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Divide the interval [0, 1] into n equal subintervals. For the resulting partition På compute U(ƒ, Pn) and L(ƒ, Pn) for each of the following functions, and determine S f (if it exists). (a) f(x) = 2

Divide the interval [0, 1] into n equal subintervals. For the resulting partition På compute U(ƒ, Pn) and L(ƒ, Pn) for each of the following functions, and determine S f (if it exists). (a) f(x) = 2

Elementary Linear Algebra (MindTap Course List)
Elementary Linear Algebra (MindTap Course List)
8th Edition
ISBN:9781305658004
Author:Ron Larson
Publisher:Ron Larson
Chapter4: Vector Spaces
Section4.2: Vector Spaces
Problem 48E: Let R be the set of all infinite sequences of real numbers, with the operations...
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Can you please refer to the notes as to how to compute U & L. Thank you!

Partitions; Upper and Lower Sums. DEFINITION 7.1. A partition of [a, b] is a finite set P = {x0, x₁,x₂,..., Xn} such that a = xo, b = xn, and x0 < x1 < x₂ < ... < Xn. REMARK 7.2. Given a partition {xo, x1, x2, ..., xn} of [a, b] and a bounded function ƒ : [a, b] → R, consider a subinterval [x-1, xi]. Denote • m₂ = inf{f(x) : f = [i-1, xi]} M₁ = sup{f(x): ƒ € [xi-1, xi]} DEFINITION 7.3. Consider a bounded function f [a, b] → R, and consider a partition P = {x0, x1, x2,..., n} of [a, b]. Define the upper sum as n U (f, P) ΣM₁ · (xi — xi-1), and i=1 the lower sum as n L(f, P) := Σmi ⋅ (xi — Xi–1). i=1 DEFINITION 7.4. Consider a partition P = {x0, x₁, x2,..., n} of [a, b]. A partition of [a, b] is called a refinement of P if PCQ. PROPOSITION 7.5. Consider a bounded function f : [a, b] → R and a partition P = {xo, x1, x2,..., n} of [a, b]. If Q is a refinement of P, then U(f, P) ≥ U(ƒ,Q) and L(f, P) ≤ L(ƒ, Q). PROPOSITION 7.6. Let f: [a, b] → R be bounded. (1) If P is a partition of [a, b], then L(ƒ, P) ≤ U(ƒ, P). (2) If P₁ and P₂ and any partitions of [a, b], then L(f, P₁) ≤U(f, P₂). :=
Transcribed Image Text:Partitions; Upper and Lower Sums. DEFINITION 7.1. A partition of [a, b] is a finite set P = {x0, x₁,x₂,..., Xn} such that a = xo, b = xn, and x0 < x1 < x₂ < ... < Xn. REMARK 7.2. Given a partition {xo, x1, x2, ..., xn} of [a, b] and a bounded function ƒ : [a, b] → R, consider a subinterval [x-1, xi]. Denote • m₂ = inf{f(x) : f = [i-1, xi]} M₁ = sup{f(x): ƒ € [xi-1, xi]} DEFINITION 7.3. Consider a bounded function f [a, b] → R, and consider a partition P = {x0, x1, x2,..., n} of [a, b]. Define the upper sum as n U (f, P) ΣM₁ · (xi — xi-1), and i=1 the lower sum as n L(f, P) := Σmi ⋅ (xi — Xi–1). i=1 DEFINITION 7.4. Consider a partition P = {x0, x₁, x2,..., n} of [a, b]. A partition of [a, b] is called a refinement of P if PCQ. PROPOSITION 7.5. Consider a bounded function f : [a, b] → R and a partition P = {xo, x1, x2,..., n} of [a, b]. If Q is a refinement of P, then U(f, P) ≥ U(ƒ,Q) and L(f, P) ≤ L(ƒ, Q). PROPOSITION 7.6. Let f: [a, b] → R be bounded. (1) If P is a partition of [a, b], then L(ƒ, P) ≤ U(ƒ, P). (2) If P₁ and P₂ and any partitions of [a, b], then L(f, P₁) ≤U(f, P₂). :=
Divide the interval [0, 1] into n equal subintervals. For the resulting partition På compute U(f, Pn) and L(f, Pn) for each of the following functions, and determine St. f (if it exists). (a) f(x) = 2
Transcribed Image Text:Divide the interval [0, 1] into n equal subintervals. For the resulting partition På compute U(f, Pn) and L(f, Pn) for each of the following functions, and determine St. f (if it exists). (a) f(x) = 2
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