Sr Examen
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- 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
-
¿Cómo usar?
- Gráfico de la función y =:
-
-sqrt(-1-x^2)
-
7-x-2*x^2
-
y=x^3+x
-
y=(x^3)/(x^2-4)
-
Expresiones idénticas
- sinx\(x(x- dos))
- seno de x\(x(x menos 2))
- seno de x\(x(x menos dos))
- sinx\xx-2
-
Expresiones semejantes
- sinx\(x(x+2))
-
Expresiones con funciones
- sinx
- sinx*sinx+cosx
- sinx+sin2x
- sinx:2
- sinx/(x^2-4)
- sinx(π/2)
Gráfico de la función y = sinx\(x(x-2))
Solución
Ha introducido
[src]
sin(x)
f(x) = ---------
x*(x - 2)
$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{x \left(x - 2\right)}$$
f = sin(x)/((x*(x - 2)))
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{2} = 2$$
$$x_{1} = 0$$
$$x_{2} = 2$$
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:
$$\frac{\sin{\left(x \right)}}{x \left(x - 2\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \pi$$
Solución numérica
$$x_{1} = -138.230076757951$$
$$x_{2} = 40.8407044966673$$
$$x_{3} = -18.8495559215388$$
$$x_{4} = -56.5486677646163$$
$$x_{5} = 97.3893722612836$$
$$x_{6} = 34.5575191894877$$
$$x_{7} = 53.4070751110265$$
$$x_{8} = 47.1238898038469$$
$$x_{9} = -97.3893722612836$$
$$x_{10} = 62.8318530717959$$
$$x_{11} = 87.9645943005142$$
$$x_{12} = 43.9822971502571$$
$$x_{13} = 37.6991118430775$$
$$x_{14} = -21.9911485751286$$
$$x_{15} = 65.9734457253857$$
$$x_{16} = 69.1150383789755$$
$$x_{17} = -50.2654824574367$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = -75.398223686155$$
$$x_{20} = -53.4070751110265$$
$$x_{21} = 12.5663706143592$$
$$x_{22} = -9.42477796076938$$
$$x_{23} = -8033.0524152291$$
$$x_{24} = -703.716754404114$$
$$x_{25} = -34.5575191894877$$
$$x_{26} = 21.9911485751286$$
$$x_{27} = -116.238928182822$$
$$x_{28} = -47.1238898038469$$
$$x_{29} = -43.9822971502571$$
$$x_{30} = 28.2743338823081$$
$$x_{31} = -31.4159265358979$$
$$x_{32} = -3.14159265358979$$
$$x_{33} = -6.28318530717959$$
$$x_{34} = -25.1327412287183$$
$$x_{35} = -62.8318530717959$$
$$x_{36} = 31.4159265358979$$
$$x_{37} = -65.9734457253857$$
$$x_{38} = 72.2566310325652$$
$$x_{39} = -59.6902604182061$$
$$x_{40} = 94.2477796076938$$
$$x_{41} = 81.6814089933346$$
$$x_{42} = -91.106186954104$$
$$x_{43} = -100.530964914873$$
$$x_{44} = 59.6902604182061$$
$$x_{45} = -40.8407044966673$$
$$x_{46} = 91.106186954104$$
$$x_{47} = 78.5398163397448$$
$$x_{48} = -12.5663706143592$$
$$x_{49} = 56.5486677646163$$
$$x_{50} = 84.8230016469244$$
$$x_{51} = 100.530964914873$$
$$x_{52} = -69.1150383789755$$
$$x_{53} = 9.42477796076938$$
$$x_{54} = 116.238928182822$$
$$x_{55} = -84.8230016469244$$
$$x_{56} = -78.5398163397448$$
$$x_{57} = -87.9645943005142$$
$$x_{58} = -81.6814089933346$$
$$x_{59} = 15.707963267949$$
$$x_{60} = -28.2743338823081$$
$$x_{61} = -15.707963267949$$
$$x_{62} = -37.6991118430775$$
$$x_{63} = 18.8495559215388$$
$$x_{64} = 25.1327412287183$$
$$x_{65} = 50.2654824574367$$
$$x_{66} = -72.2566310325652$$
$$x_{67} = 75.398223686155$$
$$x_{68} = 6.28318530717959$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(x \right)}}{x \left(x - 2\right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \pi$$
Solución numérica
$$x_{1} = -138.230076757951$$
$$x_{2} = 40.8407044966673$$
$$x_{3} = -18.8495559215388$$
$$x_{4} = -56.5486677646163$$
$$x_{5} = 97.3893722612836$$
$$x_{6} = 34.5575191894877$$
$$x_{7} = 53.4070751110265$$
$$x_{8} = 47.1238898038469$$
$$x_{9} = -97.3893722612836$$
$$x_{10} = 62.8318530717959$$
$$x_{11} = 87.9645943005142$$
$$x_{12} = 43.9822971502571$$
$$x_{13} = 37.6991118430775$$
$$x_{14} = -21.9911485751286$$
$$x_{15} = 65.9734457253857$$
$$x_{16} = 69.1150383789755$$
$$x_{17} = -50.2654824574367$$
$$x_{18} = -94.2477796076938$$
$$x_{19} = -75.398223686155$$
$$x_{20} = -53.4070751110265$$
$$x_{21} = 12.5663706143592$$
$$x_{22} = -9.42477796076938$$
$$x_{23} = -8033.0524152291$$
$$x_{24} = -703.716754404114$$
$$x_{25} = -34.5575191894877$$
$$x_{26} = 21.9911485751286$$
$$x_{27} = -116.238928182822$$
$$x_{28} = -47.1238898038469$$
$$x_{29} = -43.9822971502571$$
$$x_{30} = 28.2743338823081$$
$$x_{31} = -31.4159265358979$$
$$x_{32} = -3.14159265358979$$
$$x_{33} = -6.28318530717959$$
$$x_{34} = -25.1327412287183$$
$$x_{35} = -62.8318530717959$$
$$x_{36} = 31.4159265358979$$
$$x_{37} = -65.9734457253857$$
$$x_{38} = 72.2566310325652$$
$$x_{39} = -59.6902604182061$$
$$x_{40} = 94.2477796076938$$
$$x_{41} = 81.6814089933346$$
$$x_{42} = -91.106186954104$$
$$x_{43} = -100.530964914873$$
$$x_{44} = 59.6902604182061$$
$$x_{45} = -40.8407044966673$$
$$x_{46} = 91.106186954104$$
$$x_{47} = 78.5398163397448$$
$$x_{48} = -12.5663706143592$$
$$x_{49} = 56.5486677646163$$
$$x_{50} = 84.8230016469244$$
$$x_{51} = 100.530964914873$$
$$x_{52} = -69.1150383789755$$
$$x_{53} = 9.42477796076938$$
$$x_{54} = 116.238928182822$$
$$x_{55} = -84.8230016469244$$
$$x_{56} = -78.5398163397448$$
$$x_{57} = -87.9645943005142$$
$$x_{58} = -81.6814089933346$$
$$x_{59} = 15.707963267949$$
$$x_{60} = -28.2743338823081$$
$$x_{61} = -15.707963267949$$
$$x_{62} = -37.6991118430775$$
$$x_{63} = 18.8495559215388$$
$$x_{64} = 25.1327412287183$$
$$x_{65} = 50.2654824574367$$
$$x_{66} = -72.2566310325652$$
$$x_{67} = 75.398223686155$$
$$x_{68} = 6.28318530717959$$
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{1}{x \left(x - 2\right)} \cos{\left(x \right)} + \frac{\left(2 - 2 x\right) \sin{\left(x \right)}}{x^{2} \left(x - 2\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 32.9240947361922$$
$$x_{2} = -51.7984033793852$$
$$x_{3} = 17.1551175692195$$
$$x_{4} = 36.0713042845073$$
$$x_{5} = -98.9401572801928$$
$$x_{6} = -20.3266414701876$$
$$x_{7} = 39.2175881820314$$
$$x_{8} = -42.3653892667121$$
$$x_{9} = -80.0859487309835$$
$$x_{10} = 45.5081654611057$$
$$x_{11} = -10.8268690995624$$
$$x_{12} = 80.0853248723334$$
$$x_{13} = -70.6579261366639$$
$$x_{14} = 67.5141755390553$$
$$x_{15} = -7.62307729555873$$
$$x_{16} = 95.7974791095636$$
$$x_{17} = -17.1687916817231$$
$$x_{18} = -58.0856181381215$$
$$x_{19} = -186.914119829292$$
$$x_{20} = -92.6556292628207$$
$$x_{21} = 10.7920322357124$$
$$x_{22} = 190.055776752248$$
$$x_{23} = -76.943361762789$$
$$x_{24} = 4.08557388547682$$
$$x_{25} = -86.3709080585407$$
$$x_{26} = 83.2278838695937$$
$$x_{27} = 64.3710918861937$$
$$x_{28} = -32.927791200115$$
$$x_{29} = -256.032020175128$$
$$x_{30} = -36.0743829885085$$
$$x_{31} = -61.2289201135729$$
$$x_{32} = -83.2284614928992$$
$$x_{33} = -39.220192145926$$
$$x_{34} = -73.8006912312722$$
$$x_{35} = -23.4802919264971$$
$$x_{36} = 51.7969113967309$$
$$x_{37} = 29.7756549714707$$
$$x_{38} = 7.55102453615362$$
$$x_{39} = 48.6527219697238$$
$$x_{40} = 58.0844318525335$$
$$x_{41} = -26.6311871536774$$
$$x_{42} = -64.3720576734236$$
$$x_{43} = 76.9426858848596$$
$$x_{44} = 61.2278525685536$$
$$x_{45} = 23.4730079186956$$
$$x_{46} = -89.5132953248997$$
$$x_{47} = 73.7999565431799$$
$$x_{48} = 92.6551632265069$$
$$x_{49} = 26.6255299708041$$
$$x_{50} = -95.7979150674353$$
$$x_{51} = -29.7801761695834$$
$$x_{52} = -48.6544131811461$$
$$x_{53} = 13.9834279458844$$
$$x_{54} = 89.5127959855887$$
$$x_{55} = 98.9397485795284$$
$$x_{56} = 86.3703717136003$$
$$x_{57} = -14.0040665914265$$
$$x_{58} = 70.6571246114187$$
$$x_{59} = 20.3169083532025$$
$$x_{60} = -45.5100986876418$$
$$x_{61} = 1371.30373376177$$
$$x_{62} = -488.513571950338$$
$$x_{63} = 54.9407979981311$$
$$x_{64} = -67.5150534592896$$
$$x_{65} = 42.3631580330254$$
$$x_{66} = -54.9421240104386$$
$$x_{67} = -4.34230123285199$$
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:
$$x_{1} = -51.7984033793852$$
$$x_{2} = 17.1551175692195$$
$$x_{3} = 36.0713042845073$$
$$x_{4} = -20.3266414701876$$
$$x_{5} = 80.0853248723334$$
$$x_{6} = -70.6579261366639$$
$$x_{7} = 67.5141755390553$$
$$x_{8} = -7.62307729555873$$
$$x_{9} = -58.0856181381215$$
$$x_{10} = 10.7920322357124$$
$$x_{11} = -76.943361762789$$
$$x_{12} = 4.08557388547682$$
$$x_{13} = -32.927791200115$$
$$x_{14} = -83.2284614928992$$
$$x_{15} = -39.220192145926$$
$$x_{16} = 29.7756549714707$$
$$x_{17} = 48.6527219697238$$
$$x_{18} = -26.6311871536774$$
$$x_{19} = -64.3720576734236$$
$$x_{20} = 61.2278525685536$$
$$x_{21} = 23.4730079186956$$
$$x_{22} = -89.5132953248997$$
$$x_{23} = 73.7999565431799$$
$$x_{24} = 92.6551632265069$$
$$x_{25} = -95.7979150674353$$
$$x_{26} = 98.9397485795284$$
$$x_{27} = 86.3703717136003$$
$$x_{28} = -14.0040665914265$$
$$x_{29} = -45.5100986876418$$
$$x_{30} = 54.9407979981311$$
$$x_{31} = 42.3631580330254$$
Puntos máximos de la función:
$$x_{31} = 32.9240947361922$$
$$x_{31} = -98.9401572801928$$
$$x_{31} = 39.2175881820314$$
$$x_{31} = -42.3653892667121$$
$$x_{31} = -80.0859487309835$$
$$x_{31} = 45.5081654611057$$
$$x_{31} = -10.8268690995624$$
$$x_{31} = 95.7974791095636$$
$$x_{31} = -17.1687916817231$$
$$x_{31} = -186.914119829292$$
$$x_{31} = -92.6556292628207$$
$$x_{31} = 190.055776752248$$
$$x_{31} = -86.3709080585407$$
$$x_{31} = 83.2278838695937$$
$$x_{31} = 64.3710918861937$$
$$x_{31} = -256.032020175128$$
$$x_{31} = -36.0743829885085$$
$$x_{31} = -61.2289201135729$$
$$x_{31} = -73.8006912312722$$
$$x_{31} = -23.4802919264971$$
$$x_{31} = 51.7969113967309$$
$$x_{31} = 7.55102453615362$$
$$x_{31} = 58.0844318525335$$
$$x_{31} = 76.9426858848596$$
$$x_{31} = 26.6255299708041$$
$$x_{31} = -29.7801761695834$$
$$x_{31} = -48.6544131811461$$
$$x_{31} = 13.9834279458844$$
$$x_{31} = 89.5127959855887$$
$$x_{31} = 70.6571246114187$$
$$x_{31} = 20.3169083532025$$
$$x_{31} = 1371.30373376177$$
$$x_{31} = -488.513571950338$$
$$x_{31} = -67.5150534592896$$
$$x_{31} = -54.9421240104386$$
$$x_{31} = -4.34230123285199$$
Decrece en los intervalos
$$\left[98.9397485795284, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.7979150674353\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{1}{x \left(x - 2\right)} \cos{\left(x \right)} + \frac{\left(2 - 2 x\right) \sin{\left(x \right)}}{x^{2} \left(x - 2\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 32.9240947361922$$
$$x_{2} = -51.7984033793852$$
$$x_{3} = 17.1551175692195$$
$$x_{4} = 36.0713042845073$$
$$x_{5} = -98.9401572801928$$
$$x_{6} = -20.3266414701876$$
$$x_{7} = 39.2175881820314$$
$$x_{8} = -42.3653892667121$$
$$x_{9} = -80.0859487309835$$
$$x_{10} = 45.5081654611057$$
$$x_{11} = -10.8268690995624$$
$$x_{12} = 80.0853248723334$$
$$x_{13} = -70.6579261366639$$
$$x_{14} = 67.5141755390553$$
$$x_{15} = -7.62307729555873$$
$$x_{16} = 95.7974791095636$$
$$x_{17} = -17.1687916817231$$
$$x_{18} = -58.0856181381215$$
$$x_{19} = -186.914119829292$$
$$x_{20} = -92.6556292628207$$
$$x_{21} = 10.7920322357124$$
$$x_{22} = 190.055776752248$$
$$x_{23} = -76.943361762789$$
$$x_{24} = 4.08557388547682$$
$$x_{25} = -86.3709080585407$$
$$x_{26} = 83.2278838695937$$
$$x_{27} = 64.3710918861937$$
$$x_{28} = -32.927791200115$$
$$x_{29} = -256.032020175128$$
$$x_{30} = -36.0743829885085$$
$$x_{31} = -61.2289201135729$$
$$x_{32} = -83.2284614928992$$
$$x_{33} = -39.220192145926$$
$$x_{34} = -73.8006912312722$$
$$x_{35} = -23.4802919264971$$
$$x_{36} = 51.7969113967309$$
$$x_{37} = 29.7756549714707$$
$$x_{38} = 7.55102453615362$$
$$x_{39} = 48.6527219697238$$
$$x_{40} = 58.0844318525335$$
$$x_{41} = -26.6311871536774$$
$$x_{42} = -64.3720576734236$$
$$x_{43} = 76.9426858848596$$
$$x_{44} = 61.2278525685536$$
$$x_{45} = 23.4730079186956$$
$$x_{46} = -89.5132953248997$$
$$x_{47} = 73.7999565431799$$
$$x_{48} = 92.6551632265069$$
$$x_{49} = 26.6255299708041$$
$$x_{50} = -95.7979150674353$$
$$x_{51} = -29.7801761695834$$
$$x_{52} = -48.6544131811461$$
$$x_{53} = 13.9834279458844$$
$$x_{54} = 89.5127959855887$$
$$x_{55} = 98.9397485795284$$
$$x_{56} = 86.3703717136003$$
$$x_{57} = -14.0040665914265$$
$$x_{58} = 70.6571246114187$$
$$x_{59} = 20.3169083532025$$
$$x_{60} = -45.5100986876418$$
$$x_{61} = 1371.30373376177$$
$$x_{62} = -488.513571950338$$
$$x_{63} = 54.9407979981311$$
$$x_{64} = -67.5150534592896$$
$$x_{65} = 42.3631580330254$$
$$x_{66} = -54.9421240104386$$
$$x_{67} = -4.34230123285199$$
Signos de extremos en los puntos:
(32.92409473619218, 0.000980250092940406)
(-51.79840337938525, -0.000358593701721455)
(17.155117569219506, -0.00381697128504473)
(36.07130428450728, -0.000812350070903317)
(-98.94015728019278, 0.00010010976999049)
(-20.326641470187568, -0.00219382132146474)
(39.21758818203142, 0.000684189186675888)
(-42.365389266712086, 0.000531474734651108)
(-80.08594873098346, 0.000152069720255664)
(45.50816546110567, 0.000504546775560921)
(-10.826869099562385, 0.00709850000065266)
(80.08532487233344, -0.000159859614862926)
(-70.6579261366639, -0.000194709418827277)
(67.51417553905529, -0.000225981778238015)
(-7.623077295558733, -0.0132700859723812)
(95.79747910956362, 0.000111264878610497)
(-17.16879168172315, 0.00302019005045724)
(-58.08561813812148, -0.000286359800368288)
(-186.91411982929165, 2.83184105065328e-5)
(-92.65562926282065, 0.000113994190199678)
(10.792032235712393, -0.0103216328869677)
(190.055776752248, 2.7977438203328e-5)
(-76.94336176278898, -0.000164577420962203)
(4.085573885476822, -0.0950501114395919)
(-86.37090805854075, 0.000130981335117623)
(83.22788386959367, 0.00014787594936823)
(64.37109188619371, 0.000248948485979724)
(-32.92779120011495, -0.000867983965173453)
(-256.0320201751279, 1.51362735214327e-5)
(-36.074382988508454, 0.00072700314182954)
(-61.22892011357289, 0.000258168549826435)
(-83.22846149289921, -0.000140935711429685)
(-39.220192145926006, -0.000617793494390805)
(-73.80069123127224, 0.000178694491361609)
(-23.480291926497095, 0.00166587622836872)
(51.79691139673087, 0.000387397763643712)
(29.775654971470725, -0.00120621686240201)
(7.551024536153622, 0.0227707843461169)
(48.652721969723785, -0.000440183098167783)
(58.08443185253347, 0.00030678306311284)
(-26.631187153677416, -0.00130807441032797)
(-64.37205767342357, -0.000233945166452643)
(76.94268588485963, 0.000173361556566348)
(61.227852568553566, -0.000275604010475087)
(23.473007918695554, -0.0019761436581552)
(-89.51329532489973, -0.000122045631771871)
(73.79995654317989, -0.000188649571321626)
(92.65516322650693, -0.000119023970882221)
(26.625529970804127, 0.00152052452751166)
(-95.79791506743528, -0.000106714060670071)
(-29.78017616958336, 0.00105438602731828)
(-48.654413181146104, 0.000405422814201224)
(13.983427945884438, 0.00589729038830167)
(89.51279598558865, 0.000127624028951783)
(98.93974857952836, -0.000104240565063727)
(86.3703717136003, -0.000137191141528712)
(-14.004066591426517, -0.00442239117312082)
(70.65712461141868, 0.000206053225221036)
(20.316908353202518, 0.00267277521726431)
(-45.51009868764183, -0.000462066792988833)
(1371.3037337617723, 5.32556986595609e-7)
(-488.51357195033813, 4.17319533301501e-6)
(54.940797998131075, -0.00034357062848065)
(-67.5150534592896, 0.000212978367092361)
(42.363158033025364, -0.000584142570509146)
(-54.942124010438576, 0.000319435622287472)
(-4.3423012328519945, 0.033852194422576)
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:
$$x_{1} = -51.7984033793852$$
$$x_{2} = 17.1551175692195$$
$$x_{3} = 36.0713042845073$$
$$x_{4} = -20.3266414701876$$
$$x_{5} = 80.0853248723334$$
$$x_{6} = -70.6579261366639$$
$$x_{7} = 67.5141755390553$$
$$x_{8} = -7.62307729555873$$
$$x_{9} = -58.0856181381215$$
$$x_{10} = 10.7920322357124$$
$$x_{11} = -76.943361762789$$
$$x_{12} = 4.08557388547682$$
$$x_{13} = -32.927791200115$$
$$x_{14} = -83.2284614928992$$
$$x_{15} = -39.220192145926$$
$$x_{16} = 29.7756549714707$$
$$x_{17} = 48.6527219697238$$
$$x_{18} = -26.6311871536774$$
$$x_{19} = -64.3720576734236$$
$$x_{20} = 61.2278525685536$$
$$x_{21} = 23.4730079186956$$
$$x_{22} = -89.5132953248997$$
$$x_{23} = 73.7999565431799$$
$$x_{24} = 92.6551632265069$$
$$x_{25} = -95.7979150674353$$
$$x_{26} = 98.9397485795284$$
$$x_{27} = 86.3703717136003$$
$$x_{28} = -14.0040665914265$$
$$x_{29} = -45.5100986876418$$
$$x_{30} = 54.9407979981311$$
$$x_{31} = 42.3631580330254$$
Puntos máximos de la función:
$$x_{31} = 32.9240947361922$$
$$x_{31} = -98.9401572801928$$
$$x_{31} = 39.2175881820314$$
$$x_{31} = -42.3653892667121$$
$$x_{31} = -80.0859487309835$$
$$x_{31} = 45.5081654611057$$
$$x_{31} = -10.8268690995624$$
$$x_{31} = 95.7974791095636$$
$$x_{31} = -17.1687916817231$$
$$x_{31} = -186.914119829292$$
$$x_{31} = -92.6556292628207$$
$$x_{31} = 190.055776752248$$
$$x_{31} = -86.3709080585407$$
$$x_{31} = 83.2278838695937$$
$$x_{31} = 64.3710918861937$$
$$x_{31} = -256.032020175128$$
$$x_{31} = -36.0743829885085$$
$$x_{31} = -61.2289201135729$$
$$x_{31} = -73.8006912312722$$
$$x_{31} = -23.4802919264971$$
$$x_{31} = 51.7969113967309$$
$$x_{31} = 7.55102453615362$$
$$x_{31} = 58.0844318525335$$
$$x_{31} = 76.9426858848596$$
$$x_{31} = 26.6255299708041$$
$$x_{31} = -29.7801761695834$$
$$x_{31} = -48.6544131811461$$
$$x_{31} = 13.9834279458844$$
$$x_{31} = 89.5127959855887$$
$$x_{31} = 70.6571246114187$$
$$x_{31} = 20.3169083532025$$
$$x_{31} = 1371.30373376177$$
$$x_{31} = -488.513571950338$$
$$x_{31} = -67.5150534592896$$
$$x_{31} = -54.9421240104386$$
$$x_{31} = -4.34230123285199$$
Decrece en los intervalos
$$\left[98.9397485795284, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.7979150674353\right]$$
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 94.2048551569328$$
$$x_{2} = 37.5896554681014$$
$$x_{3} = -94.2057576185378$$
$$x_{4} = -18.6450684444873$$
$$x_{5} = 21.79808611927$$
$$x_{6} = -65.9136449449715$$
$$x_{7} = 91.0617638658387$$
$$x_{8} = 8.90587840675485$$
$$x_{9} = -28.1367815741316$$
$$x_{10} = 50.1840993820307$$
$$x_{11} = -97.3486936418635$$
$$x_{12} = 15.4285406261646$$
$$x_{13} = 81.6317880542076$$
$$x_{14} = 59.6219935715278$$
$$x_{15} = 47.0369347933021$$
$$x_{16} = -84.7763581906396$$
$$x_{17} = -59.6242503189044$$
$$x_{18} = -109.919675859272$$
$$x_{19} = -21.815268529783$$
$$x_{20} = -9.01907137188613$$
$$x_{21} = 87.9185640627454$$
$$x_{22} = 65.9117992853432$$
$$x_{23} = -50.1872883920623$$
$$x_{24} = 75.3444038224365$$
$$x_{25} = 62.7670653675399$$
$$x_{26} = -47.0405667699869$$
$$x_{27} = 97.347848573433$$
$$x_{28} = -31.2918587569744$$
$$x_{29} = -12.2619358385377$$
$$x_{30} = 84.7752435095284$$
$$x_{31} = -87.9196003632094$$
$$x_{32} = -37.5953566524211$$
$$x_{33} = -53.3334140575947$$
$$x_{34} = -116.204795746496$$
$$x_{35} = -40.7447926388754$$
$$x_{36} = 298.437853650273$$
$$x_{37} = 31.2836024135663$$
$$x_{38} = 40.7399440312312$$
$$x_{39} = 100.490753405596$$
$$x_{40} = -122.489718589867$$
$$x_{41} = -91.0627297768666$$
$$x_{42} = -78.4894819481931$$
$$x_{43} = -141.343566076059$$
$$x_{44} = 24.9653511953722$$
$$x_{45} = 43.8889486335222$$
$$x_{46} = -75.345815531326$$
$$x_{47} = 18.6213492879788$$
$$x_{48} = 56.4765261792329$$
$$x_{49} = -69.0579233879794$$
$$x_{50} = 5.2766637040869$$
$$x_{51} = 78.4881812412389$$
$$x_{52} = -34.4445173164217$$
$$x_{53} = 28.126543898197$$
$$x_{54} = -100.491546389739$$
$$x_{55} = -62.7691010739784$$
$$x_{56} = -376.980535751818$$
$$x_{57} = -5.66149669154895$$
$$x_{58} = -24.9783881327673$$
$$x_{59} = -56.4790421008596$$
$$x_{60} = -81.6329903654042$$
$$x_{61} = 69.056242322581$$
$$x_{62} = 34.4377158397736$$
$$x_{63} = -72.2019707064293$$
$$x_{64} = 12.2047300409779$$
$$x_{65} = -43.8931230440976$$
$$x_{66} = -103.63432337662$$
$$x_{67} = 53.3305915039441$$
$$x_{68} = 72.2004331376217$$
$$x_{69} = -15.4635207898039$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$x_{2} = 2$$
$$\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\right) = - \frac{1}{12}$$
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\right) = - \frac{1}{12}$$
- los límites son iguales, es decir omitimos el punto correspondiente
$$\lim_{x \to 2^-}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\right) = \infty$$
- los límites no son iguales, signo
$$x_{2} = 2$$
- es el punto de flexión
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
$$\left[298.437853650273, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -141.343566076059\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
segunda derivada
$$\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 94.2048551569328$$
$$x_{2} = 37.5896554681014$$
$$x_{3} = -94.2057576185378$$
$$x_{4} = -18.6450684444873$$
$$x_{5} = 21.79808611927$$
$$x_{6} = -65.9136449449715$$
$$x_{7} = 91.0617638658387$$
$$x_{8} = 8.90587840675485$$
$$x_{9} = -28.1367815741316$$
$$x_{10} = 50.1840993820307$$
$$x_{11} = -97.3486936418635$$
$$x_{12} = 15.4285406261646$$
$$x_{13} = 81.6317880542076$$
$$x_{14} = 59.6219935715278$$
$$x_{15} = 47.0369347933021$$
$$x_{16} = -84.7763581906396$$
$$x_{17} = -59.6242503189044$$
$$x_{18} = -109.919675859272$$
$$x_{19} = -21.815268529783$$
$$x_{20} = -9.01907137188613$$
$$x_{21} = 87.9185640627454$$
$$x_{22} = 65.9117992853432$$
$$x_{23} = -50.1872883920623$$
$$x_{24} = 75.3444038224365$$
$$x_{25} = 62.7670653675399$$
$$x_{26} = -47.0405667699869$$
$$x_{27} = 97.347848573433$$
$$x_{28} = -31.2918587569744$$
$$x_{29} = -12.2619358385377$$
$$x_{30} = 84.7752435095284$$
$$x_{31} = -87.9196003632094$$
$$x_{32} = -37.5953566524211$$
$$x_{33} = -53.3334140575947$$
$$x_{34} = -116.204795746496$$
$$x_{35} = -40.7447926388754$$
$$x_{36} = 298.437853650273$$
$$x_{37} = 31.2836024135663$$
$$x_{38} = 40.7399440312312$$
$$x_{39} = 100.490753405596$$
$$x_{40} = -122.489718589867$$
$$x_{41} = -91.0627297768666$$
$$x_{42} = -78.4894819481931$$
$$x_{43} = -141.343566076059$$
$$x_{44} = 24.9653511953722$$
$$x_{45} = 43.8889486335222$$
$$x_{46} = -75.345815531326$$
$$x_{47} = 18.6213492879788$$
$$x_{48} = 56.4765261792329$$
$$x_{49} = -69.0579233879794$$
$$x_{50} = 5.2766637040869$$
$$x_{51} = 78.4881812412389$$
$$x_{52} = -34.4445173164217$$
$$x_{53} = 28.126543898197$$
$$x_{54} = -100.491546389739$$
$$x_{55} = -62.7691010739784$$
$$x_{56} = -376.980535751818$$
$$x_{57} = -5.66149669154895$$
$$x_{58} = -24.9783881327673$$
$$x_{59} = -56.4790421008596$$
$$x_{60} = -81.6329903654042$$
$$x_{61} = 69.056242322581$$
$$x_{62} = 34.4377158397736$$
$$x_{63} = -72.2019707064293$$
$$x_{64} = 12.2047300409779$$
$$x_{65} = -43.8931230440976$$
$$x_{66} = -103.63432337662$$
$$x_{67} = 53.3305915039441$$
$$x_{68} = 72.2004331376217$$
$$x_{69} = -15.4635207898039$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$x_{2} = 2$$
$$\lim_{x \to 0^-}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\right) = - \frac{1}{12}$$
$$\lim_{x \to 0^+}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\right) = - \frac{1}{12}$$
- los límites son iguales, es decir omitimos el punto correspondiente
$$\lim_{x \to 2^-}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{- \sin{\left(x \right)} - \frac{4 \left(x - 1\right) \cos{\left(x \right)}}{x \left(x - 2\right)} + \frac{2 \left(\left(x - 1\right) \left(\frac{1}{x - 2} + \frac{1}{x}\right) - 1 + \frac{x - 1}{x - 2} + \frac{x - 1}{x}\right) \sin{\left(x \right)}}{x \left(x - 2\right)}}{x \left(x - 2\right)}\right) = \infty$$
- los límites no son iguales, signo
$$x_{2} = 2$$
- es el punto de flexión
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
$$\left[298.437853650273, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -141.343566076059\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{2} = 2$$
$$x_{1} = 0$$
$$x_{2} = 2$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(x - 2\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(x - 2\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(x - 2\right)}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(x - 2\right)}\right) = 0$$
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(x)/((x*(x - 2))), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{x \left(x - 2\right)} \sin{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{1}{x \left(x - 2\right)} \sin{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\frac{1}{x \left(x - 2\right)} \sin{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\frac{1}{x \left(x - 2\right)} \sin{\left(x \right)}}{x}\right) = 0$$
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:
$$\frac{\sin{\left(x \right)}}{x \left(x - 2\right)} = \frac{\sin{\left(x \right)}}{x \left(- x - 2\right)}$$
- No
$$\frac{\sin{\left(x \right)}}{x \left(x - 2\right)} = - \frac{\sin{\left(x \right)}}{x \left(- x - 2\right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(x \right)}}{x \left(x - 2\right)} = \frac{\sin{\left(x \right)}}{x \left(- x - 2\right)}$$
- No
$$\frac{\sin{\left(x \right)}}{x \left(x - 2\right)} = - \frac{\sin{\left(x \right)}}{x \left(- x - 2\right)}$$
- No
es decir, función
no es
par ni impar