Problem 26 - Applied Calculus

Question

In the form \(P=P_{0} a^{t}\). Which represent exponential growth and which represent exponential decay? $$P=7 e^{-\pi t}$$

Step-by-Step Solution

Answer

Exponential decay

Step 1: Identify the components of the given function

The function given is \(P = 7 e^{-\pi t}\). Here, the initial value \(P_0 = 7\) and the base of the exponential function is \(e^{-\pi}\). In the form \(P = P_0 a^t\), \(a\) corresponds to the base of the exponential term, which in this case is \(e^{-\pi}\).

Step 2: Determine the value of the exponential base

The base \(a\) here is \(e^{-\pi}\). Since \(e\) is a mathematical constant approximately equal to 2.71828, \(e^{-\pi}\) means \(e\) raised to the power of \(-\pi\), which is a negative exponent.

Step 3: Analyze the sign of the exponent

Because the exponent of \(e\) is negative (\(-\pi\)), the exponential form \(e^{-\pi}\) will result in a value that is a positive fraction less than one. This is because any positive number with a negative exponent results in a fraction.

Step 4: Determine the type of exponential function

In exponential functions, if the base \(a\) is less than 1, the function represents exponential decay. Since \(e^{-\pi}\) is a fraction less than one, the function \(P = 7 e^{-\pi t}\) represents exponential decay.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions that grow or decay at a constant percentage rate. They are unique in that a constant, usually greater than one, is raised to the power which varies over time. These functions can model a vast range of real-world phenomena such as population growth, radioactive decay, and even the cooling of a hot object. The general form of an exponential function is given by:

• \( P = P_0 a^t \)

Where:

• \( P \) is the final amount or value.
• \( P_0 \) is the initial amount or value.
• \( a \) is the base of the exponential function, determining the nature of growth or decay.
• \( t \) is the time variable.

When the base \( a \) is greater than one, the function models exponential growth, meaning it increases over time. Conversely, if the base \( a \) is between zero and one, the function will represent exponential decay, illustrating decrease over a period.

Negative Exponent

In mathematics, a negative exponent signifies that the base should be moved to the denominator of a fraction. It shows how division works in exponential terms and provides a direct connection to how values decrease in exponential decay functions.For example, \( e^{- ext{value}} \) is equivalent to \( \frac{1}{e^{ ext{value}}} \). Here's what a negative exponent does:

• Makes large numbers small: A negative power of a large constant like \( e \) results in a very small number or fraction.
• Indicates inversion: It tells you to take the reciprocal of the base raised to a positive exponent.
• In exponential decay: It ensures that the exponential function decreases over time.

So when the function is \( e^{- ext{something}} \), like \( e^{- ext{π}} \) in our example, it creates a fraction, causing the overall exponential function to decay as time \( t \) increases. This aligns with the premise that for decay, the exponential term must be constantly less than one.

Mathematical Constant e

The mathematical constant \( e \), known as Euler's number, plays a critical role in exponential functions, especially those representing natural phenomena. Approximately equal to 2.71828, \( e \) is the base of natural logarithms and is pivotal in describing continuously compounding growth or decay processes.Key traits of \( e \):

• It is an irrational number, meaning it cannot be expressed as a simple fraction.
• \( e \) originates from the concept of continuously compounded interest, which is why it naturally appears in growth and decay contexts.
• The function \( e^x \) is its own derivative, making calculus operations simpler in many exponential scenarios.

In our example \( P = 7 e^{- ext{π} t} \), \( e \) forms the base of the exponential part, capturing the natural decrease dictated by a negative coefficient of \( t \). This not only helps illustrate decay but also simplifies calculations related to changes over continuous intervals.