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Trazado el gráfico de la función/ (1-cos(x))/((x*sin(2*x)))

Gráfico de la función y = (1-cos(x))/((x*sin(2*x)))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       1 - cos(x)
f(x) = ----------
       x*sin(2*x)
f(x)=1−cos⁡(x)xsin⁡(2x)f{\left(x \right)} = \frac{1 - \cos{\left(x \right)}}{x \sin{\left(2 x \right)}} 
f = (1 - cos(x))/((x*sin(2*x)))
Gráfico de la función
02468-8-6-4-2-1010-5050
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=1.5707963267949x_{2} = 1.5707963267949
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:
1−cos⁡(x)xsin⁡(2x)=0\frac{1 - \cos{\left(x \right)}}{x \sin{\left(2 x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−100.530964914873x_{1} = -100.530964914873
x2=−25.1327412287183x_{2} = -25.1327412287183
x3=−75.398223686155x_{3} = -75.398223686155
x4=−50.2654824574367x_{4} = -50.2654824574367
x5=−18.8495559215388x_{5} = -18.8495559215388
x6=81.6814089933346x_{6} = 81.6814089933346
x7=−6.28318530717958x_{7} = -6.28318530717958
x8=6.28318530717956x_{8} = 6.28318530717956
x9=62.8318530717959x_{9} = 62.8318530717959
x10=50.2654824574367x_{10} = 50.2654824574367
x11=−6.28318530717958x_{11} = -6.28318530717958
x12=−37.6991118430775x_{12} = -37.6991118430775
x13=−43.9822971502571x_{13} = -43.9822971502571
x14=25.1327412287183x_{14} = 25.1327412287183
x15=37.6991118430775x_{15} = 37.6991118430775
x16=−75.398223686155x_{16} = -75.398223686155
x17=−18.8495559215388x_{17} = -18.8495559215388
x18=31.4159265358979x_{18} = 31.4159265358979
x19=−31.4159265358979x_{19} = -31.4159265358979
x20=−56.5486677646163x_{20} = -56.5486677646163
x21=18.8495559215388x_{21} = 18.8495559215388
x22=50.2654824574367x_{22} = 50.2654824574367
x23=−6.28318530717959x_{23} = -6.28318530717959
x24=75.398223686155x_{24} = 75.398223686155
x25=12.5663706143592x_{25} = 12.5663706143592
x26=−87.9645943005142x_{26} = -87.9645943005142
x27=56.5486677646163x_{27} = 56.5486677646163
x28=69.1150383789755x_{28} = 69.1150383789755
x29=6.28318530717959x_{29} = 6.28318530717959
x30=−62.8318530717959x_{30} = -62.8318530717959
x31=37.6991118430775x_{31} = 37.6991118430775
x32=−25.1327412287184x_{32} = -25.1327412287184
x33=31.4159265358979x_{33} = 31.4159265358979
x34=100.530964914873x_{34} = 100.530964914873
x35=−182.212373908208x_{35} = -182.212373908208
x36=94.2477796076938x_{36} = 94.2477796076938
x37=−62.8318530717959x_{37} = -62.8318530717959
x38=−12.5663706143592x_{38} = -12.5663706143592
x39=−94.2477796076938x_{39} = -94.2477796076938
x40=37.6991118430775x_{40} = 37.6991118430775
x41=43.9822971502571x_{41} = 43.9822971502571
x42=12.5663706143592x_{42} = 12.5663706143592
x43=75.398223686155x_{43} = 75.398223686155
x44=62.8318530717959x_{44} = 62.8318530717959
x45=87.9645943005142x_{45} = 87.9645943005142
x46=69.1150383789754x_{46} = 69.1150383789754
x47=−81.6814089933346x_{47} = -81.6814089933346
x48=−87.9645943005142x_{48} = -87.9645943005142
x49=−69.1150383789755x_{49} = -69.1150383789755
x50=−31.4159265358979x_{50} = -31.4159265358979
x51=25.1327412287184x_{51} = 25.1327412287184
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 (1 - cos(x))/((x*sin(2*x))).
1−cos⁡(0)0sin⁡(0⋅2)\frac{1 - \cos{\left(0 \right)}}{0 \sin{\left(0 \cdot 2 \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
1xsin⁡(2x)sin⁡(x)+(1−cos⁡(x))(−2xcos⁡(2x)−sin⁡(2x))x2sin⁡2(2x)=0\frac{1}{x \sin{\left(2 x \right)}} \sin{\left(x \right)} + \frac{\left(1 - \cos{\left(x \right)}\right) \left(- 2 x \cos{\left(2 x \right)} - \sin{\left(2 x \right)}\right)}{x^{2} \sin^{2}{\left(2 x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−21.0997786190612x_{1} = -21.0997786190612
x2=−2.35903204498278x_{2} = -2.35903204498278
x3=−14.8222285221178x_{3} = -14.8222285221178
x4=−92.0137428813551x_{4} = -92.0137428813551
x5=46.2253307504569x_{5} = 46.2253307504569
x6=85.7307769850148x_{6} = 85.7307769850148
x7=−83.9217438912508x_{7} = -83.9217438912508
x8=−33.6612081744243x_{8} = -33.6612081744243
x9=−27.3799241503631x_{9} = -27.3799241503631
x10=−48.0341830943912x_{10} = -48.0341830943912
x11=60.5993672152362x_{11} = 60.5993672152362
x12=22.9076920601837x_{12} = 22.9076920601837
x13=29.1883117520455x_{13} = 29.1883117520455
x14=4.11067205558904x_{14} = 4.11067205558904
x15=92.0137428813551x_{15} = 92.0137428813551
x16=−39.9430921063837x_{16} = -39.9430921063837
x17=−60.5993672152362x_{17} = -60.5993672152362
x18=−90.204699041177x_{18} = -90.204699041177
x19=27.3799241503631x_{19} = 27.3799241503631
x20=−35.469835821083x_{20} = -35.469835821083
x21=96.4876842084868x_{21} = 96.4876842084868
x22=41.7518578220228x_{22} = 41.7518578220228
x23=83.9217438912508x_{23} = 83.9217438912508
x24=10.3556266699783x_{24} = 10.3556266699783
x25=−4.11067205558904x_{25} = -4.11067205558904
x26=16.628990050393x_{26} = 16.628990050393
x27=33.6612081744243x_{27} = 33.6612081744243
x28=2.35903204498278x_{28} = 2.35903204498278
x29=14.8222285221178x_{29} = 14.8222285221178
x30=8.55304201163282x_{30} = 8.55304201163282
x31=66.8821260674787x_{31} = 66.8821260674787
x32=−41.7518578220228x_{32} = -41.7518578220228
x33=90.204699041177x_{33} = 90.204699041177
x34=−71.3559554197142x_{34} = -71.3559554197142
x35=−96.4876842084868x_{35} = -96.4876842084868
x36=35.469835821083x_{36} = 35.469835821083
x37=52.507796494609x_{37} = 52.507796494609
x38=77.6388260588097x_{38} = 77.6388260588097
x39=410.644754035796x_{39} = 410.644754035796
x40=58.7904163791633x_{40} = 58.7904163791633
x41=48.0341830943912x_{41} = 48.0341830943912
x42=−77.6388260588097x_{42} = -77.6388260588097
x43=−85.7307769850148x_{43} = -85.7307769850148
x44=−79.447845725128x_{44} = -79.447845725128
x45=−52.507796494609x_{45} = -52.507796494609
x46=−16.628990050393x_{46} = -16.628990050393
x47=−146.752183087622x_{47} = -146.752183087622
x48=54.3167067346688x_{48} = 54.3167067346688
x49=−65.0731456714068x_{49} = -65.0731456714068
x50=−8.55304201163282x_{50} = -8.55304201163282
x51=−54.3167067346688x_{51} = -54.3167067346688
x52=71.3559554197142x_{52} = 71.3559554197142
x53=−10.3556266699783x_{53} = -10.3556266699783
x54=−98.2967367830456x_{54} = -98.2967367830456
x55=−46.2253307504569x_{55} = -46.2253307504569
x56=39.9430921063837x_{56} = 39.9430921063837
x57=98.2967367830456x_{57} = 98.2967367830456
Signos de extremos en los puntos:
(-21.09977861906122, -0.0789400167897395)

(-2.3590320449827753, -0.724508135686241)

(-14.822228522117832, -0.112409283601198)

(-92.0137428813551, 0.0180964539367232)

(46.22533075045695, -0.0360236141212341)

(85.73077698501483, 0.019422739816553)

(-83.92174389125084, -0.0198414380991413)

(-33.66120817442426, -0.0494723651395544)

(-27.37992415036312, -0.0608257506350598)

(-48.03418309439118, 0.0346668675653127)

(60.59936721523623, 0.0274781383720191)

(22.90769206018372, 0.0727061485455461)

(29.188311752045504, 0.0570558388985918)

(4.11067205558904, 0.408210556939668)

(92.0137428813551, 0.0180964539367232)

(-39.94309210638365, -0.0416903192670335)

(-60.59936721523623, 0.0274781383720191)

(-90.204699041177, -0.0184593895515073)

(27.37992415036312, -0.0608257506350598)

(-35.46983582108299, 0.0469491101063058)

(96.48768420848683, -0.0172573330831066)

(41.75185782202277, 0.0398838951977928)

(83.92174389125084, -0.0198414380991413)

(10.355626669978271, 0.160994549148767)

(-4.11067205558904, 0.408210556939668)

(16.62899005039297, 0.100181523234274)

(33.66120817442426, -0.0494723651395544)

(2.3590320449827753, -0.724508135686241)

(14.822228522117832, -0.112409283601198)

(8.553042011632824, -0.195054444868888)

(66.88212606747874, 0.0248967359111755)

(-41.75185782202277, 0.0398838951977928)

(90.204699041177, -0.0184593895515073)

(-71.3559554197142, -0.0233356927886555)

(-96.48768420848683, -0.0172573330831066)

(35.46983582108299, 0.0469491101063058)

(52.507796494609025, -0.0317129877503068)

(77.63882605880968, -0.0214471765347026)

(410.6447540357962, -0.0040548349571471)

(58.79041637916331, -0.028323700297562)

(48.03418309439118, 0.0346668675653127)

(-77.63882605880968, -0.0214471765347026)

(-85.73077698501483, 0.019422739816553)

(-79.44784572512803, 0.0209588024678898)

(-52.507796494609025, -0.0317129877503068)

(-16.62899005039297, 0.100181523234274)

(-146.75218308762211, -0.0113463800149005)

(54.31670673466879, 0.0306567381010815)

(-65.07314567140679, -0.025588894223446)

(-8.553042011632824, -0.195054444868888)

(-54.31670673466879, 0.0306567381010815)

(71.3559554197142, -0.0233356927886555)

(-10.355626669978271, 0.160994549148767)

(-98.29673678304557, 0.0169397190694041)

(-46.22533075045695, -0.0360236141212341)

(39.94309210638365, -0.0416903192670335)

(98.29673678304557, 0.0169397190694041)


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=−92.0137428813551x_{1} = -92.0137428813551
x2=85.7307769850148x_{2} = 85.7307769850148
x3=−48.0341830943912x_{3} = -48.0341830943912
x4=60.5993672152362x_{4} = 60.5993672152362
x5=22.9076920601837x_{5} = 22.9076920601837
x6=29.1883117520455x_{6} = 29.1883117520455
x7=4.11067205558904x_{7} = 4.11067205558904
x8=92.0137428813551x_{8} = 92.0137428813551
x9=−60.5993672152362x_{9} = -60.5993672152362
x10=−35.469835821083x_{10} = -35.469835821083
x11=41.7518578220228x_{11} = 41.7518578220228
x12=10.3556266699783x_{12} = 10.3556266699783
x13=−4.11067205558904x_{13} = -4.11067205558904
x14=16.628990050393x_{14} = 16.628990050393
x15=66.8821260674787x_{15} = 66.8821260674787
x16=−41.7518578220228x_{16} = -41.7518578220228
x17=35.469835821083x_{17} = 35.469835821083
x18=48.0341830943912x_{18} = 48.0341830943912
x19=−85.7307769850148x_{19} = -85.7307769850148
x20=−79.447845725128x_{20} = -79.447845725128
x21=−16.628990050393x_{21} = -16.628990050393
x22=54.3167067346688x_{22} = 54.3167067346688
x23=−54.3167067346688x_{23} = -54.3167067346688
x24=−10.3556266699783x_{24} = -10.3556266699783
x25=−98.2967367830456x_{25} = -98.2967367830456
x26=98.2967367830456x_{26} = 98.2967367830456
Puntos máximos de la función:
x26=−21.0997786190612x_{26} = -21.0997786190612
x26=−2.35903204498278x_{26} = -2.35903204498278
x26=−14.8222285221178x_{26} = -14.8222285221178
x26=46.2253307504569x_{26} = 46.2253307504569
x26=−83.9217438912508x_{26} = -83.9217438912508
x26=−33.6612081744243x_{26} = -33.6612081744243
x26=−27.3799241503631x_{26} = -27.3799241503631
x26=−39.9430921063837x_{26} = -39.9430921063837
x26=−90.204699041177x_{26} = -90.204699041177
x26=27.3799241503631x_{26} = 27.3799241503631
x26=96.4876842084868x_{26} = 96.4876842084868
x26=83.9217438912508x_{26} = 83.9217438912508
x26=33.6612081744243x_{26} = 33.6612081744243
x26=2.35903204498278x_{26} = 2.35903204498278
x26=14.8222285221178x_{26} = 14.8222285221178
x26=8.55304201163282x_{26} = 8.55304201163282
x26=90.204699041177x_{26} = 90.204699041177
x26=−71.3559554197142x_{26} = -71.3559554197142
x26=−96.4876842084868x_{26} = -96.4876842084868
x26=52.507796494609x_{26} = 52.507796494609
x26=77.6388260588097x_{26} = 77.6388260588097
x26=410.644754035796x_{26} = 410.644754035796
x26=58.7904163791633x_{26} = 58.7904163791633
x26=−77.6388260588097x_{26} = -77.6388260588097
x26=−52.507796494609x_{26} = -52.507796494609
x26=−146.752183087622x_{26} = -146.752183087622
x26=−65.0731456714068x_{26} = -65.0731456714068
x26=−8.55304201163282x_{26} = -8.55304201163282
x26=71.3559554197142x_{26} = 71.3559554197142
x26=−46.2253307504569x_{26} = -46.2253307504569
x26=39.9430921063837x_{26} = 39.9430921063837
Decrece en los intervalos
[98.2967367830456,∞)\left[98.2967367830456, \infty\right)
Crece en los intervalos
(−∞,−98.2967367830456]\left(-\infty, -98.2967367830456\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=1.5707963267949x_{2} = 1.5707963267949
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(1−cos⁡(x)xsin⁡(2x))y = \lim_{x \to -\infty}\left(\frac{1 - \cos{\left(x \right)}}{x \sin{\left(2 x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(1−cos⁡(x)xsin⁡(2x))y = \lim_{x \to \infty}\left(\frac{1 - \cos{\left(x \right)}}{x \sin{\left(2 x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (1 - cos(x))/((x*sin(2*x))), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(1xsin⁡(2x)(1−cos⁡(x))x)y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{x \sin{\left(2 x \right)}} \left(1 - \cos{\left(x \right)}\right)}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(1xsin⁡(2x)(1−cos⁡(x))x)y = x \lim_{x \to \infty}\left(\frac{\frac{1}{x \sin{\left(2 x \right)}} \left(1 - \cos{\left(x \right)}\right)}{x}\right)
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:
1−cos⁡(x)xsin⁡(2x)=1−cos⁡(x)xsin⁡(2x)\frac{1 - \cos{\left(x \right)}}{x \sin{\left(2 x \right)}} = \frac{1 - \cos{\left(x \right)}}{x \sin{\left(2 x \right)}}
- Sí
1−cos⁡(x)xsin⁡(2x)=−1−cos⁡(x)xsin⁡(2x)\frac{1 - \cos{\left(x \right)}}{x \sin{\left(2 x \right)}} = - \frac{1 - \cos{\left(x \right)}}{x \sin{\left(2 x \right)}}
- No
es decir, función
es
par
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