Multiple Choice
Which operation can be used to eliminate the natural logarithm () from both sides of an equation such as ?
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0. Functions
Exponential Functions
3:10 minutesProblem 7.2.52
Textbook Question
"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 guidance1
Start with the given exponential function: \(y(t) = y_0 e^{k t}\), where \(y_0\) and \(k\) are constants.
Write the expression for the relative growth rate \(R_T\) over the interval \(T\) using the definition:
\(R_T = \frac{y(t + T) - y(t)}{y(t)}\).
Substitute \(y(t + T)\) and \(y(t)\) into the formula:
\(R_T = \frac{y_0 e^{k (t + T)} - y_0 e^{k t}}{y_0 e^{k t}}\).
Factor out \(y_0 e^{k t}\) from the numerator:
\(R_T = \frac{y_0 e^{k t} (e^{k T} - 1)}{y_0 e^{k t}}\).
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 \(t\).
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Video duration:
3mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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