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  # 11.3E: Fourier Series II (Exercises)
  
  $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
  
  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 $$[0,2L]$$.
explanation: The provided text is a set of exercises and problems related to Fourier series, specifically focusing on Fourier cosine series, Fourier sine series, and mixed Fourier series. The document contains multiple questions, each asking the reader to find a specific type of Fourier series for a given function over a specified interval. There are no answers provided for these questions.
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distilabel_metadata.role: system
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  # 11.3E: Fourier Series II (Exercises)
  
  $$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$
  
  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 $$[0,2L]$$.
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The provided text is a set of exercises and problems related to Fourier series, specifically focusing on Fourier cosine series, Fourier sine series, and mixed Fourier series. The document contains multiple questions, each asking the reader to find a specific type of Fourier series for a given function over a specified interval. There are no answers provided for these questions.
  
  Label: multiple-question