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