Fourier Series Partial differentiation

Diploma In Electrical and Electronic Engineering.
Unit Title: Advanced Mathematics For Engineering.
Assignment Title:Fourier Series Partial differentiation.

1(a)Shown below is the output of a half-wave rectifier. The signal is a half-wave rectified sine wave.This rectified sine wave is given as the input to a filter.

Determine the Fourier series for the half wave rectified sinusoidal voltage Vsinwt defined by:

f(x)= { Vsinwt 0 < t / π (δ)

0 π /( δ) < t < 2/ π (δ)

which is a periodic function of period 2 π /(δ).

2(a)Determine the Fourier series to represent the periodic functions given by the table below:

Angle θ

Current i

30

0

60

-1.4

90

-1.8

120

-1.9

150

-1.8

180

-1.3

210

0

240

2.2

270

3.8

300

3.9

330

3.5

360

2.5

2(b) When a pulley is turned through an angle of θ degrees, the displacement y on a point on the pulley is given in the following table.

Angle θ

Displacement y

30

3.99

60

4.01

90

3.6

120

2.84

150

1.84

180

0.88

210

0.27

240

0.13

270

0.45

300

1.25

330

2.37

360

3.41

Construct a Fourier series for the first three harmonics.

3(a)The volume of a cone of height h and base radius r is given by V=1/3 π r2h

Determine ӘV/ Әh and ӘV/Әr

3(b)For z=5x4 + 2x3y2 – 3y,find

(i)Әz/Әx

(ii) Әz/Әy

3(c)Determine the co-ordinates of the stationary values of the function f(x,y0=x3 – 6x2 -8y2) and distinguish between them.

3(d)Given z=x2 sin(x-2y),find:

(i)Ә2 z/Әy2

(ii) Ә2 zӘx2

(iii)Show that Ә2 z/Әx Әy = Ә2z/Әy Әx

3(e)Find the rate of change of the total surface area of a right circular at the instant when the base radius is 5 cm and the height is 12 cm if the radius is increasing at 5 mm/s and the height is decreasing at 15 mm/s.

4(a)Solve Ә u/Әt =2tcosq given that u=2t when q

4(b)Solve the differential equation Ә2 u/Әx Әy =8ey sin 2x, given that at y =0,

Әu/Әx = sin x, and at x=π2, u=2y2

4(c)Verify that u=e-y cos x is a solution of Ә2u/Әx2 +Ә2u/Әy2 =0.

4(d)Solve T” =c2 m T given c =3 and m =1.

5(a)An elastic string is stretched between two points 40cm apart.Its center point is displaced 1.5 cm from its position of rest at right angles to the original direction of the string and then released with zero velocity.

Determine the subsequent motion u(x,t) by applying the wave equation

Ә2u/Әx2 = 1/c2 Ә2u/Әt2 with c2 =9.