Two hours UNIVERSITY OF MANCHESTER. 21 January

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1 Two hours MATH20111 UNIVERSITY OF MANCHESTER REAL ANALYSIS 21 January Answer ALL SIX questions in Section A (50 marks in total). Answer TWO of the THREE questions in Section B (30 marks in total). If more than TWO questions from Section B are attempted, then credit will be given for the best TWO answers. The use of electronic calculators is not permitted. 1 of 6 P.T.O.

2 SECTION A MATH20111 Answer ALL six questions A1. Show that the sequence ( ( 5 n + 2 ) ) n 3 n N is a null sequence. Name or state the results that you are using. [8 marks] (a) It was stated that < 1 for n sufficiently large and therefore the sequence is null. However, n 3 as seen in the examples class, this reasoning is not sound in general. The correct reasoning says: if there is some c with 0 c < 1 and (a n ) n N is a sequence with a n c from some n on, then (a n ) n N is a null sequence (using the standard list of null sequences and the sandwich rule). (b) Results from the lectures were not quoted. A2. Find the limit of the sequence ( n 2 + n n 2 n) n N [7 marks] (a) Expression involving the limit symbol were floating about, mixed with expressions not involving the limit symbol. For example, expression like or lim n2 + n n 2 n = ( n 2 + n n 2 n) ( n 2 + n + n 2 n) n n2 + n + n 2 n n n were stated. Both expression do not make sense. (b) Various computational errors lead to the wrong result. = 1 A3. Show that for every real number c 0 the series is convergent. Justify your answer by naming or stating the results that you are using. Some answers tried to apply the limit comparison or the comparison test using n=1 n=1 c n n! [7 marks] 1. But this n 2 does not work for all c 0, or, the comparison test is not applicable. The correct answer goes via the ratio test and may be found in of the lecture notes (replace c by c 1 ). A4. Let f : R R be a function. State the definition of 2 of 6 P.T.O.

3 MATH20111 (i) f is continuous and of (ii) f is differentiable. Give an example of a differentiable function whose derivative is not differentiable. No justification is required. [8 marks] (a) The most popular mistake in both definitions was the following: It was stated what it means that a function is continuous (or differentiable) at a point x 0, but it was not stated what it means that f is continuous (or differentiable). (b) One possible example is x x. Some answers said x 2 x, but this function has a differentiable derivative. Other answers said x, but this function is not differentiable. A5. State the intermediate value theorem and use this theorem to show that for every continuous function f : [ 1 2, 1] [1, 2] there is some x [ 1 2, 1] with f(x) x = 1. (Hint: Sketch the graph of 1 x in the interval [ 1 2, 1].) (a) The statement of the intermediate value theorem was flawed. [10 marks] (b) For the application, the assumptions of the intermediate value theorem were only partially verified. (c) For the application, some attempts were made to show that f(x) = 1 as functions (which x means f(x) = 1 for all x [ 1, 1]). But this is not true in general. The question asks to show x 2 that f(x) = 1 for some x [ 1, 1]. x 2 (d) For the application, a picture was drawn and then some heuristic (and flawed) argument was given why f(x) = 1 at some point. x A6. For each point x 0 R, decide whether the function f : R R defined by f(0) = 0 and f(x) = x sin( 1 ) if x 0, x3 is differentiable at x 0. Justify your answer (you may use without proof that sin(x) is differentiable). [10 marks] Many solutions computed the derivative of x sin( 1 x 3 ) (which was actually not asked, the question asked to explain why f is differentiable/not differentiable at a point). From this computation several flawed answers were derived: 3 of 6 P.T.O.

4 MATH20111 (a) The function was labeled as differentiable, which is not true (x 0 = 0), and insufficient evidence was given for the case x 0 0. (b) For x 0 = 0, the limit of the derivative when x 0 was computed and used in subsequent arguments. However, this is not how to confirm differentiability/nondifferentiability at 0; to do this one has to analyze the limit behavior of the Newton quotient at 0. 4 of 6 P.T.O.

5 SECTION B MATH20111 Answer TWO of the THREE questions B7. (i) What is meant by saying that the sequence (a n ) n N does not converge to the real number r? (ii) Give a proof from first principles that the sequence (1 + ( 1) n ) n N is divergent. (iii) State the division rule for convergent sequences. (iv) Let n=1 a n and n=1 b n be absolutely convergent series. Prove that n=1 (a n + b n ) is absolutely convergent. [15 marks] In part (i) the negation of the definition went wrong, because the quantifiers were not negated correctly. Other mistakes were the conversion of ε > 0 into ε < 0, etc. In part (ii) a proof using subsequences and was given. But this is not from first principles. Some people actually went on proving In that case of course the proof is again from first principle and full marks were awarded. In part (iii) the majority of answers forgot to say (any, or some) assumptions in the division rule. B8. Let f : R R be a function and let g : R R be defined by g(x) := f(x). Prove or disprove the following statements: (i) If f is continuous, then g is continuous. (ii) If g is continuous, then f is continuous. (iii) If g is continuous, then f 2 is continuous. (iv) If g is continuous, then f 3 is continuous. This question went well when it was attempted. However, very few people tried this. B9. (i) State the Cauchy Mean Value Theorem and L Hôpital s Rule. (ii) Prove L Hôpital s Rule from the Cauchy Mean Value Theorem. [15 marks] 5 of 6 P.T.O.

6 MATH20111 This went more or less well. Some mistakes were as follows: [15 marks] In (i), the statements were flawed, mainly the assumption were incomplete or stated incorrectly. For example f was assumed to be continuous and differentiable on [a, b]. In (ii), some answers did not show clearly how the Cauchy mean value theorem is applied. Another mistake was the absence of the dependency of the point ξ x of x. See the proof of L Hôpital s Rule in the notes. END OF EXAMINATION PAPER 6 of 6