Calculus Homework
mirananwar
Hw21.pdf
Miran Anwar, mth 235 ss21 1: Hw21-6.2-BVP-FS. Due: 04/26/2021 at 11:00pm EDT.
See in LN, § 6.2, See Examples 6.2.3-6.2.7. 1. (10 points)
Consider the function f defined on the interval [−5,5] as follows,
f (x) =
{ −5, x ∈ [−5,0),
5, x ∈ [0,5].
Denote by fF the Fourier series expansion of f on [−5,5],
fF (x) = a0 2 +
∞
∑ n=1
[ an cos
(nπx L
) + bn sin
(nπx L
)] .
Find the coefficients a0, an, and bn, with n > 1.
a0 =
an =
bn =
Comments on the graph below: (Optional, worth no points)
• The first time running it may take a few minutes to load. • The graph is interactive, it allows you to see your solutions. • You type the answers for a0, an = a(n), and bn = b(n) inside the graph. • Use the key Tab to enter the values inside the graph; do not use Enter; if you do, the graph resets
itself. • You can move the slider for the Fourier approximation N. • Type math as in Webwork. Use * for multiplication. • You can use parenthesis in the graph, ( ), but not brackets, [ ]. • Recall, the graph part of the problem is worth no points.
1
See in LN, § 6.2, See Examples 6.2.3-6.2.7. 2. (10 points)
Consider the function f defined on the interval [−5,5] as follows,
f (x) =
{ −1, x ∈ [−5,0),
2, x ∈ [0,5].
Denote by fF the Fourier series expansion of f on [−5,5], 1
fF (x) = a0 2 +
∞
∑ n=1
[ an cos
(nπx L
) + bn sin
(nπx L
)] .
Find the coefficients a0, an, and bn, with n > 1.
a0 =
an =
bn =
See in LN, § 6.2, See Examples 6.2.3-6.2.7. 3. (10 points)
Consider the function f defined on the interval [−3,3] as follows,
f (x) =
−
4 3
x, x ∈ [−3,0),
4 3
x, x ∈ [0,3].
Denote by fF the Fourier series expansion of f on [−3,3],
fF (x) = a0 2 +
∞
∑ n=1
[ an cos
(nπx L
) + bn sin
(nπx L
)] .
Find the coefficients a0, an, and bn, with n > 1.
a0 =
an =
bn =
Comments on the graph below: (Optional, worth no points)
• The first time running it may take a few minutes to load. • The graph is interactive, it allows you to see your solutions. • You type the answers for a0, an = a(n), and bn = b(n) inside the graph. • Use the key Tab to enter the values inside the graph; do not use Enter; if you do, the graph resets
itself. • You can move the slider for the Fourier approximation N. • Type math as in Webwork. Use * for multiplication. • You can use parenthesis in the graph, ( ), but not brackets, [ ]. • Recall, the graph part of the problem is worth no points.
1
2
See in LN, § 6.2, See Examples 6.2.3-6.2.7. 4. (10 points)
Consider the function f defined on the interval [−3,3] as follows,
f (x) = 2 3
x.
Denote by fF the Fourier series expansion of f on [−3,3],
fF (x) = a0 2 +
∞
∑ n=1
[ an cos
(nπx L
) + bn sin
(nπx L
)] .
Find the coefficients a0, an, and bn, with n > 1.
a0 =
an =
bn =
See in LN, § 6.2, See Examples 6.2.3-6.2.7. 5. (10 points)
Consider the function f defined on the interval [−3,3] as follows,
f (x) =
0, x ∈ [−3,0), 1 3
x, x ∈ [0,3].
Denote by fF the Fourier series expansion of f on [−3,3],
fF (x) = a0 2 +
∞
∑ n=1
[ an cos
(nπx L
) + bn sin
(nπx L
)] .
Find the coefficients a0, an, and bn, with n > 1.
a0 =
an =
bn =
Comments on the graph below: (Optional, worth no points) 3
• The first time running it may take a few minutes to load. • The graph is interactive, it allows you to see your solutions. • You type the answers for a0, an = a(n), and bn = b(n) inside the graph. • Use the key Tab to enter the values inside the graph; do not use Enter; if you do, the graph resets
itself. • You can move the slider for the Fourier approximation N. • Type math as in Webwork. Use * for multiplication. • You can use parenthesis in the graph, ( ), but not brackets, [ ]. • Recall, the graph part of the problem is worth no points.
1
See in LN, § 6.2, See Examples 6.2.3-6.2.7. 6. (10 points)
Consider the function f defined on the interval [−5,5] as follows,
f (x) =
0, x ∈ [−5,0),
3 (
1− x 5
) , x ∈ [0,5].
Denote by fF the Fourier series expansion of f on [−5,5],
fF (x) = a0 2 +
∞
∑ n=1
[ an cos
(nπx L
) + bn sin
(nπx L
)] .
Find the coefficients a0, an, and bn, with n > 1.
a0 =
an =
bn =
See in LN, § 6.2, See Examples 6.2.3-6.2.7. 7. (10 points)
Given the function f (x) = 5x + 2 defined on the interval (0,2], denote by fo the odd extension on [−2,2] of f .
Find foF , the Fourier series expansion of fo,
foF (x) = a0 2 +
∞
∑ n=1
[ an cos
(nπx L
) + bn sin
(nπx L
)] ,
that is, find the coefficients a0, an, and bn, with n > 1. 4
a0 =
an =
bn =
See in LN, § 6.2, See Examples 6.2.3-6.2.7. 8. (10 points)
Given the function f (x) = 2x + 1 defined on the interval (0,5], denote by fe the even extension on [−5,5] of f .
Find feF , the Fourier series expansion of fe,
feF (x) = a0 2 +
∞
∑ n=1
[ an cos
(nπx L
) + bn sin
(nπx L
)] ,
that is, find the coefficients a0, an, and bn, with n > 1.
a0 =
an =
bn =
See in LN, § 6.2, See Examples 6.2.3-6.2.7. 9. (0 points)
Consider the function f defined on the interval [−3,3] as follows,
f (x) =
4 (
1 + x 3
) , x ∈ [−3,0),
4 (
1− x 3
) , x ∈ [0,3].
Denote by fF the Fourier series expansion of f on [−3,3],
fF (x) = a0 2 +
∞
∑ n=1
[ an cos
(nπx L
) + bn sin
(nπx L
)] .
Find the coefficients a0, an, and bn, with n > 1.
a0 =
an = 5
bn =
See in LN, § 6.2, See Examples 6.2.3-6.2.7. 10. (0 points)
Consider the function f defined on the interval [−5,5] as follows,
f (x) =
−1 (
1 + x 5
) , x ∈ [−5,0),
1 (
1− x 5
) , x ∈ [0,5].
Denote by fF the Fourier series expansion of f on [−5,5],
fF (x) = a0 2 +
∞
∑ n=1
[ an cos
(nπx L
) + bn sin
(nπx L
)] .
Find the coefficients a0, an, and bn, with n > 1.
a0 =
an =
bn =
Generated by c©WeBWorK, http://webwork.maa.org, Mathematical Association of America
6
