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Trazado el gráfico de la función/ -(1/2)*sin(x)*cos(x)/|sin(x)|+0.5

Gráfico de la función y = -(1/2)*sin(x)*cos(x)/|sin(x)|+0.5

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       -sin(x)            
       --------*cos(x)    
          2              1
f(x) = --------------- + -
           |sin(x)|      2
f(x)=−sin⁡(x)2cos⁡(x)∣sin⁡(x)∣+12f{\left(x \right)} = \frac{- \frac{\sin{\left(x \right)}}{2} \cos{\left(x \right)}}{\left|{\sin{\left(x \right)}}\right|} + \frac{1}{2} 
f = ((-sin(x)/2)*cos(x))/Abs(sin(x)) + 1/2
Gráfico de la función
02468-8-6-4-2-101002
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
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:
−sin⁡(x)2cos⁡(x)∣sin⁡(x)∣+12=0\frac{- \frac{\sin{\left(x \right)}}{2} \cos{\left(x \right)}}{\left|{\sin{\left(x \right)}}\right|} + \frac{1}{2} = 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=128.805298898577x_{1} = 128.805298898577
x2=−12.5663703112531x_{2} = -12.5663703112531
x3=3.14159306054457x_{3} = 3.14159306054457
x4=37.6991120311338x_{4} = 37.6991120311338
x5=1310.04413772164x_{5} = 1310.04413772164
x6=−6.2831851275477x_{6} = -6.2831851275477
x7=−53.4070750912683x_{7} = -53.4070750912683
x8=−40.8407042359622x_{8} = -40.8407042359622
x9=−72.2566308657983x_{9} = -72.2566308657983
x10=87.9645946044253x_{10} = 87.9645946044253
x11=−28.2743337069329x_{11} = -28.2743337069329
x12=−2397.03519395376x_{12} = -2397.03519395376
x13=−21.9911483955057x_{13} = -21.9911483955057
x14=−53.4070745963886x_{14} = -53.4070745963886
x15=28.2743343711514x_{15} = 28.2743343711514
x16=94.2477796093432x_{16} = 94.2477796093432
x17=21.9911489072506x_{17} = 21.9911489072506
x18=43.9822974733639x_{18} = 43.9822974733639
x19=−87.964593928489x_{19} = -87.964593928489
x20=−100.530964626003x_{20} = -100.530964626003
x21=−25.1327407505866x_{21} = -25.1327407505866
x22=62.8318535568358x_{22} = 62.8318535568358
x23=−18.8495555173448x_{23} = -18.8495555173448
x24=−43.9822967932182x_{24} = -43.9822967932182
x25=−43.9822969772045x_{25} = -43.9822969772045
x26=56.5486692415812x_{26} = 56.5486692415812
x27=−3.14159217367683x_{27} = -3.14159217367683
x28=91.1061873718352x_{28} = 91.1061873718352
x29=97.389372581711x_{29} = 97.389372581711
x30=−47.1238893275319x_{30} = -47.1238893275319
x31=−65.973445558908x_{31} = -65.973445558908
x32=56.5486682809363x_{32} = 56.5486682809363
x33=75.3982240031607x_{33} = 75.3982240031607
x34=25.1327416384075x_{34} = 25.1327416384075
x35=−40.8407044578985x_{35} = -40.8407044578985
x36=−40.8407040952604x_{36} = -40.8407040952604
x37=6.28318579821791x_{37} = 6.28318579821791
x38=−78.5398160472843x_{38} = -78.5398160472843
x39=−91.1061864815274x_{39} = -91.1061864815274
x40=−97.3893717476911x_{40} = -97.3893717476911
x41=31.4159268459961x_{41} = 31.4159268459961
x42=15.7079634518075x_{42} = 15.7079634518075
x43=−50.2654822863493x_{43} = -50.2654822863493
x44=−87.9645941406159x_{44} = -87.9645941406159
x45=9.42477826738203x_{45} = 9.42477826738203
x46=78.5398168562347x_{46} = 78.5398168562347
x47=65.9734460390947x_{47} = 65.9734460390947
x48=21.9911485852059x_{48} = 21.9911485852059
x49=40.8407049800347x_{49} = 40.8407049800347
x50=18.8495564031971x_{50} = 18.8495564031971
x51=−56.5486674685864x_{51} = -56.5486674685864
x52=34.5575197055812x_{52} = 34.5575197055812
x53=−75.3982231720141x_{53} = -75.3982231720141
x54=−9.42477744529557x_{54} = -9.42477744529557
x55=−122.522112207808x_{55} = -122.522112207808
x56=72.2566315166773x_{56} = 72.2566315166773
x57=94.2477800892631x_{57} = 94.2477800892631
x58=12.5663711301703x_{58} = 12.5663711301703
x59=59.6902606104322x_{59} = 59.6902606104322
x60=87.964594335905x_{60} = 87.964594335905
x61=81.6814091897036x_{61} = 81.6814091897036
x62=53.4070757253805x_{62} = 53.4070757253805
x63=−31.4159260208155x_{63} = -31.4159260208155
x64=−37.6991113479743x_{64} = -37.6991113479743
x65=−28.2743322419509x_{65} = -28.2743322419509
x66=−2591.81393896242x_{66} = -2591.81393896242
x67=−81.6814084945807x_{67} = -81.6814084945807
x68=43.9822971694647x_{68} = 43.9822971694647
x69=−34.5575188899093x_{69} = -34.5575188899093
x70=−94.2477794452815x_{70} = -94.2477794452815
x71=−84.8230012511693x_{71} = -84.8230012511693
x72=50.2654829439723x_{72} = 50.2654829439723
x73=47.1238902162437x_{73} = 47.1238902162437
x74=−59.6902599212271x_{74} = -59.6902599212271
x75=−15.707962774825x_{75} = -15.707962774825
x76=53.407075424589x_{76} = 53.407075424589
x77=84.8230021335997x_{77} = 84.8230021335997
x78=69.115038794053x_{78} = 69.115038794053
x79=−62.831852673202x_{79} = -62.831852673202
x80=65.9734457529812x_{80} = 65.9734457529812
x81=−65.9734453607004x_{81} = -65.9734453607004
x82=−21.991148226056x_{82} = -21.991148226056
x83=−69.1150379045123x_{83} = -69.1150379045123
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 ((-sin(x)/2)*cos(x))/Abs(sin(x)) + 1/2.
−sin⁡(0)2cos⁡(0)∣sin⁡(0)∣+12\frac{- \frac{\sin{\left(0 \right)}}{2} \cos{\left(0 \right)}}{\left|{\sin{\left(0 \right)}}\right|} + \frac{1}{2}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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⁡2(x)2−cos⁡2(x)2∣sin⁡(x)∣+cos⁡2(x)sign⁡(sin⁡(x))2sin⁡(x)=0\frac{\frac{\sin^{2}{\left(x \right)}}{2} - \frac{\cos^{2}{\left(x \right)}}{2}}{\left|{\sin{\left(x \right)}}\right|} + \frac{\cos^{2}{\left(x \right)} \operatorname{sign}{\left(\sin{\left(x \right)} \right)}}{2 \sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=40.8407044966673x_{1} = 40.8407044966673
x2=15.707963267949x_{2} = 15.707963267949
x3=−285.884931476671x_{3} = -285.884931476671
x4=97.3893722612836x_{4} = 97.3893722612836
x5=−56.5486677646163x_{5} = -56.5486677646163
x6=53.4070751110265x_{6} = 53.4070751110265
x7=87.9645943005142x_{7} = 87.9645943005142
x8=−97.3893722612836x_{8} = -97.3893722612836
x9=62.8318530717959x_{9} = 62.8318530717959
x10=43.9822971502571x_{10} = 43.9822971502571
x11=−21.9911485751286x_{11} = -21.9911485751286
x12=65.9734457253857x_{12} = 65.9734457253857
x13=−50.2654824574367x_{13} = -50.2654824574367
x14=−94.2477796076938x_{14} = -94.2477796076938
x15=−75.398223686155x_{15} = -75.398223686155
x16=12.5663706143592x_{16} = 12.5663706143592
x17=−34.5575191894877x_{17} = -34.5575191894877
x18=21.9911485751286x_{18} = 21.9911485751286
x19=−9.42477796076935x_{19} = -9.42477796076935
x20=−47.1238898038469x_{20} = -47.1238898038469
x21=−43.9822971502571x_{21} = -43.9822971502571
x22=28.2743338823081x_{22} = 28.2743338823081
x23=−6.28318530717959x_{23} = -6.28318530717959
x24=−25.1327412287183x_{24} = -25.1327412287183
x25=−3.14159265358979x_{25} = -3.14159265358979
x26=650.309679293087x_{26} = 650.309679293087
x27=59.6902604182061x_{27} = 59.6902604182061
x28=−65.9734457253857x_{28} = -65.9734457253857
x29=72.2566310325652x_{29} = 72.2566310325652
x30=−59.6902604182061x_{30} = -59.6902604182061
x31=31.4159265358979x_{31} = 31.4159265358979
x32=−427.256600888212x_{32} = -427.256600888212
x33=34.5575191894877x_{33} = 34.5575191894877
x34=94.2477796076938x_{34} = 94.2477796076938
x35=−12.5663706143592x_{35} = -12.5663706143592
x36=−53.4070751110265x_{36} = -53.4070751110265
x37=81.6814089933346x_{37} = 81.6814089933346
x38=−91.106186954104x_{38} = -91.106186954104
x39=−100.530964914873x_{39} = -100.530964914873
x40=−31.4159265358979x_{40} = -31.4159265358979
x41=78.5398163397448x_{41} = 78.5398163397448
x42=84.8230016469244x_{42} = 84.8230016469244
x43=100.530964914873x_{43} = 100.530964914873
x44=−69.1150383789755x_{44} = -69.1150383789755
x45=9.42477796076938x_{45} = 9.42477796076938
x46=−28.2743338823081x_{46} = -28.2743338823081
x47=−78.5398163397448x_{47} = -78.5398163397448
x48=−87.9645943005142x_{48} = -87.9645943005142
x49=−81.6814089933346x_{49} = -81.6814089933346
x50=56.5486677646163x_{50} = 56.5486677646163
x51=−15.707963267949x_{51} = -15.707963267949
x52=37.6991118430775x_{52} = 37.6991118430775
x53=−37.6991118430775x_{53} = -37.6991118430775
x54=18.8495559215388x_{54} = 18.8495559215388
x55=50.2654824574367x_{55} = 50.2654824574367
x56=−72.2566310325652x_{56} = -72.2566310325652
x57=75.398223686155x_{57} = 75.398223686155
x58=6.28318530717959x_{58} = 6.28318530717959
Signos de extremos en los puntos:
(40.840704496667314, 0)

(15.707963267948962, 1)

(-285.88493147667117, 0)

(97.3893722612836, 0)

(-56.548667764616276, 0)

(53.40707511102649, 0)

(87.96459430051421, 1)

(-97.3893722612836, 1)

(62.83185307179586, 1)

(43.982297150257104, 1)

(-21.991148575128552, 0)

(65.97344572538566, 0)

(-50.26548245743669, 0)

(-94.2477796076938, 1)

(-75.39822368615503, 5.55111512312578e-17)

(12.566370614359196, 0)

(-34.55751918948773, 1)

(21.991148575128552, 1)

(-9.424777960769353, 0)

(-47.1238898038469, 1)

(-43.982297150257104, 0)

(28.274333882308138, 1)

(-6.283185307179588, 1)

(-25.132741228718345, 0)

(-3.141592653589793, 0)

(650.3096792930872, 0)

(59.69026041820606, 1)

(-65.97344572538566, 1)

(72.25663103256524, 1)

(-59.69026041820607, 1)

(31.41592653589793, 1)

(-427.2566008882119, 1)

(34.557519189487735, 0)

(94.2477796076938, 0)

(-12.566370614359172, 0)

(-53.40707511102648, 0)

(81.68140899333463, 0)

(-91.106186954104, 1)

(-100.53096491487338, 0)

(-31.415926535897928, 0)

(78.53981633974483, 0)

(84.82300164692441, 1)

(100.53096491487338, 1)

(-69.11503837897546, 1)

(9.42477796076938, 1)

(-28.27433388230814, 1)

(-78.53981633974483, 1)

(-87.96459430051421, 0)

(-81.68140899333463, 1)

(56.54866776461628, 0)

(-15.707963267948966, 5.55111512312578e-17)

(37.69911184307751, 1)

(-37.69911184307752, 5.55111512312578e-17)

(18.84955592153876, 1)

(50.26548245743669, 1)

(-72.25663103256524, 0)

(75.39822368615503, 1)

(6.283185307179586, 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=40.8407044966673x_{1} = 40.8407044966673
x2=−285.884931476671x_{2} = -285.884931476671
x3=97.3893722612836x_{3} = 97.3893722612836
x4=−56.5486677646163x_{4} = -56.5486677646163
x5=53.4070751110265x_{5} = 53.4070751110265
x6=−21.9911485751286x_{6} = -21.9911485751286
x7=65.9734457253857x_{7} = 65.9734457253857
x8=−50.2654824574367x_{8} = -50.2654824574367
x9=−75.398223686155x_{9} = -75.398223686155
x10=12.5663706143592x_{10} = 12.5663706143592
x11=−9.42477796076935x_{11} = -9.42477796076935
x12=−43.9822971502571x_{12} = -43.9822971502571
x13=−25.1327412287183x_{13} = -25.1327412287183
x14=−3.14159265358979x_{14} = -3.14159265358979
x15=650.309679293087x_{15} = 650.309679293087
x16=34.5575191894877x_{16} = 34.5575191894877
x17=94.2477796076938x_{17} = 94.2477796076938
x18=−12.5663706143592x_{18} = -12.5663706143592
x19=−53.4070751110265x_{19} = -53.4070751110265
x20=81.6814089933346x_{20} = 81.6814089933346
x21=−100.530964914873x_{21} = -100.530964914873
x22=−31.4159265358979x_{22} = -31.4159265358979
x23=78.5398163397448x_{23} = 78.5398163397448
x24=−87.9645943005142x_{24} = -87.9645943005142
x25=56.5486677646163x_{25} = 56.5486677646163
x26=−15.707963267949x_{26} = -15.707963267949
x27=−37.6991118430775x_{27} = -37.6991118430775
x28=−72.2566310325652x_{28} = -72.2566310325652
Puntos máximos de la función:
x28=15.707963267949x_{28} = 15.707963267949
x28=87.9645943005142x_{28} = 87.9645943005142
x28=−97.3893722612836x_{28} = -97.3893722612836
x28=62.8318530717959x_{28} = 62.8318530717959
x28=43.9822971502571x_{28} = 43.9822971502571
x28=−94.2477796076938x_{28} = -94.2477796076938
x28=−34.5575191894877x_{28} = -34.5575191894877
x28=21.9911485751286x_{28} = 21.9911485751286
x28=−47.1238898038469x_{28} = -47.1238898038469
x28=28.2743338823081x_{28} = 28.2743338823081
x28=−6.28318530717959x_{28} = -6.28318530717959
x28=59.6902604182061x_{28} = 59.6902604182061
x28=−65.9734457253857x_{28} = -65.9734457253857
x28=72.2566310325652x_{28} = 72.2566310325652
x28=−59.6902604182061x_{28} = -59.6902604182061
x28=31.4159265358979x_{28} = 31.4159265358979
x28=−427.256600888212x_{28} = -427.256600888212
x28=−91.106186954104x_{28} = -91.106186954104
x28=84.8230016469244x_{28} = 84.8230016469244
x28=100.530964914873x_{28} = 100.530964914873
x28=−69.1150383789755x_{28} = -69.1150383789755
x28=9.42477796076938x_{28} = 9.42477796076938
x28=−28.2743338823081x_{28} = -28.2743338823081
x28=−78.5398163397448x_{28} = -78.5398163397448
x28=−81.6814089933346x_{28} = -81.6814089933346
x28=37.6991118430775x_{28} = 37.6991118430775
x28=18.8495559215388x_{28} = 18.8495559215388
x28=50.2654824574367x_{28} = 50.2654824574367
x28=75.398223686155x_{28} = 75.398223686155
x28=6.28318530717959x_{28} = 6.28318530717959
Decrece en los intervalos
[650.309679293087,∞)\left[650.309679293087, \infty\right)
Crece en los intervalos
(−∞,−285.884931476671]\left(-\infty, -285.884931476671\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=3.14159265358979x_{2} = 3.14159265358979
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−sin⁡(x)2cos⁡(x)∣sin⁡(x)∣+12)=⟨−12,12⟩∣⟨−1,1⟩∣+12\lim_{x \to -\infty}\left(\frac{- \frac{\sin{\left(x \right)}}{2} \cos{\left(x \right)}}{\left|{\sin{\left(x \right)}}\right|} + \frac{1}{2}\right) = \frac{\left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|} + \frac{1}{2}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−12,12⟩∣⟨−1,1⟩∣+12y = \frac{\left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|} + \frac{1}{2}
lim⁡x→∞(−sin⁡(x)2cos⁡(x)∣sin⁡(x)∣+12)=⟨−12,12⟩∣⟨−1,1⟩∣+12\lim_{x \to \infty}\left(\frac{- \frac{\sin{\left(x \right)}}{2} \cos{\left(x \right)}}{\left|{\sin{\left(x \right)}}\right|} + \frac{1}{2}\right) = \frac{\left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|} + \frac{1}{2}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−12,12⟩∣⟨−1,1⟩∣+12y = \frac{\left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle}{\left|{\left\langle -1, 1\right\rangle}\right|} + \frac{1}{2}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((-sin(x)/2)*cos(x))/Abs(sin(x)) + 1/2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−sin⁡(x)2cos⁡(x)∣sin⁡(x)∣+12x)=0\lim_{x \to -\infty}\left(\frac{\frac{- \frac{\sin{\left(x \right)}}{2} \cos{\left(x \right)}}{\left|{\sin{\left(x \right)}}\right|} + \frac{1}{2}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−sin⁡(x)2cos⁡(x)∣sin⁡(x)∣+12x)=0\lim_{x \to \infty}\left(\frac{\frac{- \frac{\sin{\left(x \right)}}{2} \cos{\left(x \right)}}{\left|{\sin{\left(x \right)}}\right|} + \frac{1}{2}}{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:
−sin⁡(x)2cos⁡(x)∣sin⁡(x)∣+12=sin⁡(x)cos⁡(x)2∣sin⁡(x)∣+12\frac{- \frac{\sin{\left(x \right)}}{2} \cos{\left(x \right)}}{\left|{\sin{\left(x \right)}}\right|} + \frac{1}{2} = \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2 \left|{\sin{\left(x \right)}}\right|} + \frac{1}{2}
- No
−sin⁡(x)2cos⁡(x)∣sin⁡(x)∣+12=−sin⁡(x)cos⁡(x)2∣sin⁡(x)∣−12\frac{- \frac{\sin{\left(x \right)}}{2} \cos{\left(x \right)}}{\left|{\sin{\left(x \right)}}\right|} + \frac{1}{2} = - \frac{\sin{\left(x \right)} \cos{\left(x \right)}}{2 \left|{\sin{\left(x \right)}}\right|} - \frac{1}{2}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор