Problem 10 - 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 \(p\) $$y=\frac{2 x^{2}}{10}$$

Step-by-Step Solution

Answer

The function is a power function: \( y = \frac{1}{5} x^2 \), with \( k = \frac{1}{5} \) and \( p = 2 \).

Step 1: Identify the Function Form

The given function is \( y = \frac{2x^2}{10} \). A power function is of the form \( y = kx^p \). Our task is to verify if the given function can be expressed in this form.

Step 2: Simplify the Function

Start by simplifying the given function. We can reduce \( \frac{2}{10} \) to \( \frac{1}{5} \), so the function becomes \( y = \frac{1}{5} x^2 \).

Step 3: Match with Power Function Form

Compare the simplified function \( y = \frac{1}{5} x^2 \) to the power function form \( y = kx^p \). Here, \( k = \frac{1}{5} \) and \( p = 2 \).

Step 4: Determine if it's a Power Function

Since the function can be expressed in the form \( y = kx^p \), with \( k = \frac{1}{5} \) and \( p = 2 \), the given function is indeed a power function.

Key Concepts

Function Simplification

Simplifying a function involves reducing it to its simplest, most understandable form. This often helps to reveal the underlying structure of the function more clearly. In our exercise, the function given was \[ y = \frac{2x^2}{10} \]. By focusing on the fraction \( \frac{2}{10} \), we can simplify it to \( \frac{1}{5} \), simplifying the whole expression to \[ y = \frac{1}{5} x^2 \].
Simplification makes it easier to compare the function to known forms, like the power function form. In math, we often simplify fractions by finding the greatest common divisor of the numerator and the denominator. Here, both 2 and 10 are divisible by 2.

Mathematical Notation

Mathematical notation is like the language of mathematics. It uses symbols to represent numbers, expressions, and operations in a form that is easy to analyze and manipulate.
For example, in our exercise, we use the notation \( y = \frac{2x^2}{10} \) to express the function. The fraction bar \( / \) denotes division, while \( x^2 \) indicates that \( x \) is raised to the power of 2. By transforming this notation to a simpler one, \( y = \frac{1}{5} x^2 \), it becomes clearer how the function relates to the power function form.When comparing to the power function form \( y = kx^p \), consistent notation helps us to immediately recognize \( k = \frac{1}{5} \) and \( p = 2 \). Understanding and correctly using mathematical notation is crucial for identifying function types and solving equations efficiently.

Power Functions

A power function is any function of the form \[ y = kx^p \], where \( k \) and \( p \) are constants, and \( x \) is the variable. These functions are characterized by their simplicity and the role of the exponent \( p \). The exponent indicates the degree of the function and significantly affects its shape and behavior. In the exercise, the given function ultimately takes the form \[ y = \frac{1}{5} x^2 \],
which is a power function where \( k = \frac{1}{5} \) and \( p = 2 \).
Power functions are often used in models because they describe a wide array of phenomena, especially in natural and social sciences, where relationships between variables are not linear but exhibit some exponent behavior. Recognizing a function as a power function allows us to predict its behavior and use this model in real-life applications.