Reference no: EM132758834

Assignment

In order to complete this assignment you will have to consider functions which are unique to you. The parameters of the functions are derived from your student number and so if you use the wrong number, you will get the wrong answers and will lose marks. Use the last two digits of your student number where K1 = last_but_one_digit+2 and K2 = last_digit+2. Thus the number 201345678 would give K1 = 9 and K2 = 10 and for question 3a) you would use F(s) = 9/(s+10)²

Question 1. Determine the Laplace transforms of the following functions of time.

a. f(t) = e^-k1tcos(k₂t)
b. f(t) = t³e^-k2t

Question 2. Determine the Laplace transforms of the following functions of time.

f(t) = {e-k₁t  0 ≤ t < k₂
            0         t ≥ k2

Question 3. Determine the inverse Laplace transforms of the following functions of s.

a. F(s) = k1/(s+k₂)²

b. F(s) = s+k₁+k²₂

Question 4. Using Laplace transforms, solve the following differential equation with the initial conditions indicated. Sketch the resulting function of time.

d²z + (k₁ + k₂)dz/dt + k₁k₂z = 0, with z(0) = 0, dz(0)/dt = 1

Question 5. Show that the two polynomials B0(t) = √6/6 and B₁(t) = √2/6(t-3) are orthonormal with respect to the inner product.

<Bi(t),Bj(t)> = ₀∫⁶ B_i(t)B_j(t)dt

Question 6. Calculate the symmetric Fourier series for the periodic function

f(t) = {K1  -Π ≤ t < 0
            0   0 ≤ t < Π

with the period 2Π, using Cn = 1/2Π _Π∫^-Π f(t)e^-jntdt

Question 7. Calculate the Fourier transform F(ω) = _∞∫^-∞ f(t)e^-jωtdt, for the two-sided exponential pulse function given by:

f(t) = { e^3t (t < 0)
            e^-3t (t ≥ 0)

Find the Fourier transforms of f(t - K₁) and f(K₂t).

Question 8. Consider the function f(t) = cosh(t) .

Estimate the derivative df(t)/dt at t = 4 to three decimal places using the forward difference method, the backward difference method, and the central difference method with a step size h = 0.1

Question 9. A cyclist's speed as a function of time is measured at one-second intervals as the cyclist cycles over the top of a hill, see Table 1 below. Estimate the distance covered with the Trapezium rule using 5 points, and using the Simpson's rule of 5 points to get a better estimate.

time (s)	0	1	2	3	4
speed (m/s)	15	20	23	24	25

Table 1: speed of cyclist.

Question 10. Perform five iterations of the bisection method to get the approximate zero, to one decimal place, of the following function starting with the bracketing points a = 1.0 and b = 2.0

f(t) = t tanh (t/2) - 1

Question 11. Use linear regression to find the Least Squares best fit line to the data points in Table 2:

t	1	2	3	4	5
y	2.5	5.7	8.0	8.5	10.5

Table 2: data points for linear regression.