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Trazado el gráfico de la función/ sin(2x)/(2|sinx|)

Gráfico de la función y = sin(2x)/(2|sinx|)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        sin(2*x) 
f(x) = ----------
       2*|sin(x)|
f(x)=sin⁡(2x)2∣sin⁡(x)∣f{\left(x \right)} = \frac{\sin{\left(2 x \right)}}{2 \left|{\sin{\left(x \right)}}\right|} 
f = sin(2*x)/((2*Abs(sin(x))))
Gráfico de la función
02468-8-6-4-2-10102-2
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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⁡(2x)2∣sin⁡(x)∣=0\frac{\sin{\left(2 x \right)}}{2 \left|{\sin{\left(x \right)}}\right|} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π2x_{1} = \frac{\pi}{2}
Solución numérica
x1=−29.845130209103x_{1} = -29.845130209103
x2=−67.5442420521806x_{2} = -67.5442420521806
x3=−70.6858347057703x_{3} = -70.6858347057703
x4=64.4026493985908x_{4} = 64.4026493985908
x5=−36.1283155162826x_{5} = -36.1283155162826
x6=−92.6769832808989x_{6} = -92.6769832808989
x7=−61.261056745001x_{7} = -61.261056745001
x8=−306.305283725005x_{8} = -306.305283725005
x9=−76.9690200129499x_{9} = -76.9690200129499
x10=−98.9601685880785x_{10} = -98.9601685880785
x11=−95.8185759344887x_{11} = -95.8185759344887
x12=29.845130209103x_{12} = 29.845130209103
x13=80.1106126665397x_{13} = 80.1106126665397
x14=−64.4026493985908x_{14} = -64.4026493985908
x15=36.1283155162826x_{15} = 36.1283155162826
x16=73.8274273593601x_{16} = 73.8274273593601
x17=32.9867228626928x_{17} = 32.9867228626928
x18=−4.71238898038469x_{18} = -4.71238898038469
x19=−39.2699081698724x_{19} = -39.2699081698724
x20=26.7035375555132x_{20} = 26.7035375555132
x21=−7.85398163397448x_{21} = -7.85398163397448
x22=95.8185759344887x_{22} = 95.8185759344887
x23=−17.2787595947439x_{23} = -17.2787595947439
x24=−10.9955742875643x_{24} = -10.9955742875643
x25=98.9601685880785x_{25} = 98.9601685880785
x26=−86.3937979737193x_{26} = -86.3937979737193
x27=92.6769832808989x_{27} = 92.6769832808989
x28=−48.6946861306418x_{28} = -48.6946861306418
x29=54.9778714378214x_{29} = 54.9778714378214
x30=45.553093477052x_{30} = 45.553093477052
x31=23.5619449019235x_{31} = 23.5619449019235
x32=76.9690200129499x_{32} = 76.9690200129499
x33=815.243293606551x_{33} = 815.243293606551
x34=−89.5353906273091x_{34} = -89.5353906273091
x35=4.71238898038469x_{35} = 4.71238898038469
x36=−26.7035375555132x_{36} = -26.7035375555132
x37=−80.1106126665397x_{37} = -80.1106126665397
x38=7.85398163397448x_{38} = 7.85398163397448
x39=14.1371669411541x_{39} = 14.1371669411541
x40=86.3937979737193x_{40} = 86.3937979737193
x41=−45.553093477052x_{41} = -45.553093477052
x42=−83.2522053201295x_{42} = -83.2522053201295
x43=70.6858347057703x_{43} = 70.6858347057703
x44=83.2522053201295x_{44} = 83.2522053201295
x45=237.190245346029x_{45} = 237.190245346029
x46=48.6946861306418x_{46} = 48.6946861306418
x47=−20.4203522483337x_{47} = -20.4203522483337
x48=51.8362787842316x_{48} = 51.8362787842316
x49=10.9955742875643x_{49} = 10.9955742875643
x50=20.4203522483337x_{50} = 20.4203522483337
x51=1.5707963267949x_{51} = 1.5707963267949
x52=89.5353906273091x_{52} = 89.5353906273091
x53=130044.657099023x_{53} = 130044.657099023
x54=17.2787595947439x_{54} = 17.2787595947439
x55=58.1194640914112x_{55} = 58.1194640914112
x56=61.261056745001x_{56} = 61.261056745001
x57=−32.9867228626928x_{57} = -32.9867228626928
x58=−51.8362787842316x_{58} = -51.8362787842316
x59=−14.1371669411541x_{59} = -14.1371669411541
x60=−2279.22547017939x_{60} = -2279.22547017939
x61=−58.1194640914112x_{61} = -58.1194640914112
x62=−42.4115008234622x_{62} = -42.4115008234622
x63=−54.9778714378214x_{63} = -54.9778714378214
x64=−1.5707963267949x_{64} = -1.5707963267949
x65=42.4115008234622x_{65} = 42.4115008234622
x66=39.2699081698724x_{66} = 39.2699081698724
x67=67.5442420521806x_{67} = 67.5442420521806
x68=−23.5619449019235x_{68} = -23.5619449019235
x69=−73.8274273593601x_{69} = -73.8274273593601
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(2*x)/((2*Abs(sin(x)))).
sin⁡(0⋅2)2∣sin⁡(0)∣\frac{\sin{\left(0 \cdot 2 \right)}}{2 \left|{\sin{\left(0 \right)}}\right|}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
2cos⁡(2x)12∣sin⁡(x)∣−sin⁡(2x)cos⁡(x)sign⁡(sin⁡(x))2sin⁡2(x)=02 \cos{\left(2 x \right)} \frac{1}{2 \left|{\sin{\left(x \right)}}\right|} - \frac{\sin{\left(2 x \right)} \cos{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{2 \sin^{2}{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−100.530964914873x_{1} = -100.530964914873
x2=−15.707963267949x_{2} = -15.707963267949
x3=−25.1327412287183x_{3} = -25.1327412287183
x4=84.8230016469244x_{4} = 84.8230016469244
x5=97.3893722612836x_{5} = 97.3893722612836
x6=−50.2654824574367x_{6} = -50.2654824574367
x7=12.5663706143592x_{7} = 12.5663706143592
x8=650.309679293087x_{8} = 650.309679293087
x9=−6.28318530717959x_{9} = -6.28318530717959
x10=−72.2566310325652x_{10} = -72.2566310325652
x11=50.2654824574367x_{11} = 50.2654824574367
x12=−65.9734457253857x_{12} = -65.9734457253857
x13=15.707963267949x_{13} = 15.707963267949
x14=37.6991118430775x_{14} = 37.6991118430775
x15=−37.6991118430775x_{15} = -37.6991118430775
x16=−53.4070751110265x_{16} = -53.4070751110265
x17=−75.398223686155x_{17} = -75.398223686155
x18=−59.6902604182061x_{18} = -59.6902604182061
x19=28.2743338823081x_{19} = 28.2743338823081
x20=−285.884931476671x_{20} = -285.884931476671
x21=9.42477796076938x_{21} = 9.42477796076938
x22=18.8495559215388x_{22} = 18.8495559215388
x23=−56.5486677646163x_{23} = -56.5486677646163
x24=56.5486677646163x_{24} = 56.5486677646163
x25=59.6902604182061x_{25} = 59.6902604182061
x26=40.8407044966673x_{26} = 40.8407044966673
x27=−21.9911485751286x_{27} = -21.9911485751286
x28=6.28318530717959x_{28} = 6.28318530717959
x29=72.2566310325652x_{29} = 72.2566310325652
x30=−427.256600888212x_{30} = -427.256600888212
x31=−9.42477796076937x_{31} = -9.42477796076937
x32=−78.5398163397448x_{32} = -78.5398163397448
x33=21.9911485751286x_{33} = 21.9911485751286
x34=−97.3893722612836x_{34} = -97.3893722612836
x35=100.530964914873x_{35} = 100.530964914873
x36=−47.1238898038469x_{36} = -47.1238898038469
x37=34.5575191894877x_{37} = 34.5575191894877
x38=94.2477796076938x_{38} = 94.2477796076938
x39=−12.5663706143592x_{39} = -12.5663706143592
x40=−34.5575191894877x_{40} = -34.5575191894877
x41=−43.9822971502571x_{41} = -43.9822971502571
x42=78.5398163397448x_{42} = 78.5398163397448
x43=−94.2477796076938x_{43} = -94.2477796076938
x44=−91.106186954104x_{44} = -91.106186954104
x45=−31.415926535898x_{45} = -31.415926535898
x46=43.9822971502571x_{46} = 43.9822971502571
x47=75.398223686155x_{47} = 75.398223686155
x48=62.8318530717959x_{48} = 62.8318530717959
x49=−3.14159265358979x_{49} = -3.14159265358979
x50=87.9645943005142x_{50} = 87.9645943005142
x51=53.4070751110265x_{51} = 53.4070751110265
x52=−81.6814089933346x_{52} = -81.6814089933346
x53=−87.9645943005142x_{53} = -87.9645943005142
x54=65.9734457253857x_{54} = 65.9734457253857
x55=−69.1150383789755x_{55} = -69.1150383789755
x56=31.4159265358979x_{56} = 31.4159265358979
x57=81.6814089933346x_{57} = 81.6814089933346
x58=−28.2743338823081x_{58} = -28.2743338823081
Signos de extremos en los puntos:
(-100.53096491487338, 1)

(-15.707963267948966, 1)

(-25.132741228718345, 1)

(84.82300164692441, -1)

(97.3893722612836, 1)

(-50.26548245743669, 1)

(12.566370614359196, 1)

(650.3096792930872, 1)

(-6.283185307179588, -1)

(-72.25663103256524, 1)

(50.26548245743669, -1)

(-65.97344572538566, -1)

(15.707963267948987, 1)

(37.69911184307751, -1)

(-37.69911184307752, 1)

(-53.40707511102649, -1)

(-75.39822368615505, -1)

(-59.69026041820607, -1)

(28.27433388230814, 1)

(-285.88493147667117, 1)

(9.42477796076938, -1)

(18.84955592153876, -1)

(-56.548667764616276, 1)

(56.548667764616276, -1)

(59.69026041820606, -1)

(40.840704496667314, 1)

(-21.991148575128552, 1)

(6.283185307179586, -1)

(72.25663103256524, -1)

(-427.2566008882119, -1)

(-9.424777960769367, 1)

(-78.53981633974483, -1)

(21.991148575128552, -1)

(-97.3893722612836, -1)

(100.53096491487338, -1)

(-47.1238898038469, -1)

(34.557519189487735, 1)

(94.2477796076938, 1)

(-12.566370614359172, 1)

(-34.55751918948773, -1)

(-43.98229715025711, -1)

(78.53981633974483, 1)

(-94.2477796076938, -1)

(-91.106186954104, -1)

(-31.41592653589797, -1)

(43.982297150257104, -1)

(75.39822368615503, -1)

(62.83185307179586, -1)

(-3.141592653589793, 1)

(87.96459430051421, -1)

(53.40707511102649, 1)

(-81.68140899333463, -1)

(-87.96459430051421, 1)

(65.97344572538566, 1)

(-69.11503837897546, -1)

(31.41592653589793, -1)

(81.6814089933346, -1)

(-28.27433388230814, -1)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
x1=84.8230016469244x_{1} = 84.8230016469244
x2=−6.28318530717959x_{2} = -6.28318530717959
x3=50.2654824574367x_{3} = 50.2654824574367
x4=−65.9734457253857x_{4} = -65.9734457253857
x5=37.6991118430775x_{5} = 37.6991118430775
x6=−53.4070751110265x_{6} = -53.4070751110265
x7=−75.398223686155x_{7} = -75.398223686155
x8=−59.6902604182061x_{8} = -59.6902604182061
x9=9.42477796076938x_{9} = 9.42477796076938
x10=18.8495559215388x_{10} = 18.8495559215388
x11=56.5486677646163x_{11} = 56.5486677646163
x12=59.6902604182061x_{12} = 59.6902604182061
x13=6.28318530717959x_{13} = 6.28318530717959
x14=72.2566310325652x_{14} = 72.2566310325652
x15=−427.256600888212x_{15} = -427.256600888212
x16=−78.5398163397448x_{16} = -78.5398163397448
x17=21.9911485751286x_{17} = 21.9911485751286
x18=−97.3893722612836x_{18} = -97.3893722612836
x19=100.530964914873x_{19} = 100.530964914873
x20=−47.1238898038469x_{20} = -47.1238898038469
x21=−34.5575191894877x_{21} = -34.5575191894877
x22=−43.9822971502571x_{22} = -43.9822971502571
x23=−94.2477796076938x_{23} = -94.2477796076938
x24=−91.106186954104x_{24} = -91.106186954104
x25=−31.415926535898x_{25} = -31.415926535898
x26=43.9822971502571x_{26} = 43.9822971502571
x27=75.398223686155x_{27} = 75.398223686155
x28=62.8318530717959x_{28} = 62.8318530717959
x29=87.9645943005142x_{29} = 87.9645943005142
x30=−81.6814089933346x_{30} = -81.6814089933346
x31=−69.1150383789755x_{31} = -69.1150383789755
x32=31.4159265358979x_{32} = 31.4159265358979
x33=81.6814089933346x_{33} = 81.6814089933346
x34=−28.2743338823081x_{34} = -28.2743338823081
Puntos máximos de la función:
x34=−100.530964914873x_{34} = -100.530964914873
x34=−15.707963267949x_{34} = -15.707963267949
x34=−25.1327412287183x_{34} = -25.1327412287183
x34=97.3893722612836x_{34} = 97.3893722612836
x34=−50.2654824574367x_{34} = -50.2654824574367
x34=12.5663706143592x_{34} = 12.5663706143592
x34=650.309679293087x_{34} = 650.309679293087
x34=−72.2566310325652x_{34} = -72.2566310325652
x34=15.707963267949x_{34} = 15.707963267949
x34=−37.6991118430775x_{34} = -37.6991118430775
x34=28.2743338823081x_{34} = 28.2743338823081
x34=−285.884931476671x_{34} = -285.884931476671
x34=−56.5486677646163x_{34} = -56.5486677646163
x34=40.8407044966673x_{34} = 40.8407044966673
x34=−21.9911485751286x_{34} = -21.9911485751286
x34=−9.42477796076937x_{34} = -9.42477796076937
x34=34.5575191894877x_{34} = 34.5575191894877
x34=94.2477796076938x_{34} = 94.2477796076938
x34=−12.5663706143592x_{34} = -12.5663706143592
x34=78.5398163397448x_{34} = 78.5398163397448
x34=−3.14159265358979x_{34} = -3.14159265358979
x34=53.4070751110265x_{34} = 53.4070751110265
x34=−87.9645943005142x_{34} = -87.9645943005142
x34=65.9734457253857x_{34} = 65.9734457253857
Decrece en los intervalos
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Crece en los intervalos
(−∞,−427.256600888212]\left(-\infty, -427.256600888212\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(la segunda derivada es igual a cero),
las raíces de la ecuación obtenida serán los puntos de flexión para el gráfico de la función indicado:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
(sign⁡(sin⁡(x))−2cos⁡2(x)δ(sin⁡(x))sin⁡(x)+2cos⁡2(x)sign⁡2(sin⁡(x))sin⁡(x)∣sin⁡(x)∣)sin⁡(2x)2sin⁡(x)−2sin⁡(2x)∣sin⁡(x)∣−2cos⁡(x)cos⁡(2x)sign⁡(sin⁡(x))sin⁡2(x)=0\frac{\left(\operatorname{sign}{\left(\sin{\left(x \right)} \right)} - \frac{2 \cos^{2}{\left(x \right)} \delta\left(\sin{\left(x \right)}\right)}{\sin{\left(x \right)}} + \frac{2 \cos^{2}{\left(x \right)} \operatorname{sign}^{2}{\left(\sin{\left(x \right)} \right)}}{\sin{\left(x \right)} \left|{\sin{\left(x \right)}}\right|}\right) \sin{\left(2 x \right)}}{2 \sin{\left(x \right)}} - \frac{2 \sin{\left(2 x \right)}}{\left|{\sin{\left(x \right)}}\right|} - \frac{2 \cos{\left(x \right)} \cos{\left(2 x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{\sin^{2}{\left(x \right)}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(2x)2∣sin⁡(x)∣)=⟨−12,12⟩∣⟨−1,1⟩∣\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)}}{2 \left|{\sin{\left(x \right)}}\right|}\right) = \frac{\left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−12,12⟩∣⟨−1,1⟩∣y = \frac{\left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}
lim⁡x→∞(sin⁡(2x)2∣sin⁡(x)∣)=⟨−12,12⟩∣⟨−1,1⟩∣\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)}}{2 \left|{\sin{\left(x \right)}}\right|}\right) = \frac{\left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−12,12⟩∣⟨−1,1⟩∣y = \frac{\left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(2*x)/((2*Abs(sin(x)))), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(2x)12∣sin⁡(x)∣x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} \frac{1}{2 \left|{\sin{\left(x \right)}}\right|}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(2x)12∣sin⁡(x)∣x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} \frac{1}{2 \left|{\sin{\left(x \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⁡(2x)2∣sin⁡(x)∣=−sin⁡(2x)12∣sin⁡(x)∣\frac{\sin{\left(2 x \right)}}{2 \left|{\sin{\left(x \right)}}\right|} = - \sin{\left(2 x \right)} \frac{1}{2 \left|{\sin{\left(x \right)}}\right|}
- No
sin⁡(2x)2∣sin⁡(x)∣=sin⁡(2x)12∣sin⁡(x)∣\frac{\sin{\left(2 x \right)}}{2 \left|{\sin{\left(x \right)}}\right|} = \sin{\left(2 x \right)} \frac{1}{2 \left|{\sin{\left(x \right)}}\right|}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор