Problem 22 - Applied Calculus

Question

Write the functions in Problems \(21-24\) 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

The function represents exponential decay because \( a = e^{-0.5} < 1 \).

Step 1: Identify the general form

The general form of an exponential function is given by \( P = P_0 a^t \), where \( P_0 \) is the initial value, \( a \) is the base, and \( t \) is time.

Step 2: Rewrite the given function

The given function is \( P = 2 e^{-0.5 t} \). We need to rewrite this in the form \( P = P_0 a^t \). Here, we have \( P_0 = 2 \) and \( a = e^{-0.5} \).

Step 3: Determine the type of function

Since the base \( a = e^{-0.5} \) is less than 1, the function represents exponential decay. An exponential decay occurs when the base of the exponential function is between 0 and 1.

Key Concepts

Exponential Growth

Exponential growth describes a process where a quantity increases over time at a rate proportional to its current value. This occurs when the base of the exponential function, denoted as \(a\), is greater than one. In such cases, as time \(t\) progresses, the value of the function increases rapidly.

For example, consider the function \(P = P_0 a^t\), where \(a > 1\). This kind of growth is very common in nature when looking at populations, finance, and other situations involving multiplication over time. Here are some key points about exponential growth:

• The initial value \(P_0\) represents the starting point or quantity.
• The base \(a\) determines the growth rate. The further \(a\) is from 1, the faster the growth rate.

Remember, exponential growth continues to accelerate as time goes on. If you're observing such growth in data, expect the values to multiply rather quickly as time increments.

Exponential Decay

Exponential decay occurs when a quantity decreases over time at a rate that is proportional to its current value. This happens in exponential functions where the base, \(a\), is between 0 and 1. Instead of increasing, the value of the function decreases as time \(t\) moves forward.

The rewritten function \(P = 2 e^{-0.5t}\) serves as an example of exponential decay:

• The initial value \(P_0 = 2\) indicates that the decay starts from this amount.
• The base \(a = e^{-0.5}\) shows a decay factor, as it is calculated to a value between 0 and 1.

Exponential decay models are frequently found in scenarios such as radioactive decay, depreciation of assets, and cooling of substances. An important aspect to note with exponential decay is how quickly the value can decrease. It diminishes rapidly initially, and then gradually levels out over time.

Rewriting Functions

Rewriting functions in exponential form can help identify whether they represent growth or decay. The general exponential form is \(P = P_0 a^t\), where \(P_0\) is the initial value and \(a\) is the base.

To rewrite a function, follow these steps:

• Identify \(P_0\) as the starting constant in the function.
• Recognize the exponential term and convert it to the form \(a^t\) where possible.

For example, in the function \(P = 2 e^{-0.5t}\), we identified:

• \(P_0 = 2\), the initial quantity.
• Converted \(e^{-0.5t}\) to \((e^{-0.5})^t\).

Understanding the rewritten form aids in analyzing the behavior of the function over time and predicting future outcomes. This is especially helpful when graphing or modeling exponential relationships.