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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 =:
-
x/(x^3+2)
-
x*e^(-x)^2
-
2*x^3-15*x^2+36*x-32
-
x*(x-4)
- Derivada de:
-
sin(x)*atan(x)
-
Expresiones idénticas
- sin(x)*atan(x)
- seno de (x) multiplicar por arco tangente de gente de (x)
- sin(x)atan(x)
- sinxatanx
-
Expresiones semejantes
- sin(x)*arctan(x)
- sinx*arctanx
-
Expresiones con funciones
- Seno sin
- sin(x)/1+cos(x)
- sin(-x+3)
- sin(4*x)^(2)
- sinh(tanh(x))
- sint^2
- Arcotangente arctan
- atan((4/x^2)-1)-atan((1/x^2)-1)
- atan(3^x)+7*x+5
- atan(x)^2/x^2+1
- atan(1/(x^2))
- atan(2/(x-2))
Gráfico de la función y = sin(x)*atan(x)
Solución
Ha introducido
[src]
f(x) = sin(x)*atan(x)
$$f{\left(x \right)} = \sin{\left(x \right)} \operatorname{atan}{\left(x \right)}$$
f = sin(x)*atan(x)
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:
$$\sin{\left(x \right)} \operatorname{atan}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = \pi$$
Solución numérica
$$x_{1} = 69.1150383789755$$
$$x_{2} = 65.9734457253857$$
$$x_{3} = -91.106186954104$$
$$x_{4} = 163.362817986669$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = -21.9911485751286$$
$$x_{7} = 12.5663706143592$$
$$x_{8} = 21.9911485751286$$
$$x_{9} = -69.1150383789755$$
$$x_{10} = -100.530964914873$$
$$x_{11} = 3.14159265358979$$
$$x_{12} = -3.14159265358979$$
$$x_{13} = -25.1327412287183$$
$$x_{14} = -15.707963267949$$
$$x_{15} = -53.4070751110265$$
$$x_{16} = -72.2566310325652$$
$$x_{17} = 84.8230016469244$$
$$x_{18} = -81.6814089933346$$
$$x_{19} = -94.2477796076938$$
$$x_{20} = 18.8495559215388$$
$$x_{21} = -65.9734457253857$$
$$x_{22} = 94.2477796076938$$
$$x_{23} = 9.42477796076938$$
$$x_{24} = -40.8407044966673$$
$$x_{25} = -122.522113490002$$
$$x_{26} = 34.5575191894877$$
$$x_{27} = 292.168116783851$$
$$x_{28} = 0$$
$$x_{29} = 53.4070751110265$$
$$x_{30} = 97.3893722612836$$
$$x_{31} = -62.8318530717959$$
$$x_{32} = 59.6902604182061$$
$$x_{33} = -28.2743338823081$$
$$x_{34} = -56.5486677646163$$
$$x_{35} = 91.106186954104$$
$$x_{36} = 15.707963267949$$
$$x_{37} = -18.8495559215388$$
$$x_{38} = 6.28318530717959$$
$$x_{39} = 56.5486677646163$$
$$x_{40} = 87.9645943005142$$
$$x_{41} = 31.4159265358979$$
$$x_{42} = 25.1327412287183$$
$$x_{43} = 43.9822971502571$$
$$x_{44} = -47.1238898038469$$
$$x_{45} = 72.2566310325652$$
$$x_{46} = -34.5575191894877$$
$$x_{47} = -97.3893722612836$$
$$x_{48} = -50.2654824574367$$
$$x_{49} = 100.530964914873$$
$$x_{50} = 1030.44239037745$$
$$x_{51} = 81.6814089933346$$
$$x_{52} = -75.398223686155$$
$$x_{53} = 40.8407044966673$$
$$x_{54} = -9.42477796076938$$
$$x_{55} = 78.5398163397448$$
$$x_{56} = -87.9645943005142$$
$$x_{57} = 37.6991118430775$$
$$x_{58} = -78.5398163397448$$
$$x_{59} = -6.28318530717959$$
$$x_{60} = 50.2654824574367$$
$$x_{61} = -37.6991118430775$$
$$x_{62} = -43.9822971502571$$
$$x_{63} = 47.1238898038469$$
$$x_{64} = 28.2743338823081$$
$$x_{65} = 62.8318530717959$$
$$x_{66} = -31.4159265358979$$
$$x_{67} = -12.5663706143592$$
$$x_{68} = 75.398223686155$$
$$x_{69} = -84.8230016469244$$
o sea hay que resolver la ecuación:
$$\sin{\left(x \right)} \operatorname{atan}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = 0$$
$$x_{2} = \pi$$
Solución numérica
$$x_{1} = 69.1150383789755$$
$$x_{2} = 65.9734457253857$$
$$x_{3} = -91.106186954104$$
$$x_{4} = 163.362817986669$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = -21.9911485751286$$
$$x_{7} = 12.5663706143592$$
$$x_{8} = 21.9911485751286$$
$$x_{9} = -69.1150383789755$$
$$x_{10} = -100.530964914873$$
$$x_{11} = 3.14159265358979$$
$$x_{12} = -3.14159265358979$$
$$x_{13} = -25.1327412287183$$
$$x_{14} = -15.707963267949$$
$$x_{15} = -53.4070751110265$$
$$x_{16} = -72.2566310325652$$
$$x_{17} = 84.8230016469244$$
$$x_{18} = -81.6814089933346$$
$$x_{19} = -94.2477796076938$$
$$x_{20} = 18.8495559215388$$
$$x_{21} = -65.9734457253857$$
$$x_{22} = 94.2477796076938$$
$$x_{23} = 9.42477796076938$$
$$x_{24} = -40.8407044966673$$
$$x_{25} = -122.522113490002$$
$$x_{26} = 34.5575191894877$$
$$x_{27} = 292.168116783851$$
$$x_{28} = 0$$
$$x_{29} = 53.4070751110265$$
$$x_{30} = 97.3893722612836$$
$$x_{31} = -62.8318530717959$$
$$x_{32} = 59.6902604182061$$
$$x_{33} = -28.2743338823081$$
$$x_{34} = -56.5486677646163$$
$$x_{35} = 91.106186954104$$
$$x_{36} = 15.707963267949$$
$$x_{37} = -18.8495559215388$$
$$x_{38} = 6.28318530717959$$
$$x_{39} = 56.5486677646163$$
$$x_{40} = 87.9645943005142$$
$$x_{41} = 31.4159265358979$$
$$x_{42} = 25.1327412287183$$
$$x_{43} = 43.9822971502571$$
$$x_{44} = -47.1238898038469$$
$$x_{45} = 72.2566310325652$$
$$x_{46} = -34.5575191894877$$
$$x_{47} = -97.3893722612836$$
$$x_{48} = -50.2654824574367$$
$$x_{49} = 100.530964914873$$
$$x_{50} = 1030.44239037745$$
$$x_{51} = 81.6814089933346$$
$$x_{52} = -75.398223686155$$
$$x_{53} = 40.8407044966673$$
$$x_{54} = -9.42477796076938$$
$$x_{55} = 78.5398163397448$$
$$x_{56} = -87.9645943005142$$
$$x_{57} = 37.6991118430775$$
$$x_{58} = -78.5398163397448$$
$$x_{59} = -6.28318530717959$$
$$x_{60} = 50.2654824574367$$
$$x_{61} = -37.6991118430775$$
$$x_{62} = -43.9822971502571$$
$$x_{63} = 47.1238898038469$$
$$x_{64} = 28.2743338823081$$
$$x_{65} = 62.8318530717959$$
$$x_{66} = -31.4159265358979$$
$$x_{67} = -12.5663706143592$$
$$x_{68} = 75.398223686155$$
$$x_{69} = -84.8230016469244$$
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
$$\cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(x \right)}}{x^{2} + 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -61.2612281128569$$
$$x_{2} = 73.827545153735$$
$$x_{3} = 1.79103397776014$$
$$x_{4} = -4.74359618362999$$
$$x_{5} = -1.79103397776014$$
$$x_{6} = 54.9780844551589$$
$$x_{7} = -11.0011109024118$$
$$x_{8} = 4.74359618362999$$
$$x_{9} = 17.2809654364073$$
$$x_{10} = -76.9691283508747$$
$$x_{11} = -45.5534044627264$$
$$x_{12} = 29.8458596412272$$
$$x_{13} = 95.8186457300403$$
$$x_{14} = 39.2703275097022$$
$$x_{15} = 45.5534044627264$$
$$x_{16} = -86.3938838884988$$
$$x_{17} = 11.0011109024118$$
$$x_{18} = 20.4219240188353$$
$$x_{19} = 26.7044507808002$$
$$x_{20} = 98.9602340090719$$
$$x_{21} = -80.1107126426188$$
$$x_{22} = 32.987318864864$$
$$x_{23} = -32.987318864864$$
$$x_{24} = -89.5354705986824$$
$$x_{25} = -64.4028043800929$$
$$x_{26} = -51.8365185642119$$
$$x_{27} = 67.5443828898684$$
$$x_{28} = 48.694958052076$$
$$x_{29} = -70.6859632509207$$
$$x_{30} = 51.8365185642119$$
$$x_{31} = -54.9780844551589$$
$$x_{32} = 0$$
$$x_{33} = -92.6770579048997$$
$$x_{34} = -42.4118599365484$$
$$x_{35} = 83.2522978663322$$
$$x_{36} = -14.1404840184881$$
$$x_{37} = -98.9602340090719$$
$$x_{38} = -95.8186457300403$$
$$x_{39} = -23.5631212086715$$
$$x_{40} = 7.8649958747173$$
$$x_{41} = 89.5354705986824$$
$$x_{42} = -73.827545153735$$
$$x_{43} = 80.1107126426188$$
$$x_{44} = 92.6770579048997$$
$$x_{45} = 61.2612281128569$$
$$x_{46} = 23.5631212086715$$
$$x_{47} = -36.1288116046131$$
$$x_{48} = -26.7044507808002$$
$$x_{49} = -29.8458596412272$$
$$x_{50} = -58.1196545885159$$
$$x_{51} = -7.8649958747173$$
$$x_{52} = -48.694958052076$$
$$x_{53} = -39.2703275097022$$
$$x_{54} = 76.9691283508747$$
$$x_{55} = 64.4028043800929$$
$$x_{56} = 42.4118599365484$$
$$x_{57} = -20.4219240188353$$
$$x_{58} = -67.5443828898684$$
$$x_{59} = -17.2809654364073$$
$$x_{60} = 36.1288116046131$$
$$x_{61} = 70.6859632509207$$
$$x_{62} = 86.3938838884988$$
$$x_{63} = -83.2522978663322$$
$$x_{64} = 14.1404840184881$$
$$x_{65} = 58.1196545885159$$
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:
$$x_{1} = -61.2612281128569$$
$$x_{2} = 73.827545153735$$
$$x_{3} = -4.74359618362999$$
$$x_{4} = 54.9780844551589$$
$$x_{5} = -11.0011109024118$$
$$x_{6} = 4.74359618362999$$
$$x_{7} = 17.2809654364073$$
$$x_{8} = 29.8458596412272$$
$$x_{9} = -86.3938838884988$$
$$x_{10} = 11.0011109024118$$
$$x_{11} = 98.9602340090719$$
$$x_{12} = -80.1107126426188$$
$$x_{13} = 67.5443828898684$$
$$x_{14} = 48.694958052076$$
$$x_{15} = -54.9780844551589$$
$$x_{16} = 0$$
$$x_{17} = -92.6770579048997$$
$$x_{18} = -42.4118599365484$$
$$x_{19} = -98.9602340090719$$
$$x_{20} = -23.5631212086715$$
$$x_{21} = -73.827545153735$$
$$x_{22} = 80.1107126426188$$
$$x_{23} = 92.6770579048997$$
$$x_{24} = 61.2612281128569$$
$$x_{25} = 23.5631212086715$$
$$x_{26} = -36.1288116046131$$
$$x_{27} = -29.8458596412272$$
$$x_{28} = -48.694958052076$$
$$x_{29} = 42.4118599365484$$
$$x_{30} = -67.5443828898684$$
$$x_{31} = -17.2809654364073$$
$$x_{32} = 36.1288116046131$$
$$x_{33} = 86.3938838884988$$
Puntos máximos de la función:
$$x_{33} = 1.79103397776014$$
$$x_{33} = -1.79103397776014$$
$$x_{33} = -76.9691283508747$$
$$x_{33} = -45.5534044627264$$
$$x_{33} = 95.8186457300403$$
$$x_{33} = 39.2703275097022$$
$$x_{33} = 45.5534044627264$$
$$x_{33} = 20.4219240188353$$
$$x_{33} = 26.7044507808002$$
$$x_{33} = 32.987318864864$$
$$x_{33} = -32.987318864864$$
$$x_{33} = -89.5354705986824$$
$$x_{33} = -64.4028043800929$$
$$x_{33} = -51.8365185642119$$
$$x_{33} = -70.6859632509207$$
$$x_{33} = 51.8365185642119$$
$$x_{33} = 83.2522978663322$$
$$x_{33} = -14.1404840184881$$
$$x_{33} = -95.8186457300403$$
$$x_{33} = 7.8649958747173$$
$$x_{33} = 89.5354705986824$$
$$x_{33} = -26.7044507808002$$
$$x_{33} = -58.1196545885159$$
$$x_{33} = -7.8649958747173$$
$$x_{33} = -39.2703275097022$$
$$x_{33} = 76.9691283508747$$
$$x_{33} = 64.4028043800929$$
$$x_{33} = -20.4219240188353$$
$$x_{33} = 70.6859632509207$$
$$x_{33} = -83.2522978663322$$
$$x_{33} = 14.1404840184881$$
$$x_{33} = 58.1196545885159$$
Decrece en los intervalos
$$\left[98.9602340090719, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9602340090719\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\cos{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{\sin{\left(x \right)}}{x^{2} + 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -61.2612281128569$$
$$x_{2} = 73.827545153735$$
$$x_{3} = 1.79103397776014$$
$$x_{4} = -4.74359618362999$$
$$x_{5} = -1.79103397776014$$
$$x_{6} = 54.9780844551589$$
$$x_{7} = -11.0011109024118$$
$$x_{8} = 4.74359618362999$$
$$x_{9} = 17.2809654364073$$
$$x_{10} = -76.9691283508747$$
$$x_{11} = -45.5534044627264$$
$$x_{12} = 29.8458596412272$$
$$x_{13} = 95.8186457300403$$
$$x_{14} = 39.2703275097022$$
$$x_{15} = 45.5534044627264$$
$$x_{16} = -86.3938838884988$$
$$x_{17} = 11.0011109024118$$
$$x_{18} = 20.4219240188353$$
$$x_{19} = 26.7044507808002$$
$$x_{20} = 98.9602340090719$$
$$x_{21} = -80.1107126426188$$
$$x_{22} = 32.987318864864$$
$$x_{23} = -32.987318864864$$
$$x_{24} = -89.5354705986824$$
$$x_{25} = -64.4028043800929$$
$$x_{26} = -51.8365185642119$$
$$x_{27} = 67.5443828898684$$
$$x_{28} = 48.694958052076$$
$$x_{29} = -70.6859632509207$$
$$x_{30} = 51.8365185642119$$
$$x_{31} = -54.9780844551589$$
$$x_{32} = 0$$
$$x_{33} = -92.6770579048997$$
$$x_{34} = -42.4118599365484$$
$$x_{35} = 83.2522978663322$$
$$x_{36} = -14.1404840184881$$
$$x_{37} = -98.9602340090719$$
$$x_{38} = -95.8186457300403$$
$$x_{39} = -23.5631212086715$$
$$x_{40} = 7.8649958747173$$
$$x_{41} = 89.5354705986824$$
$$x_{42} = -73.827545153735$$
$$x_{43} = 80.1107126426188$$
$$x_{44} = 92.6770579048997$$
$$x_{45} = 61.2612281128569$$
$$x_{46} = 23.5631212086715$$
$$x_{47} = -36.1288116046131$$
$$x_{48} = -26.7044507808002$$
$$x_{49} = -29.8458596412272$$
$$x_{50} = -58.1196545885159$$
$$x_{51} = -7.8649958747173$$
$$x_{52} = -48.694958052076$$
$$x_{53} = -39.2703275097022$$
$$x_{54} = 76.9691283508747$$
$$x_{55} = 64.4028043800929$$
$$x_{56} = 42.4118599365484$$
$$x_{57} = -20.4219240188353$$
$$x_{58} = -67.5443828898684$$
$$x_{59} = -17.2809654364073$$
$$x_{60} = 36.1288116046131$$
$$x_{61} = 70.6859632509207$$
$$x_{62} = 86.3938838884988$$
$$x_{63} = -83.2522978663322$$
$$x_{64} = 14.1404840184881$$
$$x_{65} = 58.1196545885159$$
Signos de extremos en los puntos:
(-61.26122811285691, -1.5544742153389)
(73.82754515373497, -1.5572520643401)
(1.7910339777601358, 1.03593336473382)
(-4.7435961836299905, -1.36236430357899)
(-1.7910339777601358, 1.03593336473382)
(54.97808445515894, -1.55260923119595)
(-11.001110902411808, -1.4801228582299)
(4.7435961836299905, -1.36236430357899)
(17.280965436407328, -1.51298997049872)
(-76.96912835087466, 1.55780482663327)
(-45.553404462726384, 1.54884752107669)
(29.84585964122719, -1.53730296241164)
(95.8186457300403, 1.56036031976415)
(39.270327509702184, 1.54533717389503)
(45.553404462726384, 1.54884752107669)
(-86.39388388849885, -1.55922194448498)
(11.001110902411808, -1.4801228582299)
(20.42192401883534, 1.52186654514573)
(26.704450780800197, 1.53336623503182)
(98.96023400907185, -1.56069159831347)
(-80.11071264261882, -1.55831424223611)
(32.98731886486398, 1.54049065473337)
(-32.98731886486398, 1.54049065473337)
(-89.53547059868244, 1.55962802821361)
(-64.4028043800929, 1.5552702816364)
(-51.83651856421186, 1.55150725579965)
(67.54438288986843, -1.55599231281641)
(48.694958052076046, -1.55026314860543)
(-70.68596325092066, 1.55665017729342)
(51.83651856421186, 1.55150725579965)
(-54.97808445515894, -1.55260923119595)
(0, 0)
(-92.6770579048997, -1.5600065842339)
(-42.41185993654844, -1.54722228447912)
(83.25229786633224, 1.55878521728754)
(-14.140484018488104, 1.50018667629801)
(-98.96023400907185, -1.56069159831347)
(-95.8186457300403, 1.56036031976415)
(-23.56312120867149, -1.52838152134177)
(7.864995874717303, 1.44424164730901)
(89.53547059868244, 1.55962802821361)
(-73.82754515373497, -1.5572520643401)
(80.11071264261882, -1.55831424223611)
(92.6770579048997, -1.5600065842339)
(61.26122811285691, -1.5544742153389)
(23.56312120867149, -1.52838152134177)
(-36.128811604613105, -1.54312446153498)
(-26.704450780800197, 1.53336623503182)
(-29.84585964122719, -1.53730296241164)
(-58.119654588515886, 1.55359211279627)
(-7.864995874717303, 1.44424164730901)
(-48.694958052076046, -1.55026314860543)
(-39.270327509702184, 1.54533717389503)
(76.96912835087466, 1.55780482663327)
(64.4028043800929, 1.5552702816364)
(42.41185993654844, -1.54722228447912)
(-20.42192401883534, 1.52186654514573)
(-67.54438288986843, -1.55599231281641)
(-17.280965436407328, -1.51298997049872)
(36.128811604613105, -1.54312446153498)
(70.68596325092066, 1.55665017729342)
(86.39388388849885, -1.55922194448498)
(-83.25229786633224, 1.55878521728754)
(14.140484018488104, 1.50018667629801)
(58.119654588515886, 1.55359211279627)
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:
$$x_{1} = -61.2612281128569$$
$$x_{2} = 73.827545153735$$
$$x_{3} = -4.74359618362999$$
$$x_{4} = 54.9780844551589$$
$$x_{5} = -11.0011109024118$$
$$x_{6} = 4.74359618362999$$
$$x_{7} = 17.2809654364073$$
$$x_{8} = 29.8458596412272$$
$$x_{9} = -86.3938838884988$$
$$x_{10} = 11.0011109024118$$
$$x_{11} = 98.9602340090719$$
$$x_{12} = -80.1107126426188$$
$$x_{13} = 67.5443828898684$$
$$x_{14} = 48.694958052076$$
$$x_{15} = -54.9780844551589$$
$$x_{16} = 0$$
$$x_{17} = -92.6770579048997$$
$$x_{18} = -42.4118599365484$$
$$x_{19} = -98.9602340090719$$
$$x_{20} = -23.5631212086715$$
$$x_{21} = -73.827545153735$$
$$x_{22} = 80.1107126426188$$
$$x_{23} = 92.6770579048997$$
$$x_{24} = 61.2612281128569$$
$$x_{25} = 23.5631212086715$$
$$x_{26} = -36.1288116046131$$
$$x_{27} = -29.8458596412272$$
$$x_{28} = -48.694958052076$$
$$x_{29} = 42.4118599365484$$
$$x_{30} = -67.5443828898684$$
$$x_{31} = -17.2809654364073$$
$$x_{32} = 36.1288116046131$$
$$x_{33} = 86.3938838884988$$
Puntos máximos de la función:
$$x_{33} = 1.79103397776014$$
$$x_{33} = -1.79103397776014$$
$$x_{33} = -76.9691283508747$$
$$x_{33} = -45.5534044627264$$
$$x_{33} = 95.8186457300403$$
$$x_{33} = 39.2703275097022$$
$$x_{33} = 45.5534044627264$$
$$x_{33} = 20.4219240188353$$
$$x_{33} = 26.7044507808002$$
$$x_{33} = 32.987318864864$$
$$x_{33} = -32.987318864864$$
$$x_{33} = -89.5354705986824$$
$$x_{33} = -64.4028043800929$$
$$x_{33} = -51.8365185642119$$
$$x_{33} = -70.6859632509207$$
$$x_{33} = 51.8365185642119$$
$$x_{33} = 83.2522978663322$$
$$x_{33} = -14.1404840184881$$
$$x_{33} = -95.8186457300403$$
$$x_{33} = 7.8649958747173$$
$$x_{33} = 89.5354705986824$$
$$x_{33} = -26.7044507808002$$
$$x_{33} = -58.1196545885159$$
$$x_{33} = -7.8649958747173$$
$$x_{33} = -39.2703275097022$$
$$x_{33} = 76.9691283508747$$
$$x_{33} = 64.4028043800929$$
$$x_{33} = -20.4219240188353$$
$$x_{33} = 70.6859632509207$$
$$x_{33} = -83.2522978663322$$
$$x_{33} = 14.1404840184881$$
$$x_{33} = 58.1196545885159$$
Decrece en los intervalos
$$\left[98.9602340090719, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -98.9602340090719\right]$$
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
$$- \frac{2 x \sin{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \sin{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x^{2} + 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -28.2759609016006$$
$$x_{2} = 21.9938531789562$$
$$x_{3} = -18.8532519435147$$
$$x_{4} = 15.7133138949299$$
$$x_{5} = -72.256877018428$$
$$x_{6} = 75.3984495204372$$
$$x_{7} = 18.8532519435147$$
$$x_{8} = -21.9938531789562$$
$$x_{9} = 43.9829646343884$$
$$x_{10} = 87.9647600263064$$
$$x_{11} = -53.4075267137478$$
$$x_{12} = -87.9647600263064$$
$$x_{13} = -56.5490703302555$$
$$x_{14} = -81.6816012996122$$
$$x_{15} = 65.9737410345762$$
$$x_{16} = 56.5490703302555$$
$$x_{17} = 40.8414794224676$$
$$x_{18} = 78.5400244008501$$
$$x_{19} = 72.256877018428$$
$$x_{20} = -43.9829646343884$$
$$x_{21} = 31.4172417711937$$
$$x_{22} = 3.27025559384994$$
$$x_{23} = -94.2479239057106$$
$$x_{24} = -15.7133138949299$$
$$x_{25} = -3.27025559384994$$
$$x_{26} = 84.8231799223075$$
$$x_{27} = -12.5747920390384$$
$$x_{28} = -37.7000223964691$$
$$x_{29} = -97.3895073712924$$
$$x_{30} = -50.2659926334542$$
$$x_{31} = -84.8231799223075$$
$$x_{32} = -78.5400244008501$$
$$x_{33} = 12.5747920390384$$
$$x_{34} = -25.134805522921$$
$$x_{35} = -59.690621520068$$
$$x_{36} = 28.2759609016006$$
$$x_{37} = -31.4172417711937$$
$$x_{38} = -75.3984495204372$$
$$x_{39} = 62.8321788004678$$
$$x_{40} = 37.7000223964691$$
$$x_{41} = 6.31756549582733$$
$$x_{42} = 147.654913369085$$
$$x_{43} = 69.1153073388908$$
$$x_{44} = -47.1244707319001$$
$$x_{45} = 91.1063414102081$$
$$x_{46} = 0.777864066729225$$
$$x_{47} = -100.531091687256$$
$$x_{48} = 34.558604347829$$
$$x_{49} = 100.531091687256$$
$$x_{50} = -62.8321788004678$$
$$x_{51} = 97.3895073712924$$
$$x_{52} = -65.9737410345762$$
$$x_{53} = 59.690621520068$$
$$x_{54} = 25.134805522921$$
$$x_{55} = 53.4075267137478$$
$$x_{56} = -34.558604347829$$
$$x_{57} = 81.6816012996122$$
$$x_{58} = 50.2659926334542$$
$$x_{59} = -40.8414794224676$$
$$x_{60} = 9.43990010438496$$
$$x_{61} = -6.31756549582733$$
$$x_{62} = 94.2479239057106$$
$$x_{63} = 47.1244707319001$$
$$x_{64} = -9.43990010438496$$
$$x_{65} = -69.1153073388908$$
$$x_{66} = -91.1063414102081$$
$$x_{67} = -103.672676751813$$
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
$$\left[147.654913369085, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.531091687256\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{2 x \sin{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \sin{\left(x \right)} \operatorname{atan}{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x^{2} + 1} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -28.2759609016006$$
$$x_{2} = 21.9938531789562$$
$$x_{3} = -18.8532519435147$$
$$x_{4} = 15.7133138949299$$
$$x_{5} = -72.256877018428$$
$$x_{6} = 75.3984495204372$$
$$x_{7} = 18.8532519435147$$
$$x_{8} = -21.9938531789562$$
$$x_{9} = 43.9829646343884$$
$$x_{10} = 87.9647600263064$$
$$x_{11} = -53.4075267137478$$
$$x_{12} = -87.9647600263064$$
$$x_{13} = -56.5490703302555$$
$$x_{14} = -81.6816012996122$$
$$x_{15} = 65.9737410345762$$
$$x_{16} = 56.5490703302555$$
$$x_{17} = 40.8414794224676$$
$$x_{18} = 78.5400244008501$$
$$x_{19} = 72.256877018428$$
$$x_{20} = -43.9829646343884$$
$$x_{21} = 31.4172417711937$$
$$x_{22} = 3.27025559384994$$
$$x_{23} = -94.2479239057106$$
$$x_{24} = -15.7133138949299$$
$$x_{25} = -3.27025559384994$$
$$x_{26} = 84.8231799223075$$
$$x_{27} = -12.5747920390384$$
$$x_{28} = -37.7000223964691$$
$$x_{29} = -97.3895073712924$$
$$x_{30} = -50.2659926334542$$
$$x_{31} = -84.8231799223075$$
$$x_{32} = -78.5400244008501$$
$$x_{33} = 12.5747920390384$$
$$x_{34} = -25.134805522921$$
$$x_{35} = -59.690621520068$$
$$x_{36} = 28.2759609016006$$
$$x_{37} = -31.4172417711937$$
$$x_{38} = -75.3984495204372$$
$$x_{39} = 62.8321788004678$$
$$x_{40} = 37.7000223964691$$
$$x_{41} = 6.31756549582733$$
$$x_{42} = 147.654913369085$$
$$x_{43} = 69.1153073388908$$
$$x_{44} = -47.1244707319001$$
$$x_{45} = 91.1063414102081$$
$$x_{46} = 0.777864066729225$$
$$x_{47} = -100.531091687256$$
$$x_{48} = 34.558604347829$$
$$x_{49} = 100.531091687256$$
$$x_{50} = -62.8321788004678$$
$$x_{51} = 97.3895073712924$$
$$x_{52} = -65.9737410345762$$
$$x_{53} = 59.690621520068$$
$$x_{54} = 25.134805522921$$
$$x_{55} = 53.4075267137478$$
$$x_{56} = -34.558604347829$$
$$x_{57} = 81.6816012996122$$
$$x_{58} = 50.2659926334542$$
$$x_{59} = -40.8414794224676$$
$$x_{60} = 9.43990010438496$$
$$x_{61} = -6.31756549582733$$
$$x_{62} = 94.2479239057106$$
$$x_{63} = 47.1244707319001$$
$$x_{64} = -9.43990010438496$$
$$x_{65} = -69.1153073388908$$
$$x_{66} = -91.1063414102081$$
$$x_{67} = -103.672676751813$$
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
$$\left[147.654913369085, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.531091687256\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle \pi$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle \pi$$
$$\lim_{x \to \infty}\left(\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle \pi$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle \pi$$
$$\lim_{x \to -\infty}\left(\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle \pi$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle \pi$$
$$\lim_{x \to \infty}\left(\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle \pi$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle \pi$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)*atan(x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} \operatorname{atan}{\left(x \right)}}{x}\right) = 0$$
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:
$$\sin{\left(x \right)} \operatorname{atan}{\left(x \right)} = \sin{\left(x \right)} \operatorname{atan}{\left(x \right)}$$
- No
$$\sin{\left(x \right)} \operatorname{atan}{\left(x \right)} = - \sin{\left(x \right)} \operatorname{atan}{\left(x \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\sin{\left(x \right)} \operatorname{atan}{\left(x \right)} = \sin{\left(x \right)} \operatorname{atan}{\left(x \right)}$$
- No
$$\sin{\left(x \right)} \operatorname{atan}{\left(x \right)} = - \sin{\left(x \right)} \operatorname{atan}{\left(x \right)}$$
- No
es decir, función
no es
par ni impar