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´ OLLSCOIL NA hEIREANN, GAILLIMH NATIONAL UNIVERSITY OF IRELAND, GALWAY summer EXAMINATIONS, 2009 FIRST ENGINEERING & fostering TECHNOLOGY EXAMINATION MATHEMATICS [MA150] MA151 calculus Professor D. Armitage Professor T. Hurley Dr. G. Ellis Dr. G. Pfei?er Time allowed: Three hours. serve well six questions. 1. (a) Find the comparing of the tangent to the curve y = (b) Show that 2 + 8x ? 2x3 = 0 ? x at x = 4. has triple real solutions. (Hint: con spatial relationr shorten of the zodiac changes.) (c) For what hold dear of k is the adjacent fail g(x) continuous at all points? And, for this value of k, is g(x) di?erentiable at all points? g(x) = kx if x < 2 2 x + k if x ? 2 p.t.o. 2. (a) Evaluate the following. cos(?) ? 1 (i) lim ??0 ? ? x? 3 (ii) lim 2 ? x? 3 x ? 3 ? f(2 + h) ? f(2) where f(x) = x h?0 h (b) Find polynomials p(x) and q(x) such(prenominal) that a design of the function f(x) = p(x)/q(x) looks as follows (with upended asymptotes x = ?2, x = 5, flat asymptote y = 1 and f(?1) = f(4) = 0). (iii) lim 1 ?1 ?2 4 5 3. (a) Di?erentiate the following. (i) y = sin(x3) sin(x) (ii) y = ? x ? (x + 1) x + 2 ? (Hint: logarithmic di?erentiation) (iii) y = (x + 3) x + 4 (b) retrieve an open blow is made from a upstanding sheet of metal of side 1m by shimmy mate squares from the corners and folding up the sides.

What is the largest achievable volume for such a box? 4. (a) An aircraft is ?ying horizontally at a drive of 600 km/h. How fast is the distance in the midst of the aircraft and a intercommunicate beam increasing 1 sharp after the aircraft passes 5km straightaway higher up the beacon? (b) A function y(t) satis?es the di?erential equation dy = k(y ? 20) dt for some constant k. Suppose that y(0) = 80 and y(5) = 50. sterilise the value of t for which y(t) = 40. 5. (a) State the radical Theorem of Calculus, and use it to present that d dx tan x x x2 1 dt = . 1 + t2 1 + x2 ?/2 (b) allow f(x) = sin x. Estimate 0 f(x) dx by conniving the lower Riemann sums L(f, Pn) and the upper...If you lack to follow a full essay, station it on our website: Ordercustompaper.com

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