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11.3E: Fourier Series II (Exercises)

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    In Exercises 11.3.2, 11.3.3, 11.3.5, 11.3.9-11.3.12, 11.3.14-11.3.16, 11.3.18, 11.3.20, 11.3.21, 11.3.24, 11.3.25, 11.3.27, 11.3.30, 11.3.36, 11.3.37, and 11.3.43 graph \(f\) and some partial sums of the required series. If the interval is \([0,L]\), choose a specific value of \(L\) for the graph.

    Q11.3.1

    In Exercises 11.3.1-11.3.10 find the Fourier cosine series.

    1. \(f(x)=x^2\); \([0,L]\)

    2. \(f(x)=1-x\); \([0,1]\)

    3. \(f(x)=x^2-2Lx\); \([0,L]\)

    4. \(f(x)=\sin kx\)  (\(k\ne\) integer);\([0,\pi]\)

    5. \(f(x)= \left\{\begin{array}{cl} 1,&0\le x\le{L\over2}\\0,&{L\over2}<x<L; \end{array}\right.\) \([0,L]\)

    6. \(f(x)=x^2-L^2\); \([0,L]\)

    7. \(f(x)=(x-1)^2\); \([0,1]\)

    8. \(f(x)=e^x\); \([0,\pi]\)

    9. \(f(x)=x(L-x)\); \([0,L]\)

    10. \(f(x)=x(x-2L)\); \([0,L]\)

    Q11.3.2

    In Exercises 11.3.11-11.3.17 find the Fourier sine series

    11. \(f(x)=1\); \([0,L]\)

    12. \(f(x)=1-x\); \([0,1]\)

    13. \(f(x)=\cos kx\)  (\(k\ne\) integer); \([0,\pi]\)

    14. \(f(x)= \left\{\begin{array}{cl} 1,&0\le x\le{L\over2}\\0,&{L\over2}<x<L; \end{array}\right.\) \([0,L]\)

    15. \(f(x)= \left\{\begin{array}{cl} x,&0\le x\le{L\over2},\\L-x,&{L\over2}\le x\le L; \end{array}\right.\) \([0,L]\).

    16. \(f(x)=x\sin x\); \([0,\pi]\)

    17. \(f(x)=e^x\); \([0,\pi]\)

    Q11.3.3

    In Exercises 11.3.18-11.3.24 find the mixed Fourier cosine series.

    18. \(f(x)=1\); \([0,L]\)

    19. \(f(x)=x^2\); \([0,L]\)

    20. \(f(x)=x\); \([0,1]\)

    21. \(f(x)= \left\{\begin{array}{cl} 1,&0\le x\le{L\over2}\\0,&{L\over2}<x<L; \end{array}\right.\) \([0,L]\)

    22. \(f(x)=\cos x\); \([0,\pi]\)

    23. \(f(x)=\sin x\); \([0,\pi]\)

    24. \(f(x)=x(L-x)\); \([0,L]\)

    Q11.3.4

    In Exercises 11.3.25-11.3.30 find the mixed Fourier sine series.

    25. \(f(x)=1\); \([0,L]\)

    26. \(f(x)=x^2\); \([0,L]\)

    27. \(f(x)= \left\{\begin{array}{cl} 1,&0\le x\le{L\over2}\\0,&{L\over2}<x<L; \end{array}\right.\)\([0,L]\)

    28. \(f(x)=\cos x\); \([0,\pi]\)

    29. \(f(x)=\sin x\); \([0,\pi]\)

    30. \(f(x)=x(L-x)\); \([0,L]\).

    Q11.3.5

    In Exercises 11.3.31-11.3.34 use Theorem 11.3.5a to find the Fourier cosine series of \(f\) on \([0,L]\).

    31. \(f(x)=3x^2(x^2-2L^2)\)

    32. \(f(x)=x^3(3x-4L)\)

    33. \(f(x)=x^2(3x^2-8Lx+6L^2)\)

    34. \(f(x)=x^2(x-L)^2\)

    Q11.3.6

    35.

    1. Prove Theorem 11.3.5b.
    2. In addition to the assumptions of Theorem 11.3.5b, suppose \(f''(0)=f''(L)=0\), \(f'''\) is continuous, and \(f^{(4)}\) is piecewise continuous on \([0,L]\). Show that \[b_n={2L^3\over n^4\pi^4}\int_0^L f^{(4)}(x)\sin{n\pi x\over L}\,dx, \quad n\ge1.\nonumber\]

    Q11.3.7

    In Exercises 11.3.36-11.3.41 use Theorem 11.3.5b or, where applicable, Exercises 11.1.35b to find the Fourier sine series of \(f\) on \([0,L]\).

    36. \(f(x)=x(L-x)\)

    37. \(f(x)=x^2(L-x)\)

    38. \(f(x)=x(L^2-x^2)\)

    39. \(f(x)=x(x^3-2Lx^2+L^3)\)

    40. \(f(x)=x(3x^4-10L^2x^2+7L^4)\)

    41. \(f(x)=x(3x^4-5Lx^3+2L^4)\)

    Q11.3.8

    42.

    1. Prove Theorem 11.3.5c.
    2. In addition to the assumptions of Theorem 11.3.5c, suppose \(f''(L)=0\), \(f''\) is continuous, and \(f'''\) is piecewise continuous on \([0,L]\). Show that \[c_n={16L^2\over(2n-1)^3\pi^3}\int_0^L f'''(x)\sin{(2n-1)\pi x\over2L} \,dx,\quad n\ge1.\nonumber \]

    Q11.3.9

    In Exercises 11.3.43-11.3.49 use Theorem 11.3.5c, or where applicable, Exercise 11.1.42b, to find the mixed Fourier cosine series of \(f\) on \([0,L]\).

    43. \(f(x)=x^2(L-x)\)

    44. \(f(x)=L^2-x^2\)

    45. \(f(x)=L^3-x^3\)

    46. \(f(x)=2x^3+3Lx^2-5L^3\)

    47. \(f(x)=4x^3+3Lx^2-7L^3\)

    48. \(f(x)=x^4-2Lx^3+L^4\)

    49. \(f(x)=x^4-4Lx^3+6L^2x^2-3L^4\)

    Q11.3.10

    50.

    1. Prove Theorem 11.3.5d.
    2. In addition to the assumptions of Theorem 11.3.5d, suppose \(f''(0)=0\), \(f''\) is continuous, and \(f'''\) is piecewise continuous on \([0,L]\). Show that \[d_n=-{16L^2\over(2n-1)^3\pi^3}\int_0^L f'''(x)\cos{(2n-1)\pi x\over2L} \,dx,\quad n\ge1. \nonumber\]

    Q11.3.11

    In Exercises 11.3.51-11.3.56 use Theorem 11.3.5d or, where applicable, Exercise 11.3.50b, to find the mixed Fourier sine series of the \(f\) on \([0,L]\).

    51. \(f(x)=x(2L -x)\)

    52. \(f(x)=x^2(3L-2x)\)

    53. \(f(x)=(x-L)^3+L^3\)

    54. \(f(x)=x(x^2-3L^2)\)

    55. \(f(x)=x^3(3x-4L)\)

    56. \(f(x)=x(x^3-2Lx^2+2L^3)\)

    Q11.3.12

    57. Show that the mixed Fourier cosine series of \(f\) on \([0,L]\) is the restriction to \([0,L]\) of the Fourier cosine series of

    \[f_3(x)= \left\{\begin{array}{cl} f(x),&0\le x\le L,\\-f(2L-x),&L< x\le 2L \end{array}\right.\nonumber\]

    on \([0,2L]\). Use this to prove Theorem 11.3.3.

    58. Show that the mixed Fourier sine series of \(f\) on \([0,L]\) is the restriction to \([0,L]\) of the Fourier sine series of

    \[f_4(x)= \left\{\begin{array}{cl} f(x),&0\le x\le L,\\f(2L-x),&L< x\le 2L \end{array}\right.\nonumber\]

    on \([0,2L]\). Use this to prove Theorem 11.3.4.

    59. Show that the Fourier sine series of \(f\) on \([0,L]\) is the restriction to \([0,L]\) of the Fourier sine series of

    \[f_3(x)= \left\{\begin{array}{cl} f(x),&0\le x\le L,\\-f(2L-x),&L< x\le 2L \end{array}\right.\nonumber\]

    on \([0,2L]\).

    60. Show that the Fourier cosine series of \(f\) on \([0,L]\) is the restriction to \([0,L]\) of the Fourier cosine series of

    \[f_4(x)= \left\{\begin{array}{cl} f(x),&0\le x\le L,\\f(2L-x),&L< x\le 2L \end{array}\right.\nonumber\]

    on \([0,2L]\).

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