question 5 Consider the partition P = {0, 1/4, 1/2, 1} of the interval [0, 1]. C

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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 f 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 Delta, use the Archimedean Property of to show that there is a partition P = {x0,...,xn} of [a, b] such that each partition interval [xi-1, xi] 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 have 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 g f.

Explanation / Answer

integration finds the area under the curve ....so if f[x] is less than g[x] in a given range ...then integration will also follow the same principle.....so as partition because an interval are made up of partitions so if in the whole interval a function is smaller than the other one then it's partition will also follow the same rule....thanx ...please rate 5 stars....

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