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- Gráfico de la función implícita
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- 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 =:
-
y=(x+1)^3
-
y=2x
-
2*x^3-3*x
-
3-x^2
-
Expresiones idénticas
- (-sinx-(dos cosx)/x+(2sinx)/x^2)/x
- ( menos seno de x menos (2 coseno de x) dividir por x más (2 seno de x) dividir por x al cuadrado ) dividir por x
- ( menos seno de x menos (dos coseno de x) dividir por x más (2 seno de x) dividir por x al cuadrado ) dividir por x
- (-sinx-(2cosx)/x+(2sinx)/x2)/x
- -sinx-2cosx/x+2sinx/x2/x
- (-sinx-(2cosx)/x+(2sinx)/x²)/x
- (-sinx-(2cosx)/x+(2sinx)/x en el grado 2)/x
- -sinx-2cosx/x+2sinx/x^2/x
- (-sinx-(2cosx) dividir por x+(2sinx) dividir por x^2) dividir por x
-
Expresiones semejantes
- (-sinx+(2cosx)/x+(2sinx)/x^2)/x
- (sinx-(2cosx)/x+(2sinx)/x^2)/x
- (-sinx-(2cosx)/x-(2sinx)/x^2)/x
-
Expresiones con funciones
- sinx
- sinx+0,5
Gráfico de la función y = (-sinx-(2cosx)/x+(2sinx)/x^2)/x
Solución
Ha introducido
[src]
2*cos(x) 2*sin(x)
-sin(x) - -------- + --------
x 2
x
f(x) = -----------------------------
x
$$f{\left(x \right)} = \frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}$$
f = (-sin(x) - 2*cos(x)/x + (2*sin(x))/x^2)/x
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 12.404445021902$$
$$x_{2} = -21.8996964794928$$
$$x_{3} = 50.2256516491831$$
$$x_{4} = 56.5132704621986$$
$$x_{5} = 62.8000005565198$$
$$x_{6} = -72.2289377620154$$
$$x_{7} = 43.9367614714198$$
$$x_{8} = -40.7916552312719$$
$$x_{9} = -62.8000005565198$$
$$x_{10} = -69.0860849466452$$
$$x_{11} = -91.0842274914688$$
$$x_{12} = -100.511065295271$$
$$x_{13} = 59.6567290035279$$
$$x_{14} = -28.2033610039524$$
$$x_{15} = -53.3695918204908$$
$$x_{16} = 31.3520917265645$$
$$x_{17} = -18.7426455847748$$
$$x_{18} = -9.20584014293667$$
$$x_{19} = -25.052825280993$$
$$x_{20} = -1790.70669566846$$
$$x_{21} = 2.0815759778181$$
$$x_{22} = 91.0842274914688$$
$$x_{23} = 9.20584014293667$$
$$x_{24} = 53.3695918204908$$
$$x_{25} = 40.7916552312719$$
$$x_{26} = 94.2265525745684$$
$$x_{27} = -94.2265525745684$$
$$x_{28} = -43.9367614714198$$
$$x_{29} = -78.5143405319308$$
$$x_{30} = 69.0860849466452$$
$$x_{31} = 81.6569138240367$$
$$x_{32} = 78.5143405319308$$
$$x_{33} = -65.9431119046552$$
$$x_{34} = -37.6459603230864$$
$$x_{35} = 97.368830362901$$
$$x_{36} = 15.5792364103872$$
$$x_{37} = -2.0815759778181$$
$$x_{38} = 28.2033610039524$$
$$x_{39} = 75.3716854092873$$
$$x_{40} = -12.404445021902$$
$$x_{41} = -59.6567290035279$$
$$x_{42} = 37.6459603230864$$
$$x_{43} = 18.7426455847748$$
$$x_{44} = 65.9431119046552$$
$$x_{45} = -47.0813974121542$$
$$x_{46} = -81.6569138240367$$
$$x_{47} = -75.3716854092873$$
$$x_{48} = -342.42775856009$$
$$x_{49} = 25.052825280993$$
$$x_{50} = -87.9418500396598$$
$$x_{51} = -1288.05143523817$$
$$x_{52} = 84.7994143922025$$
$$x_{53} = 21.8996964794928$$
$$x_{54} = -34.499514921367$$
$$x_{55} = 72.2289377620154$$
$$x_{56} = 100.511065295271$$
$$x_{57} = 47.0813974121542$$
$$x_{58} = 34.499514921367$$
$$x_{59} = -50.2256516491831$$
$$x_{60} = -56.5132704621986$$
$$x_{61} = 131.931731514843$$
$$x_{62} = -15.5792364103872$$
$$x_{63} = -84.7994143922025$$
$$x_{64} = -5.94036999057271$$
$$x_{65} = 5.94036999057271$$
$$x_{66} = 87.9418500396598$$
$$x_{67} = -31.3520917265645$$
$$x_{68} = -97.368830362901$$
o sea hay que resolver la ecuación:
$$\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución numérica
$$x_{1} = 12.404445021902$$
$$x_{2} = -21.8996964794928$$
$$x_{3} = 50.2256516491831$$
$$x_{4} = 56.5132704621986$$
$$x_{5} = 62.8000005565198$$
$$x_{6} = -72.2289377620154$$
$$x_{7} = 43.9367614714198$$
$$x_{8} = -40.7916552312719$$
$$x_{9} = -62.8000005565198$$
$$x_{10} = -69.0860849466452$$
$$x_{11} = -91.0842274914688$$
$$x_{12} = -100.511065295271$$
$$x_{13} = 59.6567290035279$$
$$x_{14} = -28.2033610039524$$
$$x_{15} = -53.3695918204908$$
$$x_{16} = 31.3520917265645$$
$$x_{17} = -18.7426455847748$$
$$x_{18} = -9.20584014293667$$
$$x_{19} = -25.052825280993$$
$$x_{20} = -1790.70669566846$$
$$x_{21} = 2.0815759778181$$
$$x_{22} = 91.0842274914688$$
$$x_{23} = 9.20584014293667$$
$$x_{24} = 53.3695918204908$$
$$x_{25} = 40.7916552312719$$
$$x_{26} = 94.2265525745684$$
$$x_{27} = -94.2265525745684$$
$$x_{28} = -43.9367614714198$$
$$x_{29} = -78.5143405319308$$
$$x_{30} = 69.0860849466452$$
$$x_{31} = 81.6569138240367$$
$$x_{32} = 78.5143405319308$$
$$x_{33} = -65.9431119046552$$
$$x_{34} = -37.6459603230864$$
$$x_{35} = 97.368830362901$$
$$x_{36} = 15.5792364103872$$
$$x_{37} = -2.0815759778181$$
$$x_{38} = 28.2033610039524$$
$$x_{39} = 75.3716854092873$$
$$x_{40} = -12.404445021902$$
$$x_{41} = -59.6567290035279$$
$$x_{42} = 37.6459603230864$$
$$x_{43} = 18.7426455847748$$
$$x_{44} = 65.9431119046552$$
$$x_{45} = -47.0813974121542$$
$$x_{46} = -81.6569138240367$$
$$x_{47} = -75.3716854092873$$
$$x_{48} = -342.42775856009$$
$$x_{49} = 25.052825280993$$
$$x_{50} = -87.9418500396598$$
$$x_{51} = -1288.05143523817$$
$$x_{52} = 84.7994143922025$$
$$x_{53} = 21.8996964794928$$
$$x_{54} = -34.499514921367$$
$$x_{55} = 72.2289377620154$$
$$x_{56} = 100.511065295271$$
$$x_{57} = 47.0813974121542$$
$$x_{58} = 34.499514921367$$
$$x_{59} = -50.2256516491831$$
$$x_{60} = -56.5132704621986$$
$$x_{61} = 131.931731514843$$
$$x_{62} = -15.5792364103872$$
$$x_{63} = -84.7994143922025$$
$$x_{64} = -5.94036999057271$$
$$x_{65} = 5.94036999057271$$
$$x_{66} = 87.9418500396598$$
$$x_{67} = -31.3520917265645$$
$$x_{68} = -97.368830362901$$
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
$$\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{4 \cos{\left(x \right)}}{x^{2}} - \frac{4 \sin{\left(x \right)}}{x^{3}}}{x} - \frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -13.9205214266357$$
$$x_{2} = -3.87023858022217$$
$$x_{3} = -98.9298409677395$$
$$x_{4} = 36.0450220033785$$
$$x_{5} = -89.5018675871561$$
$$x_{6} = -10.7130109882558$$
$$x_{7} = 7.44308705395446$$
$$x_{8} = -83.2161494163012$$
$$x_{9} = 98.9298409677395$$
$$x_{10} = 26.5905550258527$$
$$x_{11} = -42.3406072405316$$
$$x_{12} = 51.7783178293904$$
$$x_{13} = -23.4336891696933$$
$$x_{14} = -39.1933145376163$$
$$x_{15} = -17.1027407890472$$
$$x_{16} = 67.499787704256$$
$$x_{17} = -80.0731410729394$$
$$x_{18} = 89.5018675871561$$
$$x_{19} = -36.0450220033785$$
$$x_{20} = -92.644597693687$$
$$x_{21} = 86.3590546280651$$
$$x_{22} = -26.5905550258527$$
$$x_{23} = -61.2120336791686$$
$$x_{24} = 249.744603496217$$
$$x_{25} = -48.6329734602127$$
$$x_{26} = 70.6433593524237$$
$$x_{27} = -64.356022448055$$
$$x_{28} = -212.043355752614$$
$$x_{29} = 73.7867621864701$$
$$x_{30} = -95.7872531120814$$
$$x_{31} = 45.4871087849823$$
$$x_{32} = 10.7130109882558$$
$$x_{33} = 58.0677849921727$$
$$x_{34} = 23.4336891696933$$
$$x_{35} = -70.6433593524237$$
$$x_{36} = -73.7867621864701$$
$$x_{37} = 83.2161494163012$$
$$x_{38} = 3.87023858022217$$
$$x_{39} = 39.1933145376163$$
$$x_{40} = 20.2720010891386$$
$$x_{41} = 17.1027407890472$$
$$x_{42} = -20.2720010891386$$
$$x_{43} = 61.2120336791686$$
$$x_{44} = -7.44308705395446$$
$$x_{45} = 92.644597693687$$
$$x_{46} = 76.9300169373264$$
$$x_{47} = 48.6329734602127$$
$$x_{48} = -51.7783178293904$$
$$x_{49} = -67.499787704256$$
$$x_{50} = -54.9232316104305$$
$$x_{51} = 42.3406072405316$$
$$x_{52} = 95.7872531120814$$
$$x_{53} = 54.9232316104305$$
$$x_{54} = -86.3590546280651$$
$$x_{55} = -45.4871087849823$$
$$x_{56} = 29.7441555138788$$
$$x_{57} = -29.7441555138788$$
$$x_{58} = 1148.2495022125$$
$$x_{59} = 64.356022448055$$
$$x_{60} = 80.0731410729394$$
$$x_{61} = -58.0677849921727$$
$$x_{62} = 13.9205214266357$$
$$x_{63} = 32.8954402073454$$
$$x_{64} = -76.9300169373264$$
$$x_{65} = 230.894066823744$$
$$x_{66} = -32.8954402073454$$
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:
$$x_{1} = -13.9205214266357$$
$$x_{2} = -89.5018675871561$$
$$x_{3} = 7.44308705395446$$
$$x_{4} = -83.2161494163012$$
$$x_{5} = 26.5905550258527$$
$$x_{6} = 51.7783178293904$$
$$x_{7} = -39.1933145376163$$
$$x_{8} = 89.5018675871561$$
$$x_{9} = -26.5905550258527$$
$$x_{10} = 70.6433593524237$$
$$x_{11} = -64.356022448055$$
$$x_{12} = -95.7872531120814$$
$$x_{13} = 45.4871087849823$$
$$x_{14} = 58.0677849921727$$
$$x_{15} = -70.6433593524237$$
$$x_{16} = 83.2161494163012$$
$$x_{17} = 39.1933145376163$$
$$x_{18} = 20.2720010891386$$
$$x_{19} = -20.2720010891386$$
$$x_{20} = -7.44308705395446$$
$$x_{21} = 76.9300169373264$$
$$x_{22} = -51.7783178293904$$
$$x_{23} = 95.7872531120814$$
$$x_{24} = -45.4871087849823$$
$$x_{25} = 64.356022448055$$
$$x_{26} = -58.0677849921727$$
$$x_{27} = 13.9205214266357$$
$$x_{28} = 32.8954402073454$$
$$x_{29} = -76.9300169373264$$
$$x_{30} = -32.8954402073454$$
Puntos máximos de la función:
$$x_{30} = -3.87023858022217$$
$$x_{30} = -98.9298409677395$$
$$x_{30} = 36.0450220033785$$
$$x_{30} = -10.7130109882558$$
$$x_{30} = 98.9298409677395$$
$$x_{30} = -42.3406072405316$$
$$x_{30} = -23.4336891696933$$
$$x_{30} = -17.1027407890472$$
$$x_{30} = 67.499787704256$$
$$x_{30} = -80.0731410729394$$
$$x_{30} = -36.0450220033785$$
$$x_{30} = -92.644597693687$$
$$x_{30} = 86.3590546280651$$
$$x_{30} = -61.2120336791686$$
$$x_{30} = 249.744603496217$$
$$x_{30} = -48.6329734602127$$
$$x_{30} = -212.043355752614$$
$$x_{30} = 73.7867621864701$$
$$x_{30} = 10.7130109882558$$
$$x_{30} = 23.4336891696933$$
$$x_{30} = -73.7867621864701$$
$$x_{30} = 3.87023858022217$$
$$x_{30} = 17.1027407890472$$
$$x_{30} = 61.2120336791686$$
$$x_{30} = 92.644597693687$$
$$x_{30} = 48.6329734602127$$
$$x_{30} = -67.499787704256$$
$$x_{30} = -54.9232316104305$$
$$x_{30} = 42.3406072405316$$
$$x_{30} = 54.9232316104305$$
$$x_{30} = -86.3590546280651$$
$$x_{30} = 29.7441555138788$$
$$x_{30} = -29.7441555138788$$
$$x_{30} = 1148.2495022125$$
$$x_{30} = 80.0731410729394$$
$$x_{30} = 230.894066823744$$
Decrece en los intervalos
$$\left[95.7872531120814, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.7872531120814\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{4 \cos{\left(x \right)}}{x^{2}} - \frac{4 \sin{\left(x \right)}}{x^{3}}}{x} - \frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x^{2}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -13.9205214266357$$
$$x_{2} = -3.87023858022217$$
$$x_{3} = -98.9298409677395$$
$$x_{4} = 36.0450220033785$$
$$x_{5} = -89.5018675871561$$
$$x_{6} = -10.7130109882558$$
$$x_{7} = 7.44308705395446$$
$$x_{8} = -83.2161494163012$$
$$x_{9} = 98.9298409677395$$
$$x_{10} = 26.5905550258527$$
$$x_{11} = -42.3406072405316$$
$$x_{12} = 51.7783178293904$$
$$x_{13} = -23.4336891696933$$
$$x_{14} = -39.1933145376163$$
$$x_{15} = -17.1027407890472$$
$$x_{16} = 67.499787704256$$
$$x_{17} = -80.0731410729394$$
$$x_{18} = 89.5018675871561$$
$$x_{19} = -36.0450220033785$$
$$x_{20} = -92.644597693687$$
$$x_{21} = 86.3590546280651$$
$$x_{22} = -26.5905550258527$$
$$x_{23} = -61.2120336791686$$
$$x_{24} = 249.744603496217$$
$$x_{25} = -48.6329734602127$$
$$x_{26} = 70.6433593524237$$
$$x_{27} = -64.356022448055$$
$$x_{28} = -212.043355752614$$
$$x_{29} = 73.7867621864701$$
$$x_{30} = -95.7872531120814$$
$$x_{31} = 45.4871087849823$$
$$x_{32} = 10.7130109882558$$
$$x_{33} = 58.0677849921727$$
$$x_{34} = 23.4336891696933$$
$$x_{35} = -70.6433593524237$$
$$x_{36} = -73.7867621864701$$
$$x_{37} = 83.2161494163012$$
$$x_{38} = 3.87023858022217$$
$$x_{39} = 39.1933145376163$$
$$x_{40} = 20.2720010891386$$
$$x_{41} = 17.1027407890472$$
$$x_{42} = -20.2720010891386$$
$$x_{43} = 61.2120336791686$$
$$x_{44} = -7.44308705395446$$
$$x_{45} = 92.644597693687$$
$$x_{46} = 76.9300169373264$$
$$x_{47} = 48.6329734602127$$
$$x_{48} = -51.7783178293904$$
$$x_{49} = -67.499787704256$$
$$x_{50} = -54.9232316104305$$
$$x_{51} = 42.3406072405316$$
$$x_{52} = 95.7872531120814$$
$$x_{53} = 54.9232316104305$$
$$x_{54} = -86.3590546280651$$
$$x_{55} = -45.4871087849823$$
$$x_{56} = 29.7441555138788$$
$$x_{57} = -29.7441555138788$$
$$x_{58} = 1148.2495022125$$
$$x_{59} = 64.356022448055$$
$$x_{60} = 80.0731410729394$$
$$x_{61} = -58.0677849921727$$
$$x_{62} = 13.9205214266357$$
$$x_{63} = 32.8954402073454$$
$$x_{64} = -76.9300169373264$$
$$x_{65} = 230.894066823744$$
$$x_{66} = -32.8954402073454$$
Signos de extremos en los puntos:
(-13.920521426635718, -0.0716515891912541)
(-3.870238580222165, 0.248692236445789)
(-98.92984096773947, 0.0101076571692611)
(36.04502200337846, 0.027732411296845)
(-89.5018675871561, -0.0111722538908926)
(-10.713010988255775, 0.0929394035698334)
(7.443087053954458, -0.133143202113607)
(-83.21614941630125, -0.012016030684702)
(98.92984096773947, 0.0101076571692611)
(26.590555025852712, -0.0375807705734432)
(-42.34060724053156, 0.0236114046136127)
(51.77831782939038, -0.0193095024183706)
(-23.433689169693317, 0.0426347989465758)
(-39.19331453761631, -0.0255062545726498)
(-17.102740789047186, 0.0583704306894909)
(67.499787704256, 0.0148132358906579)
(-80.07314107293935, 0.012487608370541)
(89.5018675871561, -0.0111722538908926)
(-36.04502200337846, 0.027732411296845)
(-92.64459769368703, 0.0107933087887867)
(86.35905462806514, 0.0115787854311861)
(-26.590555025852712, -0.0375807705734432)
(-61.212033679168634, 0.016334477440491)
(249.7446034962173, 0.00400405842499421)
(-48.63297346021267, 0.0205578354154553)
(70.64335935242372, -0.0141541941579985)
(-64.35602244805503, -0.0155366857700825)
(-212.04335575261447, 0.00471596422453044)
(73.78676218647006, 0.0135513220378694)
(-95.7872531120814, -0.0104392335849293)
(45.48710878498235, -0.02197893994184)
(10.713010988255775, 0.0929394035698334)
(58.06778499217275, -0.0172186996097649)
(23.433689169693317, 0.0426347989465758)
(-70.64335935242372, -0.0141541941579985)
(-73.78676218647006, 0.0135513220378694)
(83.21614941630125, -0.012016030684702)
(3.870238580222165, 0.248692236445789)
(39.19331453761631, -0.0255062545726498)
(20.272001089138612, -0.0492692013251242)
(17.102740789047186, 0.0583704306894909)
(-20.272001089138612, -0.0492692013251242)
(61.212033679168634, 0.016334477440491)
(-7.443087053954458, -0.133143202113607)
(92.64459769368703, 0.0107933087887867)
(76.93001693732643, -0.0129977291788139)
(48.63297346021267, 0.0205578354154553)
(-51.77831782939038, -0.0193095024183706)
(-67.499787704256, 0.0148132358906579)
(-54.92323161043051, 0.0182042144946165)
(42.34060724053156, 0.0236114046136127)
(95.7872531120814, -0.0104392335849293)
(54.92323161043051, 0.0182042144946165)
(-86.35905462806514, 0.0115787854311861)
(-45.48710878498235, -0.02197893994184)
(29.744155513878802, 0.0336010649596176)
(-29.744155513878802, 0.0336010649596176)
(1148.2495022124986, 0.000870890532804869)
(64.35602244805503, -0.0155366857700825)
(80.07314107293935, 0.012487608370541)
(-58.06778499217275, -0.0172186996097649)
(13.920521426635718, -0.0716515891912541)
(32.89544020734542, -0.0303853130040648)
(-76.93001693732643, -0.0129977291788139)
(230.8940668237441, 0.0043309498383777)
(-32.89544020734542, -0.0303853130040648)
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:
$$x_{1} = -13.9205214266357$$
$$x_{2} = -89.5018675871561$$
$$x_{3} = 7.44308705395446$$
$$x_{4} = -83.2161494163012$$
$$x_{5} = 26.5905550258527$$
$$x_{6} = 51.7783178293904$$
$$x_{7} = -39.1933145376163$$
$$x_{8} = 89.5018675871561$$
$$x_{9} = -26.5905550258527$$
$$x_{10} = 70.6433593524237$$
$$x_{11} = -64.356022448055$$
$$x_{12} = -95.7872531120814$$
$$x_{13} = 45.4871087849823$$
$$x_{14} = 58.0677849921727$$
$$x_{15} = -70.6433593524237$$
$$x_{16} = 83.2161494163012$$
$$x_{17} = 39.1933145376163$$
$$x_{18} = 20.2720010891386$$
$$x_{19} = -20.2720010891386$$
$$x_{20} = -7.44308705395446$$
$$x_{21} = 76.9300169373264$$
$$x_{22} = -51.7783178293904$$
$$x_{23} = 95.7872531120814$$
$$x_{24} = -45.4871087849823$$
$$x_{25} = 64.356022448055$$
$$x_{26} = -58.0677849921727$$
$$x_{27} = 13.9205214266357$$
$$x_{28} = 32.8954402073454$$
$$x_{29} = -76.9300169373264$$
$$x_{30} = -32.8954402073454$$
Puntos máximos de la función:
$$x_{30} = -3.87023858022217$$
$$x_{30} = -98.9298409677395$$
$$x_{30} = 36.0450220033785$$
$$x_{30} = -10.7130109882558$$
$$x_{30} = 98.9298409677395$$
$$x_{30} = -42.3406072405316$$
$$x_{30} = -23.4336891696933$$
$$x_{30} = -17.1027407890472$$
$$x_{30} = 67.499787704256$$
$$x_{30} = -80.0731410729394$$
$$x_{30} = -36.0450220033785$$
$$x_{30} = -92.644597693687$$
$$x_{30} = 86.3590546280651$$
$$x_{30} = -61.2120336791686$$
$$x_{30} = 249.744603496217$$
$$x_{30} = -48.6329734602127$$
$$x_{30} = -212.043355752614$$
$$x_{30} = 73.7867621864701$$
$$x_{30} = 10.7130109882558$$
$$x_{30} = 23.4336891696933$$
$$x_{30} = -73.7867621864701$$
$$x_{30} = 3.87023858022217$$
$$x_{30} = 17.1027407890472$$
$$x_{30} = 61.2120336791686$$
$$x_{30} = 92.644597693687$$
$$x_{30} = 48.6329734602127$$
$$x_{30} = -67.499787704256$$
$$x_{30} = -54.9232316104305$$
$$x_{30} = 42.3406072405316$$
$$x_{30} = 54.9232316104305$$
$$x_{30} = -86.3590546280651$$
$$x_{30} = 29.7441555138788$$
$$x_{30} = -29.7441555138788$$
$$x_{30} = 1148.2495022125$$
$$x_{30} = 80.0731410729394$$
$$x_{30} = 230.894066823744$$
Decrece en los intervalos
$$\left[95.7872531120814, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.7872531120814\right]$$
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
$$\frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{x} - \frac{4 \cos{\left(x \right)}}{x^{2}} + \frac{4 \sin{\left(x \right)}}{x^{3}}\right)}{x} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}\right)}{x^{2}} - \frac{6 \sin{\left(x \right)}}{x^{2}} - \frac{12 \cos{\left(x \right)}}{x^{3}} + \frac{12 \sin{\left(x \right)}}{x^{4}}}{x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 81.6324039381741$$
$$x_{2} = -47.0388279944848$$
$$x_{3} = 84.775813998667$$
$$x_{4} = -15.4482964241208$$
$$x_{5} = -43.891130947962$$
$$x_{6} = 21.8074653570703$$
$$x_{7} = 34.4413140629846$$
$$x_{8} = 28.1320266710925$$
$$x_{9} = -94.2053159674705$$
$$x_{10} = 37.5926577196659$$
$$x_{11} = -81.6324039381741$$
$$x_{12} = 72.2012232176578$$
$$x_{13} = -12.2380147777786$$
$$x_{14} = 59.623159795497$$
$$x_{15} = -65.9127501167638$$
$$x_{16} = 144.485576756624$$
$$x_{17} = -24.9723917634929$$
$$x_{18} = 91.0622574285078$$
$$x_{19} = 15.4482964241208$$
$$x_{20} = -59.623159795497$$
$$x_{21} = -1.88055044189143$$
$$x_{22} = 78.4888481666001$$
$$x_{23} = -69.0571071984352$$
$$x_{24} = -37.5926577196659$$
$$x_{25} = 204.183931989947$$
$$x_{26} = 94.2053159674705$$
$$x_{27} = 56.4778286872598$$
$$x_{28} = -122.489456166536$$
$$x_{29} = 40.7424873426775$$
$$x_{30} = -62.7681156530028$$
$$x_{31} = -50.1857574315064$$
$$x_{32} = 254.453284801053$$
$$x_{33} = 47.0388279944848$$
$$x_{34} = -28.1320266710925$$
$$x_{35} = -91.0622574285078$$
$$x_{36} = 100.491157788754$$
$$x_{37} = 62.7681156530028$$
$$x_{38} = -21.8074653570703$$
$$x_{39} = -75.3451284134656$$
$$x_{40} = -5.54218959396937$$
$$x_{41} = -87.9190939999624$$
$$x_{42} = 65.9127501167638$$
$$x_{43} = 53.332055705137$$
$$x_{44} = -40.7424873426775$$
$$x_{45} = 87.9190939999624$$
$$x_{46} = 31.2879944766898$$
$$x_{47} = 5.54218959396937$$
$$x_{48} = -72.2012232176578$$
$$x_{49} = 24.9723917634929$$
$$x_{50} = 69.0571071984352$$
$$x_{51} = -31.2879944766898$$
$$x_{52} = -97.3482797885283$$
$$x_{53} = 18.6344820913471$$
$$x_{54} = -18.6344820913471$$
$$x_{55} = 97.3482797885283$$
$$x_{56} = -78.4888481666001$$
$$x_{57} = -53.332055705137$$
$$x_{58} = -84.775813998667$$
$$x_{59} = -34.4413140629846$$
$$x_{60} = -56.4778286872598$$
$$x_{61} = 8.97516007482073$$
$$x_{62} = -8.97516007482073$$
$$x_{63} = 75.3451284134656$$
$$x_{64} = 4156.32611831087$$
$$x_{65} = 43.891130947962$$
$$x_{66} = -100.491157788754$$
$$x_{67} = 12.2380147777786$$
$$x_{68} = 50.1857574315064$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{x} - \frac{4 \cos{\left(x \right)}}{x^{2}} + \frac{4 \sin{\left(x \right)}}{x^{3}}\right)}{x} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}\right)}{x^{2}} - \frac{6 \sin{\left(x \right)}}{x^{2}} - \frac{12 \cos{\left(x \right)}}{x^{3}} + \frac{12 \sin{\left(x \right)}}{x^{4}}}{x}\right) = \frac{1}{5}$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{x} - \frac{4 \cos{\left(x \right)}}{x^{2}} + \frac{4 \sin{\left(x \right)}}{x^{3}}\right)}{x} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}\right)}{x^{2}} - \frac{6 \sin{\left(x \right)}}{x^{2}} - \frac{12 \cos{\left(x \right)}}{x^{3}} + \frac{12 \sin{\left(x \right)}}{x^{4}}}{x}\right) = \frac{1}{5}$$
- 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
$$\left[144.485576756624, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -122.489456166536\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{x} - \frac{4 \cos{\left(x \right)}}{x^{2}} + \frac{4 \sin{\left(x \right)}}{x^{3}}\right)}{x} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}\right)}{x^{2}} - \frac{6 \sin{\left(x \right)}}{x^{2}} - \frac{12 \cos{\left(x \right)}}{x^{3}} + \frac{12 \sin{\left(x \right)}}{x^{4}}}{x} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 81.6324039381741$$
$$x_{2} = -47.0388279944848$$
$$x_{3} = 84.775813998667$$
$$x_{4} = -15.4482964241208$$
$$x_{5} = -43.891130947962$$
$$x_{6} = 21.8074653570703$$
$$x_{7} = 34.4413140629846$$
$$x_{8} = 28.1320266710925$$
$$x_{9} = -94.2053159674705$$
$$x_{10} = 37.5926577196659$$
$$x_{11} = -81.6324039381741$$
$$x_{12} = 72.2012232176578$$
$$x_{13} = -12.2380147777786$$
$$x_{14} = 59.623159795497$$
$$x_{15} = -65.9127501167638$$
$$x_{16} = 144.485576756624$$
$$x_{17} = -24.9723917634929$$
$$x_{18} = 91.0622574285078$$
$$x_{19} = 15.4482964241208$$
$$x_{20} = -59.623159795497$$
$$x_{21} = -1.88055044189143$$
$$x_{22} = 78.4888481666001$$
$$x_{23} = -69.0571071984352$$
$$x_{24} = -37.5926577196659$$
$$x_{25} = 204.183931989947$$
$$x_{26} = 94.2053159674705$$
$$x_{27} = 56.4778286872598$$
$$x_{28} = -122.489456166536$$
$$x_{29} = 40.7424873426775$$
$$x_{30} = -62.7681156530028$$
$$x_{31} = -50.1857574315064$$
$$x_{32} = 254.453284801053$$
$$x_{33} = 47.0388279944848$$
$$x_{34} = -28.1320266710925$$
$$x_{35} = -91.0622574285078$$
$$x_{36} = 100.491157788754$$
$$x_{37} = 62.7681156530028$$
$$x_{38} = -21.8074653570703$$
$$x_{39} = -75.3451284134656$$
$$x_{40} = -5.54218959396937$$
$$x_{41} = -87.9190939999624$$
$$x_{42} = 65.9127501167638$$
$$x_{43} = 53.332055705137$$
$$x_{44} = -40.7424873426775$$
$$x_{45} = 87.9190939999624$$
$$x_{46} = 31.2879944766898$$
$$x_{47} = 5.54218959396937$$
$$x_{48} = -72.2012232176578$$
$$x_{49} = 24.9723917634929$$
$$x_{50} = 69.0571071984352$$
$$x_{51} = -31.2879944766898$$
$$x_{52} = -97.3482797885283$$
$$x_{53} = 18.6344820913471$$
$$x_{54} = -18.6344820913471$$
$$x_{55} = 97.3482797885283$$
$$x_{56} = -78.4888481666001$$
$$x_{57} = -53.332055705137$$
$$x_{58} = -84.775813998667$$
$$x_{59} = -34.4413140629846$$
$$x_{60} = -56.4778286872598$$
$$x_{61} = 8.97516007482073$$
$$x_{62} = -8.97516007482073$$
$$x_{63} = 75.3451284134656$$
$$x_{64} = 4156.32611831087$$
$$x_{65} = 43.891130947962$$
$$x_{66} = -100.491157788754$$
$$x_{67} = 12.2380147777786$$
$$x_{68} = 50.1857574315064$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{x} - \frac{4 \cos{\left(x \right)}}{x^{2}} + \frac{4 \sin{\left(x \right)}}{x^{3}}\right)}{x} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}\right)}{x^{2}} - \frac{6 \sin{\left(x \right)}}{x^{2}} - \frac{12 \cos{\left(x \right)}}{x^{3}} + \frac{12 \sin{\left(x \right)}}{x^{4}}}{x}\right) = \frac{1}{5}$$
$$\lim_{x \to 0^+}\left(\frac{\sin{\left(x \right)} + \frac{2 \left(\cos{\left(x \right)} - \frac{2 \sin{\left(x \right)}}{x} - \frac{4 \cos{\left(x \right)}}{x^{2}} + \frac{4 \sin{\left(x \right)}}{x^{3}}\right)}{x} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \left(\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}\right)}{x^{2}} - \frac{6 \sin{\left(x \right)}}{x^{2}} - \frac{12 \cos{\left(x \right)}}{x^{3}} + \frac{12 \sin{\left(x \right)}}{x^{4}}}{x}\right) = \frac{1}{5}$$
- 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
$$\left[144.485576756624, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -122.489456166536\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0$$
$$x_{1} = 0$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}\right) = 0$$
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) - 2*cos(x)/x + (2*sin(x))/x^2)/x, dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x^{2}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x^{2}}\right) = 0$$
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:
$$\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x} = - \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}$$
- No
$$\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x} = \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x} = - \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}$$
- No
$$\frac{\left(- \sin{\left(x \right)} - \frac{2 \cos{\left(x \right)}}{x}\right) + \frac{2 \sin{\left(x \right)}}{x^{2}}}{x} = \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}}}{x}$$
- No
es decir, función
no es
par ni impar