Upper and Lower Darboux Sum Inequality

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• Magnetons
In summary, when comparing partitions of a bounded function on a closed interval, if the finer partition is a subset of the courser partition, then the lower sum of the finer partition will be less than or equal to the lower sum of the courser partition and the upper sum of the courser partition will be less than or equal to the upper sum of the finer partition. This is due to the fact that the finer partition will be closer to the limit in both upper and lower sums, with the supremums of the courser partition being applied to smaller intervals.
Magnetons
TL;DR Summary
L(f,P) ##\leq## L(f,Q) ##\leq## U(f,Q) ##\leq## U(f,P)
Lemma
Let f be a bounded function on [a,b]. If P & Q are partitions of [a,b] and P ##\subseteq## Q , then

L(f,P) ##\leq## L(f,Q) ##\leq## U(f,Q) ##\leq## U(f,P) .

Question is "How can P have bigger upper darboux sum than Q while it is a subset of Q"

The finer partition will be as close or closer in both upper and lower sum to the limit than the courser partition. For the upper sum, the supremums of the courser partition are still there, but they are applied to smaller ##\Delta x##s.

mathguy_1995 and Magnetons
FactChecker said:
The finer partition will be as close or closer in both upper and lower sum to the limit than the courser partition. For the upper sum, the supremums of the courser partition are still there, but they are applied to smaller ##\Delta x##s.

What is the Upper and Lower Darboux Sum Inequality?

The Upper and Lower Darboux Sum Inequality is a mathematical theorem that states that the upper Darboux sum of a function is always greater than or equal to the lower Darboux sum of the same function on a given interval.

What is the significance of the Upper and Lower Darboux Sum Inequality?

The Upper and Lower Darboux Sum Inequality is significant because it provides a way to approximate the area under a curve using rectangles. This is useful in many applications, such as calculating the work done by a variable force or finding the total distance traveled by an object.

How is the Upper and Lower Darboux Sum Inequality used in calculus?

In calculus, the Upper and Lower Darboux Sum Inequality is used to prove the existence of the definite integral. It is also used to find the upper and lower bounds of the integral, which can help in evaluating the integral.

What is the difference between the upper and lower Darboux sums?

The upper Darboux sum is the sum of the areas of the rectangles with their upper edges on the graph of the function, while the lower Darboux sum is the sum of the areas of the rectangles with their lower edges on the graph of the function. The upper Darboux sum is always greater than or equal to the lower Darboux sum.

Can the Upper and Lower Darboux Sum Inequality be applied to all functions?

Yes, the Upper and Lower Darboux Sum Inequality can be applied to all functions that are bounded on a given interval. This means that the function must have both a maximum and minimum value within the interval.

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