Problem 25 - Applied Calculus

Question

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 given equation represents exponential growth.

Step 1: Understand the Equation

The equation is given in the form \(P = P_{0} e^{0.2t}\). Here, \(e^{0.2t}\) is the exponential term.

Step 2: Identify the Base of Exponent

The base of the exponential function is \(e\), which is a constant approximately equal to 2.718.

Step 3: Analyze the Exponent

The exponent is \(0.2t\). Since 0.2 is positive, it indicates growth.

Step 4: Recognize the Rate of Change

In the equation \(P = P_{0} e^{0.2t}\), the rate of change is determined by the coefficient 0.2, which further confirms this is exponential growth.

Key Concepts

Exponential Functions

Exponential functions are fundamental in mathematics and appear across various real-life scenarios. An exponential function has the form \( f(t) = P_0 a^{t} \), where \( P_0 \) is the initial value, \( a \) is the base of the exponential, and \( t \) is the time or variable. This type of function is distinct because the variable is in the exponent, leading to rapid increases or decreases, unlike linear functions that grow or decay by a constant amount. Understanding the Roles in Exponential Functions- **\( P_0 \)**: This denotes the starting amount or initial value of the function. It sets the stage for the entire function.- **\( a \)**: The base \( a \) determines whether the function grows or decays. If \( a > 1 \), the function represents growth, while if \( 0 < a < 1 \), it indicates decay.- **\( t \)**: This represents the time or some form of progression, such as periods or cycles.When working with exponential functions, the mathematical constant \( e \) (approximately 2.718) plays an interesting part. Often used as a base in these functions, particularly for modeling natural growth and decay processes, \( e \) serves to represent a natural exponential function.

Exponential Growth

Exponential growth describes a process where the quantity increases rapidly over time. This is characterized by a constant rate of proportionality and is commonly represented as \( P = P_0 e^{kt} \) where \( k > 0 \). In this equation, the initial quantity \( P_0 \) is multiplied by \( e^{kt} \), where \( e \) is the natural exponential base, and \( k \) is a positive constant representing the growth rate.Core Characteristics of Exponential Growth- **Positive Growth Rate**: If \( k \/ \ a > 1 \), the function is growing exponentially, as each unit increase in \( t \) leads to a multiplicative increase in \( P \).- **Doubling Time**: A remarkable property of exponential growth is “doubling time,” which refers to the period it takes for a quantity to double. This is calculated using the approximation: \( T_d = \frac{\ln{2}}{k} \).Exponential growth appears in many contexts, from population growth to finance, where interest compounds, and even in some natural processes. Understanding these dynamics is crucial for predicting behavior over time.

Exponential Decay

Exponential decay is the process where a quantity decreases rapidly at a rate proportional to its current value. Represented by the equation \( P = P_0 a^{t} \), where \( 0 < a < 1 \), or equivalently \( P = P_0 e^{-kt} \) with \( k > 0 \). In these equations, \( P_0 \) signifies the initial amount, and the crucial factor is the base or exponent that indicates a downward trend.Characteristics of Exponential Decay- **Decay Factor**: This is signified by the term \( a \) when \( 0 < a < 1 \) or \( -k \) for \( e^{-kt} \), showing how fast the decay happens. The function's value diminishes rapidly as \( t \) increases.- **Half-life**: An important concept in exponential decay, the "half-life" is the time required for a quantity to reduce to half its initial value. It's given by \( T_{1/2} = \frac{\ln 2}{k} \).Exponential decay is widely observed in fields such as chemistry with radioactive decay, physics, and even in financial models where depreciation is considered. This knowledge helps to manage and anticipate the decline of resources or values over time.