Problem 7 - 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. $$[-4,5]$$

Step-by-Step Solution

Answer

Closed and bounded.

Step 1: Analyze the Type of Interval

An interval is defined by its endpoints, and whether these endpoints are included or not affects its classification. In the given interval, \([-4, 5]\), both endpoints, -4 and 5, are included. This is indicated by the square brackets around both numbers. Therefore, this is a closed interval.

Step 2: Determine If the Interval is Bounded or Unbounded

To check if an interval is bounded or unbounded, we examine the extent of the interval on the number line. Since the interval starts at -4 and ends at 5, and both are finite numbers, the interval does not extend infinitely in either direction. Thus, this interval is bounded.

Step 3: Sketch the Interval on the Real Line

To sketch the interval \([-4, 5]\) on the number line, draw a line extending from -4 to 5. Because both endpoints are included in the interval, represent them with filled (solid) dots:1. Place a solid dot on -4.2. Draw a solid line joining -4 to 5.3. Place a solid dot on 5. This visual representation confirms the interval is closed and bounded.

Key Concepts

Understanding a Closed Interval

An interval consists of all the numbers between two endpoints. When those endpoints are included in the interval, we call it a closed interval. This inclusion is shown by square brackets around the endpoints. For example, in the interval \([-4, 5]\), both -4 and 5 are included. This means every number between -4 and 5, including -4 and 5 themselves, is part of this interval. Closed intervals are often used when the value at the boundaries is important or relevant. You can think of them as a section of the number line where both ends are "closed off" or "capped" by the endpoints.

Exploring Bounded Intervals

Intervals can either be bounded or unbounded. A bounded interval has both its endpoints as finite numbers. This means the interval does not extend indefinitely to the left or right on the number line. In calculus, bounded intervals are crucial because they limit the scope of solutions and make mathematical problems more practical to solve. The interval \([-4, 5]\) is an example of a bounded interval because it starts at -4 and ends at 5, providing a clear range within the number line. Bounded intervals help in analyzing variables within specific limits.

Visualizing Intervals on a Number Line

The number line is a simple and effective way to represent intervals. It helps visually display the range of numbers included in an interval. To sketch a closed and bounded interval on the number line, like \([-4, 5]\), follow these steps:

• Identify the two endpoints, -4 and 5, on the number line.
• Place solid dots at both endpoints to show they are included.
• Draw a solid line connecting these dots, indicating all numbers in between are part of the interval.

This visualization confirms both the boundary (closed interval) and the limit (bounded interval) characteristics on the number line.