Problem 12 - 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 \(p\) $$y=3 x^{2}+4$$

Step-by-Step Solution

Answer

Not a power function; it has an extra constant term.

Step 1: Understand the Form of a Power Function

A power function is of the form \( y = k \cdot x^p \), where \( k \) and \( p \) are constants. This means the function must consist of a single term involving \( x \) raised to a constant power.

Step 2: Analyze Given Function

Look at the function \( y = 3x^2 + 4 \). It consists of two terms: \( 3x^2 \) and a constant \( 4 \). For it to be a power function, it should have a single term of the form \( kx^p \).

Step 3: Determine If It Fits Power Function Form

The given function has an additional constant (4), which means it doesn't match the power function form \( y = kx^p \) that requires a single term with \( x \) raised to a power. Therefore, it is not a power function because of the additional constant term.

Key Concepts

Understanding Function Forms

When examining mathematical functions, it's essential to distinguish between different types based on their form. The term "form" refers to the structure or the arrangement of mathematical expressions. In the context of power functions, we're specifically looking for functions that can be expressed in a specific format, typically in the form of \( y = k \cdot x^p \). Here, \( y \) represents the output of the function, while \( x \) is the input, often called the variable.

• \( k \) is a constant, which means its value does not change.
• \( p \) is the power to which \( x \) is raised.
• For a function to be classified as a power function, it must be a single term of \( k \cdot x^p \).

Other functions might include additional terms or components, which means they deviate from this power function form. Such variations in terms can indicate the presence of a different kind of function, such as polynomial functions which might include additional constants or multiple terms.

Role of Constant Term in Functions

The concept of a constant term can be slightly confusing when trying to identify the form of a function. A constant term is a fixed value that does not change with the variable \( x \). In our example, the function \( y = 3x^2 + 4 \) includes a constant term of \( 4 \).

• This means regardless of what \( x \) is, there is always an additional value of \( 4 \) added to the function's output.
• In power functions, constant terms are not present because they disrupt the single term structure required for a power function.
• The presence of a constant indicates that the function cannot be written purely in the form \( y = k \cdot x^p \).

Thus, if a function includes an extra constant term or additional terms that are not part of \( k \cdot x^p \), it does not fit the strict definition of a power function. Identifying these constants helps in understanding why a function may not be classified as a specific type.

Power Functions in Calculus

Power functions play a significant role in calculus, primarily because of their simplicity and the ease with which they can be differentiated or integrated. Calculus often deals with changes and areas under curves, making power functions a cornerstone in these calculations.

• Differentiating a power function \( y = k \cdot x^p \) involves applying the power rule, resulting in \( \frac{d}{dx}(k \cdot x^p) = k \cdot p \cdot x^{p-1} \).
• This rule simplifies the process of finding the slope of a tangent line to the curve at any point.
• Integration or finding the anti-derivative of a power function reverses this process, often used to calculate the area under a curve.

Through differentiation and integration, power functions help simplify complex mathematical models, owing to their straightforward properties. Recognizing when a function fits the power function form is crucial for employing these rules accurately within calculus.