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

Gráfico de la función y = ln*cos(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = log(x)*cos(x)
f(x)=log⁡(x)cos⁡(x)f{\left(x \right)} = \log{\left(x \right)} \cos{\left(x \right)} 
f = log(x)*cos(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⁡(x)=0\log{\left(x \right)} \cos{\left(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}
Solución numérica
x1=73.8274273593601x_{1} = 73.8274273593601
x2=−83.2522053201295x_{2} = -83.2522053201295
x3=70.6858347057703x_{3} = 70.6858347057703
x4=−10.9955742875643x_{4} = -10.9955742875643
x5=−14.1371669411541x_{5} = -14.1371669411541
x6=42.4115008234622x_{6} = 42.4115008234622
x7=−32.9867228626928x_{7} = -32.9867228626928
x8=14.1371669411541x_{8} = 14.1371669411541
x9=10.9955742875643x_{9} = 10.9955742875643
x10=−48.6946861306418x_{10} = -48.6946861306418
x11=54.9778714378214x_{11} = 54.9778714378214
x12=−67.5442420521806x_{12} = -67.5442420521806
x13=89.5353906273091x_{13} = 89.5353906273091
x14=−39.2699081698724x_{14} = -39.2699081698724
x15=36.1283155162826x_{15} = 36.1283155162826
x16=32.9867228626928x_{16} = 32.9867228626928
x17=39.2699081698724x_{17} = 39.2699081698724
x18=95.8185759344887x_{18} = 95.8185759344887
x19=45.553093477052x_{19} = 45.553093477052
x20=−92.6769832808989x_{20} = -92.6769832808989
x21=92.6769832808989x_{21} = 92.6769832808989
x22=−54.9778714378214x_{22} = -54.9778714378214
x23=1.5707963267949x_{23} = 1.5707963267949
x24=−61.261056745001x_{24} = -61.261056745001
x25=23.5619449019235x_{25} = 23.5619449019235
x26=20.4203522483337x_{26} = 20.4203522483337
x27=−76.9690200129499x_{27} = -76.9690200129499
x28=−70.6858347057703x_{28} = -70.6858347057703
x29=−36.1283155162826x_{29} = -36.1283155162826
x30=−98.9601685880785x_{30} = -98.9601685880785
x31=7.85398163397448x_{31} = 7.85398163397448
x32=98.9601685880785x_{32} = 98.9601685880785
x33=4.71238898038469x_{33} = 4.71238898038469
x34=−64.4026493985908x_{34} = -64.4026493985908
x35=26.7035375555132x_{35} = 26.7035375555132
x36=−7.85398163397448x_{36} = -7.85398163397448
x37=−42.4115008234622x_{37} = -42.4115008234622
x38=−1.5707963267949x_{38} = -1.5707963267949
x39=61.261056745001x_{39} = 61.261056745001
x40=17.2787595947439x_{40} = 17.2787595947439
x41=−20.4203522483337x_{41} = -20.4203522483337
x42=−4.71238898038469x_{42} = -4.71238898038469
x43=−26.7035375555132x_{43} = -26.7035375555132
x44=58.1194640914112x_{44} = 58.1194640914112
x45=−23.5619449019235x_{45} = -23.5619449019235
x46=48.6946861306418x_{46} = 48.6946861306418
x47=−95.8185759344887x_{47} = -95.8185759344887
x48=86.3937979737193x_{48} = 86.3937979737193
x49=−17.2787595947439x_{49} = -17.2787595947439
x50=83.2522053201295x_{50} = 83.2522053201295
x51=−29.845130209103x_{51} = -29.845130209103
x52=67.5442420521806x_{52} = 67.5442420521806
x53=−51.8362787842316x_{53} = -51.8362787842316
x54=80.1106126665397x_{54} = 80.1106126665397
x55=29.845130209103x_{55} = 29.845130209103
x56=−58.1194640914112x_{56} = -58.1194640914112
x57=−86.3937979737193x_{57} = -86.3937979737193
x58=−45.553093477052x_{58} = -45.553093477052
x59=−80.1106126665397x_{59} = -80.1106126665397
x60=51.8362787842316x_{60} = 51.8362787842316
x61=−89.5353906273091x_{61} = -89.5353906273091
x62=−73.8274273593601x_{62} = -73.8274273593601
x63=76.9690200129499x_{63} = 76.9690200129499
x64=64.4026493985908x_{64} = 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 log(x)*cos(x).
log⁡(0)cos⁡(0)\log{\left(0 \right)} \cos{\left(0 \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
−log⁡(x)sin⁡(x)+cos⁡(x)x=0- \log{\left(x \right)} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=62.8356966541501x_{1} = 62.8356966541501
x2=75.4012916642681x_{2} = 75.4012916642681
x3=28.2849113047725x_{3} = 28.2849113047725
x4=47.1293968198114x_{4} = 47.1293968198114
x5=59.6943570030875x_{5} = 59.6943570030875
x6=31.4251563350128x_{6} = 31.4251563350128
x7=50.27056033759x_{7} = 50.27056033759
x8=9.47170218677955x_{8} = 9.47170218677955
x9=100.533122377741x_{9} = 100.533122377741
x10=43.9883049460921x_{10} = 43.9883049460921
x11=84.8256564376189x_{11} = 84.8256564376189
x12=15.7310277752208x_{12} = 15.7310277752208
x13=37.7064180281721x_{13} = 37.7064180281721
x14=97.3916147574604x_{14} = 97.3916147574604
x15=12.5976921976804x_{15} = 12.5976921976804
x16=1.27285069827148x_{16} = 1.27285069827148
x17=91.1086195251935x_{17} = 91.1086195251935
x18=81.6841895128946x_{18} = 81.6841895128946
x19=40.8473034495909x_{19} = 40.8473034495909
x20=72.2598642156451x_{20} = 72.2598642156451
x21=65.9770636783598x_{21} = 65.9770636783598
x22=3.37991614208723x_{22} = 3.37991614208723
x23=34.5656848442796x_{23} = 34.5656848442796
x24=22.0058475927713x_{24} = 22.0058475927713
x25=18.8675971617309x_{25} = 18.8675971617309
x26=69.1184539759405x_{26} = 69.1184539759405
x27=25.1450734377105x_{27} = 25.1450734377105
x28=6.36781151369107x_{28} = 6.36781151369107
x29=53.4117815402062x_{29} = 53.4117815402062
x30=94.2501135627054x_{30} = 94.2501135627054
x31=78.5427340593526x_{31} = 78.5427340593526
x32=56.5530498251275x_{32} = 56.5530498251275
x33=87.967133489911x_{33} = 87.967133489911
Signos de extremos en los puntos:
(62.835696654150134, 4.14049274584816)

(75.40129166426813, 4.32280406143887)

(28.284911304772503, -3.34214152179011)

(47.1293968198114, -3.85283851894378)

(59.69435700308747, -4.08920318061012)

(31.425156335012787, 3.44746188086714)

(50.27056033759003, 3.91736911923955)

(9.471702186779549, -2.24583383410247)

(100.53312237774094, 4.61047651900993)

(43.98830494609213, 3.78385551437724)

(84.82565643761893, -4.44058240090197)

(15.731027775220827, -2.7549021263166)

(37.70641802817207, 3.62973343908044)

(97.39161475746043, -4.57872860340226)

(12.597692197680386, 2.53227099874907)

(1.2728506982714773, 0.0708232692475832)

(91.10861952519349, -4.5120390660658)

(81.68418951289463, 4.40284344444233)

(40.847303449590925, -3.70976003369716)

(72.25986421564514, -4.28024647479203)

(65.9770636783598, -4.18927974348899)

(3.3799161420872266, -1.18342849059061)

(34.56568484427963, -3.54274330777479)

(22.00584759277127, -3.09097426796676)

(18.86759716173087, 2.93696797853021)

(69.11845397594048, 4.23579704883419)

(25.14507343771052, 3.22441678455125)

(6.367811513691074, 1.8446308321891)

(53.41178154020617, -3.9779872921632)

(94.25011356270541, 4.54593964955556)

(78.5427340593526, -4.3636242855634)

(56.553049825127495, 4.03514038997721)

(87.96713348991098, 4.4769488290828)


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=28.2849113047725x_{1} = 28.2849113047725
x2=47.1293968198114x_{2} = 47.1293968198114
x3=59.6943570030875x_{3} = 59.6943570030875
x4=9.47170218677955x_{4} = 9.47170218677955
x5=84.8256564376189x_{5} = 84.8256564376189
x6=15.7310277752208x_{6} = 15.7310277752208
x7=97.3916147574604x_{7} = 97.3916147574604
x8=91.1086195251935x_{8} = 91.1086195251935
x9=40.8473034495909x_{9} = 40.8473034495909
x10=72.2598642156451x_{10} = 72.2598642156451
x11=65.9770636783598x_{11} = 65.9770636783598
x12=3.37991614208723x_{12} = 3.37991614208723
x13=34.5656848442796x_{13} = 34.5656848442796
x14=22.0058475927713x_{14} = 22.0058475927713
x15=53.4117815402062x_{15} = 53.4117815402062
x16=78.5427340593526x_{16} = 78.5427340593526
Puntos máximos de la función:
x16=62.8356966541501x_{16} = 62.8356966541501
x16=75.4012916642681x_{16} = 75.4012916642681
x16=31.4251563350128x_{16} = 31.4251563350128
x16=50.27056033759x_{16} = 50.27056033759
x16=100.533122377741x_{16} = 100.533122377741
x16=43.9883049460921x_{16} = 43.9883049460921
x16=37.7064180281721x_{16} = 37.7064180281721
x16=12.5976921976804x_{16} = 12.5976921976804
x16=1.27285069827148x_{16} = 1.27285069827148
x16=81.6841895128946x_{16} = 81.6841895128946
x16=18.8675971617309x_{16} = 18.8675971617309
x16=69.1184539759405x_{16} = 69.1184539759405
x16=25.1450734377105x_{16} = 25.1450734377105
x16=6.36781151369107x_{16} = 6.36781151369107
x16=94.2501135627054x_{16} = 94.2501135627054
x16=56.5530498251275x_{16} = 56.5530498251275
x16=87.967133489911x_{16} = 87.967133489911
Decrece en los intervalos
[97.3916147574604,∞)\left[97.3916147574604, \infty\right)
Crece en los intervalos
(−∞,3.37991614208723]\left(-\infty, 3.37991614208723\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
−(log⁡(x)cos⁡(x)+2sin⁡(x)x+cos⁡(x)x2)=0- (\log{\left(x \right)} \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{\cos{\left(x \right)}}{x^{2}}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=80.1163073549944x_{1} = 80.1163073549944
x2=89.5403599344556x_{2} = 89.5403599344556
x3=86.3989891915502x_{3} = 86.3989891915502
x4=61.2689882085312x_{4} = 61.2689882085312
x5=42.4240774344727x_{5} = 42.4240774344727
x6=67.5512693271385x_{6} = 67.5512693271385
x7=20.4527151636554x_{7} = 20.4527151636554
x8=51.8460478223206x_{8} = 51.8460478223206
x9=17.3191834025164x_{9} = 17.3191834025164
x10=23.5887477756706x_{10} = 23.5887477756706
x11=45.5645846448375x_{11} = 45.5645846448375
x12=36.1437352734116x_{12} = 36.1437352734116
x13=58.127932349539x_{13} = 58.127932349539
x14=54.9869474043708x_{14} = 54.9869474043708
x15=98.9645667863943x_{15} = 98.9645667863943
x16=48.7052521633042x_{16} = 48.7052521633042
x17=33.0040471601321x_{17} = 33.0040471601321
x18=26.7262995386281x_{18} = 26.7262995386281
x19=64.4101035740511x_{19} = 64.4101035740511
x20=7.97332415905512x_{20} = 7.97332415905512
x21=4.95364945549859x_{21} = 4.95364945549859
x22=14.1901528741925x_{22} = 14.1901528741925
x23=39.2837741382964x_{23} = 39.2837741382964
x24=83.2576375062293x_{24} = 83.2576375062293
x25=2.28203436188726x_{25} = 2.28203436188726
x26=11.0703232999089x_{26} = 11.0703232999089
x27=95.8231504264293x_{27} = 95.8231504264293
x28=70.6924780956907x_{28} = 70.6924780956907
x29=73.8337238588912x_{29} = 73.8337238588912
x30=29.8648369783603x_{30} = 29.8648369783603
x31=76.9750016538421x_{31} = 76.9750016538421
x32=92.681747618866x_{32} = 92.681747618866

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
[95.8231504264293,∞)\left[95.8231504264293, \infty\right)
Convexa en los intervalos
(−∞,2.28203436188726]\left(-\infty, 2.28203436188726\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(x)cos⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\log{\left(x \right)} \cos{\left(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⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\log{\left(x \right)} \cos{\left(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 log(x)*cos(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(x)cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} \cos{\left(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)cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} \cos{\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:
log⁡(x)cos⁡(x)=log⁡(−x)cos⁡(x)\log{\left(x \right)} \cos{\left(x \right)} = \log{\left(- x \right)} \cos{\left(x \right)}
- No
log⁡(x)cos⁡(x)=−log⁡(−x)cos⁡(x)\log{\left(x \right)} \cos{\left(x \right)} = - \log{\left(- x \right)} \cos{\left(x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = ln*cos(x)
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