Problem 24 - Applied Calculus

Question

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 expression represents exponential decay because the base is less than 1.

Step 1: Identify the Exponential Form

We need to compare the given expression to the standard form of an exponential function, which is \( P = P_0 a^{t} \). Our expression is \( P = 2 e^{-0.5 t} \), where \( P_0 = 2 \) and the base is \( e^{-0.5} \).

Step 2: Determine the Base

In the expression \( P = 2 e^{-0.5 t} \), we see the base is \( e^{-0.5} \). This tells us that \( a = e^{-0.5} \). The important part here is the exponent sign. Since the exponent is \(-0.5\), it indicates something about the direction of the growth/decay.

Step 3: Evaluate the Base Size

Since \( e^{-0.5} \) is less than 1 (because it is equivalent to \( \frac{1}{e^{0.5}} \), which is a fraction), it indicates exponential decay. If \( a < 1 \), the function represents decay.

Step 4: Conclusion

Since \( e^{-0.5} < 1 \), the function \( P = 2 e^{-0.5 t} \) represents exponential decay.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases at a rate proportional to its current value, meaning the growth becomes faster over time. It is often represented by equations of the form \( P = P_0 a^t \), where:

• \( P \) is the final quantity.
• \( P_0 \) is the initial quantity.
• \( a \) is the base of the exponential function, determining whether the function grows or decays.
• \( t \) represents time or periods of growth.

In the case of exponential growth, the base \( a \) must be greater than 1, leading to a continuous increase in \( P \). As time passes, the increment for each additional time unit becomes larger. A common real-world example is money earning interest in a bank, where the amount increases over time at an exponential rate.
High population growth or compounding interest are practical illustrations of exponential growth, clearly showing how something can "snowball" given enough time.

Exponential Function

An exponential function is a type of mathematical function that models situations where growth or decay happens exponentially. The general form is \( P = P_0 a^t \). They are characterized by having a constant base raised to a variable exponent. This unique structure allows exponential functions to model rapid changes, more effectively than linear functions can.
Key characteristics of exponential functions include:

• **Rapid Increase or Decrease:** They can drastically change values at a quick pace.
• **Non-linearity:** The rate of change is not constant.
• **Horizontal Asymptote:** The graph tends towards a horizontal line as it approaches certain values.

Exponential functions are prominent in various disciplines, from physics to social sciences. In the context of the given exercise, we see exponential decay, but exponential functions can model growth in populations, bacteria growth, or any scenario with consistent proportional changes.
This function type is incredibly versatile, enabling analyses and predictions in many scientific fields by adjusting base values and initial conditions.

Base of an Exponential Function

The base of an exponential function, denoted as \( a \) in the function \( P = P_0 a^t \), is crucial as it determines the function's behavior, such as growth or decay. If the base \( a \) is:

• Greater than 1: The function models exponential growth, meaning the quantity will get larger over time.
• Less than 1: The function represents exponential decay, indicating a decrease over time.
• Exactly 1: The function remains constant, as there is no change across time.

In the standard form, the assignment of \( a \) depends on the specific situation. For instance, when dealing with natural exponential functions, we often use the number \( e \) (approximately 2.718), a fundamental constant often associated with continuous growth or decay scenarios, such as in the exercise given which uses \( e^{-0.5} \). Understanding the base helps in predicting the behavior of the system being modeled—be it a decaying radioactive isotope, cooling of hot liquid, or even economic inflation.