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sin(2*x)*log(x)
Trazado el gráfico de la función/ sin(2*x)*log(x)

Gráfico de la función y = sin(2*x)*log(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = sin(2*x)*log(x)
f(x)=log⁡(x)sin⁡(2x)f{\left(x \right)} = \log{\left(x \right)} \sin{\left(2 x \right)} 
f = log(x)*sin(2*x)
Gráfico de la función
02468-8-6-4-2-10105-5
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:
log⁡(x)sin⁡(2x)=0\log{\left(x \right)} \sin{\left(2 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1x_{1} = 1
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
x4=πx_{4} = \pi
Solución numérica
x1=4.71238898038469x_{1} = 4.71238898038469
x2=−14.1371669411541x_{2} = -14.1371669411541
x3=−37.6991118430775x_{3} = -37.6991118430775
x4=7.85398163397448x_{4} = 7.85398163397448
x5=−21.9911485751286x_{5} = -21.9911485751286
x6=−133.517687777566x_{6} = -133.517687777566
x7=50.2654824574367x_{7} = 50.2654824574367
x8=−20.4203522483337x_{8} = -20.4203522483337
x9=−81.6814089933346x_{9} = -81.6814089933346
x10=−1.5707963267949x_{10} = -1.5707963267949
x11=95.8185759344887x_{11} = 95.8185759344887
x12=34.5575191894877x_{12} = 34.5575191894877
x13=−61.261056745001x_{13} = -61.261056745001
x14=−51.8362787842316x_{14} = -51.8362787842316
x15=26.7035375555132x_{15} = 26.7035375555132
x16=−86.3937979737193x_{16} = -86.3937979737193
x17=−73.8274273593601x_{17} = -73.8274273593601
x18=−36.1283155162826x_{18} = -36.1283155162826
x19=−84.8230016469244x_{19} = -84.8230016469244
x20=43.9822971502571x_{20} = 43.9822971502571
x21=−43.9822971502571x_{21} = -43.9822971502571
x22=48.6946861306418x_{22} = 48.6946861306418
x23=−58.1194640914112x_{23} = -58.1194640914112
x24=639.314105005523x_{24} = 639.314105005523
x25=23.5619449019235x_{25} = 23.5619449019235
x26=−59.6902604182061x_{26} = -59.6902604182061
x27=−83.2522053201295x_{27} = -83.2522053201295
x28=−64.4026493985908x_{28} = -64.4026493985908
x29=86.3937979737193x_{29} = 86.3937979737193
x30=45.553093477052x_{30} = 45.553093477052
x31=−75.398223686155x_{31} = -75.398223686155
x32=56.5486677646163x_{32} = 56.5486677646163
x33=51.8362787842316x_{33} = 51.8362787842316
x34=−87.9645943005142x_{34} = -87.9645943005142
x35=−6.28318530717959x_{35} = -6.28318530717959
x36=20.4203522483337x_{36} = 20.4203522483337
x37=87.9645943005142x_{37} = 87.9645943005142
x38=59.6902604182061x_{38} = 59.6902604182061
x39=−29.845130209103x_{39} = -29.845130209103
x40=72.2566310325652x_{40} = 72.2566310325652
x41=−23.5619449019235x_{41} = -23.5619449019235
x42=81.6814089933346x_{42} = 81.6814089933346
x43=−80.1106126665397x_{43} = -80.1106126665397
x44=64.4026493985908x_{44} = 64.4026493985908
x45=−65.9734457253857x_{45} = -65.9734457253857
x46=−28.2743338823081x_{46} = -28.2743338823081
x47=28.2743338823081x_{47} = 28.2743338823081
x48=29.845130209103x_{48} = 29.845130209103
x49=−72.2566310325652x_{49} = -72.2566310325652
x50=92.6769832808989x_{50} = 92.6769832808989
x51=−31.4159265358979x_{51} = -31.4159265358979
x52=−53.4070751110265x_{52} = -53.4070751110265
x53=−50.2654824574367x_{53} = -50.2654824574367
x54=−67.5442420521806x_{54} = -67.5442420521806
x55=100.530964914873x_{55} = 100.530964914873
x56=94.2477796076938x_{56} = 94.2477796076938
x57=21.9911485751286x_{57} = 21.9911485751286
x58=−15.707963267949x_{58} = -15.707963267949
x59=−94.2477796076938x_{59} = -94.2477796076938
x60=65.9734457253857x_{60} = 65.9734457253857
x61=−40.8407044966673x_{61} = -40.8407044966673
x62=14.1371669411541x_{62} = 14.1371669411541
x63=−7.85398163397448x_{63} = -7.85398163397448
x64=89.5353906273091x_{64} = 89.5353906273091
x65=80.1106126665397x_{65} = 80.1106126665397
x66=78.5398163397448x_{66} = 78.5398163397448
x67=15.707963267949x_{67} = 15.707963267949
x68=37.6991118430775x_{68} = 37.6991118430775
x69=42.4115008234622x_{69} = 42.4115008234622
x70=−97.3893722612836x_{70} = -97.3893722612836
x71=70.6858347057703x_{71} = 70.6858347057703
x72=36.1283155162826x_{72} = 36.1283155162826
x73=−95.8185759344887x_{73} = -95.8185759344887
x74=6.28318530717959x_{74} = 6.28318530717959
x75=−9.42477796076938x_{75} = -9.42477796076938
x76=67.5442420521806x_{76} = 67.5442420521806
x77=−39.2699081698724x_{77} = -39.2699081698724
x78=−89.5353906273091x_{78} = -89.5353906273091
x79=12.5663706143592x_{79} = 12.5663706143592
x80=−17.2787595947439x_{80} = -17.2787595947439
x81=1x_{81} = 1
x82=73.8274273593601x_{82} = 73.8274273593601
x83=−45.553093477052x_{83} = -45.553093477052
x84=58.1194640914112x_{84} = 58.1194640914112
x85=1.5707963267949x_{85} = 1.5707963267949
x86=−42.4115008234622x_{86} = -42.4115008234622
x87=−100.530964914873x_{87} = -100.530964914873
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(2*x)*log(x).
log⁡(0)sin⁡(0⋅2)\log{\left(0 \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
2log⁡(x)cos⁡(2x)+sin⁡(2x)x=02 \log{\left(x \right)} \cos{\left(2 x \right)} + \frac{\sin{\left(2 x \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=2.46662246182406x_{1} = 2.46662246182406
x2=8.65276590100135x_{2} = 8.65276590100135
x3=62.0474309910149x_{3} = 62.0474309910149
x4=47.9106365276308x_{4} = 47.9106365276308
x5=52.6228756848429x_{5} = 52.6228756848429
x6=41.6277132716067x_{6} = 41.6277132716067
x7=24.3505588482616x_{7} = 24.3505588482616
x8=96.6045402936603x_{8} = 96.6045402936603
x9=16.4987665379521x_{9} = 16.4987665379521
x10=32.2035605555116x_{10} = 32.2035605555116
x11=99.7461113017511x_{11} = 99.7461113017511
x12=69.901278642307x_{12} = 69.901278642307
x13=90.321403416883x_{13} = 90.321403416883
x14=5.524240624726x_{14} = 5.524240624726
x15=10.2206977223401x_{15} = 10.2206977223401
x16=54.1936286925819x_{16} = 54.1936286925819
x17=55.7643844954788x_{17} = 55.7643844954788
x18=91.8921869333118x_{18} = 91.8921869333118
x19=40.0569975446324x_{19} = 40.0569975446324
x20=60.4766662755471x_{20} = 60.4766662755471
x21=76.1843791520252x_{21} = 76.1843791520252
x22=11.789566393009x_{22} = 11.789566393009
x23=25.9211023317763x_{23} = 25.9211023317763
x24=33.7742240619455x_{24} = 33.7742240619455
x25=49.4813792496946x_{25} = 49.4813792496946
x26=84.0382748106232x_{26} = 84.0382748106232
x27=38.48628952834x_{27} = 38.48628952834
x28=44.7691642485154x_{28} = 44.7691642485154
x29=68.3305063083065x_{29} = 68.3305063083065
x30=30.6329132105271x_{30} = 30.6329132105271
x31=3.97248841332099x_{31} = 3.97248841332099
x32=82.4674941961375x_{32} = 82.4674941961375
x33=74.6136025037316x_{33} = 74.6136025037316
x34=63.6181974869599x_{34} = 63.6181974869599
x35=77.755156701807x_{35} = 77.755156701807
x36=19.6392292115165x_{36} = 19.6392292115165
x37=18.0689381825286x_{37} = 18.0689381825286
x38=46.3398980273589x_{38} = 46.3398980273589
x39=98.1753256022933x_{39} = 98.1753256022933
x40=85.609056077116x_{40} = 85.609056077116
x41=88.7506204145858x_{41} = 88.7506204145858
x42=66.7597352667801x_{42} = 66.7597352667801
x43=27.491679801302x_{43} = 27.491679801302
Signos de extremos en los puntos:
(2.4666224618240604, -0.880919839660843)

(8.65276590100135, -2.15710574229964)

(62.047430991014885, -4.12789124294944)

(47.910636527630835, 3.86932346304722)

(52.622875684842924, -3.9631395342942)

(41.62771327160666, 3.72874678453796)

(24.35055884826164, -3.19248877006214)

(96.6045402936603, -4.57062281060288)

(16.49876653795208, 2.80312182745805)

(32.203560555511636, 3.47204230855898)

(99.74611130175111, -4.60262534088608)

(69.90127864230699, 4.24707791803583)

(90.32140341688302, -4.50337105555731)

(5.524240624725996, -1.70675427756413)

(10.220697722340134, 2.32390022754927)

(54.19362869258188, 3.99255268966997)

(55.76438449547882, -4.02112539846379)

(91.89218693331178, 4.5206127341262)

(40.05699754463239, -3.69028226850725)

(60.476666275547075, 4.10224927799845)

(76.18437915202516, 4.33315147348226)

(11.789566393008972, -2.46685050885951)

(25.921102331776293, 3.2550002466069)

(33.774224061945475, -3.51966677633177)

(49.48137924969459, -3.9015833367826)

(84.03827481062316, -4.43126835340099)

(38.486289528339974, 3.65027894293774)

(44.769164248515445, 3.80150319881563)

(68.3305063083065, -4.2243499809984)

(30.632913210527114, -3.42203610033169)

(3.9724884133209946, 1.37368587004475)

(82.46749419613745, 4.41240004044145)

(74.61360250373158, -4.3123176230874)

(63.61819748695992, 4.15289211648199)

(77.75515670180697, -4.35356012395842)

(19.639229211516483, 2.97742021811584)

(18.06893818252863, -2.89406206363672)

(46.33989802735888, -3.83598814383295)

(98.17532560229327, 4.58675208952962)

(85.60905607711597, 4.44978723991936)

(88.75062041458582, 4.48582688119475)

(66.75973526678014, 4.20109345733287)

(27.491679801302, -3.31383349933388)


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=2.46662246182406x_{1} = 2.46662246182406
x2=8.65276590100135x_{2} = 8.65276590100135
x3=62.0474309910149x_{3} = 62.0474309910149
x4=52.6228756848429x_{4} = 52.6228756848429
x5=24.3505588482616x_{5} = 24.3505588482616
x6=96.6045402936603x_{6} = 96.6045402936603
x7=99.7461113017511x_{7} = 99.7461113017511
x8=90.321403416883x_{8} = 90.321403416883
x9=5.524240624726x_{9} = 5.524240624726
x10=55.7643844954788x_{10} = 55.7643844954788
x11=40.0569975446324x_{11} = 40.0569975446324
x12=11.789566393009x_{12} = 11.789566393009
x13=33.7742240619455x_{13} = 33.7742240619455
x14=49.4813792496946x_{14} = 49.4813792496946
x15=84.0382748106232x_{15} = 84.0382748106232
x16=68.3305063083065x_{16} = 68.3305063083065
x17=30.6329132105271x_{17} = 30.6329132105271
x18=74.6136025037316x_{18} = 74.6136025037316
x19=77.755156701807x_{19} = 77.755156701807
x20=18.0689381825286x_{20} = 18.0689381825286
x21=46.3398980273589x_{21} = 46.3398980273589
x22=27.491679801302x_{22} = 27.491679801302
Puntos máximos de la función:
x22=47.9106365276308x_{22} = 47.9106365276308
x22=41.6277132716067x_{22} = 41.6277132716067
x22=16.4987665379521x_{22} = 16.4987665379521
x22=32.2035605555116x_{22} = 32.2035605555116
x22=69.901278642307x_{22} = 69.901278642307
x22=10.2206977223401x_{22} = 10.2206977223401
x22=54.1936286925819x_{22} = 54.1936286925819
x22=91.8921869333118x_{22} = 91.8921869333118
x22=60.4766662755471x_{22} = 60.4766662755471
x22=76.1843791520252x_{22} = 76.1843791520252
x22=25.9211023317763x_{22} = 25.9211023317763
x22=38.48628952834x_{22} = 38.48628952834
x22=44.7691642485154x_{22} = 44.7691642485154
x22=3.97248841332099x_{22} = 3.97248841332099
x22=82.4674941961375x_{22} = 82.4674941961375
x22=63.6181974869599x_{22} = 63.6181974869599
x22=19.6392292115165x_{22} = 19.6392292115165
x22=98.1753256022933x_{22} = 98.1753256022933
x22=85.609056077116x_{22} = 85.609056077116
x22=88.7506204145858x_{22} = 88.7506204145858
x22=66.7597352667801x_{22} = 66.7597352667801
Decrece en los intervalos
[99.7461113017511,∞)\left[99.7461113017511, \infty\right)
Crece en los intervalos
(−∞,2.46662246182406]\left(-\infty, 2.46662246182406\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
−4log⁡(x)sin⁡(2x)+4cos⁡(2x)x−sin⁡(2x)x2=0- 4 \log{\left(x \right)} \sin{\left(2 x \right)} + \frac{4 \cos{\left(2 x \right)}}{x} - \frac{\sin{\left(2 x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=86.3950958877647x_{1} = 86.3950958877647
x2=87.9658639084669x_{2} = 87.9658639084669
x3=72.2582476506368x_{3} = 72.2582476506368
x4=73.8290016742782x_{4} = 73.8290016742782
x5=39.2733764554486x_{5} = 39.2733764554486
x6=65.9752547385099x_{6} = 65.9752547385099
x7=51.8387217072068x_{7} = 51.8387217072068
x8=14.1504925240144x_{8} = 14.1504925240144
x9=28.2796233835769x_{9} = 28.2796233835769
x10=89.5366330508377x_{10} = 89.5366330508377
x11=20.4284626971174x_{11} = 20.4284626971174
x12=43.9853012049327x_{12} = 43.9853012049327
x13=26.7092353503257x_{13} = 26.7092353503257
x14=42.414646335887x_{14} = 42.414646335887
x15=1.8916971279294x_{15} = 1.8916971279294
x16=37.7027652100405x_{16} = 37.7027652100405
x17=80.1120364813595x_{17} = 80.1120364813595
x18=59.6923087629724x_{18} = 59.6923087629724
x19=36.1321728792801x_{19} = 36.1321728792801
x20=6.32578672226969x_{20} = 6.32578672226969
x21=50.2680214943533x_{21} = 50.2680214943533
x22=15.7195028063962x_{22} = 15.7195028063962
x23=7.88459430750167x_{23} = 7.88459430750167
x24=100.532043654605x_{24} = 100.532043654605
x25=23.5686572032586x_{25} = 23.5686572032586
x26=81.6827992703194x_{26} = 81.6827992703194
x27=12.5820487825233x_{27} = 12.5820487825233
x28=94.2489465956024x_{28} = 94.2489465956024
x29=45.5559673213233x_{29} = 45.5559673213233
x30=64.4045132498211x_{30} = 64.4045132498211
x31=70.6874957746659x_{31} = 70.6874957746659
x32=67.545999130764x_{32} = 67.545999130764
x33=34.5616023940506x_{33} = 34.5616023940506
x34=78.5412752193063x_{34} = 78.5412752193063
x35=56.5508588583477x_{35} = 56.5508588583477
x36=58.1215815978423x_{36} = 58.1215815978423
x37=92.6781744511727x_{37} = 92.6781744511727
x38=48.6973284679084x_{38} = 48.6973284679084
x39=21.9985001040128x_{39} = 21.9985001040128
x40=29.8500617566647x_{40} = 29.8500617566647
x41=95.8197196338529x_{41} = 95.8197196338529

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
[95.8197196338529,∞)\left[95.8197196338529, \infty\right)
Convexa en los intervalos
(−∞,1.8916971279294]\left(-\infty, 1.8916971279294\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(x)sin⁡(2x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\log{\left(x \right)} \sin{\left(2 x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(log⁡(x)sin⁡(2x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\log{\left(x \right)} \sin{\left(2 x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(2*x)*log(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(x)sin⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} \sin{\left(2 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(log⁡(x)sin⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} \sin{\left(2 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:
log⁡(x)sin⁡(2x)=−log⁡(−x)sin⁡(2x)\log{\left(x \right)} \sin{\left(2 x \right)} = - \log{\left(- x \right)} \sin{\left(2 x \right)}
- No
log⁡(x)sin⁡(2x)=log⁡(−x)sin⁡(2x)\log{\left(x \right)} \sin{\left(2 x \right)} = \log{\left(- x \right)} \sin{\left(2 x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = sin(2*x)*log(x)
    © Sr Examen – Калькулятор