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April 18, 2006 Lecturer: Dr Martin Kurth Trinity Term 2006 Course 1E1 2005-2006 (JF Engineers & JF MSISS & JF MEMS) Problem Sheet 2 Due: in the Tutorials 28 April / Lecture 02 May Every human activity, good or bad, except mathematics, must come to an end. Paul Erdòˆs (1913–1996) 1. Calculate the following integrals, provided they exist: ¢ R∞¡ (a) 1 cosh x − 12 ex dx, R∞ (b) 1 1+x x2 dx, R2 1 (c) 1 x−1 dx, ¢ R2¡ (d) 0 sin( x1 ) − x1 cos( x1 ) dx. (4 points) Hint for (d): Calculate the derivative of x sin(1/x) first. 2. Calculate the limits of the following sequences {an }, provided they exist: (a) an = (b) an = 4 n2 , n+3 n , (c) an = (−1)n , (d) an = 1 n2 sin(n) cos(n). (4 points) 3. Consider the function f defined by f (x) = x4 − x2 . Find the critical points of f , and determine whether they are minima, maxima, or neither. (2 points) 4. Prove the Continuous Function Theorem: If {an } is a sequence with limn→∞ an = L and f is a function that is continuous at L and defined at all an , then limn→∞ f (an ) = f (L). (*) Questions 1, 2 and 3 should be answered by all students, you will get points for them. Question 4 is more challenging and meant as an exercise for the more mathematically interested students.