Problem 23 - 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? $$P=P_{0} e^{0.2 t}$$

Step-by-Step Solution

Answer

The function \( P = P_{0} e^{0.2 t} \) represents exponential growth.

Step 1: Identify the Given Expression

The function given is \( P = P_{0} e^{0.2 t} \). We need to express this in the form \( P = P_{0} a^{t} \).

Step 2: Rewrite the Function in the Exponential Form

The given function is already in exponential form because the base \( e \) can be rewritten as an equivalent exponent base \( a \) raised to the power of \( t \). Identify \( a \): \[ P = P_{0} \times e^{0.2t} \]Thus the function is \( P = P_{0} (e^{0.2})^{t} \).So, \( a = e^{0.2} \).

Step 3: Determine the Nature of the Exponential Function

To decide if it's exponential growth or decay, we need to evaluate the base \( a \): Calculate \( e^{0.2} \approx 1.2214 \). Since \( a > 1 \), this represents exponential growth.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases exponentially over time. It's an important concept in mathematics and is commonly observed in natural processes like population growth, compound interest, and radioactive decay. When we say something grows exponentially, it means the rate of growth accelerates with each passing moment.

• To model exponential growth, we use the equation \( P = P_0 a^t \), where \( P_0 \) is the initial amount, \( a \) is the growth factor, and \( t \) is time.
• In the exercise, \( a = e^{0.2} \approx 1.2214 \), indicating a growth factor greater than 1. This means the quantity is growing.

Understanding exponential growth allows us to predict future amounts based on current data. It highlights how quickly things can grow when amplified by consistent rates over periods. A small change in the growth factor \( a \) can lead to significant changes over time.

Exponential Decay

Unlike exponential growth, exponential decay describes a situation where a quantity decreases over time. You often see decay in processes like depreciation of assets, cooling of hot substances, and radioactive decay.

• The general formula for exponential decay is similar: \( P = P_0 a^t \), but here, \( a \) is between 0 and 1.
• When the base \( a \) is less than 1, it indicates a reduction, causing the value of \( P \) to decrease as time progresses.

In problems involving exponential decay, it's crucial to understand that even though the process slows, it never truly reaches zero. This concept helps in effectively managing resources and understanding long-term trends in diminishing elements.

Exponentiation

Exponentiation is a mathematical operation involving two numbers: a base and an exponent. This operation is the foundation of both exponential growth and decay functions.

• The expression \( a^t \) signifies the base \( a \) raised to the power of \( t \), where \( t \) is any real number.
• The base can be any positive number, but its value (greater than 1 or between 0 and 1) determines whether we observe growth or decay.
• In the example \( P = P_0 e^{0.2t} \), we're using exponentiation with the natural base \( e \), resulting in the calculation of \( e^{0.2} \), showcasing simplicity in representing complicated growth scenarios.

Understanding exponentiation is crucial as it applies across various domains, including but not limited to calculus, finance, and science. Recognizing how to manipulate exponents allows us to comprehensively interpret changes and predict future states.