Problem 39 - Calculus: One and Several Variables

Question

Each of the following sets is the solution of an inequality of the form \(|x-c|<\delta\). Find \(c\) and \(\delta\). $$(-3,7)$$.

Step-by-Step Solution

Answer

The values are \(c = 2\) and \(\delta = 5\).

Step 1: Find the midpoint (c)

To find the value of 'c', calculate the midpoint of -3 and 7 using the formula (a+b)/2. Thus, \(c = (-3+7)/2 = 2\). This means that 'c' is 2.

Step 2: Find the range's half (delta)

To find the value of \(\delta\), calculate half of the range of -3 and 7, which is (b-a)/2. In this case, \(\delta = (7-(-3))/2 = 5\). This means that \(\delta\) is 5.

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, irrespective of direction. In mathematical terms, it is denoted by two vertical lines, like \(|x|\). This concept is crucial in inequalities when we express a condition such as \( |x - c| < \delta\). Here, \(c\) represents the center point, and \(\delta\) is the range or tolerance from that point. In simple terms, this inequality states that the values of \(x\) are within \(\delta\) units from \(c\), covering a balanced interval around \(c\) on the number line.
Understanding absolute value helps us express intervals without specifying exact directions. Instead of saying 'within 5 units right and left of 2', we say \( |x - 2| < 5\), which neatly establishes a boundary around a central point.

Solution Sets

A solution set in the context of inequalities refers to all values that satisfy the inequality condition. For our inequality, \( |x-c| < \delta\), the solution set is an interval on the number line.
In the example given with \((-3, 7)\), this interval is the solution set. It includes all numbers between -3 and 7, but does not include -3 and 7 themselves because these are open bounds, indicated by the parentheses.

• It represents all possible \(x\) that lie \(\delta\) distance from \(c\), forming the central region around \(c\).
• Solution sets are crucial for visualizing ranges and understanding how inequalities partition the number line.

Midpoint Formula

The midpoint formula is used to find the center of two given points on a number line. For any two points \(a\) and \(b\), the midpoint \(c\) is calculated as:
\[ c = \frac{a+b}{2} \]
This calculation gives the mean or average of the two points, providing a symmetric center between them.

• Knowing the midpoint helps identify a point from which a uniform distance can be measured in both directions.
• It simplifies understanding intervals by calculating their central point directly.

For example, the midpoint between -3 and 7 is:
\[ c = \frac{-3+7}{2} = 2 \]
This means the point 2 is exactly in the middle of -3 and 7.

Range Calculation

Range calculation involves determining the span between the two endpoints of an interval, which in turn aids in calculating \(\delta\).
The range formula for endpoints \(a\) and \(b\) is simple:
\[ \text{Range} = b-a \]
To find \(\delta\), which represents half of this range, divide by 2:
\[ \delta = \frac{b-a}{2} \]
This gives us the extent from the midpoint that our values can stretch before exceeding the bounds.

• It's important because it tells us exactly how far each side deviates from the central midpoint to form the total interval.
• In our example, with endpoints -3 and 7, the total range is 10, and half of that is 5, our \(\delta\).

Thus, \(\delta = 5\) informs us that values are within 5 units of the midpoint, 2, completing the interval of interest.