Testing quiz: FourierSeries

Subject: MAEE309
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Index of questions


1 fourierSeries_1
2 fourierSeries_2
3 fourierSeries_3
4 fourierSeries_4
5 fourierSeries_5
6 fourierSeries_6
7 fourierSeries_7

Question 1 (1 mark)
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Question name: fourierSeries_1 Try separately Edit source file
 
Define the function τ of two arguments, τ(u,c) with c > 0 and u a real number by:
τ(u,c) = sign(u) min
(|u|,c)
(The letter τ was chosen as it is like the first letter of `truncation'.)
Select those of the following functions which are periodic.
A sin(x2)
B sin(sin(x))
C (sin(x))2
D sin(τ(x,1/2))
None of the above
I do not want to answer this yet
Your last answer was {B,C}
Question note: [["rpnlis_", "[\"$\\\\sin(x^2)$\", \"$\\\\sin(\\\\sin(x))$\", \"$(\\\\sin(x))^2$\", \"$\\\\sin(\\\\tau(x,1/2))$\"]"], ["lisTrue_", "[\"$(\\\\sin(x))^2$\", \"$\\\\tau(\\\\sin(x),1/2)$\", \"$\\\\tau(\\\\tan(x),1)$\", \"$\\\\sin(\\\\sin(x))$\", \"$((\\\\sin(x))^2)^{(\\\\sin(x))^2}$\"]"], ["lisTruec_", "[\"$(\\\\sin(x))^2$\", \"$\\\\sin(\\\\sin(x))$\"]"], ["lisFalse_", "[\"$\\\\sin(x^2)$\", \"$\\\\sin(\\\\tau(x,1/2))$\"]"]]
Answer note: {BC}
Your answer is correct.
The mark for your last attempt was 1.00
The correct answer is {B,C}
Question 2 (2 marks)
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Question name: fourierSeries_2 Try separately Edit source file
 
The function below (defined on the whole real line) has Fourier series consisting of just two terms.
Find the Fourier series (in the usual form for 2π-periodic functions) for
f(x) = cos( 2 x+1/6 π)
The Fourier series of f will have a form involving just 2 terms from a sum like
aj cos(j x) + bj sin(j x) + ak cos(k x) + bk sin(k x)
for distinct integers j and k.

2.1 (1 mark)
 

a2=
Your last answer was:
1/2 √3

Question note: []
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
1/2 √3

This can be entered as: 
1/2*3^(1/2)


2.2 (1 mark)
 

b2=
Your last answer was:
−1/2

Question note: []
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
−1/2
Question note: [["x"], ["b"], ["a"], ["fLiss1_", "[cos(x+1/3*Pi), sin(x+1/3*Pi)]"], ["fLis_", "[cos(x)^2, sin(x)^2, cos(x)^3, sin(x)^3, cos(x+1/3*Pi), sin(x+1/3*Pi), -sin(2*x+1/6*Pi), cos(2*x+1/6*Pi)]"], ["rLis_", "[1/4, 1/3, 2/3, 3/4]"], ["solLis_", "[[[a, 0, 1/2], [a, 2, 1/2]], [[a, 0, 1/2], [a, 2, -1/2]], [[a, 1, 3/4], [a, 3, 1/4]], [[b, 1, 3/4], [b, 3, -1/4]], [[a, 1, 1/2], [b, 1, -1/2*3^(1/2)]], [[a, 1, 1/2*3^(1/2)], [b, 1, 1/2]], [[a, 2, -1/2], [b, 2, -1/2*3^(1/2)]], [[a, 2, 1/2*3^(1/2)], [b, 2, -1/2]]]"], ["fLiss2_", "[-sin(2*x+1/6*Pi), cos(2*x+1/6*Pi)]"], ["nf_", "8"], ["f_", "cos(2*x+1/6*Pi)"], ["nr_", "2"], ["r_", "1/3"], ["fLisp_", "[cos(x)^2, sin(x)^2, cos(x)^3, sin(x)^3]"], ["sol_", "[[a, 2, 1/2*3^(1/2)], [b, 2, -1/2]]"], ["solLisp_", "[[[a, 0, 1/2], [a, 2, 1/2]], [[a, 0, 1/2], [a, 2, -1/2]], [[a, 1, 3/4], [a, 3, 1/4]], [[b, 1, 3/4], [b, 3, -1/4]]]"], ["solLiss1_", "[[[a, 1, 1/2], [b, 1, -1/2*3^(1/2)]], [[a, 1, 1/2*3^(1/2)], [b, 1, 1/2]]]"], ["solLiss2_", "[[[a, 2, -1/2], [b, 2, -1/2*3^(1/2)]], [[a, 2, 1/2*3^(1/2)], [b, 2, -1/2]]]"]]
Question 3 (1 mark)
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Question name: fourierSeries_3 Try separately Edit source file
 
Let
f(x) = sin(x)7
Suppose that the Fourier series of (the 2π-periodic) function f is written
s(x)=a0 + ∞
∑
k=1 
 (ak cos(k x)+ bk sin(k x))
What is b1?
b1=
Your last answer was:
35/64

Question note: [["ne_", "3"], ["n_", "7"], ["m_", "1"], ["x_"], ["truesol_", "35/64"]]
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
35/64
Question 4 (4 marks)
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Question name: fourierSeries_4 Try separately Edit source file
 
Let L2 denote the inner-product space of 2π-periodic square-integrable functions with the inner-product
〈f, g〉 = ⌠
⌡ 		π

−π 
 f(x) g(x) dx
Let W1 be the subspace of L2 consisting of functions of the form
W1 = { a0 + a1 cos(x) + b1sin(x) |  a0, a1, b1 real numbers }
What is the dimension of the vector space W1?
4.1 (1 mark)
 

Answer:
Your last answer was:
3

Question note: []
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
3

4.2 (1 mark)
 
Let the function f in L2 be given by
f(x) = | x | ( π− | x | )   for  −π < x ≤ π
and its 2π-periodic extension.
Find the function s1 in W1,
s1(x) = a0 + a1 cos(x) + b1sin(x)
such that the distance between s1 and f is minimized, i.e.
E2 = 〈(f−s1),(f−s1)〉
is minimized.
Give your answers exactly. Enter π as Pi. Remove trig functions from your expressions for the coefficients, using facts like sin(π)=0, etc..
a0=
Your last answer was:
1/6 π2

Question note: []
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
1/6 π2

This can be entered as: 
1/6*Pi^2


4.3 (1 mark)
 

a1=
Your last answer was:
0

Question note: []
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
0

4.4 (1 mark)
 

b1=
Your last answer was:
0

Question note: []
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
0
Question note: [["m"], ["x"], ["f_", "abs(x)*(Pi-abs(x))"], ["nr_", "3"], ["a2m_", "-1/m^2"], ["a2mm1_", "0"], ["b2mm1_", "0"], ["b2m_", "0"], ["a0Lis_", "[1/2*Pi, 1/3*Pi^2, 1/6*Pi^2, 2/Pi]"], ["a2mLis_", "[0, 1/m^2, -1/m^2, -4/Pi/(2*m-1)/(1+2*m)]"], ["b2mLis_", "[0, -1/m, 8*m/Pi/(2*m-1)/(1+2*m), -3/2/m^3, 0]"], ["d_", "\"discontinuous at integer multiples of $\\\\pi$\""], ["de_", "\"discontinuous at even multiples of $\\\\pi$\""], ["do_", "\"discontinuous at odd multiples of $\\\\pi$\""], ["c_", "\"continuous on the whole real line\""], ["na_", "4"], ["no_"], ["a0_", "1/6*Pi^2"], ["a1_", "0"], ["b1_", "0"], ["evnFnLis_", "[abs(x), x^2, abs(x)*(Pi-abs(x)), abs(sin(x))]"], ["oddFnLis_", "[signum(x), x, signum(x)*cos(x), x*(Pi^2-x^2), x*(Pi-abs(x))]"], ["noeFnLis_", "[x, max(0,sin(x))]"], ["a2mm1Lis_", "[-4/Pi/(2*m-1)^2, -4/(2*m-1)^2, 0, 0]"], ["b2mm1Lis_", "[4/Pi/(2*m-1), 2/(2*m-1), 0, 12/(2*m-1)^3, 8/Pi/(2*m-1)^3]"], ["oddcttyLis_", "[\"discontinuous at integer multiples of $\\\\pi$\", \"discontinuous at odd multiples of $\\\\pi$\", \"discontinuous at integer multiples of $\\\\pi$\", \"continuous on the whole real line\", \"continuous on the whole real line\"]"], ["evncttyLis_", "[\"continuous on the whole real line\", \"continuous on the whole real line\", \"continuous on the whole real line\", \"continuous on the whole real line\"]"], ["noecttyLis_", "[\"discontinuous at even multiples of $\\\\pi$\", \"continuous on the whole real line\"]"], ["niceEvnFns_", "[x, x^2, x*(Pi-x), sin(x)]"], ["niceOddFns_", "[1, x, cos(x), x*(Pi^2-x^2), x*(Pi-x)]"], ["nb_", "5"], ["nn_"]]
Question 5 (5 marks)
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Question name: fourierSeries_5 Try separately Edit source file
 
You might find the lab on Fourier Series useful in answering this question. The Matlab lab sheet is here. Find the Fourier series of the 2π-periodic function
f(x) = x        for −π < x ≤ π
In entering your answers, we will denote the coefficients in the series by
f(x) = a0 + ∞
∑
m=1 
 ( a2m−1 cos((2m−1) x) + b2m−1 sin((2m−1) x) + a2m cos(2m x) + b2m sin(2m x) )
In this sum, m goes from 1 to infinity.
The various coefficients for which you are asked are rational functions of m, and should be entered as such. Use the facts that sin(mπ) = 0, cos(2mπ) = 1, cos((2m−1)π) = −1 to remove trig functions from your expressions for the coefficients. Enter π as Pi.
5.1 (1 mark)
 

a0 =
Your last answer was:
0

Question note: []
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
0

5.2 (1 mark)
 
For the remainder of the question, m ≥ 1:
a2m−1 =
Your last answer was:
0

Question note: []
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
0

5.3 (1 mark)
 

a2m =
Your last answer was:
0

Question note: []
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
0

5.4 (1 mark)
 

b2m−1 =
Your last answer was:
2 ( 2 m−1 ) −1

Question note: []
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
2 ( 2 m−1 ) −1

This can be entered as: 
2/(2*m-1)


5.5 (1 mark)
 

b2m =
Your last answer was:
−m−1

Question note: []
Your answer is correct.
The mark for your last attempt was 1.00
The teacher's answer was:
−m−1

This can be entered as: 
-1/m

Solution:
While there are general formulae for Fourier coefficients, it is usual, and saves effort, to check, at the beginning, whether the function is even or odd. Why? ...
  • Even functions necessarily have all their sine series coefficients equal to zero.

  • Odd functions necessarily have all their cosine series coefficients equal to zero.

In this problem, the function f is odd, and hence all an = 0.

To find the sine coefficients, use the formulae:
bn =  2
π
 ⌠
⌡ 		π

0 
 f(x) sin(n x) dx, n ≥ 1.
Matlab can be used to evaluate the integrals. While it is possible to give Matlab the information that m, n or whatever are integers and hence that
sin(n π) = 0,    cos(2 m π) = 1,    cos((2 m − 1)π) = −1,    etc.
it is probably easier in this problem to find a few coefficients and look for a pattern.

If the problem arose in a real engineering situation, one would do both this first step - numeric values of m as examples - and the general m case. In any event, Matlab such as the following should find the integrals: 
syms x f s0 s1 s2 Pi c sm 
s0 = sym(0); s1 = sym(1); s2 = sym(2); Pi = sym(pi); c = s2/Pi; 
f = x 
for m = 1:5 
  sm = sym(m) 
  b2mMinus1 = simplify(c*int(f*sin((s2*sm-s1)*x), x, s0, Pi)) 
  b2m =  simplify(c*int(f*sin(s2*sm*x), x, s0, Pi)) 
end
Question note: [["m"], ["x"], ["n"], ["f_", "x"], ["nr_", "2"], ["s_", "\"syms x f s0 s1 s2 Pi c sm \\ns0 = sym(0); s1 = sym(1); s2 = sym(2); Pi = sym(pi); c = s2/Pi; \\nf = x \\nfor m = 1:5 \\n sm = sym(m) \\n b2mMinus1 = simplify(c*int(f*sin((s2*sm-s1)*x), x, s0, Pi)) \\n b2m = simplify(c*int(f*sin(s2*sm*x), x, s0, Pi)) \\nend\""], ["a2m_", "0"], ["a2mm1_", "0"], ["b2mm1_", "2/(2*m-1)"], ["b2m_", "-1/m"], ["a0Lis_", "[1/2*Pi, 1/3*Pi^2, 1/6*Pi^2, 2/Pi]"], ["a2mLis_", "[0, 1/m^2, -1/m^2, -4/Pi/(2*m-1)/(1+2*m)]"], ["b2mLis_", "[0, -1/m, 8*m/Pi/(2*m-1)/(1+2*m), -3/2/m^3, 0]"], ["d_", "\"discontinuous at integer multiples of $\\\\pi$\""], ["de_", "\"discontinuous at even multiples of $\\\\pi$\""], ["do_", "\"discontinuous at odd multiples of $\\\\pi$\""], ["c_", "\"continuous on the whole real line\""], ["i_", "2"], ["nicef_", "x"], ["na_", "4"], ["a0_", "0"], ["evnFnLis_", "[abs(x), x^2, abs(x)*(Pi-abs(x)), abs(sin(x))]"], ["oddFnLis_", "[signum(x), x, signum(x)*cos(x), x*(Pi^2-x^2), x*(Pi-abs(x))]"], ["noeFnLis_", "[x, max(0,sin(x))]"], ["a2mm1Lis_", "[-4/Pi/(2*m-1)^2, -4/(2*m-1)^2, 0, 0]"], ["b2mm1Lis_", "[4/Pi/(2*m-1), 2/(2*m-1), 0, 12/(2*m-1)^3, 8/Pi/(2*m-1)^3]"], ["oddcttyLis_", "[\"discontinuous at integer multiples of $\\\\pi$\", \"discontinuous at odd multiples of $\\\\pi$\", \"discontinuous at integer multiples of $\\\\pi$\", \"continuous on the whole real line\", \"continuous on the whole real line\"]"], ["evncttyLis_", "[\"continuous on the whole real line\", \"continuous on the whole real line\", \"continuous on the whole real line\", \"continuous on the whole real line\"]"], ["noecttyLis_", "[\"discontinuous at even multiples of $\\\\pi$\", \"continuous on the whole real line\"]"], ["niceEvnFns_", "[x, x^2, x*(Pi-x), sin(x)]"], ["niceOddFns_", "[1, x, cos(x), x*(Pi^2-x^2), x*(Pi-x)]"], ["nb_", "5"], ["nn_"]]
Question 6 (2 marks)
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Question name: fourierSeries_6 Try separately Edit source file
 
Select the properties possessed by the 2π-periodic function f defined, as below, in the interval shown.
f(x) = max
( 0,sin( x ) )    for  0 < x < 2 π .

6.1 (1 mark)
 
A even
B odd
C None of the above
I do not want to answer this yet
Your last answer was C
Question note: []
Answer note: C
Your answer is correct.
The mark for your last attempt was 1.00
The correct answer is C

6.2 (1 mark)
 
The 2π-periodic function f has continuity properties which can be described as one of the following. Select the best choice.
A discontinuous at integer multiples of π
B discontinuous at even multiples of π
C discontinuous at odd multiples of π
D continuous on the whole real line
I do not want to answer this yet
Your last answer was D
Question note: []
Answer note: D
Your answer is correct.
The mark for your last attempt was 1.00
The correct answer is D
Question note: [["m"], ["x"], ["eoPropLis_", "[\"even\", \"odd\", \"None of the above\"]"], ["cttysol_", "\"continuous on the whole real line\""], ["f_", "max(0,sin(x))"], ["nr_", "2"], ["a0Lis_", "[1/2*Pi, 1/3*Pi^2, 1/6*Pi^2, 2/Pi]"], ["a2mLis_", "[0, 1/m^2, -1/m^2, -4/Pi/(2*m-1)/(1+2*m)]"], ["b2mLis_", "[0, -1/m, 8*m/Pi/(2*m-1)/(1+2*m), -3/2/m^3, 0]"], ["d_", "\"discontinuous at integer multiples of $\\\\pi$\""], ["de_", "\"discontinuous at even multiples of $\\\\pi$\""], ["do_", "\"discontinuous at odd multiples of $\\\\pi$\""], ["c_", "\"continuous on the whole real line\""], ["eotype_", "3"], ["eosol_", "\"None of the above\""], ["intvl_", "[0, 2*Pi]"], ["na_", "4"], ["no_", "5"], ["ne_", "4"], ["evnFnLis_", "[abs(x), x^2, abs(x)*(Pi-abs(x)), abs(sin(x))]"], ["oddFnLis_", "[signum(x), x, signum(x)*cos(x), x*(Pi^2-x^2), x*(Pi-abs(x))]"], ["noeFnLis_", "[x, max(0,sin(x))]"], ["a2mm1Lis_", "[-4/Pi/(2*m-1)^2, -4/(2*m-1)^2, 0, 0]"], ["b2mm1Lis_", "[4/Pi/(2*m-1), 2/(2*m-1), 0, 12/(2*m-1)^3, 8/Pi/(2*m-1)^3]"], ["oddcttyLis_", "[\"discontinuous at integer multiples of $\\\\pi$\", \"discontinuous at odd multiples of $\\\\pi$\", \"discontinuous at integer multiples of $\\\\pi$\", \"continuous on the whole real line\", \"continuous on the whole real line\"]"], ["evncttyLis_", "[\"continuous on the whole real line\", \"continuous on the whole real line\", \"continuous on the whole real line\", \"continuous on the whole real line\"]"], ["noecttyLis_", "[\"discontinuous at even multiples of $\\\\pi$\", \"continuous on the whole real line\"]"], ["niceEvnFns_", "[x, x^2, x*(Pi-x), sin(x)]"], ["niceOddFns_", "[1, x, cos(x), x*(Pi^2-x^2), x*(Pi-x)]"], ["nb_", "5"], ["nn_", "2"]]
Question 7 (2 marks)
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Question name: fourierSeries_7 Try separately Edit source file
 
You are given the following two Fourier series of odd 2π-periodic functions
s1(x)
=
∞
∑
m=1 
 ( 12   sin( ( 2 m−1 ) x )
( 2 m−1 ) 3
 −3/2   sin( 2 mx )
m3
 )
s2(x)
=
∞
∑
m=1 
 ( 8   msin( 2 mx )
π ( 2 m−1 ) ( 1+2 m )
) )
The functions next to the choice buttons below are both defined on the interval −π < x ≤ π and periodically extended 2π-periodic. One corresponds to Fourier series s1: the other to Fourier series s2.
Choose the function which corresponds to the Fourier series s1.
7.1 (1 mark)
 
A x ( π2−x2 )
B signum ( x ) cos( x )
I do not want to answer this yet
Your last answer was A
Question note: []
Answer note: A
Your answer is correct.
The mark for your last attempt was 1.00
The correct answer is A

7.2 (1 mark)
 
The function corresponding to summing the Fourier series s1 has continuity properties which can be described as one of the following. Select the best choice.
A discontinuous at integer multiples of π
B discontinuous at even multiples of π
C discontinuous at odd multiples of π
D continuous on the whole real line
I do not want to answer this yet
Your last answer was D
Question note: []
Answer note: D
Your answer is correct.
The mark for your last attempt was 1.00
The correct answer is D
Question note: [["b2m3_", "8*m/Pi/(2*m-1)/(1+2*m)"], ["b2mm13_", "0"], ["b2m1_", "-3/2/m^3"], ["b2mm11_", "12/(2*m-1)^3"], ["nswap_", "0"], ["m"], ["x"], ["f1_", "x*(Pi^2-x^2)"], ["p1_", "\"continuous on the whole real line\""], ["f_"], ["nr_"], ["f3_", "signum(x)*cos(x)"], ["a2m_"], ["a2mm1_"], ["b2mm1_"], ["b2m_"], ["a0Lis_", "[1/2*Pi, 1/3*Pi^2, 1/6*Pi^2, 2/Pi]"], ["a2mLis_", "[0, 1/m^2, -1/m^2, -4/Pi/(2*m-1)/(1+2*m)]"], ["b2mLis_", "[0, -1/m, 8*m/Pi/(2*m-1)/(1+2*m), -3/2/m^3, 0]"], ["d_", "\"discontinuous at integer multiples of $\\\\pi$\""], ["de_", "\"discontinuous at even multiples of $\\\\pi$\""], ["do_", "\"discontinuous at odd multiples of $\\\\pi$\""], ["c_", "\"continuous on the whole real line\""], ["p3_", "\"discontinuous at integer multiples of $\\\\pi$\""], ["nd_", "4"], ["nc_", "3"], ["na_", "4"], ["a0_"], ["nt_", "4"], ["evnFnLis_", "[abs(x), x^2, abs(x)*(Pi-abs(x)), abs(sin(x))]"], ["oddFnLis_", "[signum(x), x, signum(x)*cos(x), x*(Pi^2-x^2), x*(Pi-abs(x))]"], ["noeFnLis_", "[x, max(0,sin(x))]"], ["a2mm1Lis_", "[-4/Pi/(2*m-1)^2, -4/(2*m-1)^2, 0, 0]"], ["b2mm1Lis_", "[4/Pi/(2*m-1), 2/(2*m-1), 0, 12/(2*m-1)^3, 8/Pi/(2*m-1)^3]"], ["oddcttyLis_", "[\"discontinuous at integer multiples of $\\\\pi$\", \"discontinuous at odd multiples of $\\\\pi$\", \"discontinuous at integer multiples of $\\\\pi$\", \"continuous on the whole real line\", \"continuous on the whole real line\"]"], ["evncttyLis_", "[\"continuous on the whole real line\", \"continuous on the whole real line\", \"continuous on the whole real line\", \"continuous on the whole real line\"]"], ["noecttyLis_", "[\"discontinuous at even multiples of $\\\\pi$\", \"continuous on the whole real line\"]"], ["niceEvnFns_", "[x, x^2, x*(Pi-x), sin(x)]"], ["niceOddFns_", "[1, x, cos(x), x*(Pi^2-x^2), x*(Pi-x)]"], ["nb_", "5"], ["nn_"], ["trueprop_", "\"continuous on the whole real line\""], ["fchoicesc_", "[\"$x \\\\left( {\\\\pi }^{2}-{x}^{2} \\\\right) $\", \"${\\\\it signum} \\\\left( x \\\\right) \\\\cos \\\\left( x \\\\right) $\"]"], ["truesol_", "\"$x \\\\left( {\\\\pi }^{2}-{x}^{2} \\\\right) $\""], ["fchoices_", "[\"$x \\\\left( {\\\\pi }^{2}-{x}^{2} \\\\right) $\", \"${\\\\it signum} \\\\left( x \\\\right) \\\\cos \\\\left( x \\\\right) $\"]"]]

Mark summary:
Question Value Your mark
1 1.00 1.00
2.1 1.00 1.00
2.2 1.00 1.00
3 1.00 1.00
4.1 1.00 1.00
4.2 1.00 1.00
4.3 1.00 1.00
4.4 1.00 1.00
5.1 1.00 1.00
5.2 1.00 1.00
5.3 1.00 1.00
5.4 1.00 1.00
5.5 1.00 1.00
6.1 1.00 1.00
6.2 1.00 1.00
7.1 1.00 1.00
7.2 1.00 1.00
Total 17.00 17.00

Your mark of 17.00 out of 17.00 is equivalent to 100%.
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