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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 =:
-
x^3+3*x^2+2
-
-sqrt(3*x+1)
-
x*(-1-log(x))
-
x+1/(x-1)^2
-
Expresiones idénticas
- sinx/(uno -x)
- seno de x dividir por (1 menos x)
- seno de x dividir por (uno menos x)
- sinx/1-x
- sinx dividir por (1-x)
-
Expresiones semejantes
- sinx/(1-x^(sin2x))
- sinx/(1+x)
- sin(x)/(1-x^(2))
-
Expresiones con funciones
- sinx
- sinx-log(e,sin(x))
- sinx/(1-x^(sin2x))
- sinx/|x|
- sinx-|sinx|
- sinx+x*e^-x
Gráfico de la función y = sinx/(1-x)
Solución
Ha introducido
[src]
sin(x)
f(x) = ------
1 - x
$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{1 - x}$$
f = sin(x)/(1 - x)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 1$$
$$x_{1} = 1$$
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:
$$\frac{\sin{\left(x \right)}}{1 - x} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = \pi$$
Solución numérica
$$x_{1} = 62.8318530717959$$
$$x_{2} = -50.2654824574367$$
$$x_{3} = 47.1238898038469$$
$$x_{4} = 84.8230016469244$$
$$x_{5} = -53.4070751110265$$
$$x_{6} = 91.106186954104$$
$$x_{7} = -84.8230016469244$$
$$x_{8} = 113.097335529233$$
$$x_{9} = 25.1327412287183$$
$$x_{10} = -3.14159265358979$$
$$x_{11} = -6.28318530717959$$
$$x_{12} = -40.8407044966673$$
$$x_{13} = -18.8495559215388$$
$$x_{14} = -194.778744522567$$
$$x_{15} = 78.5398163397448$$
$$x_{16} = -75.398223686155$$
$$x_{17} = -9.42477796076938$$
$$x_{18} = 72.2566310325652$$
$$x_{19} = -43.9822971502571$$
$$x_{20} = 31.4159265358979$$
$$x_{21} = 9.42477796076938$$
$$x_{22} = 157.07963267949$$
$$x_{23} = 40.8407044966673$$
$$x_{24} = -69.1150383789755$$
$$x_{25} = -103.672557568463$$
$$x_{26} = 12.5663706143592$$
$$x_{27} = 87.9645943005142$$
$$x_{28} = 59.6902604182061$$
$$x_{29} = -37.6991118430775$$
$$x_{30} = -100.530964914873$$
$$x_{31} = -91.106186954104$$
$$x_{32} = 97.3893722612836$$
$$x_{33} = 0$$
$$x_{34} = -12.5663706143592$$
$$x_{35} = -78.5398163397448$$
$$x_{36} = 18.8495559215388$$
$$x_{37} = 34.5575191894877$$
$$x_{38} = -94.2477796076938$$
$$x_{39} = 43.9822971502571$$
$$x_{40} = -31.4159265358979$$
$$x_{41} = -81.6814089933346$$
$$x_{42} = -65.9734457253857$$
$$x_{43} = 75.398223686155$$
$$x_{44} = -188.495559215388$$
$$x_{45} = 56.5486677646163$$
$$x_{46} = 3.14159265358979$$
$$x_{47} = 15.707963267949$$
$$x_{48} = -56.5486677646163$$
$$x_{49} = -21.9911485751286$$
$$x_{50} = 50.2654824574367$$
$$x_{51} = -15.707963267949$$
$$x_{52} = 28.2743338823081$$
$$x_{53} = -320.442450666159$$
$$x_{54} = 94.2477796076938$$
$$x_{55} = -59.6902604182061$$
$$x_{56} = -62.8318530717959$$
$$x_{57} = 69.1150383789755$$
$$x_{58} = -34.5575191894877$$
$$x_{59} = -97.3893722612836$$
$$x_{60} = 21.9911485751286$$
$$x_{61} = 65.9734457253857$$
$$x_{62} = 37.6991118430775$$
$$x_{63} = -87.9645943005142$$
$$x_{64} = -72.2566310325652$$
$$x_{65} = 251.327412287183$$
$$x_{66} = -25.1327412287183$$
$$x_{67} = -307.8760800518$$
$$x_{68} = -28.2743338823081$$
$$x_{69} = 81.6814089933346$$
$$x_{70} = 6.28318530717959$$
$$x_{71} = 100.530964914873$$
$$x_{72} = 53.4070751110265$$
$$x_{73} = -47.1238898038469$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(x \right)}}{1 - x} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = \pi$$
Solución numérica
$$x_{1} = 62.8318530717959$$
$$x_{2} = -50.2654824574367$$
$$x_{3} = 47.1238898038469$$
$$x_{4} = 84.8230016469244$$
$$x_{5} = -53.4070751110265$$
$$x_{6} = 91.106186954104$$
$$x_{7} = -84.8230016469244$$
$$x_{8} = 113.097335529233$$
$$x_{9} = 25.1327412287183$$
$$x_{10} = -3.14159265358979$$
$$x_{11} = -6.28318530717959$$
$$x_{12} = -40.8407044966673$$
$$x_{13} = -18.8495559215388$$
$$x_{14} = -194.778744522567$$
$$x_{15} = 78.5398163397448$$
$$x_{16} = -75.398223686155$$
$$x_{17} = -9.42477796076938$$
$$x_{18} = 72.2566310325652$$
$$x_{19} = -43.9822971502571$$
$$x_{20} = 31.4159265358979$$
$$x_{21} = 9.42477796076938$$
$$x_{22} = 157.07963267949$$
$$x_{23} = 40.8407044966673$$
$$x_{24} = -69.1150383789755$$
$$x_{25} = -103.672557568463$$
$$x_{26} = 12.5663706143592$$
$$x_{27} = 87.9645943005142$$
$$x_{28} = 59.6902604182061$$
$$x_{29} = -37.6991118430775$$
$$x_{30} = -100.530964914873$$
$$x_{31} = -91.106186954104$$
$$x_{32} = 97.3893722612836$$
$$x_{33} = 0$$
$$x_{34} = -12.5663706143592$$
$$x_{35} = -78.5398163397448$$
$$x_{36} = 18.8495559215388$$
$$x_{37} = 34.5575191894877$$
$$x_{38} = -94.2477796076938$$
$$x_{39} = 43.9822971502571$$
$$x_{40} = -31.4159265358979$$
$$x_{41} = -81.6814089933346$$
$$x_{42} = -65.9734457253857$$
$$x_{43} = 75.398223686155$$
$$x_{44} = -188.495559215388$$
$$x_{45} = 56.5486677646163$$
$$x_{46} = 3.14159265358979$$
$$x_{47} = 15.707963267949$$
$$x_{48} = -56.5486677646163$$
$$x_{49} = -21.9911485751286$$
$$x_{50} = 50.2654824574367$$
$$x_{51} = -15.707963267949$$
$$x_{52} = 28.2743338823081$$
$$x_{53} = -320.442450666159$$
$$x_{54} = 94.2477796076938$$
$$x_{55} = -59.6902604182061$$
$$x_{56} = -62.8318530717959$$
$$x_{57} = 69.1150383789755$$
$$x_{58} = -34.5575191894877$$
$$x_{59} = -97.3893722612836$$
$$x_{60} = 21.9911485751286$$
$$x_{61} = 65.9734457253857$$
$$x_{62} = 37.6991118430775$$
$$x_{63} = -87.9645943005142$$
$$x_{64} = -72.2566310325652$$
$$x_{65} = 251.327412287183$$
$$x_{66} = -25.1327412287183$$
$$x_{67} = -307.8760800518$$
$$x_{68} = -28.2743338823081$$
$$x_{69} = 81.6814089933346$$
$$x_{70} = 6.28318530717959$$
$$x_{71} = 100.530964914873$$
$$x_{72} = 53.4070751110265$$
$$x_{73} = -47.1238898038469$$
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
$$\frac{\cos{\left(x \right)}}{1 - x} + \frac{\sin{\left(x \right)}}{\left(1 - x\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -54.9600034336304$$
$$x_{2} = -51.8173478727486$$
$$x_{3} = 10.8948536303641$$
$$x_{4} = 76.9558552315675$$
$$x_{5} = -29.8126874326337$$
$$x_{6} = -17.2239416992227$$
$$x_{7} = 51.8166027158994$$
$$x_{8} = 48.6737132861388$$
$$x_{9} = -39.2450655150134$$
$$x_{10} = 80.0979707906851$$
$$x_{11} = -14.0709110733623$$
$$x_{12} = 20.3687685412497$$
$$x_{13} = -98.9501639357993$$
$$x_{14} = 64.3868745704895$$
$$x_{15} = 42.3873435477815$$
$$x_{16} = 58.1019533456208$$
$$x_{17} = -95.8082466033645$$
$$x_{18} = -48.6745578196793$$
$$x_{19} = 98.9499596481$$
$$x_{20} = -67.5296508640044$$
$$x_{21} = 86.3820864505975$$
$$x_{22} = 61.2444592323791$$
$$x_{23} = 67.5292121928599$$
$$x_{24} = 111.517491107874$$
$$x_{25} = 23.5175642862085$$
$$x_{26} = -20.3735996512825$$
$$x_{27} = 45.5306408032197$$
$$x_{28} = -26.667409674396$$
$$x_{29} = 70.6714826164982$$
$$x_{30} = -32.9572826072601$$
$$x_{31} = -80.0982825740805$$
$$x_{32} = 73.8136945424488$$
$$x_{33} = -83.2403350796909$$
$$x_{34} = -64.3873571144002$$
$$x_{35} = -92.6663074889768$$
$$x_{36} = -36.1013688569472$$
$$x_{37} = -70.6718831382477$$
$$x_{38} = 32.9554394950363$$
$$x_{39} = 26.6645930692902$$
$$x_{40} = -42.3884572986485$$
$$x_{41} = -89.524344324422$$
$$x_{42} = -23.5211864259599$$
$$x_{43} = 14.0607507547713$$
$$x_{44} = -61.2449925776205$$
$$x_{45} = -73.8140616841155$$
$$x_{46} = -1.13226772527289$$
$$x_{47} = -7.74006134563749$$
$$x_{48} = 17.2171745487221$$
$$x_{49} = 89.5240947482585$$
$$x_{50} = 95.808028695694$$
$$x_{51} = -4.53360450162482$$
$$x_{52} = -76.9561929992537$$
$$x_{53} = -58.1025459608263$$
$$x_{54} = 39.2437660743313$$
$$x_{55} = 36.0998330569238$$
$$x_{56} = 29.8104344913478$$
$$x_{57} = 92.6660745522577$$
$$x_{58} = -45.5316060155777$$
$$x_{59} = 4.42859686541094$$
$$x_{60} = 7.705951184346$$
$$x_{61} = -86.3823545152058$$
$$x_{62} = -10.9118204503674$$
$$x_{63} = 124.084785514558$$
$$x_{64} = 83.2400463931404$$
$$x_{65} = 54.9593410866482$$
$$x_{66} = -127.226703940932$$
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:
$$x_{1} = -51.8173478727486$$
$$x_{2} = 76.9558552315675$$
$$x_{3} = 51.8166027158994$$
$$x_{4} = -39.2450655150134$$
$$x_{5} = -14.0709110733623$$
$$x_{6} = 20.3687685412497$$
$$x_{7} = 64.3868745704895$$
$$x_{8} = 58.1019533456208$$
$$x_{9} = -95.8082466033645$$
$$x_{10} = -20.3735996512825$$
$$x_{11} = 45.5306408032197$$
$$x_{12} = -26.667409674396$$
$$x_{13} = 70.6714826164982$$
$$x_{14} = -32.9572826072601$$
$$x_{15} = -83.2403350796909$$
$$x_{16} = -64.3873571144002$$
$$x_{17} = -70.6718831382477$$
$$x_{18} = 32.9554394950363$$
$$x_{19} = 26.6645930692902$$
$$x_{20} = -89.524344324422$$
$$x_{21} = 14.0607507547713$$
$$x_{22} = -1.13226772527289$$
$$x_{23} = -7.74006134563749$$
$$x_{24} = 89.5240947482585$$
$$x_{25} = 95.808028695694$$
$$x_{26} = -76.9561929992537$$
$$x_{27} = -58.1025459608263$$
$$x_{28} = 39.2437660743313$$
$$x_{29} = -45.5316060155777$$
$$x_{30} = 7.705951184346$$
$$x_{31} = 83.2400463931404$$
$$x_{32} = -127.226703940932$$
Puntos máximos de la función:
$$x_{32} = -54.9600034336304$$
$$x_{32} = 10.8948536303641$$
$$x_{32} = -29.8126874326337$$
$$x_{32} = -17.2239416992227$$
$$x_{32} = 48.6737132861388$$
$$x_{32} = 80.0979707906851$$
$$x_{32} = -98.9501639357993$$
$$x_{32} = 42.3873435477815$$
$$x_{32} = -48.6745578196793$$
$$x_{32} = 98.9499596481$$
$$x_{32} = -67.5296508640044$$
$$x_{32} = 86.3820864505975$$
$$x_{32} = 61.2444592323791$$
$$x_{32} = 67.5292121928599$$
$$x_{32} = 111.517491107874$$
$$x_{32} = 23.5175642862085$$
$$x_{32} = -80.0982825740805$$
$$x_{32} = 73.8136945424488$$
$$x_{32} = -92.6663074889768$$
$$x_{32} = -36.1013688569472$$
$$x_{32} = -42.3884572986485$$
$$x_{32} = -23.5211864259599$$
$$x_{32} = -61.2449925776205$$
$$x_{32} = -73.8140616841155$$
$$x_{32} = 17.2171745487221$$
$$x_{32} = -4.53360450162482$$
$$x_{32} = 36.0998330569238$$
$$x_{32} = 29.8104344913478$$
$$x_{32} = 92.6660745522577$$
$$x_{32} = 4.42859686541094$$
$$x_{32} = -86.3823545152058$$
$$x_{32} = -10.9118204503674$$
$$x_{32} = 124.084785514558$$
$$x_{32} = 54.9593410866482$$
Decrece en los intervalos
$$\left[95.808028695694, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -127.226703940932\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\cos{\left(x \right)}}{1 - x} + \frac{\sin{\left(x \right)}}{\left(1 - x\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -54.9600034336304$$
$$x_{2} = -51.8173478727486$$
$$x_{3} = 10.8948536303641$$
$$x_{4} = 76.9558552315675$$
$$x_{5} = -29.8126874326337$$
$$x_{6} = -17.2239416992227$$
$$x_{7} = 51.8166027158994$$
$$x_{8} = 48.6737132861388$$
$$x_{9} = -39.2450655150134$$
$$x_{10} = 80.0979707906851$$
$$x_{11} = -14.0709110733623$$
$$x_{12} = 20.3687685412497$$
$$x_{13} = -98.9501639357993$$
$$x_{14} = 64.3868745704895$$
$$x_{15} = 42.3873435477815$$
$$x_{16} = 58.1019533456208$$
$$x_{17} = -95.8082466033645$$
$$x_{18} = -48.6745578196793$$
$$x_{19} = 98.9499596481$$
$$x_{20} = -67.5296508640044$$
$$x_{21} = 86.3820864505975$$
$$x_{22} = 61.2444592323791$$
$$x_{23} = 67.5292121928599$$
$$x_{24} = 111.517491107874$$
$$x_{25} = 23.5175642862085$$
$$x_{26} = -20.3735996512825$$
$$x_{27} = 45.5306408032197$$
$$x_{28} = -26.667409674396$$
$$x_{29} = 70.6714826164982$$
$$x_{30} = -32.9572826072601$$
$$x_{31} = -80.0982825740805$$
$$x_{32} = 73.8136945424488$$
$$x_{33} = -83.2403350796909$$
$$x_{34} = -64.3873571144002$$
$$x_{35} = -92.6663074889768$$
$$x_{36} = -36.1013688569472$$
$$x_{37} = -70.6718831382477$$
$$x_{38} = 32.9554394950363$$
$$x_{39} = 26.6645930692902$$
$$x_{40} = -42.3884572986485$$
$$x_{41} = -89.524344324422$$
$$x_{42} = -23.5211864259599$$
$$x_{43} = 14.0607507547713$$
$$x_{44} = -61.2449925776205$$
$$x_{45} = -73.8140616841155$$
$$x_{46} = -1.13226772527289$$
$$x_{47} = -7.74006134563749$$
$$x_{48} = 17.2171745487221$$
$$x_{49} = 89.5240947482585$$
$$x_{50} = 95.808028695694$$
$$x_{51} = -4.53360450162482$$
$$x_{52} = -76.9561929992537$$
$$x_{53} = -58.1025459608263$$
$$x_{54} = 39.2437660743313$$
$$x_{55} = 36.0998330569238$$
$$x_{56} = 29.8104344913478$$
$$x_{57} = 92.6660745522577$$
$$x_{58} = -45.5316060155777$$
$$x_{59} = 4.42859686541094$$
$$x_{60} = 7.705951184346$$
$$x_{61} = -86.3823545152058$$
$$x_{62} = -10.9118204503674$$
$$x_{63} = 124.084785514558$$
$$x_{64} = 83.2400463931404$$
$$x_{65} = 54.9593410866482$$
$$x_{66} = -127.226703940932$$
Signos de extremos en los puntos:
(-54.96000343363045, 0.0178670534330114)
(-51.81734787274865, -0.0189297807617202)
(10.894853630364137, 0.100550447576556)
(76.95585523156745, -0.0131644011178422)
(-29.812687432633666, 0.0324370855827002)
(-17.22394169922269, 0.0547904450013511)
(51.81660271589941, -0.0196747987661075)
(48.67371328613878, 0.0209713070168763)
(-39.24506551501339, -0.0248400996327527)
(80.09797079068508, 0.0126415391261771)
(-14.07091107336234, -0.0662074029872918)
(20.368768541249715, -0.0515608337948906)
(-98.95016393579934, 0.0100044853805955)
(64.38687457048954, -0.0157741738609847)
(42.38734354778151, 0.0241549261563828)
(58.101953345620764, -0.0175098509285069)
(-95.8082466033645, -0.0103291474437141)
(-48.674557819679315, 0.0201269518294962)
(98.94995964810002, 0.0102087626459196)
(-67.52965086400441, 0.0145906704309162)
(86.38208645059747, 0.0117112553987014)
(61.24445923237913, 0.016596750592326)
(67.52921219285986, 0.0150292934612251)
(111.51749110787372, 0.0090479711061809)
(23.517564286208504, 0.0443660481840196)
(-20.373599651282518, -0.0467355668998296)
(45.5306408032197, -0.0224507873966093)
(-26.667409674395977, -0.0361200224684724)
(70.6714826164982, -0.014351596564099)
(-32.957282607260126, -0.0294360028315749)
(-80.09828257408046, 0.0123297800352262)
(73.8136945424488, 0.0137323852694885)
(-83.24033507969094, -0.0118699616825748)
(-64.38735711440016, -0.0152916881707175)
(-92.66630748897677, 0.0106755891321149)
(-36.101368856947204, 0.0269433983581047)
(-70.6718831382477, -0.0139511149237106)
(32.95543949503625, -0.0312782653330243)
(26.664593069290245, -0.0389346426276655)
(-42.38845729864853, 0.0230414855003745)
(-89.52434432442197, -0.0110460782420367)
(-23.521186425959854, 0.0407471918750439)
(14.06075075477126, -0.0763418368871013)
(-61.24499257762051, 0.0160634764762972)
(-73.81406168411554, 0.0133652773046089)
(-1.1322677252728852, -0.424607754243884)
(-7.740061345637487, -0.113674041782936)
(17.217174548722074, 0.0615461242858119)
(89.52409474825846, -0.0112956388323449)
(95.80802869569398, -0.0105470432425303)
(-4.53360450162482, 0.177833558557402)
(-76.95619299925373, -0.0128266619561237)
(-58.102545960826276, -0.0169173235363655)
(39.2437660743313, -0.0261391180182751)
(36.09983305692385, 0.0284786084467854)
(29.810434491347824, 0.0346887570981381)
(92.66607455225765, 0.0109085122854225)
(-45.53160601557773, -0.0214858080128856)
(4.428596865410944, 0.279998085534244)
(7.705951184345997, -0.147490409443988)
(-86.38235451520583, 0.0114432087563988)
(-10.911820450367392, 0.0836559534542903)
(124.08478551455839, 0.00812421286560894)
(83.24004639314036, -0.0121586273964131)
(54.959341086648216, 0.0185292907181696)
(-127.22670394093191, -0.00779845040767736)
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:
$$x_{1} = -51.8173478727486$$
$$x_{2} = 76.9558552315675$$
$$x_{3} = 51.8166027158994$$
$$x_{4} = -39.2450655150134$$
$$x_{5} = -14.0709110733623$$
$$x_{6} = 20.3687685412497$$
$$x_{7} = 64.3868745704895$$
$$x_{8} = 58.1019533456208$$
$$x_{9} = -95.8082466033645$$
$$x_{10} = -20.3735996512825$$
$$x_{11} = 45.5306408032197$$
$$x_{12} = -26.667409674396$$
$$x_{13} = 70.6714826164982$$
$$x_{14} = -32.9572826072601$$
$$x_{15} = -83.2403350796909$$
$$x_{16} = -64.3873571144002$$
$$x_{17} = -70.6718831382477$$
$$x_{18} = 32.9554394950363$$
$$x_{19} = 26.6645930692902$$
$$x_{20} = -89.524344324422$$
$$x_{21} = 14.0607507547713$$
$$x_{22} = -1.13226772527289$$
$$x_{23} = -7.74006134563749$$
$$x_{24} = 89.5240947482585$$
$$x_{25} = 95.808028695694$$
$$x_{26} = -76.9561929992537$$
$$x_{27} = -58.1025459608263$$
$$x_{28} = 39.2437660743313$$
$$x_{29} = -45.5316060155777$$
$$x_{30} = 7.705951184346$$
$$x_{31} = 83.2400463931404$$
$$x_{32} = -127.226703940932$$
Puntos máximos de la función:
$$x_{32} = -54.9600034336304$$
$$x_{32} = 10.8948536303641$$
$$x_{32} = -29.8126874326337$$
$$x_{32} = -17.2239416992227$$
$$x_{32} = 48.6737132861388$$
$$x_{32} = 80.0979707906851$$
$$x_{32} = -98.9501639357993$$
$$x_{32} = 42.3873435477815$$
$$x_{32} = -48.6745578196793$$
$$x_{32} = 98.9499596481$$
$$x_{32} = -67.5296508640044$$
$$x_{32} = 86.3820864505975$$
$$x_{32} = 61.2444592323791$$
$$x_{32} = 67.5292121928599$$
$$x_{32} = 111.517491107874$$
$$x_{32} = 23.5175642862085$$
$$x_{32} = -80.0982825740805$$
$$x_{32} = 73.8136945424488$$
$$x_{32} = -92.6663074889768$$
$$x_{32} = -36.1013688569472$$
$$x_{32} = -42.3884572986485$$
$$x_{32} = -23.5211864259599$$
$$x_{32} = -61.2449925776205$$
$$x_{32} = -73.8140616841155$$
$$x_{32} = 17.2171745487221$$
$$x_{32} = -4.53360450162482$$
$$x_{32} = 36.0998330569238$$
$$x_{32} = 29.8104344913478$$
$$x_{32} = 92.6660745522577$$
$$x_{32} = 4.42859686541094$$
$$x_{32} = -86.3823545152058$$
$$x_{32} = -10.9118204503674$$
$$x_{32} = 124.084785514558$$
$$x_{32} = 54.9593410866482$$
Decrece en los intervalos
$$\left[95.808028695694, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -127.226703940932\right]$$
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
$$\frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x - 1} - \frac{2 \sin{\left(x \right)}}{\left(x - 1\right)^{2}}}{x - 1} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x - 1} - \frac{2 \sin{\left(x \right)}}{\left(x - 1\right)^{2}}}{x - 1} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
$$x_{1} = 1$$
$$x_{1} = 1$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{1 - x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{1 - x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{1 - x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{1 - x}\right) = 0$$
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)/(1 - x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - x\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - x\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - x\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(1 - x\right)}\right) = 0$$
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:
$$\frac{\sin{\left(x \right)}}{1 - x} = - \frac{\sin{\left(x \right)}}{x + 1}$$
- No
$$\frac{\sin{\left(x \right)}}{1 - x} = \frac{\sin{\left(x \right)}}{x + 1}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(x \right)}}{1 - x} = - \frac{\sin{\left(x \right)}}{x + 1}$$
- No
$$\frac{\sin{\left(x \right)}}{1 - x} = \frac{\sin{\left(x \right)}}{x + 1}$$
- No
es decir, función
no es
par ni impar