Problem 11 - Calculus

Question

Write the functions 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 represents exponential growth, as the base \( a = e^{0.2} > 1 \).

Step 1: Identify the Given Function Structure

The given function is \(P = P_{0} e^{0.2t} \). This is the standard form for exponential functions, which can represent either growth or decay.

Step 2: Compare to Exponential Form Formula

An exponential function is generally expressed in the form \(P = P_{0} a^{t}\), where \(a\) is a positive constant. The base \(a\) determines if the function is growth or decay: if \(a > 1\), it's exponential growth; if \(0 < a < 1\), it's exponential decay.

Step 3: Transform Given Function to Standard Form

To match the given function to \(P = P_{0} a^{t}\), we need to identify \(a\). The function \(P = P_{0} e^{0.2t}\) can be written in the form \(P = P_{0} (e^{0.2})^{t}\).

Step 4: Calculate the Base \(a\)

In the expression \((e^{0.2})^{t}\), the base \(a\) is calculated as \(e^{0.2}\). We know that \(e\) is approximately 2.71828, so \(e^{0.2} \approx 1.2214\).

Step 5: Determine Growth or Decay from Base

Since \(a \approx 1.2214\) and \(a > 1\), the function represents exponential growth.

Key Concepts

Exponential Growth

Exponential growth occurs when a quantity increases at a consistent rate over time. In mathematical terms, this is expressed with a function in the form \( P = P_{0} \cdot a^{t} \), where

• \( P \) represents the quantity at time \( t \)
• \( P_{0} \) is the initial quantity or value
• \( a \) is the growth factor (where \( a > 1 \))
• \( t \) is the time variable

This formula reflects a situation where the rate of increase of \( P \) is proportional to its current value, leading to rapid growth over time.
In the context of the given problem, the function \( P = P_{0} e^{0.2t} \) represents exponential growth because its growth factor, calculated as \( e^{0.2} \), is greater than 1. Specifically, \( e^{0.2} \approx 1.2214 \), which means the quantity increases by approximately 22.14% per time period. Exponential growth is commonly observed in populations, finance with compound interest, and other natural phenomena.

Exponential Decay

Exponential decay is characterized by a decreasing quantity over time, represented mathematically with the formula \( P = P_{0} \cdot a^{t} \), where

• \( P \) is the quantity at time \( t \)
• \( P_{0} \) is the initial quantity
• \( a \) is the decay factor (where \( 0 < a < 1 \))
• \( t \) is the time

When \( a \) is between 0 and 1, the function describes a situation where the quantity decreases over time at a consistent rate.
Although the given function \( P = P_{0} e^{0.2t} \) is an example of exponential growth, understanding exponential decay is crucial for situations such as radioactive decay, depreciation in value, or cooling processes. In these cases, the typical approach is to use a decay factor \( a \) that indicates how much of the quantity remains after each time interval. Thus, while the current exercise does not feature decay, grasping both principles is essential for a full understanding of exponential functions.

Function Transformation

Function transformation often involves altering the basic form of a function to shift, stretch, compress, or reflect it on a graph. Understanding transformations helps to identify and visualize changes in a mathematical model.
The transformation of an exponential function can affect its parameters or the entire expression. Types of transformations include:

• Translation: Shifting the graph vertically or horizontally
• Reflection: Flipping the graph over an axis
• Stretching or Compressing: Changing the rate of growth or decay by adjusting the growth factor \( a \)

In the exercise, the function \( P = P_{0} e^{0.2t} \) stays in its exponential form but can be interpreted as a transformation from the base model \( P = P_{0} e^{t} \) by affecting the rate of growth. The term \( 0.2 \) in the exponent affects the speed at which \( P \) grows relative to time. Grasping these transformations aids in comprehending shifts in function behavior and their graphical representations.