Reference no: EM133043749

2010ODL-N Further Maths - Teesside University

Question 1. The output response, f(t) of a feedback control system is given by:

f(t) = k/1+k [1 - e^-(1+k)t], where k is a constant.

Determine:

a) The value of f(3), taking k = 5. Give your answer accurate to 2d.p.

b) The inverse function f^-1(t), where k is a constant.

Question 2. The functions g and h are defined by:

g(x) = 6/3 - x; x ≠ 3

h(x) = 5x + 2

Determine:

(a) The function g(h(x).

(b) The function h(g(x)).

Question 3. Solve the equations correct to 2 decimal places accuracy.

(a) Estimate using the bisection method, the positive real root of:

ln(x⁴) = 0.7

Take the initial guess as x_L = 0.5, x_IJ = 2

(b) Estimate using Newton Raphson method, one root of:

2x³ - 11. 7x² + 17. 7x - 5 = 0

Take the initial guess as x₀ = 3.

Question 4. Evaluate ₀∫^Π/3 √sin (x) dx correct to 3 decimal places using n= 6 equal intervals, applying:

(a) Trapezoidal rule

(b) Mid-ordinate rule

(c) Simpsons rule

Question 5. Determine the general solution of the differential equations. Write out the solution y explicitly as a function of x.

(a) 3x²y² dy/dx = 2x - 1

(b) 2dy/dx + 3y = e^-2x - 5

Question 6. Determine the particular solution of the equation:

d²y/dx² + 3dy/dx + 2y = 10cos(2x) satisfying the initial conditions y(0) = 1, y′(0) = 0.

Question 7. Find the inverse Laplace transform of:

(a) 3-2s/s²-4s+9

(b) s+2/(s²+9)(s-2)

Question 8. Solve using the Laplace transform method:

d²y/dt² - 2 dy/dt + y = 5e^t and y(0) = 1, y'(0) = 1

Attachment:- Assignment - Further Mathematics.rar