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x^3*sin(x)
Trazado el gráfico de la función/ x^3*sin(x)

Gráfico de la función y = x^3*sin(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        3       
f(x) = x *sin(x)
f(x)=x3sin⁡(x)f{\left(x \right)} = x^{3} \sin{\left(x \right)} 
f = x^3*sin(x)
Gráfico de la función
02468-8-6-4-2-1010-10001000
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:
x3sin⁡(x)=0x^{3} \sin{\left(x \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=−59.6902604182061x_{1} = -59.6902604182061
x2=−62.8318530717959x_{2} = -62.8318530717959
x3=−97.3893722612836x_{3} = -97.3893722612836
x4=87.9645943005142x_{4} = 87.9645943005142
x5=−56.5486677646163x_{5} = -56.5486677646163
x6=31.4159265358979x_{6} = 31.4159265358979
x7=69.1150383789755x_{7} = 69.1150383789755
x8=−37.6991118430775x_{8} = -37.6991118430775
x9=−81.6814089933346x_{9} = -81.6814089933346
x10=−84.8230016469244x_{10} = -84.8230016469244
x11=−21.9911485751286x_{11} = -21.9911485751286
x12=47.1238898038469x_{12} = 47.1238898038469
x13=−15.707963267949x_{13} = -15.707963267949
x14=−12.5663706143592x_{14} = -12.5663706143592
x15=12.5663706143592x_{15} = 12.5663706143592
x16=−87.9645943005142x_{16} = -87.9645943005142
x17=53.4070751110265x_{17} = 53.4070751110265
x18=72.2566310325652x_{18} = 72.2566310325652
x19=−100.530964914873x_{19} = -100.530964914873
x20=−3.14159265358979x_{20} = -3.14159265358979
x21=34.5575191894877x_{21} = 34.5575191894877
x22=−94.2477796076938x_{22} = -94.2477796076938
x23=6.28318530717959x_{23} = 6.28318530717959
x24=−69.1150383789755x_{24} = -69.1150383789755
x25=97.3893722612836x_{25} = 97.3893722612836
x26=0x_{26} = 0
x27=65.9734457253857x_{27} = 65.9734457253857
x28=−50.2654824574367x_{28} = -50.2654824574367
x29=15.707963267949x_{29} = 15.707963267949
x30=3.14159265358979x_{30} = 3.14159265358979
x31=−25.1327412287183x_{31} = -25.1327412287183
x32=−18.8495559215388x_{32} = -18.8495559215388
x33=40.8407044966673x_{33} = 40.8407044966673
x34=−53.4070751110265x_{34} = -53.4070751110265
x35=37.6991118430775x_{35} = 37.6991118430775
x36=−43.9822971502571x_{36} = -43.9822971502571
x37=18.8495559215388x_{37} = 18.8495559215388
x38=−78.5398163397448x_{38} = -78.5398163397448
x39=−6.28318530717959x_{39} = -6.28318530717959
x40=−135.088484104361x_{40} = -135.088484104361
x41=−40.8407044966673x_{41} = -40.8407044966673
x42=43.9822971502571x_{42} = 43.9822971502571
x43=−125.663706143592x_{43} = -125.663706143592
x44=56.5486677646163x_{44} = 56.5486677646163
x45=−65.9734457253857x_{45} = -65.9734457253857
x46=25.1327412287183x_{46} = 25.1327412287183
x47=78.5398163397448x_{47} = 78.5398163397448
x48=−28.2743338823081x_{48} = -28.2743338823081
x49=75.398223686155x_{49} = 75.398223686155
x50=59.6902604182061x_{50} = 59.6902604182061
x51=−34.5575191894877x_{51} = -34.5575191894877
x52=81.6814089933346x_{52} = 81.6814089933346
x53=−47.1238898038469x_{53} = -47.1238898038469
x54=100.530964914873x_{54} = 100.530964914873
x55=−9.42477796076938x_{55} = -9.42477796076938
x56=−75.398223686155x_{56} = -75.398223686155
x57=−72.2566310325652x_{57} = -72.2566310325652
x58=−31.4159265358979x_{58} = -31.4159265358979
x59=28.2743338823081x_{59} = 28.2743338823081
x60=−91.106186954104x_{60} = -91.106186954104
x61=21.9911485751286x_{61} = 21.9911485751286
x62=62.8318530717959x_{62} = 62.8318530717959
x63=9.42477796076938x_{63} = 9.42477796076938
x64=50.2654824574367x_{64} = 50.2654824574367
x65=94.2477796076938x_{65} = 94.2477796076938
x66=91.106186954104x_{66} = 91.106186954104
x67=84.8230016469244x_{67} = 84.8230016469244
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 x^3*sin(x).
03sin⁡(0)0^{3} \sin{\left(0 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
x3cos⁡(x)+3x2sin⁡(x)=0x^{3} \cos{\left(x \right)} + 3 x^{2} \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−73.8680180276454x_{1} = -73.8680180276454
x2=80.1480259413025x_{2} = 80.1480259413025
x3=64.4491641378738x_{3} = 64.4491641378738
x4=83.2882092591146x_{4} = 83.2882092591146
x5=5.23293845351241x_{5} = 5.23293845351241
x6=73.8680180276454x_{6} = 73.8680180276454
x7=−58.170990540028x_{7} = -58.170990540028
x8=−29.9449807735163x_{8} = -29.9449807735163
x9=51.894024636399x_{9} = 51.894024636399
x10=−48.75613936684x_{10} = -48.75613936684
x11=48.75613936684x_{11} = 48.75613936684
x12=98.9904652640992x_{12} = 98.9904652640992
x13=95.8498646688189x_{13} = 95.8498646688189
x14=−64.4491641378738x_{14} = -64.4491641378738
x15=61.3099494475655x_{15} = 61.3099494475655
x16=−55.0323309441547x_{16} = -55.0323309441547
x17=−92.7093311956205x_{17} = -92.7093311956205
x18=−77.0079573362515x_{18} = -77.0079573362515
x19=0x_{19} = 0
x20=92.7093311956205x_{20} = 92.7093311956205
x21=29.9449807735163x_{21} = 29.9449807735163
x22=−51.894024636399x_{22} = -51.894024636399
x23=−26.814952130975x_{23} = -26.814952130975
x24=−17.4490243427188x_{24} = -17.4490243427188
x25=23.6879210560017x_{25} = 23.6879210560017
x26=39.3460075465194x_{26} = 39.3460075465194
x27=−2.45564386287944x_{27} = -2.45564386287944
x28=−42.4820019253669x_{28} = -42.4820019253669
x29=−8.20453136258127x_{29} = -8.20453136258127
x30=8.20453136258127x_{30} = 8.20453136258127
x31=−80.1480259413025x_{31} = -80.1480259413025
x32=77.0079573362515x_{32} = 77.0079573362515
x33=−83.2882092591146x_{33} = -83.2882092591146
x34=−98.9904652640992x_{34} = -98.9904652640992
x35=−45.6187613383417x_{35} = -45.6187613383417
x36=−95.8498646688189x_{36} = -95.8498646688189
x37=−14.3433507883915x_{37} = -14.3433507883915
x38=14.3433507883915x_{38} = 14.3433507883915
x39=26.814952130975x_{39} = 26.814952130975
x40=−70.7282251775385x_{40} = -70.7282251775385
x41=33.0771723843072x_{41} = 33.0771723843072
x42=−39.3460075465194x_{42} = -39.3460075465194
x43=−89.5688718899173x_{43} = -89.5688718899173
x44=−20.5652079398333x_{44} = -20.5652079398333
x45=11.2560430143535x_{45} = 11.2560430143535
x46=17.4490243427188x_{46} = 17.4490243427188
x47=45.6187613383417x_{47} = 45.6187613383417
x48=−67.5885991217338x_{48} = -67.5885991217338
x49=−61.3099494475655x_{49} = -61.3099494475655
x50=−36.2109745555852x_{50} = -36.2109745555852
x51=36.2109745555852x_{51} = 36.2109745555852
x52=−33.0771723843072x_{52} = -33.0771723843072
x53=58.170990540028x_{53} = 58.170990540028
x54=−5.23293845351241x_{54} = -5.23293845351241
x55=42.4820019253669x_{55} = 42.4820019253669
x56=−23.6879210560017x_{56} = -23.6879210560017
x57=89.5688718899173x_{57} = 89.5688718899173
x58=2.45564386287944x_{58} = 2.45564386287944
x59=86.4284948180722x_{59} = 86.4284948180722
x60=20.5652079398333x_{60} = 20.5652079398333
x61=70.7282251775385x_{61} = 70.7282251775385
x62=55.0323309441547x_{62} = 55.0323309441547
x63=67.5885991217338x_{63} = 67.5885991217338
x64=−11.2560430143535x_{64} = -11.2560430143535
x65=−86.4284948180722x_{65} = -86.4284948180722
Signos de extremos en los puntos:
(-73.86801802764536, -402727.669491498)

(80.14802594130248, -514487.072547109)

(64.44916413787378, 267412.604455205)

(83.28820925911458, 577389.695139745)

(5.232938453512406, -124.316680634702)

(73.86801802764536, -402727.669491498)

(-58.17099054002796, 196581.480455827)

(-29.944980773516342, -26717.9738988985)

(51.894024636399, 139517.139252855)

(-48.756139366839975, -115682.417566907)

(48.756139366839975, -115682.417566907)

(98.99046526409923, -969573.526679447)

(95.84986466881885, 880160.538929613)

(-64.44916413787378, 267412.604455205)

(61.309949447565465, -230183.175698878)

(-55.032330944154715, -166421.48092055)

(-92.70933119562048, -796421.699586266)

(-77.00795733625147, 456328.409900699)

(0, 0)

(92.70933119562048, -796421.699586266)

(29.944980773516342, -26717.9738988985)

(-51.894024636399, 139517.139252855)

(-26.81495213097502, 19161.5214252829)

(-17.449024342718843, -5235.85577950966)

(23.687921056001688, -13186.37925766)

(39.34600754651944, 60735.5924841558)

(-2.45564386287944, 9.37949248744233)

(-42.48200192536688, -76477.6822699254)

(-8.204531362581267, 518.694993552911)

(8.204531362581267, 518.694993552911)

(-80.14802594130248, -514487.072547109)

(77.00795733625147, 456328.409900699)

(-83.28820925911458, 577389.695139745)

(-98.99046526409923, -969573.526679447)

(-45.6187613383417, 94731.2779158677)

(-95.84986466881885, 880160.538929613)

(-14.34335078839151, 2888.3803804149)

(14.34335078839151, 2888.3803804149)

(26.81495213097502, 19161.5214252829)

(-70.72822517753846, 353498.813601871)

(33.07717238430719, 36041.7770225777)

(-39.34600754651944, 60735.5924841558)

(-89.56887188991735, 718170.970965642)

(-20.56520793983334, 8606.50554943)

(11.256043014353493, -1378.01976203725)

(17.449024342718843, -5235.85577950966)

(45.6187613383417, 94731.2779158677)

(-67.5885991217338, -308455.804503574)

(-61.309949447565465, -230183.175698878)

(-36.21097455558523, -47318.9702503321)

(36.21097455558523, -47318.9702503321)

(-33.07717238430719, 36041.7770225777)

(58.17099054002796, 196581.480455827)

(-5.232938453512406, -124.316680634702)

(42.48200192536688, -76477.6822699254)

(-23.687921056001688, -13186.37925766)

(89.56887188991735, 718170.970965642)

(2.45564386287944, 9.37949248744233)

(86.42849481807224, -645222.315380553)

(20.56520793983334, 8606.50554943)

(70.72822517753846, 353498.813601871)

(55.032330944154715, -166421.48092055)

(67.5885991217338, -308455.804503574)

(-11.256043014353493, -1378.01976203725)

(-86.42849481807224, -645222.315380553)


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=−73.8680180276454x_{1} = -73.8680180276454
x2=80.1480259413025x_{2} = 80.1480259413025
x3=5.23293845351241x_{3} = 5.23293845351241
x4=73.8680180276454x_{4} = 73.8680180276454
x5=−29.9449807735163x_{5} = -29.9449807735163
x6=−48.75613936684x_{6} = -48.75613936684
x7=48.75613936684x_{7} = 48.75613936684
x8=98.9904652640992x_{8} = 98.9904652640992
x9=61.3099494475655x_{9} = 61.3099494475655
x10=−55.0323309441547x_{10} = -55.0323309441547
x11=−92.7093311956205x_{11} = -92.7093311956205
x12=0x_{12} = 0
x13=92.7093311956205x_{13} = 92.7093311956205
x14=29.9449807735163x_{14} = 29.9449807735163
x15=−17.4490243427188x_{15} = -17.4490243427188
x16=23.6879210560017x_{16} = 23.6879210560017
x17=−42.4820019253669x_{17} = -42.4820019253669
x18=−80.1480259413025x_{18} = -80.1480259413025
x19=−98.9904652640992x_{19} = -98.9904652640992
x20=11.2560430143535x_{20} = 11.2560430143535
x21=17.4490243427188x_{21} = 17.4490243427188
x22=−67.5885991217338x_{22} = -67.5885991217338
x23=−61.3099494475655x_{23} = -61.3099494475655
x24=−36.2109745555852x_{24} = -36.2109745555852
x25=36.2109745555852x_{25} = 36.2109745555852
x26=−5.23293845351241x_{26} = -5.23293845351241
x27=42.4820019253669x_{27} = 42.4820019253669
x28=−23.6879210560017x_{28} = -23.6879210560017
x29=86.4284948180722x_{29} = 86.4284948180722
x30=55.0323309441547x_{30} = 55.0323309441547
x31=67.5885991217338x_{31} = 67.5885991217338
x32=−11.2560430143535x_{32} = -11.2560430143535
x33=−86.4284948180722x_{33} = -86.4284948180722
Puntos máximos de la función:
x33=64.4491641378738x_{33} = 64.4491641378738
x33=83.2882092591146x_{33} = 83.2882092591146
x33=−58.170990540028x_{33} = -58.170990540028
x33=51.894024636399x_{33} = 51.894024636399
x33=95.8498646688189x_{33} = 95.8498646688189
x33=−64.4491641378738x_{33} = -64.4491641378738
x33=−77.0079573362515x_{33} = -77.0079573362515
x33=−51.894024636399x_{33} = -51.894024636399
x33=−26.814952130975x_{33} = -26.814952130975
x33=39.3460075465194x_{33} = 39.3460075465194
x33=−2.45564386287944x_{33} = -2.45564386287944
x33=−8.20453136258127x_{33} = -8.20453136258127
x33=8.20453136258127x_{33} = 8.20453136258127
x33=77.0079573362515x_{33} = 77.0079573362515
x33=−83.2882092591146x_{33} = -83.2882092591146
x33=−45.6187613383417x_{33} = -45.6187613383417
x33=−95.8498646688189x_{33} = -95.8498646688189
x33=−14.3433507883915x_{33} = -14.3433507883915
x33=14.3433507883915x_{33} = 14.3433507883915
x33=26.814952130975x_{33} = 26.814952130975
x33=−70.7282251775385x_{33} = -70.7282251775385
x33=33.0771723843072x_{33} = 33.0771723843072
x33=−39.3460075465194x_{33} = -39.3460075465194
x33=−89.5688718899173x_{33} = -89.5688718899173
x33=−20.5652079398333x_{33} = -20.5652079398333
x33=45.6187613383417x_{33} = 45.6187613383417
x33=−33.0771723843072x_{33} = -33.0771723843072
x33=58.170990540028x_{33} = 58.170990540028
x33=89.5688718899173x_{33} = 89.5688718899173
x33=2.45564386287944x_{33} = 2.45564386287944
x33=20.5652079398333x_{33} = 20.5652079398333
x33=70.7282251775385x_{33} = 70.7282251775385
Decrece en los intervalos
[98.9904652640992,∞)\left[98.9904652640992, \infty\right)
Crece en los intervalos
(−∞,−98.9904652640992]\left(-\infty, -98.9904652640992\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
x(−x2sin⁡(x)+6xcos⁡(x)+6sin⁡(x))=0x \left(- x^{2} \sin{\left(x \right)} + 6 x \cos{\left(x \right)} + 6 \sin{\left(x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−1.81453497473617x_{1} = -1.81453497473617
x2=7.05208859318106x_{2} = 7.05208859318106
x3=69.2016332025015x_{3} = 69.2016332025015
x4=−19.1578008101567x_{4} = -19.1578008101567
x5=34.7294331656689x_{5} = 34.7294331656689
x6=84.893619603402x_{6} = 84.893619603402
x7=9.99322916735933x_{7} = 9.99322916735933
x8=−72.3394784349932x_{8} = -72.3394784349932
x9=37.85694574059x_{9} = 37.85694574059
x10=50.3842871966612x_{10} = 50.3842871966612
x11=−59.7904430977076x_{11} = -59.7904430977076
x12=−75.4776339039269x_{12} = -75.4776339039269
x13=62.9270575685711x_{13} = 62.9270575685711
x14=−50.3842871966612x_{14} = -50.3842871966612
x15=−97.4509028811532x_{15} = -97.4509028811532
x16=78.6160626870951x_{16} = 78.6160626870951
x17=4.26739380712901x_{17} = 4.26739380712901
x18=−16.0730074093467x_{18} = -16.0730074093467
x19=13.012298102066x_{19} = 13.012298102066
x20=−34.7294331656689x_{20} = -34.7294331656689
x21=−56.6543759268167x_{21} = -56.6543759268167
x22=28.4834495364194x_{22} = 28.4834495364194
x23=16.0730074093467x_{23} = 16.0730074093467
x24=−62.9270575685711x_{24} = -62.9270575685711
x25=−44.1178800161513x_{25} = -44.1178800161513
x26=−4.26739380712901x_{26} = -4.26739380712901
x27=22.2575395263247x_{27} = 22.2575395263247
x28=31.6046464917248x_{28} = 31.6046464917248
x29=−78.6160626870951x_{29} = -78.6160626870951
x30=0x_{30} = 0
x31=−25.3671067403905x_{31} = -25.3671067403905
x32=66.064142073588x_{32} = 66.064142073588
x33=−28.4834495364194x_{33} = -28.4834495364194
x34=−31.6046464917248x_{34} = -31.6046464917248
x35=1.81453497473617x_{35} = 1.81453497473617
x36=−7.05208859318106x_{36} = -7.05208859318106
x37=53.5189511635138x_{37} = 53.5189511635138
x38=−69.2016332025015x_{38} = -69.2016332025015
x39=−53.5189511635138x_{39} = -53.5189511635138
x40=−40.9865751123733x_{40} = -40.9865751123733
x41=−88.0326981155368x_{41} = -88.0326981155368
x42=−100.590577325313x_{42} = -100.590577325313
x43=81.7547334875609x_{43} = 81.7547334875609
x44=40.9865751123733x_{44} = 40.9865751123733
x45=−66.064142073588x_{45} = -66.064142073588
x46=100.590577325313x_{46} = 100.590577325313
x47=72.3394784349932x_{47} = 72.3394784349932
x48=−22.2575395263247x_{48} = -22.2575395263247
x49=56.6543759268167x_{49} = 56.6543759268167
x50=−37.85694574059x_{50} = -37.85694574059
x51=−91.1719492416891x_{51} = -91.1719492416891
x52=44.1178800161513x_{52} = 44.1178800161513
x53=75.4776339039269x_{53} = 75.4776339039269
x54=−9.99322916735933x_{54} = -9.99322916735933
x55=−81.7547334875609x_{55} = -81.7547334875609
x56=−47.2505332434495x_{56} = -47.2505332434495
x57=88.0326981155368x_{57} = 88.0326981155368
x58=−94.3113558182004x_{58} = -94.3113558182004
x59=−13.012298102066x_{59} = -13.012298102066
x60=97.4509028811532x_{60} = 97.4509028811532
x61=19.1578008101567x_{61} = 19.1578008101567
x62=25.3671067403905x_{62} = 25.3671067403905
x63=94.3113558182004x_{63} = 94.3113558182004
x64=47.2505332434495x_{64} = 47.2505332434495
x65=91.1719492416891x_{65} = 91.1719492416891
x66=−84.893619603402x_{66} = -84.893619603402
x67=59.7904430977076x_{67} = 59.7904430977076

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[97.4509028811532,∞)\left[97.4509028811532, \infty\right)
Convexa en los intervalos
(−∞,−100.590577325313]\left(-\infty, -100.590577325313\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x3sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x^{3} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(x3sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x^{3} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x^3*sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(x2sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x^{2} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
lim⁡x→∞(x2sin⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x^{2} \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
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:
x3sin⁡(x)=x3sin⁡(x)x^{3} \sin{\left(x \right)} = x^{3} \sin{\left(x \right)}
- Sí
x3sin⁡(x)=−x3sin⁡(x)x^{3} \sin{\left(x \right)} = - x^{3} \sin{\left(x \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = x^3*sin(x)
    © Sr Examen – Калькулятор