Problem 32 - College Algebra

Question

For the following exercises, find the average rate of change of each function on the interval specified. $$y=\frac{1}{x} \text { on }[1,3]$$

Step-by-Step Solution

Answer

The average rate of change is \(-\frac{1}{3}\).

Step 1: Identify the Function and Interval

The function given is \( y = \frac{1}{x} \) and the interval is \([1, 3]\). We need to find the average rate of change of this function over this interval.

Step 2: Calculate the Function Values at the Interval's Endpoints

Calculate the function value at the lower endpoint: \( f(1) = \frac{1}{1} = 1 \). Calculate the function value at the upper endpoint: \( f(3) = \frac{1}{3} \).

Step 3: Use the Average Rate of Change Formula

The average rate of change formula for a function \( y = f(x) \) on an interval \([a, b]\) is given by \( \frac{f(b) - f(a)}{b-a} \). Substitute \( a = 1 \), \( b = 3 \), \( f(1) = 1 \), and \( f(3) = \frac{1}{3} \) into the formula.

Step 4: Perform the Calculation

Substitute the known values into the formula: \( \frac{\frac{1}{3} - 1}{3 - 1} = \frac{-\frac{2}{3}}{2} \). Simplify this expression to find the average rate of change: \( \frac{-\frac{2}{3}}{2} = -\frac{1}{3} \).

Key Concepts

Rational Functions

Rational functions are a fascinating type of mathematical expression. They are defined as the ratio of two polynomials. Think of it like a fraction where both the numerator and denominator are polynomials. The function we are inspecting, \( y = \frac{1}{x} \), is a classic example of a rational function. Here:

• The numerator is just \(1\), which is a simple constant polynomial.
• The denominator is \(x\), which is a linear polynomial.

Rational functions can exhibit quite different behaviors such as asymptotes and discontinuities, compared to simpler polynomial functions.
Due to the division by \(x\), it is important to remember that \(x\) cannot be zero in our rational function because division by zero is undefined. This plays a crucial role when determining the domain of the function and setting any intervals for analysis.

Interval Notation

Interval notation is a concise way to indicate a range of numbers. It is used to specify the domain over which we are interested in analyzing a function or conducting calculations, such as the average rate of change in our example. You will frequently encounter it when working with calculus and functions.
In general, intervals can be expressed in different ways:

• Square brackets \([a, b]\) include both endpoints \(a\) and \(b\).
• Parentheses \((a, b)\) exclude these endpoints.
• Mixed notations \((a, b]\) or \([a, b)\) include one endpoint and exclude the other.

For the exercise at hand, the interval is given as \([1, 3]\). This means that the analysis includes both \(x = 1\) and \(x = 3\), making them part of the domain we consider when evaluating the rational function.

Function Evaluation

In mathematics, function evaluation is the process of determining the output of a function given specific input values. When faced with a task involving functions, such as finding the average rate of change, evaluating the function at particular points is crucial.
The function mentioned in our exercise, \( y = \frac{1}{x} \), requires evaluation at two specific points: \(x = 1\) and \(x = 3\), the endpoints of the interval. Here's how it's done:

• For \(f(1)\): Substitute \(x = 1\) into the function, resulting in \(y = \frac{1}{1} = 1\).
• For \(f(3)\): Substitute \(x = 3\) into the function, resulting in \(y = \frac{1}{3}\).

These steps allow us to gather the necessary values to calculate the average rate of change, effectively linking the elements of function evaluation with the broader analysis.

Average Rate of Change Formula

The average rate of change provides insight into how a function's output changes, on average, across a specified interval. It's a critical concept in calculus, helping to characterize the function's overall behavior over that interval.
The average rate of change formula is derived from the slope formula used in algebra for lines, expanded for functions across intervals. For a function \( y = f(x) \), the average rate of change over an interval \([a, b]\) is:\[\frac{f(b) - f(a)}{b-a}\]This formula essentially gives the slope of the "secant line" that passes through the points \((a, f(a))\) and \((b, f(b))\). It measures how much \(f(x)\) changes per unit change in \(x\) across the interval.
In our case, using \(f(1) = 1\) and \(f(3) = \frac{1}{3}\), the formula becomes:\[\frac{\frac{1}{3} - 1}{3 - 1} = \frac{-\frac{2}{3}}{2} = -\frac{1}{3}\]This result tells us that, on average, for each unit increase in \(x\) from 1 to 3, the function \( f(x) \) decreases by \( \frac{1}{3} \).