Exponential Functions

What is an Exponential Function?

An exponential function is a mathematical function in the form:

$$f(x) = a \times b^x$$

Where:

• $a$ is the initial value (the function's value when $x = 0$).
• $b$ is the base. For exponential growth, $b > 1$. For exponential decay, $0 < b < 1$
• $x$ is the exponent, usually a real number.

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Key Characteristics

Domain and Range:

• Domain: All real numbers ($x \in \mathbb{R}$)
• Range: Positive real numbers ($f(x) > 0$ if $a > 0$)

Graph Features:

• Exponential growth curves upwards as $x$ increases ($b>1$)
• Exponential decay curves downwards as $x$ increases ($0 < b < 1$)
• The graph passes through the point $(0, a)$, since $f(0) = a \times b^0 = a$

Asymptote:

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

Rate of Change:

• The rate of change increases (for growth) or decreases (for decay) exponentially.

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Integration and Applications

Integration allows us to compute areas under exponential curves and model cumulative effects.

• For $e^x$

$$\int e^x \, dx = e^x + C$$

• For $a \times b^x$

$$\int a \times b^x \, dx = \frac{a}{\ln(b)} \times b^x + C \quad \text{where } b > 0, b \neq 1$$

These are essential for solving problems in biology, finance, and physics.

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Examples

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Summary

• Exponential functions are of the form .

$$f(x) = a \times b^x$$

• They exhibit exponential growth ($b > 1$) or decay ($0 < b < 1$).
• The $x-axis$ ($y = 0$) is a horizontal asymptote.
• Integration:
  • $\int e^x dx = e^x + C$
  • $\int a \times b^x dx = \frac{a}{\ln(b)} \times b^x + C$
• Applications include population growth, radioactive decay, and compound interest.
• Key formulas:

$$P(t) = P_0 \times b^t \quad \text{(growth)}$$ $$A(t) = A_0 \times b^t \quad \text{(decay)}$$