Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ absolute(cosx-1)

Gráfico de la función y = absolute(cosx-1)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = |cos(x) - 1|
f(x)=∣cos⁡(x)−1∣f{\left(x \right)} = \left|{\cos{\left(x \right)} - 1}\right| 
f = Abs(cos(x) - 1)
Gráfico de la función
02468-8-6-4-2-101004
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)−1∣=0\left|{\cos{\left(x \right)} - 1}\right| = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=2πx_{2} = 2 \pi
Solución numérica
x1=62.8318535568358x_{1} = 62.8318535568358
x2=37.6991115173992x_{2} = 37.6991115173992
x3=−31.4159260507536x_{3} = -31.4159260507536
x4=−37.6991113479743x_{4} = -37.6991113479743
x5=−50.2654829667315x_{5} = -50.2654829667315
x6=−18.8495563230046x_{6} = -18.8495563230046
x7=94.2477800892631x_{7} = 94.2477800892631
x8=100.530965157364x_{8} = 100.530965157364
x9=31.4159268459961x_{9} = 31.4159268459961
x10=−94.2477801171671x_{10} = -94.2477801171671
x11=12.5663710110881x_{11} = 12.5663710110881
x12=−43.9822976246252x_{12} = -43.9822976246252
x13=−94.2477797298079x_{13} = -94.2477797298079
x14=−87.964593928489x_{14} = -87.964593928489
x15=−62.8318534787248x_{15} = -62.8318534787248
x16=81.6814084860076x_{16} = 81.6814084860076
x17=6.28318579821791x_{17} = 6.28318579821791
x18=100.530964759815x_{18} = 100.530964759815
x19=−75.3982238741744x_{19} = -75.3982238741744
x20=56.5486668532011x_{20} = 56.5486668532011
x21=87.964594335905x_{21} = 87.964594335905
x22=−56.5486682426592x_{22} = -56.5486682426592
x23=−12.5663710889626x_{23} = -12.5663710889626
x24=−43.9822971745392x_{24} = -43.9822971745392
x25=81.6814085860518x_{25} = 81.6814085860518
x26=−56.5486674685864x_{26} = -56.5486674685864
x27=−69.1150379045123x_{27} = -69.1150379045123
x28=−62.831852673202x_{28} = -62.831852673202
x29=−25.1327407505866x_{29} = -25.1327407505866
x30=56.5486682809363x_{30} = 56.5486682809363
x31=50.2654824463392x_{31} = 50.2654824463392
x32=−31.4159267157965x_{32} = -31.4159267157965
x33=56.5486680806249x_{33} = 56.5486680806249
x34=6.28318626747926x_{34} = 6.28318626747926
x35=37.6991120311338x_{35} = 37.6991120311338
x36=−18.8495552124105x_{36} = -18.8495552124105
x37=69.115038794053x_{37} = 69.115038794053
x38=−87.9645947692094x_{38} = -87.9645947692094
x39=0x_{39} = 0
x40=87.9645946044253x_{40} = 87.9645946044253
x41=−50.265482641087x_{41} = -50.265482641087
x42=25.1327416384075x_{42} = 25.1327416384075
x43=−12.5663703112531x_{43} = -12.5663703112531
x44=−6.2831858160515x_{44} = -6.2831858160515
x45=−75.3982231720141x_{45} = -75.3982231720141
x46=−69.1150386869085x_{46} = -69.1150386869085
x47=−25.1327415297174x_{47} = -25.1327415297174
x48=37.6991113348642x_{48} = 37.6991113348642
x49=69.1150379887504x_{49} = 69.1150379887504
x50=12.5663704426592x_{50} = 12.5663704426592
x51=−6.2831851275477x_{51} = -6.2831851275477
x52=87.9645938121814x_{52} = 87.9645938121814
x53=−37.6991121287155x_{53} = -37.6991121287155
x54=−81.6814075578313x_{54} = -81.6814075578313
x55=−100.530964626003x_{55} = -100.530964626003
x56=−81.6814092565354x_{56} = -81.6814092565354
x57=−18.8495555173448x_{57} = -18.8495555173448
x58=−37.6991118772631x_{58} = -37.6991118772631
x59=75.3982232188727x_{59} = 75.3982232188727
x60=−43.9822967932182x_{60} = -43.9822967932182
x61=−31.4159260208155x_{61} = -31.4159260208155
x62=−94.2477794452815x_{62} = -94.2477794452815
x63=6.28318528420851x_{63} = 6.28318528420851
x64=94.2477792651059x_{64} = 94.2477792651059
x65=−81.6814090382277x_{65} = -81.6814090382277
x66=94.2477796093523x_{66} = 94.2477796093523
x67=−6.28318555849548x_{67} = -6.28318555849548
x68=50.2654821322586x_{68} = 50.2654821322586
x69=25.1327408328211x_{69} = 25.1327408328211
x70=−50.2654822863493x_{70} = -50.2654822863493
x71=62.8318527849002x_{71} = 62.8318527849002
x72=43.9822974733639x_{72} = 43.9822974733639
x73=81.6814091897036x_{73} = 81.6814091897036
x74=6.28318500093652x_{74} = 6.28318500093652
x75=−81.6814084945807x_{75} = -81.6814084945807
x76=12.5663711301703x_{76} = 12.5663711301703
x77=43.9822971694647x_{77} = 43.9822971694647
x78=75.3982240031607x_{78} = 75.3982240031607
x79=−75.3982231045728x_{79} = -75.3982231045728
x80=43.9822966661001x_{80} = 43.9822966661001
x81=31.4159260648825x_{81} = 31.4159260648825
x82=18.8495556275525x_{82} = 18.8495556275525
x83=18.8495564031971x_{83} = 18.8495564031971
x84=50.2654829439723x_{84} = 50.2654829439723
x85=56.5486676011951x_{85} = 56.5486676011951
x86=−87.9645943586158x_{86} = -87.9645943586158
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 Abs(cos(x) - 1).
∣−1+cos⁡(0)∣\left|{-1 + \cos{\left(0 \right)}}\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)−1)=0- \sin{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} - 1 \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−267.035375555132x_{1} = -267.035375555132
x2=−31.4159265358979x_{2} = -31.4159265358979
x3=47.1238898038469x_{3} = 47.1238898038469
x4=−75.398223686155x_{4} = -75.398223686155
x5=9.42477796076938x_{5} = 9.42477796076938
x6=97.3893722612836x_{6} = 97.3893722612836
x7=−34.5575191894877x_{7} = -34.5575191894877
x8=−97.3893722612836x_{8} = -97.3893722612836
x9=−62.8318530717959x_{9} = -62.8318530717959
x10=18.8495559215388x_{10} = 18.8495559215388
x11=87.9645943005142x_{11} = 87.9645943005142
x12=−3.14159265358979x_{12} = -3.14159265358979
x13=−87.9645943005142x_{13} = -87.9645943005142
x14=6.28318530717959x_{14} = 6.28318530717959
x15=59.6902604182061x_{15} = 59.6902604182061
x16=−47.1238898038469x_{16} = -47.1238898038469
x17=100.530964914873x_{17} = 100.530964914873
x18=−40.8407044966673x_{18} = -40.8407044966673
x19=62.8318530717959x_{19} = 62.8318530717959
x20=−75.398223686155x_{20} = -75.398223686155
x21=3.14159265358979x_{21} = 3.14159265358979
x22=28.2743338823081x_{22} = 28.2743338823081
x23=−69.1150383789755x_{23} = -69.1150383789755
x24=94.2477796076938x_{24} = 94.2477796076938
x25=−50.2654824574367x_{25} = -50.2654824574367
x26=6.28318530717959x_{26} = 6.28318530717959
x27=31.4159265358979x_{27} = 31.4159265358979
x28=43.9822971502571x_{28} = 43.9822971502571
x29=−37.6991118430775x_{29} = -37.6991118430775
x30=−94.2477796076938x_{30} = -94.2477796076938
x31=−59.6902604182061x_{31} = -59.6902604182061
x32=−31.4159265358979x_{32} = -31.4159265358979
x33=−56.5486677646163x_{33} = -56.5486677646163
x34=25.1327412287183x_{34} = 25.1327412287183
x35=81.6814089933346x_{35} = 81.6814089933346
x36=21.9911485751286x_{36} = 21.9911485751286
x37=69.1150383789755x_{37} = 69.1150383789755
x38=15.707963267949x_{38} = 15.707963267949
x39=34.5575191894877x_{39} = 34.5575191894877
x40=−91.106186954104x_{40} = -91.106186954104
x41=40.8407044966673x_{41} = 40.8407044966673
x42=37.6991118430775x_{42} = 37.6991118430775
x43=65.9734457253857x_{43} = 65.9734457253857
x44=−72.2566310325652x_{44} = -72.2566310325652
x45=56.5486677646163x_{45} = 56.5486677646163
x46=−21.9911485751286x_{46} = -21.9911485751286
x47=53.4070751110265x_{47} = 53.4070751110265
x48=91.106186954104x_{48} = 91.106186954104
x49=−28.2743338823081x_{49} = -28.2743338823081
x50=56.5486677646163x_{50} = 56.5486677646163
x51=−65.9734457253857x_{51} = -65.9734457253857
x52=−53.4070751110265x_{52} = -53.4070751110265
x53=−100.530964914873x_{53} = -100.530964914873
x54=−18.8495559215388x_{54} = -18.8495559215388
x55=−113.097335529233x_{55} = -113.097335529233
x56=−15.707963267949x_{56} = -15.707963267949
x57=−6.28318530717957x_{57} = -6.28318530717957
x58=84.8230016469244x_{58} = 84.8230016469244
x59=18.8495559215388x_{59} = 18.8495559215388
x60=72.2566310325652x_{60} = 72.2566310325652
x61=−87.9645943005142x_{61} = -87.9645943005142
x62=−232.477856365645x_{62} = -232.477856365645
x63=0x_{63} = 0
x64=−43.9822971502571x_{64} = -43.9822971502571
x65=−78.5398163397448x_{65} = -78.5398163397448
x66=12.5663706143592x_{66} = 12.5663706143592
x67=−12.5663706143592x_{67} = -12.5663706143592
x68=75.398223686155x_{68} = 75.398223686155
x69=−84.8230016469244x_{69} = -84.8230016469244
x70=−25.1327412287183x_{70} = -25.1327412287183
x71=78.5398163397448x_{71} = 78.5398163397448
x72=−81.6814089933346x_{72} = -81.6814089933346
x73=−9.42477796076938x_{73} = -9.42477796076938
x74=50.2654824574367x_{74} = 50.2654824574367
x75=−2642.07942166902x_{75} = -2642.07942166902
x76=−25.1327412287183x_{76} = -25.1327412287183
Signos de extremos en los puntos:
(-267.0353755551324, 2)

(-31.41592653589793, 0)

(47.1238898038469, 2)

(-75.39822368615503, 0)

(9.42477796076938, 2)

(97.3893722612836, 2)

(-34.55751918948773, 2)

(-97.3893722612836, 2)

(-62.83185307179586, 0)

(18.849555921538755, 0)

(87.96459430051421, 0)

(-3.141592653589793, 2)

(-87.96459430051421, 0)

(6.283185307179586, 0)

(59.69026041820607, 2)

(-47.1238898038469, 2)

(100.53096491487338, 0)

(-40.840704496667314, 2)

(62.83185307179586, 0)

(-75.39822368615505, 0)

(3.141592653589793, 2)

(28.274333882308138, 2)

(-69.11503837897546, 0)

(94.2477796076938, 0)

(-50.26548245743668, 0)

(6.283185307179591, 0)

(31.41592653589793, 0)

(43.98229715025708, 0)

(-37.69911184307752, 0)

(-94.2477796076938, 0)

(-59.69026041820607, 2)

(-31.41592653589794, 0)

(-56.548667764616276, 0)

(25.132741228718345, 0)

(81.68140899333463, 0)

(21.991148575128552, 2)

(69.11503837897546, 0)

(15.707963267948966, 2)

(34.55751918948773, 2)

(-91.106186954104, 2)

(40.840704496667314, 2)

(37.69911184307752, 0)

(65.97344572538566, 2)

(-72.25663103256524, 2)

(56.5486677646163, 0)

(-21.991148575128552, 2)

(53.40707511102649, 2)

(91.106186954104, 2)

(-28.274333882308138, 2)

(56.548667764616276, 0)

(-65.97344572538566, 2)

(-53.40707511102649, 2)

(-100.53096491487338, 0)

(-18.84955592153876, 0)

(-113.09733552923255, 0)

(-15.707963267948966, 2)

(-6.283185307179572, 0)

(84.82300164692441, 2)

(18.84955592153876, 0)

(72.25663103256524, 2)

(-87.9645943005142, 0)

(-232.4778563656447, 0)

(0, 0)

(-43.982297150257104, 0)

(-78.53981633974483, 2)

(12.56637061435917, 0)

(-12.566370614359172, 0)

(75.39822368615503, 0)

(-84.82300164692441, 2)

(-25.13274122871832, 0)

(78.53981633974483, 2)

(-81.68140899333463, 0)

(-9.42477796076938, 2)

(50.26548245743669, 0)

(-2642.079421669016, 2)

(-25.132741228718345, 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=−31.4159265358979x_{1} = -31.4159265358979
x2=−75.398223686155x_{2} = -75.398223686155
x3=−62.8318530717959x_{3} = -62.8318530717959
x4=18.8495559215388x_{4} = 18.8495559215388
x5=87.9645943005142x_{5} = 87.9645943005142
x6=−87.9645943005142x_{6} = -87.9645943005142
x7=6.28318530717959x_{7} = 6.28318530717959
x8=100.530964914873x_{8} = 100.530964914873
x9=62.8318530717959x_{9} = 62.8318530717959
x10=−75.398223686155x_{10} = -75.398223686155
x11=−69.1150383789755x_{11} = -69.1150383789755
x12=94.2477796076938x_{12} = 94.2477796076938
x13=−50.2654824574367x_{13} = -50.2654824574367
x14=6.28318530717959x_{14} = 6.28318530717959
x15=31.4159265358979x_{15} = 31.4159265358979
x16=43.9822971502571x_{16} = 43.9822971502571
x17=−37.6991118430775x_{17} = -37.6991118430775
x18=−94.2477796076938x_{18} = -94.2477796076938
x19=−31.4159265358979x_{19} = -31.4159265358979
x20=−56.5486677646163x_{20} = -56.5486677646163
x21=25.1327412287183x_{21} = 25.1327412287183
x22=81.6814089933346x_{22} = 81.6814089933346
x23=69.1150383789755x_{23} = 69.1150383789755
x24=37.6991118430775x_{24} = 37.6991118430775
x25=56.5486677646163x_{25} = 56.5486677646163
x26=56.5486677646163x_{26} = 56.5486677646163
x27=−100.530964914873x_{27} = -100.530964914873
x28=−18.8495559215388x_{28} = -18.8495559215388
x29=−113.097335529233x_{29} = -113.097335529233
x30=−6.28318530717957x_{30} = -6.28318530717957
x31=18.8495559215388x_{31} = 18.8495559215388
x32=−87.9645943005142x_{32} = -87.9645943005142
x33=−232.477856365645x_{33} = -232.477856365645
x34=0x_{34} = 0
x35=−43.9822971502571x_{35} = -43.9822971502571
x36=12.5663706143592x_{36} = 12.5663706143592
x37=−12.5663706143592x_{37} = -12.5663706143592
x38=75.398223686155x_{38} = 75.398223686155
x39=−25.1327412287183x_{39} = -25.1327412287183
x40=−81.6814089933346x_{40} = -81.6814089933346
x41=50.2654824574367x_{41} = 50.2654824574367
x42=−25.1327412287183x_{42} = -25.1327412287183
Puntos máximos de la función:
x42=−267.035375555132x_{42} = -267.035375555132
x42=47.1238898038469x_{42} = 47.1238898038469
x42=9.42477796076938x_{42} = 9.42477796076938
x42=97.3893722612836x_{42} = 97.3893722612836
x42=−34.5575191894877x_{42} = -34.5575191894877
x42=−97.3893722612836x_{42} = -97.3893722612836
x42=−3.14159265358979x_{42} = -3.14159265358979
x42=59.6902604182061x_{42} = 59.6902604182061
x42=−47.1238898038469x_{42} = -47.1238898038469
x42=−40.8407044966673x_{42} = -40.8407044966673
x42=3.14159265358979x_{42} = 3.14159265358979
x42=28.2743338823081x_{42} = 28.2743338823081
x42=−59.6902604182061x_{42} = -59.6902604182061
x42=21.9911485751286x_{42} = 21.9911485751286
x42=15.707963267949x_{42} = 15.707963267949
x42=34.5575191894877x_{42} = 34.5575191894877
x42=−91.106186954104x_{42} = -91.106186954104
x42=40.8407044966673x_{42} = 40.8407044966673
x42=65.9734457253857x_{42} = 65.9734457253857
x42=−72.2566310325652x_{42} = -72.2566310325652
x42=−21.9911485751286x_{42} = -21.9911485751286
x42=53.4070751110265x_{42} = 53.4070751110265
x42=91.106186954104x_{42} = 91.106186954104
x42=−28.2743338823081x_{42} = -28.2743338823081
x42=−65.9734457253857x_{42} = -65.9734457253857
x42=−53.4070751110265x_{42} = -53.4070751110265
x42=−15.707963267949x_{42} = -15.707963267949
x42=84.8230016469244x_{42} = 84.8230016469244
x42=72.2566310325652x_{42} = 72.2566310325652
x42=−78.5398163397448x_{42} = -78.5398163397448
x42=−84.8230016469244x_{42} = -84.8230016469244
x42=78.5398163397448x_{42} = 78.5398163397448
x42=−9.42477796076938x_{42} = -9.42477796076938
x42=−2642.07942166902x_{42} = -2642.07942166902
Decrece en los intervalos
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Crece en los intervalos
(−∞,−232.477856365645]\left(-\infty, -232.477856365645\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)−1)−cos⁡(x)sign⁡(cos⁡(x)−1)=02 \sin^{2}{\left(x \right)} \delta\left(\cos{\left(x \right)} - 1\right) - \cos{\left(x \right)} \operatorname{sign}{\left(\cos{\left(x \right)} - 1 \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)−1∣=∣⟨−2,0⟩∣\lim_{x \to -\infty} \left|{\cos{\left(x \right)} - 1}\right| = \left|{\left\langle -2, 0\right\rangle}\right|
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=∣⟨−2,0⟩∣y = \left|{\left\langle -2, 0\right\rangle}\right|
lim⁡x→∞∣cos⁡(x)−1∣=∣⟨−2,0⟩∣\lim_{x \to \infty} \left|{\cos{\left(x \right)} - 1}\right| = \left|{\left\langle -2, 0\right\rangle}\right|
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=∣⟨−2,0⟩∣y = \left|{\left\langle -2, 0\right\rangle}\right|
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función Abs(cos(x) - 1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(∣cos⁡(x)−1∣x)=0\lim_{x \to -\infty}\left(\frac{\left|{\cos{\left(x \right)} - 1}\right|}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(∣cos⁡(x)−1∣x)=0\lim_{x \to \infty}\left(\frac{\left|{\cos{\left(x \right)} - 1}\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)−1∣=∣cos⁡(x)−1∣\left|{\cos{\left(x \right)} - 1}\right| = \left|{\cos{\left(x \right)} - 1}\right|
- Sí
∣cos⁡(x)−1∣=−∣cos⁡(x)−1∣\left|{\cos{\left(x \right)} - 1}\right| = - \left|{\cos{\left(x \right)} - 1}\right|
- No
es decir, función
es
par
    © Sr Examen – Калькулятор