Reference no: EM132653242

EMT4801 Engineering Mathematics - University of South Africa

QUESTION 1:

Consider the series
∑^∞_n=1 2n + 6/(n + 2)³.
For each of the following convergence tests state with justification whether the test proves convergence, divergence, or does not confirm either:

1.1 Ratio test.
1.2 Comparison test.

QUESTION 2:

2.1 Estimate the maximum error if the first four terms of the series

∑^∞_n=1 5n - 4/4ⁿ

is used to estimate the sum.

2.2 Determine the interval of convergence of the power series

∑^∞_n=1

(n+2)(x+2)ⁿ/2n(3n+2)

QUESTION 3:

3.1 Consider the transformation

w = 2/z + 2

from the z-plane onto the w-plane, where z = x + iy and w = u + iv.

3.1.1 Determine the equation of the image of the line y = x + 1 under this transformation.

3.1.2 Now compute the image of the points
A (-1, 0) ; B (0, 1) ; C (1, 2)
on the line in question 3.1.1.

3.2 Consider u(x, y) = (x + 1)² - y²

3.2.1 Show that u is harmonic.

3.2.2 Determine the harmonic conjugate v of u.

3.3.1 Show how the real integral

_-∞∫^∞ x²/(x²+1)(x²+4).dx

may be converted to a contour integral for a suitable path. Make sure you describe or sketch the path used.

3.3.2 Now determine

_-∞∫^∞ x²/(x²+1)(x²+4).dx

using the contour integral described in (3.3.1).

QUESTION 4
Suppose we are given a system with input u (t) and output (t) described by the equation
x"+4x' + 7x = 5u' - 3u.

Assume also that the system is initially at rest (i.e. x' (0) = x(0) = 0 = u (0)).
4.1 Write down the transfer function G(p) of the system.
4.2 Now write down a state-space model for the system (yielding the same transfer function).
4.3 Use the initial and final value theorems to determine g (0₊) and lim_t→∞ g (t) where g (t) = L^-1(G (P)).

QUESTION 5
Use the method of convolution to find

L^-1 [P²/(P²+1)²]

QUESTION 6

Suppose we have a system described by the difference equation with input {u_k}

3Y_k+2 + 4y_k+1 + Y_k = u_k+1 - u_k

which is initially in a quiescent state (y₀ = y₁ = 0 = u₀)

QUESTION 7

7.1 Solve the following state-space equations by taking a Z-transform and using an inverse ma¬trix, given that

0 = x₁(0) = x₂ (0) and u_k = {1, 0, 0,.....}.

7.2 Determine the values y₀; y₁; y₂.