Analysis 2 Notes 3 Flashcards
(3D-N10-3.1) Function bounded on a closed interval. Set of all bounded functions denotation.
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(3D-N10-3.2) Definition of Partition, Mesh (norm, width), Refinement
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(3L-N11-3.1) If P, Q are partitions, and P is a subset of Q, then the mesh of P is greater than or equal to mesh of Q
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(3D-N11-3.3) Lower Darboux Sum, Upper Darboux Sum
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(3L-N12-3.2) If the partition Q is a refinement of the partition P with one extra point then L(f,P) ≤ L(f,Q), U(f,Q) ≤ U(f,P)
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(3L-N12-3.3) If the partition Q is a refinement of the partition P then L(f,P) ≤ L(f,Q), U(f,Q) ≤ U(f,P)
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(3T-N12-3.1) For any P, Q ∈ P[a, b], L(f,P) ≤ U(f,Q)
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(3D-N13-3.4) Lower, Upper Riemann Integral of f over a closed interval
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(3C-N13-3.1) Lower Riemann Intergral less than or equal to Upper Riemann Integral
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(3D-N13-3.5) Function Riemann Integrable
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(3T-N14-3.2) Riemann’s Criterion for Integrability.
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(3T-N15-3.3) Attainment of bounds theorem
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(3T-N15-3.4) Continuous functions subset of Riemann Integrable functions
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(3D-N15-3.6) Function increasing decreasing
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(3D-N16-3.8) Function monotonic
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(3T-N16-3.5) If f : [a, b] → R is monotonic on [a, b] then it is Riemann integrable on [a, b].
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(3L-N17-3.4) Let the partition Q be a refinement of the partition P with one extra point, and let M be an upper bound for |f| on [a,b]. Then L(f,Q)−2M∥P∥ ≤ L(f,P) ≤ L(f,Q), U(f,Q) ≤ U(f,P) ≤ U(f,Q)+2M∥P∥
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(3L-N17-3.5) Let the partition Q be a refinement of the partition P with k extra points, and let M be an upper bound for |f| on [a,b]. Then L(f,Q)−2kM∥P∥ ≤ L(f,P) ≤ L(f,Q), U(f,Q) ≤ U(f,P) ≤ U(f,Q)+2kM∥P∥
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(3T-N17-3.6) Let. f∈B[a,b]. Then∀ε>0 ∃δ>0 such that ∥P∥
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(3D-N18-3.9) Limiting sequence.
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(3T-N18-3.7) Let f ∈ B[ab] and let {Pn} be a limiting sequence of partitions. Then lim n to infinity L(f,Pn) is the LRI. Same with upper sum and Integral (see HW 3)
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(3T-N18-3.8) Let f,g ∈ B[a,b] and suppose that f(x) ̸= g(x) only at a finite number of points. Then LRI f(x) = LRI g(x). Same with URI. In particular, f ∈ R[a, b] iff g ∈ R[a, b].
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(3T-N19-3.9) Linear Properties. Let f,g ∈ R[a,b]. Then f+g ∈ R[a,b] and (as expected), Then αf ∈ R[a,b] and (as expected). Order Properties.Let f,g ∈ R[a,b] and f ≥ g. Then RI (f) greater than or equal to RI (g). Domain splitting property. Triangle Inequality. Cauchy-Schwarz inequality.
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(3O-N20-) What I as an Interval means, plus notation
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(3D-N20-3.10) Definition of a function being locally Riemann Integrable.
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(3D-N21-3.11) Definition of a function being a primitive
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(3T-N21-3.10) Fundamental Theorem of Calculus
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(3R-N22-3.1) Fundamental Theorem of Calculus can be proven without __. It suffices to use __
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(3D-N22-3.12) Definition of Indefinite Integral
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(3T-N23-3.11) Let f ∈ Rloc(I) and let F : I → R be an indefinite integral of f. Then (i) F is continuous on I; (ii) F is differentiable at each interior point x0 ∈ I at which f is continuous; (iii) if f is continuous on I then F is a primitive of f.
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(3C-N24-3.2) Let f,g ∈ C[a,b] and let F, G be primitives of f, g respectively. Then …(fill in)
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(3C-N25-3.3) Let I be an open interval and let φ : I → R be continuously differen- tiable. Let J be an interval of positive length such that φ(I) ⊂ J and let f : J → R be continuous. Then for any a, b ∈ I
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(3T-N25-3.12) Let r be a natural number and let f : [r, +∞) → [0, +∞) be a decreasing function. Then the infinite series (E) f(k) converges iff lim f(x)dx exists.
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