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- Puntos de cruce de las funciones
-
¿Cómo usar?
- Gráfico de la función y =:
-
x^4-x^3
-
x*e^x
-
x^2-3*x+1
-
x^3/(x-1)^2
-
Expresiones idénticas
- atan(sin(x)/x)
- arco tangente de gente de ( seno de (x) dividir por x)
- atansinx/x
- atan(sin(x) dividir por x)
-
Expresiones semejantes
- arctan(sin(x)/x)
- atan(sinx/x)
-
Expresiones con funciones
- Arcotangente arctan
- atan(sqrt(x))
- atan(x)/x
- atan(x)*log(x)
- atan(x)/3
- atan(x^2)
- Seno sin
- sin(x)^(3)
- sin(x/5)*exp(x/10)+5*exp(-x/2)
- sin(x)/(2*x+16)
- sin(2*x+3)^(2)
- sin(x^3)
Gráfico de la función y = atan(sin(x)/x)
Solución
Ha introducido
[src]
/sin(x)\
f(x) = atan|------|
\ x /
$$f{\left(x \right)} = \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)}$$
f = atan(sin(x)/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:
$$\operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \pi$$
Solución numérica
$$x_{1} = -59.6902604182061$$
$$x_{2} = -62.8318530717959$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 87.9645943005142$$
$$x_{5} = -56.5486677646163$$
$$x_{6} = 31.4159265358979$$
$$x_{7} = 69.1150383789755$$
$$x_{8} = 772.831792783089$$
$$x_{9} = -37.6991118430775$$
$$x_{10} = -81.6814089933346$$
$$x_{11} = -84.8230016469244$$
$$x_{12} = -21.9911485751286$$
$$x_{13} = 47.1238898038469$$
$$x_{14} = -15.707963267949$$
$$x_{15} = -12.5663706143592$$
$$x_{16} = 12.5663706143592$$
$$x_{17} = -87.9645943005142$$
$$x_{18} = 53.4070751110265$$
$$x_{19} = 72.2566310325652$$
$$x_{20} = -100.530964914873$$
$$x_{21} = -3.14159265358979$$
$$x_{22} = 307.8760800518$$
$$x_{23} = 34.5575191894877$$
$$x_{24} = -94.2477796076938$$
$$x_{25} = 6.28318530717959$$
$$x_{26} = -69.1150383789755$$
$$x_{27} = 97.3893722612836$$
$$x_{28} = 65.9734457253857$$
$$x_{29} = 15.707963267949$$
$$x_{30} = -50.2654824574367$$
$$x_{31} = -25.1327412287183$$
$$x_{32} = 3.14159265358979$$
$$x_{33} = -18.8495559215388$$
$$x_{34} = 40.8407044966673$$
$$x_{35} = 18.8495559215388$$
$$x_{36} = -53.4070751110265$$
$$x_{37} = 37.6991118430775$$
$$x_{38} = -43.9822971502571$$
$$x_{39} = -78.5398163397448$$
$$x_{40} = -6.28318530717959$$
$$x_{41} = 135.088484104361$$
$$x_{42} = -40.8407044966673$$
$$x_{43} = 43.9822971502571$$
$$x_{44} = 56.5486677646163$$
$$x_{45} = -65.9734457253857$$
$$x_{46} = 25.1327412287183$$
$$x_{47} = 78.5398163397448$$
$$x_{48} = -28.2743338823081$$
$$x_{49} = 75.398223686155$$
$$x_{50} = 59.6902604182061$$
$$x_{51} = -34.5575191894877$$
$$x_{52} = 81.6814089933346$$
$$x_{53} = -47.1238898038469$$
$$x_{54} = 100.530964914873$$
$$x_{55} = -9.42477796076938$$
$$x_{56} = -75.398223686155$$
$$x_{57} = -72.2566310325652$$
$$x_{58} = -31.4159265358979$$
$$x_{59} = 28.2743338823081$$
$$x_{60} = -91.106186954104$$
$$x_{61} = 21.9911485751286$$
$$x_{62} = 62.8318530717959$$
$$x_{63} = 235.619449019234$$
$$x_{64} = 9.42477796076938$$
$$x_{65} = 50.2654824574367$$
$$x_{66} = 94.2477796076938$$
$$x_{67} = 91.106186954104$$
$$x_{68} = 84.8230016469244$$
o sea hay que resolver la ecuación:
$$\operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \pi$$
Solución numérica
$$x_{1} = -59.6902604182061$$
$$x_{2} = -62.8318530717959$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 87.9645943005142$$
$$x_{5} = -56.5486677646163$$
$$x_{6} = 31.4159265358979$$
$$x_{7} = 69.1150383789755$$
$$x_{8} = 772.831792783089$$
$$x_{9} = -37.6991118430775$$
$$x_{10} = -81.6814089933346$$
$$x_{11} = -84.8230016469244$$
$$x_{12} = -21.9911485751286$$
$$x_{13} = 47.1238898038469$$
$$x_{14} = -15.707963267949$$
$$x_{15} = -12.5663706143592$$
$$x_{16} = 12.5663706143592$$
$$x_{17} = -87.9645943005142$$
$$x_{18} = 53.4070751110265$$
$$x_{19} = 72.2566310325652$$
$$x_{20} = -100.530964914873$$
$$x_{21} = -3.14159265358979$$
$$x_{22} = 307.8760800518$$
$$x_{23} = 34.5575191894877$$
$$x_{24} = -94.2477796076938$$
$$x_{25} = 6.28318530717959$$
$$x_{26} = -69.1150383789755$$
$$x_{27} = 97.3893722612836$$
$$x_{28} = 65.9734457253857$$
$$x_{29} = 15.707963267949$$
$$x_{30} = -50.2654824574367$$
$$x_{31} = -25.1327412287183$$
$$x_{32} = 3.14159265358979$$
$$x_{33} = -18.8495559215388$$
$$x_{34} = 40.8407044966673$$
$$x_{35} = 18.8495559215388$$
$$x_{36} = -53.4070751110265$$
$$x_{37} = 37.6991118430775$$
$$x_{38} = -43.9822971502571$$
$$x_{39} = -78.5398163397448$$
$$x_{40} = -6.28318530717959$$
$$x_{41} = 135.088484104361$$
$$x_{42} = -40.8407044966673$$
$$x_{43} = 43.9822971502571$$
$$x_{44} = 56.5486677646163$$
$$x_{45} = -65.9734457253857$$
$$x_{46} = 25.1327412287183$$
$$x_{47} = 78.5398163397448$$
$$x_{48} = -28.2743338823081$$
$$x_{49} = 75.398223686155$$
$$x_{50} = 59.6902604182061$$
$$x_{51} = -34.5575191894877$$
$$x_{52} = 81.6814089933346$$
$$x_{53} = -47.1238898038469$$
$$x_{54} = 100.530964914873$$
$$x_{55} = -9.42477796076938$$
$$x_{56} = -75.398223686155$$
$$x_{57} = -72.2566310325652$$
$$x_{58} = -31.4159265358979$$
$$x_{59} = 28.2743338823081$$
$$x_{60} = -91.106186954104$$
$$x_{61} = 21.9911485751286$$
$$x_{62} = 62.8318530717959$$
$$x_{63} = 235.619449019234$$
$$x_{64} = 9.42477796076938$$
$$x_{65} = 50.2654824574367$$
$$x_{66} = 94.2477796076938$$
$$x_{67} = 91.106186954104$$
$$x_{68} = 84.8230016469244$$
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
$$\frac{\frac{\cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}}{1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 17.2207552719308$$
$$x_{2} = -80.0981286289451$$
$$x_{3} = -83.2401924707234$$
$$x_{4} = 98.9500628243319$$
$$x_{5} = -45.5311340139913$$
$$x_{6} = 347.143107573282$$
$$x_{7} = -86.3822220347287$$
$$x_{8} = 7.72525183693771$$
$$x_{9} = -4.49340945790906$$
$$x_{10} = 4.49340945790906$$
$$x_{11} = -114.659410595023$$
$$x_{12} = 39.2444323611642$$
$$x_{13} = -70.6716857116195$$
$$x_{14} = 10.9041216594289$$
$$x_{15} = -42.3879135681319$$
$$x_{16} = 80.0981286289451$$
$$x_{17} = 89.5242209304172$$
$$x_{18} = -48.6741442319544$$
$$x_{19} = 14.0661939128315$$
$$x_{20} = -36.1006222443756$$
$$x_{21} = -95.8081387868617$$
$$x_{22} = 64.3871195905574$$
$$x_{23} = 61.2447302603744$$
$$x_{24} = -54.9596782878889$$
$$x_{25} = 76.9560263103312$$
$$x_{26} = -76.9560263103312$$
$$x_{27} = -98.9500628243319$$
$$x_{28} = 6.6311787278932 \cdot 10^{-18}$$
$$x_{29} = -7.72525183693771$$
$$x_{30} = 1.19665054005695 \cdot 10^{-16}$$
$$x_{31} = -39.2444323611642$$
$$x_{32} = 111.517572246131$$
$$x_{33} = -20.3713029592876$$
$$x_{34} = -108.375719651675$$
$$x_{35} = -14.0661939128315$$
$$x_{36} = -32.9563890398225$$
$$x_{37} = 54.9596782878889$$
$$x_{38} = 73.8138806006806$$
$$x_{39} = 26.6660542588127$$
$$x_{40} = -444.533110935535$$
$$x_{41} = -26.6660542588127$$
$$x_{42} = -61.2447302603744$$
$$x_{43} = -67.5294347771441$$
$$x_{44} = 29.811598790893$$
$$x_{45} = 51.8169824872797$$
$$x_{46} = 23.519452498689$$
$$x_{47} = -58.1022547544956$$
$$x_{48} = 67.5294347771441$$
$$x_{49} = -10.9041216594289$$
$$x_{50} = -89.5242209304172$$
$$x_{51} = -5.35366042711037 \cdot 10^{-17}$$
$$x_{52} = 86.3822220347287$$
$$x_{53} = -23.519452498689$$
$$x_{54} = -17.2207552719308$$
$$x_{55} = 58.1022547544956$$
$$x_{56} = -92.6661922776228$$
$$x_{57} = -29.811598790893$$
$$x_{58} = 92.6661922776228$$
$$x_{59} = -64.3871195905574$$
$$x_{60} = 32.9563890398225$$
$$x_{61} = 20.3713029592876$$
$$x_{62} = 48.6741442319544$$
$$x_{63} = 45.5311340139913$$
$$x_{64} = 36.1006222443756$$
$$x_{65} = 70.6716857116195$$
$$x_{66} = 83.2401924707234$$
$$x_{67} = 95.8081387868617$$
$$x_{68} = -73.8138806006806$$
$$x_{69} = 42.3879135681319$$
$$x_{70} = -51.8169824872797$$
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:
$$x_{1} = 17.2207552719308$$
$$x_{2} = -80.0981286289451$$
$$x_{3} = 98.9500628243319$$
$$x_{4} = -86.3822220347287$$
$$x_{5} = -4.49340945790906$$
$$x_{6} = 4.49340945790906$$
$$x_{7} = 10.9041216594289$$
$$x_{8} = -42.3879135681319$$
$$x_{9} = 80.0981286289451$$
$$x_{10} = -48.6741442319544$$
$$x_{11} = -36.1006222443756$$
$$x_{12} = 61.2447302603744$$
$$x_{13} = -54.9596782878889$$
$$x_{14} = -98.9500628243319$$
$$x_{15} = 111.517572246131$$
$$x_{16} = 54.9596782878889$$
$$x_{17} = 73.8138806006806$$
$$x_{18} = -444.533110935535$$
$$x_{19} = -61.2447302603744$$
$$x_{20} = -67.5294347771441$$
$$x_{21} = 29.811598790893$$
$$x_{22} = 23.519452498689$$
$$x_{23} = 67.5294347771441$$
$$x_{24} = -10.9041216594289$$
$$x_{25} = 86.3822220347287$$
$$x_{26} = -23.519452498689$$
$$x_{27} = -17.2207552719308$$
$$x_{28} = -92.6661922776228$$
$$x_{29} = -29.811598790893$$
$$x_{30} = 92.6661922776228$$
$$x_{31} = 48.6741442319544$$
$$x_{32} = 36.1006222443756$$
$$x_{33} = -73.8138806006806$$
$$x_{34} = 42.3879135681319$$
Puntos máximos de la función:
$$x_{34} = -83.2401924707234$$
$$x_{34} = -45.5311340139913$$
$$x_{34} = 347.143107573282$$
$$x_{34} = 7.72525183693771$$
$$x_{34} = -114.659410595023$$
$$x_{34} = 39.2444323611642$$
$$x_{34} = -70.6716857116195$$
$$x_{34} = 89.5242209304172$$
$$x_{34} = 14.0661939128315$$
$$x_{34} = -95.8081387868617$$
$$x_{34} = 64.3871195905574$$
$$x_{34} = 76.9560263103312$$
$$x_{34} = -76.9560263103312$$
$$x_{34} = 6.6311787278932 \cdot 10^{-18}$$
$$x_{34} = -7.72525183693771$$
$$x_{34} = 1.19665054005695 \cdot 10^{-16}$$
$$x_{34} = -39.2444323611642$$
$$x_{34} = -20.3713029592876$$
$$x_{34} = -108.375719651675$$
$$x_{34} = -14.0661939128315$$
$$x_{34} = -32.9563890398225$$
$$x_{34} = 26.6660542588127$$
$$x_{34} = -26.6660542588127$$
$$x_{34} = 51.8169824872797$$
$$x_{34} = -58.1022547544956$$
$$x_{34} = -89.5242209304172$$
$$x_{34} = -5.35366042711037 \cdot 10^{-17}$$
$$x_{34} = 58.1022547544956$$
$$x_{34} = -64.3871195905574$$
$$x_{34} = 32.9563890398225$$
$$x_{34} = 20.3713029592876$$
$$x_{34} = 45.5311340139913$$
$$x_{34} = 70.6716857116195$$
$$x_{34} = 83.2401924707234$$
$$x_{34} = 95.8081387868617$$
$$x_{34} = -51.8169824872797$$
Decrece en los intervalos
$$\left[111.517572246131, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -444.533110935535\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$\frac{\frac{\cos{\left(x \right)}}{x} - \frac{\sin{\left(x \right)}}{x^{2}}}{1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 17.2207552719308$$
$$x_{2} = -80.0981286289451$$
$$x_{3} = -83.2401924707234$$
$$x_{4} = 98.9500628243319$$
$$x_{5} = -45.5311340139913$$
$$x_{6} = 347.143107573282$$
$$x_{7} = -86.3822220347287$$
$$x_{8} = 7.72525183693771$$
$$x_{9} = -4.49340945790906$$
$$x_{10} = 4.49340945790906$$
$$x_{11} = -114.659410595023$$
$$x_{12} = 39.2444323611642$$
$$x_{13} = -70.6716857116195$$
$$x_{14} = 10.9041216594289$$
$$x_{15} = -42.3879135681319$$
$$x_{16} = 80.0981286289451$$
$$x_{17} = 89.5242209304172$$
$$x_{18} = -48.6741442319544$$
$$x_{19} = 14.0661939128315$$
$$x_{20} = -36.1006222443756$$
$$x_{21} = -95.8081387868617$$
$$x_{22} = 64.3871195905574$$
$$x_{23} = 61.2447302603744$$
$$x_{24} = -54.9596782878889$$
$$x_{25} = 76.9560263103312$$
$$x_{26} = -76.9560263103312$$
$$x_{27} = -98.9500628243319$$
$$x_{28} = 6.6311787278932 \cdot 10^{-18}$$
$$x_{29} = -7.72525183693771$$
$$x_{30} = 1.19665054005695 \cdot 10^{-16}$$
$$x_{31} = -39.2444323611642$$
$$x_{32} = 111.517572246131$$
$$x_{33} = -20.3713029592876$$
$$x_{34} = -108.375719651675$$
$$x_{35} = -14.0661939128315$$
$$x_{36} = -32.9563890398225$$
$$x_{37} = 54.9596782878889$$
$$x_{38} = 73.8138806006806$$
$$x_{39} = 26.6660542588127$$
$$x_{40} = -444.533110935535$$
$$x_{41} = -26.6660542588127$$
$$x_{42} = -61.2447302603744$$
$$x_{43} = -67.5294347771441$$
$$x_{44} = 29.811598790893$$
$$x_{45} = 51.8169824872797$$
$$x_{46} = 23.519452498689$$
$$x_{47} = -58.1022547544956$$
$$x_{48} = 67.5294347771441$$
$$x_{49} = -10.9041216594289$$
$$x_{50} = -89.5242209304172$$
$$x_{51} = -5.35366042711037 \cdot 10^{-17}$$
$$x_{52} = 86.3822220347287$$
$$x_{53} = -23.519452498689$$
$$x_{54} = -17.2207552719308$$
$$x_{55} = 58.1022547544956$$
$$x_{56} = -92.6661922776228$$
$$x_{57} = -29.811598790893$$
$$x_{58} = 92.6661922776228$$
$$x_{59} = -64.3871195905574$$
$$x_{60} = 32.9563890398225$$
$$x_{61} = 20.3713029592876$$
$$x_{62} = 48.6741442319544$$
$$x_{63} = 45.5311340139913$$
$$x_{64} = 36.1006222443756$$
$$x_{65} = 70.6716857116195$$
$$x_{66} = 83.2401924707234$$
$$x_{67} = 95.8081387868617$$
$$x_{68} = -73.8138806006806$$
$$x_{69} = 42.3879135681319$$
$$x_{70} = -51.8169824872797$$
Signos de extremos en los puntos:
(17.22075527193077, -0.0579069904626494)
(-80.09812862894512, -0.0124830648822234)
(-83.2401924707234, 0.0120119827214717)
(98.95006282433188, -0.0101052477531854)
(-45.53113401399128, 0.0219541703503676)
(347.14310757328207, 0.00288063643813538)
(-86.38222203472871, -0.0115751634669471)
(7.725251836937707, 0.127676240235727)
(-4.493409457909064, -0.213910117849372)
(4.493409457909064, -0.213910117849372)
(-114.65941059502308, 0.00872092935202151)
(39.24443236116419, 0.0254675456156415)
(-70.6716857116195, 0.0141475780913362)
(10.904121659428899, -0.0910725728546308)
(-42.38791356813192, -0.0235806965804474)
(80.09812862894512, -0.0124830648822234)
(89.52422093041719, 0.0111690001795289)
(-48.674144231954386, -0.0205375660300922)
(14.066193912831473, 0.0707949488720282)
(-36.10062224437561, -0.0276826587859623)
(-95.8081387868617, 0.0104365791930517)
(64.38711959055742, 0.0155279356717673)
(61.2447302603744, -0.0163243091157563)
(-54.959678287888934, -0.0181901397978919)
(76.95602631033118, 0.0129926058532411)
(-76.95602631033118, 0.0129926058532411)
(-98.95006282433188, -0.0101052477531854)
(6.631178727893204e-18, 0.785398163397448)
(-7.725251836937707, 0.127676240235727)
(1.1966505400569458e-16, 0.785398163397448)
(-39.24443236116419, 0.0254675456156415)
(111.51757224613101, -0.00896659582847264)
(-20.37130295928756, 0.0489903930793876)
(-108.37571965167469, 0.00922650442939595)
(-14.066193912831473, 0.0707949488720282)
(-32.956389039822476, 0.030319876798918)
(54.959678287888934, -0.0181901397978919)
(73.81388060068065, -0.0135455158360732)
(26.666054258812675, 0.0374569924422619)
(-444.5331109355349, -0.00224954172897443)
(-26.666054258812675, 0.0374569924422619)
(-61.2447302603744, -0.0163243091157563)
(-67.52943477714412, -0.0148056520158585)
(29.81159879089296, -0.0335125834639765)
(51.81698248727967, 0.0192927054949316)
(23.519452498689006, -0.0424540928646622)
(-58.10225475449559, 0.0172067891118748)
(67.52943477714412, -0.0148056520158585)
(-10.904121659428899, -0.0910725728546308)
(-89.52422093041719, 0.0111690001795289)
(-5.3536604271103745e-17, 0.785398163397448)
(86.38222203472871, -0.0115751634669471)
(-23.519452498689006, -0.0424540928646622)
(-17.22075527193077, -0.0579069904626494)
(58.10225475449559, 0.0172067891118748)
(-92.66619227762284, -0.0107903750476836)
(-29.81159879089296, -0.0335125834639765)
(92.66619227762284, -0.0107903750476836)
(-64.38711959055742, 0.0155279356717673)
(32.956389039822476, 0.030319876798918)
(20.37130295928756, 0.0489903930793876)
(48.674144231954386, -0.0205375660300922)
(45.53113401399128, 0.0219541703503676)
(36.10062224437561, -0.0276826587859623)
(70.6716857116195, 0.0141475780913362)
(83.2401924707234, 0.0120119827214717)
(95.8081387868617, 0.0104365791930517)
(-73.81388060068065, -0.0135455158360732)
(42.38791356813192, -0.0235806965804474)
(-51.81698248727967, 0.0192927054949316)
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:
$$x_{1} = 17.2207552719308$$
$$x_{2} = -80.0981286289451$$
$$x_{3} = 98.9500628243319$$
$$x_{4} = -86.3822220347287$$
$$x_{5} = -4.49340945790906$$
$$x_{6} = 4.49340945790906$$
$$x_{7} = 10.9041216594289$$
$$x_{8} = -42.3879135681319$$
$$x_{9} = 80.0981286289451$$
$$x_{10} = -48.6741442319544$$
$$x_{11} = -36.1006222443756$$
$$x_{12} = 61.2447302603744$$
$$x_{13} = -54.9596782878889$$
$$x_{14} = -98.9500628243319$$
$$x_{15} = 111.517572246131$$
$$x_{16} = 54.9596782878889$$
$$x_{17} = 73.8138806006806$$
$$x_{18} = -444.533110935535$$
$$x_{19} = -61.2447302603744$$
$$x_{20} = -67.5294347771441$$
$$x_{21} = 29.811598790893$$
$$x_{22} = 23.519452498689$$
$$x_{23} = 67.5294347771441$$
$$x_{24} = -10.9041216594289$$
$$x_{25} = 86.3822220347287$$
$$x_{26} = -23.519452498689$$
$$x_{27} = -17.2207552719308$$
$$x_{28} = -92.6661922776228$$
$$x_{29} = -29.811598790893$$
$$x_{30} = 92.6661922776228$$
$$x_{31} = 48.6741442319544$$
$$x_{32} = 36.1006222443756$$
$$x_{33} = -73.8138806006806$$
$$x_{34} = 42.3879135681319$$
Puntos máximos de la función:
$$x_{34} = -83.2401924707234$$
$$x_{34} = -45.5311340139913$$
$$x_{34} = 347.143107573282$$
$$x_{34} = 7.72525183693771$$
$$x_{34} = -114.659410595023$$
$$x_{34} = 39.2444323611642$$
$$x_{34} = -70.6716857116195$$
$$x_{34} = 89.5242209304172$$
$$x_{34} = 14.0661939128315$$
$$x_{34} = -95.8081387868617$$
$$x_{34} = 64.3871195905574$$
$$x_{34} = 76.9560263103312$$
$$x_{34} = -76.9560263103312$$
$$x_{34} = 6.6311787278932 \cdot 10^{-18}$$
$$x_{34} = -7.72525183693771$$
$$x_{34} = 1.19665054005695 \cdot 10^{-16}$$
$$x_{34} = -39.2444323611642$$
$$x_{34} = -20.3713029592876$$
$$x_{34} = -108.375719651675$$
$$x_{34} = -14.0661939128315$$
$$x_{34} = -32.9563890398225$$
$$x_{34} = 26.6660542588127$$
$$x_{34} = -26.6660542588127$$
$$x_{34} = 51.8169824872797$$
$$x_{34} = -58.1022547544956$$
$$x_{34} = -89.5242209304172$$
$$x_{34} = -5.35366042711037 \cdot 10^{-17}$$
$$x_{34} = 58.1022547544956$$
$$x_{34} = -64.3871195905574$$
$$x_{34} = 32.9563890398225$$
$$x_{34} = 20.3713029592876$$
$$x_{34} = 45.5311340139913$$
$$x_{34} = 70.6716857116195$$
$$x_{34} = 83.2401924707234$$
$$x_{34} = 95.8081387868617$$
$$x_{34} = -51.8169824872797$$
Decrece en los intervalos
$$\left[111.517572246131, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -444.533110935535\right]$$
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
$$- \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}} + \frac{2 \left(\cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}\right)^{2} \sin{\left(x \right)}}{x^{2} \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}}{x \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 56.5132926241755$$
$$x_{2} = 18.7432530945386$$
$$x_{3} = 34.4996123350132$$
$$x_{4} = 157.066899940015$$
$$x_{5} = -53.3696181339615$$
$$x_{6} = -97.3688346960149$$
$$x_{7} = -50.2256832197934$$
$$x_{8} = 40.7917141624847$$
$$x_{9} = 106.795425016936$$
$$x_{10} = -9.21096438740149$$
$$x_{11} = 100.511069234565$$
$$x_{12} = 28.2035393053095$$
$$x_{13} = 59.6567478435559$$
$$x_{14} = 21.9000773156394$$
$$x_{15} = 91.0842327848165$$
$$x_{16} = -94.2265573558031$$
$$x_{17} = 37.6460352959305$$
$$x_{18} = 5.95939190757933$$
$$x_{19} = -31.3522215217643$$
$$x_{20} = -34.4996123350132$$
$$x_{21} = 69.0860970774096$$
$$x_{22} = -2.45871417599962$$
$$x_{23} = -47.0814357397523$$
$$x_{24} = 25.053079662454$$
$$x_{25} = 65.9431258539286$$
$$x_{26} = 75.3716947511882$$
$$x_{27} = -100.511069234565$$
$$x_{28} = 31.3522215217643$$
$$x_{29} = -81.6569211705466$$
$$x_{30} = -56.5132926241755$$
$$x_{31} = -75.3716947511882$$
$$x_{32} = 53.3696181339615$$
$$x_{33} = 50.2256832197934$$
$$x_{34} = 81.6569211705466$$
$$x_{35} = -59.6567478435559$$
$$x_{36} = -78.5143487963623$$
$$x_{37} = 87.9418559209576$$
$$x_{38} = -65.9431258539286$$
$$x_{39} = -72.2289483771681$$
$$x_{40} = 72.2289483771681$$
$$x_{41} = -5.95939190757933$$
$$x_{42} = -87.9418559209576$$
$$x_{43} = -62.8000167068325$$
$$x_{44} = -84.7994209518635$$
$$x_{45} = -15.5802941824244$$
$$x_{46} = 9.21096438740149$$
$$x_{47} = 97.3688346960149$$
$$x_{48} = 78.5143487963623$$
$$x_{49} = 62.8000167068325$$
$$x_{50} = 47.0814357397523$$
$$x_{51} = -28.2035393053095$$
$$x_{52} = 84.7994209518635$$
$$x_{53} = -69.0860970774096$$
$$x_{54} = -21.9000773156394$$
$$x_{55} = -18.7432530945386$$
$$x_{56} = -40.7917141624847$$
$$x_{57} = 2.45871417599962$$
$$x_{58} = -12.4065403639626$$
$$x_{59} = -91.0842327848165$$
$$x_{60} = 12.4065403639626$$
$$x_{61} = -43.9368086315937$$
$$x_{62} = 94.2265573558031$$
$$x_{63} = -25.053079662454$$
$$x_{64} = 15.5802941824244$$
$$x_{65} = 43.9368086315937$$
$$x_{66} = -37.6460352959305$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}} + \frac{2 \left(\cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}\right)^{2} \sin{\left(x \right)}}{x^{2} \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}}{x \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}\right) = - \frac{1}{6}$$
$$\lim_{x \to 0^+}\left(- \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}} + \frac{2 \left(\cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}\right)^{2} \sin{\left(x \right)}}{x^{2} \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}}{x \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}\right) = - \frac{1}{6}$$
- 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
$$\left[97.3688346960149, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.511069234565\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}} + \frac{2 \left(\cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}\right)^{2} \sin{\left(x \right)}}{x^{2} \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}}{x \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 56.5132926241755$$
$$x_{2} = 18.7432530945386$$
$$x_{3} = 34.4996123350132$$
$$x_{4} = 157.066899940015$$
$$x_{5} = -53.3696181339615$$
$$x_{6} = -97.3688346960149$$
$$x_{7} = -50.2256832197934$$
$$x_{8} = 40.7917141624847$$
$$x_{9} = 106.795425016936$$
$$x_{10} = -9.21096438740149$$
$$x_{11} = 100.511069234565$$
$$x_{12} = 28.2035393053095$$
$$x_{13} = 59.6567478435559$$
$$x_{14} = 21.9000773156394$$
$$x_{15} = 91.0842327848165$$
$$x_{16} = -94.2265573558031$$
$$x_{17} = 37.6460352959305$$
$$x_{18} = 5.95939190757933$$
$$x_{19} = -31.3522215217643$$
$$x_{20} = -34.4996123350132$$
$$x_{21} = 69.0860970774096$$
$$x_{22} = -2.45871417599962$$
$$x_{23} = -47.0814357397523$$
$$x_{24} = 25.053079662454$$
$$x_{25} = 65.9431258539286$$
$$x_{26} = 75.3716947511882$$
$$x_{27} = -100.511069234565$$
$$x_{28} = 31.3522215217643$$
$$x_{29} = -81.6569211705466$$
$$x_{30} = -56.5132926241755$$
$$x_{31} = -75.3716947511882$$
$$x_{32} = 53.3696181339615$$
$$x_{33} = 50.2256832197934$$
$$x_{34} = 81.6569211705466$$
$$x_{35} = -59.6567478435559$$
$$x_{36} = -78.5143487963623$$
$$x_{37} = 87.9418559209576$$
$$x_{38} = -65.9431258539286$$
$$x_{39} = -72.2289483771681$$
$$x_{40} = 72.2289483771681$$
$$x_{41} = -5.95939190757933$$
$$x_{42} = -87.9418559209576$$
$$x_{43} = -62.8000167068325$$
$$x_{44} = -84.7994209518635$$
$$x_{45} = -15.5802941824244$$
$$x_{46} = 9.21096438740149$$
$$x_{47} = 97.3688346960149$$
$$x_{48} = 78.5143487963623$$
$$x_{49} = 62.8000167068325$$
$$x_{50} = 47.0814357397523$$
$$x_{51} = -28.2035393053095$$
$$x_{52} = 84.7994209518635$$
$$x_{53} = -69.0860970774096$$
$$x_{54} = -21.9000773156394$$
$$x_{55} = -18.7432530945386$$
$$x_{56} = -40.7917141624847$$
$$x_{57} = 2.45871417599962$$
$$x_{58} = -12.4065403639626$$
$$x_{59} = -91.0842327848165$$
$$x_{60} = 12.4065403639626$$
$$x_{61} = -43.9368086315937$$
$$x_{62} = 94.2265573558031$$
$$x_{63} = -25.053079662454$$
$$x_{64} = 15.5802941824244$$
$$x_{65} = 43.9368086315937$$
$$x_{66} = -37.6460352959305$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}} + \frac{2 \left(\cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}\right)^{2} \sin{\left(x \right)}}{x^{2} \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}}{x \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}\right) = - \frac{1}{6}$$
$$\lim_{x \to 0^+}\left(- \frac{\sin{\left(x \right)} + \frac{2 \cos{\left(x \right)}}{x} - \frac{2 \sin{\left(x \right)}}{x^{2}} + \frac{2 \left(\cos{\left(x \right)} - \frac{\sin{\left(x \right)}}{x}\right)^{2} \sin{\left(x \right)}}{x^{2} \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}}{x \left(1 + \frac{\sin^{2}{\left(x \right)}}{x^{2}}\right)}\right) = - \frac{1}{6}$$
- 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
$$\left[97.3688346960149, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -100.511069234565\right]$$
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
$$\lim_{x \to -\infty} \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty} \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = 0$$
$$\lim_{x \to -\infty} \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)} = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
$$\lim_{x \to \infty} \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)} = 0$$
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 atan(sin(x)/x), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)}}{x}\right) = 0$$
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:
$$\operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)} = \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)}$$
- No
$$\operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)} = - \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)} = \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)}$$
- No
$$\operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)} = - \operatorname{atan}{\left(\frac{\sin{\left(x \right)}}{x} \right)}$$
- No
es decir, función
no es
par ni impar
Gráfico