Advance Calculus
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tutorial_6.pdf
Please use 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 :
