Download 2008 Lancaster Uni Math & Stats Exam: Integration (Part II) - Section A & B Questions and more Exams Information Integration in PDF only on Docsity!
LANCASTER UNIVERSITY
2008 EXAMINATIONS
PART II (Third or Fourth Year) MATHEMATICS & STATISTICS 2 hours Math 314: Integration
You should answer ALL Section A questions and TWO Section B questions. In Section A there are questions worth a total of 50 marks, but the maximum mark that you can gain there is capped at 40. SECTION A
A1. (i) Show with the aid of a suitable diagram that^ N^ ×^ N^ is countable. (ii) Deduce that the set of rational numbers is countable. (iii) A real number is said to be algebraic if it is the root of a non–zero polynomial that has integral coefficients. Show that 2 + √3 and 2 − √3 are algebraic. [14]
A2. (i) Explain briefly why sin^ x^ is a measurable function. (ii) Let f be a bounded and measurable function on a bounded interval (a, b) such that L ≤ f (x) ≤ M for all x ∈ (a, b). With reference to a suitable diagram, show that L(b − a) ≤
∫ (^) b a^ f^ (x)dx^ ≤^ M^ (b^ −^ a). (iii) Deduce that − 1 ≤ cos^ a b −− acos b≤ 1 (a < b). [12]
A3. (i) State the Bounded Convergence Theorem. (ii) Show that the set S = {x ∈ (0, 1) : sin^2 (1/x) = 1} is countable. (iii) Deduce that (^) ∫ (^1) 0 sin
2 n(1/x)dx → 0 (n → ∞). You may assume that the integral of a function is zero over any countable set.
[14]
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SECTION A continued
A4. (i) State the Heine–Borel Theorem on covers. (ii) Show that the collection of intervals (−n, n) (n = 1, 2 ,.. .) gives a cover of R that has no finite subcover. [10]
SECTION B
B1. (i) Define what is meant by an open subset of R. Show that (a, b) is open, and that any union of open sets Ej (j ∈ J) is again open. [8] (ii) Let (Ej ) (j ∈ J) be a collection of disjoint non-empty open intervals. Show using Archimedes’ Axiom that J is countable. [6] (iii) State the form of a typical open subset E of R, and show how to define the measure m(E). [6] (iv) Starting with the definition of outer measure me, show that any countable set F has outer measure me(F ) = 0. You should state clearly any general results that you use. [10]
B2. (i) State the Monotone Convergence Theorem. [8] (ii) By applying the inequality of the means to the n + 1 numbers 1 , (1 − t/n), (1 − t/n),... , (1 − t/n), show that fn(t) = tα−^1 (1 − t/n)nI(0,n)(t) gives an increasing sequence of functions for any α > 0. [8] (iii) Deduce that (^) ∫ (^) n 0 t
α− (^1) (1 − t/n)n (^) dt →^ ∫^ ∞ 0 t
α− (^1) e−t (^) dt as n → ∞. [6] (iv) Let t = n sin^2 θ in the left-hand side of (iii) and deduce a limit formula. [8]
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