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UNIVERSITY OF DUBLIN XMA1213 TRINITY COLLEGE Faculty of Engineering, Mathematics and Science school of mathematics JF Maths, JF TP JF TSM Hilary Term 2009 Course 121 Monday, March 9 Regent House 14:00 – 16:00 Dr. P. Karageorgis Attempt all questions. All questions are weighted equally. Log tables are available from the invigilators, if required. Page 2 of 2 1. Compute each of the following integrals: Z 3x + 5 dx, x3 − x XMA1213 Z x cos x dx. 1 if x 6= 0 2. Let f be defined by f (x) = . Show that f is integrable on [0, 1]. 0 if x = 0 3. Define a sequence {an } by setting a1 = 4 and an+1 = 1 5 − an for each n ≥ 1. Show that 0 ≤ an+1 ≤ an ≤ 4 for each n ≥ 1, use this fact to conclude that the sequence converges and then find its limit. 4. Compute each of the following limits: lim x→0 ex − x − 1 , x2 lim x sin(1/x). x→∞ 5. Test each of the following series for convergence: ∞ X (−1)n−1 n2 n=1 1 + n4 , ¶n ∞ µ X 2n . 1 + 3n n=1 6. Find the radius of convergence of the power series ∞ X (n!)2 n f (x) = ·x . (2n)! n=0 7. Suppose f is a differentiable function such that f 0 (x) = 2x · f (x) for all x ∈ R. Show that 2 there exists some constant C such that f (x) = Cex for all x ∈ R. 8. Suppose that f is a function with |f (x) − f (y)| ≤ |x − y|2 for all x, y ∈ R. Using the limit definition of the derivative, show that f is actually constant. c UNIVERSITY OF DUBLIN 2009 °