Problem 11 - 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. $$ [0, \infty) $$

Step-by-Step Solution

Answer

The interval \([0, \infty)\) is half-open and unbounded.

Step 1: Identify Interval Type

To determine if the interval \([0, \infty)\) is open, closed, or half-open, we look at the inclusion of endpoints \(0\) and \(\infty\). The bracket \([\) at 0 means that the interval includes 0, while the parenthesis \()\) at infinity indicates that infinity is not included. This makes the interval half-open (closed on the left).

Step 2: Determine Boundedness

An interval is bounded if there are real numbers \(a\) and \(b\) such that all elements \(x\) in the interval satisfy \(a < x < b\). In this case, the interval is \([0, \infty)\), where it starts at 0 and extends indefinitely in the positive direction. Since it has no upper bound, the interval is unbounded.

Step 3: Sketch the Interval

To sketch this interval on a real number line, plot a solid dot at 0 to indicate that it is included in the interval. Draw a line or arrow extending to the right towards infinity, without an endpoint, indicating that the interval extends infinitely in the positive direction. This visual representation helps in understanding the interval's structure.

Key Concepts

Open Interval

In mathematics, an open interval refers to a range of numbers on the number line where the endpoint numbers are not included in the interval itself. For instance, in the interval \((a, b)\), neither \(a\) nor \(b\) is part of the interval. This means that the interval includes all numbers greater than \(a\) and less than \(b\). While discussing intervals:

• Open intervals are denoted using parentheses \(( )\).
• Common examples include \((2, 5)\) or \((-3, 7)\).
• These intervals do not include the endpoints themselves.

Open intervals are often used in calculus and analysis since they don't bound the function at the endpoints. When sketching, open intervals use hollow points (or circles) to mark the ends of lines on a graph.

Closed Interval

A closed interval includes both of its endpoints. This means all the numbers between the two endpoints, as well as the endpoints themselves, form part of the interval. An example would be the interval \[a, b\]. This interval includes every number \(x\) such that \(a \leq x \leq b\). The characteristics of closed intervals include:

• Square brackets \([ ]\) are used to denote closed intervals.
• Examples include \[1, 3\] or \[-2, 6\].
• They are very common in situations involving real-world measurements because they include boundary values.

In diagrams, closed intervals have filled or solid dots at the endpoints, indicating inclusion of these points in the interval. This type is often used in situations where no ambiguity about the boundary values is required.

Bounded Interval

Bounded intervals are intervals that have both upper and lower limits that are real numbers. These constraints ensure that the interval doesn't extend to infinity in either direction. Bounded intervals can be either open or closed, and even half-open (one endpoint is included, the other is not).

Characteristics of Bounded Intervals

• They are defined within two finite numbers, \(a\) and \(b\).
• They can include intervals like \([2,5]\), \((1,3)\), or \[2,5)\].
• Bounded intervals are practical when dealing with fixed limits.

Diagrams illustrating bounded intervals on a number line will show endpoints at both sides. The finite range is what makes these intervals 'bounded' as they are confined within two limits on a real line.

Unbounded Interval

Unbounded intervals extend indefinitely in one or both directions on the number line. They either have no upper limit, no lower limit, or no limits at all. Thus, an unbounded interval might look like an infinite stretch of values.

Types of Unbounded Intervals

• These intervals might begin at a point and extend infinitely, such as \[0, \infty)\].
• Conversely, they can start from infinity and end at a real number, for example, \((-\infty, 5]\).
• Some might be endless in both directions, such as \((-\infty, \infty)\).

The way to represent these on a graph involves arrows extending either left or right or in both directions to show indefinite continuation. Understanding unbounded intervals is important, particularly in calculus, where behaviors near infinity are often analyzed.