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  3. 1 a derive the first three nonzero terms for the...

Question: 1 a derive the first three nonzero terms for the...

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1 (a) Derive the first three non-zero terms for the Maclaurin series expansion of [5 marks] exp(-2x) (b) Use the series expansion in (a) to find lim 2 marks (c) Use rHopitals rule to find liexpl-2x) [3 marks] Turn over
2 The motion of a particle oscillating in an electric field subject to air resistance may be modelled by the initial-value problem (IVP) 翼+2i + 5x = 0 , x(0) =1, i(0)--1 C) The following questions refer to the solution of this IVP. (a) Use the Shift Theorem to find the Laplace transform of e cos(2). 13 marks] (b) Show, by taking Laplace transforms, that the Laplace transform of the solution 7 marks s of the lP C) s +1 Continued over
(e) Find a and b such that s + 2s+ (s+a +b 2 marks] (d) Use the result in (c) to find the inverse Laplace transform of x =-st s +2s+5 13 marks Turn over
(c) Calculate the work done on the particle moving along part of the spiral x(t)-cos, y)-sint,z) e0ss2, given by the line integral 6 marks] described parametrically by the equations Fdr. Turn over
4Verify Gausss divergence theorem for the surface integral JFds, where F_xy2i-2pzj+ark and Siste outside of the unit ate (a) Compute the surface integral here. 10 marks) Continued over 7
(b) Compute the volume integral here 5 marks Turn over
5 Aseries of 20 flow diverters in a pipe operates in such a way that 3% of the flow is diverted at each junction (see Figure Q5 below) Diverter 1 Diverter 2 Diverter 3 Diverter 20 3% of flow 3% of flow 3% of flow 3% of flow Flow Figure Q5 Flow diverters schematic If the flow coming into the series of diverters on the left is Fms, show that the flow through diverter jis 0.03(0.97y F ms (a) [5 marks] Continued over 9
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