Sr Examen

Gráfico de la función y = cos(x)-|cos(x)|

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

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

Solución numérica
$$x_{1} = 18$$
$$x_{2} = -72080.5334319336$$
$$x_{3} = 1.50932280296592$$
$$x_{4} = 50$$
$$x_{5} = 4.78730945922003$$
$$x_{6} = -1.56268866060787$$
$$x_{7} = 31.970260808218$$
$$x_{8} = 70$$
$$x_{9} = 30$$
$$x_{10} = 24$$
$$x_{11} = 99.2240852394006$$
$$x_{12} = 94$$
$$x_{13} = -100$$
$$x_{14} = -56$$
$$x_{15} = -6$$
$$x_{16} = 38$$
$$x_{17} = 11.3055197466968$$
$$x_{18} = 89.459497911885$$
$$x_{19} = -31.9859091024297$$
$$x_{20} = 6$$
$$x_{21} = -25.5177327391909$$
$$x_{22} = 82$$
$$x_{23} = -32.6511545137895$$
$$x_{24} = -55.8434778964253$$
$$x_{25} = -88$$
$$x_{26} = -83.1606077106582$$
$$x_{27} = -44$$
$$x_{28} = -74$$
$$x_{29} = 0$$
$$x_{30} = 95.8102604397009$$
$$x_{31} = -14$$
$$x_{32} = -89.5235445780902$$
$$x_{33} = -36.1325268906033$$
$$x_{34} = -62$$
$$x_{35} = -55.8027447145845$$
$$x_{36} = 62.2672274377015$$
$$x_{37} = 56$$
$$x_{38} = -24$$
$$x_{39} = 64$$
$$x_{40} = 12$$
$$x_{41} = -50$$
$$x_{42} = 243.880276274012$$
$$x_{43} = -69.3961229186548$$
$$x_{44} = 48.762186268866$$
$$x_{45} = 18.370450489588$$
$$x_{46} = -70$$
$$x_{47} = 45.4846942555662$$
$$x_{48} = 75.8660188579398$$
$$x_{49} = -68$$
$$x_{50} = 55.2639771162362$$
$$x_{51} = 74$$
$$x_{52} = -45.5432321851046$$
$$x_{53} = -99.7598323488752$$
$$x_{54} = -76.6058993602428$$
$$x_{55} = 86.3990282625544$$
$$x_{56} = -76$$
$$x_{57} = -82$$
$$x_{58} = 76$$
$$x_{59} = 62.6154246585311$$
$$x_{60} = 7.84859385700983$$
$$x_{61} = -95.75$$
$$x_{62} = -26$$
$$x_{63} = -38$$
$$x_{64} = -58$$
$$x_{65} = -7.75$$
$$x_{66} = 51.8295279395865$$
$$x_{67} = 26$$
$$x_{68} = -42.4665825346695$$
$$x_{69} = -48.9404554284412$$
$$x_{70} = -18$$
$$x_{71} = -11.9331970991102$$
$$x_{72} = 100$$
$$x_{73} = 88$$
$$x_{74} = -20$$
$$x_{75} = -94$$
$$x_{76} = -92.9033431485983$$
$$x_{77} = -12$$
$$x_{78} = -51.75$$
$$x_{79} = 80.25$$
$$x_{80} = 92.7376264990321$$
$$x_{81} = -30$$
$$x_{82} = 42.4180699604306$$
$$x_{83} = 20$$
$$x_{84} = 14$$
$$x_{85} = 36.25$$
$$x_{86} = -4.97900239166155$$
$$x_{87} = 32$$
$$x_{88} = -32$$
$$x_{89} = -86.4429899562409$$
$$x_{90} = 44$$
$$x_{91} = 62$$
$$x_{92} = -13.3487755613934$$
$$x_{93} = -214.475232071327$$
$$x_{94} = -80.11381990843$$
$$x_{95} = 38.9809055076008$$
$$x_{96} = -64$$
$$x_{97} = -20.0681158886759$$
$$x_{98} = 68$$
$$x_{99} = 58$$
$$x_{100} = -39.1869680091833$$
$$x_{101} = 82.9391467589083$$
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) - Abs(cos(x)).
$$- \left|{\cos{\left(0 \right)}}\right| + \cos{\left(0 \right)}$$
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
$$\sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)} - \sin{\left(x \right)} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = 18$$
$$x_{2} = 50$$
$$x_{3} = -72.2566310325652$$
$$x_{4} = 91.106186954104$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = 70$$
$$x_{7} = 30$$
$$x_{8} = 24$$
$$x_{9} = 94$$
$$x_{10} = -100$$
$$x_{11} = -56$$
$$x_{12} = 40.8407044966673$$
$$x_{13} = -6$$
$$x_{14} = 38$$
$$x_{15} = 72.2566310325652$$
$$x_{16} = -47.1238898038469$$
$$x_{17} = 6$$
$$x_{18} = 82$$
$$x_{19} = -88$$
$$x_{20} = 59.6902604182061$$
$$x_{21} = -44$$
$$x_{22} = 84.8230016469244$$
$$x_{23} = 0$$
$$x_{24} = -74$$
$$x_{25} = -14$$
$$x_{26} = 38.7483330486114$$
$$x_{27} = -53.4070751110265$$
$$x_{28} = -62$$
$$x_{29} = -21.9911485751286$$
$$x_{30} = 99.2691857797929$$
$$x_{31} = 56$$
$$x_{32} = -24$$
$$x_{33} = 64$$
$$x_{34} = -34.5575191894877$$
$$x_{35} = 12$$
$$x_{36} = -50$$
$$x_{37} = -3.14159265358979$$
$$x_{38} = 53.4070751110265$$
$$x_{39} = -70$$
$$x_{40} = -75.7312933201993$$
$$x_{41} = -68$$
$$x_{42} = 74$$
$$x_{43} = -92.6907188312909$$
$$x_{44} = -65.9734457253857$$
$$x_{45} = -76$$
$$x_{46} = -82$$
$$x_{47} = 82.5136949447994$$
$$x_{48} = 76$$
$$x_{49} = -95.75$$
$$x_{50} = -26$$
$$x_{51} = -38$$
$$x_{52} = -58$$
$$x_{53} = 3.14159265358979$$
$$x_{54} = -7.75$$
$$x_{55} = -78.5398163397448$$
$$x_{56} = 26$$
$$x_{57} = 47.1238898038469$$
$$x_{58} = -18$$
$$x_{59} = 28.2743338823081$$
$$x_{60} = 100$$
$$x_{61} = 88$$
$$x_{62} = -20$$
$$x_{63} = -91.106186954104$$
$$x_{64} = -94$$
$$x_{65} = -12$$
$$x_{66} = -51.75$$
$$x_{67} = -48.8578078716517$$
$$x_{68} = 80.25$$
$$x_{69} = -30$$
$$x_{70} = -9.42477796076938$$
$$x_{71} = 65.9734457253857$$
$$x_{72} = 20$$
$$x_{73} = 14$$
$$x_{74} = 36.25$$
$$x_{75} = 32$$
$$x_{76} = -15.707963267949$$
$$x_{77} = 97.3893722612836$$
$$x_{78} = -32$$
$$x_{79} = 44$$
$$x_{80} = 62$$
$$x_{81} = 15.707963267949$$
$$x_{82} = 9.42477796076938$$
$$x_{83} = -84.8230016469244$$
$$x_{84} = -64$$
$$x_{85} = 68$$
$$x_{86} = 21.9911485751286$$
$$x_{87} = 34.5575191894877$$
$$x_{88} = -97.3893722612836$$
$$x_{89} = -32.0322958555426$$
$$x_{90} = -5.04363182667898$$
$$x_{91} = -40.8407044966673$$
$$x_{92} = -28.2743338823081$$
$$x_{93} = 58$$
$$x_{94} = 78.5398163397448$$
$$x_{95} = 11.7052375414856$$
$$x_{96} = 55.4742156532357$$
Signos de extremos en los puntos:
(18, 0)

(50, 0)

(-72.25663103256524, -2)

(91.106186954104, -2)

(-59.69026041820607, -2)

(70, 0)

(30, 0)

(24, 0)

(94, 0)

(-100, 0)

(-56, 0)

(40.840704496667314, -2)

(-6, 0)

(38, 0)

(72.25663103256524, -2)

(-47.1238898038469, -2)

(6, 0)

(82, 0)

(-88, 0)

(59.69026041820607, -2)

(-44, 0)

(84.82300164692441, -2)

(0, 0)

(-74, 0)

(-14, 0)

(38.748333048611386, 0)

(-53.40707511102649, -2)

(-62, 0)

(-21.991148575128552, -2)

(99.2691857797929, 0)

(56, 0)

(-24, 0)

(64, 0)

(-34.55751918948773, -2)

(12, 0)

(-50, 0)

(-3.141592653589793, -2)

(53.40707511102649, -2)

(-70, 0)

(-75.73129332019926, 0)

(-68, 0)

(74, 0)

(-92.69071883129092, 0)

(-65.97344572538566, -2)

(-76, 0)

(-82, 0)

(82.51369494479937, 0)

(76, 0)

(-95.75, 0)

(-26, 0)

(-38, 0)

(-58, 0)

(3.141592653589793, -2)

(-7.75, 0)

(-78.53981633974483, -2)

(26, 0)

(47.1238898038469, -2)

(-18, 0)

(28.274333882308138, -2)

(100, 0)

(88, 0)

(-20, 0)

(-91.106186954104, -2)

(-94, 0)

(-12, 0)

(-51.75, 0)

(-48.8578078716517, 0)

(80.25, 0)

(-30, 0)

(-9.42477796076938, -2)

(65.97344572538566, -2)

(20, 0)

(14, 0)

(36.25, 0)

(32, 0)

(-15.707963267948966, -2)

(97.3893722612836, -2)

(-32, 0)

(44, 0)

(62, 0)

(15.707963267948966, -2)

(9.42477796076938, -2)

(-84.82300164692441, -2)

(-64, 0)

(68, 0)

(21.991148575128552, -2)

(34.55751918948773, -2)

(-97.3893722612836, -2)

(-32.03229585554262, 0)

(-5.043631826678977, 0)

(-40.840704496667314, -2)

(-28.274333882308138, -2)

(58, 0)

(78.53981633974483, -2)

(11.705237541485587, 0)

(55.47421565323565, 0)


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:
$$x_{1} = -72.2566310325652$$
$$x_{2} = 91.106186954104$$
$$x_{3} = -59.6902604182061$$
$$x_{4} = 40.8407044966673$$
$$x_{5} = 72.2566310325652$$
$$x_{6} = -47.1238898038469$$
$$x_{7} = 59.6902604182061$$
$$x_{8} = 84.8230016469244$$
$$x_{9} = -53.4070751110265$$
$$x_{10} = -21.9911485751286$$
$$x_{11} = -34.5575191894877$$
$$x_{12} = -3.14159265358979$$
$$x_{13} = 53.4070751110265$$
$$x_{14} = -65.9734457253857$$
$$x_{15} = 3.14159265358979$$
$$x_{16} = -78.5398163397448$$
$$x_{17} = 47.1238898038469$$
$$x_{18} = 28.2743338823081$$
$$x_{19} = -91.106186954104$$
$$x_{20} = -9.42477796076938$$
$$x_{21} = 65.9734457253857$$
$$x_{22} = -15.707963267949$$
$$x_{23} = 97.3893722612836$$
$$x_{24} = 15.707963267949$$
$$x_{25} = 9.42477796076938$$
$$x_{26} = -84.8230016469244$$
$$x_{27} = 21.9911485751286$$
$$x_{28} = 34.5575191894877$$
$$x_{29} = -97.3893722612836$$
$$x_{30} = -40.8407044966673$$
$$x_{31} = -28.2743338823081$$
$$x_{32} = 78.5398163397448$$
La función no tiene puntos máximos
Decrece en los intervalos
$$\left[97.3893722612836, \infty\right)$$
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
$$- 2 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)}\right) + \cos{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)} - \cos{\left(x \right)} = 0$$
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\cos{\left(x \right)} - \left|{\cos{\left(x \right)}}\right|\right) = \left\langle -1, 1\right\rangle - \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = \left\langle -1, 1\right\rangle - \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to \infty}\left(\cos{\left(x \right)} - \left|{\cos{\left(x \right)}}\right|\right) = \left\langle -1, 1\right\rangle - \left|{\left\langle -1, 1\right\rangle}\right|$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \left\langle -1, 1\right\rangle - \left|{\left\langle -1, 1\right\rangle}\right|$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x) - Abs(cos(x)), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)} - \left|{\cos{\left(x \right)}}\right|}{x}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
$$\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)} - \left|{\cos{\left(x \right)}}\right|}{x}\right) = 0$$
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:
$$\cos{\left(x \right)} - \left|{\cos{\left(x \right)}}\right| = \cos{\left(x \right)} - \left|{\cos{\left(x \right)}}\right|$$
- Sí
$$\cos{\left(x \right)} - \left|{\cos{\left(x \right)}}\right| = - \cos{\left(x \right)} + \left|{\cos{\left(x \right)}}\right|$$
- No
es decir, función
es
par