Problem 30 - College Algebra with Corequisite Support

Question

For the following exercises, find the average rate of change of each function on the interval specified. \(q(x)=x^{3}\) on [-4,2]

Step-by-Step Solution

Answer

The average rate of change is 12.

Step 1: Understanding the Problem

We need to find the average rate of change of the function \(q(x) = x^3\) over the interval \([-4, 2]\). The average rate of change provides information on how much the function's output changes, per unit change in the input, across the given interval.

Step 2: Formula for Average Rate of Change

The average rate of change of a function \(q(x)\) over the interval \([a, b]\) is given by the formula: \[\text{Average Rate of Change} = \frac{q(b) - q(a)}{b - a}\] Here, \(a = -4\) and \(b = 2\).

Step 3: Calculate Function Values at Endpoints

Calculate \(q(2)\) and \(q(-4)\) using the function \(q(x) = x^3\):- \(q(2) = 2^3 = 8\)- \(q(-4) = (-4)^3 = -64\)

Step 4: Apply the Formula

Substitute the calculated values into the average rate of change formula:\[\text{Average Rate of Change} = \frac{q(2) - q(-4)}{2 - (-4)} = \frac{8 - (-64)}{2 + 4} = \frac{8 + 64}{6}\]

Step 5: Simplify the Expression

Calculate the result from the expression:\[\frac{72}{6} = 12\]Thus, the average rate of change of the function \(q(x) = x^3\) on the interval \([-4, 2]\) is 12.

Key Concepts

Polynomial Functions

Polynomial functions are mathematical expressions involving a sum of powers of one or more variables multiplied by coefficients. A polynomial function can be expressed generally as \( f(x) = a_n x^n + a_{n-1} x^{n-1} + ... + a_1 x + a_0 \), where each \( a_i \) is a constant, and \( n \) denotes the degree of the polynomial, which is the highest power of the variable \( x \).
In the given problem, the function \( q(x) = x^3 \) is a simple polynomial of degree 3, also known as a cubic polynomial. The highest power of \( x \) is 3, which makes it a cubic function. Cubic polynomials can have differing shapes – they might curve in a wavy fashion or steeply rise or fall, depending on the coefficients and the number of real roots they possess.
Polynomial functions are continuous and smooth, which means they have no breaks, holes, or sharp corners. This makes them convenient for solving real-world problems where smoothness and continuity are required. Understanding the behavior and characteristics of polynomial functions is key to analyzing their rate of change.

Interval Analysis

Interval analysis involves studying a function's behavior between two points, often denoted as \([a, b]\). When an exercise asks you to determine the average rate of change over an interval, it is essentially asking you to evaluate how the function behaves between these two x-values.
In this context, the interval \([-4, 2]\) defines the range within which we examine the changes in the function \( q(x) = x^3 \). Interval analysis helps in identifying how the function values increase or decrease over specific sections of the input variable, offering insights into the general trends of the function over specified ranges.
By determining the average rate of change over an interval, you gauge how much, on average, the output of the function changes per unit increase in \( x \). This methodology provides a broad overview of the function's behavior, which can be crucial in applications where understanding the overall trend of the function is more important than focusing on specific points.

Function Evaluation

Function evaluation involves calculating the value of a function for specific inputs. This is a critical step in analyzing and interpreting the behavior of mathematical functions. To find the average rate of change, you need to evaluate the function at the endpoints of the interval.
Given the function \( q(x) = x^3 \), we evaluate \( q(2) \) and \( q(-4) \). This means substituting these x-values into the polynomial:

• For \( x = 2: q(2) = 2^3 = 8 \)
• For \( x = -4: q(-4) = (-4)^3 = -64 \)

After evaluating these, you use the calculated function values to apply the average rate of change formula \( \frac{q(b) - q(a)}{b - a} \). This technique is pivotal not just for this problem but also broadly in calculus and other math-related fields, as it forms the basis for more complex types of function analysis. By mastering function evaluation, you're better equipped to tackle an array of mathematical challenges that involve understanding how input changes affect output results.