Problem 17 - Calculus for the Life Sciences: A Modeling Approach Volume I

Question

What is the norm of the partition \\{0.0,0.2,0.3,0.6,0.7,0.9,1.0\\} of [0,1]\(?\) Write a partition of [0,1] whose norm is 0.15 .

Step-by-Step Solution

Answer

The norm of the original partition is 0.3. A partition with norm 0.15: \(\{0.0, 0.15, 0.3, 0.45, 0.6, 0.75, 0.9, 1.0\}\).

Step 1: Understand the Norm of a Partition

The norm of a partition is the size of the largest subinterval in the partition. Given the partition \(\{0.0, 0.2, 0.3, 0.6, 0.7, 0.9, 1.0\}\), calculate the lengths of each subinterval.

Step 2: Compute Lengths of Subintervals

Calculate the difference between consecutive points in the partition to find the length of each subinterval:- \([0.0, 0.2] = 0.2 - 0.0 = 0.2\)- \([0.2, 0.3] = 0.3 - 0.2 = 0.1\)- \([0.3, 0.6] = 0.6 - 0.3 = 0.3\)- \([0.6, 0.7] = 0.7 - 0.6 = 0.1\)- \([0.7, 0.9] = 0.9 - 0.7 = 0.2\)- \([0.9, 1.0] = 1.0 - 0.9 = 0.1\)

Step 3: Identify the Norm

The norm is the maximum length of the subintervals computed in Step 2. The lengths are \(0.2, 0.1, 0.3, 0.1, 0.2, 0.1\). Therefore, the norm of the partition is \(0.3\).

Step 4: Create a New Partition

To create a partition of \([0, 1]\) with norm \(0.15\), ensure no subinterval exceeds \(0.15\) in length. A possible partition is \(\{0.0, 0.15, 0.3, 0.45, 0.6, 0.75, 0.9, 1.0\}\).

Step 5: Verify the New Partition's Norm

Calculate the lengths of the new partition's subintervals:- \([0.0, 0.15] = 0.15\)- \([0.15, 0.3] = 0.15\)- \([0.3, 0.45] = 0.15\)- \([0.45, 0.6] = 0.15\)- \([0.6, 0.75] = 0.15\)- \([0.75, 0.9] = 0.15\)- \([0.9, 1.0] = 0.1\)All subintervals are \(0.15\) or shorter, confirming the norm is \(0.15\).

Key Concepts

Partition Norm

The concept of a "partition norm" in calculus is closely tied to an understanding of intervals and their sizes. When you partition an interval like \([0, 1]\), you break it into smaller pieces. Each piece is called a subinterval. The partition norm is a single number that tells you how large the largest subinterval is in your partition. It's as if you're looking for the biggest slice in a cake!
This norm is significant because in mathematical analysis and integration, it's often used to measure how 'fine' a partition is. A smaller norm generally means a finer partition, which can lead to more accurate approximations in calculus computations. If you aim for precision, using a partition with a smaller norm is beneficial. That's relevant especially when considering Riemann sums, which approximate the area under a curve.

In our example, with the partition \({0.0, 0.2, 0.3, 0.6, 0.7, 0.9, 1.0}\), the largest subinterval is from \(0.3\) to \(0.6\), which gives the norm of \(0.3\). This step is crucial for any exercise involving partitions, as identifying the norm is often the goal.

Subintervals

Subintervals are the fundamental building blocks of a partition. Each subinterval is simply the segment between two consecutive points in your partition. Think of them as the steps on a ladder that help you climb across the entire interval.
To find each subinterval, you subtract the start of a pair from the end. For example, in the partition \({0.0, 0.2, 0.3, 0.6, 0.7, 0.9, 1.0}\), you have the subinterval \([0.0, 0.2]\), where the length is \(0.2 - 0.0 = 0.2\). This process is repeated through the partition:

• \([0.0, 0.2] = 0.2\)
• \([0.2, 0.3] = 0.1\)
• \([0.3, 0.6] = 0.3\)
• \([0.6, 0.7] = 0.1\)
• \([0.7, 0.9] = 0.2\)
• \([0.9, 1.0] = 0.1\)

Understanding these subintervals is crucial for when you want to determine the norm, as the largest subinterval will dictate the norm of the partition. Knowing how to break down an interval into subintervals is a critical skill in many calculus applications including integration and solving differential equations.

Calculating Norms

Calculating a norm involves identifying the largest subinterval from your partition. It's a straightforward but essential task in mathematical analysis. To calculate the norm:

• First, find the length of each subinterval by subtracting the starting point from the ending point.
• Then, compare these lengths and select the largest one.

For your partition \({0.0, 0.2, 0.3, 0.6, 0.7, 0.9, 1.0}\), calculating the subinterval lengths gives you: \(0.2, 0.1, 0.3, 0.1, 0.2, 0.1\). The largest among these is \(0.3\), making it the norm of this partition.
When creating a partition with a specific norm, the key is to ensure that none of your subintervals exceed the desired norm. For instance, if you want a partition on \([0,1]\) with a norm of \(0.15\), you'll ensure that all subintervals measure \(0.15\) or less. A possible partition could be \({0.0, 0.15, 0.3, 0.45, 0.6, 0.75, 0.9, 1.0}\), where each subinterval respects the norm condition. This skill is vital for adjusting partitions to meet the precise requirements needed for specific analytical tasks.