Question

Exponential Growth and Decay Give the differential equation that models exponential growth and decay.

Short answer

The differential equation that models exponential growth and decay is: \[ \frac{dN}{dt} = kN \], where N is the quantity, t is time, and k is a constant of proportionality.

Step-by-step solution

Understand Exponential Growth and Decay

Exponential growth and decay are modeled by certain types of functions that occur frequently in real-world applications. Exponential growth occurs when a quantity increases by the same ratio in each unit of time. Conversely, decay occurs when a quantity decreases at a rate proportional to its current value.

Formulate the Differential Equation for Exponential Growth and Decay

To model exponential growth or decay, we will use a differential equation. This is a way to describe how a quantity evolves over time, based on a rate of change that is proportional to the current value of the quantity. The differential equation that is used to model exponential growth or decay will take the form of \[ \frac{dN}{dt} = kN \], where N is the quantity we're observing, t is time, and k is a constant of proportionality. If k is positive, we get exponential growth; if k is negative, we get exponential decay.

Understand the Role of The Constant k

The constant \( k \) in the differential equation is crucial in determining whether the function models growth or decay. If \( k \) is greater than zero, the function models exponential growth. If \( k \) is less than zero, the function models exponential decay. This constant is related to the 'rate' or 'speed' of the growth or decay.

Key concepts

Differential Equation

Differential equations are powerful mathematical tools used to model how quantities change over time. In the case of exponential growth or decay, a differential equation connects the change in a quantity to the quantity itself. Specifically, this type of equation expresses how a small change in time affects the overall change in the quantity. A standard form of the differential equation for exponential growth and decay is given by \[ \frac{dN}{dt} = kN \],where:

• \( \frac{dN}{dt} \) is the derivative of \( N \) with respect to \( t \), or essentially, how \( N \) changes as time progresses.
• \( N \) is the quantity in question, for example, population size or radioactive material.
• \( t \) represents time.
• \( k \) is the constant of proportionality, which we will discuss later.

This equation tells us that the rate at which \( N \) changes is directly proportional to its current size, giving it the unique property of growing or decaying exponentially.

Proportional Change

Proportional change is a key concept in understanding exponential functions. In the context of exponential growth and decay, proportional change means that the rate of change of a quantity is directly proportional to the quantity itself. This creates a situation where if the quantity doubles, the rate at which it grows or decays also doubles. The relationship can be visualized with the equation \( kN \) in \[ \frac{dN}{dt} = kN \].In simple terms:

• The bigger the existing quantity, the faster it grows or shrinks.
• This relationship facilitates the rapid growth seen in some real-world scenarios, like population growth.

Conversely, it also explains how quickly some materials decay, like radioactive substances. The idea of proportional change underpins the entire concept of exponential processes.

Rate of Change

The rate of change is vital in understanding how a quantity evolves over time. In exponential growth and decay, the rate of change is expressed within the differential equation as \( \frac{dN}{dt} \). This notation states how \( N \) changes with respect to time \( t \). If this rate is positive, the quantity is increasing (growth), and if negative, it is decreasing (decay).

Understanding Rate of Change Deeper

Since in these scenarios, the rate of change itself depends on the current amount:

• When \( N \) is large, the changes are more pronounced.
• When \( N \) is small, the changes become less dramatic.

This inherent link between current value and its rate of change is what gives exponential equations their unique accelerating or decelerating nature.

Constant of Proportionality

The constant of proportionality, denoted as \( k \) in the equation \( \frac{dN}{dt} = kN \), plays a pivotal role in dictating whether a process involves growth or decay. More importantly:

• If \( k > 0 \), the system is experiencing growth, meaning the quantity increases over time.
• If \( k < 0 \), the system is undergoing decay, thus the quantity diminishes over time.

How Does \( k \) Affect Growth and Decay?

The value of \( k \) determines the rate or speed at which growth or decay occurs:

• A large positive \( k \) results in faster growth.
• A small positive \( k \) indicates slower growth.
• Similarly, a more negative \( k \) leads to quicker decay.
• Whereas a less negative \( k \) results in slower decay.

Thus, \( k \) effectively serves as a measure of the intensity of exponential changes within a system.