Exponential Functions and the Number e

Exponential Functions and the Number e

The Number e

The mathematical constant e is a fundamental number in mathematics, especially in the study of exponential functions. It is an irrational number approximately equal to 2.718281828459, and it serves as the base for natural exponential functions.

• Definition: The number e is defined such that the function $F(x) = e^x$ has unique properties in calculus and mathematical modeling.

• Exponential Functions: Functions of the form $F(x) = a^x$ (where $a > 0$, $a \neq 1$) and $F(x) = e^x$ are called exponential functions.

• Calculator Use: Calculators should be used to evaluate exponential functions with base e.

Evaluating Exponential Functions

To evaluate exponential functions, substitute the given value of x into the function and calculate the result.

• Example 1: $F(x) = 2^x$, for $x = 3$ $2^3 = 8$

• Example 2: $F(x) = e^x$, for $x = 2$ $e^2 \approx 7.389$

Graphing Exponential Functions

The graph of $F(x) = a^x$ is an exponential curve. The graph of $F(x) = e^x$ is a specific exponential curve with base e. These graphs share common characteristics:

• Domain: $(-\infty, \infty)$

• Range: $(0, \infty)$

• Y-intercept: At $x = 0$, $F(0) = a^0 = 1$

• Asymptote: The x-axis ($y = 0$) is a horizontal asymptote.

• Growth: For $a > 1$, the function increases rapidly as $x$ increases.

Applications of e

The number e arises naturally in many mathematical and real-world contexts, such as:

• Compound Interest: Used in continuous compounding formulas.

• Population Growth: Models exponential growth in biology.

• Radioactive Decay: Describes exponential decay in physics and chemistry.

Practice: Graphing Exponential Functions

To graph $F(x) = e^x$:

• Plot several points by evaluating $e^x$ for different values of $x$.

• Draw a smooth curve through the points, approaching the x-axis as $x$ decreases.

• Note the y-intercept at $(0, 1)$ and the horizontal asymptote at $y = 0$.

Summary Table: Properties of Exponential Functions

Property	Description
Domain	$(-\infty, \infty)$
Range	$(0, \infty)$
Y-intercept	$(0, 1)$
Horizontal Asymptote	$y = 0$
Growth/Decay	Growth for $a > 1$, decay for $0 < a < 1$

Additional info: The notes include visual representations of compounding interest, population growth, and radioactive decay, which are classic applications of exponential functions with base e.