Problem 27 - Calculus Single Variable

Question

Write the interval in the form \(\\{x:|x-c|< r\\}\) or \(\\{x:|x-c| \leq r\\}\).\((-\pi, \pi+2)\)

Step-by-Step Solution

Answer

The interval can be expressed as \(\{x:|x-1| < \pi + 1\}\).

Step 1: Identify the Center and Radius

To express the interval (-\pi, \pi+2) in the form \(\{x:|x-c| < r\}\), we must first find the midpoint (or center) of the interval, which will be \(c\). To do this, calculate the midpoint using \(c = \frac{a+b}{2}\), where \(a\) and \(b\) are the endpoints of the interval. So, \(c = \frac{-\pi + (\pi + 2)}{2}\).

Step 2: Calculate the Center

Substitute the values into the midpoint formula: \(c = \frac{-\pi + \pi + 2}{2} = \frac{2}{2} = 1\). The center \(c\) of the interval is therefore 1.

Step 3: Calculate the Radius

The radius \(r\) is given by half the length of the interval. Calculate the length of the interval by subtracting the lower endpoint from the upper endpoint: \(\pi + 2 - (-\pi) = 2\pi + 2\). Now, divide this length by 2 to find the radius: \(r = \frac{2\pi + 2}{2}\), which simplifies to \(\pi + 1\).

Step 4: Express the Interval in Absolute Value Form

Now that we know the center \(c = 1\) and the radius \(r = \pi + 1\), we can write the interval in the form \(\{x:|x-c| < r\}\). This gives \(\{x:|x-1| < \pi + 1\}\). Thus, the interval (-\pi, \pi+2) can be expressed in the desired form.

Key Concepts

Understanding Absolute Value in Interval Notation

When you see the term "absolute value," it refers to the distance of a number from zero on the number line, regardless of direction. In our context, it helps describe how far numbers within an interval can stray from a central point, known as the center of the interval.

For example, if you're given the expression \(|x-1| < \pi + 1\), it means that the numbers "x" can differ from 1 by less than \(\pi + 1\). Key things to remember about absolute values in interval notation include:

• The absolute value expression \(|x-c|\) allows all "x" such that their distance from "c" is less than a specified radius \(r\).
• It represents a range of numbers forming an interval centered around "c" with a reach defined by "r."
• Absolute value intervals can represent both open and closed sets, shown by inequalities using "<" or "\leq" respectively.

Identifying Interval Endpoints

In order to express an interval using absolute value, it's crucial to identify the interval's endpoints. Endpoints provide the bounds of the interval and are essential for other calculations.

Consider the interval (-\pi, \pi+2):

• The lower endpoint is the smaller number. In this case, it is \(-\pi\).
• The upper endpoint is the larger number, which is \(\pi + 2\).
• These endpoints determine the initial size of the interval and the midpoint used to find the center ("c").

Knowing these bounds is the first step to several more calculations necessary to express the interval in its absolute value form.

Utilizing the Midpoint Formula

Intervals require a center point to utilize the absolute value format properly. The midpoint (or center) can be efficiently found using the midpoint formula:

\[ c = \frac{a + b}{2} \]

where \(a\) and \(b\) are the endpoints of the interval.

Let's revisit our example with the interval (-\pi, \pi+2):

• Substitute the interval endpoints into the midpoint formula.
• Calculate: \( c = \frac{-\pi + (\pi + 2)}{2} = 1 \).
• This shows that our center or midpoint "c" is 1.

The midpoint provides a foundation around which the rest of the interval is symmetrically distributed.

Radius Calculation for Intervals

Radius is a critical element when describing intervals using absolute values. It reflects how far the interval extends around its center.

To find the radius, first determine the interval's total length. Use the difference between the two endpoints:

• For our example, \(\pi + 2 - (-\pi) = 2\pi + 2\).
• The interval's length, 2\pi + 2, is then halved to discover the radius.
• Calculate: \( r = \frac{2\pi + 2}{2} = \pi + 1 \).

The radius \(r\) of \(\pi + 1\) informs us how much beyond "c" the interval can reach. Together with "c," this radius encapsulates the interval's absolute distance in both directions.