"General relative growth rates Define the relative growth rate of the function f over the time interval T to be the relative change in f over an interval of length T:
R_T = [f(t + T) − f(t)] / f(t)
Show that for the exponential function y(t) = y₀ e^{kt}, the relative growth rate R_T, for fixed T, is constant for all t."

Verified step by step guidance

1. Start with the given exponential function: \(y(t) = y_0 e^{k t}\), where \(y_0\) and \(k\) are constants.
2. Write the expression for the relative growth rate \(R_T\) over the interval \(T\) using the definition: \(R_T = \frac{y(t + T) - y(t)}{y(t)}\).
3. Substitute \(y(t + T)\) and \(y(t)\) into the formula: \(R_T = \frac{y_0 e^{k (t + T)} - y_0 e^{k t}}{y_0 e^{k t}}\).
4. Factor out \(y_0 e^{k t}\) from the numerator: \(R_T = \frac{y_0 e^{k t} (e^{k T} - 1)}{y_0 e^{k t}}\).
5. Simplify the expression by canceling \(y_0 e^{k t}\): \(R_T = e^{k T} - 1\). Notice that this expression depends only on \(T\) and \(k\), and not on \(t\), which shows that \(R_T\) is constant for all \(t\).

Key Concepts

Relative Growth Rate

The relative growth rate measures how much a function changes relative to its current value over a time interval. It is defined as the change in the function's value divided by the original value, capturing proportional growth rather than absolute change.

Exponential Functions

An exponential function has the form y(t) = y₀ e^{kt}, where y₀ is the initial value and k is the growth constant. Such functions model continuous growth or decay, and their rate of change is proportional to their current value.

Properties of Exponential Growth

For exponential functions, the relative growth rate over any fixed interval T is constant because the ratio [y(t+T) - y(t)] / y(t) simplifies to e^{kT} - 1, which depends only on T and k, not on t. This reflects the memoryless, consistent proportional growth characteristic.