An example of an exponential growth function is:
\[ f(t) = a \cdot e^{bt} \]
where:
- \( f(t) \) is the quantity at time \( t \),
- \( a \) is the initial quantity (the value of \( f(0) \)),
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828),
- \( b \) is a positive constant that represents the growth rate,
- \( t \) is time.
Explanation of Exponential Growth
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Definition of Exponential Growth: Exponential growth occurs when the growth rate of a quantity is proportional to its current value. This means that as the quantity increases, the amount that it grows also increases, leading to an accelerating rate of growth over time.
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Characteristics of the Function:
- The function is defined such that \( b > 0 \). This ensures that as time \( t \) increases, \( f(t) \) grows without bound.
- At \( t = 0 \), the function assumes the initial value, \( f(0) = a \).
- The term \( e^{bt} \) increases rapidly for positive values of \( t \) since the exponent \( bt \) grows as \( t \) increases, leading to a rapid increase in \( f(t) \).
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Graph Behavior:
- When graphed, exponential growth functions produce a J-shaped curve. Initially, the growth appears slow, but as time progresses, the slope of the curve increases dramatically, demonstrating acceleration in growth.
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Real-World Applications:
- Exponential growth can be seen in various real-world phenomena, such as population growth, compound interest in finance, the spread of information or diseases, and certain chemical reactions. In these contexts, the quantity grows relative to its current size, emphasizing the nature of exponential growth.
Overall, the function \( f(t) = a \cdot e^{bt} \) exemplifies exponential growth because it accurately describes how a quantity grows at a rate proportional to its existing amount, leading to increasingly rapid growth over time.