Exponential decay occurs when a quantity decreases at a rate proportional to its current value. This can be mathematically represented by the equation:
\[ y(t) = y_0 e^{-kt} \]
Where:
- \( y(t) \) is the quantity at time \( t \)
- \( y_0 \) is the initial quantity (when \( t = 0 \))
- \( k \) is a positive constant that determines the rate of decay
- \( e \) is the base of the natural logarithm (approximately equal to 2.71828)
Examples of Exponential Decay Equations:
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Basic Form: \[ y = 5 e^{-0.1t} \] This represents a quantity starting at 5 and decaying at a rate determined by the constant \( 0.1 \).
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Half-Life Form: \[ N(t) = N_0 \left(\frac{1}{2}\right)^{\frac{t}{T_{1/2}}} \] This form is often used in contexts like radioactive decay, where \( T_{1/2} \) is the half-life of the substance.
Graph Characteristics:
- The graph of an exponential decay function will show a curve that starts at the value of \( y_0 \) when \( t = 0 \) and approaches zero as \( t \) increases.
- The graph will never actually reach zero; this is known as an asymptotic behavior.
Key Features of the Graph:
- Y-Intercept: The y-intercept of the graph will be at \( (0, y_0) \).
- Decay Rate: A larger value of \( k \)/smaller \( T_{1/2} \) leads to a steeper slope and faster decay.
- Asymptote: The x-axis (y = 0) serves as a horizontal asymptote.
Graph Example:
Graphing the function \( y(t) = 5 e^{-0.1t} \) will show a decreasing curve starting at \( y = 5 \) at \( t = 0 \), gradually approaching the x-axis but never actually reaching it.
In summary, exponential decay is represented by equations of the form \( y(t) = y_0 e^{-kt} \), and the graphs will depict a continuous decrease towards zero over time, characterized by a smooth, downward-sloping curve.
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