Problem 8 - Calculus with Concepts in Calculus

Question

State whether the interval is open, half-open, or closed and whether it is bounded or unbounded. Then sketch the interval on the real line. $$ (-2,-1) $$

Step-by-Step Solution

Answer

The interval (-2, -1) is open and bounded.

Step 1: Determine the Openness or Closedness of the Interval

An interval is categorized as open if it does not include its endpoints. The interval (-2, -1) uses parentheses, indicating that neither endpoint is included. Thus, it is an open interval.

Step 2: Determine if the Interval is Bounded or Unbounded

Bounded intervals have both a finite start and end. Since the interval (-2, -1) has finite numbers for both its endpoints, it is considered bounded.

Step 3: Sketch the Interval on the Real Line

To sketch (-2, -1) on the real line, draw a number line, mark the points -2 and -1, and represent the interval with an open circle at both -2 and -1, showing that these endpoints are not included in the interval. Shade the line segment between -2 and -1.

Key Concepts

Open Interval

An open interval is a key concept in mathematics, typically represented using parentheses. When you see an interval written as \((-2, -1)\), the parentheses signify an open interval.
This means the interval includes all the numbers between -2 and -1, but not the endpoints themselves.
In other words:

• If \( x \) is within this interval, \(-2 < x < -1\).
  
• Neither -2 nor -1 are part of the interval, which distinguishes open intervals from closed or half-open ones.

Remember, when sketching an open interval on a number line, you use open circles at the endpoints to show they aren't included. This visualization helps make the concept more tangible.

Bounded Interval

A bounded interval has two very important characteristics: it starts and ends with finite numbers.
In our example, \((-2, -1)\), both -2 and -1 are finite, making the interval bounded.
Here's what you need to know about bounded intervals:

• Bounded intervals have a definitive start and end within the real number line.
• They contain all the points between the specified numbers.

Bounded intervals are useful when you need to consider only a specific segment of the real line, without venturing into infinity.
This differentiates them from unbounded intervals, which stretch indefinitely in one or both directions.

Real Line

The real line is essentially a visual representation of the set of all real numbers. Imagine it as a horizontal line extending infinitely in both directions.
When sketching intervals like \((-2, -1)\) on the real line, this is what you need to do:

• Identify the points on the line corresponding to the interval endpoints.
  In this example, these are -2 and -1.
• Use open circles to indicate that these endpoints are not part of the interval.
  Shade the segment between these points to show which numbers are included.

The real line helps us to visually grasp various mathematical concepts, by providing a clear and simple way to display intervals.
It essentially bridges abstract ideas with practical visuals.