The number e

A base that is frequently used with exponential functions in calculus is the number e. e is the number that approaches as n approaches .

Let's evaluate for several values of n. When n = 1, we have raised to the power of 1. The result is two. When n = 2 we have or 2.25. If n = 4 we get or 2.44140625.

The values we obtained get larger each time but by a decreasing amount. If we look at the graph of for n from 0 to 10000, we will see that it has a horizontal asymptote at a value around 2.718. The actual value is the number e.

Listed below are the first forty digits of e.

e =

Example 1

Graph

Solution .

Since e > 1, the graph is increasing.

Example 2

On the same coordinate system graph in blue, in red, and in black.

Solution .

Since 2.5 < e < 3 the graph should show the blue curve between the black and red curve when x > 0.

Example 3

If interest were compounded continuously then the formula would be where A is the compound amount, P is the principal, r is the rate as a decimal, and t is time in years. If P = 100 and r = 0.05 (corresponding to 5%) then Find the compound amount in five years.

Solution .

Letting t = 5, we get which simplifies to $128.40