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Trazado el gráfico de la función/ cos|x|

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(x)=cos⁡(∣x∣)f{\left(x \right)} = \cos{\left(\left|{x}\right| \right)} 
f = cos(|x|)
Gráfico de la función
02468-8-6-4-2-10102-2
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∣)=0\cos{\left(\left|{x}\right| \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Solución numérica
x1=−1.5707963267949x_{1} = -1.5707963267949
x2=−64.4026493985908x_{2} = -64.4026493985908
x3=76.9690200129499x_{3} = 76.9690200129499
x4=−23.5619449019235x_{4} = -23.5619449019235
x5=−58.1194640914112x_{5} = -58.1194640914112
x6=61.261056745001x_{6} = 61.261056745001
x7=80.1106126665397x_{7} = 80.1106126665397
x8=−48.6946861306418x_{8} = -48.6946861306418
x9=−29.845130209103x_{9} = -29.845130209103
x10=−4.71238898038469x_{10} = -4.71238898038469
x11=−86.3937979737193x_{11} = -86.3937979737193
x12=−36.1283155162826x_{12} = -36.1283155162826
x13=−98.9601685880785x_{13} = -98.9601685880785
x14=1.5707963267949x_{14} = 1.5707963267949
x15=−39.2699081698724x_{15} = -39.2699081698724
x16=73.8274273593601x_{16} = 73.8274273593601
x17=−92.6769832808989x_{17} = -92.6769832808989
x18=42.4115008234622x_{18} = 42.4115008234622
x19=67.5442420521806x_{19} = 67.5442420521806
x20=−32.9867228626928x_{20} = -32.9867228626928
x21=14.1371669411541x_{21} = 14.1371669411541
x22=4.71238898038469x_{22} = 4.71238898038469
x23=32.9867228626928x_{23} = 32.9867228626928
x24=−10.9955742875643x_{24} = -10.9955742875643
x25=70.6858347057703x_{25} = 70.6858347057703
x26=36.1283155162826x_{26} = 36.1283155162826
x27=20.4203522483337x_{27} = 20.4203522483337
x28=−70.6858347057703x_{28} = -70.6858347057703
x29=−26.7035375555132x_{29} = -26.7035375555132
x30=10.9955742875643x_{30} = 10.9955742875643
x31=23.5619449019235x_{31} = 23.5619449019235
x32=45.553093477052x_{32} = 45.553093477052
x33=83.2522053201295x_{33} = 83.2522053201295
x34=−67.5442420521806x_{34} = -67.5442420521806
x35=−89.5353906273091x_{35} = -89.5353906273091
x36=−54.9778714378214x_{36} = -54.9778714378214
x37=95.8185759344887x_{37} = 95.8185759344887
x38=−17.2787595947439x_{38} = -17.2787595947439
x39=26.7035375555132x_{39} = 26.7035375555132
x40=17.2787595947439x_{40} = 17.2787595947439
x41=−42.4115008234622x_{41} = -42.4115008234622
x42=54.9778714378214x_{42} = 54.9778714378214
x43=−2266.65909956504x_{43} = -2266.65909956504
x44=−7.85398163397448x_{44} = -7.85398163397448
x45=48.6946861306418x_{45} = 48.6946861306418
x46=−51.8362787842316x_{46} = -51.8362787842316
x47=89.5353906273091x_{47} = 89.5353906273091
x48=92.6769832808989x_{48} = 92.6769832808989
x49=58.1194640914112x_{49} = 58.1194640914112
x50=−80.1106126665397x_{50} = -80.1106126665397
x51=−73.8274273593601x_{51} = -73.8274273593601
x52=86.3937979737193x_{52} = 86.3937979737193
x53=−76.9690200129499x_{53} = -76.9690200129499
x54=51.8362787842316x_{54} = 51.8362787842316
x55=39.2699081698724x_{55} = 39.2699081698724
x56=−20.4203522483337x_{56} = -20.4203522483337
x57=64.4026493985908x_{57} = 64.4026493985908
x58=−83.2522053201295x_{58} = -83.2522053201295
x59=98.9601685880785x_{59} = 98.9601685880785
x60=7.85398163397448x_{60} = 7.85398163397448
x61=−95.8185759344887x_{61} = -95.8185759344887
x62=−14.1371669411541x_{62} = -14.1371669411541
x63=−387.986692718339x_{63} = -387.986692718339
x64=29.845130209103x_{64} = 29.845130209103
x65=−45.553093477052x_{65} = -45.553093477052
x66=−61.261056745001x_{66} = -61.261056745001
x67=−168.075206967054x_{67} = -168.075206967054
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⁡(∣0∣)\cos{\left(\left|{0}\right| \right)}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
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⁡(x)=0- \sin{\left(\left|{x}\right| \right)} \operatorname{sign}{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=31.4159265358979x_{1} = 31.4159265358979
x2=3.14159265358979x_{2} = 3.14159265358979
x3=−267.035375555132x_{3} = -267.035375555132
x4=−47.1238898038469x_{4} = -47.1238898038469
x5=−12.5663706143592x_{5} = -12.5663706143592
x6=−34.5575191894877x_{6} = -34.5575191894877
x7=−69.1150383789755x_{7} = -69.1150383789755
x8=−2642.07942166902x_{8} = -2642.07942166902
x9=−65.9734457253857x_{9} = -65.9734457253857
x10=75.398223686155x_{10} = 75.398223686155
x11=−56.5486677646163x_{11} = -56.5486677646163
x12=−50.2654824574367x_{12} = -50.2654824574367
x13=59.6902604182061x_{13} = 59.6902604182061
x14=72.2566310325652x_{14} = 72.2566310325652
x15=91.106186954104x_{15} = 91.106186954104
x16=−91.106186954104x_{16} = -91.106186954104
x17=−62.8318530717959x_{17} = -62.8318530717959
x18=−6.28318530717959x_{18} = -6.28318530717959
x19=−232.477856365645x_{19} = -232.477856365645
x20=62.8318530717959x_{20} = 62.8318530717959
x21=6.28318530717959x_{21} = 6.28318530717959
x22=−25.1327412287183x_{22} = -25.1327412287183
x23=94.2477796076938x_{23} = 94.2477796076938
x24=−9.42477796076938x_{24} = -9.42477796076938
x25=−37.6991118430775x_{25} = -37.6991118430775
x26=65.9734457253857x_{26} = 65.9734457253857
x27=−100.530964914873x_{27} = -100.530964914873
x28=−43.9822971502571x_{28} = -43.9822971502571
x29=25.1327412287183x_{29} = 25.1327412287183
x30=21.9911485751286x_{30} = 21.9911485751286
x31=87.9645943005142x_{31} = 87.9645943005142
x32=−40.8407044966673x_{32} = -40.8407044966673
x33=−97.3893722612836x_{33} = -97.3893722612836
x34=43.9822971502571x_{34} = 43.9822971502571
x35=−53.4070751110265x_{35} = -53.4070751110265
x36=97.3893722612836x_{36} = 97.3893722612836
x37=100.530964914873x_{37} = 100.530964914873
x38=−94.2477796076938x_{38} = -94.2477796076938
x39=−31.4159265358979x_{39} = -31.4159265358979
x40=18.8495559215388x_{40} = 18.8495559215388
x41=78.5398163397448x_{41} = 78.5398163397448
x42=−18.8495559215388x_{42} = -18.8495559215388
x43=53.4070751110265x_{43} = 53.4070751110265
x44=47.1238898038469x_{44} = 47.1238898038469
x45=12.5663706143592x_{45} = 12.5663706143592
x46=81.6814089933346x_{46} = 81.6814089933346
x47=34.5575191894877x_{47} = 34.5575191894877
x48=−75.398223686155x_{48} = -75.398223686155
x49=−15.707963267949x_{49} = -15.707963267949
x50=50.2654824574367x_{50} = 50.2654824574367
x51=−81.6814089933346x_{51} = -81.6814089933346
x52=−3.14159265358979x_{52} = -3.14159265358979
x53=−59.6902604182061x_{53} = -59.6902604182061
x54=−28.2743338823081x_{54} = -28.2743338823081
x55=−87.9645943005142x_{55} = -87.9645943005142
x56=9.42477796076938x_{56} = 9.42477796076938
x57=−21.9911485751286x_{57} = -21.9911485751286
x58=−113.097335529233x_{58} = -113.097335529233
x59=56.5486677646163x_{59} = 56.5486677646163
x60=15.707963267949x_{60} = 15.707963267949
x61=84.8230016469244x_{61} = 84.8230016469244
x62=−78.5398163397448x_{62} = -78.5398163397448
x63=37.6991118430775x_{63} = 37.6991118430775
x64=−72.2566310325652x_{64} = -72.2566310325652
x65=−84.8230016469244x_{65} = -84.8230016469244
x66=69.1150383789755x_{66} = 69.1150383789755
x67=0x_{67} = 0
x68=28.2743338823081x_{68} = 28.2743338823081
x69=40.8407044966673x_{69} = 40.8407044966673
Signos de extremos en los puntos:
(31.41592653589793, 1)

(3.141592653589793, -1)

(-267.0353755551324, -1)

(-47.1238898038469, -1)

(-12.566370614359172, 1)

(-34.55751918948773, -1)

(-69.11503837897546, 1)

(-2642.079421669016, -1)

(-65.97344572538566, -1)

(75.39822368615503, 1)

(-56.548667764616276, 1)

(-50.26548245743669, 1)

(59.69026041820607, -1)

(72.25663103256524, -1)

(91.106186954104, -1)

(-91.106186954104, -1)

(-62.83185307179586, 1)

(-6.283185307179586, 1)

(-232.4778563656447, 1)

(62.83185307179586, 1)

(6.283185307179586, 1)

(-25.132741228718345, 1)

(94.2477796076938, 1)

(-9.42477796076938, -1)

(-37.69911184307752, 1)

(65.97344572538566, -1)

(-100.53096491487338, 1)

(-43.982297150257104, 1)

(25.132741228718345, 1)

(21.991148575128552, -1)

(87.96459430051421, 1)

(-40.840704496667314, -1)

(-97.3893722612836, -1)

(43.982297150257104, 1)

(-53.40707511102649, -1)

(97.3893722612836, -1)

(100.53096491487338, 1)

(-94.2477796076938, 1)

(-31.41592653589793, 1)

(18.84955592153876, 1)

(78.53981633974483, -1)

(-18.84955592153876, 1)

(53.40707511102649, -1)

(47.1238898038469, -1)

(12.566370614359172, 1)

(81.68140899333463, 1)

(34.55751918948773, -1)

(-75.39822368615503, 1)

(-15.707963267948966, -1)

(50.26548245743669, 1)

(-81.68140899333463, 1)

(-3.141592653589793, -1)

(-59.69026041820607, -1)

(-28.274333882308138, -1)

(-87.96459430051421, 1)

(9.42477796076938, -1)

(-21.991148575128552, -1)

(-113.09733552923255, 1)

(56.548667764616276, 1)

(15.707963267948966, -1)

(84.82300164692441, -1)

(-78.53981633974483, -1)

(37.69911184307752, 1)

(-72.25663103256524, -1)

(-84.82300164692441, -1)

(69.11503837897546, 1)

(0, 1)

(28.274333882308138, -1)

(40.840704496667314, -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:
x1=3.14159265358979x_{1} = 3.14159265358979
x2=−267.035375555132x_{2} = -267.035375555132
x3=−47.1238898038469x_{3} = -47.1238898038469
x4=−34.5575191894877x_{4} = -34.5575191894877
x5=−2642.07942166902x_{5} = -2642.07942166902
x6=−65.9734457253857x_{6} = -65.9734457253857
x7=59.6902604182061x_{7} = 59.6902604182061
x8=72.2566310325652x_{8} = 72.2566310325652
x9=91.106186954104x_{9} = 91.106186954104
x10=−91.106186954104x_{10} = -91.106186954104
x11=−9.42477796076938x_{11} = -9.42477796076938
x12=65.9734457253857x_{12} = 65.9734457253857
x13=21.9911485751286x_{13} = 21.9911485751286
x14=−40.8407044966673x_{14} = -40.8407044966673
x15=−97.3893722612836x_{15} = -97.3893722612836
x16=−53.4070751110265x_{16} = -53.4070751110265
x17=97.3893722612836x_{17} = 97.3893722612836
x18=78.5398163397448x_{18} = 78.5398163397448
x19=53.4070751110265x_{19} = 53.4070751110265
x20=47.1238898038469x_{20} = 47.1238898038469
x21=34.5575191894877x_{21} = 34.5575191894877
x22=−15.707963267949x_{22} = -15.707963267949
x23=−3.14159265358979x_{23} = -3.14159265358979
x24=−59.6902604182061x_{24} = -59.6902604182061
x25=−28.2743338823081x_{25} = -28.2743338823081
x26=9.42477796076938x_{26} = 9.42477796076938
x27=−21.9911485751286x_{27} = -21.9911485751286
x28=15.707963267949x_{28} = 15.707963267949
x29=84.8230016469244x_{29} = 84.8230016469244
x30=−78.5398163397448x_{30} = -78.5398163397448
x31=−72.2566310325652x_{31} = -72.2566310325652
x32=−84.8230016469244x_{32} = -84.8230016469244
x33=28.2743338823081x_{33} = 28.2743338823081
x34=40.8407044966673x_{34} = 40.8407044966673
Puntos máximos de la función:
x34=31.4159265358979x_{34} = 31.4159265358979
x34=−12.5663706143592x_{34} = -12.5663706143592
x34=−69.1150383789755x_{34} = -69.1150383789755
x34=75.398223686155x_{34} = 75.398223686155
x34=−56.5486677646163x_{34} = -56.5486677646163
x34=−50.2654824574367x_{34} = -50.2654824574367
x34=−62.8318530717959x_{34} = -62.8318530717959
x34=−6.28318530717959x_{34} = -6.28318530717959
x34=−232.477856365645x_{34} = -232.477856365645
x34=62.8318530717959x_{34} = 62.8318530717959
x34=6.28318530717959x_{34} = 6.28318530717959
x34=−25.1327412287183x_{34} = -25.1327412287183
x34=94.2477796076938x_{34} = 94.2477796076938
x34=−37.6991118430775x_{34} = -37.6991118430775
x34=−100.530964914873x_{34} = -100.530964914873
x34=−43.9822971502571x_{34} = -43.9822971502571
x34=25.1327412287183x_{34} = 25.1327412287183
x34=87.9645943005142x_{34} = 87.9645943005142
x34=43.9822971502571x_{34} = 43.9822971502571
x34=100.530964914873x_{34} = 100.530964914873
x34=−94.2477796076938x_{34} = -94.2477796076938
x34=−31.4159265358979x_{34} = -31.4159265358979
x34=18.8495559215388x_{34} = 18.8495559215388
x34=−18.8495559215388x_{34} = -18.8495559215388
x34=12.5663706143592x_{34} = 12.5663706143592
x34=81.6814089933346x_{34} = 81.6814089933346
x34=−75.398223686155x_{34} = -75.398223686155
x34=50.2654824574367x_{34} = 50.2654824574367
x34=−81.6814089933346x_{34} = -81.6814089933346
x34=−87.9645943005142x_{34} = -87.9645943005142
x34=−113.097335529233x_{34} = -113.097335529233
x34=56.5486677646163x_{34} = 56.5486677646163
x34=37.6991118430775x_{34} = 37.6991118430775
x34=69.1150383789755x_{34} = 69.1150383789755
x34=0x_{34} = 0
Decrece en los intervalos
[97.3893722612836,∞)\left[97.3893722612836, \infty\right)
Crece en los intervalos
(−∞,−2642.07942166902]\left(-\infty, -2642.07942166902\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⁡(∣x∣)δ(x)+cos⁡(∣x∣)sign⁡2(x))=0- (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
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{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
(−∞,−π2]∪[π2,∞)\left(-\infty, - \frac{\pi}{2}\right] \cup \left[\frac{\pi}{2}, \infty\right)
Convexa en los intervalos
[−π2,π2]\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→−∞cos⁡(∣x∣)=⟨−1,1⟩\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=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞cos⁡(∣x∣)=⟨−1,1⟩\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=⟨−1,1⟩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→−∞(cos⁡(∣x∣)x)=0\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→∞(cos⁡(∣x∣)x)=0\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⁡(∣x∣)=cos⁡(∣x∣)\cos{\left(\left|{x}\right| \right)} = \cos{\left(\left|{x}\right| \right)}
- Sí
cos⁡(∣x∣)=−cos⁡(∣x∣)\cos{\left(\left|{x}\right| \right)} = - \cos{\left(\left|{x}\right| \right)}
- No
es decir, función
es
par
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