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Exponential Functions - Introduction

Exponential Functions: (lesson 1 of 3)

Exponential Functions - Introductions

If $a > 0$ and $a \ne 1$ , then the exponential function with base $_a$ is given by $f( x ) = a^x$ .

Natural Exponential Functions

1. The irrational number $e = \mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{1}{n}} \right)^n} \approx 2.71828 \cdot \cdot \cdot$ or $e = \mathop {\lim }\limits_{x \to 0} {\left( {1 + x} \right)^{\frac{1}{x}}}$ .

2. For each real $x$ , $\mathop {\lim }\limits_{x \to \infty } {\left( {1 + \frac{x}{n}} \right)^n} = e^x$ .

3. $f(x) = e^x$ is a natural exponential function.

Derivatives of Exponential Functions

1. $\frac{d}{{dx}}[{e^x}] = e^x$

2. $\frac{d}{{dx}}[e^{f(x)}] = {e^{f(x)}} \cdot f'(x)$ .

The hyperbolic cosine and sine functions

The hyperbolic cosine function is defined as

$\cosh (x) = \frac{{\exp (x) + \exp ( - x)}}{2}$ , $- \infty < x < \infty$ ,

while the hyperbolic sine function is defined as

$\sin (x) = \frac{{\exp (x) - \exp ( - x)}}{2}$ , $- \infty < x < \infty$ .

Properties of hyperbolic cosine and sine functions

1. $(\sinh (x))' = \cosh (x)$ , $(\cosh (x))' = \sinh (x)$ .

2. $\cosh (x) > \sinh (x)$ .

3. ${\cosh ^2}(x) - {\sinh ^2}(x) = 1$

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God created the natural number, and all the rest is the work of man.

Leopold Kronecker

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I am ever more convinced that the necessity of our geometry cannot be proved -- at least not by human reason for human reason.

Carl Friedrich Gauss