Problem 40 - College Algebra with Modeling and Visualization

Question

Exercises \(39-44\) : Average Rate of Change Find the average rate of change of \(f\) from \(-2\) to \(2 .\) What is the average rate of change of \(f\) from \(x_{1}\) to \(x_{2}\), where \(x_{1} \neq x_{2} ?\) $$ f(x)=-5 $$

Step-by-Step Solution

Answer

The average rate of change is 0.

Step 1: Introduction to the Problem

The exercise asks for the average rate of change of the function \( f(x) = -5 \) over the interval from \( x_1 = -2 \) to \( x_2 = 2 \).

Step 2: Formula for Average Rate of Change

The average rate of change of a function \( f(x) \) from \( x_1 \) to \( x_2 \) is given by the formula \[ \frac{f(x_2) - f(x_1)}{x_2 - x_1} \].

Step 3: Evaluate Function at Endpoints

Determine \( f(x_1) \) and \( f(x_2) \). For \( f(x) = -5 \), no matter what \( x \) is, \( f(x) = -5 \). Thus, \( f(-2) = -5 \) and \( f(2) = -5 \).

Step 4: Apply the Formula

Substitute \( f(-2) = -5 \) and \( f(2) = -5 \) into the formula: \[ \frac{-5 - (-5)}{2 - (-2)} = \frac{0}{4} = 0 \].

Step 5: Conclusion

The average rate of change of the constant function \( f(x) = -5 \) from \( x = -2 \) to \( x = 2 \) is \( 0 \). This occurs because the function is constant and does not change regardless of \( x \).

Key Concepts

Understanding Constant Functions

In mathematics, a constant function is a very straightforward but foundational concept. A constant function is defined as a function that always returns the same value, no matter what the input is. It's like setting the volume on a speaker and keeping it static. For example:

• If you have a function expressed as \( f(x) = -5 \), it means no matter what value you choose for \( x \), the output will always be \(-5\).

There are no hills or valleys in the graph of a constant function, just a flat line. This means there is no change or variation, making its average rate of change zero over any interval. Like our example, when asked to find the average rate of change from \( -2 \) to \( 2 \), you will find that this rate is always zero. This is because there is no difference in the function's value at any point.

Using the Rate of Change Formula

The rate of change formula is a useful tool in understanding how a quantity changes over a certain interval. For any given function \( f(x) \), the average rate of change from \( x_1 \) to \( x_2 \) is given by the formula: \[ \frac{f(x_2) - f(x_1)}{x_2 - x_1} \]This formula works like finding the slope of a line between two points on a graph.

• For constant functions, like \( f(x) = -5 \), no matter what two points \( x_1 \) and \( x_2 \) you select, the function value \( f(x) \) will be the same at both points.
• This results in the numerator \( f(x_2) - f(x_1) \) being zero, and hence, the average rate of change is zero.

Understanding this formula helps in realizing why constant functions have zero change. They are pivotal in recognition that some quantities remain unchanged over time or various conditions.

Introduction to Mathematical Intervals

Mathematical intervals are a way to describe a range of numbers between two endpoints. They specify the domain over which we evaluate functions or perform operations. Here’s how to understand them better:

• An interval might be closed, where both endpoints are included, written as \([a, b]\).
• Or an interval might be open, where one or both endpoints are excluded, noted as \((a, b)\), \([a, b)\), or \((a, b]\).

In exercises about the average rate of change, like finding this rate from \( -2 \) to \( 2 \), it’s likely working with a closed interval to surely include the endpoints. When dealing with constant functions in these intervals, the calculation reveals consistency. The function's lack of change inside the interval, regardless of how wide or narrow, illustrates how intervals don't affect a constant function’s output.