Problem 65 - Introductory and Intermediate Algebra for College Students 4th

Question

What is the natural exponential function?

Step-by-Step Solution

Answer

The natural exponential function is denoted as \( e^x \), where \( e \) is Euler's number, approximately equal to 2.71828. It has unique properties such as always being positive, its rate of increase is directly proportional to the function's current value. The derivative of e^x is itself.

Step 1: Definition

The natural exponential function is a mathematical function denoted as \( e^x \), where \( e \) is Euler's number, approximately equal to 2.71828. This number is the base of the natural logarithm.

Step 2: Representation of Natural Exponential Function

It can be represented as an infinite series: \( e^x = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \frac{x^4}{4!} + ....+ \frac{x^n}{n!}\), for all complex numbers x.

Step 3: Properties of the natural exponential function

The natural exponential function, \( e^x \) has some unique properties : 1. It's always positive, 2. The rate of increase of the function is directly proportional to the function's current value, 3. \( e^{x+y} = e^x \cdot e^y \), 4. \( e^{x-y} = \frac{e^x}{e^y} \), 5. \( e^{-x} = \frac{1}{e^x} \), 6. \( (e^x)' = e^x \), the derivative of e^x is itself.