Problem 9 - 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 \(p\) $$y=3 \cdot 5^{x}$$

Step-by-Step Solution

Answer

The function is not a power function; it's exponential.

Step 1: Understanding the Function

Examine the given function \( y = 3 \cdot 5^{x} \). We need to determine if it fits the form \( y = kx^{p} \), which defines a power function. A power function has a variable base raised to a constant exponent.

Step 2: Comparing with Power Function Form

The form \( y = k x^{p} \) suggests the base is a variable \( x \) raised to a constant power \( p \). However, in \( y = 3 \cdot 5^{x} \), the base is a constant \( 5 \) and the exponent is the variable \( x \).

Step 3: Conclusion on Function Type

Since the function \( y = 3 \cdot 5^{x} \) has a constant base raised to the variable power, it is not a power function. It fits the form of an exponential function \( f(x) = a \cdot b^{x} \), where \( a = 3 \) and \( b = 5 \).

Key Concepts

Power Functions

When it comes to understanding different types of functions, power functions play a crucial role. A power function is generally expressed in the form:

• \( y = kx^p \)

where \( k \) and \( p \) are constants, and \( x \) is the variable. The constants \( k \) and \( p \) must be real numbers, with \( x \) serving as the base. In a true power function, the variable is always in the base, and it's raised to a constant exponent.
Power functions are easily recognizable:

• The function has a form like \( y = x^2 \), \( y = 5x^3 \), or even \( y = 0.5x^{-1} \).
• They feature a consistent operation on the variable \( x \), which is exponentiation.

These functions are fundamental in calculus and other areas because of their simple form and predictable behavior, making them easier to differentiate and integrate. When determining if a function is a power function, always ensure that the variable \( x \) is the base, and it is raised to a fixed power.

Function Types

Functions are the backbone of calculus and other mathematical analyses. Understanding different types of functions helps in identifying their properties and behavioral patterns. Two broad categories of functions often encountered are power functions and exponential functions.

• Power Functions: As previously explained, involve a variable base raised to a constant power.
• Exponential Functions: Unlike power functions, have a constant base and a variable exponent. Notable form: \( y = ab^x \), like \( y = 3 \cdot 5^x \).

Distinguishing between these function types is crucial:

• In power functions, \( x \) is the base. In exponential functions, \( x \) appears in the exponent.
• Exponential functions often model growth and decay, where the rate of change is proportional to the value of the function at any point.

Recognizing these distinctions aids in tackling various mathematical problems, especially in calculus-where derivatives and integrals behave differently for different function types.

Calculus

Calculus is a vast field of mathematics centered around the concepts of change and motion. It's essential when studying functions like power and exponential types, as it allows us to understand their behavior through derivatives and integrals.
In calculus:

• The derivative measures how a function changes as its input changes. For power functions, the derivative of \( y = kx^p \) follows the rule \( y' = pkx^{p-1} \).
• In contrast, exponential functions, like \( y = ab^x \), have a derivative that takes the form \( y' = ab^x \ln(b) \).
• Integrals, on the other hand, help calculate areas under curves as well as accumulated quantities and can be found using rules specific to the function type.

Calculus provides the tools needed to model and predict behaviors in natural phenomena and real-world applications. Mastering power and exponential functions within calculus is key to unlocking more complex mathematical concepts.