Rate of Exponential Growth
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Theorem
Let $y = a e^{b t}$ be an exponential growth function.
Then the rate of growth of $y$ is proportional to the value of $y$ such that:
- $\dfrac {\d y} {\d t} = b y$
Proof
\(\ds y\) | \(=\) | \(\ds a e^{b t}\) | ||||||||||||
\(\ds \leadsto \ \ \) | \(\ds \dfrac {\d y} {\d t}\) | \(=\) | \(\ds a b e^{b t}\) | Derivative of Exponential Function: Corollary $1$ | ||||||||||
\(\ds \) | \(=\) | \(\ds b y\) | Definition of $y$ |
$\blacksquare$
Sources
- 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): exponential growth or decay
- 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): exponential growth or decay