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Chapter 12: Problem 65

What is the natural exponential function?

Short Answer

Expert verified
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 by step solution

01

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.
02

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.
03

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.

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Most popular questions from this chapter

The percentage of adult height attained by a girl who is \(x\) years old can be modeled by $$f(x)=62+35 \log (x-4)$$ where \(x\) represents the girl's age (from 5 to 15 ) and \(f(x)\) represents the percentage of her adult height. Use the formula to solve Exercises. Round answers to the nearest tenth of a percent. Approximately what percentage of her adult height has a girl attained at age \(13 ?\)

Graph: \(5 x-2 y>10\)

Simplify: \(\left(-2 x^{3} y^{-2}\right)^{-4}\)

Explain how to solve an exponential equation when both sides can be written as a power of the same base.

Explain why the logarithm of 1 with base \(b\) is \(0 .\)
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