(b) Prove that the following function does not tend to a limit as x tends. is continuous at 1. [6] you use. (i) f(x) = x 4 4x+7, I = [1,2]

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1 TMA M Cut-off date 28 April 2014 (Analysis Block B) Question 1 (Unit AB1) 25 marks This question tests your understanding of limits, the ε δ definition of continuity and uniform continuity, and your ability to give careful proofs and to apply results involving these concepts. (a) In this part of the question you may assume any results that are proved in Unit AB1, but you should refer to any result that you use. Prove each of the following statements. x 3 x 2 +2x 2 (i) lim x 1 x 2 = (ii) lim x 0 cos(x+x 3 ) 1 x+x 3 = 0 [8] (b) Prove that the following function does not tend to a limit as x tends to zero: f(x) = 5x2 +2 x. [4] x (c) Use the ε δ definition of continuity to prove that the function f(x) = x 2 +3x is continuous at 1. [6] (d) Determine whether each of the following functions is uniformly continuous on the given interval. State any results from M208 that you use. (i) f(x) = x 4 4x+7, I = [1,2] (ii) f(x) = x+2, I = (0,1] [7] x page 5 of 11

2 Question 2 (Unit AB2) 25 marks This question tests your ability to determine whether a function is differentiable at a given point, and to apply theorems relating to differentiability. (a) (i) Prove from the definition of differentiability that the function f(x) = x 1 x+1 is differentiable at 2, and find f (2). (ii) Sketch the graph of the function { x 2 +x, x 1, f(x) = 2sin ( π 2 x), x > 1. Determine whether or not the function f is differentiable at 1. [9] (b) The function f is differentiable on [2,6]. Use the Mean Value Theorem to prove that if f(2) = 3 and f (x) 2 for x (2,6), then 5 f(6) 11. [4] (c) Prove the following inequality: tan 1 x x 1 3 x3, for x [0,1]. [4] (d) Prove that the following limit exists, and evaluate it: lim x 2 1 cos(πx) x 3 5x 2 +8x 4. [8] Question 3 (Unit AB3) 25 marks This question tests your understanding of the ideas behind Riemann integration, and your ability to evaluate integrals using various techniques and to apply results relating to integration. (a) Let f be the function 1, x = 1, 3x+3, 1 < x 0, f(x) = 2 2x, 0 < x < 1, 1, x = 1. (b) Let Sketch the graph of f, and evaluate L(f,P) and U(f,P) for each of the following partitions P of [ 1, 1]. (i) P = {[ 1, 1 2], [ 1 2,1]} (ii) P = {[ 1, 1 3], [ 1 3,0], [ 0, 1 2], [ 1 2,1]} [7] I n = 1 0 (i) Evaluate I 0. e x (1 x) n dx, n = 0,1,2,... (ii) Show that a reduction formula for I n is I n = 1+nI n 1, for n 1. (iii) Deduce the values of I 1, I 2 and I 3. [6] page 6 of 11

3 (c) Prove that dx x(log e x) = 8(log ex) 1/8, 7/8 and hence determine whether the series 1 n(log e n) 7/8 n=2 is convergent or divergent. [7] (d) Using Stirling s Formula, determine a number λ such that ((3n)!) 2 n λ (6n)! 2 6n as n. [5] Question 4 (Unit AB4) 25 marks This question tests your ability to determine a Taylor polynomial and a remainder estimate, to determine the interval of convergence for a power series, and to apply the General Binomial Theorem. (a) (i) Calculate the Taylor polynomial T 2 (x) at 2 for the function f(x) = x x+2. (ii) Show that T 2 (x) approximates f(x) with an error of less than on the interval [2, 2.5]. [9] (b) (i) Determine the interval of convergence of the power series ( 1) n+1 (n+3)3 n (x+3)n. n=1 (ii) State the radius of convergence of the power series ( 1) n+1 (n 2 +4n+3)3 n (x+3)n+1, n=1 briefly explaining your answer. [10] (c) Use the General Binomial Theorem to determine the first four terms of the Taylor series at 0 for the function f(x) = (1 3x) 2/3. State the radius of convergence of this power series. [6] page 7 of 11

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5 is uniformly continuous on [1,2]

6 f is differentiable & continuous at x=1, so f is continuous at x = 1 but is not smooth at that point You MUST check the conditions for the MVT are met before use and don't omit the existence of the point c

7 Using tables helps avoid silly errors

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9 When X = -6 X = -6 and X= 0 L

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