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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*exp(-x)
-
(1/2)^x
-
-x^2+3*x
-
(x-3)^2
-
Expresiones idénticas
- dos *sin(pi/ cuatro + uno)^ dos *cos(pi*x/ tres + uno)/atan(x/ tres)
- 2 multiplicar por seno de ( número pi dividir por 4 más 1) al cuadrado multiplicar por coseno de ( número pi multiplicar por x dividir por 3 más 1) dividir por arco tangente de gente de (x dividir por 3)
- dos multiplicar por seno de ( número pi dividir por cuatro más uno) en el grado dos multiplicar por coseno de ( número pi multiplicar por x dividir por tres más uno) dividir por arco tangente de gente de (x dividir por tres)
- 2*sin(pi/4+1)2*cos(pi*x/3+1)/atan(x/3)
- 2*sinpi/4+12*cospi*x/3+1/atanx/3
- 2*sin(pi/4+1)²*cos(pi*x/3+1)/atan(x/3)
- 2*sin(pi/4+1) en el grado 2*cos(pi*x/3+1)/atan(x/3)
- 2sin(pi/4+1)^2cos(pix/3+1)/atan(x/3)
- 2sin(pi/4+1)2cos(pix/3+1)/atan(x/3)
- 2sinpi/4+12cospix/3+1/atanx/3
- 2sinpi/4+1^2cospix/3+1/atanx/3
- 2*sin(pi dividir por 4+1)^2*cos(pi*x dividir por 3+1) dividir por atan(x dividir por 3)
-
Expresiones semejantes
- 2*sin(pi/4+1)^2*cos(pi*x/3-1)/atan(x/3)
- 2*sin(pi/4-1)^2*cos(pi*x/3+1)/atan(x/3)
- 2*sin(pi/4+1)^2*cos(pi*x/3+1)/arctan(x/3)
-
Expresiones con funciones
- Seno sin
- sin(2*x)/3
- sin(x)+sin(2*x)+sin(3*x)
- sin(x)-x^2
- sin(x)*e^x
- sin(x)+x/2
- Número Pi pi
- pi/2-arcsin(absolute((2x-1)/3))
- pi/2+arcsin(x/2)
- pi^x
- pi^(1/5x+2)
- pi+x
- Coseno cos
- cosx*cos2x
- cos(x)-x^2
- cos(x)^(6)+sin(x)^(6)
- cos(7*x)+cos(3*x)
- cos(x)^2+sin(x)
- Número Pi pi
- pi/2-arcsin(absolute((2x-1)/3))
- pi/2+arcsin(x/2)
- pi^x
- pi^(1/5x+2)
- pi+x
- Arcotangente arctan
- atan(2)^(3)*x*log(x+5)
- atan((x+2)/(1-2*x))
- atan(6*x)*cot(2*x)
- atan((1+5*x^2)^(1/3))
- atan(36/(x-158))-atan(102/(x-113))-157*2/50/180
Gráfico de la función y = 2*sin(pi/4+1)^2*cos(pi*x/3+1)/atan(x/3)
Solución
Ha introducido
[src]
2/pi \ /pi*x \
2*sin |-- + 1|*cos|---- + 1|
\4 / \ 3 /
f(x) = ----------------------------
/x\
atan|-|
\3/
$$f{\left(x \right)} = \frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}$$
f = ((2*sin(pi/4 + 1)^2)*cos((pi*x)/3 + 1))/atan(x/3)
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:
$$\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{3}{2} - \frac{3}{\pi}$$
$$x_{2} = \frac{9}{2} - \frac{3}{\pi}$$
Solución numérica
$$x_{1} = -35.4549296585514$$
$$x_{2} = 78.5450703414486$$
$$x_{3} = 66.5450703414486$$
$$x_{4} = 57.5450703414486$$
$$x_{5} = -341.454929658551$$
$$x_{6} = -56.4549296585514$$
$$x_{7} = -83.4549296585514$$
$$x_{8} = 1089.54507034145$$
$$x_{9} = 81.5450703414486$$
$$x_{10} = 36.5450703414486$$
$$x_{11} = 99.5450703414486$$
$$x_{12} = -23.4549296585514$$
$$x_{13} = -89.4549296585514$$
$$x_{14} = -47.4549296585514$$
$$x_{15} = 60.5450703414486$$
$$x_{16} = -98.4549296585514$$
$$x_{17} = 30.5450703414486$$
$$x_{18} = -305.454929658551$$
$$x_{19} = 42.5450703414486$$
$$x_{20} = 3258.54507034145$$
$$x_{21} = 15.5450703414486$$
$$x_{22} = -68.4549296585514$$
$$x_{23} = -41.4549296585514$$
$$x_{24} = 27.5450703414486$$
$$x_{25} = -20.4549296585514$$
$$x_{26} = -71.4549296585514$$
$$x_{27} = -74.4549296585514$$
$$x_{28} = -8.45492965855137$$
$$x_{29} = -14.4549296585514$$
$$x_{30} = 48.5450703414486$$
$$x_{31} = 87.5450703414486$$
$$x_{32} = 69.5450703414486$$
$$x_{33} = 63.5450703414486$$
$$x_{34} = 54.5450703414486$$
$$x_{35} = 72.5450703414486$$
$$x_{36} = -77.4549296585514$$
$$x_{37} = 150.545070341449$$
$$x_{38} = -32.4549296585514$$
$$x_{39} = 84.5450703414486$$
$$x_{40} = -53.4549296585514$$
$$x_{41} = -59.4549296585514$$
$$x_{42} = -80.4549296585514$$
$$x_{43} = -62.4549296585514$$
$$x_{44} = -65.4549296585514$$
$$x_{45} = -11.4549296585514$$
$$x_{46} = -338.454929658551$$
$$x_{47} = 93.5450703414486$$
$$x_{48} = 96.5450703414486$$
$$x_{49} = -5.45492965855137$$
$$x_{50} = -17.4549296585514$$
$$x_{51} = 90.5450703414486$$
$$x_{52} = 3.54507034144863$$
$$x_{53} = -44.4549296585514$$
$$x_{54} = -50.4549296585514$$
$$x_{55} = -29.4549296585514$$
$$x_{56} = 9.54507034144863$$
$$x_{57} = 75.5450703414486$$
$$x_{58} = 33.5450703414486$$
$$x_{59} = 18.5450703414486$$
$$x_{60} = 6.54507034144863$$
$$x_{61} = -2.45492965855137$$
$$x_{62} = 123.545070341449$$
$$x_{63} = 24.5450703414486$$
$$x_{64} = 12.5450703414486$$
$$x_{65} = 21.5450703414486$$
$$x_{66} = 764037.545070341$$
$$x_{67} = 39.5450703414486$$
$$x_{68} = -95.4549296585514$$
$$x_{69} = -26.4549296585514$$
$$x_{70} = 51.5450703414486$$
$$x_{71} = -92.4549296585514$$
$$x_{72} = 45.5450703414486$$
$$x_{73} = -86.4549296585514$$
$$x_{74} = 135.545070341449$$
$$x_{75} = -38.4549296585514$$
o sea hay que resolver la ecuación:
$$\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = \frac{3}{2} - \frac{3}{\pi}$$
$$x_{2} = \frac{9}{2} - \frac{3}{\pi}$$
Solución numérica
$$x_{1} = -35.4549296585514$$
$$x_{2} = 78.5450703414486$$
$$x_{3} = 66.5450703414486$$
$$x_{4} = 57.5450703414486$$
$$x_{5} = -341.454929658551$$
$$x_{6} = -56.4549296585514$$
$$x_{7} = -83.4549296585514$$
$$x_{8} = 1089.54507034145$$
$$x_{9} = 81.5450703414486$$
$$x_{10} = 36.5450703414486$$
$$x_{11} = 99.5450703414486$$
$$x_{12} = -23.4549296585514$$
$$x_{13} = -89.4549296585514$$
$$x_{14} = -47.4549296585514$$
$$x_{15} = 60.5450703414486$$
$$x_{16} = -98.4549296585514$$
$$x_{17} = 30.5450703414486$$
$$x_{18} = -305.454929658551$$
$$x_{19} = 42.5450703414486$$
$$x_{20} = 3258.54507034145$$
$$x_{21} = 15.5450703414486$$
$$x_{22} = -68.4549296585514$$
$$x_{23} = -41.4549296585514$$
$$x_{24} = 27.5450703414486$$
$$x_{25} = -20.4549296585514$$
$$x_{26} = -71.4549296585514$$
$$x_{27} = -74.4549296585514$$
$$x_{28} = -8.45492965855137$$
$$x_{29} = -14.4549296585514$$
$$x_{30} = 48.5450703414486$$
$$x_{31} = 87.5450703414486$$
$$x_{32} = 69.5450703414486$$
$$x_{33} = 63.5450703414486$$
$$x_{34} = 54.5450703414486$$
$$x_{35} = 72.5450703414486$$
$$x_{36} = -77.4549296585514$$
$$x_{37} = 150.545070341449$$
$$x_{38} = -32.4549296585514$$
$$x_{39} = 84.5450703414486$$
$$x_{40} = -53.4549296585514$$
$$x_{41} = -59.4549296585514$$
$$x_{42} = -80.4549296585514$$
$$x_{43} = -62.4549296585514$$
$$x_{44} = -65.4549296585514$$
$$x_{45} = -11.4549296585514$$
$$x_{46} = -338.454929658551$$
$$x_{47} = 93.5450703414486$$
$$x_{48} = 96.5450703414486$$
$$x_{49} = -5.45492965855137$$
$$x_{50} = -17.4549296585514$$
$$x_{51} = 90.5450703414486$$
$$x_{52} = 3.54507034144863$$
$$x_{53} = -44.4549296585514$$
$$x_{54} = -50.4549296585514$$
$$x_{55} = -29.4549296585514$$
$$x_{56} = 9.54507034144863$$
$$x_{57} = 75.5450703414486$$
$$x_{58} = 33.5450703414486$$
$$x_{59} = 18.5450703414486$$
$$x_{60} = 6.54507034144863$$
$$x_{61} = -2.45492965855137$$
$$x_{62} = 123.545070341449$$
$$x_{63} = 24.5450703414486$$
$$x_{64} = 12.5450703414486$$
$$x_{65} = 21.5450703414486$$
$$x_{66} = 764037.545070341$$
$$x_{67} = 39.5450703414486$$
$$x_{68} = -95.4549296585514$$
$$x_{69} = -26.4549296585514$$
$$x_{70} = 51.5450703414486$$
$$x_{71} = -92.4549296585514$$
$$x_{72} = 45.5450703414486$$
$$x_{73} = -86.4549296585514$$
$$x_{74} = 135.545070341449$$
$$x_{75} = -38.4549296585514$$
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
$$- \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)} \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{3 \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{3 \left(\frac{x^{2}}{9} + 1\right) \operatorname{atan}^{2}{\left(\frac{x}{3} \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 65.0446471551009$$
$$x_{2} = -78.9546437734811$$
$$x_{3} = 101.04489663439$$
$$x_{4} = -30.9530107803312$$
$$x_{5} = -39.9537904111553$$
$$x_{6} = 59.0445554304911$$
$$x_{7} = 80.0447922733148$$
$$x_{8} = -12.9433916147356$$
$$x_{9} = 86.0448300650018$$
$$x_{10} = -33.9533415610825$$
$$x_{11} = -9.93505088725385$$
$$x_{12} = 62.0446046591783$$
$$x_{13} = 14.0353070912986$$
$$x_{14} = -45.9540729018243$$
$$x_{15} = -75.9546204775596$$
$$x_{16} = -90.9547148584446$$
$$x_{17} = 56.0444979691161$$
$$x_{18} = 53.0444303333317$$
$$x_{19} = 20.0403858521832$$
$$x_{20} = 8.01427314241055$$
$$x_{21} = 38.0438114021911$$
$$x_{22} = 92.0448606470826$$
$$x_{23} = -36.9535937642116$$
$$x_{24} = 4.9661486177045$$
$$x_{25} = -42.9539466795558$$
$$x_{26} = -84.9546831120399$$
$$x_{27} = 95.0448737941732$$
$$x_{28} = 47.0442534911438$$
$$x_{29} = -54.9543340044467$$
$$x_{30} = -21.9510457285375$$
$$x_{31} = -57.9543949106738$$
$$x_{32} = -96.9547408472594$$
$$x_{33} = -51.9542620784985$$
$$x_{34} = -81.9546645332808$$
$$x_{35} = 50.0443499739057$$
$$x_{36} = -69.9545644606477$$
$$x_{37} = 23.0415549303514$$
$$x_{38} = -66.9545305666873$$
$$x_{39} = 98.0448857428364$$
$$x_{40} = -48.9541763078526$$
$$x_{41} = -3.82803929263352$$
$$x_{42} = -24.9519448835217$$
$$x_{43} = 74.0447448152697$$
$$x_{44} = -87.9546998048222$$
$$x_{45} = 71.044716397544$$
$$x_{46} = 29.042884150723$$
$$x_{47} = 44.0441362706937$$
$$x_{48} = -72.9545942151565$$
$$x_{49} = 26.0423364267544$$
$$x_{50} = -27.9525651538481$$
$$x_{51} = 68.0446840916865$$
$$x_{52} = 35.0435815529748$$
$$x_{53} = -99.9547521079561$$
$$x_{54} = -60.9544469371416$$
$$x_{55} = 83.0448122029392$$
$$x_{56} = -93.9547284806413$$
$$x_{57} = 11.0290280072741$$
$$x_{58} = 41.0439919237837$$
$$x_{59} = 77.0447699444161$$
$$x_{60} = -15.9474278264858$$
$$x_{61} = 89.0448461358262$$
$$x_{62} = -63.954491728768$$
$$x_{63} = 1.59695520836124$$
$$x_{64} = -18.9496728516901$$
$$x_{65} = -6.91348234301887$$
$$x_{66} = 32.0432826687215$$
$$x_{67} = 17.0385256432214$$
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:
$$x_{1} = -78.9546437734811$$
$$x_{2} = -30.9530107803312$$
$$x_{3} = 80.0447922733148$$
$$x_{4} = -12.9433916147356$$
$$x_{5} = 86.0448300650018$$
$$x_{6} = 62.0446046591783$$
$$x_{7} = 14.0353070912986$$
$$x_{8} = -90.9547148584446$$
$$x_{9} = 56.0444979691161$$
$$x_{10} = 20.0403858521832$$
$$x_{11} = 8.01427314241055$$
$$x_{12} = 38.0438114021911$$
$$x_{13} = 92.0448606470826$$
$$x_{14} = -36.9535937642116$$
$$x_{15} = -42.9539466795558$$
$$x_{16} = -84.9546831120399$$
$$x_{17} = -54.9543340044467$$
$$x_{18} = -96.9547408472594$$
$$x_{19} = 50.0443499739057$$
$$x_{20} = -66.9545305666873$$
$$x_{21} = 98.0448857428364$$
$$x_{22} = -48.9541763078526$$
$$x_{23} = -24.9519448835217$$
$$x_{24} = 74.0447448152697$$
$$x_{25} = 44.0441362706937$$
$$x_{26} = -72.9545942151565$$
$$x_{27} = 26.0423364267544$$
$$x_{28} = 68.0446840916865$$
$$x_{29} = -60.9544469371416$$
$$x_{30} = 1.59695520836124$$
$$x_{31} = -18.9496728516901$$
$$x_{32} = -6.91348234301887$$
$$x_{33} = 32.0432826687215$$
Puntos máximos de la función:
$$x_{33} = 65.0446471551009$$
$$x_{33} = 101.04489663439$$
$$x_{33} = -39.9537904111553$$
$$x_{33} = 59.0445554304911$$
$$x_{33} = -33.9533415610825$$
$$x_{33} = -9.93505088725385$$
$$x_{33} = -45.9540729018243$$
$$x_{33} = -75.9546204775596$$
$$x_{33} = 53.0444303333317$$
$$x_{33} = 4.9661486177045$$
$$x_{33} = 95.0448737941732$$
$$x_{33} = 47.0442534911438$$
$$x_{33} = -21.9510457285375$$
$$x_{33} = -57.9543949106738$$
$$x_{33} = -51.9542620784985$$
$$x_{33} = -81.9546645332808$$
$$x_{33} = -69.9545644606477$$
$$x_{33} = 23.0415549303514$$
$$x_{33} = -3.82803929263352$$
$$x_{33} = -87.9546998048222$$
$$x_{33} = 71.044716397544$$
$$x_{33} = 29.042884150723$$
$$x_{33} = -27.9525651538481$$
$$x_{33} = 35.0435815529748$$
$$x_{33} = -99.9547521079561$$
$$x_{33} = 83.0448122029392$$
$$x_{33} = -93.9547284806413$$
$$x_{33} = 11.0290280072741$$
$$x_{33} = 41.0439919237837$$
$$x_{33} = 77.0447699444161$$
$$x_{33} = -15.9474278264858$$
$$x_{33} = 89.0448461358262$$
$$x_{33} = -63.954491728768$$
$$x_{33} = 17.0385256432214$$
Decrece en los intervalos
$$\left[98.0448857428364, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -96.9547408472594\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)} \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{3 \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{3 \left(\frac{x^{2}}{9} + 1\right) \operatorname{atan}^{2}{\left(\frac{x}{3} \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 65.0446471551009$$
$$x_{2} = -78.9546437734811$$
$$x_{3} = 101.04489663439$$
$$x_{4} = -30.9530107803312$$
$$x_{5} = -39.9537904111553$$
$$x_{6} = 59.0445554304911$$
$$x_{7} = 80.0447922733148$$
$$x_{8} = -12.9433916147356$$
$$x_{9} = 86.0448300650018$$
$$x_{10} = -33.9533415610825$$
$$x_{11} = -9.93505088725385$$
$$x_{12} = 62.0446046591783$$
$$x_{13} = 14.0353070912986$$
$$x_{14} = -45.9540729018243$$
$$x_{15} = -75.9546204775596$$
$$x_{16} = -90.9547148584446$$
$$x_{17} = 56.0444979691161$$
$$x_{18} = 53.0444303333317$$
$$x_{19} = 20.0403858521832$$
$$x_{20} = 8.01427314241055$$
$$x_{21} = 38.0438114021911$$
$$x_{22} = 92.0448606470826$$
$$x_{23} = -36.9535937642116$$
$$x_{24} = 4.9661486177045$$
$$x_{25} = -42.9539466795558$$
$$x_{26} = -84.9546831120399$$
$$x_{27} = 95.0448737941732$$
$$x_{28} = 47.0442534911438$$
$$x_{29} = -54.9543340044467$$
$$x_{30} = -21.9510457285375$$
$$x_{31} = -57.9543949106738$$
$$x_{32} = -96.9547408472594$$
$$x_{33} = -51.9542620784985$$
$$x_{34} = -81.9546645332808$$
$$x_{35} = 50.0443499739057$$
$$x_{36} = -69.9545644606477$$
$$x_{37} = 23.0415549303514$$
$$x_{38} = -66.9545305666873$$
$$x_{39} = 98.0448857428364$$
$$x_{40} = -48.9541763078526$$
$$x_{41} = -3.82803929263352$$
$$x_{42} = -24.9519448835217$$
$$x_{43} = 74.0447448152697$$
$$x_{44} = -87.9546998048222$$
$$x_{45} = 71.044716397544$$
$$x_{46} = 29.042884150723$$
$$x_{47} = 44.0441362706937$$
$$x_{48} = -72.9545942151565$$
$$x_{49} = 26.0423364267544$$
$$x_{50} = -27.9525651538481$$
$$x_{51} = 68.0446840916865$$
$$x_{52} = 35.0435815529748$$
$$x_{53} = -99.9547521079561$$
$$x_{54} = -60.9544469371416$$
$$x_{55} = 83.0448122029392$$
$$x_{56} = -93.9547284806413$$
$$x_{57} = 11.0290280072741$$
$$x_{58} = 41.0439919237837$$
$$x_{59} = 77.0447699444161$$
$$x_{60} = -15.9474278264858$$
$$x_{61} = 89.0448461358262$$
$$x_{62} = -63.954491728768$$
$$x_{63} = 1.59695520836124$$
$$x_{64} = -18.9496728516901$$
$$x_{65} = -6.91348234301887$$
$$x_{66} = 32.0432826687215$$
$$x_{67} = 17.0385256432214$$
Signos de extremos en los puntos:
2/pi \
(65.04464715510095, 1.31172758508052*sin |-- + 1|*cos(1 + 21.6815490517003*pi))
\4 / 2/pi \
(-78.95464377348107, -1.3047862642285*sin |-- + 1|*cos(1 - 26.3182145911604*pi))
\4 / 2/pi \
(101.04489663438974, 1.29776145539671*sin |-- + 1|*cos(1 + 33.6816322114632*pi))
\4 / 2/pi \
(-30.95301078033117, -1.35668918485075*sin |-- + 1|*cos(1 - 10.3176702601104*pi))
\4 / 2/pi \
(-39.95379041115532, -1.33703225834123*sin |-- + 1|*cos(1 - 13.3179301370518*pi))
\4 / 2/pi \
(59.044555430491144, 1.31576273106049*sin |-- + 1|*cos(1 + 19.6815184768304*pi))
\4 / 2/pi \
(80.0447922733148, 1.304346539933*sin |-- + 1|*cos(1 + 26.6815974244383*pi))
\4 / 2/pi \
(-12.943391614735598, -1.48915953919604*sin |-- + 1|*cos(1 - 4.31446387157853*pi))
\4 / 2/pi \
(86.04483006500179, 1.30213004566513*sin |-- + 1|*cos(1 + 28.6816100216673*pi))
\4 / 2/pi \
(-33.95334156108247, -1.34891906894198*sin |-- + 1|*cos(1 - 11.3177805203608*pi))
\4 / 2/pi \
(-9.93505088725385, -1.56550720658075*sin |-- + 1|*cos(1 - 3.31168362908462*pi))
\4 / 2/pi \
(62.04460465917826, 1.31364473687044*sin |-- + 1|*cos(1 + 20.6815348863928*pi))
\4 / 2/pi \
(14.03530709129865, 1.470352071004*sin |-- + 1|*cos(1 + 4.67843569709955*pi))
\4 / 2/pi \
(-45.95407290182433, -1.32836854643405*sin |-- + 1|*cos(1 - 15.3180243006081*pi))
\4 / 2/pi \
(-75.95462047755957, -1.30606310263924*sin |-- + 1|*cos(1 - 25.3182068258532*pi))
\4 / 2/pi \
(-90.95471485844465, -1.30053823873251*sin |-- + 1|*cos(1 - 30.3182382861482*pi))
\4 / 2/pi \
(56.044497969116144, 1.31811484408972*sin |-- + 1|*cos(1 + 18.6814993230387*pi))
\4 / 2/pi \
(53.04443033333174, 1.32074212613324*sin |-- + 1|*cos(1 + 17.6814767777772*pi))
\4 / 2/pi \
(20.040385852183245, 1.40626997160643*sin |-- + 1|*cos(1 + 6.68012861739441*pi))
\4 / 2/pi \
(8.01427314241055, 1.64933312677169*sin |-- + 1|*cos(1 + 2.67142438080352*pi))
\4 / 2/pi \
(38.04381140219114, 1.34039029216093*sin |-- + 1|*cos(1 + 12.681270467397*pi))
\4 / 2/pi \
(92.04486064708261, 1.30020830972951*sin |-- + 1|*cos(1 + 30.6816202156942*pi))
\4 / 2/pi \
(-36.95359376421159, -1.34247012048457*sin |-- + 1|*cos(1 - 12.3178645880705*pi))
\4 / 2/pi \
(4.9661486177045, 1.94670883527179*sin |-- + 1|*cos(1 + 1.65538287256817*pi))
\4 / 2/pi \
(-42.95394667955585, -1.33238528415755*sin |-- + 1|*cos(1 - 14.3179822265186*pi))
\4 / 2/pi \
(-84.95468311203993, -1.30250898080417*sin |-- + 1|*cos(1 - 28.3182277040133*pi))
\4 / 2/pi \
(95.04487379417324, 1.29934019869217*sin |-- + 1|*cos(1 + 31.6816245980577*pi))
\4 / 2/pi \
(47.04425349114377, 1.3270406709147*sin |-- + 1|*cos(1 + 15.6814178303813*pi))
\4 / 2/pi \
(-54.95433400444668, -1.3190352733355*sin |-- + 1|*cos(1 - 18.3181113348156*pi))
\4 / 2/pi \
(-21.9510457285375, -1.39375729818924*sin |-- + 1|*cos(1 - 7.31701524284583*pi))
\4 / 2/pi \
(-57.95439491067376, -1.31658839287297*sin |-- + 1|*cos(1 - 19.3181316368913*pi))
\4 / 2/pi \
(-96.95474084725943, -1.29881606368722*sin |-- + 1|*cos(1 - 32.3182469490865*pi))
\4 / 2/pi \
(-51.95426207849849, -1.32177460951675*sin |-- + 1|*cos(1 - 17.3180873594995*pi))
\4 / 2/pi \
(-81.95466453328078, -1.30360500382754*sin |-- + 1|*cos(1 - 27.3182215110936*pi))
\4 / 2/pi \
(50.04434997390568, 1.32369580955496*sin |-- + 1|*cos(1 + 16.6814499913019*pi))
\4 / 2/pi \
(-69.95456446064769, -1.30895397013167*sin |-- + 1|*cos(1 - 23.3181881535492*pi))
\4 / 2/pi \
(23.041554930351445, 1.38761199902279*sin |-- + 1|*cos(1 + 7.68051831011715*pi))
\4 / 2/pi \
(-66.95453056668727, -1.310599028721*sin |-- + 1|*cos(1 - 22.3181768555624*pi))
\4 / 2/pi \
(98.04488574283641, 1.29852621665345*sin |-- + 1|*cos(1 + 32.6816285809455*pi))
\4 / 2/pi \
(-48.95417630785262, -1.32486210408683*sin |-- + 1|*cos(1 - 16.3180587692842*pi))
\4 / 2/pi \
(-3.8280392926335174, -2.20731231649318*sin |-- + 1|*cos(1 - 1.27601309754451*pi))
\4 / 2/pi \
(-24.95194488352173, -1.37822717334108*sin |-- + 1|*cos(1 - 8.31731496117391*pi))
\4 / 2/pi \
(74.0447448152697, 1.30693120356289*sin |-- + 1|*cos(1 + 24.6815816050899*pi))
\4 / 2/pi \
(-87.95469980482221, -1.30148929378165*sin |-- + 1|*cos(1 - 29.3182332682741*pi))
\4 / 2/pi \
(71.04471639754401, 1.30839149060686*sin |-- + 1|*cos(1 + 23.6815721325147*pi))
\4 / 2/pi \
(29.042884150723005, 1.36252232855745*sin |-- + 1|*cos(1 + 9.68096138357433*pi))
\4 / 2/pi \
(44.04413627069369, 1.33085981755581*sin |-- + 1|*cos(1 + 14.6813787568979*pi))
\4 / 2/pi \
(-72.95459421515648, -1.30744760194208*sin |-- + 1|*cos(1 - 24.3181980717188*pi))
\4 / 2/pi \
(26.042336426754414, 1.37352747312723*sin |-- + 1|*cos(1 + 8.68077880891814*pi))
\4 / 2/pi \
(-27.952565153848056, -1.36623136279072*sin |-- + 1|*cos(1 - 9.31752171794935*pi))
\4 / 2/pi \
(68.04468409168655, 1.30998401694728*sin |-- + 1|*cos(1 + 22.6815613638955*pi))
\4 / 2/pi \
(35.04358155297482, 1.34644152121067*sin |-- + 1|*cos(1 + 11.6811938509916*pi))
\4 / 2/pi \
(-99.95475210795614, -1.2980339460358*sin |-- + 1|*cos(1 - 33.318250702652*pi))
\4 / 2/pi \
(-60.954446937141576, -1.31438953882109*sin |-- + 1|*cos(1 - 20.3181489790472*pi))
\4 / 2/pi \
(83.04481220293916, 1.3031973677048*sin |-- + 1|*cos(1 + 27.6816040676464*pi))
\4 / 2/pi \
(-93.95472848064131, -1.29964911426273*sin |-- + 1|*cos(1 - 31.3182428268804*pi))
\4 / 2/pi \
(11.029028007274148, 1.53231751888211*sin |-- + 1|*cos(1 + 3.67634266909138*pi))
\4 / 2/pi \
(41.04399192378365, 1.33526165554065*sin |-- + 1|*cos(1 + 13.6813306412612*pi))
\4 / 2/pi \
(77.04476994441606, 1.30558734849638*sin |-- + 1|*cos(1 + 25.681589981472*pi))
\4 / 2/pi \
(-15.947427826485766, -1.44419835099217*sin |-- + 1|*cos(1 - 5.31580927549525*pi))
\4 / 2/pi \
(89.04484613582618, 1.30113613277213*sin |-- + 1|*cos(1 + 29.6816153786087*pi))
\4 / 2/pi \
(-63.95449172876801, -1.31240284224882*sin |-- + 1|*cos(1 - 21.3181639095893*pi))
\4 / 2/pi \
(1.5969552083612446, 4.08858472866807*sin |-- + 1|*cos(1 + 0.532318402787082*pi))
\4 / 2/pi \
(-18.949672851690053, -1.41464191988161*sin |-- + 1|*cos(1 - 6.31655761723002*pi))
\4 / 2/pi \
(-6.913482343018871, -1.72208582895412*sin |-- + 1|*cos(1 - 2.30449411433962*pi))
\4 / 2/pi \
(32.04328266872153, 1.35368822220233*sin |-- + 1|*cos(1 + 10.6810942229072*pi))
\4 / 2/pi \
(17.038525643221387, 1.43214040501933*sin |-- + 1|*cos(1 + 5.67950854774046*pi))
\4 / 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:
$$x_{1} = -78.9546437734811$$
$$x_{2} = -30.9530107803312$$
$$x_{3} = 80.0447922733148$$
$$x_{4} = -12.9433916147356$$
$$x_{5} = 86.0448300650018$$
$$x_{6} = 62.0446046591783$$
$$x_{7} = 14.0353070912986$$
$$x_{8} = -90.9547148584446$$
$$x_{9} = 56.0444979691161$$
$$x_{10} = 20.0403858521832$$
$$x_{11} = 8.01427314241055$$
$$x_{12} = 38.0438114021911$$
$$x_{13} = 92.0448606470826$$
$$x_{14} = -36.9535937642116$$
$$x_{15} = -42.9539466795558$$
$$x_{16} = -84.9546831120399$$
$$x_{17} = -54.9543340044467$$
$$x_{18} = -96.9547408472594$$
$$x_{19} = 50.0443499739057$$
$$x_{20} = -66.9545305666873$$
$$x_{21} = 98.0448857428364$$
$$x_{22} = -48.9541763078526$$
$$x_{23} = -24.9519448835217$$
$$x_{24} = 74.0447448152697$$
$$x_{25} = 44.0441362706937$$
$$x_{26} = -72.9545942151565$$
$$x_{27} = 26.0423364267544$$
$$x_{28} = 68.0446840916865$$
$$x_{29} = -60.9544469371416$$
$$x_{30} = 1.59695520836124$$
$$x_{31} = -18.9496728516901$$
$$x_{32} = -6.91348234301887$$
$$x_{33} = 32.0432826687215$$
Puntos máximos de la función:
$$x_{33} = 65.0446471551009$$
$$x_{33} = 101.04489663439$$
$$x_{33} = -39.9537904111553$$
$$x_{33} = 59.0445554304911$$
$$x_{33} = -33.9533415610825$$
$$x_{33} = -9.93505088725385$$
$$x_{33} = -45.9540729018243$$
$$x_{33} = -75.9546204775596$$
$$x_{33} = 53.0444303333317$$
$$x_{33} = 4.9661486177045$$
$$x_{33} = 95.0448737941732$$
$$x_{33} = 47.0442534911438$$
$$x_{33} = -21.9510457285375$$
$$x_{33} = -57.9543949106738$$
$$x_{33} = -51.9542620784985$$
$$x_{33} = -81.9546645332808$$
$$x_{33} = -69.9545644606477$$
$$x_{33} = 23.0415549303514$$
$$x_{33} = -3.82803929263352$$
$$x_{33} = -87.9546998048222$$
$$x_{33} = 71.044716397544$$
$$x_{33} = 29.042884150723$$
$$x_{33} = -27.9525651538481$$
$$x_{33} = 35.0435815529748$$
$$x_{33} = -99.9547521079561$$
$$x_{33} = 83.0448122029392$$
$$x_{33} = -93.9547284806413$$
$$x_{33} = 11.0290280072741$$
$$x_{33} = 41.0439919237837$$
$$x_{33} = 77.0447699444161$$
$$x_{33} = -15.9474278264858$$
$$x_{33} = 89.0448461358262$$
$$x_{33} = -63.954491728768$$
$$x_{33} = 17.0385256432214$$
Decrece en los intervalos
$$\left[98.0448857428364, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -96.9547408472594\right]$$
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
$$\frac{2 \left(\frac{6 \left(x + \frac{3}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right)^{2} \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{\pi^{2} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{9} + \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right) \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 36.5423366072492$$
$$x_{2} = -428.454910600311$$
$$x_{3} = 93.5446644153194$$
$$x_{4} = 57.5439852935404$$
$$x_{5} = -41.4528158380405$$
$$x_{6} = 90.5446368012048$$
$$x_{7} = 453.545053337612$$
$$x_{8} = -17.4424483862453$$
$$x_{9} = -50.4535125284486$$
$$x_{10} = 24.5388910312524$$
$$x_{11} = 42.5430654217194$$
$$x_{12} = 423.5450508378$$
$$x_{13} = 9.50130361412559$$
$$x_{14} = -5.31252777895664$$
$$x_{15} = -53.4536694345123$$
$$x_{16} = -65.4540939665356$$
$$x_{17} = 3.15993387709981$$
$$x_{18} = 87.5446062713843$$
$$x_{19} = 123.544838690268$$
$$x_{20} = -74.4542858252128$$
$$x_{21} = 213.544993284613$$
$$x_{22} = 114.544800550632$$
$$x_{23} = 75.5444451642865$$
$$x_{24} = -59.4539141487404$$
$$x_{25} = -35.4520214834876$$
$$x_{26} = -77.4543352671467$$
$$x_{27} = -8.39853211686134$$
$$x_{28} = -32.4514452992605$$
$$x_{29} = -86.4544536916344$$
$$x_{30} = 12.5203077359387$$
$$x_{31} = -38.4524659053982$$
$$x_{32} = -47.4533244194099$$
$$x_{33} = 84.5445724005981$$
$$x_{34} = -71.4542299508964$$
$$x_{35} = 48.5435375651596$$
$$x_{36} = -83.4544184863111$$
$$x_{37} = -95.4545399602143$$
$$x_{38} = 30.54112538342$$
$$x_{39} = 354.545042483703$$
$$x_{40} = 15.5292045102585$$
$$x_{41} = 27.5401941516671$$
$$x_{42} = 51.5437134521548$$
$$x_{43} = 81.5445346831614$$
$$x_{44} = 6.44847053079324$$
$$x_{45} = -122.454693834022$$
$$x_{46} = -1688.45492843539$$
$$x_{47} = 33.5418136674159$$
$$x_{48} = -98.4545635534746$$
$$x_{49} = -29.4506797592818$$
$$x_{50} = -92.454514011289$$
$$x_{51} = 63.5441829711077$$
$$x_{52} = 60.544091569207$$
$$x_{53} = -80.4543792271114$$
$$x_{54} = 96.5446894734333$$
$$x_{55} = 66.5442621493015$$
$$x_{56} = -89.4544853824358$$
$$x_{57} = -26.449632087741$$
$$x_{58} = -23.4481450893327$$
$$x_{59} = 72.5443917497405$$
$$x_{60} = -20.4459351998821$$
$$x_{61} = 54.5438607355483$$
$$x_{62} = 69.5443311891507$$
$$x_{63} = -62.4540106139345$$
$$x_{64} = 21.5369890404614$$
$$x_{65} = 240.545009671203$$
$$x_{66} = -11.4250099333197$$
$$x_{67} = 99.544712281458$$
$$x_{68} = 78.5444925140042$$
$$x_{69} = -56.4538016713076$$
$$x_{70} = -14.436481697601$$
$$x_{71} = 18.5340584217125$$
$$x_{72} = -68.4541664786334$$
$$x_{73} = -44.4530962608714$$
$$x_{74} = 39.5427431560573$$
$$x_{75} = 45.5433251674318$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(\frac{6 \left(x + \frac{3}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right)^{2} \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{\pi^{2} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{9} + \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right) \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(\frac{6 \left(x + \frac{3}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right)^{2} \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{\pi^{2} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{9} + \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right) \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = 0$$
- es el punto de flexión
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
$$\left[354.545042483703, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -1688.45492843539\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \left(\frac{6 \left(x + \frac{3}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right)^{2} \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{\pi^{2} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{9} + \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right) \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 36.5423366072492$$
$$x_{2} = -428.454910600311$$
$$x_{3} = 93.5446644153194$$
$$x_{4} = 57.5439852935404$$
$$x_{5} = -41.4528158380405$$
$$x_{6} = 90.5446368012048$$
$$x_{7} = 453.545053337612$$
$$x_{8} = -17.4424483862453$$
$$x_{9} = -50.4535125284486$$
$$x_{10} = 24.5388910312524$$
$$x_{11} = 42.5430654217194$$
$$x_{12} = 423.5450508378$$
$$x_{13} = 9.50130361412559$$
$$x_{14} = -5.31252777895664$$
$$x_{15} = -53.4536694345123$$
$$x_{16} = -65.4540939665356$$
$$x_{17} = 3.15993387709981$$
$$x_{18} = 87.5446062713843$$
$$x_{19} = 123.544838690268$$
$$x_{20} = -74.4542858252128$$
$$x_{21} = 213.544993284613$$
$$x_{22} = 114.544800550632$$
$$x_{23} = 75.5444451642865$$
$$x_{24} = -59.4539141487404$$
$$x_{25} = -35.4520214834876$$
$$x_{26} = -77.4543352671467$$
$$x_{27} = -8.39853211686134$$
$$x_{28} = -32.4514452992605$$
$$x_{29} = -86.4544536916344$$
$$x_{30} = 12.5203077359387$$
$$x_{31} = -38.4524659053982$$
$$x_{32} = -47.4533244194099$$
$$x_{33} = 84.5445724005981$$
$$x_{34} = -71.4542299508964$$
$$x_{35} = 48.5435375651596$$
$$x_{36} = -83.4544184863111$$
$$x_{37} = -95.4545399602143$$
$$x_{38} = 30.54112538342$$
$$x_{39} = 354.545042483703$$
$$x_{40} = 15.5292045102585$$
$$x_{41} = 27.5401941516671$$
$$x_{42} = 51.5437134521548$$
$$x_{43} = 81.5445346831614$$
$$x_{44} = 6.44847053079324$$
$$x_{45} = -122.454693834022$$
$$x_{46} = -1688.45492843539$$
$$x_{47} = 33.5418136674159$$
$$x_{48} = -98.4545635534746$$
$$x_{49} = -29.4506797592818$$
$$x_{50} = -92.454514011289$$
$$x_{51} = 63.5441829711077$$
$$x_{52} = 60.544091569207$$
$$x_{53} = -80.4543792271114$$
$$x_{54} = 96.5446894734333$$
$$x_{55} = 66.5442621493015$$
$$x_{56} = -89.4544853824358$$
$$x_{57} = -26.449632087741$$
$$x_{58} = -23.4481450893327$$
$$x_{59} = 72.5443917497405$$
$$x_{60} = -20.4459351998821$$
$$x_{61} = 54.5438607355483$$
$$x_{62} = 69.5443311891507$$
$$x_{63} = -62.4540106139345$$
$$x_{64} = 21.5369890404614$$
$$x_{65} = 240.545009671203$$
$$x_{66} = -11.4250099333197$$
$$x_{67} = 99.544712281458$$
$$x_{68} = 78.5444925140042$$
$$x_{69} = -56.4538016713076$$
$$x_{70} = -14.436481697601$$
$$x_{71} = 18.5340584217125$$
$$x_{72} = -68.4541664786334$$
$$x_{73} = -44.4530962608714$$
$$x_{74} = 39.5427431560573$$
$$x_{75} = 45.5433251674318$$
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \left(\frac{6 \left(x + \frac{3}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right)^{2} \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{\pi^{2} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{9} + \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right) \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(\frac{6 \left(x + \frac{3}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right)^{2} \operatorname{atan}{\left(\frac{x}{3} \right)}} - \frac{\pi^{2} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{9} + \frac{2 \pi \sin{\left(\frac{\pi x}{3} + 1 \right)}}{\left(x^{2} + 9\right) \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = \infty$$
- los límites no son iguales, signo
$$x_{1} = 0$$
- es el punto de flexión
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
$$\left[354.545042483703, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -1688.45492843539\right]$$
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
$$\lim_{x \to -\infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}$$
$$\lim_{x \to \infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}$$
$$\lim_{x \to -\infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}$$
$$\lim_{x \to \infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \frac{\left\langle -4, 4\right\rangle \sin^{2}{\left(\frac{\pi}{4} + 1 \right)}}{\pi}$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((2*sin(pi/4 + 1)^2)*cos((pi*x)/3 + 1))/atan(x/3), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{x \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{x \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{x \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{x \operatorname{atan}{\left(\frac{x}{3} \right)}}\right) = 0$$
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:
$$\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = - \frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} - 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}$$
- No
$$\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = \frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} - 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}$$
- No
es decir, función
no es
par ni impar
Pues, comprobamos:
$$\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = - \frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} - 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}$$
- No
$$\frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} + 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}} = \frac{2 \sin^{2}{\left(\frac{\pi}{4} + 1 \right)} \cos{\left(\frac{\pi x}{3} - 1 \right)}}{\operatorname{atan}{\left(\frac{x}{3} \right)}}$$
- No
es decir, función
no es
par ni impar