Sr Examen
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- 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
- cos(x)*cos(sin(x))
- coseno de (x) multiplicar por coseno de ( seno de (x))
- cos(x)cos(sin(x))
- cosxcossinx
-
Expresiones semejantes
- cosx*cos(sinx)
-
Expresiones con funciones
- 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)
- 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)
- 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
Gráfico de la función y = cos(x)*cos(sin(x))
Solución
Ha introducido
[src]
f(x) = cos(x)*cos(sin(x))
$$f{\left(x \right)} = \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}$$
f = cos(x)*cos(sin(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:
$$\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
Solución numérica
$$x_{1} = -29.845130209103$$
$$x_{2} = -67.5442420521806$$
$$x_{3} = -70.6858347057703$$
$$x_{4} = 64.4026493985908$$
$$x_{5} = -36.1283155162826$$
$$x_{6} = -92.6769832808989$$
$$x_{7} = -61.261056745001$$
$$x_{8} = -76.9690200129499$$
$$x_{9} = -98.9601685880785$$
$$x_{10} = -95.8185759344887$$
$$x_{11} = 29.845130209103$$
$$x_{12} = 80.1106126665397$$
$$x_{13} = -64.4026493985908$$
$$x_{14} = 36.1283155162826$$
$$x_{15} = 73.8274273593601$$
$$x_{16} = 32.9867228626928$$
$$x_{17} = -4.71238898038469$$
$$x_{18} = -39.2699081698724$$
$$x_{19} = 26.7035375555132$$
$$x_{20} = -5244.88893516816$$
$$x_{21} = 95.8185759344887$$
$$x_{22} = -7.85398163397448$$
$$x_{23} = -17.2787595947439$$
$$x_{24} = -10.9955742875643$$
$$x_{25} = 98.9601685880785$$
$$x_{26} = -86.3937979737193$$
$$x_{27} = 92.6769832808989$$
$$x_{28} = -48.6946861306418$$
$$x_{29} = 54.9778714378214$$
$$x_{30} = 45.553093477052$$
$$x_{31} = 23.5619449019235$$
$$x_{32} = 76.9690200129499$$
$$x_{33} = -89.5353906273091$$
$$x_{34} = 4.71238898038469$$
$$x_{35} = -26.7035375555132$$
$$x_{36} = -80.1106126665397$$
$$x_{37} = 7.85398163397448$$
$$x_{38} = 14.1371669411541$$
$$x_{39} = 86.3937979737193$$
$$x_{40} = -45.553093477052$$
$$x_{41} = -83.2522053201295$$
$$x_{42} = 70.6858347057703$$
$$x_{43} = 83.2522053201295$$
$$x_{44} = 48.6946861306418$$
$$x_{45} = -20.4203522483337$$
$$x_{46} = 51.8362787842316$$
$$x_{47} = 10.9955742875643$$
$$x_{48} = 20.4203522483337$$
$$x_{49} = 1.5707963267949$$
$$x_{50} = 89.5353906273091$$
$$x_{51} = 17.2787595947439$$
$$x_{52} = 58.1194640914112$$
$$x_{53} = 61.261056745001$$
$$x_{54} = -51.8362787842316$$
$$x_{55} = -32.9867228626928$$
$$x_{56} = -14.1371669411541$$
$$x_{57} = -58.1194640914112$$
$$x_{58} = -42.4115008234622$$
$$x_{59} = -54.9778714378214$$
$$x_{60} = -1.5707963267949$$
$$x_{61} = 42.4115008234622$$
$$x_{62} = 39.2699081698724$$
$$x_{63} = 67.5442420521806$$
$$x_{64} = -23.5619449019235$$
$$x_{65} = -73.8274273593601$$
o sea hay que resolver la ecuación:
$$\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:
Solución analítica
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
Solución numérica
$$x_{1} = -29.845130209103$$
$$x_{2} = -67.5442420521806$$
$$x_{3} = -70.6858347057703$$
$$x_{4} = 64.4026493985908$$
$$x_{5} = -36.1283155162826$$
$$x_{6} = -92.6769832808989$$
$$x_{7} = -61.261056745001$$
$$x_{8} = -76.9690200129499$$
$$x_{9} = -98.9601685880785$$
$$x_{10} = -95.8185759344887$$
$$x_{11} = 29.845130209103$$
$$x_{12} = 80.1106126665397$$
$$x_{13} = -64.4026493985908$$
$$x_{14} = 36.1283155162826$$
$$x_{15} = 73.8274273593601$$
$$x_{16} = 32.9867228626928$$
$$x_{17} = -4.71238898038469$$
$$x_{18} = -39.2699081698724$$
$$x_{19} = 26.7035375555132$$
$$x_{20} = -5244.88893516816$$
$$x_{21} = 95.8185759344887$$
$$x_{22} = -7.85398163397448$$
$$x_{23} = -17.2787595947439$$
$$x_{24} = -10.9955742875643$$
$$x_{25} = 98.9601685880785$$
$$x_{26} = -86.3937979737193$$
$$x_{27} = 92.6769832808989$$
$$x_{28} = -48.6946861306418$$
$$x_{29} = 54.9778714378214$$
$$x_{30} = 45.553093477052$$
$$x_{31} = 23.5619449019235$$
$$x_{32} = 76.9690200129499$$
$$x_{33} = -89.5353906273091$$
$$x_{34} = 4.71238898038469$$
$$x_{35} = -26.7035375555132$$
$$x_{36} = -80.1106126665397$$
$$x_{37} = 7.85398163397448$$
$$x_{38} = 14.1371669411541$$
$$x_{39} = 86.3937979737193$$
$$x_{40} = -45.553093477052$$
$$x_{41} = -83.2522053201295$$
$$x_{42} = 70.6858347057703$$
$$x_{43} = 83.2522053201295$$
$$x_{44} = 48.6946861306418$$
$$x_{45} = -20.4203522483337$$
$$x_{46} = 51.8362787842316$$
$$x_{47} = 10.9955742875643$$
$$x_{48} = 20.4203522483337$$
$$x_{49} = 1.5707963267949$$
$$x_{50} = 89.5353906273091$$
$$x_{51} = 17.2787595947439$$
$$x_{52} = 58.1194640914112$$
$$x_{53} = 61.261056745001$$
$$x_{54} = -51.8362787842316$$
$$x_{55} = -32.9867228626928$$
$$x_{56} = -14.1371669411541$$
$$x_{57} = -58.1194640914112$$
$$x_{58} = -42.4115008234622$$
$$x_{59} = -54.9778714378214$$
$$x_{60} = -1.5707963267949$$
$$x_{61} = 42.4115008234622$$
$$x_{62} = 39.2699081698724$$
$$x_{63} = 67.5442420521806$$
$$x_{64} = -23.5619449019235$$
$$x_{65} = -73.8274273593601$$
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
$$- \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 62.8318530717959$$
$$x_{2} = -50.2654824574367$$
$$x_{3} = 84.8230016469244$$
$$x_{4} = -53.4070751110265$$
$$x_{5} = -84.8230016469244$$
$$x_{6} = -3.14159265358979$$
$$x_{7} = -6.28318530717959$$
$$x_{8} = 2899.69001926338$$
$$x_{9} = 78.5398163397448$$
$$x_{10} = -75.398223686155$$
$$x_{11} = -9.42477796076938$$
$$x_{12} = 72.2566310325652$$
$$x_{13} = -43.9822971502571$$
$$x_{14} = 40.8407044966673$$
$$x_{15} = -69.1150383789755$$
$$x_{16} = 12.5663706143592$$
$$x_{17} = 87.9645943005142$$
$$x_{18} = 59.6902604182061$$
$$x_{19} = -37.6991118430775$$
$$x_{20} = -91.106186954104$$
$$x_{21} = 97.3893722612836$$
$$x_{22} = 0$$
$$x_{23} = 18.8495559215388$$
$$x_{24} = -78.5398163397448$$
$$x_{25} = 34.5575191894877$$
$$x_{26} = -94.2477796076938$$
$$x_{27} = 2582.38916125081$$
$$x_{28} = 43.9822971502571$$
$$x_{29} = 153.9380400259$$
$$x_{30} = -31.4159265358979$$
$$x_{31} = -81.6814089933346$$
$$x_{32} = -65.9734457253857$$
$$x_{33} = 56.5486677646163$$
$$x_{34} = 3.14159265358979$$
$$x_{35} = 15.707963267949$$
$$x_{36} = -21.9911485751286$$
$$x_{37} = 50.2654824574367$$
$$x_{38} = -15.707963267949$$
$$x_{39} = 28.2743338823081$$
$$x_{40} = 94.2477796076938$$
$$x_{41} = -2199.11485751286$$
$$x_{42} = -59.6902604182061$$
$$x_{43} = -62.8318530717959$$
$$x_{44} = -34.5575191894877$$
$$x_{45} = -97.3893722612836$$
$$x_{46} = 21.9911485751286$$
$$x_{47} = 65.9734457253857$$
$$x_{48} = 37.6991118430775$$
$$x_{49} = -87.9645943005142$$
$$x_{50} = -72.2566310325652$$
$$x_{51} = -25.1327412287183$$
$$x_{52} = -28.2743338823081$$
$$x_{53} = 81.6814089933346$$
$$x_{54} = 6.28318530717959$$
$$x_{55} = 100.530964914873$$
$$x_{56} = -47.1238898038469$$
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:
$$x_{1} = 84.8230016469244$$
$$x_{2} = -53.4070751110265$$
$$x_{3} = -84.8230016469244$$
$$x_{4} = -3.14159265358979$$
$$x_{5} = 2899.69001926338$$
$$x_{6} = 78.5398163397448$$
$$x_{7} = -9.42477796076938$$
$$x_{8} = 72.2566310325652$$
$$x_{9} = 40.8407044966673$$
$$x_{10} = 59.6902604182061$$
$$x_{11} = -91.106186954104$$
$$x_{12} = 97.3893722612836$$
$$x_{13} = -78.5398163397448$$
$$x_{14} = 34.5575191894877$$
$$x_{15} = 153.9380400259$$
$$x_{16} = -65.9734457253857$$
$$x_{17} = 3.14159265358979$$
$$x_{18} = 15.707963267949$$
$$x_{19} = -21.9911485751286$$
$$x_{20} = -15.707963267949$$
$$x_{21} = 28.2743338823081$$
$$x_{22} = -59.6902604182061$$
$$x_{23} = -34.5575191894877$$
$$x_{24} = -97.3893722612836$$
$$x_{25} = 21.9911485751286$$
$$x_{26} = 65.9734457253857$$
$$x_{27} = -72.2566310325652$$
$$x_{28} = -28.2743338823081$$
$$x_{29} = -47.1238898038469$$
Puntos máximos de la función:
$$x_{29} = 62.8318530717959$$
$$x_{29} = -50.2654824574367$$
$$x_{29} = -6.28318530717959$$
$$x_{29} = -75.398223686155$$
$$x_{29} = -43.9822971502571$$
$$x_{29} = -69.1150383789755$$
$$x_{29} = 12.5663706143592$$
$$x_{29} = 87.9645943005142$$
$$x_{29} = -37.6991118430775$$
$$x_{29} = 0$$
$$x_{29} = 18.8495559215388$$
$$x_{29} = -94.2477796076938$$
$$x_{29} = 2582.38916125081$$
$$x_{29} = 43.9822971502571$$
$$x_{29} = -31.4159265358979$$
$$x_{29} = -81.6814089933346$$
$$x_{29} = 56.5486677646163$$
$$x_{29} = 50.2654824574367$$
$$x_{29} = 94.2477796076938$$
$$x_{29} = -2199.11485751286$$
$$x_{29} = -62.8318530717959$$
$$x_{29} = 37.6991118430775$$
$$x_{29} = -87.9645943005142$$
$$x_{29} = -25.1327412287183$$
$$x_{29} = 81.6814089933346$$
$$x_{29} = 6.28318530717959$$
$$x_{29} = 100.530964914873$$
Decrece en los intervalos
$$\left[2899.69001926338, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.3893722612836\right]$$
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(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
$$- \sin{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \sin{\left(\sin{\left(x \right)} \right)} \cos^{2}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 62.8318530717959$$
$$x_{2} = -50.2654824574367$$
$$x_{3} = 84.8230016469244$$
$$x_{4} = -53.4070751110265$$
$$x_{5} = -84.8230016469244$$
$$x_{6} = -3.14159265358979$$
$$x_{7} = -6.28318530717959$$
$$x_{8} = 2899.69001926338$$
$$x_{9} = 78.5398163397448$$
$$x_{10} = -75.398223686155$$
$$x_{11} = -9.42477796076938$$
$$x_{12} = 72.2566310325652$$
$$x_{13} = -43.9822971502571$$
$$x_{14} = 40.8407044966673$$
$$x_{15} = -69.1150383789755$$
$$x_{16} = 12.5663706143592$$
$$x_{17} = 87.9645943005142$$
$$x_{18} = 59.6902604182061$$
$$x_{19} = -37.6991118430775$$
$$x_{20} = -91.106186954104$$
$$x_{21} = 97.3893722612836$$
$$x_{22} = 0$$
$$x_{23} = 18.8495559215388$$
$$x_{24} = -78.5398163397448$$
$$x_{25} = 34.5575191894877$$
$$x_{26} = -94.2477796076938$$
$$x_{27} = 2582.38916125081$$
$$x_{28} = 43.9822971502571$$
$$x_{29} = 153.9380400259$$
$$x_{30} = -31.4159265358979$$
$$x_{31} = -81.6814089933346$$
$$x_{32} = -65.9734457253857$$
$$x_{33} = 56.5486677646163$$
$$x_{34} = 3.14159265358979$$
$$x_{35} = 15.707963267949$$
$$x_{36} = -21.9911485751286$$
$$x_{37} = 50.2654824574367$$
$$x_{38} = -15.707963267949$$
$$x_{39} = 28.2743338823081$$
$$x_{40} = 94.2477796076938$$
$$x_{41} = -2199.11485751286$$
$$x_{42} = -59.6902604182061$$
$$x_{43} = -62.8318530717959$$
$$x_{44} = -34.5575191894877$$
$$x_{45} = -97.3893722612836$$
$$x_{46} = 21.9911485751286$$
$$x_{47} = 65.9734457253857$$
$$x_{48} = 37.6991118430775$$
$$x_{49} = -87.9645943005142$$
$$x_{50} = -72.2566310325652$$
$$x_{51} = -25.1327412287183$$
$$x_{52} = -28.2743338823081$$
$$x_{53} = 81.6814089933346$$
$$x_{54} = 6.28318530717959$$
$$x_{55} = 100.530964914873$$
$$x_{56} = -47.1238898038469$$
Signos de extremos en los puntos:
(62.83185307179586, 1)
(-50.26548245743669, 1)
(84.82300164692441, -1)
(-53.40707511102649, -1)
(-84.82300164692441, -1)
(-3.141592653589793, -1)
(-6.283185307179586, 1)
(2899.6900192633793, -1)
(78.53981633974483, -1)
(-75.39822368615503, 1)
(-9.42477796076938, -1)
(72.25663103256524, -1)
(-43.982297150257104, 1)
(40.840704496667314, -1)
(-69.11503837897546, 1)
(12.566370614359172, 1)
(87.96459430051421, 1)
(59.69026041820607, -1)
(-37.69911184307752, 1)
(-91.106186954104, -1)
(97.3893722612836, -1)
(0, 1)
(18.84955592153876, 1)
(-78.53981633974483, -1)
(34.55751918948773, -1)
(-94.2477796076938, 1)
(2582.38916125081, 1)
(43.982297150257104, 1)
(153.93804002589988, -1)
(-31.41592653589793, 1)
(-81.68140899333463, 1)
(-65.97344572538566, -1)
(56.548667764616276, 1)
(3.141592653589793, -1)
(15.707963267948966, -1)
(-21.991148575128552, -1)
(50.26548245743669, 1)
(-15.707963267948966, -1)
(28.274333882308138, -1)
(94.2477796076938, 1)
(-2199.1148575128555, 1)
(-59.69026041820607, -1)
(-62.83185307179586, 1)
(-34.55751918948773, -1)
(-97.3893722612836, -1)
(21.991148575128552, -1)
(65.97344572538566, -1)
(37.69911184307752, 1)
(-87.96459430051421, 1)
(-72.25663103256524, -1)
(-25.132741228718345, 1)
(-28.274333882308138, -1)
(81.68140899333463, 1)
(6.283185307179586, 1)
(100.53096491487338, 1)
(-47.1238898038469, -1)
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:
$$x_{1} = 84.8230016469244$$
$$x_{2} = -53.4070751110265$$
$$x_{3} = -84.8230016469244$$
$$x_{4} = -3.14159265358979$$
$$x_{5} = 2899.69001926338$$
$$x_{6} = 78.5398163397448$$
$$x_{7} = -9.42477796076938$$
$$x_{8} = 72.2566310325652$$
$$x_{9} = 40.8407044966673$$
$$x_{10} = 59.6902604182061$$
$$x_{11} = -91.106186954104$$
$$x_{12} = 97.3893722612836$$
$$x_{13} = -78.5398163397448$$
$$x_{14} = 34.5575191894877$$
$$x_{15} = 153.9380400259$$
$$x_{16} = -65.9734457253857$$
$$x_{17} = 3.14159265358979$$
$$x_{18} = 15.707963267949$$
$$x_{19} = -21.9911485751286$$
$$x_{20} = -15.707963267949$$
$$x_{21} = 28.2743338823081$$
$$x_{22} = -59.6902604182061$$
$$x_{23} = -34.5575191894877$$
$$x_{24} = -97.3893722612836$$
$$x_{25} = 21.9911485751286$$
$$x_{26} = 65.9734457253857$$
$$x_{27} = -72.2566310325652$$
$$x_{28} = -28.2743338823081$$
$$x_{29} = -47.1238898038469$$
Puntos máximos de la función:
$$x_{29} = 62.8318530717959$$
$$x_{29} = -50.2654824574367$$
$$x_{29} = -6.28318530717959$$
$$x_{29} = -75.398223686155$$
$$x_{29} = -43.9822971502571$$
$$x_{29} = -69.1150383789755$$
$$x_{29} = 12.5663706143592$$
$$x_{29} = 87.9645943005142$$
$$x_{29} = -37.6991118430775$$
$$x_{29} = 0$$
$$x_{29} = 18.8495559215388$$
$$x_{29} = -94.2477796076938$$
$$x_{29} = 2582.38916125081$$
$$x_{29} = 43.9822971502571$$
$$x_{29} = -31.4159265358979$$
$$x_{29} = -81.6814089933346$$
$$x_{29} = 56.5486677646163$$
$$x_{29} = 50.2654824574367$$
$$x_{29} = 94.2477796076938$$
$$x_{29} = -2199.11485751286$$
$$x_{29} = -62.8318530717959$$
$$x_{29} = 37.6991118430775$$
$$x_{29} = -87.9645943005142$$
$$x_{29} = -25.1327412287183$$
$$x_{29} = 81.6814089933346$$
$$x_{29} = 6.28318530717959$$
$$x_{29} = 100.530964914873$$
Decrece en los intervalos
$$\left[2899.69001926338, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -97.3893722612836\right]$$
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
$$\left(3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} - \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \cos{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 3.86780572583686$$
$$x_{2} = 60.4164734904531$$
$$x_{3} = -29.845130209103$$
$$x_{4} = 50.9916955296838$$
$$x_{5} = -82.4076220655817$$
$$x_{6} = -10.1509910330164$$
$$x_{7} = -19.5757689937858$$
$$x_{8} = 29.845130209103$$
$$x_{9} = -93.5215665354467$$
$$x_{10} = 95.8185759344887$$
$$x_{11} = -84.0967885746773$$
$$x_{12} = -77.8136032674978$$
$$x_{13} = -5.55697223493252$$
$$x_{14} = -85.5492147191715$$
$$x_{15} = 62.1056399995488$$
$$x_{16} = 54.1332881832735$$
$$x_{17} = -41.5669175689144$$
$$x_{18} = 70.6858347057703$$
$$x_{19} = -76.1244367584021$$
$$x_{20} = -49.5392693851896$$
$$x_{21} = -14.1371669411541$$
$$x_{22} = 42.4115008234622$$
$$x_{23} = -40.1144914244202$$
$$x_{24} = 47.850102876094$$
$$x_{25} = -3.86780572583686$$
$$x_{26} = -60.4164734904531$$
$$x_{27} = 33.8313061172407$$
$$x_{28} = 99.8047518426263$$
$$x_{29} = 84.0967885746773$$
$$x_{30} = -80.1106126665397$$
$$x_{31} = 55.8224546923692$$
$$x_{32} = 76.1244367584021$$
$$x_{33} = -32.142139608145$$
$$x_{34} = -54.1332881832735$$
$$x_{35} = 48.6946861306418$$
$$x_{36} = 46.3976767315998$$
$$x_{37} = -20.4203522483337$$
$$x_{38} = 10.1509910330164$$
$$x_{39} = 20.4203522483337$$
$$x_{40} = 40.1144914244202$$
$$x_{41} = 11.8401575421121$$
$$x_{42} = 90.3799738818569$$
$$x_{43} = 77.8136032674978$$
$$x_{44} = -71.5304179603182$$
$$x_{45} = 2.41537958134273$$
$$x_{46} = 32.142139608145$$
$$x_{47} = -73.8274273593601$$
$$x_{48} = -67.5442420521806$$
$$x_{49} = 98.1155853335307$$
$$x_{50} = -36.1283155162826$$
$$x_{51} = -27.5481208100611$$
$$x_{52} = 82.4076220655817$$
$$x_{53} = 36.1283155162826$$
$$x_{54} = 16.434176340196$$
$$x_{55} = 26.7035375555132$$
$$x_{56} = -18.1233428492917$$
$$x_{57} = -89.5353906273091$$
$$x_{58} = -91.8324000263511$$
$$x_{59} = -99.8047518426263$$
$$x_{60} = 7.85398163397448$$
$$x_{61} = 14.1371669411541$$
$$x_{62} = 86.3937979737193$$
$$x_{63} = 18.1233428492917$$
$$x_{64} = -55.8224546923692$$
$$x_{65} = 51.8362787842316$$
$$x_{66} = -51.8362787842316$$
$$x_{67} = -33.8313061172407$$
$$x_{68} = -38.4253249153246$$
$$x_{69} = 5.55697223493252$$
$$x_{70} = 25.8589543009654$$
$$x_{71} = -1.5707963267949$$
$$x_{72} = 24.4065281564713$$
$$x_{73} = -23.5619449019235$$
$$x_{74} = -16.434176340196$$
$$x_{75} = 69.8412514512225$$
$$x_{76} = 64.4026493985908$$
$$x_{77} = -62.1056399995488$$
$$x_{78} = -47.850102876094$$
$$x_{79} = 38.4253249153246$$
$$x_{80} = -95.8185759344887$$
$$x_{81} = -98.1155853335307$$
$$x_{82} = 80.1106126665397$$
$$x_{83} = -64.4026493985908$$
$$x_{84} = 88.6908073727613$$
$$x_{85} = 73.8274273593601$$
$$x_{86} = -11.8401575421121$$
$$x_{87} = 68.3888253067284$$
$$x_{88} = -7.85398163397448$$
$$x_{89} = 23.5619449019235$$
$$x_{90} = 92.6769832808989$$
$$x_{91} = 91.8324000263511$$
$$x_{92} = -63.5580661440429$$
$$x_{93} = -69.8412514512225$$
$$x_{94} = -45.553093477052$$
$$x_{95} = -25.8589543009654$$
$$x_{96} = 1.5707963267949$$
$$x_{97} = -58.1194640914112$$
$$x_{98} = 58.1194640914112$$
$$x_{99} = -42.4115008234622$$
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
$$\left[92.6769832808989, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -99.8047518426263\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\left(3 \sin{\left(x \right)} \sin{\left(\sin{\left(x \right)} \right)} - \cos^{2}{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} - \cos{\left(\sin{\left(x \right)} \right)}\right) \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 3.86780572583686$$
$$x_{2} = 60.4164734904531$$
$$x_{3} = -29.845130209103$$
$$x_{4} = 50.9916955296838$$
$$x_{5} = -82.4076220655817$$
$$x_{6} = -10.1509910330164$$
$$x_{7} = -19.5757689937858$$
$$x_{8} = 29.845130209103$$
$$x_{9} = -93.5215665354467$$
$$x_{10} = 95.8185759344887$$
$$x_{11} = -84.0967885746773$$
$$x_{12} = -77.8136032674978$$
$$x_{13} = -5.55697223493252$$
$$x_{14} = -85.5492147191715$$
$$x_{15} = 62.1056399995488$$
$$x_{16} = 54.1332881832735$$
$$x_{17} = -41.5669175689144$$
$$x_{18} = 70.6858347057703$$
$$x_{19} = -76.1244367584021$$
$$x_{20} = -49.5392693851896$$
$$x_{21} = -14.1371669411541$$
$$x_{22} = 42.4115008234622$$
$$x_{23} = -40.1144914244202$$
$$x_{24} = 47.850102876094$$
$$x_{25} = -3.86780572583686$$
$$x_{26} = -60.4164734904531$$
$$x_{27} = 33.8313061172407$$
$$x_{28} = 99.8047518426263$$
$$x_{29} = 84.0967885746773$$
$$x_{30} = -80.1106126665397$$
$$x_{31} = 55.8224546923692$$
$$x_{32} = 76.1244367584021$$
$$x_{33} = -32.142139608145$$
$$x_{34} = -54.1332881832735$$
$$x_{35} = 48.6946861306418$$
$$x_{36} = 46.3976767315998$$
$$x_{37} = -20.4203522483337$$
$$x_{38} = 10.1509910330164$$
$$x_{39} = 20.4203522483337$$
$$x_{40} = 40.1144914244202$$
$$x_{41} = 11.8401575421121$$
$$x_{42} = 90.3799738818569$$
$$x_{43} = 77.8136032674978$$
$$x_{44} = -71.5304179603182$$
$$x_{45} = 2.41537958134273$$
$$x_{46} = 32.142139608145$$
$$x_{47} = -73.8274273593601$$
$$x_{48} = -67.5442420521806$$
$$x_{49} = 98.1155853335307$$
$$x_{50} = -36.1283155162826$$
$$x_{51} = -27.5481208100611$$
$$x_{52} = 82.4076220655817$$
$$x_{53} = 36.1283155162826$$
$$x_{54} = 16.434176340196$$
$$x_{55} = 26.7035375555132$$
$$x_{56} = -18.1233428492917$$
$$x_{57} = -89.5353906273091$$
$$x_{58} = -91.8324000263511$$
$$x_{59} = -99.8047518426263$$
$$x_{60} = 7.85398163397448$$
$$x_{61} = 14.1371669411541$$
$$x_{62} = 86.3937979737193$$
$$x_{63} = 18.1233428492917$$
$$x_{64} = -55.8224546923692$$
$$x_{65} = 51.8362787842316$$
$$x_{66} = -51.8362787842316$$
$$x_{67} = -33.8313061172407$$
$$x_{68} = -38.4253249153246$$
$$x_{69} = 5.55697223493252$$
$$x_{70} = 25.8589543009654$$
$$x_{71} = -1.5707963267949$$
$$x_{72} = 24.4065281564713$$
$$x_{73} = -23.5619449019235$$
$$x_{74} = -16.434176340196$$
$$x_{75} = 69.8412514512225$$
$$x_{76} = 64.4026493985908$$
$$x_{77} = -62.1056399995488$$
$$x_{78} = -47.850102876094$$
$$x_{79} = 38.4253249153246$$
$$x_{80} = -95.8185759344887$$
$$x_{81} = -98.1155853335307$$
$$x_{82} = 80.1106126665397$$
$$x_{83} = -64.4026493985908$$
$$x_{84} = 88.6908073727613$$
$$x_{85} = 73.8274273593601$$
$$x_{86} = -11.8401575421121$$
$$x_{87} = 68.3888253067284$$
$$x_{88} = -7.85398163397448$$
$$x_{89} = 23.5619449019235$$
$$x_{90} = 92.6769832808989$$
$$x_{91} = 91.8324000263511$$
$$x_{92} = -63.5580661440429$$
$$x_{93} = -69.8412514512225$$
$$x_{94} = -45.553093477052$$
$$x_{95} = -25.8589543009654$$
$$x_{96} = 1.5707963267949$$
$$x_{97} = -58.1194640914112$$
$$x_{98} = 58.1194640914112$$
$$x_{99} = -42.4115008234622$$
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
$$\left[92.6769832808989, \infty\right)$$
Convexa en los intervalos
$$\left(-\infty, -99.8047518426263\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}\right) = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}\right) = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to -\infty}\left(\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}\right) = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}\right) = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -1, 1\right\rangle$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x)*cos(sin(x)), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}}{x}\right) = 0$$
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:
$$\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} = \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}$$
- Sí
$$\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} = - \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}$$
- No
es decir, función
es
par
Pues, comprobamos:
$$\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} = \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}$$
- Sí
$$\cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)} = - \cos{\left(x \right)} \cos{\left(\sin{\left(x \right)} \right)}$$
- No
es decir, función
es
par