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Gráfico de la función y = (x*cos(x)^(2))/(sin(x)+cos(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src]
               2      
          x*cos (x)   
f(x) = ---------------
       sin(x) + cos(x)
$$f{\left(x \right)} = \frac{x \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}$$
f = (x*cos(x)^2)/(sin(x) + cos(x))
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = -0.785398163397448$$
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:
$$\frac{x \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
Solución numérica
$$x_{1} = -39.2699083391782$$
$$x_{2} = 20.4203521658248$$
$$x_{3} = -1.57079642651137$$
$$x_{4} = 29.8451303681156$$
$$x_{5} = 64.4026493218244$$
$$x_{6} = -4.71238929978881$$
$$x_{7} = 42.4115007435938$$
$$x_{8} = -45.5530935703464$$
$$x_{9} = -51.8362786641962$$
$$x_{10} = 4.71238884138504$$
$$x_{11} = -29.8451300572295$$
$$x_{12} = -95.8185758567722$$
$$x_{13} = 17.2787593717088$$
$$x_{14} = -10.9955745202325$$
$$x_{15} = 26.7035373983183$$
$$x_{16} = -14.1371668040033$$
$$x_{17} = -17.2787597655124$$
$$x_{18} = 7.85398134664551$$
$$x_{19} = 0$$
$$x_{20} = -58.1194639708633$$
$$x_{21} = -89.5353907262057$$
$$x_{22} = -80.1106125547134$$
$$x_{23} = -36.1283153870415$$
$$x_{24} = -23.5619449926575$$
$$x_{25} = 10.9955740978569$$
$$x_{26} = -67.5442421482662$$
$$x_{27} = 48.6946859735536$$
$$x_{28} = 7.85398178608886$$
$$x_{29} = -7.85398143932466$$
$$x_{30} = -73.8274272630371$$
$$x_{31} = 86.3937979002247$$
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando x es igual a 0:
sustituimos x = 0 en (x*cos(x)^2)/(sin(x) + cos(x)).
$$\frac{0 \cos^{2}{\left(0 \right)}}{\sin{\left(0 \right)} + \cos{\left(0 \right)}}$$
Resultado:
$$f{\left(0 \right)} = 0$$
Punto:
(0, 0)
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
$$\frac{x \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) \cos^{2}{\left(x \right)}}{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)^{2}} + \frac{- 2 x \sin{\left(x \right)} \cos{\left(x \right)} + \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 17.2787595947439$$
$$x_{2} = -23.5619449019235$$
$$x_{3} = -14.1371669411541$$
$$x_{4} = 51.8362787842316$$
$$x_{5} = -17.2787595947439$$
$$x_{6} = -10.9955742875643$$
$$x_{7} = -36.1283155162826$$
$$x_{8} = -95.8185759344887$$
$$x_{9} = -26.7035375555132$$
$$x_{10} = 26.7035375555132$$
$$x_{11} = 89.5353906273091$$
$$x_{12} = 23.5619449019235$$
$$x_{13} = 14.1371669411541$$
$$x_{14} = 42.4115008234622$$
$$x_{15} = 95.8185759344887$$
$$x_{16} = -61.261056745001$$
$$x_{17} = 58.1194640914112$$
$$x_{18} = 36.1283155162826$$
$$x_{19} = 102.101761241668$$
$$x_{20} = 29.845130209103$$
$$x_{21} = -73.8274273593601$$
$$x_{22} = 48.6946861306418$$
$$x_{23} = -4.71238898038469$$
$$x_{24} = 70.6858347057703$$
$$x_{25} = -7.85398163397448$$
$$x_{26} = -51.8362787842316$$
$$x_{27} = -76.9690200129499$$
$$x_{28} = -89.5353906273091$$
$$x_{29} = -39.2699081698724$$
$$x_{30} = 80.1106126665397$$
$$x_{31} = -42.4115008234622$$
$$x_{32} = 45.553093477052$$
$$x_{33} = 20.4203522483337$$
$$x_{34} = 64.4026493985908$$
$$x_{35} = -32.9867228626928$$
$$x_{36} = 67.5442420521806$$
$$x_{37} = -20.4203522483337$$
$$x_{38} = -80.1106126665397$$
$$x_{39} = 7.85398163397448$$
$$x_{40} = -45.553093477052$$
$$x_{41} = 76.9690200129499$$
$$x_{42} = -1.5707963267949$$
$$x_{43} = 39.2699081698724$$
$$x_{44} = -67.5442420521806$$
$$x_{45} = -105.243353895258$$
$$x_{46} = -83.2522053201295$$
$$x_{47} = -29.845130209103$$
$$x_{48} = -86.3937979737193$$
$$x_{49} = 98.9601685880785$$
$$x_{50} = 73.8274273593601$$
$$x_{51} = -58.1194640914112$$
$$x_{52} = 92.6769832808989$$
$$x_{53} = 54.9778714378214$$
$$x_{54} = 86.3937979737193$$
$$x_{55} = 0.623960630317383$$
$$x_{56} = 1.5707963267949$$
$$x_{57} = -54.9778714378214$$
$$x_{58} = -64.4026493985908$$
Signos de extremos en los puntos:
(17.278759594743864, -2.10139136502907e-29)

(-23.56194490192345, -1.73402701495235e-29)

(-14.137166941154069, 4.29347676987173e-30)

(51.83627878423159, 3.09398107171563e-30)

(-17.278759594743864, -2.10139136502906e-29)

(-10.995574287564276, -2.02011321271057e-30)

(-36.12831551628262, -3.66424875021483e-28)

(-95.81857593448869, 3.67652016504401e-28)

(-26.703537555513243, 1.44419018202913e-29)

(26.703537555513243, 1.44419018202913e-29)

(89.53539062730911, 2.60267852044684e-27)

(23.56194490192345, -1.73402701495236e-29)

(14.137166941154069, 4.29347676987172e-30)

(42.411500823462205, -4.98859428281589e-28)

(95.81857593448869, 3.676520165044e-28)

(-61.26105674500097, -5.29879683037423e-28)

(58.119464091411174, 1.39112146798308e-29)

(36.12831551628262, -3.6642487502148e-28)

(102.10176124166829, 2.45314668072883e-27)

(29.845130209103036, -1.12127665170555e-29)

(-73.82742735936014, -4.43565443427592e-28)

(48.6946861306418, -5.73178094238609e-28)

(-4.71238898038469, -1.59017658143397e-31)

(70.68583470577035, 6.77618297499812e-29)

(-7.853981633974483, 7.36192861774987e-31)

(-51.83627878423159, 3.09398107171563e-30)

(-76.96902001294994, 2.66242463704609e-27)

(-89.53539062730911, 2.60267852044681e-27)

(-39.269908169872416, 2.36773935254175e-30)

(80.11061266653972, -1.9228326430437e-27)

(-42.411500823462205, -4.98859428281592e-28)

(45.553093477052, 1.74530768724744e-35)

(20.420352248333657, 1.96251734458305e-29)

(64.40264939859077, 2.61429235475569e-27)

(-32.98672286269283, 7.93556229873748e-30)

(67.54424205218055, -1.3132184568469e-27)

(-20.420352248333657, 1.96251734458305e-29)

(-80.11061266653972, -1.92283264304372e-27)

(7.853981633974483, 7.36192861774987e-31)

(-45.553093477052, 1.74530768724744e-35)

(76.96902001294994, 2.66242463704612e-27)

(-1.5707963267948966, 5.8895428941999e-33)

(39.269908169872416, 2.36773935254175e-30)

(-67.54424205218055, -1.31321845684691e-27)

(-105.24335389525807, -3.63739373812053e-27)

(-83.25220532012952, 1.79644538434511e-28)

(-29.845130209103036, -1.12127665170554e-29)

(-86.39379797371932, -3.32328051180094e-28)

(98.96016858807849, -2.14260182065912e-28)

(73.82742735936014, -4.43565443427594e-28)

(-58.119464091411174, 1.39112146798308e-29)

(92.6769832808989, -2.6915268448779e-27)

(54.977871437821385, -5.58036659424785e-28)

(86.39379797371932, -3.32328051180096e-28)

(0.6239606303173832, 0.294427989456615)

(1.5707963267948966, 5.8895428941999e-33)

(-54.977871437821385, -5.58036659424781e-28)

(-64.40264939859077, 2.61429235475566e-27)


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:
$$x_{1} = -14.1371669411541$$
$$x_{2} = 51.8362787842316$$
$$x_{3} = -95.8185759344887$$
$$x_{4} = -26.7035375555132$$
$$x_{5} = 26.7035375555132$$
$$x_{6} = 89.5353906273091$$
$$x_{7} = 14.1371669411541$$
$$x_{8} = 95.8185759344887$$
$$x_{9} = 58.1194640914112$$
$$x_{10} = 102.101761241668$$
$$x_{11} = 70.6858347057703$$
$$x_{12} = -7.85398163397448$$
$$x_{13} = -51.8362787842316$$
$$x_{14} = -76.9690200129499$$
$$x_{15} = -89.5353906273091$$
$$x_{16} = -39.2699081698724$$
$$x_{17} = 45.553093477052$$
$$x_{18} = 20.4203522483337$$
$$x_{19} = 64.4026493985908$$
$$x_{20} = -32.9867228626928$$
$$x_{21} = -20.4203522483337$$
$$x_{22} = 7.85398163397448$$
$$x_{23} = -45.553093477052$$
$$x_{24} = 76.9690200129499$$
$$x_{25} = -1.5707963267949$$
$$x_{26} = 39.2699081698724$$
$$x_{27} = -83.2522053201295$$
$$x_{28} = -58.1194640914112$$
$$x_{29} = 1.5707963267949$$
$$x_{30} = -64.4026493985908$$
Puntos máximos de la función:
$$x_{30} = 17.2787595947439$$
$$x_{30} = -23.5619449019235$$
$$x_{30} = -17.2787595947439$$
$$x_{30} = -10.9955742875643$$
$$x_{30} = -36.1283155162826$$
$$x_{30} = 23.5619449019235$$
$$x_{30} = 42.4115008234622$$
$$x_{30} = -61.261056745001$$
$$x_{30} = 36.1283155162826$$
$$x_{30} = 29.845130209103$$
$$x_{30} = -73.8274273593601$$
$$x_{30} = 48.6946861306418$$
$$x_{30} = -4.71238898038469$$
$$x_{30} = 80.1106126665397$$
$$x_{30} = -42.4115008234622$$
$$x_{30} = 67.5442420521806$$
$$x_{30} = -80.1106126665397$$
$$x_{30} = -67.5442420521806$$
$$x_{30} = -105.243353895258$$
$$x_{30} = -29.845130209103$$
$$x_{30} = -86.3937979737193$$
$$x_{30} = 98.9601685880785$$
$$x_{30} = 73.8274273593601$$
$$x_{30} = 92.6769832808989$$
$$x_{30} = 54.9778714378214$$
$$x_{30} = 86.3937979737193$$
$$x_{30} = 0.623960630317383$$
$$x_{30} = -54.9778714378214$$
Decrece en los intervalos
$$\left[102.101761241668, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -95.8185759344887\right]$$
Asíntotas verticales
Hay:
$$x_{1} = -0.785398163397448$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \lim_{x \to -\infty}\left(\frac{x \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\right)$$
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{x \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x*cos(x)^2)/(sin(x) + cos(x)), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
$$y = x \lim_{x \to -\infty}\left(\frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\right)$$
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}}\right)$$
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:
$$\frac{x \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} = - \frac{x \cos^{2}{\left(x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}}$$
- No
$$\frac{x \cos^{2}{\left(x \right)}}{\sin{\left(x \right)} + \cos{\left(x \right)}} = \frac{x \cos^{2}{\left(x \right)}}{- \sin{\left(x \right)} + \cos{\left(x \right)}}$$
- No
es decir, función
no es
par ni impar