Problem 40 - Calculus: One and Several Variables

Question

Each of the following sets is the solution of an inequality of the form \(|x-c|<\delta\). Find \(c\) and \(\delta\). $$(0,4)$$.

Step-by-Step Solution

Answer

The pair \((c, \delta)\) that satisfies the given inequality with the interval \((0, 4)\) as its solution is \((2, 2)\).

Step 1: Identifying the center of interval

To find \(c\), we calculate the center of the interval \((0, 4)\) by taking the average of \(0\) and \(4\). This yields \(c=2\).

Step 2: Identifying delta

To find \(\delta\), we calculate the difference between any one end of the interval and the center from Step 1. Hence \(\delta=2\).

Key Concepts

Absolute Value Inequality

Absolute value inequalities tell us about the distance from a point on the number line. For the inequality \(|x - c| < \delta\), it means that the distance between \(x\) and \(c\) is less than \(\delta\). This creates an interval centered at \(c\) with a radius of \(\delta\).
If we take the inequality \(|x - c| < 2\), this defines all the points within 2 units of \(c\). Whenever you see an absolute value inequality like this, imagine a flexible boundary stretching equally in both directions from the center point \(c\).

• "Less than" (\(<\)) generates an open interval.
• "Greater than" (\(>\)) creates a different kind of solution, usually involving multiple intervals.

Interval Notation

Interval notation is a shorthand way to describe a set of numbers along the number line. It uses brackets and parentheses to denote closed and open intervals. In our example, the interval \((0, 4)\), the parentheses signify that 0 and 4 are not included in the set.
- Use \((a, b)\) for open intervals where endpoints \(a\) and \(b\) are not included.- Use \([a, b]\) for closed intervals where both endpoints are included.
In inequalities, switching between expressions like \(|x - c| < \delta\) and interval notation \((c - \delta, c + \delta)\) helps visualize solutions.

Solution Sets

A solution set indicates all the possible values of \(x\) that satisfy the inequality. From the original problem, the solution set is \((0, 4)\), meaning any \(x\) between these numbers satisfies the inequality.
The concept of solution sets:

• Helps understand what values work for your inequality.
• Describes these values using interval notation.
• Visualizes them on a number line, providing a clear picture of inclusivity or exclusivity.

Understanding solution sets ultimately offers insight into what the inequality is actually conveying.

Center of Interval

To find the center \(c\) of an interval, you calculate the average of its endpoints. In our exercise, the interval is \((0, 4)\), so the center is determined by the formula \(c = \frac{0 + 4}{2}\), yielding \(c = 2\).
The center is crucial in absolute value inequalities as it represents the midpoint around which the distance (\(\delta\)) is measured. Hence, finding this center correctly informs the rest of the problem:

• It tells us where the "heart" of the interval lies.
• It provides a reference point for calculating \(\delta\).

The center simplifies understanding how far \(x\) can move from this midpoint and still remain part of the solution set.