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The may/june 2009 exam for the mathematical analysis course at the university of wales, aberystwyth. The exam covers various topics in mathematical analysis, including factorising cubics, limits, series, and continuity. Students are required to solve problems related to unbounded sets, supremum and infimum, and monotone sequences.
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MA11110 - Mathematical Analysis
Time allowed - 2 hours
Section A
Factorise the cubic x^3 − 7 x + 6, and determine the values of x which satisfy the inequality x^3 − 7 x + 6 < 0. [6 marks]
Let α be a real number that satisfies |α| < ε for every ε > 0. Prove that α = 0. [4 marks]
(a) Let A ⊆ R. What is meant by saying that A is unbounded below? [2 marks] (b) Prove that the set (−∞, 1] is unbounded below. [2 marks]
Classify the following subsets of R as bounded or unbounded. If the set is bounded, write down the supremum and infimum. (a) A 1 = {x : |x + 2| < 2 } (b) A 2 = {n^2 : n is prime} (c) A 3 = ⋂∞ n=1(1 − 1 /n, 3 + 1/n). (There is no need to provide proofs of your assertions.) [5 marks]
(a) Give the (N, ǫ)-definition of convergence for sequences. [2 marks] (b) Using this definition, prove that
nlim→∞^28 nn^ + 1+ 2 =^14. [3 marks]
(a) (^) nlim→∞^53 nn^ + 2− 7 ; (b) (^) nlim→∞^3 n
(^5) − 4 n (^4) + 7 12 n^5 + 3n^2 − 1. [3,3 marks]
(a)
n=
(−1)n 8 ;^ (b)
n=
7 n+ 8 n^. If the series is convergent, find its sum. [5 marks]
xlim→ 0 sin^ x^ + cos x x^ −^1. [5 marks]
g(x) =
{ (^) − 1 if x ≤ 0 , 1 if x > 0. Prove that g is discontinuous at x = 0, but continuous at all other points of R. [7 marks] (b) Use the (ε, δ)-definition of continuity to show that the function f : R+^ → R given by f (x) = (^1) x is continuous on R+. [7 marks] (c) State the Intermediate Value Theorem. [3 marks] (d) Show that the equation x^5 −x^3 −x^2 −1 = 0 has at least one real root.[3 marks]
(a) Let x 0 ∈ (a, b). What does it mean for f fo be differentiable at x 0? [2 marks] (b) Use the definition of the derivative to show that for f (x) = x−^2 we have f ′(x) = − 2 x−^3 for any x 6 = 0. [4 marks] (c) State and prove Rolle’s Theorem. (You may use without proof that every continuous function on [a, b] achieves its maximum and minimum).[3,8 marks] (d) Let g(x) = x^3 + 3x − 5. Show that there are no two distinct points x 1 , x 2 in R such that g(x 1 ) = g(x 2 ). [3 marks]
