Problem 10 - 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. $$ \left[\frac{3}{2}, \frac{5}{2}\right) $$

Step-by-Step Solution

Answer

The interval \(\left[\frac{3}{2}, \frac{5}{2}\right)\) is half-open and bounded.

Step 1: Identify the Type of Interval

An interval is characterized by its endpoints. Here, we have \( \left[ \frac{3}{2}, \frac{5}{2} \right) \). The square bracket \([\ ]\) indicates that the interval is closed on the left at \( \frac{3}{2} \), and the parenthesis \((\ ]\) shows it is open on the right at \( \frac{5}{2} \). Thus, this interval is half-open.

Step 2: Determine If It Is Bounded or Unbounded

An interval is bounded if both endpoints are real numbers. In this case, \( \frac{3}{2} \) and \( \frac{5}{2} \) are real numbers. Therefore, the interval \( \left[ \frac{3}{2}, \frac{5}{2} \right) \) is bounded because it does not extend to infinity on either side.

Step 3: Sketch the Interval on the Real Line

To sketch the interval \( \left[ \frac{3}{2}, \frac{5}{2} \right) \) on a real line, draw a solid circle or dot at \( \frac{3}{2} \) to indicate it is included in the interval. Then draw an open circle at \( \frac{5}{2} \) to show it is not included in the interval. Finally, shade the line segment connecting these two points, indicating all numbers between and including \( \frac{3}{2} \) but excluding \( \frac{5}{2} \).

Key Concepts

Open and Closed Intervals

Intervals are a crucial concept in calculus, helping define a range of real numbers. An interval can either be open, closed, or half-open, depending on its endpoints.
Open intervals exclude their endpoints. For example, \( (a, b) \) means that the set includes all numbers between \(a\) and \(b\), but not \(a\) or \(b\) themselves. Think of it as starting from just above \(a\) and ending just below \(b\).
Closed intervals include their endpoints. The notation \( [a, b] \) means that the set contains everything between \(a\) and \(b\), including \(a\) and \(b\). This is a complete range, from \(a\) to \(b\).
Half-open or half-closed intervals contain one endpoint but not the other. For example, \( [a, b) \) includes all numbers between \(a\) and \(b\), including \(a\) but not \(b\). This gives a flexible way to work with ranges where one boundary is not fixed.
Remember, it's the brackets and parentheses that tell us which numbers are included and which are not!

Bounded and Unbounded Intervals

Understanding whether an interval is bounded or unbounded can help us know if the set of numbers stretches to infinity.
Bounded intervals are those which do not extend infinitely in any direction. They are enclosed within finite bounds with both endpoints being real numbers. For instance, the interval \( [rac{3}{2}, rac{5}{2}] \) is bounded because it has clear, finite limits.
Unbounded intervals reach out to infinity in one or both directions. This happens when at least one of the endpoints is infinite. For example, \( (-\infty, b) \) extends infinitely to the left, and \( (a, \infty) \) extends infinitely to the right.
By identifying these characteristics, you can quickly determine if the interval is manageable in terms of size or if it requires consideration of infinite boundaries.

Real Number Line

The real number line is a fundamental tool in calculus, offering a visual representation of all possible values an interval can take.
Imagine a straight line stretching infinitely in both directions, covering every possible real number. Each point on this line corresponds to a real number, helping depict numerical relationships and inequalities.
When interpreting an interval on the real number line, include points by marking them with dots or circles. A solid circle or filled dot indicates that a particular point (or endpoint) is included in the interval, which corresponds to a closed interval edge. An open circle, on the other hand, indicates an excluded point, typical for open interval edges.
Drawing intervals on the real number line not only helps visualize included and excluded numbers but also allows a better understanding of how these numbers connect or jump from one to another.