Advance Calculus

Pleaseuse radian mode and π = 3.142

Question 1

A function f(t) is defined as,

f(t) = π – t for 0 < t < π

Write down the odd extension of f(t) for -π < t < 0.

Determine the Fourier sine series, and hence, calculate the Fourier series approximation for

f(t) up to the 3rd harmonics when t = 1.11. Use π = 3.142. Give your answer to 3 decimal

places.

Answer :

Question 2

The Fourier series expansion for the periodic function, f(t) = |sin t| is defined in its
fundamental interval. Taking π = 3.142, calculate the Fourier cosine series
approximation of f(t), up to the 6th harmonics when t = 2.16. Give your answer to 3
decimal places.

Answer :

Question 3

An infinite cosine series is given by,

Compute the sum to r = 3 and t = 0.94. Use π = 3.142. Give your answer to 3
decimal places.

Answer :

Question 4

Consider the temperature distribution in a perfectly insulated rod of length L, where one end,

at x = 0, is maintained at a temperature of 0
0
C and the other end, at x = L, is insulated. This is

well modeled by the diffusion equation,

where a is a constant. It is subjected to boundary conditions,

t

The method of separation of variables, with Ө(x, t) = X(x)T(t), is used to determine the

boundary-value problem satisfied by T(t) which is given by, T(t) = exp(-kt)
where k is in terms of a, L and n = 1,2,3,……

Calculate the value of T at t = 0.32, when a = 0.5 n = 1 and L = 1.61, giving your answer to 3 decimal

places. You may assume that π = 3.142.

Answer :

Question 5

Consider the temperature distribution in a perfectly insulated rod of length L, where one end,
at x = 0, is maintained at a temperature of 0
0
C and the other end, at x = L, is insulated. This is
well modeled by the diffusion equation,

where a is a constant. It is subjected to boundary conditions,

t

The method of separation of variables, with Ө(x, t) = X(x)T(t), is used to determine the

boundary-value problem satisfied by X(x) which is given by. X(x) = sin(βx) where β is in
terms of L and n = 1,2,3,……

Calculate the value of β when n = 2 and L = 0.81, giving your answer to 3 decimal places. You may

assume that π = 3.142.

Answer :

Question 6

A periodic function f(t), with period 2π is defined as,

f(t) = 0 for -π < t < 0f(t) = π for 0 < t < π

Taking π = 3.142, calculate the Fourier series approximation up to the 5th harmonics
when t = 0.75. Give your answer to 3 decimal places.

Answer :

Question 7

A periodic function f(t), with period 2π is defined as

,f(t) = c for 0 < t < πf(t) = -c for -π < t < 0

where c = 1.6, Taking π = 3.142, calculate the Fourier sine series approximation up
to the 5th harmonics when t = 0.46. Give your answer to 3 decimal places.

Answer :

Question 8

Find the Fourier series expansion for the periodic function,

f(t) = t in the interval -π < t < π.

Taking π = 3.142, calculate the Fourier sine series approximation of f(t), up to the 3rd
harmonics when t = 0.14. Give your answer to 3 decimal places.

Answer :

Question 9

Find the Fourier series expansion for the periodic function,

f(t) = t
2
in the interval -π < t < π.

Taking π = 3.142, calculate the Fourier cosine series approximation of f(t), up to the
3rd harmonics when t = 2.92. Give your answer to 3 decimal places.

Answer :

Question 10

A function f(t) is defined as,
f(t) = π – t for 0 < t < π

Write down the even extension of f(t) for -π < t < 0. Determine the Fourier cosine series, and

hence, calculate the Fourier series approximation for f(t) up to the 5th harmonics when t =

0.79. Use π = 3.142. Give your answer to 3 decimal places.

Answer :