Problem 1 - Calculus Early Transcendentals

Question

In terms of relative growth rate, what is the defining property of exponential growth?

Step-by-Step Solution

Answer

The defining property of exponential growth in terms of relative growth rate is that the relative growth rate remains constant over time, meaning the percentage increase of the quantity remains the same throughout the entire process, regardless of the current value of the quantity.

Step 1: Define Exponential Growth

Exponential growth occurs when a quantity increases by a fixed percentage over regular intervals of time. Mathematically, exponential growth can be represented as: \[ y(t) = y_0e^{rt} \] where \(y(t)\) represents the amount at time \(t\), \(y_0\) is the initial amount, \(r\) is the growth rate, and \(e\) is the base of the natural logarithm.

Step 2: Define Relative Growth Rate

Relative growth rate is the percentage increase in a quantity over a specific time period, often expressed as a decimal or percentage. Mathematically, relative growth rate can be found by dividing the absolute growth rate (i.e., the increase in the quantity) by the initial quantity: \[ \text{Relative growth rate} = \frac{\text{Absolute growth rate}}{\text{Initial quantity}} \]

Step 3: Identify the Defining Property of Exponential Growth

In terms of relative growth rate, the defining property of exponential growth is that the relative growth rate remains constant over time. This means that the percentage increase of the quantity remains the same throughout the entire process, regardless of the current value of the quantity. In summary, the defining property of exponential growth, in terms of relative growth rate, is that the relative growth rate remains constant over time.