Question

помогите решить задание

Answer 1

$f(x)=\left\{\begin{array}{l}-2\ ,\ -1\leq x\leq 0\ ,\\{}\ \ \, 3\ ,\ \ \ 0<x\leq 2\ .\end{array}\right\qquad \qquad T=4\ \ ,\ \ 2l=4\ \ \to \ \ l=2\\\\\\f(x)\sim \dfrac{a_0}{2}+\sum \limits _{n=1}^{\infty }\Big(a_{n}\cdot cos\dfrac{\pi nx}{l}+b_{n}\cdot sin\dfrac{\pi nx}{l}}\Big)$

$a_0=\dfrac{1}{2}\int \limits _{-2}^{2}\, f(x)\, dx=\dfrac{1}{2}\int\limits^0_{-2}(-2)\, dx+\dfrac{1}{2}\int\limits^2_0\, 3\, dx=-x\Big|_{-2}^0+\dfrac{3}{2}\cdpt \, x\Big|_0^2=\\\\\\=-(0+2)+\dfrac{3}{2}\cdot (2-0)=-2+3=1$

$a_{n}=\dfrac{1}{2}\int \limits _{-2}^{2}\, f(x)\cdot cos\dfrac{\pi nx}{2}\, dx=\dfrac{1}{2}\int\limits^0_{-2}(-2)\cdot cos\dfrac{\pi nx}{2}\, dx+\dfrac{1}{2}\int\limits^2_0\, 3\cdot cos\dfrac{\pi nx}{2}\, dx=$

$=-\dfrac{2}{n}\cdot sin\dfrac{\pi nx}{2}\Big|_{-2}^0+\dfrac{3}{2}\cdot \dfrac{2}{n}\cdot sin\dfrac{\pi nx}{2}\Big|_0^2=\\\\\\=-\dfrac{2}{n}\cdot (sin0-sin(-\pi n))+\dfrac{3}{n}\cdot (sin\pi n-sin0)=0\ ;\ \ [\ sin0=0\ ,\ \ sin\pi n=0\ ]$

$b_{n}=\dfrac{1}{2}\int \limits _{-2}^{2}\, f(x)\cdot sin\dfrac{\pi nx}{2}\, dx=\dfrac{1}{2}\int\limits^0_{-2}(-2)\cdot sin\dfrac{\pi nx}{2}\, dx+\dfrac{1}{2}\int\limits^2_0\, 3\cdot sin\dfrac{\pi nx}{2}\, dx=\\\\\\=\dfrac{2}{n}\cdot cos\dfrac{\pi nx}{2}\, \Big|_{-2}^0-\dfrac{3}{2}\cdot \dfrac{2}{n}\cdot cos\dfrac{\pi nx}{2}\, \Big|_0^2=\\\\\\=\dfrac{2}{n}\cdot (cos\, 0-cos(-\pi n)\, )-\dfrac{3}{n}\cdot (cos\pi n-cos\, 0\, )=\dfrac{2}{n}\cdot (1-cos\pi n)-\dfrac{3}{n}\cdot (cos\pi n-1)=$

$=\dfrac{2}{n}\cdot (cos\, 0-cos\pi n)+\dfrac{3}{n}\cdot (cos\, 0-cos\pi n)=(cos\, 0-cos\pi n)\cdot \Big(\dfrac{2}{n}+\dfrac{3}{n}\Big)=\\\\\\=\dfrac{5}{n}\cdot (cos0-cos\pi n)=\Big[\ cos\, 0=1\ ,\ cos\pi n=(-1)^{n}\ \Big]=\dfrac{5}{n}\cdot \Big(\, 1-(-1)^{n}\, \Big)$

$f(x)\sim \dfrac{1}{2}+\sum \limits _{n=1}^{\infty }\, \dfrac{5}{n}\cdot \Big(\, 1-(-1)^{n}\, \Big)\cdot sin\dfrac{\pi nx}{2}=\\\\\\=\dfrac{1}{2}+\dfrac{10}{1}\, sin\dfrac{\pi x}{2}+\dfrac{10}{3}\, sin\dfrac{3\pi x}{2}+\dfrac{10}{5}\, sin\dfrac{5\pi x}{2}+\dfrac{10}{7}\, sin\dfrac{7\pi x}{2}+\ .\, .\, .$