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Trazado el gráfico de la función/ exp(log(sin(x))+(cos(0.1x)))

Gráfico de la función y = exp(log(sin(x))+(cos(0.1x)))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                         /x \
        log(sin(x)) + cos|--|
                         \10/
f(x) = e
f(x)=elog⁡(sin⁡(x))+cos⁡(x10)f{\left(x \right)} = e^{\log{\left(\sin{\left(x \right)} \right)} + \cos{\left(\frac{x}{10} \right)}} 
f = exp(log(sin(x)) + cos(x/10))
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:
elog⁡(sin⁡(x))+cos⁡(x10)=0e^{\log{\left(\sin{\left(x \right)} \right)} + \cos{\left(\frac{x}{10} \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=πx_{2} = \pi
Solución numérica
x1=59.6902604182061x_{1} = 59.6902604182061
x2=−100.530964914873x_{2} = -100.530964914873
x3=−15.707963267949x_{3} = -15.707963267949
x4=−25.1327412287183x_{4} = -25.1327412287183
x5=84.8230016469244x_{5} = 84.8230016469244
x6=0x_{6} = 0
x7=−75.398223686155x_{7} = -75.398223686155
x8=97.3893722612836x_{8} = 97.3893722612836
x9=−50.2654824574367x_{9} = -50.2654824574367
x10=744.557458900781x_{10} = 744.557458900781
x11=81.6814089933346x_{11} = 81.6814089933346
x12=−72.2566310325652x_{12} = -72.2566310325652
x13=91.106186954104x_{13} = 91.106186954104
x14=50.2654824574367x_{14} = 50.2654824574367
x15=−43.9822971502571x_{15} = -43.9822971502571
x16=−37.6991118430775x_{16} = -37.6991118430775
x17=25.1327412287183x_{17} = 25.1327412287183
x18=−65.9734457253857x_{18} = -65.9734457253857
x19=−53.4070751110265x_{19} = -53.4070751110265
x20=−18.8495559215388x_{20} = -18.8495559215388
x21=−59.6902604182061x_{21} = -59.6902604182061
x22=15.707963267949x_{22} = 15.707963267949
x23=9.42477796076938x_{23} = 9.42477796076938
x24=18.8495559215388x_{24} = 18.8495559215388
x25=−56.5486677646163x_{25} = -56.5486677646163
x26=−6.28318530717959x_{26} = -6.28318530717959
x27=−62.8318530717959x_{27} = -62.8318530717959
x28=12.5663706143592x_{28} = 12.5663706143592
x29=−119.380520836412x_{29} = -119.380520836412
x30=56.5486677646163x_{30} = 56.5486677646163
x31=40.8407044966673x_{31} = 40.8407044966673
x32=3.14159265358979x_{32} = 3.14159265358979
x33=−21.9911485751286x_{33} = -21.9911485751286
x34=−84.8230016469244x_{34} = -84.8230016469244
x35=6.28318530717959x_{35} = 6.28318530717959
x36=69.1150383789755x_{36} = 69.1150383789755
x37=72.2566310325652x_{37} = 72.2566310325652
x38=−78.5398163397448x_{38} = -78.5398163397448
x39=37.6991118430775x_{39} = 37.6991118430775
x40=21.9911485751286x_{40} = 21.9911485751286
x41=47.1238898038469x_{41} = 47.1238898038469
x42=34.5575191894877x_{42} = 34.5575191894877
x43=−97.3893722612836x_{43} = -97.3893722612836
x44=−31.4159265358979x_{44} = -31.4159265358979
x45=100.530964914873x_{45} = 100.530964914873
x46=−47.1238898038469x_{46} = -47.1238898038469
x47=28.2743338823081x_{47} = 28.2743338823081
x48=94.2477796076938x_{48} = 94.2477796076938
x49=−40.8407044966673x_{49} = -40.8407044966673
x50=−12.5663706143592x_{50} = -12.5663706143592
x51=−34.5575191894877x_{51} = -34.5575191894877
x52=−28.2743338823081x_{52} = -28.2743338823081
x53=78.5398163397448x_{53} = 78.5398163397448
x54=−94.2477796076938x_{54} = -94.2477796076938
x55=−91.106186954104x_{55} = -91.106186954104
x56=43.9822971502571x_{56} = 43.9822971502571
x57=75.398223686155x_{57} = 75.398223686155
x58=62.8318530717959x_{58} = 62.8318530717959
x59=−3.14159265358979x_{59} = -3.14159265358979
x60=87.9645943005142x_{60} = 87.9645943005142
x61=53.4070751110265x_{61} = 53.4070751110265
x62=−81.6814089933346x_{62} = -81.6814089933346
x63=−87.9645943005142x_{63} = -87.9645943005142
x64=65.9734457253857x_{64} = 65.9734457253857
x65=−69.1150383789755x_{65} = -69.1150383789755
x66=31.4159265358979x_{66} = 31.4159265358979
x67=−9.42477796076938x_{67} = -9.42477796076938
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 exp(log(sin(x)) + cos(x/10)).
elog⁡(sin⁡(0))+cos⁡(010)e^{\log{\left(\sin{\left(0 \right)} \right)} + \cos{\left(\frac{0}{10} \right)}}
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
ecos⁡(x10)sin⁡(x)(−sin⁡(x10)10+cos⁡(x)sin⁡(x))=0e^{\cos{\left(\frac{x}{10} \right)}} \sin{\left(x \right)} \left(- \frac{\sin{\left(\frac{x}{10} \right)}}{10} + \frac{\cos{\left(x \right)}}{\sin{\left(x \right)}}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−17.1801620751675x_{1} = -17.1801620751675
x2=95.8343740834416x_{2} = 95.8343740834416
x3=26.657763126617x_{3} = 26.657763126617
x4=−86.3227064018565x_{4} = -86.3227064018565
x5=51.9247428423613x_{5} = 51.9247428423613
x6=61.2765459489209x_{6} = 61.2765459489209
x7=−80.0120151469633x_{7} = -80.0120151469633
x8=42.5007653239139x_{8} = 42.5007653239139
x9=−4.66742138622181x_{9} = -4.66742138622181
x10=−45.6516909966284x_{10} = -45.6516909966284
x11=99.0059430169747x_{11} = 99.0059430169747
x12=48.7929786559929x_{12} = 48.7929786559929
x13=1.55530712287499x_{13} = 1.55530712287499
x14=−70.6157364717051x_{14} = -70.6157364717051
x15=−83.1629408196778x_{15} = -83.1629408196778
x16=4.66742138622181x_{16} = 4.66742138622181
x17=23.4908533300607x_{17} = 23.4908533300607
x18=20.3310877478819x_{18} = 20.3310877478819
x19=−10.9071102294346x_{19} = -10.9071102294346
x20=−36.1740899451789x_{20} = -36.1740899451789
x21=−89.4896161984129x_{21} = -89.4896161984129
x22=29.8293320601501x_{22} = 29.8293320601501
x23=−29.8293320601501x_{23} = -29.8293320601501
x24=−23.4908533300607x_{24} = -23.4908533300607
x25=7.78388339990927x_{25} = 7.78388339990927
x26=45.6516909966284x_{26} = 45.6516909966284
x27=−20.3310877478819x_{27} = -20.3310877478819
x28=−48.7929786559929x_{28} = -48.7929786559929
x29=−14.0388744158029x_{29} = -14.0388744158029
x30=−99.0059430169747x_{30} = -99.0059430169747
x31=−26.657763126617x_{31} = -26.657763126617
x32=14.0388744158029x_{32} = 14.0388744158029
x33=−42.5007653239139x_{33} = -42.5007653239139
x34=17.1801620751675x_{34} = 17.1801620751675
x35=−7.78388339990927x_{35} = -7.78388339990927
x36=−92.661185131946x_{36} = -92.661185131946
x37=−102.172852813531x_{37} = -102.172852813531
x38=80.0120151469633x_{38} = 80.0120151469633
x39=86.3227064018565x_{39} = 86.3227064018565
x40=−51.9247428423613x_{40} = -51.9247428423613
x41=−61.2765459489209x_{41} = -61.2765459489209
x42=−95.8343740834416x_{42} = -95.8343740834416
x43=33.0025210116458x_{43} = 33.0025210116458
x44=55.0479696718866x_{44} = 55.0479696718866
x45=73.7389633012305x_{45} = 73.7389633012305
x46=10.9071102294346x_{46} = 10.9071102294346
x47=−58.1644316855741x_{47} = -58.1644316855741
x48=−76.8707274875988x_{48} = -76.8707274875988
x49=92.661185131946x_{49} = 92.661185131946
x50=−39.3409997417352x_{50} = -39.3409997417352
x51=105.33261839571x_{51} = 105.33261839571
x52=−1.55530712287499x_{52} = -1.55530712287499
x53=76.8707274875988x_{53} = 76.8707274875988
x54=67.4992744580177x_{54} = 67.4992744580177
x55=−33.0025210116458x_{55} = -33.0025210116458
x56=39.3409997417352x_{56} = 39.3409997417352
x57=−64.3871601946709x_{57} = -64.3871601946709
x58=−73.7389633012305x_{58} = -73.7389633012305
x59=58.1644316855741x_{59} = 58.1644316855741
x60=64.3871601946709x_{60} = 64.3871601946709
x61=−67.4992744580177x_{61} = -67.4992744580177
x62=70.6157364717051x_{62} = 70.6157364717051
x63=−55.0479696718866x_{63} = -55.0479696718866
x64=89.4896161984129x_{64} = 89.4896161984129
x65=83.1629408196778x_{65} = 83.1629408196778
x66=36.1740899451789x_{66} = 36.1740899451789
Signos de extremos en los puntos:
(-17.180162075167463, 0.859368648952312)

(95.83437408344163, 0.372482669676623)

(26.657763126617002, 0.410669264992146)

(-86.32270640185654, 0.494310627412953)

(51.924742842361276, 1.58080905920172)

(61.27654594892088, -2.68534576903038)

(-80.01201514696332, 0.859368648952312)

(42.500765323913924, -0.63762281426652)

(-4.667421386221807, 2.44007215781738)

(-45.6516909966284, -0.859368648952312)

(99.00594301697473, -0.410669264992147)

(48.792978655992925, -1.17503329227924)

(1.5553071228749864, 2.68534576903038)

(-70.61573647170513, -2.03315173042758)

(-83.1629408196778, -0.63762281426652)

(4.667421386221807, -2.44007215781738)

(23.49085333006067, -0.494310627412953)

(20.331087747881938, 0.63762281426652)

(-10.90711022943459, 1.58080905920172)

(-36.174089945178864, 0.410669264992147)

(-89.48961619841286, -0.410669264992146)

(29.8293320601501, -0.372482669676622)

(-29.8293320601501, 0.372482669676622)

(-23.49085333006067, 0.494310627412953)

(7.783883399909268, 2.03315173042758)

(45.6516909966284, 0.859368648952312)

(-20.331087747881938, -0.63762281426652)

(-48.792978655992925, 1.17503329227924)

(-14.038874415802939, -1.17503329227924)

(-99.00594301697473, 0.410669264992147)

(-26.657763126617002, -0.410669264992146)

(14.038874415802939, 1.17503329227924)

(-42.500765323913924, 0.63762281426652)

(17.180162075167463, -0.859368648952312)

(-7.783883399909268, -2.03315173042758)

(-92.66118513194597, 0.372482669676623)

(-102.17285281353107, -0.494310627412953)

(80.01201514696332, -0.859368648952312)

(86.32270640185654, -0.494310627412953)

(-51.924742842361276, -1.58080905920172)

(-61.27654594892088, 2.68534576903038)

(-95.83437408344163, -0.372482669676623)

(33.00252101164576, 0.372482669676622)

(55.0479696718866, -2.03315173042758)

(73.73896330123046, -1.58080905920172)

(10.90711022943459, -1.58080905920172)

(-58.16443168557406, -2.44007215781738)

(-76.8707274875988, -1.17503329227924)

(92.66118513194597, -0.372482669676623)

(-39.340999741735196, -0.494310627412953)

(105.33261839570979, -0.637622814266521)

(-1.5553071228749864, -2.68534576903038)

(76.8707274875988, 1.17503329227924)

(67.49927445801767, -2.44007215781738)

(-33.00252101164576, -0.372482669676622)

(39.340999741735196, 0.494310627412953)

(-64.38716019467086, -2.68534576903037)

(-73.73896330123046, 1.58080905920172)

(58.16443168557406, 2.44007215781738)

(64.38716019467086, 2.68534576903037)

(-67.49927445801767, 2.44007215781738)

(70.61573647170513, 2.03315173042758)

(-55.0479696718866, 2.03315173042758)

(89.48961619841286, 0.410669264992146)

(83.1629408196778, 0.63762281426652)

(36.174089945178864, -0.410669264992147)


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=61.2765459489209x_{1} = 61.2765459489209
x2=42.5007653239139x_{2} = 42.5007653239139
x3=−45.6516909966284x_{3} = -45.6516909966284
x4=99.0059430169747x_{4} = 99.0059430169747
x5=48.7929786559929x_{5} = 48.7929786559929
x6=−70.6157364717051x_{6} = -70.6157364717051
x7=−83.1629408196778x_{7} = -83.1629408196778
x8=4.66742138622181x_{8} = 4.66742138622181
x9=23.4908533300607x_{9} = 23.4908533300607
x10=−89.4896161984129x_{10} = -89.4896161984129
x11=29.8293320601501x_{11} = 29.8293320601501
x12=−20.3310877478819x_{12} = -20.3310877478819
x13=−14.0388744158029x_{13} = -14.0388744158029
x14=−26.657763126617x_{14} = -26.657763126617
x15=17.1801620751675x_{15} = 17.1801620751675
x16=−7.78388339990927x_{16} = -7.78388339990927
x17=−102.172852813531x_{17} = -102.172852813531
x18=80.0120151469633x_{18} = 80.0120151469633
x19=86.3227064018565x_{19} = 86.3227064018565
x20=−51.9247428423613x_{20} = -51.9247428423613
x21=−95.8343740834416x_{21} = -95.8343740834416
x22=55.0479696718866x_{22} = 55.0479696718866
x23=73.7389633012305x_{23} = 73.7389633012305
x24=10.9071102294346x_{24} = 10.9071102294346
x25=−58.1644316855741x_{25} = -58.1644316855741
x26=−76.8707274875988x_{26} = -76.8707274875988
x27=92.661185131946x_{27} = 92.661185131946
x28=−39.3409997417352x_{28} = -39.3409997417352
x29=105.33261839571x_{29} = 105.33261839571
x30=−1.55530712287499x_{30} = -1.55530712287499
x31=67.4992744580177x_{31} = 67.4992744580177
x32=−33.0025210116458x_{32} = -33.0025210116458
x33=−64.3871601946709x_{33} = -64.3871601946709
x34=36.1740899451789x_{34} = 36.1740899451789
Puntos máximos de la función:
x34=−17.1801620751675x_{34} = -17.1801620751675
x34=95.8343740834416x_{34} = 95.8343740834416
x34=26.657763126617x_{34} = 26.657763126617
x34=−86.3227064018565x_{34} = -86.3227064018565
x34=51.9247428423613x_{34} = 51.9247428423613
x34=−80.0120151469633x_{34} = -80.0120151469633
x34=−4.66742138622181x_{34} = -4.66742138622181
x34=1.55530712287499x_{34} = 1.55530712287499
x34=20.3310877478819x_{34} = 20.3310877478819
x34=−10.9071102294346x_{34} = -10.9071102294346
x34=−36.1740899451789x_{34} = -36.1740899451789
x34=−29.8293320601501x_{34} = -29.8293320601501
x34=−23.4908533300607x_{34} = -23.4908533300607
x34=7.78388339990927x_{34} = 7.78388339990927
x34=45.6516909966284x_{34} = 45.6516909966284
x34=−48.7929786559929x_{34} = -48.7929786559929
x34=−99.0059430169747x_{34} = -99.0059430169747
x34=14.0388744158029x_{34} = 14.0388744158029
x34=−42.5007653239139x_{34} = -42.5007653239139
x34=−92.661185131946x_{34} = -92.661185131946
x34=−61.2765459489209x_{34} = -61.2765459489209
x34=33.0025210116458x_{34} = 33.0025210116458
x34=76.8707274875988x_{34} = 76.8707274875988
x34=39.3409997417352x_{34} = 39.3409997417352
x34=−73.7389633012305x_{34} = -73.7389633012305
x34=58.1644316855741x_{34} = 58.1644316855741
x34=64.3871601946709x_{34} = 64.3871601946709
x34=−67.4992744580177x_{34} = -67.4992744580177
x34=70.6157364717051x_{34} = 70.6157364717051
x34=−55.0479696718866x_{34} = -55.0479696718866
x34=89.4896161984129x_{34} = 89.4896161984129
x34=83.1629408196778x_{34} = 83.1629408196778
Decrece en los intervalos
[105.33261839571,∞)\left[105.33261839571, \infty\right)
Crece en los intervalos
(−∞,−102.172852813531]\left(-\infty, -102.172852813531\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞elog⁡(sin⁡(x))+cos⁡(x10)=⟨−e,e⟩\lim_{x \to -\infty} e^{\log{\left(\sin{\left(x \right)} \right)} + \cos{\left(\frac{x}{10} \right)}} = \left\langle - e, e\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−e,e⟩y = \left\langle - e, e\right\rangle
lim⁡x→∞elog⁡(sin⁡(x))+cos⁡(x10)=⟨−e,e⟩\lim_{x \to \infty} e^{\log{\left(\sin{\left(x \right)} \right)} + \cos{\left(\frac{x}{10} \right)}} = \left\langle - e, e\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−e,e⟩y = \left\langle - e, e\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función exp(log(sin(x)) + cos(x/10)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(ecos⁡(x10)sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{e^{\cos{\left(\frac{x}{10} \right)}} \sin{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(ecos⁡(x10)sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{e^{\cos{\left(\frac{x}{10} \right)}} \sin{\left(x \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:
elog⁡(sin⁡(x))+cos⁡(x10)=−ecos⁡(x10)sin⁡(x)e^{\log{\left(\sin{\left(x \right)} \right)} + \cos{\left(\frac{x}{10} \right)}} = - e^{\cos{\left(\frac{x}{10} \right)}} \sin{\left(x \right)}
- No
elog⁡(sin⁡(x))+cos⁡(x10)=ecos⁡(x10)sin⁡(x)e^{\log{\left(\sin{\left(x \right)} \right)} + \cos{\left(\frac{x}{10} \right)}} = e^{\cos{\left(\frac{x}{10} \right)}} \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор