Sr Examen
Otras calculadoras
- Gráfico de la función implícita
- Derivadas paso a paso
- Integrales paso a paso
- Gráfico de la función paramétrica
- Gráfico en coordenadas polares
- Superficie especificada paramétricamente
- Superficie dada por la ecuación
- Curva en el espacio especificada paramétricamente
- Superficie de una figura limitada con líneas
- Puntos de cruce de las funciones
3/(x^2+1)
(1/3)^x
x/(x^3+2)
y=2x-3
Gráfico de la función y = siny/y^3
Solución
Ha introducido
[src]
sin(y)
f(y) = ------
3
y
$$f{\left(y \right)} = \frac{\sin{\left(y \right)}}{y^{3}}$$
f = sin(y)/y^3
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$y_{1} = 0$$
$$y_{1} = 0$$
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje Y con f = 0
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(y \right)}}{y^{3}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Y:
Solución analítica
$$y_{1} = \pi$$
Solución numérica
$$y_{1} = 31.4159265358979$$
$$y_{2} = 3.14159265358979$$
$$y_{3} = -47.1238898038469$$
$$y_{4} = -12.5663706143592$$
$$y_{5} = -34.5575191894877$$
$$y_{6} = -69.1150383789755$$
$$y_{7} = 75.398223686155$$
$$y_{8} = -65.9734457253857$$
$$y_{9} = -50.2654824574367$$
$$y_{10} = -56.5486677646163$$
$$y_{11} = 59.6902604182061$$
$$y_{12} = 141.371669411541$$
$$y_{13} = 72.2566310325652$$
$$y_{14} = 91.106186954104$$
$$y_{15} = -91.106186954104$$
$$y_{16} = -109.955742875643$$
$$y_{17} = -62.8318530717959$$
$$y_{18} = -6.28318530717959$$
$$y_{19} = 6.28318530717959$$
$$y_{20} = 62.8318530717959$$
$$y_{21} = -25.1327412287183$$
$$y_{22} = 94.2477796076938$$
$$y_{23} = -9.42477796076938$$
$$y_{24} = -37.6991118430775$$
$$y_{25} = 65.9734457253857$$
$$y_{26} = -100.530964914873$$
$$y_{27} = -43.9822971502571$$
$$y_{28} = 25.1327412287183$$
$$y_{29} = 21.9911485751286$$
$$y_{30} = 87.9645943005142$$
$$y_{31} = -40.8407044966673$$
$$y_{32} = -97.3893722612836$$
$$y_{33} = 43.9822971502571$$
$$y_{34} = -53.4070751110265$$
$$y_{35} = 97.3893722612836$$
$$y_{36} = 100.530964914873$$
$$y_{37} = -94.2477796076938$$
$$y_{38} = -31.4159265358979$$
$$y_{39} = 18.8495559215388$$
$$y_{40} = 78.5398163397448$$
$$y_{41} = -18.8495559215388$$
$$y_{42} = 53.4070751110265$$
$$y_{43} = 47.1238898038469$$
$$y_{44} = 12.5663706143592$$
$$y_{45} = 81.6814089933346$$
$$y_{46} = 34.5575191894877$$
$$y_{47} = -75.398223686155$$
$$y_{48} = -15.707963267949$$
$$y_{49} = 50.2654824574367$$
$$y_{50} = -81.6814089933346$$
$$y_{51} = -3.14159265358979$$
$$y_{52} = -59.6902604182061$$
$$y_{53} = -28.2743338823081$$
$$y_{54} = -87.9645943005142$$
$$y_{55} = 9.42477796076938$$
$$y_{56} = -21.9911485751286$$
$$y_{57} = 56.5486677646163$$
$$y_{58} = 15.707963267949$$
$$y_{59} = 84.8230016469244$$
$$y_{60} = -78.5398163397448$$
$$y_{61} = 37.6991118430775$$
$$y_{62} = -72.2566310325652$$
$$y_{63} = -84.8230016469244$$
$$y_{64} = 69.1150383789755$$
$$y_{65} = 28.2743338823081$$
$$y_{66} = 40.8407044966673$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(y \right)}}{y^{3}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje Y:
Solución analítica
$$y_{1} = \pi$$
Solución numérica
$$y_{1} = 31.4159265358979$$
$$y_{2} = 3.14159265358979$$
$$y_{3} = -47.1238898038469$$
$$y_{4} = -12.5663706143592$$
$$y_{5} = -34.5575191894877$$
$$y_{6} = -69.1150383789755$$
$$y_{7} = 75.398223686155$$
$$y_{8} = -65.9734457253857$$
$$y_{9} = -50.2654824574367$$
$$y_{10} = -56.5486677646163$$
$$y_{11} = 59.6902604182061$$
$$y_{12} = 141.371669411541$$
$$y_{13} = 72.2566310325652$$
$$y_{14} = 91.106186954104$$
$$y_{15} = -91.106186954104$$
$$y_{16} = -109.955742875643$$
$$y_{17} = -62.8318530717959$$
$$y_{18} = -6.28318530717959$$
$$y_{19} = 6.28318530717959$$
$$y_{20} = 62.8318530717959$$
$$y_{21} = -25.1327412287183$$
$$y_{22} = 94.2477796076938$$
$$y_{23} = -9.42477796076938$$
$$y_{24} = -37.6991118430775$$
$$y_{25} = 65.9734457253857$$
$$y_{26} = -100.530964914873$$
$$y_{27} = -43.9822971502571$$
$$y_{28} = 25.1327412287183$$
$$y_{29} = 21.9911485751286$$
$$y_{30} = 87.9645943005142$$
$$y_{31} = -40.8407044966673$$
$$y_{32} = -97.3893722612836$$
$$y_{33} = 43.9822971502571$$
$$y_{34} = -53.4070751110265$$
$$y_{35} = 97.3893722612836$$
$$y_{36} = 100.530964914873$$
$$y_{37} = -94.2477796076938$$
$$y_{38} = -31.4159265358979$$
$$y_{39} = 18.8495559215388$$
$$y_{40} = 78.5398163397448$$
$$y_{41} = -18.8495559215388$$
$$y_{42} = 53.4070751110265$$
$$y_{43} = 47.1238898038469$$
$$y_{44} = 12.5663706143592$$
$$y_{45} = 81.6814089933346$$
$$y_{46} = 34.5575191894877$$
$$y_{47} = -75.398223686155$$
$$y_{48} = -15.707963267949$$
$$y_{49} = 50.2654824574367$$
$$y_{50} = -81.6814089933346$$
$$y_{51} = -3.14159265358979$$
$$y_{52} = -59.6902604182061$$
$$y_{53} = -28.2743338823081$$
$$y_{54} = -87.9645943005142$$
$$y_{55} = 9.42477796076938$$
$$y_{56} = -21.9911485751286$$
$$y_{57} = 56.5486677646163$$
$$y_{58} = 15.707963267949$$
$$y_{59} = 84.8230016469244$$
$$y_{60} = -78.5398163397448$$
$$y_{61} = 37.6991118430775$$
$$y_{62} = -72.2566310325652$$
$$y_{63} = -84.8230016469244$$
$$y_{64} = 69.1150383789755$$
$$y_{65} = 28.2743338823081$$
$$y_{66} = 40.8407044966673$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d y} f{\left(y \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d y} f{\left(y \right)} = $$
primera derivada
$$\frac{\cos{\left(y \right)}}{y^{3}} - \frac{3 \sin{\left(y \right)}}{y^{4}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$y_{1} = -108.357267428671$$
$$y_{2} = -92.6446127847888$$
$$y_{3} = -86.359073259985$$
$$y_{4} = -45.4872362867621$$
$$y_{5} = 17.1051395364267$$
$$y_{6} = 58.0678462801751$$
$$y_{7} = -186.908713650658$$
$$y_{8} = -13.924969952549$$
$$y_{9} = 92.6446127847888$$
$$y_{10} = 89.5018843244389$$
$$y_{11} = 64.3560674689022$$
$$y_{12} = -17.1051395364267$$
$$y_{13} = -42.3407653325706$$
$$y_{14} = -51.7784042739041$$
$$y_{15} = 48.6330777853047$$
$$y_{16} = -89.5018843244389$$
$$y_{17} = -83.2161702400252$$
$$y_{18} = -70.6433933906731$$
$$y_{19} = 13.924969952549$$
$$y_{20} = -7.47219265966058$$
$$y_{21} = 54.9233040395155$$
$$y_{22} = 7.47219265966058$$
$$y_{23} = 86.359073259985$$
$$y_{24} = 76.930043294192$$
$$y_{25} = -76.930043294192$$
$$y_{26} = -10.7227710626892$$
$$y_{27} = -95.7872667660245$$
$$y_{28} = -29.7446115259422$$
$$y_{29} = 80.0731644462726$$
$$y_{30} = 29.7446115259422$$
$$y_{31} = -32.8957773192946$$
$$y_{32} = 95.7872667660245$$
$$y_{33} = 32.8957773192946$$
$$y_{34} = -39.1935138550425$$
$$y_{35} = -23.4346216921802$$
$$y_{36} = 51.7784042739041$$
$$y_{37} = 70.6433933906731$$
$$y_{38} = -4.07814976485137$$
$$y_{39} = 36.0452782424582$$
$$y_{40} = 10.7227710626892$$
$$y_{41} = 83.2161702400252$$
$$y_{42} = -26.591193287969$$
$$y_{43} = 4.07814976485137$$
$$y_{44} = 98.9298533613919$$
$$y_{45} = 42.3407653325706$$
$$y_{46} = 39.1935138550425$$
$$y_{47} = -48.6330777853047$$
$$y_{48} = 61.2120859995403$$
$$y_{49} = 3215.41914794509$$
$$y_{50} = 67.4998267230665$$
$$y_{51} = 23.4346216921802$$
$$y_{52} = -54.9233040395155$$
$$y_{53} = 45.4872362867621$$
$$y_{54} = -80.0731644462726$$
$$y_{55} = -36.0452782424582$$
$$y_{56} = -73.7867920572034$$
$$y_{57} = -61.2120859995403$$
$$y_{58} = 73.7867920572034$$
$$y_{59} = -58.0678462801751$$
$$y_{60} = 26.591193287969$$
$$y_{61} = 20.2734415170608$$
$$y_{62} = -98.9298533613919$$
$$y_{63} = -20.2734415170608$$
$$y_{64} = -64.3560674689022$$
$$y_{65} = -67.4998267230665$$
Signos de extremos en los puntos:
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:
$$y_{1} = -92.6446127847888$$
$$y_{2} = -86.359073259985$$
$$y_{3} = 17.1051395364267$$
$$y_{4} = -186.908713650658$$
$$y_{5} = 92.6446127847888$$
$$y_{6} = -17.1051395364267$$
$$y_{7} = -42.3407653325706$$
$$y_{8} = 48.6330777853047$$
$$y_{9} = 54.9233040395155$$
$$y_{10} = 86.359073259985$$
$$y_{11} = -10.7227710626892$$
$$y_{12} = -29.7446115259422$$
$$y_{13} = 80.0731644462726$$
$$y_{14} = 29.7446115259422$$
$$y_{15} = -23.4346216921802$$
$$y_{16} = -4.07814976485137$$
$$y_{17} = 36.0452782424582$$
$$y_{18} = 10.7227710626892$$
$$y_{19} = 4.07814976485137$$
$$y_{20} = 98.9298533613919$$
$$y_{21} = 42.3407653325706$$
$$y_{22} = -48.6330777853047$$
$$y_{23} = 61.2120859995403$$
$$y_{24} = 3215.41914794509$$
$$y_{25} = 67.4998267230665$$
$$y_{26} = 23.4346216921802$$
$$y_{27} = -54.9233040395155$$
$$y_{28} = -80.0731644462726$$
$$y_{29} = -36.0452782424582$$
$$y_{30} = -73.7867920572034$$
$$y_{31} = -61.2120859995403$$
$$y_{32} = 73.7867920572034$$
$$y_{33} = -98.9298533613919$$
$$y_{34} = -67.4998267230665$$
Puntos máximos de la función:
$$y_{34} = -108.357267428671$$
$$y_{34} = -45.4872362867621$$
$$y_{34} = 58.0678462801751$$
$$y_{34} = -13.924969952549$$
$$y_{34} = 89.5018843244389$$
$$y_{34} = 64.3560674689022$$
$$y_{34} = -51.7784042739041$$
$$y_{34} = -89.5018843244389$$
$$y_{34} = -83.2161702400252$$
$$y_{34} = -70.6433933906731$$
$$y_{34} = 13.924969952549$$
$$y_{34} = -7.47219265966058$$
$$y_{34} = 7.47219265966058$$
$$y_{34} = 76.930043294192$$
$$y_{34} = -76.930043294192$$
$$y_{34} = -95.7872667660245$$
$$y_{34} = -32.8957773192946$$
$$y_{34} = 95.7872667660245$$
$$y_{34} = 32.8957773192946$$
$$y_{34} = -39.1935138550425$$
$$y_{34} = 51.7784042739041$$
$$y_{34} = 70.6433933906731$$
$$y_{34} = 83.2161702400252$$
$$y_{34} = -26.591193287969$$
$$y_{34} = 39.1935138550425$$
$$y_{34} = 45.4872362867621$$
$$y_{34} = -58.0678462801751$$
$$y_{34} = 26.591193287969$$
$$y_{34} = 20.2734415170608$$
$$y_{34} = -20.2734415170608$$
$$y_{34} = -64.3560674689022$$
Decrece en los intervalos
$$\left[3215.41914794509, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -186.908713650658\right]$$
$$\frac{d}{d y} f{\left(y \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d y} f{\left(y \right)} = $$
primera derivada
$$\frac{\cos{\left(y \right)}}{y^{3}} - \frac{3 \sin{\left(y \right)}}{y^{4}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$y_{1} = -108.357267428671$$
$$y_{2} = -92.6446127847888$$
$$y_{3} = -86.359073259985$$
$$y_{4} = -45.4872362867621$$
$$y_{5} = 17.1051395364267$$
$$y_{6} = 58.0678462801751$$
$$y_{7} = -186.908713650658$$
$$y_{8} = -13.924969952549$$
$$y_{9} = 92.6446127847888$$
$$y_{10} = 89.5018843244389$$
$$y_{11} = 64.3560674689022$$
$$y_{12} = -17.1051395364267$$
$$y_{13} = -42.3407653325706$$
$$y_{14} = -51.7784042739041$$
$$y_{15} = 48.6330777853047$$
$$y_{16} = -89.5018843244389$$
$$y_{17} = -83.2161702400252$$
$$y_{18} = -70.6433933906731$$
$$y_{19} = 13.924969952549$$
$$y_{20} = -7.47219265966058$$
$$y_{21} = 54.9233040395155$$
$$y_{22} = 7.47219265966058$$
$$y_{23} = 86.359073259985$$
$$y_{24} = 76.930043294192$$
$$y_{25} = -76.930043294192$$
$$y_{26} = -10.7227710626892$$
$$y_{27} = -95.7872667660245$$
$$y_{28} = -29.7446115259422$$
$$y_{29} = 80.0731644462726$$
$$y_{30} = 29.7446115259422$$
$$y_{31} = -32.8957773192946$$
$$y_{32} = 95.7872667660245$$
$$y_{33} = 32.8957773192946$$
$$y_{34} = -39.1935138550425$$
$$y_{35} = -23.4346216921802$$
$$y_{36} = 51.7784042739041$$
$$y_{37} = 70.6433933906731$$
$$y_{38} = -4.07814976485137$$
$$y_{39} = 36.0452782424582$$
$$y_{40} = 10.7227710626892$$
$$y_{41} = 83.2161702400252$$
$$y_{42} = -26.591193287969$$
$$y_{43} = 4.07814976485137$$
$$y_{44} = 98.9298533613919$$
$$y_{45} = 42.3407653325706$$
$$y_{46} = 39.1935138550425$$
$$y_{47} = -48.6330777853047$$
$$y_{48} = 61.2120859995403$$
$$y_{49} = 3215.41914794509$$
$$y_{50} = 67.4998267230665$$
$$y_{51} = 23.4346216921802$$
$$y_{52} = -54.9233040395155$$
$$y_{53} = 45.4872362867621$$
$$y_{54} = -80.0731644462726$$
$$y_{55} = -36.0452782424582$$
$$y_{56} = -73.7867920572034$$
$$y_{57} = -61.2120859995403$$
$$y_{58} = 73.7867920572034$$
$$y_{59} = -58.0678462801751$$
$$y_{60} = 26.591193287969$$
$$y_{61} = 20.2734415170608$$
$$y_{62} = -98.9298533613919$$
$$y_{63} = -20.2734415170608$$
$$y_{64} = -64.3560674689022$$
$$y_{65} = -67.4998267230665$$
Signos de extremos en los puntos:
(-108.35726742867121, 7.85704936475449e-7)
(-92.64461278478876, -1.25693237122389e-6)
(-86.359073259985, -1.55172297524868e-6)
(-45.48723628676209, 1.06020259212848e-5)
(17.105139536426744, -0.000196807350078387)
(58.067846280175104, 5.1005149012716e-6)
(-186.90871365065752, -1.53128285331065e-7)
(-13.92496995254897, 0.000362047304723488)
(92.64461278478876, -1.25693237122389e-6)
(89.50188432443886, 1.39398991793862e-6)
(64.35606746890217, 3.7476597286644e-6)
(-17.105139536426744, -0.000196807350078387)
(-42.34076533257061, -1.31412441116291e-5)
(-51.77840427390411, 7.1916125507828e-6)
(48.63307778530466, -8.67720806398467e-6)
(-89.50188432443886, 1.39398991793862e-6)
(-83.21617024002518, 1.73418247352317e-6)
(-70.64339339067311, 2.83396240646379e-6)
(13.92496995254897, 0.000362047304723488)
(-7.472192659660579, 0.00222435242575847)
(54.92330403951548, -6.02674942320184e-6)
(7.472192659660579, 0.00222435242575847)
(86.359073259985, -1.55172297524868e-6)
(76.93004329419203, 2.19473504875366e-6)
(-76.93004329419203, 2.19473504875366e-6)
(-10.722771062689203, -0.00078111312666525)
(-95.78726676602449, 1.1372704641271e-6)
(-29.744611525942226, -3.78074454700618e-5)
(80.07316444627257, -1.94641047170913e-6)
(29.744611525942226, -3.78074454700618e-5)
(-32.89577731929462, 2.79757033726946e-5)
(95.78726676602449, 1.1372704641271e-6)
(32.89577731929462, 2.79757033726946e-5)
(-39.193513855042454, 1.65610881918558e-5)
(-23.4346216921802, -7.70719574291125e-5)
(51.77840427390411, 7.1916125507828e-6)
(70.64339339067311, 2.83396240646379e-6)
(-4.078149764851372, -0.0118764951343876)
(36.04527824245817, -2.12792275158681e-5)
(10.722771062689203, -0.00078111312666525)
(83.21617024002518, 1.73418247352317e-6)
(-26.59119328796898, 5.28494009082656e-5)
(4.078149764851372, -0.0118764951343876)
(98.9298533613919, -1.03232943885669e-6)
(42.34076533257061, -1.31412441116291e-5)
(39.193513855042454, 1.65610881918558e-5)
(-48.63307778530466, -8.67720806398467e-6)
(61.21208599954033, -4.35479288166118e-6)
(3215.41914794509, -3.00806370783767e-11)
(67.49982672306646, -3.24835521431348e-6)
(23.4346216921802, -7.70719574291125e-5)
(-54.92330403951548, -6.02674942320184e-6)
(45.48723628676209, 1.06020259212848e-5)
(-80.07316444627257, -1.94641047170913e-6)
(-36.04527824245817, -2.12792275158681e-5)
(-73.78679205720341, -2.48716990126569e-6)
(-61.21208599954033, -4.35479288166118e-6)
(73.78679205720341, -2.48716990126569e-6)
(-58.067846280175104, 5.1005149012716e-6)
(26.59119328796898, 5.28494009082656e-5)
(20.27344151706078, 0.000118717289027919)
(-98.9298533613919, -1.03232943885669e-6)
(-20.27344151706078, 0.000118717289027919)
(-64.35606746890217, 3.7476597286644e-6)
(-67.49982672306646, -3.24835521431348e-6)
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:
$$y_{1} = -92.6446127847888$$
$$y_{2} = -86.359073259985$$
$$y_{3} = 17.1051395364267$$
$$y_{4} = -186.908713650658$$
$$y_{5} = 92.6446127847888$$
$$y_{6} = -17.1051395364267$$
$$y_{7} = -42.3407653325706$$
$$y_{8} = 48.6330777853047$$
$$y_{9} = 54.9233040395155$$
$$y_{10} = 86.359073259985$$
$$y_{11} = -10.7227710626892$$
$$y_{12} = -29.7446115259422$$
$$y_{13} = 80.0731644462726$$
$$y_{14} = 29.7446115259422$$
$$y_{15} = -23.4346216921802$$
$$y_{16} = -4.07814976485137$$
$$y_{17} = 36.0452782424582$$
$$y_{18} = 10.7227710626892$$
$$y_{19} = 4.07814976485137$$
$$y_{20} = 98.9298533613919$$
$$y_{21} = 42.3407653325706$$
$$y_{22} = -48.6330777853047$$
$$y_{23} = 61.2120859995403$$
$$y_{24} = 3215.41914794509$$
$$y_{25} = 67.4998267230665$$
$$y_{26} = 23.4346216921802$$
$$y_{27} = -54.9233040395155$$
$$y_{28} = -80.0731644462726$$
$$y_{29} = -36.0452782424582$$
$$y_{30} = -73.7867920572034$$
$$y_{31} = -61.2120859995403$$
$$y_{32} = 73.7867920572034$$
$$y_{33} = -98.9298533613919$$
$$y_{34} = -67.4998267230665$$
Puntos máximos de la función:
$$y_{34} = -108.357267428671$$
$$y_{34} = -45.4872362867621$$
$$y_{34} = 58.0678462801751$$
$$y_{34} = -13.924969952549$$
$$y_{34} = 89.5018843244389$$
$$y_{34} = 64.3560674689022$$
$$y_{34} = -51.7784042739041$$
$$y_{34} = -89.5018843244389$$
$$y_{34} = -83.2161702400252$$
$$y_{34} = -70.6433933906731$$
$$y_{34} = 13.924969952549$$
$$y_{34} = -7.47219265966058$$
$$y_{34} = 7.47219265966058$$
$$y_{34} = 76.930043294192$$
$$y_{34} = -76.930043294192$$
$$y_{34} = -95.7872667660245$$
$$y_{34} = -32.8957773192946$$
$$y_{34} = 95.7872667660245$$
$$y_{34} = 32.8957773192946$$
$$y_{34} = -39.1935138550425$$
$$y_{34} = 51.7784042739041$$
$$y_{34} = 70.6433933906731$$
$$y_{34} = 83.2161702400252$$
$$y_{34} = -26.591193287969$$
$$y_{34} = 39.1935138550425$$
$$y_{34} = 45.4872362867621$$
$$y_{34} = -58.0678462801751$$
$$y_{34} = 26.591193287969$$
$$y_{34} = 20.2734415170608$$
$$y_{34} = -20.2734415170608$$
$$y_{34} = -64.3560674689022$$
Decrece en los intervalos
$$\left[3215.41914794509, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -186.908713650658\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0$$
(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:
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = $$
segunda derivada
$$\frac{- \sin{\left(y \right)} - \frac{6 \cos{\left(y \right)}}{y} + \frac{12 \sin{\left(y \right)}}{y^{2}}}{y^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$y_{1} = -141.329215347444$$
$$y_{2} = 72.1734981126022$$
$$y_{3} = 46.996220714232$$
$$y_{4} = -56.4423649364807$$
$$y_{5} = 5.15216592622293$$
$$y_{6} = 94.1840745935286$$
$$y_{7} = -113.044258982077$$
$$y_{8} = -78.4633475726977$$
$$y_{9} = 37.5392815729796$$
$$y_{10} = 87.8963320992957$$
$$y_{11} = -15.3164244547999$$
$$y_{12} = -18.5257597776303$$
$$y_{13} = -34.3830180083391$$
$$y_{14} = -12.0699013399661$$
$$y_{15} = 24.8917150033836$$
$$y_{16} = -59.5895718893518$$
$$y_{17} = -53.2944935242974$$
$$y_{18} = -94.1840745935286$$
$$y_{19} = -100.47124635336$$
$$y_{20} = 34.3830180083391$$
$$y_{21} = -69.028117387504$$
$$y_{22} = 373.833475850202$$
$$y_{23} = 31.223771021093$$
$$y_{24} = -65.8823744655118$$
$$y_{25} = 81.6078867352652$$
$$y_{26} = 40.6932614780489$$
$$y_{27} = 50.1458319794503$$
$$y_{28} = -84.7522070698598$$
$$y_{29} = 59.5895718893518$$
$$y_{30} = 53.2944935242974$$
$$y_{31} = 131.901402932105$$
$$y_{32} = -40.6932614780489$$
$$y_{33} = 21.7148754289157$$
$$y_{34} = -37.5392815729796$$
$$y_{35} = 75.31856211974$$
$$y_{36} = -81.6078867352652$$
$$y_{37} = 15.3164244547999$$
$$y_{38} = 43.8454539221714$$
$$y_{39} = 65.8823744655118$$
$$y_{40} = 56.4423649364807$$
$$y_{41} = -72.1734981126022$$
$$y_{42} = -75.31856211974$$
$$y_{43} = 8.74140008690353$$
$$y_{44} = -46.996220714232$$
$$y_{45} = -24.8917150033836$$
$$y_{46} = 69.028117387504$$
$$y_{47} = -28.0605201580983$$
$$y_{48} = 18.5257597776303$$
$$y_{49} = 62.7362147088964$$
$$y_{50} = -8.74140008690353$$
$$y_{51} = -21.7148754289157$$
$$y_{52} = -50.1458319794503$$
$$y_{53} = 84.7522070698598$$
$$y_{54} = 12.0699013399661$$
$$y_{55} = 28.0605201580983$$
$$y_{56} = 78.4633475726977$$
$$y_{57} = 91.0402820892519$$
$$y_{58} = -43.8454539221714$$
$$y_{59} = 97.3277248932537$$
$$y_{60} = -87.8963320992957$$
$$y_{61} = -31.223771021093$$
$$y_{62} = -5.15216592622293$$
$$y_{63} = 100.47124635336$$
$$y_{64} = -97.3277248932537$$
$$y_{65} = 116.187287429474$$
$$y_{66} = -62.7362147088964$$
$$y_{67} = -91.0402820892519$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$y_{1} = 0$$
$$\lim_{y \to 0^-}\left(\frac{- \sin{\left(y \right)} - \frac{6 \cos{\left(y \right)}}{y} + \frac{12 \sin{\left(y \right)}}{y^{2}}}{y^{3}}\right) = \infty$$
$$\lim_{y \to 0^+}\left(\frac{- \sin{\left(y \right)} - \frac{6 \cos{\left(y \right)}}{y} + \frac{12 \sin{\left(y \right)}}{y^{2}}}{y^{3}}\right) = \infty$$
- los límites son iguales, es decir omitimos el punto correspondiente
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
$$\left[373.833475850202, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -113.044258982077\right]$$
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = 0$$
(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:
$$\frac{d^{2}}{d y^{2}} f{\left(y \right)} = $$
segunda derivada
$$\frac{- \sin{\left(y \right)} - \frac{6 \cos{\left(y \right)}}{y} + \frac{12 \sin{\left(y \right)}}{y^{2}}}{y^{3}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$y_{1} = -141.329215347444$$
$$y_{2} = 72.1734981126022$$
$$y_{3} = 46.996220714232$$
$$y_{4} = -56.4423649364807$$
$$y_{5} = 5.15216592622293$$
$$y_{6} = 94.1840745935286$$
$$y_{7} = -113.044258982077$$
$$y_{8} = -78.4633475726977$$
$$y_{9} = 37.5392815729796$$
$$y_{10} = 87.8963320992957$$
$$y_{11} = -15.3164244547999$$
$$y_{12} = -18.5257597776303$$
$$y_{13} = -34.3830180083391$$
$$y_{14} = -12.0699013399661$$
$$y_{15} = 24.8917150033836$$
$$y_{16} = -59.5895718893518$$
$$y_{17} = -53.2944935242974$$
$$y_{18} = -94.1840745935286$$
$$y_{19} = -100.47124635336$$
$$y_{20} = 34.3830180083391$$
$$y_{21} = -69.028117387504$$
$$y_{22} = 373.833475850202$$
$$y_{23} = 31.223771021093$$
$$y_{24} = -65.8823744655118$$
$$y_{25} = 81.6078867352652$$
$$y_{26} = 40.6932614780489$$
$$y_{27} = 50.1458319794503$$
$$y_{28} = -84.7522070698598$$
$$y_{29} = 59.5895718893518$$
$$y_{30} = 53.2944935242974$$
$$y_{31} = 131.901402932105$$
$$y_{32} = -40.6932614780489$$
$$y_{33} = 21.7148754289157$$
$$y_{34} = -37.5392815729796$$
$$y_{35} = 75.31856211974$$
$$y_{36} = -81.6078867352652$$
$$y_{37} = 15.3164244547999$$
$$y_{38} = 43.8454539221714$$
$$y_{39} = 65.8823744655118$$
$$y_{40} = 56.4423649364807$$
$$y_{41} = -72.1734981126022$$
$$y_{42} = -75.31856211974$$
$$y_{43} = 8.74140008690353$$
$$y_{44} = -46.996220714232$$
$$y_{45} = -24.8917150033836$$
$$y_{46} = 69.028117387504$$
$$y_{47} = -28.0605201580983$$
$$y_{48} = 18.5257597776303$$
$$y_{49} = 62.7362147088964$$
$$y_{50} = -8.74140008690353$$
$$y_{51} = -21.7148754289157$$
$$y_{52} = -50.1458319794503$$
$$y_{53} = 84.7522070698598$$
$$y_{54} = 12.0699013399661$$
$$y_{55} = 28.0605201580983$$
$$y_{56} = 78.4633475726977$$
$$y_{57} = 91.0402820892519$$
$$y_{58} = -43.8454539221714$$
$$y_{59} = 97.3277248932537$$
$$y_{60} = -87.8963320992957$$
$$y_{61} = -31.223771021093$$
$$y_{62} = -5.15216592622293$$
$$y_{63} = 100.47124635336$$
$$y_{64} = -97.3277248932537$$
$$y_{65} = 116.187287429474$$
$$y_{66} = -62.7362147088964$$
$$y_{67} = -91.0402820892519$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$y_{1} = 0$$
$$\lim_{y \to 0^-}\left(\frac{- \sin{\left(y \right)} - \frac{6 \cos{\left(y \right)}}{y} + \frac{12 \sin{\left(y \right)}}{y^{2}}}{y^{3}}\right) = \infty$$
$$\lim_{y \to 0^+}\left(\frac{- \sin{\left(y \right)} - \frac{6 \cos{\left(y \right)}}{y} + \frac{12 \sin{\left(y \right)}}{y^{2}}}{y^{3}}\right) = \infty$$
- los límites son iguales, es decir omitimos el punto correspondiente
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
$$\left[373.833475850202, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -113.044258982077\right]$$
Asíntotas verticales
Hay:
$$y_{1} = 0$$
$$y_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con y->+oo y y->-oo
$$\lim_{y \to -\infty}\left(\frac{\sin{\left(y \right)}}{y^{3}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{y \to \infty}\left(\frac{\sin{\left(y \right)}}{y^{3}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{y \to -\infty}\left(\frac{\sin{\left(y \right)}}{y^{3}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{y \to \infty}\left(\frac{\sin{\left(y \right)}}{y^{3}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(y)/y^3, dividida por y con y->+oo y y ->-oo
$$\lim_{y \to -\infty}\left(\frac{\sin{\left(y \right)}}{y y^{3}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{y \to \infty}\left(\frac{\sin{\left(y \right)}}{y y^{3}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{y \to -\infty}\left(\frac{\sin{\left(y \right)}}{y y^{3}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{y \to \infty}\left(\frac{\sin{\left(y \right)}}{y y^{3}}\right) = 0$$
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(-y) и f = -f(-y).
Pues, comprobamos:
$$\frac{\sin{\left(y \right)}}{y^{3}} = \frac{\sin{\left(y \right)}}{y^{3}}$$
- No
$$\frac{\sin{\left(y \right)}}{y^{3}} = - \frac{\sin{\left(y \right)}}{y^{3}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(y \right)}}{y^{3}} = \frac{\sin{\left(y \right)}}{y^{3}}$$
- No
$$\frac{\sin{\left(y \right)}}{y^{3}} = - \frac{\sin{\left(y \right)}}{y^{3}}$$
- No
es decir, función
no es
par ni impar