In Exercises 33-36, graph each function f(x) over the given...

In Exercises $33-36,$ graph each function $f(x)$ over the given interval.Partition the interval into four sub intervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum$\Sigma_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},$ given that $c_{k}$ is the (a) left-hand endpoint, (b) right-hand endpoint, (c) midpoint of the $k$ the sub interval. (Make a separate sketch for each set of rectangles.) Find the norm of the partition $P=\{0,1.2,1.5,2.3,2.6,3\}$

In Exercises $33-36,$ graph each function $f(x)$ over the given interval.Partition the interval into four sub intervals of equal length. Then add to your sketch the rectangles associated with the Riemann sum$\Sigma_{k=1}^{4} f\left(c_{k}\right) \Delta x_{k},$ given that $c_{k}$ is the (a) left-hand endpoint, (b) right-hand endpoint, (c) midpoint of the $k$ the sub interval. (Make a separate sketch for each set of rectangles.)
Find the norm of the partition $P=\{0,1.2,1.5,2.3,2.6,3\}$

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Key Concepts

Partition of an Interval

A partition of an interval divides the interval into a finite number of subintervals. This process allows one to analyze or approximate properties of functions over the interval by focusing on smaller, more manageable segments. It is a foundational concept used in defining integrals and in numerical approximation methods.

Norm (or Mesh) of a Partition

The norm or mesh of a partition is defined as the length of the longest subinterval in the partition. This measure is crucial in the study of limits of Riemann sums as the partition is refined, with the norm approaching zero, which leads to an accurate definition of the definite integral.

Riemann Sums

Riemann sums are used to approximate the area under a curve by summing the areas of rectangles defined over each subinterval of a partition. The height of each rectangle is determined by the value of the function at specific sample points, which can be chosen in various ways such as at the left endpoint, right endpoint, or midpoint of each subinterval. This concept is central in understanding the transition from discrete sums to the continuous integral.

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