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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 $$[0,2L]$$.