Question

Given that: f(x) = { cos x , 0 < x < 2π

(a) Sketch extended periodic f(x) from -4π <x < 4π, and justify whether f (x) is even function, odd function or neither.

(b) Produce f (x) in a Fourier series.

Answer #1

6

f(x) = 2x - 7
x ∈ (0,7)
Draw a plot of the periodic Fourier Series expansion of f(x).
What is its value at x=0 and why? Is it odd or even?
Expand the given function in a Fourier Series also

1. Find the Fourier cosine series for f(x) = x on the interval 0
≤ x ≤ π in terms of cos(kx). Hint: Use the even extension.
2. Find the Fourier sine series for f(x) = x on the interval 0 ≤
x ≤ 1 in terms of sin(kπx). Hint: Use the odd extension.

Given the function f(x) =cosh(x) with period of 2π , determine
its Fourier series for interval of (-π, π) ( Please write clearly
:) )

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?

Consider the function on the interval (0, 2π). f(x) = sin(x)
cos(x) + 4. (A) Find the open interval(s) on which the function is
increasing or decreasing. (Enter your answers using interval
notation.) (B) Apply the First Derivative Test to identify all
relative extrema.

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.

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

a. Let f be an odd function. Find the Fourier series of f on
[-1, 1]
b. Let f be an even function. Find the Fourier series of f on
[-1, 1].
c. At what condition for f would make the series converge to f
at x=0 and x=1?

Consider the function on the interval (0, 2π).
f(x) =
sin(x)/
2 + (cos(x))2
(a) Find the open intervals on which the function is increasing
or decreasing. (Enter your answers using interval notation.)
increasing
decreasing
(b) Apply the First Derivative Test to identify the relative
extrema.
relative maximum
(x, y) =
relative minimum
(x, y) =

Fourier Series Expand each function into its cosine series and
sine series for the given period P = 2π f(x) = cos x

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