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Trazado el gráfico de la función/ cosx-|cosx|

Gráfico de la función y = cosx-|cosx|

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = cos(x) - |cos(x)|
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
02468-8-6-4-2-10102-4
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⁡(x)−∣cos⁡(x)∣=0\cos{\left(x \right)} - \left|{\cos{\left(x \right)}}\right| = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=18x_{1} = 18
x2=−72080.5334319336x_{2} = -72080.5334319336
x3=1.50932280296592x_{3} = 1.50932280296592
x4=50x_{4} = 50
x5=4.78730945922003x_{5} = 4.78730945922003
x6=−1.56268866060787x_{6} = -1.56268866060787
x7=31.970260808218x_{7} = 31.970260808218
x8=70x_{8} = 70
x9=30x_{9} = 30
x10=24x_{10} = 24
x11=99.2240852394006x_{11} = 99.2240852394006
x12=94x_{12} = 94
x13=−100x_{13} = -100
x14=−56x_{14} = -56
x15=−6x_{15} = -6
x16=38x_{16} = 38
x17=11.3055197466968x_{17} = 11.3055197466968
x18=89.459497911885x_{18} = 89.459497911885
x19=−31.9859091024297x_{19} = -31.9859091024297
x20=6x_{20} = 6
x21=−25.5177327391909x_{21} = -25.5177327391909
x22=82x_{22} = 82
x23=−32.6511545137895x_{23} = -32.6511545137895
x24=−55.8434778964253x_{24} = -55.8434778964253
x25=−88x_{25} = -88
x26=−83.1606077106582x_{26} = -83.1606077106582
x27=−44x_{27} = -44
x28=−74x_{28} = -74
x29=0x_{29} = 0
x30=95.8102604397009x_{30} = 95.8102604397009
x31=−14x_{31} = -14
x32=−89.5235445780902x_{32} = -89.5235445780902
x33=−36.1325268906033x_{33} = -36.1325268906033
x34=−62x_{34} = -62
x35=−55.8027447145845x_{35} = -55.8027447145845
x36=62.2672274377015x_{36} = 62.2672274377015
x37=56x_{37} = 56
x38=−24x_{38} = -24
x39=64x_{39} = 64
x40=12x_{40} = 12
x41=−50x_{41} = -50
x42=243.880276274012x_{42} = 243.880276274012
x43=−69.3961229186548x_{43} = -69.3961229186548
x44=48.762186268866x_{44} = 48.762186268866
x45=18.370450489588x_{45} = 18.370450489588
x46=−70x_{46} = -70
x47=45.4846942555662x_{47} = 45.4846942555662
x48=75.8660188579398x_{48} = 75.8660188579398
x49=−68x_{49} = -68
x50=55.2639771162362x_{50} = 55.2639771162362
x51=74x_{51} = 74
x52=−45.5432321851046x_{52} = -45.5432321851046
x53=−99.7598323488752x_{53} = -99.7598323488752
x54=−76.6058993602428x_{54} = -76.6058993602428
x55=86.3990282625544x_{55} = 86.3990282625544
x56=−76x_{56} = -76
x57=−82x_{57} = -82
x58=76x_{58} = 76
x59=62.6154246585311x_{59} = 62.6154246585311
x60=7.84859385700983x_{60} = 7.84859385700983
x61=−95.75x_{61} = -95.75
x62=−26x_{62} = -26
x63=−38x_{63} = -38
x64=−58x_{64} = -58
x65=−7.75x_{65} = -7.75
x66=51.8295279395865x_{66} = 51.8295279395865
x67=26x_{67} = 26
x68=−42.4665825346695x_{68} = -42.4665825346695
x69=−48.9404554284412x_{69} = -48.9404554284412
x70=−18x_{70} = -18
x71=−11.9331970991102x_{71} = -11.9331970991102
x72=100x_{72} = 100
x73=88x_{73} = 88
x74=−20x_{74} = -20
x75=−94x_{75} = -94
x76=−92.9033431485983x_{76} = -92.9033431485983
x77=−12x_{77} = -12
x78=−51.75x_{78} = -51.75
x79=80.25x_{79} = 80.25
x80=92.7376264990321x_{80} = 92.7376264990321
x81=−30x_{81} = -30
x82=42.4180699604306x_{82} = 42.4180699604306
x83=20x_{83} = 20
x84=14x_{84} = 14
x85=36.25x_{85} = 36.25
x86=−4.97900239166155x_{86} = -4.97900239166155
x87=32x_{87} = 32
x88=−32x_{88} = -32
x89=−86.4429899562409x_{89} = -86.4429899562409
x90=44x_{90} = 44
x91=62x_{91} = 62
x92=−13.3487755613934x_{92} = -13.3487755613934
x93=−214.475232071327x_{93} = -214.475232071327
x94=−80.11381990843x_{94} = -80.11381990843
x95=38.9809055076008x_{95} = 38.9809055076008
x96=−64x_{96} = -64
x97=−20.0681158886759x_{97} = -20.0681158886759
x98=68x_{98} = 68
x99=58x_{99} = 58
x100=−39.1869680091833x_{100} = -39.1869680091833
x101=82.9391467589083x_{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)).
−∣cos⁡(0)∣+cos⁡(0)- \left|{\cos{\left(0 \right)}}\right| + \cos{\left(0 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
primera derivada
sin⁡(x)sign⁡(cos⁡(x))−sin⁡(x)=0\sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} \right)} - \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=18x_{1} = 18
x2=50x_{2} = 50
x3=−72.2566310325652x_{3} = -72.2566310325652
x4=91.106186954104x_{4} = 91.106186954104
x5=−59.6902604182061x_{5} = -59.6902604182061
x6=70x_{6} = 70
x7=30x_{7} = 30
x8=24x_{8} = 24
x9=94x_{9} = 94
x10=−100x_{10} = -100
x11=−56x_{11} = -56
x12=40.8407044966673x_{12} = 40.8407044966673
x13=−6x_{13} = -6
x14=38x_{14} = 38
x15=72.2566310325652x_{15} = 72.2566310325652
x16=−47.1238898038469x_{16} = -47.1238898038469
x17=6x_{17} = 6
x18=82x_{18} = 82
x19=−88x_{19} = -88
x20=59.6902604182061x_{20} = 59.6902604182061
x21=−44x_{21} = -44
x22=84.8230016469244x_{22} = 84.8230016469244
x23=0x_{23} = 0
x24=−74x_{24} = -74
x25=−14x_{25} = -14
x26=38.7483330486114x_{26} = 38.7483330486114
x27=−53.4070751110265x_{27} = -53.4070751110265
x28=−62x_{28} = -62
x29=−21.9911485751286x_{29} = -21.9911485751286
x30=99.2691857797929x_{30} = 99.2691857797929
x31=56x_{31} = 56
x32=−24x_{32} = -24
x33=64x_{33} = 64
x34=−34.5575191894877x_{34} = -34.5575191894877
x35=12x_{35} = 12
x36=−50x_{36} = -50
x37=−3.14159265358979x_{37} = -3.14159265358979
x38=53.4070751110265x_{38} = 53.4070751110265
x39=−70x_{39} = -70
x40=−75.7312933201993x_{40} = -75.7312933201993
x41=−68x_{41} = -68
x42=74x_{42} = 74
x43=−92.6907188312909x_{43} = -92.6907188312909
x44=−65.9734457253857x_{44} = -65.9734457253857
x45=−76x_{45} = -76
x46=−82x_{46} = -82
x47=82.5136949447994x_{47} = 82.5136949447994
x48=76x_{48} = 76
x49=−95.75x_{49} = -95.75
x50=−26x_{50} = -26
x51=−38x_{51} = -38
x52=−58x_{52} = -58
x53=3.14159265358979x_{53} = 3.14159265358979
x54=−7.75x_{54} = -7.75
x55=−78.5398163397448x_{55} = -78.5398163397448
x56=26x_{56} = 26
x57=47.1238898038469x_{57} = 47.1238898038469
x58=−18x_{58} = -18
x59=28.2743338823081x_{59} = 28.2743338823081
x60=100x_{60} = 100
x61=88x_{61} = 88
x62=−20x_{62} = -20
x63=−91.106186954104x_{63} = -91.106186954104
x64=−94x_{64} = -94
x65=−12x_{65} = -12
x66=−51.75x_{66} = -51.75
x67=−48.8578078716517x_{67} = -48.8578078716517
x68=80.25x_{68} = 80.25
x69=−30x_{69} = -30
x70=−9.42477796076938x_{70} = -9.42477796076938
x71=65.9734457253857x_{71} = 65.9734457253857
x72=20x_{72} = 20
x73=14x_{73} = 14
x74=36.25x_{74} = 36.25
x75=32x_{75} = 32
x76=−15.707963267949x_{76} = -15.707963267949
x77=97.3893722612836x_{77} = 97.3893722612836
x78=−32x_{78} = -32
x79=44x_{79} = 44
x80=62x_{80} = 62
x81=15.707963267949x_{81} = 15.707963267949
x82=9.42477796076938x_{82} = 9.42477796076938
x83=−84.8230016469244x_{83} = -84.8230016469244
x84=−64x_{84} = -64
x85=68x_{85} = 68
x86=21.9911485751286x_{86} = 21.9911485751286
x87=34.5575191894877x_{87} = 34.5575191894877
x88=−97.3893722612836x_{88} = -97.3893722612836
x89=−32.0322958555426x_{89} = -32.0322958555426
x90=−5.04363182667898x_{90} = -5.04363182667898
x91=−40.8407044966673x_{91} = -40.8407044966673
x92=−28.2743338823081x_{92} = -28.2743338823081
x93=58x_{93} = 58
x94=78.5398163397448x_{94} = 78.5398163397448
x95=11.7052375414856x_{95} = 11.7052375414856
x96=55.4742156532357x_{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:
x1=−72.2566310325652x_{1} = -72.2566310325652
x2=91.106186954104x_{2} = 91.106186954104
x3=−59.6902604182061x_{3} = -59.6902604182061
x4=40.8407044966673x_{4} = 40.8407044966673
x5=72.2566310325652x_{5} = 72.2566310325652
x6=−47.1238898038469x_{6} = -47.1238898038469
x7=59.6902604182061x_{7} = 59.6902604182061
x8=84.8230016469244x_{8} = 84.8230016469244
x9=−53.4070751110265x_{9} = -53.4070751110265
x10=−21.9911485751286x_{10} = -21.9911485751286
x11=−34.5575191894877x_{11} = -34.5575191894877
x12=−3.14159265358979x_{12} = -3.14159265358979
x13=53.4070751110265x_{13} = 53.4070751110265
x14=−65.9734457253857x_{14} = -65.9734457253857
x15=3.14159265358979x_{15} = 3.14159265358979
x16=−78.5398163397448x_{16} = -78.5398163397448
x17=47.1238898038469x_{17} = 47.1238898038469
x18=28.2743338823081x_{18} = 28.2743338823081
x19=−91.106186954104x_{19} = -91.106186954104
x20=−9.42477796076938x_{20} = -9.42477796076938
x21=65.9734457253857x_{21} = 65.9734457253857
x22=−15.707963267949x_{22} = -15.707963267949
x23=97.3893722612836x_{23} = 97.3893722612836
x24=15.707963267949x_{24} = 15.707963267949
x25=9.42477796076938x_{25} = 9.42477796076938
x26=−84.8230016469244x_{26} = -84.8230016469244
x27=21.9911485751286x_{27} = 21.9911485751286
x28=34.5575191894877x_{28} = 34.5575191894877
x29=−97.3893722612836x_{29} = -97.3893722612836
x30=−40.8407044966673x_{30} = -40.8407044966673
x31=−28.2743338823081x_{31} = -28.2743338823081
x32=78.5398163397448x_{32} = 78.5398163397448
La función no tiene puntos máximos
Decrece en los intervalos
[97.3893722612836,∞)\left[97.3893722612836, \infty\right)
Crece en los intervalos
(−∞,−97.3893722612836]\left(-\infty, -97.3893722612836\right]
Puntos de flexiones
Hallemos los puntos de flexiones, para eso hay que resolver la ecuación
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
segunda derivada
−2sin⁡2(x)δ(cos⁡(x))+cos⁡(x)sign⁡(cos⁡(x))−cos⁡(x)=0- 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→−∞(cos⁡(x)−∣cos⁡(x)∣)=⟨−1,1⟩−∣⟨−1,1⟩∣\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=⟨−1,1⟩−∣⟨−1,1⟩∣y = \left\langle -1, 1\right\rangle - \left|{\left\langle -1, 1\right\rangle}\right|
lim⁡x→∞(cos⁡(x)−∣cos⁡(x)∣)=⟨−1,1⟩−∣⟨−1,1⟩∣\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=⟨−1,1⟩−∣⟨−1,1⟩∣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→−∞(cos⁡(x)−∣cos⁡(x)∣x)=0\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→∞(cos⁡(x)−∣cos⁡(x)∣x)=0\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⁡(x)−∣cos⁡(x)∣=cos⁡(x)−∣cos⁡(x)∣\cos{\left(x \right)} - \left|{\cos{\left(x \right)}}\right| = \cos{\left(x \right)} - \left|{\cos{\left(x \right)}}\right|
- Sí
cos⁡(x)−∣cos⁡(x)∣=−cos⁡(x)+∣cos⁡(x)∣\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
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