Problem 42 - College Algebra with Corequisite Support

Question

Let \(f(x)=\frac{1}{x}\). Find a number \(c\) such that the average rate of change of the function \(f\) on the interval \((1, c)\) is \(-\frac{1}{4}\).

Step-by-Step Solution

Answer

The value of \(c\) is 4.

Step 1: Understand the Average Rate of Change Formula

The average rate of change of a function from point \(x=a\) to \(x=b\) is given by \[\frac{f(b) - f(a)}{b-a}.\] We need to apply this formula to the function \(f(x) = \frac{1}{x}\) over the interval \((1, c)\).

Step 2: Set Up the Equation for the Interval

For the function \(f(x) = \frac{1}{x}\), the points are \(a = 1\), \(b = c\), with corresponding values \(f(1) = 1\) and \(f(c) = \frac{1}{c}\). This gives us the equation for the average rate of change as \[\frac{1/c - 1}{c - 1} = -\frac{1}{4}.\]

Step 3: Simplify the Left-Hand Side

The expression \(\frac{1/c - 1}{c - 1}\) can be simplified: \[\frac{\frac{1-c}{c}}{c - 1} = \frac{-\frac{c-1}{c}}{c-1} = -\frac{1}{c}.\]

Step 4: Set the Equation to Match the Given Average Rate of Change

Set the simplified expression equal to the given average rate of change: \[-\frac{1}{c} = -\frac{1}{4}.\] Then solve for \(c\).

Step 5: Solve for c

By equating and solving \(-\frac{1}{c} = -\frac{1}{4}\), multiply both sides by \(c\): \[\frac{1}{c} = \frac{1}{4} \implies c = 4.\] Thus, the value of \(c\) is 4.

Key Concepts

Functions

A function is a mathematical relationship between a set of inputs and a set of possible outputs. In simpler terms, a function is like a machine where you input a value and the machine gives you an output according to a specific rule. This rule defines how the input and output are related.
In the given exercise, we are working with the function \(f(x) = \frac{1}{x}\). This is a type of function known as a rational function because it is the ratio of two polynomials. Here, the input is \(x\), and the output is \(\frac{1}{x}\), meaning that for any number \(x\) (except zero), the function outputs the reciprocal of \(x\).
Understanding functions is crucial in Calculus and many other mathematical areas. Functions allow us to model real-world phenomena and make predictions. They provide a systematic way of explaining the relationship between quantities and are foundational to understanding mathematical changes, such as rate of change, which we see here in this problem.

Interval

In mathematics, an interval is a range of numbers between two endpoints. Intervals can be either closed, open, or a mix (half-open).

• Closed Interval: Includes its endpoints, denoted by square brackets, e.g., \([a, b]\).
• Open Interval: Does not include its endpoints, denoted by parentheses, e.g., \((a, b)\).
• Half-open Interval: Includes one endpoint but not the other, e.g., \([a, b)\) or \((a, b]\).

In the exercise, the interval in question is \((1, c)\), which means it is an open interval from 1 to \(c\), excluding both 1 and \(c\). The goal is to find a value of \(c\) such that the average rate of change of the function over this interval is a specific value. The notion of intervals helps us specify the portion of the domain we are analyzing, which is essential for exploring how functions behave over specific ranges.
Understanding intervals is key when discussing limits, continuity, and derivatives in calculus and other advanced mathematical subjects.

Solving Equations

Solving equations is about finding the unknown values that make a given equation true. It involves manipulating the equation using various algebraic methods to isolate the unknown variable.
In this exercise, the key step was setting up an equation based on the average rate of change formula for the interval \((1, c)\). The equation derived is \(-\frac{1}{c} = -\frac{1}{4}\). Solving this involves:
1. Understanding that the terms on each side with negatives are equal if their absolute values are equal. Thus, \(\frac{1}{c} = \frac{1}{4}\).
2. Solving for \(c\) by reciprocating both sides to get \(c = 4\).
These steps use core algebraic techniques, such as reciprocals and simplification, which are crucial in handling rational equations.
Solving equations is a fundamental skill in math used to find unknowns in various types of problems, making it a valuable tool in everyday mathematical reasoning.