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Trazado el gráfico de la función/ log(x)*cos(3*x)

Gráfico de la función y = log(x)*cos(3*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = log(x)*cos(3*x)
f(x)=log⁡(x)cos⁡(3x)f{\left(x \right)} = \log{\left(x \right)} \cos{\left(3 x \right)} 
f = log(x)*cos(3*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⁡(3x)=0\log{\left(x \right)} \cos{\left(3 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1x_{1} = 1
x2=−5π6x_{2} = - \frac{5 \pi}{6}
x3=−π2x_{3} = - \frac{\pi}{2}
x4=−π6x_{4} = - \frac{\pi}{6}
x5=π6x_{5} = \frac{\pi}{6}
x6=π2x_{6} = \frac{\pi}{2}
x7=5π6x_{7} = \frac{5 \pi}{6}
Solución numérica
x1=−62.3082542961976x_{1} = -62.3082542961976
x2=−73.8274273593601x_{2} = -73.8274273593601
x3=−23.5619449019235x_{3} = -23.5619449019235
x4=−65.4498469497874x_{4} = -65.4498469497874
x5=42.4115008234622x_{5} = 42.4115008234622
x6=−91.6297857297023x_{6} = -91.6297857297023
x7=22.5147473507269x_{7} = 22.5147473507269
x8=93.7241808320955x_{8} = 93.7241808320955
x9=−42.4115008234622x_{9} = -42.4115008234622
x10=−108.384946548848x_{10} = -108.384946548848
x11=−97.9129710368819x_{11} = -97.9129710368819
x12=−82.2050077689329x_{12} = -82.2050077689329
x13=66.497044500984x_{13} = 66.497044500984
x14=−36.1283155162826x_{14} = -36.1283155162826
x15=86.3937979737193x_{15} = 86.3937979737193
x16=60.2138591938044x_{16} = 60.2138591938044
x17=−84.2994028713261x_{17} = -84.2994028713261
x18=97.9129710368819x_{18} = 97.9129710368819
x19=58.1194640914112x_{19} = 58.1194640914112
x20=−31.9395253114962x_{20} = -31.9395253114962
x21=34.0339204138894x_{21} = 34.0339204138894
x22=100.007366139275x_{22} = 100.007366139275
x23=−51.8362787842316x_{23} = -51.8362787842316
x24=−45.553093477052x_{24} = -45.553093477052
x25=−27.7507351067098x_{25} = -27.7507351067098
x26=12.0427718387609x_{26} = 12.0427718387609
x27=75.9218224617533x_{27} = 75.9218224617533
x28=62.3082542961976x_{28} = 62.3082542961976
x29=71.733032256967x_{29} = 71.733032256967
x30=18.3259571459405x_{30} = 18.3259571459405
x31=−56.025068989018x_{31} = -56.025068989018
x32=−58.1194640914112x_{32} = -58.1194640914112
x33=−14.1371669411541x_{33} = -14.1371669411541
x34=1.5707963267949x_{34} = 1.5707963267949
x35=36.1283155162826x_{35} = 36.1283155162826
x36=−7.85398163397448x_{36} = -7.85398163397448
x37=38.2227106186758x_{37} = 38.2227106186758
x38=−3.66519142918809x_{38} = -3.66519142918809
x39=31.9395253114962x_{39} = 31.9395253114962
x40=−5.75958653158129x_{40} = -5.75958653158129
x41=−9.94837673636768x_{41} = -9.94837673636768
x42=73.8274273593601x_{42} = 73.8274273593601
x43=68.5914396033772x_{43} = 68.5914396033772
x44=−100.007366139275x_{44} = -100.007366139275
x45=44.5058959258554x_{45} = 44.5058959258554
x46=−34.0339204138894x_{46} = -34.0339204138894
x47=699.004365423729x_{47} = 699.004365423729
x48=20.4203522483337x_{48} = 20.4203522483337
x49=−88.4881930761125x_{49} = -88.4881930761125
x50=−43.4586983746588x_{50} = -43.4586983746588
x51=−93.7241808320955x_{51} = -93.7241808320955
x52=82.2050077689329x_{52} = 82.2050077689329
x53=−12.0427718387609x_{53} = -12.0427718387609
x54=84.2994028713261x_{54} = 84.2994028713261
x55=95.8185759344887x_{55} = 95.8185759344887
x56=−16.2315620435473x_{56} = -16.2315620435473
x57=−71.733032256967x_{57} = -71.733032256967
x58=−25.6563400043166x_{58} = -25.6563400043166
x59=−38.2227106186758x_{59} = -38.2227106186758
x60=−69.6386371545737x_{60} = -69.6386371545737
x61=−89.5353906273091x_{61} = -89.5353906273091
x62=3.66519142918809x_{62} = 3.66519142918809
x63=80.1106126665397x_{63} = 80.1106126665397
x64=−6.80678408277789x_{64} = -6.80678408277789
x65=49.7418836818384x_{65} = 49.7418836818384
x66=7.85398163397448x_{66} = 7.85398163397448
x67=40.317105721069x_{67} = 40.317105721069
x68=−95.8185759344887x_{68} = -95.8185759344887
x69=29.845130209103x_{69} = 29.845130209103
x70=−78.0162175641465x_{70} = -78.0162175641465
x71=14.1371669411541x_{71} = 14.1371669411541
x72=2.61799387799149x_{72} = 2.61799387799149
x73=−1.5707963267949x_{73} = -1.5707963267949
x74=27.7507351067098x_{74} = 27.7507351067098
x75=−75.9218224617533x_{75} = -75.9218224617533
x76=88.4881930761125x_{76} = 88.4881930761125
x77=−19.3731546971371x_{77} = -19.3731546971371
x78=−80.1106126665397x_{78} = -80.1106126665397
x79=51.8362787842316x_{79} = 51.8362787842316
x80=56.025068989018x_{80} = 56.025068989018
x81=91.6297857297023x_{81} = 91.6297857297023
x82=78.0162175641465x_{82} = 78.0162175641465
x83=−47.6474885794452x_{83} = -47.6474885794452
x84=−67.5442420521806x_{84} = -67.5442420521806
x85=64.4026493985908x_{85} = 64.4026493985908
x86=−49.7418836818384x_{86} = -49.7418836818384
x87=−21.4675497995303x_{87} = -21.4675497995303
x88=−15.1843644923507x_{88} = -15.1843644923507
x89=−53.9306738866248x_{89} = -53.9306738866248
x90=16.2315620435473x_{90} = 16.2315620435473
x91=−29.845130209103x_{91} = -29.845130209103
x92=5.75958653158129x_{92} = 5.75958653158129
x93=9.94837673636768x_{93} = 9.94837673636768
x94=−60.2138591938044x_{94} = -60.2138591938044
x95=53.9306738866248x_{95} = 53.9306738866248
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(3*x).
log⁡(0)cos⁡(0⋅3)\log{\left(0 \right)} \cos{\left(0 \cdot 3 \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
−3log⁡(x)sin⁡(3x)+cos⁡(3x)x=0- 3 \log{\left(x \right)} \sin{\left(3 x \right)} + \frac{\cos{\left(3 x \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=92.1536510526427x_{1} = 92.1536510526427
x2=74.3513729610722x_{2} = 74.3513729610722
x3=76.445756400651x_{3} = 76.445756400651
x4=100.531204638865x_{4} = 100.531204638865
x5=96.3424271874236x_{5} = 96.3424271874236
x6=43.9829647927364x_{6} = 43.9829647927364
x7=41.8886122281726x_{7} = 41.8886122281726
x8=6.29278191723044x_{8} = 6.29278191723044
x9=15.7105309536845x_{9} = 15.7105309536845
x10=19.8986205587611x_{10} = 19.8986205587611
x11=65.9738477459929x_{11} = 65.9738477459929
x12=56.5491547048483x_{12} = 56.5491547048483
x13=78.5401405447888x_{13} = 78.5401405447888
x14=98.4368157600501x_{14} = 98.4368157600501
x15=85.8704897852191x_{15} = 85.8704897852191
x16=46.0773218089678x_{16} = 46.0773218089678
x17=2.16029784897009x_{17} = 2.16029784897009
x18=68.0682275933998x_{18} = 68.0682275933998
x19=24.0869935064117x_{19} = 24.0869935064117
x20=79.5873328619807x_{20} = 79.5873328619807
x21=26.1812385788794x_{21} = 26.1812385788794
x22=10.4764899884892x_{22} = 10.4764899884892
x23=39.7942649107512x_{23} = 39.7942649107512
x24=21.9927831881195x_{24} = 21.9927831881195
x25=30.3698008017668x_{25} = 30.3698008017668
x26=61.7850916250074x_{26} = 61.7850916250074
x27=13.6166929275579x_{27} = 13.6166929275579
x28=48.171682632084x_{28} = 48.171682632084
x29=63.8794690465696x_{29} = 63.8794690465696
x30=52.3604136794562x_{30} = 52.3604136794562
x31=94.2480389434111x_{31} = 94.2480389434111
x32=4.20715294221441x_{32} = 4.20715294221441
x33=32.4641075469997x_{33} = 32.4641075469997
x34=50.2660467342498x_{34} = 50.2660467342498
x35=90.0592635422263x_{35} = 90.0592635422263
x36=28.2755096981709x_{36} = 28.2755096981709
x37=70.1626084756746x_{37} = 70.1626084756746
x38=8.38381259282155x_{38} = 8.38381259282155
x39=83.7761036083348x_{39} = 83.7761036083348
x40=59.6907156311258x_{40} = 59.6907156311258
x41=17.8045256303582x_{41} = 17.8045256303582
x42=87.9648764420645x_{42} = 87.9648764420645
x43=54.4547831045677x_{43} = 54.4547831045677
x44=33.511265754867x_{44} = 33.511265754867
x45=37.6999238326345x_{45} = 37.6999238326345
x46=72.2569902938691x_{46} = 72.2569902938691
Signos de extremos en los puntos:
(92.15365105264269, 4.52345585787558)

(74.35137296107223, -4.30879980657959)

(76.445756400651, -4.33657923050441)

(100.53120463886512, 4.6104669809387)

(96.34242718742355, 4.56790748454639)

(43.98296479273636, 3.7837948053774)

(41.888612228172626, 3.73500552846426)

(6.292781917230444, 1.83864100311292)

(15.710530953684493, -2.75424953227859)

(19.898620558761056, -2.9906034963246)

(65.97384774599286, -4.18925537041119)

(56.549154704848284, 4.03510594924633)

(78.54014054478876, -4.36360777467183)

(98.43681576005011, 4.58941362872088)

(85.8704897852191, 4.45283853645973)

(46.07732180896777, 3.83031406272333)

(2.160297848970094, 0.755241176694825)

(68.06822759339975, -4.220507708109)

(24.086993506411748, -3.18164191083909)

(79.5873328619807, 4.37685294138224)

(26.181238578879434, -3.26501824653837)

(10.476489988489204, 2.34891825700492)

(39.79426491075122, 3.6837132806588)

(21.992783188119542, -3.09067720052523)

(30.369800801766818, -3.41343107396627)

(61.785091625007404, -4.12365857031863)

(13.6166929275579, -2.61118172530022)

(48.171682632084014, 3.87476517245795)

(63.8794690465696, -4.15699473648039)

(52.36041367945622, 3.95814572218027)

(94.24803894341115, 4.54592864333061)

(4.207152942214409, 1.43460659397474)

(32.464107546999664, -3.48011994906059)

(50.26604673424984, 3.91732422106114)

(90.0592635422263, 4.50046641543387)

(28.275509698170943, -3.34197525628304)

(70.1626084756746, -4.25081287145983)

(8.383812592821554, 2.12593114896635)

(83.77610360833481, 4.42814601943997)

(59.690715631125755, -4.0891726781444)

(17.80452563035817, -2.87939181233473)

(87.96487644206448, 4.47693599974731)

(54.45478310456767, 3.99736600265705)

(33.51126575486701, 3.51186758697996)

(37.69992383263446, 3.62964730501)

(72.25699029386908, -4.28022658778899)


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=74.3513729610722x_{1} = 74.3513729610722
x2=76.445756400651x_{2} = 76.445756400651
x3=15.7105309536845x_{3} = 15.7105309536845
x4=19.8986205587611x_{4} = 19.8986205587611
x5=65.9738477459929x_{5} = 65.9738477459929
x6=78.5401405447888x_{6} = 78.5401405447888
x7=68.0682275933998x_{7} = 68.0682275933998
x8=24.0869935064117x_{8} = 24.0869935064117
x9=26.1812385788794x_{9} = 26.1812385788794
x10=21.9927831881195x_{10} = 21.9927831881195
x11=30.3698008017668x_{11} = 30.3698008017668
x12=61.7850916250074x_{12} = 61.7850916250074
x13=13.6166929275579x_{13} = 13.6166929275579
x14=63.8794690465696x_{14} = 63.8794690465696
x15=32.4641075469997x_{15} = 32.4641075469997
x16=28.2755096981709x_{16} = 28.2755096981709
x17=70.1626084756746x_{17} = 70.1626084756746
x18=59.6907156311258x_{18} = 59.6907156311258
x19=17.8045256303582x_{19} = 17.8045256303582
x20=72.2569902938691x_{20} = 72.2569902938691
Puntos máximos de la función:
x20=92.1536510526427x_{20} = 92.1536510526427
x20=100.531204638865x_{20} = 100.531204638865
x20=96.3424271874236x_{20} = 96.3424271874236
x20=43.9829647927364x_{20} = 43.9829647927364
x20=41.8886122281726x_{20} = 41.8886122281726
x20=6.29278191723044x_{20} = 6.29278191723044
x20=56.5491547048483x_{20} = 56.5491547048483
x20=98.4368157600501x_{20} = 98.4368157600501
x20=85.8704897852191x_{20} = 85.8704897852191
x20=46.0773218089678x_{20} = 46.0773218089678
x20=2.16029784897009x_{20} = 2.16029784897009
x20=79.5873328619807x_{20} = 79.5873328619807
x20=10.4764899884892x_{20} = 10.4764899884892
x20=39.7942649107512x_{20} = 39.7942649107512
x20=48.171682632084x_{20} = 48.171682632084
x20=52.3604136794562x_{20} = 52.3604136794562
x20=94.2480389434111x_{20} = 94.2480389434111
x20=4.20715294221441x_{20} = 4.20715294221441
x20=50.2660467342498x_{20} = 50.2660467342498
x20=90.0592635422263x_{20} = 90.0592635422263
x20=8.38381259282155x_{20} = 8.38381259282155
x20=83.7761036083348x_{20} = 83.7761036083348
x20=87.9648764420645x_{20} = 87.9648764420645
x20=54.4547831045677x_{20} = 54.4547831045677
x20=33.511265754867x_{20} = 33.511265754867
x20=37.6999238326345x_{20} = 37.6999238326345
Decrece en los intervalos
[78.5401405447888,∞)\left[78.5401405447888, \infty\right)
Crece en los intervalos
(−∞,13.6166929275579]\left(-\infty, 13.6166929275579\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
−(9log⁡(x)cos⁡(3x)+6sin⁡(3x)x+cos⁡(3x)x2)=0- (9 \log{\left(x \right)} \cos{\left(3 x \right)} + \frac{6 \sin{\left(3 x \right)}}{x} + \frac{\cos{\left(3 x \right)}}{x^{2}}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=3.71031523340446x_{1} = 3.71031523340446
x2=82.2056208554033x_{2} = 82.2056208554033
x3=71.7337572470289x_{3} = 71.7337572470289
x4=53.9317071553088x_{4} = 53.9317071553088
x5=60.214759762845x_{5} = 60.214759762845
x6=22.5179158502217x_{6} = 22.5179158502217
x7=5.78141981025447x_{7} = 5.78141981025447
x8=56.0260542206658x_{8} = 56.0260542206658
x9=64.403477802222x_{9} = 64.403477802222
x10=86.3943748334274x_{10} = 86.3943748334274
x11=16.2364713628348x_{11} = 16.2364713628348
x12=95.8190842516045x_{12} = 95.8190842516045
x13=25.6590087988046x_{13} = 25.6590087988046
x14=91.6303225429824x_{14} = 91.6303225429824
x15=80.1112454848742x_{15} = 80.1112454848742
x16=20.4239585238892x_{16} = 20.4239585238892
x17=19.3770232316731x_{17} = 19.3770232316731
x18=34.035771298825x_{18} = 34.035771298825
x19=88.4887532733364x_{19} = 88.4887532733364
x20=75.9224984752237x_{20} = 75.9224984752237
x21=14.1430960173866x_{21} = 14.1430960173866
x22=36.130030099593x_{22} = 36.130030099593
x23=84.2999973343513x_{23} = 84.2999973343513
x24=100.007848641105x_{24} = 100.007848641105
x25=49.7430271381783x_{25} = 49.7430271381783
x26=9.95807859605126x_{26} = 9.95807859605126
x27=51.8373645824136x_{27} = 51.8373645824136
x28=66.4978406970204x_{28} = 66.4978406970204
x29=40.3185966045144x_{29} = 40.3185966045144
x30=29.8473224084856x_{30} = 29.8473224084856
x31=78.0168713213474x_{31} = 78.0168713213474
x32=97.9134661347637x_{32} = 97.9134661347637
x33=27.7531443070156x_{33} = 27.7531443070156
x34=12.0501772700338x_{34} = 12.0501772700338
x35=62.3091173945134x_{35} = 62.3091173945134
x36=58.1204052417674x_{36} = 58.1204052417674
x37=18.3301246319229x_{37} = 18.3301246319229
x38=7.86765489287871x_{38} = 7.86765489287871
x39=93.7247030375304x_{39} = 93.7247030375304
x40=1.75852871774408x_{40} = 1.75852871774408
x41=31.9415336946425x_{41} = 31.9415336946425
x42=73.8281270705241x_{42} = 73.8281270705241
x43=44.5072113388584x_{43} = 44.5072113388584
x44=38.2243062042622x_{44} = 38.2243062042622
x45=42.412898940997x_{45} = 42.412898940997

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
[88.4887532733364,∞)\left[88.4887532733364, \infty\right)
Convexa en los intervalos
(−∞,19.3770232316731]\left(-\infty, 19.3770232316731\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(x)cos⁡(3x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\log{\left(x \right)} \cos{\left(3 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⁡(3x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\log{\left(x \right)} \cos{\left(3 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(3*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(x)cos⁡(3x)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} \cos{\left(3 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⁡(3x)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} \cos{\left(3 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⁡(3x)=log⁡(−x)cos⁡(3x)\log{\left(x \right)} \cos{\left(3 x \right)} = \log{\left(- x \right)} \cos{\left(3 x \right)}
- No
log⁡(x)cos⁡(3x)=−log⁡(−x)cos⁡(3x)\log{\left(x \right)} \cos{\left(3 x \right)} = - \log{\left(- x \right)} \cos{\left(3 x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор