91Ó°ÊÓ

In precalculus, the concept of an interval is fundamental to understanding sets of numbers on the number line. An open interval refers to a set of real numbers that are within a certain range but do not include the end points of that range.

For instance, the open interval \( (a, b) \) is represented mathematically as all points x that satisfy the inequality \( a < x < b \). It's like a segment of a line that has its endpoints removed. To visualize it, you could think of the points a and b as being 'open' or not covered; no matter how close you get to these points, they are never actually part of the interval.

This is distinguishable from a closed interval \( [a, b] \), where the endpoints are included, or a half-open or half-closed interval where only one endpoint is included, expressed as \( [a, b) \) or \( (a, b] \). The open interval is essential for various topics in calculus and analysis, where it's used to define continuity, limits, and amid interval-based proofs.

The midpoint of an interval is the value exactly in the middle of the two endpoints of the interval. It's the average of the endpoints. For any interval \( (a, b) \) that has real numbers a and b as endpoints, the midpoint \( m \) is calculated by the formula:

\[ m = \frac{a + b}{2} \
\]In the context of an open interval, the midpoint does not depend on whether the endpoints are included in the set. Even if the ends are 'open' as described earlier, the midpoint remains the same. The midpoint itself might or might not be part of the interval, since in an open interval the ends are not included. However, in the example exercise given, where the interval contains 0, the midpoint falls within the interval, illustrating that open interval centered at zero does indeed exist and reinforcing the significance of midpoints in interval discussions.

The idea of interval containment is another fundamental concept in mathematics. It revolves around the question of whether all points of one interval are contained within another. To say that interval \( (c, d) \) is contained within interval \( (a, b) \), denoted \( (c, d) \subseteq (a, b) \), means every element x in \( (c, d) \) is also an element of \( (a, b) \).

Practically, for this containment to hold true, we need \( c > a \) and \( d < b \) since none of the intervals include their endpoints. Interval containment is crucial when discussing functions, sequences, and series where one seeks to understand the bounds within which values lie. In the provided exercise solution, when crafting a new interval \( (-p, p) \) contained within the original interval \( (a, b) \) and centered at zero, the concept of interval containment shows that despite \( a \) and \( b \) being kept out of the open interval, the newly formed interval adheres to these rules of containment.