Problem 3 - Applied Calculus

Question

Determine whether or not the function is a power function. If it is a power function, write it in the form \(y=k x^{p}\) and give the values of \(k\) and \(p\). $$y=2^{x}$$

Step-by-Step Solution

Answer

The function is not a power function because it is in the form of an exponential function.

Step 1: Understand Power Functions

A power function is defined as a function that can be written in the form \( y = kx^p \), where \( k \) and \( p \) are constants and \( k eq 0 \).

Step 2: Analyze Given Function

The given function is \( y = 2^{x} \). This function is an exponential function because the variable \( x \) is in the exponent rather than the base.

Step 3: Compare with Power Function Form

Since a power function has the form \( y = k x^p \) and our given function has the form \( y = a^x \), it is clear that these forms do not match because the variable is the exponent, not the base. Hence, \( y = 2^x \) is not a power function.

Key Concepts

Exponential Functions

Exponential functions are a special type of function where the variable appears in the exponent. This means the function has the general form \( y = a^x \). The base \( a \) is a constant, and the exponent is the variable \( x \). Exponential functions are known for their rapid growth or decay, depending on the value of \( a \). For instance, when \( a > 1 \), the function increases exponentially, and when \( 0 < a < 1 \), the function decreases or decays exponentially.

Some common real-world applications of exponential functions include:

• Population growth, which can be modeled using functions where the population grows by a fixed percentage annually.
• Radioactive decay, where substances decrease at a consistent rate over time.
• Interest calculations in finance, such as compound interest.

Understanding the characteristics of exponential functions, like their domain, range, and asymptotes, helps in analyzing different real-world phenomena.

Power Function Analysis

Power functions have a distinct structure and are defined by the formula \( y = k x^p \). Here, \( k \) is a constant that stretches or compresses the graph of the function, while \( p \) determines the degree or the power of the function. The variable \( x \) is the base, and unlike exponential functions, the power \( p \) is a constant.

The behavior of a power function depends significantly on the value of \( p \):

• If \( p \) is a positive integer, the function is a polynomial.
• If \( p \) is a negative integer, the function can represent scenarios like inverse relationships.
• If \( p \) is a fraction (like 1/2), the function can represent roots, such as square roots or cube roots.

This variability in \( p \) makes power functions versatile for modeling different mathematical and physical phenomena. They behave differently based on the sign and magnitude of \( p \), making their analysis crucial in understanding their graphs and real-world applications.

Function Forms

The form of a function plays a critical role in classifying and understanding it. While both exponential and power functions involve constants and variables, they are structured differently and serve different purposes.

When analyzing functions, recognizing their form is key to understanding their behavior and applications:

• **Exponential form**: As seen before, exhibits growth or decay depending on the base \( a \).
• **Power form**: Characterized by \( y = k x^p \), where the exponent is a fixed number, defines various curves depending on the power \( p \).

To determine whether a given function is a power function, one must check if it can be written in the form \( y = k x^p \). If the function has a variable in the exponent, as in \( y = 2^x \), it is an exponential function, not a power function.

Understanding these forms allows us to apply the correct analytical techniques, predict function behavior, and solve problems effectively.