Problem 34 - Calculus: One and Several Variables

Question

Sketch the set on a number line. \([3, \infty)\).

Step-by-Step Solution

Answer

The set [3, ∞) depicted on a number line consists of a closed (filled) dot at point 3, with a line extending from this point towards positive infinity.

Step 1: Message of the set

Recognize the meaning of the set notation [3, ∞). This is interval notation which represents all numbers from 3 to positive infinity. The [ symbol shows that the number it is attached to (3), is included in the set (also known as 'closed'). The ) symbol indicates that ∞, or 'infinity', is not included in the set (also known as 'open').

Step 2: Draw a number line

Draw a straight horizontal line, which will represent our number line. Now pick a point on the line to be our 'zero' and mark it as such. Ensure there is enough space to the right of zero to mark off a few numbers greater than 3.

Step 3: Mark the given point

On the number line, mark the point corresponding to 3. As 3 is included in the set, this point should be a closed (filled) dot.

Step 4: Indicate to positive infinity

Plot a line on the number line that leads from 3 (included) to the right edge of our drawn number line. This line represents all numbers greater than 3, up to positive infinity. As ∞ is not included in the set, the end of this line should remain open.

Key Concepts

Number Line

In mathematics, the number line is a visual representation of numbers laid out on a straight, horizontal line. Think of it as a street that numbers live on, where each point corresponds to a specific number. Starting from the 'zero' point, you can move right to find positive numbers or left for negative ones. Our number line is limitless in both directions, suggesting that numbers go on forever.

When working with a number line, it's crucial to map out the numbers accurately. For instance, in our exercise, your zero is the anchor and you should ensure there are equal spaces between the marks to reflect the true scale of numbers. If you were to place '3' on this line, for example, it would be three equally spaced units to the right of 'zero'. Utilizing a number line allows students to better visualize concepts such as 'intervals' which represent a range of numbers between two endpoints.

Closed Interval

A closed interval is like a closed door—it includes everything within its boundaries. In interval notation, we use square brackets, like these: \[ \], to indicate that endpoints are part of the interval. For example, if you have \[3, 5\], this includes the numbers 3 and 5, and all the numbers in between.

Imagine you're lining up for a movie. If you're told you're standing in spots 3 to 5 in line, a closed interval means you and the people in spots 4 and 5 are all included. On a number line, we depict this by drawing solid dots over the numbers 3 and 5 to show that these numbers are included in the set. This specificity helps you clearly understand which numbers are part of the solution set in a problem.

Infinite Intervals

In contrast to closed intervals, which have defined start and end points, infinite intervals go on indefinitely in at least one direction. Think of them like a road that stretches out into the horizon with no end in sight. In the exercise, \(3, \infty)\ represents all numbers greater than '3' right up to 'infinity'. Since infinity is a concept rather than a reachable number, it's depicted with an open parenthesis, like this: \)\, to show that it’s not included.

When you sketch this interval, you would start with a solid dot on '3' (since '3' is included due to the closed interval indicated by the square bracket) and then draw a line to the right that extends beyond the boundary of your number line, stopping without a dot to signify that the interval never truly ends. This way of understanding infinite intervals can be extremely useful in mathematical and real-world contexts where we talk about things that have a beginning but no perceivable end.