining $t$ years later.
In a lake, the population of a species of fish is 400. The population is expected to grow by 25% in the next decade. The function gives the number of fish in the lake $t$ years after it was 400.

A spot of mold is found on a basement wall. Its area is about 10 square centimeters. Here are three representations of a function that models how the mold is growing.

time (weeks)	area of mold (sq cm)
0	10
1	27
2	74
3	201
4	546

$a(t) = 10 \boldcdot e^t$

Coordinate plane, horizontal, time in weeks, 0 to 4 by 1, vertical area of mold in square centimeters, 0 to 300 by 50. A curve through points 0 comma 10, 1 comma 27, 2 comma 74 and 3 comma 201.

What do you notice? What do you wonder?

Here are some functions. For each function, describe, in words, the outputs for very tiny, positive values of $x$ and for very large values of $x$.

$a(x) = 1^x$

$b(x) = \text-x$

$d(x) = \frac{1}{x}$

$f(x) = \left(\frac{1}{x}\right)^x$

$g(x) = \left(1 + \frac{1}{x}\right)^x$

$h(x) = e^{x}$

$k(x) = 1 + x$
Remember that $e \approx 2.718$. What does the function $g$ have to do with the number $e$?
What do you notice about the relationship between $h$ and $k$ for very small, positive values of $x$?

Are you ready for more?

Complete the table to show the value of each expression to the nearest hundred-thousandth. Two entries have already been completed as an example.

$x$	$2^x$	$e^x$	$3^x$
0.1	1.07177	1.10517	$\phantom{1.11612}$
0.01
0.001
0.0001

What do you notice about the values in the table?

Scientists, economists, engineers, and others often use the number $e$ in their mathematical models. What is $e$?

$e$ is an important constant in mathematics, just like the constant $\pi$, which is important in geometry. The value of $e$ is approximately 2.718. Just like $\pi$, the number $e$ is irrational, so it can’t be represented as a fraction, and its decimal representation never repeats or terminates. The number is named after the 18th-century mathematician Leonhard Euler and is sometimes called Euler’s number.

$e$ has many useful properties and it arises in situations involving exponential growth or decay, so $e$ often appears in exponential functions. In upcoming lessons, we will work with functions that are expressed using $e$.

$e$ (mathematical constant)

The number $e$ is an irrational number with an infinite decimal expansion that starts $2.71828182845\ .\ .\ .$, which is used in finance and science as the base for an exponential function.

Lesson 12

12.1: Matching Situations and Equations

12.2: Notice and Wonder: Moldy Growth

12.3: $(1 + \text{tiny})^{\text{huge}}$

Summary

Glossary Entries