question 5 Consider the partition P = {0,1/4,1/2,1} of the interval [0,1] comput
ID: 3535100 • Letter: Q
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question 5
Consider the partition P = {0,1/4,1/2,1} of the interval [0,1] compute L(f,P) and U(f,P) for the following three choices of function f:[0,1] rightarrow R f(x) = x for all x in [0, 1]. f(x) = 10 for all x in [0, 1]. f(x) = -x2 for all x in [0. 1]. For an interval [a, b] and a positive number S. use the Archimedean Property of R to show that there is a partition P = {x0,...,xn} of [a,b] such that each partition interval [xi-1, x] of P has length less than Delta. Suppose that the bounded function f: [a, b] rightarrow R has the property that for each rational number x in the interval [a, b], f(x) = 0. Prove that f 0 f. Suppose that the bounded function f: [a, b] rightarrow R has the property that f(x) 0 for all x in [a, b]. Prove that f > 0 . Suppose that the two bounded functions f: [a, b] rightarrow R and g : [a, b] rightarrow R has the property that g(x) f(x) for all x in [a,b] For P a partition of [a, b], show that L(g, P) L(f,P). Use part (a) to show that .Explanation / Answer
Between two points of the partition P, the lower Riemann sum L(f, P) approximates the function by its infimum between those two points. Since there must be a rational number between them, that infimum will be less than or equal to 0. Each term in the lower sum is then negative or 0, so the whole thing has L(f, P) <= 0. Taking the supremum of the L(f, P)'s, it must be <= 0, so the lower Riemann integral is <= 0.
The terms of the upper Riemann sums are similarly all positive or 0, and the reasoning is entirely analogous to get the fact that the upper Riemann integral is >= 0.
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