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Trazado el gráfico de la función/ y=cos(|x|)

Gráfico de la función y = 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=7.85398163397448x_{1} = 7.85398163397448
x2=−73.8274273593601x_{2} = -73.8274273593601
x3=−54.9778714378214x_{3} = -54.9778714378214
x4=73.8274273593601x_{4} = 73.8274273593601
x5=−26.7035375555132x_{5} = -26.7035375555132
x6=−1.5707963267949x_{6} = -1.5707963267949
x7=−95.8185759344887x_{7} = -95.8185759344887
x8=−39.2699081698724x_{8} = -39.2699081698724
x9=−4.71238898038469x_{9} = -4.71238898038469
x10=14.1371669411541x_{10} = 14.1371669411541
x11=10.9955742875643x_{11} = 10.9955742875643
x12=58.1194640914112x_{12} = 58.1194640914112
x13=70.6858347057703x_{13} = 70.6858347057703
x14=−36.1283155162826x_{14} = -36.1283155162826
x15=54.9778714378214x_{15} = 54.9778714378214
x16=23.5619449019235x_{16} = 23.5619449019235
x17=−387.986692718339x_{17} = -387.986692718339
x18=−168.075206967054x_{18} = -168.075206967054
x19=−92.6769832808989x_{19} = -92.6769832808989
x20=−86.3937979737193x_{20} = -86.3937979737193
x21=−10.9955742875643x_{21} = -10.9955742875643
x22=92.6769832808989x_{22} = 92.6769832808989
x23=39.2699081698724x_{23} = 39.2699081698724
x24=−32.9867228626928x_{24} = -32.9867228626928
x25=98.9601685880785x_{25} = 98.9601685880785
x26=36.1283155162826x_{26} = 36.1283155162826
x27=−7.85398163397448x_{27} = -7.85398163397448
x28=−58.1194640914112x_{28} = -58.1194640914112
x29=−67.5442420521806x_{29} = -67.5442420521806
x30=−61.261056745001x_{30} = -61.261056745001
x31=26.7035375555132x_{31} = 26.7035375555132
x32=86.3937979737193x_{32} = 86.3937979737193
x33=−48.6946861306418x_{33} = -48.6946861306418
x34=51.8362787842316x_{34} = 51.8362787842316
x35=−42.4115008234622x_{35} = -42.4115008234622
x36=−89.5353906273091x_{36} = -89.5353906273091
x37=−98.9601685880785x_{37} = -98.9601685880785
x38=−14.1371669411541x_{38} = -14.1371669411541
x39=80.1106126665397x_{39} = 80.1106126665397
x40=−64.4026493985908x_{40} = -64.4026493985908
x41=95.8185759344887x_{41} = 95.8185759344887
x42=1.5707963267949x_{42} = 1.5707963267949
x43=45.553093477052x_{43} = 45.553093477052
x44=−17.2787595947439x_{44} = -17.2787595947439
x45=−2266.65909956504x_{45} = -2266.65909956504
x46=4.71238898038469x_{46} = 4.71238898038469
x47=48.6946861306418x_{47} = 48.6946861306418
x48=76.9690200129499x_{48} = 76.9690200129499
x49=−45.553093477052x_{49} = -45.553093477052
x50=20.4203522483337x_{50} = 20.4203522483337
x51=17.2787595947439x_{51} = 17.2787595947439
x52=−83.2522053201295x_{52} = -83.2522053201295
x53=−20.4203522483337x_{53} = -20.4203522483337
x54=−80.1106126665397x_{54} = -80.1106126665397
x55=61.261056745001x_{55} = 61.261056745001
x56=32.9867228626928x_{56} = 32.9867228626928
x57=64.4026493985908x_{57} = 64.4026493985908
x58=−23.5619449019235x_{58} = -23.5619449019235
x59=29.845130209103x_{59} = 29.845130209103
x60=42.4115008234622x_{60} = 42.4115008234622
x61=89.5353906273091x_{61} = 89.5353906273091
x62=−51.8362787842316x_{62} = -51.8362787842316
x63=−70.6858347057703x_{63} = -70.6858347057703
x64=83.2522053201295x_{64} = 83.2522053201295
x65=67.5442420521806x_{65} = 67.5442420521806
x66=−29.845130209103x_{66} = -29.845130209103
x67=−76.9690200129499x_{67} = -76.9690200129499
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=72.2566310325652x_{1} = 72.2566310325652
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=3.14159265358979x_{3} = 3.14159265358979
x4=−43.9822971502571x_{4} = -43.9822971502571
x5=81.6814089933346x_{5} = 81.6814089933346
x6=−100.530964914873x_{6} = -100.530964914873
x7=28.2743338823081x_{7} = 28.2743338823081
x8=65.9734457253857x_{8} = 65.9734457253857
x9=−31.4159265358979x_{9} = -31.4159265358979
x10=−9.42477796076938x_{10} = -9.42477796076938
x11=40.8407044966673x_{11} = 40.8407044966673
x12=56.5486677646163x_{12} = 56.5486677646163
x13=−56.5486677646163x_{13} = -56.5486677646163
x14=12.5663706143592x_{14} = 12.5663706143592
x15=43.9822971502571x_{15} = 43.9822971502571
x16=100.530964914873x_{16} = 100.530964914873
x17=−3.14159265358979x_{17} = -3.14159265358979
x18=−2642.07942166902x_{18} = -2642.07942166902
x19=−15.707963267949x_{19} = -15.707963267949
x20=59.6902604182061x_{20} = 59.6902604182061
x21=−267.035375555132x_{21} = -267.035375555132
x22=6.28318530717959x_{22} = 6.28318530717959
x23=9.42477796076938x_{23} = 9.42477796076938
x24=−53.4070751110265x_{24} = -53.4070751110265
x25=−47.1238898038469x_{25} = -47.1238898038469
x26=−87.9645943005142x_{26} = -87.9645943005142
x27=69.1150383789755x_{27} = 69.1150383789755
x28=21.9911485751286x_{28} = 21.9911485751286
x29=87.9645943005142x_{29} = 87.9645943005142
x30=18.8495559215388x_{30} = 18.8495559215388
x31=−84.8230016469244x_{31} = -84.8230016469244
x32=−72.2566310325652x_{32} = -72.2566310325652
x33=25.1327412287183x_{33} = 25.1327412287183
x34=37.6991118430775x_{34} = 37.6991118430775
x35=−25.1327412287183x_{35} = -25.1327412287183
x36=0x_{36} = 0
x37=50.2654824574367x_{37} = 50.2654824574367
x38=−6.28318530717959x_{38} = -6.28318530717959
x39=−65.9734457253857x_{39} = -65.9734457253857
x40=−21.9911485751286x_{40} = -21.9911485751286
x41=−62.8318530717959x_{41} = -62.8318530717959
x42=75.398223686155x_{42} = 75.398223686155
x43=84.8230016469244x_{43} = 84.8230016469244
x44=53.4070751110265x_{44} = 53.4070751110265
x45=34.5575191894877x_{45} = 34.5575191894877
x46=−28.2743338823081x_{46} = -28.2743338823081
x47=15.707963267949x_{47} = 15.707963267949
x48=−91.106186954104x_{48} = -91.106186954104
x49=47.1238898038469x_{49} = 47.1238898038469
x50=97.3893722612836x_{50} = 97.3893722612836
x51=−69.1150383789755x_{51} = -69.1150383789755
x52=−232.477856365645x_{52} = -232.477856365645
x53=94.2477796076938x_{53} = 94.2477796076938
x54=−18.8495559215388x_{54} = -18.8495559215388
x55=−50.2654824574367x_{55} = -50.2654824574367
x56=−37.6991118430775x_{56} = -37.6991118430775
x57=−81.6814089933346x_{57} = -81.6814089933346
x58=62.8318530717959x_{58} = 62.8318530717959
x59=78.5398163397448x_{59} = 78.5398163397448
x60=31.4159265358979x_{60} = 31.4159265358979
x61=−78.5398163397448x_{61} = -78.5398163397448
x62=−40.8407044966673x_{62} = -40.8407044966673
x63=−97.3893722612836x_{63} = -97.3893722612836
x64=−113.097335529233x_{64} = -113.097335529233
x65=−75.398223686155x_{65} = -75.398223686155
x66=91.106186954104x_{66} = 91.106186954104
x67=−12.5663706143592x_{67} = -12.5663706143592
x68=−94.2477796076938x_{68} = -94.2477796076938
x69=−34.5575191894877x_{69} = -34.5575191894877
Signos de extremos en los puntos:
(72.25663103256524, -1)

(-59.69026041820607, -1)

(3.141592653589793, -1)

(-43.982297150257104, 1)

(81.68140899333463, 1)

(-100.53096491487338, 1)

(28.274333882308138, -1)

(65.97344572538566, -1)

(-31.41592653589793, 1)

(-9.42477796076938, -1)

(40.840704496667314, -1)

(56.548667764616276, 1)

(-56.548667764616276, 1)

(12.566370614359172, 1)

(43.982297150257104, 1)

(100.53096491487338, 1)

(-3.141592653589793, -1)

(-2642.079421669016, -1)

(-15.707963267948966, -1)

(59.69026041820607, -1)

(-267.0353755551324, -1)

(6.283185307179586, 1)

(9.42477796076938, -1)

(-53.40707511102649, -1)

(-47.1238898038469, -1)

(-87.96459430051421, 1)

(69.11503837897546, 1)

(21.991148575128552, -1)

(87.96459430051421, 1)

(18.84955592153876, 1)

(-84.82300164692441, -1)

(-72.25663103256524, -1)

(25.132741228718345, 1)

(37.69911184307752, 1)

(-25.132741228718345, 1)

(0, 1)

(50.26548245743669, 1)

(-6.283185307179586, 1)

(-65.97344572538566, -1)

(-21.991148575128552, -1)

(-62.83185307179586, 1)

(75.39822368615503, 1)

(84.82300164692441, -1)

(53.40707511102649, -1)

(34.55751918948773, -1)

(-28.274333882308138, -1)

(15.707963267948966, -1)

(-91.106186954104, -1)

(47.1238898038469, -1)

(97.3893722612836, -1)

(-69.11503837897546, 1)

(-232.4778563656447, 1)

(94.2477796076938, 1)

(-18.84955592153876, 1)

(-50.26548245743669, 1)

(-37.69911184307752, 1)

(-81.68140899333463, 1)

(62.83185307179586, 1)

(78.53981633974483, -1)

(31.41592653589793, 1)

(-78.53981633974483, -1)

(-40.840704496667314, -1)

(-97.3893722612836, -1)

(-113.09733552923255, 1)

(-75.39822368615503, 1)

(91.106186954104, -1)

(-12.566370614359172, 1)

(-94.2477796076938, 1)

(-34.55751918948773, -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=72.2566310325652x_{1} = 72.2566310325652
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=3.14159265358979x_{3} = 3.14159265358979
x4=28.2743338823081x_{4} = 28.2743338823081
x5=65.9734457253857x_{5} = 65.9734457253857
x6=−9.42477796076938x_{6} = -9.42477796076938
x7=40.8407044966673x_{7} = 40.8407044966673
x8=−3.14159265358979x_{8} = -3.14159265358979
x9=−2642.07942166902x_{9} = -2642.07942166902
x10=−15.707963267949x_{10} = -15.707963267949
x11=59.6902604182061x_{11} = 59.6902604182061
x12=−267.035375555132x_{12} = -267.035375555132
x13=9.42477796076938x_{13} = 9.42477796076938
x14=−53.4070751110265x_{14} = -53.4070751110265
x15=−47.1238898038469x_{15} = -47.1238898038469
x16=21.9911485751286x_{16} = 21.9911485751286
x17=−84.8230016469244x_{17} = -84.8230016469244
x18=−72.2566310325652x_{18} = -72.2566310325652
x19=−65.9734457253857x_{19} = -65.9734457253857
x20=−21.9911485751286x_{20} = -21.9911485751286
x21=84.8230016469244x_{21} = 84.8230016469244
x22=53.4070751110265x_{22} = 53.4070751110265
x23=34.5575191894877x_{23} = 34.5575191894877
x24=−28.2743338823081x_{24} = -28.2743338823081
x25=15.707963267949x_{25} = 15.707963267949
x26=−91.106186954104x_{26} = -91.106186954104
x27=47.1238898038469x_{27} = 47.1238898038469
x28=97.3893722612836x_{28} = 97.3893722612836
x29=78.5398163397448x_{29} = 78.5398163397448
x30=−78.5398163397448x_{30} = -78.5398163397448
x31=−40.8407044966673x_{31} = -40.8407044966673
x32=−97.3893722612836x_{32} = -97.3893722612836
x33=91.106186954104x_{33} = 91.106186954104
x34=−34.5575191894877x_{34} = -34.5575191894877
Puntos máximos de la función:
x34=−43.9822971502571x_{34} = -43.9822971502571
x34=81.6814089933346x_{34} = 81.6814089933346
x34=−100.530964914873x_{34} = -100.530964914873
x34=−31.4159265358979x_{34} = -31.4159265358979
x34=56.5486677646163x_{34} = 56.5486677646163
x34=−56.5486677646163x_{34} = -56.5486677646163
x34=12.5663706143592x_{34} = 12.5663706143592
x34=43.9822971502571x_{34} = 43.9822971502571
x34=100.530964914873x_{34} = 100.530964914873
x34=6.28318530717959x_{34} = 6.28318530717959
x34=−87.9645943005142x_{34} = -87.9645943005142
x34=69.1150383789755x_{34} = 69.1150383789755
x34=87.9645943005142x_{34} = 87.9645943005142
x34=18.8495559215388x_{34} = 18.8495559215388
x34=25.1327412287183x_{34} = 25.1327412287183
x34=37.6991118430775x_{34} = 37.6991118430775
x34=−25.1327412287183x_{34} = -25.1327412287183
x34=0x_{34} = 0
x34=50.2654824574367x_{34} = 50.2654824574367
x34=−6.28318530717959x_{34} = -6.28318530717959
x34=−62.8318530717959x_{34} = -62.8318530717959
x34=75.398223686155x_{34} = 75.398223686155
x34=−69.1150383789755x_{34} = -69.1150383789755
x34=−232.477856365645x_{34} = -232.477856365645
x34=94.2477796076938x_{34} = 94.2477796076938
x34=−18.8495559215388x_{34} = -18.8495559215388
x34=−50.2654824574367x_{34} = -50.2654824574367
x34=−37.6991118430775x_{34} = -37.6991118430775
x34=−81.6814089933346x_{34} = -81.6814089933346
x34=62.8318530717959x_{34} = 62.8318530717959
x34=31.4159265358979x_{34} = 31.4159265358979
x34=−113.097335529233x_{34} = -113.097335529233
x34=−75.398223686155x_{34} = -75.398223686155
x34=−12.5663706143592x_{34} = -12.5663706143592
x34=−94.2477796076938x_{34} = -94.2477796076938
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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