Sr Examen

Gráfico de la función y = cos(|x|)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src]
f(x) = cos(|x|)
$$f{\left(x \right)} = \cos{\left(\left|{x}\right| \right)}$$
f = cos(|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:
$$\cos{\left(\left|{x}\right| \right)} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
$$x_{1} = \frac{\pi}{2}$$
$$x_{2} = \frac{3 \pi}{2}$$
Solución numérica
$$x_{1} = 7.85398163397448$$
$$x_{2} = -2266.65909956504$$
$$x_{3} = -86.3937979737193$$
$$x_{4} = 58.1194640914112$$
$$x_{5} = 23.5619449019235$$
$$x_{6} = -67.5442420521806$$
$$x_{7} = -4.71238898038469$$
$$x_{8} = -20.4203522483337$$
$$x_{9} = 83.2522053201295$$
$$x_{10} = -29.845130209103$$
$$x_{11} = -39.2699081698724$$
$$x_{12} = -98.9601685880785$$
$$x_{13} = 98.9601685880785$$
$$x_{14} = 86.3937979737193$$
$$x_{15} = 26.7035375555132$$
$$x_{16} = -48.6946861306418$$
$$x_{17} = -89.5353906273091$$
$$x_{18} = -17.2787595947439$$
$$x_{19} = 20.4203522483337$$
$$x_{20} = 48.6946861306418$$
$$x_{21} = -64.4026493985908$$
$$x_{22} = 67.5442420521806$$
$$x_{23} = 14.1371669411541$$
$$x_{24} = -26.7035375555132$$
$$x_{25} = 42.4115008234622$$
$$x_{26} = -70.6858347057703$$
$$x_{27} = -32.9867228626928$$
$$x_{28} = 39.2699081698724$$
$$x_{29} = 4.71238898038469$$
$$x_{30} = 73.8274273593601$$
$$x_{31} = 89.5353906273091$$
$$x_{32} = 45.553093477052$$
$$x_{33} = 70.6858347057703$$
$$x_{34} = -168.075206967054$$
$$x_{35} = -7.85398163397448$$
$$x_{36} = -95.8185759344887$$
$$x_{37} = 76.9690200129499$$
$$x_{38} = 32.9867228626928$$
$$x_{39} = -23.5619449019235$$
$$x_{40} = 64.4026493985908$$
$$x_{41} = -36.1283155162826$$
$$x_{42} = -83.2522053201295$$
$$x_{43} = -1.5707963267949$$
$$x_{44} = -58.1194640914112$$
$$x_{45} = -387.986692718339$$
$$x_{46} = -10.9955742875643$$
$$x_{47} = 1.5707963267949$$
$$x_{48} = 29.845130209103$$
$$x_{49} = -73.8274273593601$$
$$x_{50} = -92.6769832808989$$
$$x_{51} = -54.9778714378214$$
$$x_{52} = 80.1106126665397$$
$$x_{53} = 54.9778714378214$$
$$x_{54} = -76.9690200129499$$
$$x_{55} = 36.1283155162826$$
$$x_{56} = 61.261056745001$$
$$x_{57} = 92.6769832808989$$
$$x_{58} = -61.261056745001$$
$$x_{59} = 17.2787595947439$$
$$x_{60} = 10.9955742875643$$
$$x_{61} = -51.8362787842316$$
$$x_{62} = -45.553093477052$$
$$x_{63} = -42.4115008234622$$
$$x_{64} = -80.1106126665397$$
$$x_{65} = 51.8362787842316$$
$$x_{66} = 95.8185759344887$$
$$x_{67} = -14.1371669411541$$
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 cos(|x|).
$$\cos{\left(\left|{0}\right| \right)}$$
Resultado:
$$f{\left(0 \right)} = 1$$
Punto:
(0, 1)
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
$$- \sin{\left(\left|{x}\right| \right)} \operatorname{sign}{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 12.5663706143592$$
$$x_{2} = 53.4070751110265$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 37.6991118430775$$
$$x_{5} = 97.3893722612836$$
$$x_{6} = 78.5398163397448$$
$$x_{7} = -59.6902604182061$$
$$x_{8} = -65.9734457253857$$
$$x_{9} = 0$$
$$x_{10} = -31.4159265358979$$
$$x_{11} = -113.097335529233$$
$$x_{12} = -50.2654824574367$$
$$x_{13} = -21.9911485751286$$
$$x_{14} = 6.28318530717959$$
$$x_{15} = -34.5575191894877$$
$$x_{16} = -94.2477796076938$$
$$x_{17} = -69.1150383789755$$
$$x_{18} = -15.707963267949$$
$$x_{19} = 21.9911485751286$$
$$x_{20} = 69.1150383789755$$
$$x_{21} = 62.8318530717959$$
$$x_{22} = 50.2654824574367$$
$$x_{23} = 81.6814089933346$$
$$x_{24} = 100.530964914873$$
$$x_{25} = -40.8407044966673$$
$$x_{26} = 9.42477796076938$$
$$x_{27} = -87.9645943005142$$
$$x_{28} = 34.5575191894877$$
$$x_{29} = 65.9734457253857$$
$$x_{30} = -62.8318530717959$$
$$x_{31} = -18.8495559215388$$
$$x_{32} = -28.2743338823081$$
$$x_{33} = -267.035375555132$$
$$x_{34} = -232.477856365645$$
$$x_{35} = -56.5486677646163$$
$$x_{36} = -53.4070751110265$$
$$x_{37} = -37.6991118430775$$
$$x_{38} = -25.1327412287183$$
$$x_{39} = -100.530964914873$$
$$x_{40} = -9.42477796076938$$
$$x_{41} = 40.8407044966673$$
$$x_{42} = -91.106186954104$$
$$x_{43} = -2642.07942166902$$
$$x_{44} = -75.398223686155$$
$$x_{45} = 18.8495559215388$$
$$x_{46} = 87.9645943005142$$
$$x_{47} = 59.6902604182061$$
$$x_{48} = -6.28318530717959$$
$$x_{49} = 25.1327412287183$$
$$x_{50} = 47.1238898038469$$
$$x_{51} = 91.106186954104$$
$$x_{52} = 28.2743338823081$$
$$x_{53} = 56.5486677646163$$
$$x_{54} = -43.9822971502571$$
$$x_{55} = -47.1238898038469$$
$$x_{56} = -3.14159265358979$$
$$x_{57} = 31.4159265358979$$
$$x_{58} = 94.2477796076938$$
$$x_{59} = -12.5663706143592$$
$$x_{60} = 75.398223686155$$
$$x_{61} = -72.2566310325652$$
$$x_{62} = -84.8230016469244$$
$$x_{63} = 84.8230016469244$$
$$x_{64} = 72.2566310325652$$
$$x_{65} = -81.6814089933346$$
$$x_{66} = 43.9822971502571$$
$$x_{67} = -78.5398163397448$$
$$x_{68} = 15.707963267949$$
$$x_{69} = 3.14159265358979$$
Signos de extremos en los puntos:
(12.566370614359172, 1)

(53.40707511102649, -1)

(-97.3893722612836, -1)

(37.69911184307752, 1)

(97.3893722612836, -1)

(78.53981633974483, -1)

(-59.69026041820607, -1)

(-65.97344572538566, -1)

(0, 1)

(-31.41592653589793, 1)

(-113.09733552923255, 1)

(-50.26548245743669, 1)

(-21.991148575128552, -1)

(6.283185307179586, 1)

(-34.55751918948773, -1)

(-94.2477796076938, 1)

(-69.11503837897546, 1)

(-15.707963267948966, -1)

(21.991148575128552, -1)

(69.11503837897546, 1)

(62.83185307179586, 1)

(50.26548245743669, 1)

(81.68140899333463, 1)

(100.53096491487338, 1)

(-40.840704496667314, -1)

(9.42477796076938, -1)

(-87.96459430051421, 1)

(34.55751918948773, -1)

(65.97344572538566, -1)

(-62.83185307179586, 1)

(-18.84955592153876, 1)

(-28.274333882308138, -1)

(-267.0353755551324, -1)

(-232.4778563656447, 1)

(-56.548667764616276, 1)

(-53.40707511102649, -1)

(-37.69911184307752, 1)

(-25.132741228718345, 1)

(-100.53096491487338, 1)

(-9.42477796076938, -1)

(40.840704496667314, -1)

(-91.106186954104, -1)

(-2642.079421669016, -1)

(-75.39822368615503, 1)

(18.84955592153876, 1)

(87.96459430051421, 1)

(59.69026041820607, -1)

(-6.283185307179586, 1)

(25.132741228718345, 1)

(47.1238898038469, -1)

(91.106186954104, -1)

(28.274333882308138, -1)

(56.548667764616276, 1)

(-43.982297150257104, 1)

(-47.1238898038469, -1)

(-3.141592653589793, -1)

(31.41592653589793, 1)

(94.2477796076938, 1)

(-12.566370614359172, 1)

(75.39822368615503, 1)

(-72.25663103256524, -1)

(-84.82300164692441, -1)

(84.82300164692441, -1)

(72.25663103256524, -1)

(-81.68140899333463, 1)

(43.982297150257104, 1)

(-78.53981633974483, -1)

(15.707963267948966, -1)

(3.141592653589793, -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:
$$x_{1} = 53.4070751110265$$
$$x_{2} = -97.3893722612836$$
$$x_{3} = 97.3893722612836$$
$$x_{4} = 78.5398163397448$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = -65.9734457253857$$
$$x_{7} = -21.9911485751286$$
$$x_{8} = -34.5575191894877$$
$$x_{9} = -15.707963267949$$
$$x_{10} = 21.9911485751286$$
$$x_{11} = -40.8407044966673$$
$$x_{12} = 9.42477796076938$$
$$x_{13} = 34.5575191894877$$
$$x_{14} = 65.9734457253857$$
$$x_{15} = -28.2743338823081$$
$$x_{16} = -267.035375555132$$
$$x_{17} = -53.4070751110265$$
$$x_{18} = -9.42477796076938$$
$$x_{19} = 40.8407044966673$$
$$x_{20} = -91.106186954104$$
$$x_{21} = -2642.07942166902$$
$$x_{22} = 59.6902604182061$$
$$x_{23} = 47.1238898038469$$
$$x_{24} = 91.106186954104$$
$$x_{25} = 28.2743338823081$$
$$x_{26} = -47.1238898038469$$
$$x_{27} = -3.14159265358979$$
$$x_{28} = -72.2566310325652$$
$$x_{29} = -84.8230016469244$$
$$x_{30} = 84.8230016469244$$
$$x_{31} = 72.2566310325652$$
$$x_{32} = -78.5398163397448$$
$$x_{33} = 15.707963267949$$
$$x_{34} = 3.14159265358979$$
Puntos máximos de la función:
$$x_{34} = 12.5663706143592$$
$$x_{34} = 37.6991118430775$$
$$x_{34} = 0$$
$$x_{34} = -31.4159265358979$$
$$x_{34} = -113.097335529233$$
$$x_{34} = -50.2654824574367$$
$$x_{34} = 6.28318530717959$$
$$x_{34} = -94.2477796076938$$
$$x_{34} = -69.1150383789755$$
$$x_{34} = 69.1150383789755$$
$$x_{34} = 62.8318530717959$$
$$x_{34} = 50.2654824574367$$
$$x_{34} = 81.6814089933346$$
$$x_{34} = 100.530964914873$$
$$x_{34} = -87.9645943005142$$
$$x_{34} = -62.8318530717959$$
$$x_{34} = -18.8495559215388$$
$$x_{34} = -232.477856365645$$
$$x_{34} = -56.5486677646163$$
$$x_{34} = -37.6991118430775$$
$$x_{34} = -25.1327412287183$$
$$x_{34} = -100.530964914873$$
$$x_{34} = -75.398223686155$$
$$x_{34} = 18.8495559215388$$
$$x_{34} = 87.9645943005142$$
$$x_{34} = -6.28318530717959$$
$$x_{34} = 25.1327412287183$$
$$x_{34} = 56.5486677646163$$
$$x_{34} = -43.9822971502571$$
$$x_{34} = 31.4159265358979$$
$$x_{34} = 94.2477796076938$$
$$x_{34} = -12.5663706143592$$
$$x_{34} = 75.398223686155$$
$$x_{34} = -81.6814089933346$$
$$x_{34} = 43.9822971502571$$
Decrece en los intervalos
$$\left[97.3893722612836, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -2642.07942166902\right]$$
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
$$- (2 \sin{\left(\left|{x}\right| \right)} \delta\left(x\right) + \cos{\left(\left|{x}\right| \right)} \operatorname{sign}^{2}{\left(x \right)}) = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$

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
$$\left(-\infty, - \frac{\pi}{2}\right] \cup \left[\frac{\pi}{2}, \infty\right)$$
Convexa en los intervalos
$$\left[- \frac{\pi}{2}, \frac{\pi}{2}\right]$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty} \cos{\left(\left|{x}\right| \right)} = \left\langle -1, 1\right\rangle$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty} \cos{\left(\left|{x}\right| \right)} = \left\langle -1, 1\right\rangle$$
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|), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(\left|{x}\right| \right)}}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(\left|{x}\right| \right)}}{x}\right) = 0$$
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:
$$\cos{\left(\left|{x}\right| \right)} = \cos{\left(\left|{x}\right| \right)}$$
- Sí
$$\cos{\left(\left|{x}\right| \right)} = - \cos{\left(\left|{x}\right| \right)}$$
- No
es decir, función
es
par