$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

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 $$[0,2L]$$.