201-NYB-05-2018-3.pdf - Final Examination(5(5(5(5(5(5(5...

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Final Examination 201-NYB-05 December 2018 1. Evaluate the following integrals. (a) (5) Z 5 x 2 - 3 x + 10 ( x - 2)( x 2 + 4) dx (b) (5) Z / 4 0 sin 4 x cos 6 x dx (c) (5) Z p x 2 - 25 x 4 dx (d) (5) Z e - x cos(3 x ) dx (e) (5) Z 9 0 1 p p x + 1 dx (f) (5) Z ln 3 0 e x p 15 + 2 e x - e 2 x dx (g) (5) Z x arcsin x dx 2. Evaluate the following limits. (a) (3) lim x ! sin 2 (2 x ) 1 + cos x (b) (3) lim x !1 x 2 - arctan x (c) (3) lim x ! 0 + 1 + x 2 4 /x 3. (7) Determine whether the following improper integrals converge or diverge. If an integral converges, give its exact value. (a) Z 11 - 1 1 3 p ( x - 3) 4 dx (b) Z 1 0 e - p x p x dx 4. (4) Solve the di erential equation: p 1 - x 2 dy dx = x + xy 2 , with initial condition y (0) = 1. 5. (3) A rumour starts in a town with a population of 1000. The rumour spreads at a rate proportional to the number of people who at time t (in weeks) have not heard the rumour. Initially, 25 people heard the rumour and at the end of 3 weeks, 675 people had heard it. How many people will have heard the rumour at the end of 6 weeks?

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