Problem 37 - University Calculus: Early Transcendentals

Question

Find the norm of the partition \(P=\\{0,1.2,1.5,2.3,2.6,3\\}\).

Step-by-Step Solution

Answer

The norm of the partition is 1.2.

Step 1: Determine Partition Intervals

Firstly, identify the intervals created by the partition \(P=\{0,1.2,1.5,2.3,2.6,3\}\). The intervals are \([0, 1.2], [1.2, 1.5], [1.5, 2.3], [2.3, 2.6], [2.6, 3]\).

Step 2: Calculate Lengths of Intervals

Next, calculate the length of each interval:\[ [0, 1.2] : 1.2 - 0 = 1.2, \]\[ [1.2, 1.5] : 1.5 - 1.2 = 0.3, \]\[ [1.5, 2.3] : 2.3 - 1.5 = 0.8, \]\[ [2.3, 2.6] : 2.6 - 2.3 = 0.3, \]\[ [2.6, 3] : 3 - 2.6 = 0.4 \].

Step 3: Identify the Largest Interval

Identify the interval among the calculated lengths with the largest value. The lengths are \(1.2\), \(0.3\), \(0.8\), \(0.3\), \(0.4\). The largest length is \(1.2\).

Step 4: State the Norm of the Partition

The norm of the partition, which is the largest length among the intervals, is \(1.2\).

Key Concepts

Interval Lengths

An interval length is simply the distance between the starting point and the endpoint in an interval. For instance, if you have an interval \([a, b]\), its length is calculated by subtracting the lower number from the higher number, i.e., \(b - a\). This tells us how wide the interval is.
In our problem, the partition \(P=\{0,1.2,1.5,2.3,2.6,3\}\) forms several intervals like \([0, 1.2]\).

• The length is calculated as: \(1.2 - 0 = 1.2\).
• This process is repeated for all intervals to capture the distance between each pair of adjacent numbers in our partition.

Interval lengths are crucial as they help us determine the norm of a partition by identifying its largest length.

Partition Intervals

When we talk about partition intervals, we're referring to splitting a segment into smaller chunks according to a given set of points. This set is known as a partition. It acts as a guide, dividing the main interval into smaller parts.
In our exercise, we have the partition \(P=\{0,1.2,1.5,2.3,2.6,3\}\), which divides the main range from 0 to 3 into specific intervals:

• \([0, 1.2]\)
• \([1.2, 1.5]\)
• \([1.5, 2.3]\)
• \([2.3, 2.6]\)
• \([2.6, 3]\)

Partitioning helps in various areas of math and real life, such as breaking down complex problems into manageable parts by observing how values change across intervals.

Largest Interval

Identifying the largest interval in a partition is key in finding the partition norm. The largest interval is the one with the longest distance between its beginning and end in the context of the partition intervals.
In our example, after calculating each interval's length, we compare them:

• \([0, 1.2]\) is \(1.2\)
• \([1.2, 1.5]\) is \(0.3\)
• \([1.5, 2.3]\) is \(0.8\)
• \([2.3, 2.6]\) is \(0.3\)
• \([2.6, 3]\) is \(0.4\)

The largest value here is \(1.2\), making \([0, 1.2]\) the largest interval. In mathematics, this value is effectively the 'norm' of the partition, which offers insight into the extent of variation across the partition.