Problem 10 - Calculus

Question

Write the functions in the form \(P=P_{0} a^{t}\) Which represent exponential growth and which represent exponential decay? $$P=2 e^{-0.5 t}$$

Step-by-Step Solution

Answer

It's an exponential decay function, with \(a = e^{-0.5}\).

Step 1: Identify the Function Form

The general form for exponential functions is \(P = P_0 a^t\). This means you need to express the function \(P = 2 e^{-0.5t}\) in this form.

Step 2: Express Given Function as Exponential

Given the function \(P = 2 e^{-0.5t}\), recognize that \(e^{-0.5} = a\). Thus, the function can be rewritten as \(P = 2 (e^{-0.5})^t\). Here, \(P_0 = 2\) and \(a = e^{-0.5}\).

Step 3: Analyze the Base for Decay or Growth

If the base \(a\) is greater than 1, the function represents exponential growth. If the base \(a\) is between 0 and 1, it represents exponential decay. Since \(e^{-0.5}\) is between 0 and 1, this indicates exponential decay.

Key Concepts

Exponential Growth

Exponential growth describes a process where the quantity increases rapidly over time. It occurs when the base of the exponential function, typically denoted as \(a\), is greater than 1. In such cases, as time \(t\) passes, each subsequent value of \(P\) becomes larger, as it continuously multiplies by \(a\), which is greater than 1.
To visualize exponential growth, imagine a scenario where you're observing a population of rabbits. If the growth rate is such that each pair of rabbits produces more than one additional pair over time, the population will then grow increasingly faster. This compounding effect leads to the steep upward curve characteristic of exponential growth.

• The formula used is \(P = P_0 a^t\).
• If \(a \gt 1\), it indicates exponential growth.
• The value of \(P_0\) represents the initial or starting amount.

This fundamental behavior is what makes exponential growth a significant concept in various fields like finance, biology, and physics.

Exponential Decay

Exponential decay, on the other hand, describes a process where the quantity decreases over time. This occurs when the base \(a\) is a number between 0 and 1. Here, as time \(t\) progresses, the value of \(P\) becomes smaller.
A simple way to think of exponential decay is by considering the concept of depreciation. For instance, the value of a new car decreases over time as it is used and ages. Each year, its value becomes a fixed fraction of what it was the year before. This type of reduction follows the exponential decay pattern.

• The function takes the form \(P = P_0 a^t\).
• When \(0 \lt a \lt 1\), the process represents exponential decay.
• \(P_0\) again is the initial amount before decay begins.

In the earlier mentioned exercise, we identified that \(a = e^{-0.5}\), which is approximately 0.6065. Since this value is less than 1, it confirms the process is exponential decay. This decay behavior is critical in areas like radioactive decay, cooling laws, and economic devaluation.

Exponential Form

The exponential form of a function is a powerful representation method used to model growth and decay processes. In its standard form, it is written as \(P = P_0 a^t\) where:

• \(P\) is the resulting quantity after time \(t\).
• \(P_0\) is the initial amount.
• \(a\) is the base, dictating the rate of growth or decay.

Understanding this form allows us to easily determine the behavior of a function over time. For example, exponential functions can describe real-world phenomena such as population growth, bank interest accumulation, or radioactive substance decay.
In the solution for the exercise, we transformed the specific function \(P = 2 e^{-0.5t}\) into the common exponential form by recognizing that \(e^{-0.5} = a\). By expressing the function this way, it becomes clearer that the function involves exponential decay, as \(a\) is between 0 and 1.
Overall, the clarity provided by exponential form is beneficial for both mathematical analysis and practical applications, making it a key concept in understanding exponential phenomena.