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(-x-1/(2*tan(x)))*sin(2*x)-cos(2*x)*log(sin(x))
Trazado el gráfico de la función/ (-x-1/(2*tan(x)))*sin(2*x)-cos(2*x)*log(sin(x))

Gráfico de la función y = (-x-1/(2*tan(x)))*sin(2*x)-cos(2*x)*log(sin(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /        1    \                                
f(x) = |-x - --------|*sin(2*x) - cos(2*x)*log(sin(x))
       \     2*tan(x)/
f(x)=(−x−12tan⁡(x))sin⁡(2x)−log⁡(sin⁡(x))cos⁡(2x)f{\left(x \right)} = \left(- x - \frac{1}{2 \tan{\left(x \right)}}\right) \sin{\left(2 x \right)} - \log{\left(\sin{\left(x \right)} \right)} \cos{\left(2 x \right)} 
f = (-x - 1/(2*tan(x)))*sin(2*x) - log(sin(x))*cos(2*x)
Gráfico de la función
02468-8-6-4-2-1010-2020
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
(−x−12tan⁡(x))sin⁡(2x)−log⁡(sin⁡(x))cos⁡(2x)=0\left(- x - \frac{1}{2 \tan{\left(x \right)}}\right) \sin{\left(2 x \right)} - \log{\left(\sin{\left(x \right)} \right)} \cos{\left(2 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−72.2767290171467x_{1} = -72.2767290171467
x2=−28.3136968003845x_{2} = -28.3136968003845
x3=44.0111689701573x_{3} = 44.0111689701573
x4=95.8185759344887x_{4} = 95.8185759344887
x5=−9.50435888742869x_{5} = -9.50435888742869
x6=45.553093477052x_{6} = 45.553093477052
x7=−15.7661128954257x_{7} = -15.7661128954257
x8=81.6997503628731x_{8} = 81.6997503628731
x9=−23.5619449019235x_{9} = -23.5619449019235
x10=−59.7134018599931x_{10} = -59.7134018599931
x11=70.6858347057703x_{11} = 70.6858347057703
x12=26.7035375555132x_{12} = 26.7035375555132
x13=−22.0378159420962x_{13} = -22.0378159420962
x14=−17.2787595947439x_{14} = -17.2787595947439
x15=12.6332779881221x_{15} = 12.6332779881221
x16=−42.4115008234622x_{16} = -42.4115008234622
x17=−34.5917675661531x_{17} = -34.5917675661531
x18=−103.687879068283x_{18} = -103.687879068283
x19=−61.261056745001x_{19} = -61.261056745001
x20=14.1371669411541x_{20} = 14.1371669411541
x21=−65.9949445163871x_{21} = -65.9949445163871
x22=1.5707963267949x_{22} = 1.5707963267949
x23=−53.4321738268354x_{23} = -53.4321738268354
x24=−29.845130209103x_{24} = -29.845130209103
x25=−73.8274273593601x_{25} = -73.8274273593601
x26=37.731327381393x_{26} = 37.731327381393
x27=20.4203522483337x_{27} = 20.4203522483337
x28=−86.3937979737193x_{28} = -86.3937979737193
x29=58.1194640914112x_{29} = 58.1194640914112
x30=−97.4054379725272x_{30} = -97.4054379725272
x31=51.8362787842316x_{31} = 51.8362787842316
x32=−67.5442420521806x_{32} = -67.5442420521806
x33=56.5727428493051x_{33} = 56.5727428493051
x34=100.546648639401x_{34} = 100.546648639401
x35=−78.5587037926658x_{35} = -78.5587037926658
x36=18.9012329459417x_{36} = 18.9012329459417
x37=89.5353906273091x_{37} = 89.5353906273091
x38=50.2917092306181x_{38} = 50.2917092306181
x39=94.2642488087273x_{39} = 94.2642488087273
x40=6.38319152569166x_{40} = 6.38319152569166
x41=−36.1283155162826x_{41} = -36.1283155162826
x42=7.85398163397448x_{42} = 7.85398163397448
x43=−80.1106126665397x_{43} = -80.1106126665397
x44=87.9819430706475x_{44} = 87.9819430706475
x45=64.4026493985908x_{45} = 64.4026493985908
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 (-x - 1/(2*tan(x)))*sin(2*x) - cos(2*x)*log(sin(x)).
−log⁡(sin⁡(0))cos⁡(0⋅2)+(−0−12tan⁡(0))sin⁡(0⋅2)- \log{\left(\sin{\left(0 \right)} \right)} \cos{\left(0 \cdot 2 \right)} + \left(- 0 - \frac{1}{2 \tan{\left(0 \right)}}\right) \sin{\left(0 \cdot 2 \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
2(−x−12tan⁡(x))cos⁡(2x)+(−−2tan⁡2(x)−24tan⁡2(x)−1)sin⁡(2x)+2log⁡(sin⁡(x))sin⁡(2x)−cos⁡(x)cos⁡(2x)sin⁡(x)=02 \left(- x - \frac{1}{2 \tan{\left(x \right)}}\right) \cos{\left(2 x \right)} + \left(- \frac{- 2 \tan^{2}{\left(x \right)} - 2}{4 \tan^{2}{\left(x \right)}} - 1\right) \sin{\left(2 x \right)} + 2 \log{\left(\sin{\left(x \right)} \right)} \sin{\left(2 x \right)} - \frac{\cos{\left(x \right)} \cos{\left(2 x \right)}}{\sin{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=32.2065441906759x_{1} = 32.2065441906759
x2=−60.4784774032523x_{2} = -60.4784774032523
x3=77.7566758727491x_{3} = 77.7566758727491
x4=38.4888988647805x_{4} = 38.4888988647805
x5=46.3423137958712x_{5} = 46.3423137958712
x6=−85.610400630876x_{6} = -85.610400630876
x7=−99.7473216416723x_{7} = -99.7473216416723
x8=27.4954780163039x_{8} = 27.4954780163039
x9=−35.3476856227695x_{9} = -35.3476856227695
x10=−80.8981797500793x_{10} = -80.8981797500793
x11=−91.8934506010817x_{11} = -91.8934506010817
x12=−24.354765832387x_{12} = -24.354765832387
x13=−10.2256396362646x_{13} = -10.2256396362646
x14=−43.2010057193843x_{14} = -43.2010057193843
x15=−11.797052556412x_{15} = -11.797052556412
x16=52.6250338678515x_{16} = 52.6250338678515
x17=−3.96222385983294x_{17} = -3.96222385983294
x18=25.9245773469686x_{18} = 25.9245773469686
x19=84.0396903449635x_{19} = 84.0396903449635
x20=82.4688832868053x_{20} = 82.4688832868053
x21=−68.3322139496356x_{21} = -68.3322139496356
x22=−47.9128310599597x_{22} = -47.9128310599597
x23=−5.53641251076009x_{23} = -5.53641251076009
x24=40.0597434294054x_{24} = 40.0597434294054
x25=−55.7664339394812x_{25} = -55.7664339394812
x26=−49.4836587569263x_{26} = -49.4836587569263
x27=−62.0492935495265x_{27} = -62.0492935495265
x28=69.9028806513307x_{28} = 69.9028806513307
x29=8.66208235664511x_{29} = 8.66208235664511
x30=33.7774088869088x_{30} = 33.7774088869088
x31=96.6057866552025x_{31} = 96.6057866552025
x32=−87.181206922733x_{32} = -87.181206922733
x33=72.2497040947944x_{33} = 72.2497040947944
x34=−79.3273718337047x_{34} = -79.3273718337047
x35=−98.1765177200059x_{35} = -98.1765177200059
x36=63.6199330069073x_{36} = 63.6199330069073
x37=76.185866990933x_{37} = 76.185866990933
x38=−54.1956130037313x_{38} = -54.1956130037313
x39=−16.5032687642151x_{39} = -16.5032687642151
x40=−18.0743144415669x_{40} = -18.0743144415669
x41=−93.4642555875047x_{41} = -93.4642555875047
x42=19.6433527164568x_{42} = 19.6433527164568
x43=90.3227288520815x_{43} = 90.3227288520815
x44=−41.6301680374936x_{44} = -41.6301680374936
x45=44.7714816221779x_{45} = 44.7714816221779
x46=2.48887313385325x_{46} = 2.48887313385325
x47=88.7519232472825x_{47} = 88.7519232472825
Signos de extremos en los puntos:
(32.20654419067588, -32.7031336066818)

(-60.47847740325234, -60.9766355059062)

(77.75667587274908, 77.2552006317712)

(38.488898864780474, -38.9860310506875)

(46.34231379587117, 45.8398162862715)

(-85.610400630876, -86.1090932401509)

(-99.74732164167227, 99.2461749488883)

(27.49547801630387, 26.9912030441814)

(-35.34768562276948, -35.844569886027)

(-80.89817975007932, 80.3967625181407)

(-91.89345060108171, -92.3922316434375)

(-24.354765832387034, 23.8499155092257)

(-10.22563963626459, -10.7155346008751)

(-43.20100571938429, 42.6983222566185)

(-11.797052556411955, 11.286544439457)

(52.62503386785151, 52.1228403591982)

(-3.9622238598329447, -4.43919310418988)

(25.924577346968555, -26.4203705602574)

(84.03969034496355, 83.5383267325953)

(82.46888328680531, -82.9675266866487)

(-68.33221394963557, 67.830532200088)

(-47.91283105995975, -48.4105159000241)

(-5.536412510760091, 5.01115954118519)

(40.059743429405366, 39.5568440943206)

(-55.76643393948123, 55.2643662703229)

(-49.483658756926275, 48.9813230984185)

(-62.04929354952648, 61.5474387014202)

(69.90288065133068, -70.4012836028331)

(8.66208235664511, 8.14724517236757)

(33.77740888690882, 33.2739536262689)

(96.60578665520248, 96.1046022810786)

(-87.18120692273303, 86.6798930205253)

(72.24970409479438, 4.97282264959927)

(-79.32737183370475, -79.8259621747683)

(-98.17651772000593, -98.6753759899394)

(63.6199330069073, -64.118180662278)

(76.18586699093296, -76.6843999538133)

(-54.195613003731346, -54.6935614150055)

(-16.503268764215058, -16.9967948457799)

(-18.074314441566887, 17.5676775609595)

(-93.46425558750465, 92.9630309713947)

(19.643352716456842, -20.1378647030548)

(90.32272885208147, 89.8214611631427)

(-41.63016803749358, -42.1275115980625)

(44.771481622177916, -45.2690075365067)

(2.4888731338532533, 1.90140608777966)

(88.7519232472825, -89.250661620878)


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=32.2065441906759x_{1} = 32.2065441906759
x2=−60.4784774032523x_{2} = -60.4784774032523
x3=38.4888988647805x_{3} = 38.4888988647805
x4=−85.610400630876x_{4} = -85.610400630876
x5=−35.3476856227695x_{5} = -35.3476856227695
x6=−91.8934506010817x_{6} = -91.8934506010817
x7=−10.2256396362646x_{7} = -10.2256396362646
x8=−3.96222385983294x_{8} = -3.96222385983294
x9=25.9245773469686x_{9} = 25.9245773469686
x10=82.4688832868053x_{10} = 82.4688832868053
x11=−47.9128310599597x_{11} = -47.9128310599597
x12=69.9028806513307x_{12} = 69.9028806513307
x13=72.2497040947944x_{13} = 72.2497040947944
x14=−79.3273718337047x_{14} = -79.3273718337047
x15=−98.1765177200059x_{15} = -98.1765177200059
x16=63.6199330069073x_{16} = 63.6199330069073
x17=76.185866990933x_{17} = 76.185866990933
x18=−54.1956130037313x_{18} = -54.1956130037313
x19=−16.5032687642151x_{19} = -16.5032687642151
x20=19.6433527164568x_{20} = 19.6433527164568
x21=−41.6301680374936x_{21} = -41.6301680374936
x22=44.7714816221779x_{22} = 44.7714816221779
x23=88.7519232472825x_{23} = 88.7519232472825
Puntos máximos de la función:
x23=77.7566758727491x_{23} = 77.7566758727491
x23=46.3423137958712x_{23} = 46.3423137958712
x23=−99.7473216416723x_{23} = -99.7473216416723
x23=27.4954780163039x_{23} = 27.4954780163039
x23=−80.8981797500793x_{23} = -80.8981797500793
x23=−24.354765832387x_{23} = -24.354765832387
x23=−43.2010057193843x_{23} = -43.2010057193843
x23=−11.797052556412x_{23} = -11.797052556412
x23=52.6250338678515x_{23} = 52.6250338678515
x23=84.0396903449635x_{23} = 84.0396903449635
x23=−68.3322139496356x_{23} = -68.3322139496356
x23=−5.53641251076009x_{23} = -5.53641251076009
x23=40.0597434294054x_{23} = 40.0597434294054
x23=−55.7664339394812x_{23} = -55.7664339394812
x23=−49.4836587569263x_{23} = -49.4836587569263
x23=−62.0492935495265x_{23} = -62.0492935495265
x23=8.66208235664511x_{23} = 8.66208235664511
x23=33.7774088869088x_{23} = 33.7774088869088
x23=96.6057866552025x_{23} = 96.6057866552025
x23=−87.181206922733x_{23} = -87.181206922733
x23=−18.0743144415669x_{23} = -18.0743144415669
x23=−93.4642555875047x_{23} = -93.4642555875047
x23=90.3227288520815x_{23} = 90.3227288520815
x23=2.48887313385325x_{23} = 2.48887313385325
Decrece en los intervalos
[88.7519232472825,∞)\left[88.7519232472825, \infty\right)
Crece en los intervalos
(−∞,−98.1765177200059]\left(-\infty, -98.1765177200059\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞((−x−12tan⁡(x))sin⁡(2x)−log⁡(sin⁡(x))cos⁡(2x))y = \lim_{x \to -\infty}\left(\left(- x - \frac{1}{2 \tan{\left(x \right)}}\right) \sin{\left(2 x \right)} - \log{\left(\sin{\left(x \right)} \right)} \cos{\left(2 x \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞((−x−12tan⁡(x))sin⁡(2x)−log⁡(sin⁡(x))cos⁡(2x))y = \lim_{x \to \infty}\left(\left(- x - \frac{1}{2 \tan{\left(x \right)}}\right) \sin{\left(2 x \right)} - \log{\left(\sin{\left(x \right)} \right)} \cos{\left(2 x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-x - 1/(2*tan(x)))*sin(2*x) - cos(2*x)*log(sin(x)), dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞((−x−12tan⁡(x))sin⁡(2x)−log⁡(sin⁡(x))cos⁡(2x)x)y = x \lim_{x \to -\infty}\left(\frac{\left(- x - \frac{1}{2 \tan{\left(x \right)}}\right) \sin{\left(2 x \right)} - \log{\left(\sin{\left(x \right)} \right)} \cos{\left(2 x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((−x−12tan⁡(x))sin⁡(2x)−log⁡(sin⁡(x))cos⁡(2x)x)y = x \lim_{x \to \infty}\left(\frac{\left(- x - \frac{1}{2 \tan{\left(x \right)}}\right) \sin{\left(2 x \right)} - \log{\left(\sin{\left(x \right)} \right)} \cos{\left(2 x \right)}}{x}\right)
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:
(−x−12tan⁡(x))sin⁡(2x)−log⁡(sin⁡(x))cos⁡(2x)=−(x+12tan⁡(x))sin⁡(2x)−log⁡(−sin⁡(x))cos⁡(2x)\left(- x - \frac{1}{2 \tan{\left(x \right)}}\right) \sin{\left(2 x \right)} - \log{\left(\sin{\left(x \right)} \right)} \cos{\left(2 x \right)} = - \left(x + \frac{1}{2 \tan{\left(x \right)}}\right) \sin{\left(2 x \right)} - \log{\left(- \sin{\left(x \right)} \right)} \cos{\left(2 x \right)}
- No
(−x−12tan⁡(x))sin⁡(2x)−log⁡(sin⁡(x))cos⁡(2x)=(x+12tan⁡(x))sin⁡(2x)+log⁡(−sin⁡(x))cos⁡(2x)\left(- x - \frac{1}{2 \tan{\left(x \right)}}\right) \sin{\left(2 x \right)} - \log{\left(\sin{\left(x \right)} \right)} \cos{\left(2 x \right)} = \left(x + \frac{1}{2 \tan{\left(x \right)}}\right) \sin{\left(2 x \right)} + \log{\left(- \sin{\left(x \right)} \right)} \cos{\left(2 x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (-x-1/(2*tan(x)))*sin(2*x)-cos(2*x)*log(sin(x))
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