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Gráfico de la función y = -20*(-sin(pi*x/2)/5+sin(2*pi*x/5)/4)/pi-20*(-sin(pi*x/2)/5+sin(3*pi*x/5)/6)/pi-20*(sin(pi*x/5)/2-sin(pi*x/10)-sin(3*pi*x/10)/3)/pi

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src]
           /    /pi*x\       /2*pi*x\\      /    /pi*x\       /3*pi*x\\      /   /pi*x\                  /3*pi*x\\
           |-sin|----|    sin|------||      |-sin|----|    sin|------||      |sin|----|               sin|------||
           |    \ 2  /       \  5   /|      |    \ 2  /       \  5   /|      |   \ 5  /      /pi*x\      \  10  /|
       -20*|----------- + -----------|   20*|----------- + -----------|   20*|--------- - sin|----| - -----------|
           \     5             4     /      \     5             6     /      \    2          \ 10 /        3     /
f(x) = ------------------------------- - ------------------------------ - ----------------------------------------
                      pi                               pi                                    pi                   
$$f{\left(x \right)} = \left(\frac{\left(-1\right) 20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{2 \pi x}{5} \right)}}{4}\right)}{\pi} - \frac{20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{3 \pi x}{5} \right)}}{6}\right)}{\pi}\right) - \frac{20 \left(\left(- \sin{\left(\frac{\pi x}{10} \right)} + \frac{\sin{\left(\frac{\pi x}{5} \right)}}{2}\right) - \frac{\sin{\left(\frac{3 \pi x}{10} \right)}}{3}\right)}{\pi}$$
f = (-20*((-sin((pi*x)/2))/5 + sin(((2*pi)*x)/5)/4))/pi - 20*((-sin((pi*x)/2))/5 + sin(((3*pi)*x)/5)/6)/pi - 20*(-sin((pi*x)/10) + sin((pi*x)/5)/2 - sin(((3*pi)*x)/10)/3)/pi
Gráfico de la función
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en (-20*((-sin((pi*x)/2))/5 + sin(((2*pi)*x)/5)/4))/pi - 20*((-sin((pi*x)/2))/5 + sin(((3*pi)*x)/5)/6)/pi - 20*(sin((pi*x)/5)/2 - sin((pi*x)/10) - sin(((3*pi)*x)/10)/3)/pi.
$$\left(\frac{\left(-1\right) 20 \left(\frac{\left(-1\right) \sin{\left(\frac{0 \pi}{2} \right)}}{5} + \frac{\sin{\left(\frac{0 \cdot 2 \pi}{5} \right)}}{4}\right)}{\pi} - \frac{20 \left(\frac{\left(-1\right) \sin{\left(\frac{0 \pi}{2} \right)}}{5} + \frac{\sin{\left(\frac{0 \cdot 3 \pi}{5} \right)}}{6}\right)}{\pi}\right) - \frac{20 \left(\left(\frac{\sin{\left(\frac{0 \pi}{5} \right)}}{2} - \sin{\left(\frac{0 \pi}{10} \right)}\right) - \frac{\sin{\left(\frac{0 \cdot 3 \pi}{10} \right)}}{3}\right)}{\pi}$$
Resultado:
$$f{\left(0 \right)} = 0$$
Punto:
(0, 0)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\left(\frac{\left(-1\right) 20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{2 \pi x}{5} \right)}}{4}\right)}{\pi} - \frac{20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{3 \pi x}{5} \right)}}{6}\right)}{\pi}\right) - \frac{20 \left(\left(- \sin{\left(\frac{\pi x}{10} \right)} + \frac{\sin{\left(\frac{\pi x}{5} \right)}}{2}\right) - \frac{\sin{\left(\frac{3 \pi x}{10} \right)}}{3}\right)}{\pi}\right) = \frac{\left\langle - \frac{110}{3}, \frac{110}{3}\right\rangle}{\pi} + \frac{\left\langle - \frac{22}{3}, \frac{22}{3}\right\rangle}{\pi} + \frac{\left\langle -9, 9\right\rangle}{\pi}$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \frac{\left\langle - \frac{110}{3}, \frac{110}{3}\right\rangle}{\pi} + \frac{\left\langle - \frac{22}{3}, \frac{22}{3}\right\rangle}{\pi} + \frac{\left\langle -9, 9\right\rangle}{\pi}$$
$$\lim_{x \to \infty}\left(\left(\frac{\left(-1\right) 20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{2 \pi x}{5} \right)}}{4}\right)}{\pi} - \frac{20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{3 \pi x}{5} \right)}}{6}\right)}{\pi}\right) - \frac{20 \left(\left(- \sin{\left(\frac{\pi x}{10} \right)} + \frac{\sin{\left(\frac{\pi x}{5} \right)}}{2}\right) - \frac{\sin{\left(\frac{3 \pi x}{10} \right)}}{3}\right)}{\pi}\right) = \frac{\left\langle - \frac{110}{3}, \frac{110}{3}\right\rangle}{\pi} + \frac{\left\langle - \frac{22}{3}, \frac{22}{3}\right\rangle}{\pi} + \frac{\left\langle -9, 9\right\rangle}{\pi}$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \frac{\left\langle - \frac{110}{3}, \frac{110}{3}\right\rangle}{\pi} + \frac{\left\langle - \frac{22}{3}, \frac{22}{3}\right\rangle}{\pi} + \frac{\left\langle -9, 9\right\rangle}{\pi}$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-20*((-sin((pi*x)/2))/5 + sin(((2*pi)*x)/5)/4))/pi - 20*((-sin((pi*x)/2))/5 + sin(((3*pi)*x)/5)/6)/pi - 20*(sin((pi*x)/5)/2 - sin((pi*x)/10) - sin(((3*pi)*x)/10)/3)/pi, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(\frac{\left(-1\right) 20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{2 \pi x}{5} \right)}}{4}\right)}{\pi} - \frac{20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{3 \pi x}{5} \right)}}{6}\right)}{\pi}\right) - \frac{20 \left(\left(- \sin{\left(\frac{\pi x}{10} \right)} + \frac{\sin{\left(\frac{\pi x}{5} \right)}}{2}\right) - \frac{\sin{\left(\frac{3 \pi x}{10} \right)}}{3}\right)}{\pi}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(\frac{\left(-1\right) 20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{2 \pi x}{5} \right)}}{4}\right)}{\pi} - \frac{20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{3 \pi x}{5} \right)}}{6}\right)}{\pi}\right) - \frac{20 \left(\left(- \sin{\left(\frac{\pi x}{10} \right)} + \frac{\sin{\left(\frac{\pi x}{5} \right)}}{2}\right) - \frac{\sin{\left(\frac{3 \pi x}{10} \right)}}{3}\right)}{\pi}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Paridad e imparidad de la función
Comprobemos si la función es par o impar mediante las relaciones f = f(-x) и f = -f(-x).
Pues, comprobamos:
$$\left(\frac{\left(-1\right) 20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{2 \pi x}{5} \right)}}{4}\right)}{\pi} - \frac{20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{3 \pi x}{5} \right)}}{6}\right)}{\pi}\right) - \frac{20 \left(\left(- \sin{\left(\frac{\pi x}{10} \right)} + \frac{\sin{\left(\frac{\pi x}{5} \right)}}{2}\right) - \frac{\sin{\left(\frac{3 \pi x}{10} \right)}}{3}\right)}{\pi} = \frac{5 \sin{\left(\frac{2 \pi x}{5} \right)} - 4 \sin{\left(\frac{\pi x}{2} \right)}}{\pi} - \frac{4 \sin{\left(\frac{\pi x}{2} \right)} - \frac{10 \sin{\left(\frac{3 \pi x}{5} \right)}}{3}}{\pi} - \frac{20 \sin{\left(\frac{\pi x}{10} \right)} - 10 \sin{\left(\frac{\pi x}{5} \right)} + \frac{20 \sin{\left(\frac{3 \pi x}{10} \right)}}{3}}{\pi}$$
- No
$$\left(\frac{\left(-1\right) 20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{2 \pi x}{5} \right)}}{4}\right)}{\pi} - \frac{20 \left(\frac{\left(-1\right) \sin{\left(\frac{\pi x}{2} \right)}}{5} + \frac{\sin{\left(\frac{3 \pi x}{5} \right)}}{6}\right)}{\pi}\right) - \frac{20 \left(\left(- \sin{\left(\frac{\pi x}{10} \right)} + \frac{\sin{\left(\frac{\pi x}{5} \right)}}{2}\right) - \frac{\sin{\left(\frac{3 \pi x}{10} \right)}}{3}\right)}{\pi} = - \frac{5 \sin{\left(\frac{2 \pi x}{5} \right)} - 4 \sin{\left(\frac{\pi x}{2} \right)}}{\pi} + \frac{4 \sin{\left(\frac{\pi x}{2} \right)} - \frac{10 \sin{\left(\frac{3 \pi x}{5} \right)}}{3}}{\pi} + \frac{20 \sin{\left(\frac{\pi x}{10} \right)} - 10 \sin{\left(\frac{\pi x}{5} \right)} + \frac{20 \sin{\left(\frac{3 \pi x}{10} \right)}}{3}}{\pi}$$
- No
es decir, función
no es
par ni impar