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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = cos(2*x)*log(x)
f(x)=log⁡(x)cos⁡(2x)f{\left(x \right)} = \log{\left(x \right)} \cos{\left(2 x \right)} 
f = log(x)*cos(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)cos⁡(2x)=0\log{\left(x \right)} \cos{\left(2 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1x_{1} = 1
x2=−3π4x_{2} = - \frac{3 \pi}{4}
x3=−π4x_{3} = - \frac{\pi}{4}
x4=π4x_{4} = \frac{\pi}{4}
x5=3π4x_{5} = \frac{3 \pi}{4}
Solución numérica
x1=−82.4668071567321x_{1} = -82.4668071567321
x2=66.7588438887831x_{2} = 66.7588438887831
x3=−5.49778714378214x_{3} = -5.49778714378214
x4=−19.6349540849362x_{4} = -19.6349540849362
x5=25.9181393921158x_{5} = 25.9181393921158
x6=18.0641577581413x_{6} = 18.0641577581413
x7=55.7632696012188x_{7} = 55.7632696012188
x8=−62.0464549083984x_{8} = -62.0464549083984
x9=−93.4623814442964x_{9} = -93.4623814442964
x10=−98.174770424681x_{10} = -98.174770424681
x11=96.6039740978861x_{11} = 96.6039740978861
x12=52.621676947629x_{12} = 52.621676947629
x13=76.1836218495525x_{13} = 76.1836218495525
x14=69.9004365423729x_{14} = 69.9004365423729
x15=30.6305283725005x_{15} = 30.6305283725005
x16=85.6083998103219x_{16} = 85.6083998103219
x17=−55.7632696012188x_{17} = -55.7632696012188
x18=−10.2101761241668x_{18} = -10.2101761241668
x19=−69.9004365423729x_{19} = -69.9004365423729
x20=−3.92699081698724x_{20} = -3.92699081698724
x21=−91.8915851175014x_{21} = -91.8915851175014
x22=−32.2013246992954x_{22} = -32.2013246992954
x23=38.484510006475x_{23} = 38.484510006475
x24=32.2013246992954x_{24} = 32.2013246992954
x25=−76.1836218495525x_{25} = -76.1836218495525
x26=27.4889357189107x_{26} = 27.4889357189107
x27=84.037603483527x_{27} = 84.037603483527
x28=−13.3517687777566x_{28} = -13.3517687777566
x29=98.174770424681x_{29} = 98.174770424681
x30=−99.7455667514759x_{30} = -99.7455667514759
x31=44.7676953136546x_{31} = 44.7676953136546
x32=−85.6083998103219x_{32} = -85.6083998103219
x33=41.6261026600648x_{33} = 41.6261026600648
x34=91.8915851175014x_{34} = 91.8915851175014
x35=−18.0641577581413x_{35} = -18.0641577581413
x36=16.4933614313464x_{36} = 16.4933614313464
x37=46.3384916404494x_{37} = 46.3384916404494
x38=63.6172512351933x_{38} = 63.6172512351933
x39=54.1924732744239x_{39} = 54.1924732744239
x40=−71.4712328691678x_{40} = -71.4712328691678
x41=47.9092879672443x_{41} = 47.9092879672443
x42=−27.4889357189107x_{42} = -27.4889357189107
x43=24.3473430653209x_{43} = 24.3473430653209
x44=−84.037603483527x_{44} = -84.037603483527
x45=68.329640215578x_{45} = 68.329640215578
x46=−79.3252145031423x_{46} = -79.3252145031423
x47=62.0464549083984x_{47} = 62.0464549083984
x48=−46.3384916404494x_{48} = -46.3384916404494
x49=−38.484510006475x_{49} = -38.484510006475
x50=74.6128255227576x_{50} = 74.6128255227576
x51=58.9048622548086x_{51} = 58.9048622548086
x52=5.49778714378214x_{52} = 5.49778714378214
x53=−60.4756585816035x_{53} = -60.4756585816035
x54=−54.1924732744239x_{54} = -54.1924732744239
x55=−47.9092879672443x_{55} = -47.9092879672443
x56=40.0553063332699x_{56} = 40.0553063332699
x57=−25.9181393921158x_{57} = -25.9181393921158
x58=−57.3340659280137x_{58} = -57.3340659280137
x59=49.4800842940392x_{59} = 49.4800842940392
x60=88.7499924639117x_{60} = 88.7499924639117
x61=99.7455667514759x_{61} = 99.7455667514759
x62=−40.0553063332699x_{62} = -40.0553063332699
x63=60.4756585816035x_{63} = 60.4756585816035
x64=−33.7721210260903x_{64} = -33.7721210260903
x65=77.7544181763474x_{65} = 77.7544181763474
x66=−68.329640215578x_{66} = -68.329640215578
x67=−63.6172512351933x_{67} = -63.6172512351933
x68=1x_{68} = 1
x69=8.63937979737193x_{69} = 8.63937979737193
x70=−2.35619449019234x_{70} = -2.35619449019234
x71=3.92699081698724x_{71} = 3.92699081698724
x72=82.4668071567321x_{72} = 82.4668071567321
x73=33.7721210260903x_{73} = 33.7721210260903
x74=11.7809724509617x_{74} = 11.7809724509617
x75=10.2101761241668x_{75} = 10.2101761241668
x76=19.6349540849362x_{76} = 19.6349540849362
x77=−77.7544181763474x_{77} = -77.7544181763474
x78=−41.6261026600648x_{78} = -41.6261026600648
x79=2.35619449019234x_{79} = 2.35619449019234
x80=−35.3429173528852x_{80} = -35.3429173528852
x81=−80.8960108299372x_{81} = -80.8960108299372
x82=−24.3473430653209x_{82} = -24.3473430653209
x83=−11.7809724509617x_{83} = -11.7809724509617
x84=90.3207887907066x_{84} = 90.3207887907066
x85=−90.3207887907066x_{85} = -90.3207887907066
x86=−16.4933614313464x_{86} = -16.4933614313464
x87=−49.4800842940392x_{87} = -49.4800842940392
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 cos(2*x)*log(x).
log⁡(0)cos⁡(0⋅2)\log{\left(0 \right)} \cos{\left(0 \cdot 2 \right)}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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)sin⁡(2x)+cos⁡(2x)x=0- 2 \log{\left(x \right)} \sin{\left(2 x \right)} + \frac{\cos{\left(2 x \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=37.7009387523316x_{1} = 37.7009387523316
x2=26.7063872456959x_{2} = 26.7063872456959
x3=42.4130737274375x_{3} = 42.4130737274375
x4=15.71373881069x_{4} = 15.71373881069
x5=72.2574393639103x_{5} = 72.2574393639103
x6=95.8191477924924x_{6} = 95.8191477924924
x7=34.5595611010583x_{7} = 34.5595611010583
x8=70.686665264318x_{8} = 70.686665264318
x9=519.933661056469x_{9} = 519.933661056469
x10=50.266752056388x_{10} = 50.266752056388
x11=80.1113245894887x_{11} = 80.1113245894887
x12=43.9837993072908x_{12} = 43.9837993072908
x13=7.86937620788298x_{13} = 7.86937620788298
x14=94.2483631104622x_{14} = 94.2483631104622
x15=67.5451206197507x_{15} = 67.5451206197507
x16=36.1302444610848x_{16} = 36.1302444610848
x17=21.9948259654327x_{17} = 21.9948259654327
x18=65.9743502626692x_{18} = 65.9743502626692
x19=86.3944469426778x_{19} = 86.3944469426778
x20=1.79344629148962x_{20} = 1.79344629148962
x21=73.8282145375064x_{21} = 73.8282145375064
x22=12.5742233259103x_{22} = 12.5742233259103
x23=87.9652291156992x_{23} = 87.9652291156992
x24=58.1205228926947x_{24} = 58.1205228926947
x25=81.6821041463459x_{25} = 81.6821041463459
x26=20.4244096164874x_{26} = 20.4244096164874
x27=28.2769792756552x_{27} = 28.2769792756552
x28=56.5497633644494x_{28} = 56.5497633644494
x29=6.30470732585947x_{29} = 6.30470732585947
x30=48.6960073894124x_{30} = 48.6960073894124
x31=45.5545305135714x_{31} = 45.5545305135714
x32=64.4035813576509x_{32} = 64.4035813576509
x33=29.8475965102186x_{33} = 29.8475965102186
x34=14.1438383758612x_{34} = 14.1438383758612
x35=89.5360118496113x_{35} = 89.5360118496113
x36=92.6775788753809x_{36} = 92.6775788753809
x37=23.5653023109496x_{37} = 23.5653023109496
x38=78.5405457961797x_{38} = 78.5405457961797
x39=59.6912846343285x_{39} = 59.6912846343285
x40=51.8375003178612x_{40} = 51.8375003178612
x41=100.531504291782x_{41} = 100.531504291782
Signos de extremos en los puntos:
(37.70093875233156, 3.62966076590869)

(26.706387245695893, -3.28484940798273)

(42.41307372743752, -3.74743811473636)

(15.713738810689954, 2.75435164972731)

(72.25743936391025, 4.28022969525982)

(95.8191477924924, -4.56245955353089)

(34.559561101058314, 3.54265470251846)

(70.68666526431795, -4.25825107006646)

(519.9336610564694, -6.25370115460611)

(50.26675205638797, 3.91733123705741)

(80.11132458948872, -4.3834127813923)

(43.98379930729084, 3.78380429236582)

(7.869376207882981, -2.06200105182902)

(94.24836311046224, 4.54593036309164)

(67.54512061975066, -4.21278932463658)

(36.13024446108483, -3.58710361713197)

(21.994825965432728, 3.09072364748017)

(65.97435026266925, 4.1892591788955)

(86.39444694267779, -4.45891964640157)

(1.7934462914896176, -0.527174715728299)

(73.82821453750635, -4.30173563820898)

(12.574223325910289, 2.53133672848549)

(87.96522911569919, 4.47693800438326)

(58.1205228926947, -4.06250972678627)

(81.68210414634589, 4.402830679145)

(20.42440961648742, -3.01663141165916)

(28.276979275655165, 3.34200124455635)

(56.549763364449355, 4.03511133099434)

(6.304707325859471, 1.83959104518412)

(48.69600738941237, -3.88558347657952)

(45.55453051357137, -3.81889430853472)

(64.40358135765085, -4.16516200741356)

(29.847596510218597, -3.39606300319187)

(14.14383837586117, -2.64904325383873)

(89.53601184961128, -4.4946374422701)

(92.67757887538087, -4.52912336247863)

(23.565302310949573, -3.15970415416183)

(78.54054579617974, 4.36361035458561)

(59.691284634328504, 4.08917744445498)

(51.83750031786118, -3.94810204939828)

(100.53150429178214, 4.61046847129109)


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=26.7063872456959x_{1} = 26.7063872456959
x2=42.4130737274375x_{2} = 42.4130737274375
x3=95.8191477924924x_{3} = 95.8191477924924
x4=70.686665264318x_{4} = 70.686665264318
x5=519.933661056469x_{5} = 519.933661056469
x6=80.1113245894887x_{6} = 80.1113245894887
x7=7.86937620788298x_{7} = 7.86937620788298
x8=67.5451206197507x_{8} = 67.5451206197507
x9=36.1302444610848x_{9} = 36.1302444610848
x10=86.3944469426778x_{10} = 86.3944469426778
x11=1.79344629148962x_{11} = 1.79344629148962
x12=73.8282145375064x_{12} = 73.8282145375064
x13=58.1205228926947x_{13} = 58.1205228926947
x14=20.4244096164874x_{14} = 20.4244096164874
x15=48.6960073894124x_{15} = 48.6960073894124
x16=45.5545305135714x_{16} = 45.5545305135714
x17=64.4035813576509x_{17} = 64.4035813576509
x18=29.8475965102186x_{18} = 29.8475965102186
x19=14.1438383758612x_{19} = 14.1438383758612
x20=89.5360118496113x_{20} = 89.5360118496113
x21=92.6775788753809x_{21} = 92.6775788753809
x22=23.5653023109496x_{22} = 23.5653023109496
x23=51.8375003178612x_{23} = 51.8375003178612
Puntos máximos de la función:
x23=37.7009387523316x_{23} = 37.7009387523316
x23=15.71373881069x_{23} = 15.71373881069
x23=72.2574393639103x_{23} = 72.2574393639103
x23=34.5595611010583x_{23} = 34.5595611010583
x23=50.266752056388x_{23} = 50.266752056388
x23=43.9837993072908x_{23} = 43.9837993072908
x23=94.2483631104622x_{23} = 94.2483631104622
x23=21.9948259654327x_{23} = 21.9948259654327
x23=65.9743502626692x_{23} = 65.9743502626692
x23=12.5742233259103x_{23} = 12.5742233259103
x23=87.9652291156992x_{23} = 87.9652291156992
x23=81.6821041463459x_{23} = 81.6821041463459
x23=28.2769792756552x_{23} = 28.2769792756552
x23=56.5497633644494x_{23} = 56.5497633644494
x23=6.30470732585947x_{23} = 6.30470732585947
x23=78.5405457961797x_{23} = 78.5405457961797
x23=59.6912846343285x_{23} = 59.6912846343285
x23=100.531504291782x_{23} = 100.531504291782
Decrece en los intervalos
[519.933661056469,∞)\left[519.933661056469, \infty\right)
Crece en los intervalos
(−∞,1.79344629148962]\left(-\infty, 1.79344629148962\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)cos⁡(2x)+4sin⁡(2x)x+cos⁡(2x)x2)=0- (4 \log{\left(x \right)} \cos{\left(2 x \right)} + \frac{4 \sin{\left(2 x \right)}}{x} + \frac{\cos{\left(2 x \right)}}{x^{2}}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=68.3313723467331x_{1} = 68.3313723467331
x2=55.7654992784186x_{2} = 55.7654992784186
x3=63.6191436688783x_{3} = 63.6191436688783
x4=40.0586883930762x_{4} = 40.0586883930762
x5=69.9021206921109x_{5} = 69.9021206921109
x6=10.2311577758593x_{6} = 10.2311577758593
x7=85.6097123192654x_{7} = 85.6097123192654
x8=74.6143794448388x_{8} = 74.6143794448388
x9=90.3220180226155x_{9} = 90.3220180226155
x10=5.54991836172646x_{10} = 5.54991836172646
x11=16.5041620545571x_{11} = 16.5041620545571
x12=82.4681812074604x_{12} = 82.4681812074604
x13=30.635297089318x_{13} = 30.635297089318
x14=52.624074285286x_{14} = 52.624074285286
x15=33.7763264254241x_{15} = 33.7763264254241
x16=24.3537724021593x_{16} = 24.3537724021593
x17=96.605106473257x_{17} = 96.605106473257
x18=41.6293235671071x_{18} = 41.6293235671071
x19=4.01469280137619x_{19} = 4.01469280137619
x20=60.4776738859635x_{20} = 60.4776738859635
x21=77.7558951927652x_{21} = 77.7558951927652
x22=27.4944224583325x_{22} = 27.4944224583325
x23=98.1758807646101x_{23} = 98.1758807646101
x24=62.048406997339x_{24} = 62.048406997339
x25=8.66603204353938x_{25} = 8.66603204353938
x26=66.7606265858428x_{26} = 66.7606265858428
x27=99.7466558375515x_{27} = 99.7466558375515
x28=19.6434993720682x_{28} = 19.6434993720682
x29=54.1947839875403x_{29} = 54.1947839875403
x30=38.4880686310298x_{30} = 38.4880686310298
x31=32.2057956122887x_{31} = 32.2057956122887
x32=148.440926523636x_{32} = 148.440926523636
x33=47.9119848969741x_{33} = 47.9119848969741
x34=88.7512483435269x_{34} = 88.7512483435269
x35=18.0737118111598x_{35} = 18.0737118111598
x36=25.9240635017332x_{36} = 25.9240635017332
x37=2.54777651922349x_{37} = 2.54777651922349
x38=93.4635604085711x_{38} = 93.4635604085711
x39=91.8927887298698x_{39} = 91.8927887298698
x40=76.1851364174393x_{40} = 76.1851364174393
x41=84.0389461114285x_{41} = 84.0389461114285
x42=11.7981252868266x_{42} = 11.7981252868266
x43=49.4826740350722x_{43} = 49.4826740350722
x44=46.3413041990665x_{44} = 46.3413041990665

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
[148.440926523636,∞)\left[148.440926523636, \infty\right)
Convexa en los intervalos
(−∞,4.01469280137619]\left(-\infty, 4.01469280137619\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(x)cos⁡(2x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\log{\left(x \right)} \cos{\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)cos⁡(2x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\log{\left(x \right)} \cos{\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
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)cos⁡(2x)=log⁡(−x)cos⁡(2x)\log{\left(x \right)} \cos{\left(2 x \right)} = \log{\left(- x \right)} \cos{\left(2 x \right)}
- No
log⁡(x)cos⁡(2x)=−log⁡(−x)cos⁡(2x)\log{\left(x \right)} \cos{\left(2 x \right)} = - \log{\left(- x \right)} \cos{\left(2 x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = cos(2*x)*log(x)
    © Sr Examen – Калькулятор