Reference no: EM131852597
Question 1 (a) Derive the first three non-zero terms for the Maclaurin series expansion of exp(-2x)
(b) Use the series expansion in (a) to find limx→0((1 - exp(- 2x))/x).
(c) Use l'HOpital's rule to find limx→0((1 - exp(- 2x))/x).
Question 2 The motion of a particle oscillating in an electric field subject to air resistance may be modelled by the initial-value problem (IVP)
x¨+ 2x·+ 5x = 0 , x(0) = 1 , x·(0) = -1 (*)
The following questions refer to the solution of this IVP.
(a) Use the Shift Theorem to find the Laplace transform of e-t cos(2t).
(b) Show, by taking Laplace transforms, that the Laplace transform of the solution x‾ of the IVP (*) is x‾ = (s +1)/s2 + 2s + 5
(c) Find a and b such that s2 + 2s +5 = (s+ a)2 +b.
(d) Use the result in (c) to find the inverse Laplace transform of x‾ = (s+1)/(s2 + 2s + 5)
Question 3 A charged particle moving in a spiral in a magnetic field is subject to the force field
F = -xi - yj + (z -1)k.
(a) Show by considering ∇ x F that the force field F = -xi - yj + (z -1)k is conservative.
(b) Show that φ = -x2/2 - y2/2+ (z - 1)2/2 is a scalar potential for F.
(c) Calculate the work done on the particle moving along part of the spiral described parametrically by the equations:
x(t) = cost, y(t)= sin t, z(t) = e-t , 0 ≤ t ≤ 2Π , given by the line integral c∫F.dr
Question 4. Verify Gauss's divergence theorem for the surface integral s∫F.dS, where F = xy2i - 2xyzj + zyk and S is the outside of the unit cube 0≤ x ≤ 1, 0 ≤ y ≤ 1, 0 ≤ z ≤ 1.
(a) Compute the surface integral here.
(b) Compute the volume integral here.
Question 5. A series 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).
Figure - Flow diverters schematic
(a) If the flow coming into the series of diverters on the left is F m2s-1, show that the flow through diverter j is 0.03(0.97)j-1 Fm3s-1
(b) Find the percentage of flow diverted through all 20 diverters.