Question

If the interval [4,18] is partitioned into \(n=28\) subintervals of equal length, what is \(\Delta x ?\)

Short answer

Answer: The length of each subinterval is 1/2.

Step-by-step solution

Identify the interval and the number of subintervals

In this exercise, we are given the interval [4,18] and the number of subintervals \(n = 28\).

Apply the formula for the length of a subinterval

Using the formula \(\Delta x = \frac{b - a}{n}\), we can calculate the length of each subinterval. Here, \(a = 4\), \(b = 18\), and \(n = 28\).

Calculate the length of each subinterval

Plugging in the given values into the formula, we find: \(\Delta x = \frac{18 - 4}{28} = \frac{14}{28}\)

Simplify the result

Simplify the fraction \(\frac{14}{28}\) to get the final answer: \(\Delta x = \frac{14}{28} = \frac{1}{2}\) So, the length of each subinterval is \(\Delta x = \frac{1}{2}\).

Key concepts

Partitioning Intervals

In calculus, partitioning an interval is a fundamental concept used to break down a larger interval into smaller, more manageable parts. This is essential for processes such as evaluating integrals, where we need to consider each small part of the interval separately.
To partition an interval, you first identify the start and end points, which are commonly denoted as \(a\) and \(b\) respectively. In our example, the interval is \([4, 18]\). Partitioning means dividing this interval into equal segments, which are the subintervals.
Let's consider our key steps:

• Determine the interval: Know the starting and ending points of the overall span.
• Decide the number of partitions: This is the number of equal parts you want.

When you partition an interval, each subinterval is assumed to have the same length, making calculations straightforward and consistent.

Subinterval Length

Once you've partitioned an interval, the next step is to calculate the length of each subinterval. This is where the concept of subinterval length comes in handy.
The length of each subinterval is determined by dividing the total length of the interval by the number of subintervals \(n\). Using our exercise, we have the interval \([4, 18]\) partitioned into \(n = 28\) subintervals.
Here is what we do:

• Calculate the total length: Subtract the starting point \(a = 4\) from the endpoint \(b = 18\). This gives us the interval length: \(18 - 4 = 14\).
• Divide by the number of subintervals: Divide this length by \(n\) to find the length of each subinterval.

This gives a thorough understanding of how each part fits within the whole interval, allowing precise computations and insights when handling calculus problems.

Delta x Calculation

The symbol \(\Delta x\) is commonly used to denote the length of a subinterval in the partitioned interval. It is vital in understanding how each small piece of the interval contributes to the entire length.
To find \(\Delta x\), use the formula:
\[\Delta x = \frac{b - a}{n}\]In our specific problem, we identified that the interval is \([4, 18]\) and it is divided into \(n = 28\) subintervals. Thus, substituting in the values we have:

• \(b = 18\)
• \(a = 4\)
• \(n = 28\)

Now, substitute these into our formula:
\[\Delta x = \frac{18 - 4}{28} = \frac{14}{28} = \frac{1}{2}\]This calculation shows that each subinterval is \(\frac{1}{2}\) units long. Understanding this step is crucial for tasks like approximating areas under curves or finding precise values in integrals.