Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ sin(sin(sin(sin(x))))

Gráfico de la función y = sin(sin(sin(sin(x))))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = sin(sin(sin(sin(x))))
f(x)=sin⁡(sin⁡(sin⁡(sin⁡(x))))f{\left(x \right)} = \sin{\left(\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} \right)} 
f = sin(sin(sin(sin(x))))
Gráfico de la función
02468-8-6-4-2-10101-1
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje X con f = 0
o sea hay que resolver la ecuación:
sin⁡(sin⁡(sin⁡(sin⁡(x))))=0\sin{\left(\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=40.8407044966673x_{1} = 40.8407044966673
x2=0x_{2} = 0
x3=−18.8495559215388x_{3} = -18.8495559215388
x4=−56.5486677646163x_{4} = -56.5486677646163
x5=97.3893722612836x_{5} = 97.3893722612836
x6=34.5575191894877x_{6} = 34.5575191894877
x7=53.4070751110265x_{7} = 53.4070751110265
x8=47.1238898038469x_{8} = 47.1238898038469
x9=−97.3893722612836x_{9} = -97.3893722612836
x10=62.8318530717959x_{10} = 62.8318530717959
x11=87.9645943005142x_{11} = 87.9645943005142
x12=43.9822971502571x_{12} = 43.9822971502571
x13=37.6991118430775x_{13} = 37.6991118430775
x14=−21.9911485751286x_{14} = -21.9911485751286
x15=3.14159265358979x_{15} = 3.14159265358979
x16=−144.51326206513x_{16} = -144.51326206513
x17=69.1150383789755x_{17} = 69.1150383789755
x18=65.9734457253857x_{18} = 65.9734457253857
x19=232.477856365645x_{19} = 232.477856365645
x20=−50.2654824574367x_{20} = -50.2654824574367
x21=−94.2477796076938x_{21} = -94.2477796076938
x22=−75.398223686155x_{22} = -75.398223686155
x23=−53.4070751110265x_{23} = -53.4070751110265
x24=12.5663706143592x_{24} = 12.5663706143592
x25=−9.42477796076938x_{25} = -9.42477796076938
x26=−34.5575191894877x_{26} = -34.5575191894877
x27=21.9911485751286x_{27} = 21.9911485751286
x28=−47.1238898038469x_{28} = -47.1238898038469
x29=−43.9822971502571x_{29} = -43.9822971502571
x30=28.2743338823081x_{30} = 28.2743338823081
x31=−31.4159265358979x_{31} = -31.4159265358979
x32=−3.14159265358979x_{32} = -3.14159265358979
x33=−6.28318530717959x_{33} = -6.28318530717959
x34=−25.1327412287183x_{34} = -25.1327412287183
x35=−62.8318530717959x_{35} = -62.8318530717959
x36=31.4159265358979x_{36} = 31.4159265358979
x37=−65.9734457253857x_{37} = -65.9734457253857
x38=72.2566310325652x_{38} = 72.2566310325652
x39=−59.6902604182061x_{39} = -59.6902604182061
x40=94.2477796076938x_{40} = 94.2477796076938
x41=81.6814089933346x_{41} = 81.6814089933346
x42=−91.106186954104x_{42} = -91.106186954104
x43=−100.530964914873x_{43} = -100.530964914873
x44=59.6902604182061x_{44} = 59.6902604182061
x45=−40.8407044966673x_{45} = -40.8407044966673
x46=91.106186954104x_{46} = 91.106186954104
x47=78.5398163397448x_{47} = 78.5398163397448
x48=−12.5663706143592x_{48} = -12.5663706143592
x49=56.5486677646163x_{49} = 56.5486677646163
x50=84.8230016469244x_{50} = 84.8230016469244
x51=100.530964914873x_{51} = 100.530964914873
x52=−69.1150383789755x_{52} = -69.1150383789755
x53=9.42477796076938x_{53} = 9.42477796076938
x54=−84.8230016469244x_{54} = -84.8230016469244
x55=−78.5398163397448x_{55} = -78.5398163397448
x56=119.380520836412x_{56} = 119.380520836412
x57=−87.9645943005142x_{57} = -87.9645943005142
x58=−81.6814089933346x_{58} = -81.6814089933346
x59=15.707963267949x_{59} = 15.707963267949
x60=−28.2743338823081x_{60} = -28.2743338823081
x61=−15.707963267949x_{61} = -15.707963267949
x62=−37.6991118430775x_{62} = -37.6991118430775
x63=18.8495559215388x_{63} = 18.8495559215388
x64=25.1327412287183x_{64} = 25.1327412287183
x65=50.2654824574367x_{65} = 50.2654824574367
x66=−72.2566310325652x_{66} = -72.2566310325652
x67=75.398223686155x_{67} = 75.398223686155
x68=6.28318530717959x_{68} = 6.28318530717959
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 sin(sin(sin(sin(x)))).
sin⁡(sin⁡(sin⁡(sin⁡(0))))\sin{\left(\sin{\left(\sin{\left(\sin{\left(0 \right)} \right)} \right)} \right)}
Resultado:
f(0)=0f{\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→−∞sin⁡(sin⁡(sin⁡(sin⁡(x))))=⟨−sin⁡(sin⁡(sin⁡(1))),sin⁡(sin⁡(sin⁡(1)))⟩\lim_{x \to -\infty} \sin{\left(\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} \right)} = \left\langle - \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}, \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−sin⁡(sin⁡(sin⁡(1))),sin⁡(sin⁡(sin⁡(1)))⟩y = \left\langle - \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}, \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}\right\rangle
lim⁡x→∞sin⁡(sin⁡(sin⁡(sin⁡(x))))=⟨−sin⁡(sin⁡(sin⁡(1))),sin⁡(sin⁡(sin⁡(1)))⟩\lim_{x \to \infty} \sin{\left(\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} \right)} = \left\langle - \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}, \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−sin⁡(sin⁡(sin⁡(1))),sin⁡(sin⁡(sin⁡(1)))⟩y = \left\langle - \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}, \sin{\left(\sin{\left(\sin{\left(1 \right)} \right)} \right)}\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(sin(sin(sin(x)))), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(sin⁡(sin⁡(sin⁡(x))))x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(sin⁡(sin⁡(sin⁡(x))))x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} \right)}}{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:
sin⁡(sin⁡(sin⁡(sin⁡(x))))=−sin⁡(sin⁡(sin⁡(sin⁡(x))))\sin{\left(\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} \right)} = - \sin{\left(\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} \right)}
- No
sin⁡(sin⁡(sin⁡(sin⁡(x))))=sin⁡(sin⁡(sin⁡(sin⁡(x))))\sin{\left(\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} \right)} = \sin{\left(\sin{\left(\sin{\left(\sin{\left(x \right)} \right)} \right)} \right)}
- Sí
es decir, función
es
impar
    © Sr Examen – Калькулятор