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