Textbook Question
Identifying Riemann sums Fill in the blanks with an interval and a value of n.
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β Ζ (1.5 + k) β’ 1 is a midpoint Riemann sum for f on the interval [ ___ , ___ ]
k = 1
with n = ________ .
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8. Definite Integrals
Riemann Sums
1:43 minutesProblem 5.1.59c
Textbook Question
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(c) For an increasing or decreasing nonconstant function on an interval [a,b] and a given value of n, the value of the midpoint Riemann sum always lies between the values of the left and right Riemann sums.
Verified step by step guidance1
Understand the problem: The question asks us to determine whether the midpoint Riemann sum for a nonconstant, increasing or decreasing function on an interval [a, b] always lies between the left and right Riemann sums. We need to analyze this statement and provide reasoning or a counterexample.
Recall the definitions: The left Riemann sum (LRS) uses the left endpoints of subintervals to approximate the integral, while the right Riemann sum (RRS) uses the right endpoints. The midpoint Riemann sum (MRS) uses the midpoints of subintervals. For an increasing function, LRS underestimates the integral, and RRS overestimates it. For a decreasing function, the opposite is true.
Analyze the behavior for an increasing function: For an increasing function, the function values at the midpoints of subintervals are greater than the left endpoints but less than the right endpoints. This suggests that the midpoint Riemann sum should lie between the left and right Riemann sums.
Analyze the behavior for a decreasing function: For a decreasing function, the function values at the midpoints of subintervals are less than the left endpoints but greater than the right endpoints. This also suggests that the midpoint Riemann sum should lie between the left and right Riemann sums.
Conclude and justify: Based on the analysis, the statement is true. For a nonconstant, increasing or decreasing function, the midpoint Riemann sum always lies between the left and right Riemann sums because the midpoint values are intermediate between the left and right endpoint values for each subinterval.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Riemann Sums
Riemann sums are a method for approximating the integral of a function over an interval by dividing the interval into subintervals and summing the areas of rectangles formed. The left Riemann sum uses the left endpoints of the subintervals, while the right Riemann sum uses the right endpoints. The midpoint Riemann sum, on the other hand, uses the midpoints of the subintervals, which can provide a more accurate approximation of the area under the curve.
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Monotonic Functions
A monotonic function is one that is either entirely non-increasing or non-decreasing over a given interval. For an increasing function, as the input values increase, the output values also increase, while for a decreasing function, the output values decrease. Understanding the behavior of monotonic functions is crucial when analyzing the relationships between different types of Riemann sums, as it affects the placement of the rectangles used in the approximations.
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Comparison of Riemann Sums
When comparing Riemann sums for a monotonic function, the midpoint Riemann sum typically lies between the left and right Riemann sums. This is because the midpoint captures the average height of the function over each subinterval, while the left and right sums can overestimate or underestimate the area depending on the function's behavior. This property is particularly important in understanding the accuracy of numerical integration methods.
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