Partition

Definition

A partition of an interval $[a, b]$ is a finite sequence of points that divide the interval into smaller subintervals. These points are used to approximate areas under curves in numerical integration.

5 Must Know Facts For Your Next Test

1. A partition divides the interval $[a, b]$ into $n$ subintervals.
2. The partition points are denoted as $\{x_0, x_1, x_2, ..., x_n\}$ with $a = x_0 < x_1 < ... < x_n = b$.
3. Each subinterval is defined by $[x_{i-1}, x_i]$ where $i = 1, 2, ..., n$.
4. The norm of a partition is the length of the longest subinterval and is denoted by $||P||$.
5. Partitions are essential for Riemann sums and other approximation methods.

Review Questions

• What is a partition in the context of an interval $[a,b]$?
• How do you denote the points that form a partition?
• What role does the norm of a partition play in numerical integration?

Related terms

Riemann Sum: An approximation of the integral by summing up areas of rectangles formed using function values at specific points within subintervals.

Subinterval: A smaller division within an interval created by partition points; defined as $[x_{i-1}, x_i]$.

$||P||$ (Norm of Partition): The length of the longest subinterval in a given partition.