The function shown in the graph below is increasing on the...

The function shown in the graph below is increasing on the interval $[1,4] .$ The interval will be divided into 12 subintervals. (a) What are the left endpoints of the first and last subintervals? (b) What are the right endpoints of the first two subintervals? (c) When using the right endpoints, do the rectangles lie above or below the graph of the function? (d) What can you conclude about the heights of the rectangles when the function is constant on the given interval?

The function shown in the graph below is increasing on the interval $[1,4] .$ The interval will be divided into 12 subintervals.
(a) What are the left endpoints of the first and last subintervals?
(b) What are the right endpoints of the first two subintervals?
(c) When using the right endpoints, do the rectangles lie above or below the graph of the function?
(d) What can you conclude about the heights of the rectangles when the function is constant on the given interval?

Key Concepts

Constant Functions in Riemann Sums

For a function that is constant on an interval, the height of the function remains the same throughout. This condition means that regardless of whether left or right endpoints are used, all the rectangles in the Riemann sum have the same height, leading to an exact approximation of the area under the curve.

Monotonicity of Functions

Monotonicity, especially when a function is increasing or decreasing, plays a significant role in Riemann sum approximations. For an increasing function, using left endpoints will typically provide an underestimate while using right endpoints results in an overestimate, due to the progressive change in function values across subintervals.

Riemann Sums

A Riemann sum is a method for approximating the area under a curve by dividing the region into a finite number of rectangles and adding up their areas. This fundamental concept is a precursor to the definite integral and illustrates how areas under complex curves can be approximated by simpler, manageable shapes.

Partitioning an Interval

Partitioning an interval involves dividing it into equal or unequal subintervals, which is crucial for setting up Riemann sums. This process determines the width of each subinterval and the distribution of points (endpoints) used to evaluate the function, impacting the accuracy of the approximation.

Left and Right Endpoints

The use of left and right endpoints refers to the selection of sample points at either the beginning (left) or the end (right) of each subinterval for constructing the approximating rectangles in a Riemann sum. This choice affects whether the approximation underestimates or overestimates the true value of the integral, particularly depending on the function’s behavior.

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