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 lim_x→0((1 - exp(- 2x))/x).

(c) Use l'HOpital's rule to find lim_x→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.