Question

For the periodic function y(t) with period 12, y(t) = (0 if −6 < t < −3, 4 if −3 < t < 3, 0 if 3 < t < 6)

(a) determine the real Fourier series of y(t): y(t) = a0 + ∞ Sum n=1 (an cos(2πfnt) + bn sin (2πfnt))

The ns are subscripts

Answer #1

The sketch of the following periodic function f(t)
given in one period,
f(t) = {(3t+1), -1 < t <= 1 and
0, -3 < t <= -1
a) Find period of the function, 2p?
b) Find Fourier coeff, a0, an (n
=>1), bn?
c) Fourier series representation of f(t)?
d) Result from (c), find the
first four non-zero term?

In the interval −π < t <
0, f(t) = 1; and for 0 < t
< π, f(t) = 0. f(t) = f(t+2 π)
Find the following for f(t) as associated with the Fourier
series:
a0 =?
an =?
bn =?
ωo =?

Determine the amplitude, the period and the phase shift of the
function
1. y = 1/3 cos ( -3 x ) + 1
2. y = cos ( - 2 PIE x ) + 2
3. y = ½ sin (2 PIE x + PIE )
4. y = - ¼ cos (PIE x - 4 )
5. y = 2 sin (2 PIE x + 1 )
6. y = ½ sin ( 2 x - PIE/4 )

Find the length of the curve
1) x=2sin t+2t, y=2cos t, 0≤t≤pi
2) x=6 cos t, y=6 sin t, 0≤t≤pi
3) x=7sin t- 7t cos t, y=7cos t+ 7 t sin t, 0≤t≤pi/4

Find the Fourier series of the periodic function given on one
period of length 2 by
f(x) = x2, - 1 < x < 1:

f(t)is an odd, periodic function with period 1 and
f(t)=3−6t
for 0≤t≤0.5
(I)
Find the Fourier coefficient an.
Put in this value only ie. Omit the "an = "
This question accepts numbers or formulas.
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(II)
Find the Fourier coefficient b5.
Put in this value only ie. Omit the "b5 = "
Give your answer to AT LEAST FOUR PLACES OF DECIMALS.
This question accepts numbers or formulas.
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Find the fourier series representation of each periodic
function
f(x) = 0, -4 <x<0
f(x) = 8, 0<=x<=1
f(x) = 0, 1<x<4

If y(t) is an even function, and y(t − 1) is also even, is y(t)
periodic? If so, what is the fundamental period T0? Ignore the case
where y(t) is constant, and assume that T = 1 is the smallest shift
for which y(t − T ) is even.

Solve the following initial/boundary value problem:
∂u(t,x)/∂t = ∂^2u(t,x)/∂x^2 for t>0, 0<x<π,
u(t,0)=u(t,π)=0 for t>0,
u(0,x)=sin^2x for 0≤x≤ π.
if you like, you can use/cite the solution of Fourier sine
series of sin^2(x) on [0,pi] = 1/4-(1/4)cos(2x)
please show all steps and work clearly so I can follow your
logic and learn to solve similar ones myself.

Consider the first full period of the sine function:
sin(x), 0 < x < 2π.
(1) Plot the original function and your
four-term approximation using a computer for the range −2π < x
< 0. Comment.
(2) Expand sin(x), 0 < x < 2π, in a
Fourier sine series.

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