Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
(b) If ƒ is a linear function on the interval [a,b] , then a midpoint Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.

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Step 1: Recall the definition of a linear function. A linear function is of the form ƒ(𝓍) = m𝓍 + c, where m and c are constants. Linear functions have a constant rate of change and their graphs are straight lines.

Step 2: Understand the midpoint Riemann sum. The midpoint Riemann sum approximates the integral by dividing the interval [a, b] into n subintervals, calculating the function value at the midpoint of each subinterval, and summing the areas of the rectangles formed.

Step 3: Note that for a linear function, the integral ∫ₐᵇ ƒ(𝓍) d𝓍 represents the exact area under the straight line between x = a and x = b. Since the graph of a linear function is a straight line, the midpoint Riemann sum will perfectly capture the area under the curve, regardless of the number of subintervals n.

Step 4: Explain why this works. The midpoint Riemann sum is exact for linear functions because the function's rate of change is constant, and the midpoints of the subintervals perfectly represent the average value of the function over each subinterval. This eliminates any approximation error.

Step 5: Conclude that the statement is true. For any positive integer n, the midpoint Riemann sum will give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍 when ƒ is a linear function on the interval [a, b].

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주요 개념

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Linear Functions

A linear function is a polynomial function of degree one, which can be expressed in the form ƒ(x) = mx + b, where m is the slope and b is the y-intercept. Linear functions graph as straight lines, and their properties include constant rates of change and predictable behavior over any interval. Understanding linear functions is crucial for evaluating their integrals and Riemann sums.

Riemann Sums

Riemann sums are a method for approximating the definite integral of a function over an interval by dividing the interval into subintervals, calculating the function's value at specific points within those subintervals, and summing the products of these values and the widths of the subintervals. The midpoint Riemann sum specifically uses the midpoint of each subinterval to evaluate the function, which can yield exact results for certain types of functions, such as linear functions.

Definite Integrals

A definite integral represents the signed area under a curve defined by a function ƒ(x) over a specific interval [a, b]. It is denoted as ∫ₐᵇ ƒ(x) dx and can be interpreted as the limit of Riemann sums as the number of subintervals approaches infinity. For linear functions, the definite integral can be calculated exactly, and Riemann sums will converge to this exact value, making them particularly useful for understanding the relationship between area and integration.

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.(b) If ƒ is a linear function on the interval [a,b] , then a midpoint Riemann sums give the exact value of ∫ₐᵇ ƒ(𝓍) d𝓍, for any positive integer n.