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
-
¿Cómo usar?
- Gráfico de la función y =:
-
x/(1+x^2)
-
y^2+1
-
x/(x-1)
-
x/(x^2-1)
-
Expresiones idénticas
- sin(x)/(x- tres)
- seno de (x) dividir por (x menos 3)
- seno de (x) dividir por (x menos tres)
- sinx/x-3
- sin(x) dividir por (x-3)
-
Expresiones semejantes
- (sin(x)/x)-3
- sin(x)/(x+3)
- sinx/(x-3)
-
Expresiones con funciones
- Seno sin
- sin(x)^(3)
- sin(x/2)
- sin(2*x)+3
- sin(2*x+1)
- sinx-tgx
Gráfico de la función y = sin(x)/(x-3)
Solución
Ha introducido
[src]
sin(x)
f(x) = ------
x - 3
$$f{\left(x \right)} = \frac{\sin{\left(x \right)}}{x - 3}$$
f = sin(x)/(x - 3)
Gráfico de la función
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:
$$\frac{\sin{\left(x \right)}}{x - 3} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = \pi$$
Solución numérica
$$x_{1} = -59.6902604182061$$
$$x_{2} = -62.8318530717959$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 87.9645943005142$$
$$x_{5} = -56.5486677646163$$
$$x_{6} = 31.4159265358979$$
$$x_{7} = 69.1150383789755$$
$$x_{8} = -37.6991118430775$$
$$x_{9} = -81.6814089933346$$
$$x_{10} = -84.8230016469244$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = 47.1238898038469$$
$$x_{13} = -15.707963267949$$
$$x_{14} = -12.5663706143592$$
$$x_{15} = 12.5663706143592$$
$$x_{16} = -87.9645943005142$$
$$x_{17} = 53.4070751110265$$
$$x_{18} = 72.2566310325652$$
$$x_{19} = -100.530964914873$$
$$x_{20} = -3.14159265358979$$
$$x_{21} = 34.5575191894877$$
$$x_{22} = -94.2477796076938$$
$$x_{23} = 6.28318530717959$$
$$x_{24} = -69.1150383789755$$
$$x_{25} = 97.3893722612836$$
$$x_{26} = 0$$
$$x_{27} = 65.9734457253857$$
$$x_{28} = -50.2654824574367$$
$$x_{29} = 15.707963267949$$
$$x_{30} = -25.1327412287183$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = 40.8407044966673$$
$$x_{33} = 119.380520836412$$
$$x_{34} = -53.4070751110265$$
$$x_{35} = 37.6991118430775$$
$$x_{36} = -43.9822971502571$$
$$x_{37} = 18.8495559215388$$
$$x_{38} = -78.5398163397448$$
$$x_{39} = -6.28318530717959$$
$$x_{40} = -769.690200129499$$
$$x_{41} = -40.8407044966673$$
$$x_{42} = 43.9822971502571$$
$$x_{43} = 56.5486677646163$$
$$x_{44} = -65.9734457253857$$
$$x_{45} = 188.495559215388$$
$$x_{46} = 78.5398163397448$$
$$x_{47} = 25.1327412287183$$
$$x_{48} = -28.2743338823081$$
$$x_{49} = 75.398223686155$$
$$x_{50} = 59.6902604182061$$
$$x_{51} = -34.5575191894877$$
$$x_{52} = 81.6814089933346$$
$$x_{53} = -47.1238898038469$$
$$x_{54} = 100.530964914873$$
$$x_{55} = -9.42477796076938$$
$$x_{56} = -75.398223686155$$
$$x_{57} = -72.2566310325652$$
$$x_{58} = -31.4159265358979$$
$$x_{59} = 28.2743338823081$$
$$x_{60} = -91.106186954104$$
$$x_{61} = 21.9911485751286$$
$$x_{62} = 62.8318530717959$$
$$x_{63} = 9.42477796076938$$
$$x_{64} = 50.2654824574367$$
$$x_{65} = 94.2477796076938$$
$$x_{66} = 91.106186954104$$
$$x_{67} = 84.8230016469244$$
o sea hay que resolver la ecuación:
$$\frac{\sin{\left(x \right)}}{x - 3} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = \pi$$
Solución numérica
$$x_{1} = -59.6902604182061$$
$$x_{2} = -62.8318530717959$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 87.9645943005142$$
$$x_{5} = -56.5486677646163$$
$$x_{6} = 31.4159265358979$$
$$x_{7} = 69.1150383789755$$
$$x_{8} = -37.6991118430775$$
$$x_{9} = -81.6814089933346$$
$$x_{10} = -84.8230016469244$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = 47.1238898038469$$
$$x_{13} = -15.707963267949$$
$$x_{14} = -12.5663706143592$$
$$x_{15} = 12.5663706143592$$
$$x_{16} = -87.9645943005142$$
$$x_{17} = 53.4070751110265$$
$$x_{18} = 72.2566310325652$$
$$x_{19} = -100.530964914873$$
$$x_{20} = -3.14159265358979$$
$$x_{21} = 34.5575191894877$$
$$x_{22} = -94.2477796076938$$
$$x_{23} = 6.28318530717959$$
$$x_{24} = -69.1150383789755$$
$$x_{25} = 97.3893722612836$$
$$x_{26} = 0$$
$$x_{27} = 65.9734457253857$$
$$x_{28} = -50.2654824574367$$
$$x_{29} = 15.707963267949$$
$$x_{30} = -25.1327412287183$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = 40.8407044966673$$
$$x_{33} = 119.380520836412$$
$$x_{34} = -53.4070751110265$$
$$x_{35} = 37.6991118430775$$
$$x_{36} = -43.9822971502571$$
$$x_{37} = 18.8495559215388$$
$$x_{38} = -78.5398163397448$$
$$x_{39} = -6.28318530717959$$
$$x_{40} = -769.690200129499$$
$$x_{41} = -40.8407044966673$$
$$x_{42} = 43.9822971502571$$
$$x_{43} = 56.5486677646163$$
$$x_{44} = -65.9734457253857$$
$$x_{45} = 188.495559215388$$
$$x_{46} = 78.5398163397448$$
$$x_{47} = 25.1327412287183$$
$$x_{48} = -28.2743338823081$$
$$x_{49} = 75.398223686155$$
$$x_{50} = 59.6902604182061$$
$$x_{51} = -34.5575191894877$$
$$x_{52} = 81.6814089933346$$
$$x_{53} = -47.1238898038469$$
$$x_{54} = 100.530964914873$$
$$x_{55} = -9.42477796076938$$
$$x_{56} = -75.398223686155$$
$$x_{57} = -72.2566310325652$$
$$x_{58} = -31.4159265358979$$
$$x_{59} = 28.2743338823081$$
$$x_{60} = -91.106186954104$$
$$x_{61} = 21.9911485751286$$
$$x_{62} = 62.8318530717959$$
$$x_{63} = 9.42477796076938$$
$$x_{64} = 50.2654824574367$$
$$x_{65} = 94.2477796076938$$
$$x_{66} = 91.106186954104$$
$$x_{67} = 84.8230016469244$$
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
$$\frac{\cos{\left(x \right)}}{x - 3} - \frac{\sin{\left(x \right)}}{\left(x - 3\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 105.233572684616$$
$$x_{2} = -20.3776023235733$$
$$x_{3} = -76.9565138665894$$
$$x_{4} = -86.3826105819672$$
$$x_{5} = 64.3863605756806$$
$$x_{6} = -61.2454927075855$$
$$x_{7} = -54.9606200558715$$
$$x_{8} = -58.1030997704766$$
$$x_{9} = -1.34456392370501$$
$$x_{10} = -42.38947284675$$
$$x_{11} = -67.5300646513164$$
$$x_{12} = 54.958627731809$$
$$x_{13} = -92.6665306850573$$
$$x_{14} = 29.8078449830189$$
$$x_{15} = -45.5324916393212$$
$$x_{16} = 20.3628214780858$$
$$x_{17} = 67.528746323277$$
$$x_{18} = 86.3818055246036$$
$$x_{19} = 89.5238336327072$$
$$x_{20} = -39.2462418447496$$
$$x_{21} = 45.5295847647311$$
$$x_{22} = -32.9589205172515$$
$$x_{23} = -48.6753369544401$$
$$x_{24} = 42.3861166198228$$
$$x_{25} = -36.1027474386634$$
$$x_{26} = 98.9497468430673$$
$$x_{27} = -7.76132234788669$$
$$x_{28} = 23.5132344474784$$
$$x_{29} = -89.5245831096533$$
$$x_{30} = 95.807801395196$$
$$x_{31} = -95.8084556886757$$
$$x_{32} = 51.8157964767284$$
$$x_{33} = 7.6417905860037$$
$$x_{34} = 14.0468897783104$$
$$x_{35} = 32.9533500104745$$
$$x_{36} = -51.8180386369842$$
$$x_{37} = 48.672794756222$$
$$x_{38} = -23.5242614169388$$
$$x_{39} = -29.814665458057$$
$$x_{40} = -10.923878316627$$
$$x_{41} = -73.814409703991$$
$$x_{42} = -83.2406103745349$$
$$x_{43} = 3.84061003933616$$
$$x_{44} = 80.0976428286763$$
$$x_{45} = 26.6612995861405$$
$$x_{46} = -64.3878110088085$$
$$x_{47} = -98.9503602075132$$
$$x_{48} = 325.151735529204$$
$$x_{49} = 58.1013176971777$$
$$x_{50} = 70.6710584152442$$
$$x_{51} = 76.9554991920151$$
$$x_{52} = -80.0985793473126$$
$$x_{53} = -26.6698460561676$$
$$x_{54} = 92.6658312229758$$
$$x_{55} = 73.8133066583003$$
$$x_{56} = 17.2084950274544$$
$$x_{57} = -70.6722619096271$$
$$x_{58} = 39.2423231185819$$
$$x_{59} = -14.0786811872199$$
$$x_{60} = 83.2397433132615$$
$$x_{61} = -4.58124156980892$$
$$x_{62} = 10.8691736442252$$
$$x_{63} = 61.2438892486337$$
$$x_{64} = 36.0981115003435$$
$$x_{65} = -17.229366716415$$
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:
$$x_{1} = 105.233572684616$$
$$x_{2} = -86.3826105819672$$
$$x_{3} = -61.2454927075855$$
$$x_{4} = -54.9606200558715$$
$$x_{5} = -42.38947284675$$
$$x_{6} = -67.5300646513164$$
$$x_{7} = 54.958627731809$$
$$x_{8} = -92.6665306850573$$
$$x_{9} = 29.8078449830189$$
$$x_{10} = 67.528746323277$$
$$x_{11} = 86.3818055246036$$
$$x_{12} = -48.6753369544401$$
$$x_{13} = 42.3861166198228$$
$$x_{14} = -36.1027474386634$$
$$x_{15} = 98.9497468430673$$
$$x_{16} = 23.5132344474784$$
$$x_{17} = 48.672794756222$$
$$x_{18} = -23.5242614169388$$
$$x_{19} = -29.814665458057$$
$$x_{20} = -10.923878316627$$
$$x_{21} = -73.814409703991$$
$$x_{22} = 3.84061003933616$$
$$x_{23} = 80.0976428286763$$
$$x_{24} = -98.9503602075132$$
$$x_{25} = 325.151735529204$$
$$x_{26} = -80.0985793473126$$
$$x_{27} = 92.6658312229758$$
$$x_{28} = 73.8133066583003$$
$$x_{29} = 17.2084950274544$$
$$x_{30} = -4.58124156980892$$
$$x_{31} = 10.8691736442252$$
$$x_{32} = 61.2438892486337$$
$$x_{33} = 36.0981115003435$$
$$x_{34} = -17.229366716415$$
Puntos máximos de la función:
$$x_{34} = -20.3776023235733$$
$$x_{34} = -76.9565138665894$$
$$x_{34} = 64.3863605756806$$
$$x_{34} = -58.1030997704766$$
$$x_{34} = -1.34456392370501$$
$$x_{34} = -45.5324916393212$$
$$x_{34} = 20.3628214780858$$
$$x_{34} = 89.5238336327072$$
$$x_{34} = -39.2462418447496$$
$$x_{34} = 45.5295847647311$$
$$x_{34} = -32.9589205172515$$
$$x_{34} = -7.76132234788669$$
$$x_{34} = -89.5245831096533$$
$$x_{34} = 95.807801395196$$
$$x_{34} = -95.8084556886757$$
$$x_{34} = 51.8157964767284$$
$$x_{34} = 7.6417905860037$$
$$x_{34} = 14.0468897783104$$
$$x_{34} = 32.9533500104745$$
$$x_{34} = -51.8180386369842$$
$$x_{34} = -83.2406103745349$$
$$x_{34} = 26.6612995861405$$
$$x_{34} = -64.3878110088085$$
$$x_{34} = 58.1013176971777$$
$$x_{34} = 70.6710584152442$$
$$x_{34} = 76.9554991920151$$
$$x_{34} = -26.6698460561676$$
$$x_{34} = -70.6722619096271$$
$$x_{34} = 39.2423231185819$$
$$x_{34} = -14.0786811872199$$
$$x_{34} = 83.2397433132615$$
Decrece en los intervalos
$$\left[325.151735529204, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9503602075132\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\cos{\left(x \right)}}{x - 3} - \frac{\sin{\left(x \right)}}{\left(x - 3\right)^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 105.233572684616$$
$$x_{2} = -20.3776023235733$$
$$x_{3} = -76.9565138665894$$
$$x_{4} = -86.3826105819672$$
$$x_{5} = 64.3863605756806$$
$$x_{6} = -61.2454927075855$$
$$x_{7} = -54.9606200558715$$
$$x_{8} = -58.1030997704766$$
$$x_{9} = -1.34456392370501$$
$$x_{10} = -42.38947284675$$
$$x_{11} = -67.5300646513164$$
$$x_{12} = 54.958627731809$$
$$x_{13} = -92.6665306850573$$
$$x_{14} = 29.8078449830189$$
$$x_{15} = -45.5324916393212$$
$$x_{16} = 20.3628214780858$$
$$x_{17} = 67.528746323277$$
$$x_{18} = 86.3818055246036$$
$$x_{19} = 89.5238336327072$$
$$x_{20} = -39.2462418447496$$
$$x_{21} = 45.5295847647311$$
$$x_{22} = -32.9589205172515$$
$$x_{23} = -48.6753369544401$$
$$x_{24} = 42.3861166198228$$
$$x_{25} = -36.1027474386634$$
$$x_{26} = 98.9497468430673$$
$$x_{27} = -7.76132234788669$$
$$x_{28} = 23.5132344474784$$
$$x_{29} = -89.5245831096533$$
$$x_{30} = 95.807801395196$$
$$x_{31} = -95.8084556886757$$
$$x_{32} = 51.8157964767284$$
$$x_{33} = 7.6417905860037$$
$$x_{34} = 14.0468897783104$$
$$x_{35} = 32.9533500104745$$
$$x_{36} = -51.8180386369842$$
$$x_{37} = 48.672794756222$$
$$x_{38} = -23.5242614169388$$
$$x_{39} = -29.814665458057$$
$$x_{40} = -10.923878316627$$
$$x_{41} = -73.814409703991$$
$$x_{42} = -83.2406103745349$$
$$x_{43} = 3.84061003933616$$
$$x_{44} = 80.0976428286763$$
$$x_{45} = 26.6612995861405$$
$$x_{46} = -64.3878110088085$$
$$x_{47} = -98.9503602075132$$
$$x_{48} = 325.151735529204$$
$$x_{49} = 58.1013176971777$$
$$x_{50} = 70.6710584152442$$
$$x_{51} = 76.9554991920151$$
$$x_{52} = -80.0985793473126$$
$$x_{53} = -26.6698460561676$$
$$x_{54} = 92.6658312229758$$
$$x_{55} = 73.8133066583003$$
$$x_{56} = 17.2084950274544$$
$$x_{57} = -70.6722619096271$$
$$x_{58} = 39.2423231185819$$
$$x_{59} = -14.0786811872199$$
$$x_{60} = 83.2397433132615$$
$$x_{61} = -4.58124156980892$$
$$x_{62} = 10.8691736442252$$
$$x_{63} = 61.2438892486337$$
$$x_{64} = 36.0981115003435$$
$$x_{65} = -17.229366716415$$
Signos de extremos en los puntos:
(105.23357268461616, -0.00978105467785378)
(-20.37760232357331, 0.0427369046360932)
(-76.95651386658943, 0.0125058203617939)
(-86.3826105819672, -0.0111871583888076)
(64.38636057568057, 0.0162881026123812)
(-61.245492707585505, -0.0155634090529215)
(-54.96062005587145, -0.0172505262648511)
(-58.103099770476625, 0.016363590574684)
(-1.3445639237050093, 0.22430753166831)
(-42.38947284674999, -0.0220261953097968)
(-67.5300646513164, -0.014176925929091)
(54.95862773180898, -0.0192425183121482)
(-92.66653068505734, -0.0104524055064709)
(29.807844983018946, -0.0372765877717194)
(-45.532491639321215, 0.0206003804024146)
(20.362821478085806, 0.0574990397092516)
(67.52874632327696, -0.0154951087781249)
(86.38180552460362, -0.0119921616610818)
(89.52383363270721, 0.0115567373369735)
(-39.24624184474963, 0.0236641159531375)
(45.52958476473112, 0.0235065469949681)
(-32.95892051725151, 0.0277987638480111)
(-48.675336954440134, -0.01934796886592)
(42.38611661982284, -0.0253814776422808)
(-36.10274743866336, -0.0255652919549382)
(98.94974684306735, -0.0104215563563964)
(-7.761322347886692, 0.0925267515218825)
(23.51323444747835, -0.0486911941130189)
(-89.52458310965334, 0.0108073072662695)
(95.807801395196, 0.0107743308232794)
(-95.80845568867566, 0.0101200730623707)
(51.815796476728416, 0.0204808753934717)
(7.641790586003701, 0.210602310406987)
(14.046889778310359, 0.0901545868339596)
(32.9533500104745, 0.0333666577428133)
(-51.818038636984234, 0.0182391358389777)
(48.672794756222046, -0.0218896259527883)
(-23.524261416938753, -0.0376745669102614)
(-29.81466545805702, -0.0304600388700841)
(-10.92387831662703, -0.0716345634424574)
(-73.81440970399099, -0.0130172877116949)
(-83.24061037453485, 0.0115946857869956)
(3.840610039336164, -0.765474835425238)
(80.09764282867631, -0.0129694742426031)
(26.661299586140455, 0.0422254114125872)
(-64.38781100880854, 0.014837845274178)
(-98.95036020751319, -0.00980822329794267)
(325.1517355292043, -0.00310411235435045)
(58.10131769717772, 0.0181453983405802)
(70.67105841524423, 0.0147757528258227)
(76.95549919201505, 0.0135204089759034)
(-80.09857934731257, -0.0120330288236168)
(-26.669846056167565, 0.0336851257410975)
(92.66583122297577, -0.0111518267639758)
(73.81330665830026, -0.0141202318002497)
(17.208495027454397, -0.0702067642508414)
(-70.67226190962707, 0.0135723794152411)
(39.24232311858189, 0.0275815530182016)
(-14.078681187219923, 0.0584524170695983)
(83.23974331326149, 0.0124616843089039)
(-4.581241569808925, -0.13077178563845)
(10.869173644225215, -0.126064325690938)
(61.243889248633685, -0.017166653103903)
(36.098111500343514, -0.030199423715659)
(-17.229366716414972, -0.0493727971697397)
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:
$$x_{1} = 105.233572684616$$
$$x_{2} = -86.3826105819672$$
$$x_{3} = -61.2454927075855$$
$$x_{4} = -54.9606200558715$$
$$x_{5} = -42.38947284675$$
$$x_{6} = -67.5300646513164$$
$$x_{7} = 54.958627731809$$
$$x_{8} = -92.6665306850573$$
$$x_{9} = 29.8078449830189$$
$$x_{10} = 67.528746323277$$
$$x_{11} = 86.3818055246036$$
$$x_{12} = -48.6753369544401$$
$$x_{13} = 42.3861166198228$$
$$x_{14} = -36.1027474386634$$
$$x_{15} = 98.9497468430673$$
$$x_{16} = 23.5132344474784$$
$$x_{17} = 48.672794756222$$
$$x_{18} = -23.5242614169388$$
$$x_{19} = -29.814665458057$$
$$x_{20} = -10.923878316627$$
$$x_{21} = -73.814409703991$$
$$x_{22} = 3.84061003933616$$
$$x_{23} = 80.0976428286763$$
$$x_{24} = -98.9503602075132$$
$$x_{25} = 325.151735529204$$
$$x_{26} = -80.0985793473126$$
$$x_{27} = 92.6658312229758$$
$$x_{28} = 73.8133066583003$$
$$x_{29} = 17.2084950274544$$
$$x_{30} = -4.58124156980892$$
$$x_{31} = 10.8691736442252$$
$$x_{32} = 61.2438892486337$$
$$x_{33} = 36.0981115003435$$
$$x_{34} = -17.229366716415$$
Puntos máximos de la función:
$$x_{34} = -20.3776023235733$$
$$x_{34} = -76.9565138665894$$
$$x_{34} = 64.3863605756806$$
$$x_{34} = -58.1030997704766$$
$$x_{34} = -1.34456392370501$$
$$x_{34} = -45.5324916393212$$
$$x_{34} = 20.3628214780858$$
$$x_{34} = 89.5238336327072$$
$$x_{34} = -39.2462418447496$$
$$x_{34} = 45.5295847647311$$
$$x_{34} = -32.9589205172515$$
$$x_{34} = -7.76132234788669$$
$$x_{34} = -89.5245831096533$$
$$x_{34} = 95.807801395196$$
$$x_{34} = -95.8084556886757$$
$$x_{34} = 51.8157964767284$$
$$x_{34} = 7.6417905860037$$
$$x_{34} = 14.0468897783104$$
$$x_{34} = 32.9533500104745$$
$$x_{34} = -51.8180386369842$$
$$x_{34} = -83.2406103745349$$
$$x_{34} = 26.6612995861405$$
$$x_{34} = -64.3878110088085$$
$$x_{34} = 58.1013176971777$$
$$x_{34} = 70.6710584152442$$
$$x_{34} = 76.9554991920151$$
$$x_{34} = -26.6698460561676$$
$$x_{34} = -70.6722619096271$$
$$x_{34} = 39.2423231185819$$
$$x_{34} = -14.0786811872199$$
$$x_{34} = 83.2397433132615$$
Decrece en los intervalos
$$\left[325.151735529204, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9503602075132\right]$$
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
$$\frac{- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x - 3} + \frac{2 \sin{\left(x \right)}}{\left(x - 3\right)^{2}}}{x - 3} = 0$$
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
$$\frac{- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x - 3} + \frac{2 \sin{\left(x \right)}}{\left(x - 3\right)^{2}}}{x - 3} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
$$x_{1} = 3$$
$$x_{1} = 3$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x - 3}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x - 3}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x - 3}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x - 3}\right) = 0$$
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(x)/(x - 3), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(x - 3\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(x - 3\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{x \left(x - 3\right)}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{x \left(x - 3\right)}\right) = 0$$
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:
$$\frac{\sin{\left(x \right)}}{x - 3} = - \frac{\sin{\left(x \right)}}{- x - 3}$$
- No
$$\frac{\sin{\left(x \right)}}{x - 3} = \frac{\sin{\left(x \right)}}{- x - 3}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\sin{\left(x \right)}}{x - 3} = - \frac{\sin{\left(x \right)}}{- x - 3}$$
- No
$$\frac{\sin{\left(x \right)}}{x - 3} = \frac{\sin{\left(x \right)}}{- x - 3}$$
- No
es decir, función
no es
par ni impar
Gráfico