Chapter 1.5, Problem 6E

(a)

To determine

How the number e is defined.

Explanation of Solution

The number e can be defined in the following ways.

$\underset{x\to \infty }{\lim }{\left(1+\frac{1}{x}\right)}^x=e\text{ and }{\displaystyle \sum _{n=0}^{\infty }\frac{x^n}{n!}=e}$.

(b)

To determine

An approximate value of e.

Explanation of Solution

An approximate value of e, rounded off to five decimal places is $e=2.71828$.

(c)

To determine

What is the natural exponential function?

Explanation of Solution

The function $y=e^x$ is said to be the natural exponential function with base e and an exponent x.

The graph of $y=e^x$ is lies between the graph of $y=2^x$ and $y=3^x$ as the value of e is 2.71828.

The graph of exponential function always passes through the point (0,1) as any number raised to the power 0 is 1.