Problem 25 - Calculus Single Variable

Question

In Exercises \(25-28,\) write the interval in the form \(\\{x:|x-c|< r\\}\) or \(\\{x:|x-c| \leq r\\}\). [-1,3]

Step-by-Step Solution

Answer

The interval [-1, 3] can be written as \({x: |x - 1| \leq 2}\).

Step 1: Identify the Interval Endpoints

The given interval is \([-1, 3]\). This interval includes all numbers from \(-1\) to \(3\), inclusive.

Step 2: Determine the Center ( \(c\)) and Radius ( \(r\))

The center, \(c\), of the interval is the midpoint of \(-1\) and \(3\). To find the midpoint, use the formula: \(c = \frac{a + b}{2}\), where \(a = -1\) and \(b = 3\). Compute: \(c = \frac{-1 + 3}{2} = 1\). The radius \(r\) is half the distance between the endpoints, which can be found using: \(r = \frac{b - a}{2}\), so \(r = \frac{3 - (-1)}{2} = 2\).

Step 3: Express the Interval Using Absolute Value Notation

Now that you have \(c = 1\) and \(r = 2\), express the interval \([-1, 3]\) in the form \({x: |x - c| \leq r}\). Substitute the values: \({x: |x - 1| \leq 2}\).

Key Concepts

Midpoint of an Interval

Understanding the midpoint of an interval is a fundamental concept in mathematics. To find the midpoint between two endpoints of an interval, utilize the formula: \[ c = \frac{a + b}{2} \]where \(a\) and \(b\) are the endpoints of the interval. The midpoint is essentially the "center" point that evenly divides the interval into two halves.
For example, consider the interval \([-1, 3]\). Plugging into the formula, we get:

• \(a = -1\)
• \(b = 3\)
• \(c = \frac{-1 + 3}{2} = 1\)

Thus, the midpoint of the interval \([-1, 3]\) is 1. Calculating the midpoint is particularly useful when trying to find the average or the center within any given set of values.
The midpoint plays a pivotal role in locating both the center and the distance of any interval, and it is crucial for expressing intervals using absolute value notation.

Interval Endpoints

Endpoints define the boundaries of an interval, marking where it begins and ends. When given an interval like \([-1,3]\), the numbers -1 and 3 are known as the endpoints.
Endpoints can be:

• Inclusive, when both endpoints are part of the interval, denoted by brackets: \([-1, 3]\)
• Exclusive, indicated through parentheses: \((-1, 3)\)

The nature of endpoints affects how we visualize and work with intervals. If an interval includes its endpoints, like in our example, it sometimes becomes necessary to demonstrate this by using the absolute value notation.
It is vital to clearly identify these endpoints, as they determine both the length of the interval and how we express it in equations and expressions. Correctly recognizing and interpreting endpoints is fundamental to solving many mathematical problems involving intervals.

Radius of an Interval

The radius of an interval represents half the length of the interval or the distance from the midpoint to either endpoint. Calculating the radius involves determining the difference between the endpoints and then dividing by two:\[ r = \frac{b - a}{2} \]
In our example with the interval \([-1, 3]\), the calculation is as follows:

• \(a = -1\)
• \(b = 3\)
• \(r = \frac{3 - (-1)}{2} = 2\)

Thus, the radius of this interval is 2. The radius helps us understand the "spread" or "distance" within an interval, and it's fundamental when expressing an interval using absolute value notation.
Knowing the radius ensures we can correctly depict the interval and interpret various mathematical contexts, such as determining how far the points in the interval are spread around the midpoint.

Interval Notation

Interval notation is a concise way of writing intervals, showing the set of numbers between endpoints. In mathematics, it helps represent bounded sections of the number line effectively.
This notation uses brackets \([ \text{and} ]\) to include endpoints, parentheses \(( \text{and} )\) to exclude them. For the interval \([-1, 3]\), the notation tells us:

• The interval includes -1, using \([\)
• It includes 3, using \(]\)

Expressing intervals with absolute value notation, such as \(|x-c| \leq r\), is another method undertaken to simplify and standardize interpretation. It involves stating the conditions for \(x\) regarding a midpoint and radius.
Understanding interval notation is vital in various fields, including calculus and algebra, to handle inequalities, functions, and more efficiently. It's a universal language for representing ranges and is essential for any mathematical discussion involving intervals.