Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ log(x)*sin(8*x)

Gráfico de la función y = log(x)*sin(8*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = log(x)*sin(8*x)
f(x)=log⁡(x)sin⁡(8x)f{\left(x \right)} = \log{\left(x \right)} \sin{\left(8 x \right)} 
f = log(x)*sin(8*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⁡(8x)=0\log{\left(x \right)} \sin{\left(8 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1x_{1} = 1
x2=−7π8x_{2} = - \frac{7 \pi}{8}
x3=−3π4x_{3} = - \frac{3 \pi}{4}
x4=−5π8x_{4} = - \frac{5 \pi}{8}
x5=−π2x_{5} = - \frac{\pi}{2}
x6=−3π8x_{6} = - \frac{3 \pi}{8}
x7=−π4x_{7} = - \frac{\pi}{4}
x8=−π8x_{8} = - \frac{\pi}{8}
x9=π8x_{9} = \frac{\pi}{8}
x10=π4x_{10} = \frac{\pi}{4}
x11=3π8x_{11} = \frac{3 \pi}{8}
x12=π2x_{12} = \frac{\pi}{2}
x13=5π8x_{13} = \frac{5 \pi}{8}
x14=3π4x_{14} = \frac{3 \pi}{4}
x15=7π8x_{15} = \frac{7 \pi}{8}
x16=πx_{16} = \pi
Solución numérica
x1=18.0641577581413x_{1} = 18.0641577581413
x2=14.1371669411541x_{2} = 14.1371669411541
x3=−53.7997741927252x_{3} = -53.7997741927252
x4=48.6946861306418x_{4} = 48.6946861306418
x5=8.24668071567321x_{5} = 8.24668071567321
x6=−35.7356164345839x_{6} = -35.7356164345839
x7=−80.1106126665397x_{7} = -80.1106126665397
x8=−93.8550805259951x_{8} = -93.8550805259951
x9=−64.009950316892x_{9} = -64.009950316892
x10=−19.2422550032375x_{10} = -19.2422550032375
x11=32.2013246992954x_{11} = 32.2013246992954
x12=72.2566310325652x_{12} = 72.2566310325652
x13=−9.8174770424681x_{13} = -9.8174770424681
x14=80.1106126665397x_{14} = 80.1106126665397
x15=94.2477796076938x_{15} = 94.2477796076938
x16=−91.8915851175014x_{16} = -91.8915851175014
x17=−20.0276531666349x_{17} = -20.0276531666349
x18=−57.7267650097125x_{18} = -57.7267650097125
x19=−86.0010988920206x_{19} = -86.0010988920206
x20=−42.0188017417635x_{20} = -42.0188017417635
x21=27.096236637212x_{21} = 27.096236637212
x22=23.9546439836222x_{22} = 23.9546439836222
x23=78.1471172580461x_{23} = 78.1471172580461
x24=−55.7632696012188x_{24} = -55.7632696012188
x25=30.2378292908018x_{25} = 30.2378292908018
x26=54.1924732744239x_{26} = 54.1924732744239
x27=38.0918109247762x_{27} = 38.0918109247762
x28=64.009950316892x_{28} = 64.009950316892
x29=50.2654824574367x_{29} = 50.2654824574367
x30=−33.7721210260903x_{30} = -33.7721210260903
x31=−7.85398163397448x_{31} = -7.85398163397448
x32=−79.717913584841x_{32} = -79.717913584841
x33=45.553093477052x_{33} = 45.553093477052
x34=−73.8274273593601x_{34} = -73.8274273593601
x35=−21.9911485751286x_{35} = -21.9911485751286
x36=74.2201264410589x_{36} = 74.2201264410589
x37=58.1194640914112x_{37} = 58.1194640914112
x38=−23.9546439836222x_{38} = -23.9546439836222
x39=−99.7455667514759x_{39} = -99.7455667514759
x40=−97.7820713429823x_{40} = -97.7820713429823
x41=−37.6991118430775x_{41} = -37.6991118430775
x42=−40.8407044966673x_{42} = -40.8407044966673
x43=−47.9092879672443x_{43} = -47.9092879672443
x44=40.0553063332699x_{44} = 40.0553063332699
x45=42.0188017417635x_{45} = 42.0188017417635
x46=56.1559686829176x_{46} = 56.1559686829176
x47=87.9645943005142x_{47} = 87.9645943005142
x48=−75.7909227678538x_{48} = -75.7909227678538
x49=−29.845130209103x_{49} = -29.845130209103
x50=60.0829594999048x_{50} = 60.0829594999048
x51=−84.037603483527x_{51} = -84.037603483527
x52=100.138265833175x_{52} = 100.138265833175
x53=−25.9181393921158x_{53} = -25.9181393921158
x54=−95.8185759344887x_{54} = -95.8185759344887
x55=86.0010988920206x_{55} = 86.0010988920206
x56=−43.9822971502571x_{56} = -43.9822971502571
x57=52.2289778659303x_{57} = 52.2289778659303
x58=−89.9280897090078x_{58} = -89.9280897090078
x59=10.2101761241668x_{59} = 10.2101761241668
x60=−3.92699081698724x_{60} = -3.92699081698724
x61=76.1836218495525x_{61} = 76.1836218495525
x62=6.28318530717959x_{62} = 6.28318530717959
x63=−5.89048622548086x_{63} = -5.89048622548086
x64=28.2743338823081x_{64} = 28.2743338823081
x65=20.0276531666349x_{65} = 20.0276531666349
x66=−62.0464549083984x_{66} = -62.0464549083984
x67=43.9822971502571x_{67} = 43.9822971502571
x68=65.9734457253857x_{68} = 65.9734457253857
x69=−71.8639319508665x_{69} = -71.8639319508665
x70=−31.8086256175967x_{70} = -31.8086256175967
x71=−51.8362787842316x_{71} = -51.8362787842316
x72=70.2931356240716x_{72} = 70.2931356240716
x73=−69.9004365423729x_{73} = -69.9004365423729
x74=12.1736715326604x_{74} = 12.1736715326604
x75=−13.7444678594553x_{75} = -13.7444678594553
x76=16.1006623496477x_{76} = 16.1006623496477
x77=1.96349540849362x_{77} = 1.96349540849362
x78=−87.9645943005142x_{78} = -87.9645943005142
x79=−11.7809724509617x_{79} = -11.7809724509617
x80=−49.872783375738x_{80} = -49.872783375738
x81=36.1283155162826x_{81} = 36.1283155162826
x82=34.164820107789x_{82} = 34.164820107789
x83=−1.96349540849362x_{83} = -1.96349540849362
x84=−67.9369411338793x_{84} = -67.9369411338793
x85=96.2112750161874x_{85} = 96.2112750161874
x86=−60.8683576633022x_{86} = -60.8683576633022
x87=−45.9457925587507x_{87} = -45.9457925587507
x88=82.0741080750334x_{88} = 82.0741080750334
x89=21.9911485751286x_{89} = 21.9911485751286
x90=62.0464549083984x_{90} = 62.0464549083984
x91=−65.9734457253857x_{91} = -65.9734457253857
x92=−77.7544181763474x_{92} = -77.7544181763474
x93=92.2842841992002x_{93} = 92.2842841992002
x94=98.174770424681x_{94} = 98.174770424681
x95=−15.707963267949x_{95} = -15.707963267949
x96=−27.8816348006094x_{96} = -27.8816348006094
x97=84.037603483527x_{97} = 84.037603483527
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)*sin(8*x).
log⁡(0)sin⁡(0⋅8)\log{\left(0 \right)} \sin{\left(0 \cdot 8 \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
8log⁡(x)cos⁡(8x)+sin⁡(8x)x=08 \log{\left(x \right)} \cos{\left(8 x \right)} + \frac{\sin{\left(8 x \right)}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=18.2608018697284x_{1} = 18.2608018697284
x2=50.0692126598985x_{2} = 50.0692126598985
x3=74.0238259387079x_{3} = 74.0238259387079
x4=20.2242596466284x_{4} = 20.2242596466284
x5=46.1422304740309x_{5} = 46.1422304740309
x6=52.0327043123183x_{6} = 52.0327043123183
x7=8.05126157419965x_{7} = 8.05126157419965
x8=94.0514666289506x_{8} = 94.0514666289506
x9=42.2152501734583x_{9} = 42.2152501734583
x10=33.9686010425383x_{10} = 33.9686010425383
x11=6.08825647051074x_{11} = 6.08825647051074
x12=75.9873197912678x_{12} = 75.9873197912678
x13=88.160983409589x_{13} = 88.160983409589
x14=4.12601187626081x_{14} = 4.12601187626081
x15=44.1787400527981x_{15} = 44.1787400527981
x16=28.0781512030948x_{16} = 28.0781512030948
x17=26.1146723269008x_{17} = 26.1146723269008
x18=24.151196695783x_{18} = 24.151196695783
x19=64.2063583272187x_{19} = 64.2063583272187
x20=55.9596885195547x_{20} = 55.9596885195547
x21=96.0149611276345x_{21} = 96.0149611276345
x22=2.16913055800871x_{22} = 2.16913055800871
x23=11.9778473453662x_{23} = 11.9778473453662
x24=32.0051160175556x_{24} = 32.0051160175556
x25=48.1057213634398x_{25} = 48.1057213634398
x26=77.9508137326498x_{26} = 77.9508137326498
x27=22.1877253166599x_{27} = 22.1877253166599
x28=62.2428652166406x_{28} = 62.2428652166406
x29=70.0968385331794x_{29} = 70.0968385331794
x30=30.0416326076808x_{30} = 30.0416326076808
x31=59.4939751552921x_{31} = 59.4939751552921
x32=84.2339948628895x_{32} = 84.2339948628895
x33=82.2705006820316x_{33} = 82.2705006820316
x34=97.9784556673887x_{34} = 97.9784556673887
x35=79.9143077556694x_{35} = 79.9143077556694
x36=19.0461838505964x_{36} = 19.0461838505964
x37=72.0603321830063x_{37} = 72.0603321830063
x38=92.0879721742245x_{38} = 92.0879721742245
x39=40.2517609256839x_{39} = 40.2517609256839
x40=53.9961962776884x_{40} = 53.9961962776884
x41=10.0145037528058x_{41} = 10.0145037528058
x42=90.1244777666158x_{42} = 90.1244777666158
x43=86.19748910695x_{43} = 86.19748910695
x44=66.1698515930582x_{44} = 66.1698515930582
x45=60.2793722782135x_{45} = 60.2793722782135
x46=16.2973554021834x_{46} = 16.2973554021834
x47=68.133344999376x_{47} = 68.133344999376
x48=38.2882724195937x_{48} = 38.2882724195937
Signos de extremos en los puntos:
(18.260801869728397, 2.90474872256889)

(50.06921265989852, -3.91340550510024)

(74.02382593870792, 4.30438668229243)

(20.224259646628404, -3.00687650418207)

(46.142230474030924, -3.83172763538649)

(52.03270431231832, 3.9518717197843)

(8.05126157419965, 2.08577101805123)

(94.0514666289506, -4.54384195539348)

(42.21525017345833, -3.74278036300954)

(33.96860104253829, 3.52543467864806)

(6.0882564705107445, -1.80624507738833)

(75.98731979126781, -4.33056616907704)

(88.16098340958898, 4.47916427576092)

(4.126011876260811, 1.41698761365426)

(44.178740052798084, 3.78824262245249)

(28.078151203094833, -3.33498876526801)

(26.11467232690082, 3.26249380317905)

(24.151196695783007, -3.18432972525899)

(64.2063583272187, -4.16210178983199)

(55.959688519554746, 4.02463096376719)

(96.01496112763455, 4.56450383874468)

(2.1691305580087143, -0.772190944810158)

(11.977847345366225, 2.48303695912207)

(32.00511601755564, -3.465893564999)

(48.10572136343984, 3.87340024572955)

(77.95081373264978, 4.35607773945398)

(22.187725316659915, 3.09953410234481)

(62.24286521664063, 4.13104342563488)

(70.09683853317941, 4.24987731951455)

(30.0416326076808, 3.40258162911481)

(59.493975155292084, -4.08587450932505)

(84.23399486288952, 4.43359833088473)

(82.2705006820316, -4.41001234526697)

(97.97845566738872, -4.58474743686736)

(79.91430775566938, -4.38095462828886)

(19.04618385059643, 2.94685945047606)

(72.06033218300631, -4.27750336339258)

(92.08797217422446, 4.52274413575183)

(40.25176092568393, 3.69515244768311)

(53.996196277688384, -3.98891293302926)

(10.014503752805776, -2.30400060847168)

(90.12447776661581, -4.50119158735671)

(86.19748910695003, -4.45664081262032)

(66.16985159305824, 4.19222451958539)

(60.27937227821351, -4.0989894357339)

(16.297355402183374, -2.79099231053969)

(68.13334499937596, -4.221466342221)

(38.28827241959366, -3.64514218416585)


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=50.0692126598985x_{1} = 50.0692126598985
x2=20.2242596466284x_{2} = 20.2242596466284
x3=46.1422304740309x_{3} = 46.1422304740309
x4=94.0514666289506x_{4} = 94.0514666289506
x5=42.2152501734583x_{5} = 42.2152501734583
x6=6.08825647051074x_{6} = 6.08825647051074
x7=75.9873197912678x_{7} = 75.9873197912678
x8=28.0781512030948x_{8} = 28.0781512030948
x9=24.151196695783x_{9} = 24.151196695783
x10=64.2063583272187x_{10} = 64.2063583272187
x11=2.16913055800871x_{11} = 2.16913055800871
x12=32.0051160175556x_{12} = 32.0051160175556
x13=59.4939751552921x_{13} = 59.4939751552921
x14=82.2705006820316x_{14} = 82.2705006820316
x15=97.9784556673887x_{15} = 97.9784556673887
x16=79.9143077556694x_{16} = 79.9143077556694
x17=72.0603321830063x_{17} = 72.0603321830063
x18=53.9961962776884x_{18} = 53.9961962776884
x19=10.0145037528058x_{19} = 10.0145037528058
x20=90.1244777666158x_{20} = 90.1244777666158
x21=86.19748910695x_{21} = 86.19748910695
x22=60.2793722782135x_{22} = 60.2793722782135
x23=16.2973554021834x_{23} = 16.2973554021834
x24=68.133344999376x_{24} = 68.133344999376
x25=38.2882724195937x_{25} = 38.2882724195937
Puntos máximos de la función:
x25=18.2608018697284x_{25} = 18.2608018697284
x25=74.0238259387079x_{25} = 74.0238259387079
x25=52.0327043123183x_{25} = 52.0327043123183
x25=8.05126157419965x_{25} = 8.05126157419965
x25=33.9686010425383x_{25} = 33.9686010425383
x25=88.160983409589x_{25} = 88.160983409589
x25=4.12601187626081x_{25} = 4.12601187626081
x25=44.1787400527981x_{25} = 44.1787400527981
x25=26.1146723269008x_{25} = 26.1146723269008
x25=55.9596885195547x_{25} = 55.9596885195547
x25=96.0149611276345x_{25} = 96.0149611276345
x25=11.9778473453662x_{25} = 11.9778473453662
x25=48.1057213634398x_{25} = 48.1057213634398
x25=77.9508137326498x_{25} = 77.9508137326498
x25=22.1877253166599x_{25} = 22.1877253166599
x25=62.2428652166406x_{25} = 62.2428652166406
x25=70.0968385331794x_{25} = 70.0968385331794
x25=30.0416326076808x_{25} = 30.0416326076808
x25=84.2339948628895x_{25} = 84.2339948628895
x25=19.0461838505964x_{25} = 19.0461838505964
x25=92.0879721742245x_{25} = 92.0879721742245
x25=40.2517609256839x_{25} = 40.2517609256839
x25=66.1698515930582x_{25} = 66.1698515930582
Decrece en los intervalos
[97.9784556673887,∞)\left[97.9784556673887, \infty\right)
Crece en los intervalos
(−∞,2.16913055800871]\left(-\infty, 2.16913055800871\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
−64log⁡(x)sin⁡(8x)+16cos⁡(8x)x−sin⁡(8x)x2=0- 64 \log{\left(x \right)} \sin{\left(8 x \right)} + \frac{16 \cos{\left(8 x \right)}}{x} - \frac{\sin{\left(8 x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=94.2478525460185x_{1} = 94.2478525460185
x2=62.0465769208081x_{2} = 62.0465769208081
x3=70.2932401619726x_{3} = 70.2932401619726
x4=20.0281737538408x_{4} = 20.0281737538408
x5=92.2843590359881x_{5} = 92.2843590359881
x6=30.2381324367757x_{6} = 30.2381324367757
x7=8.24847590755737x_{7} = 8.24847590755737
x8=100.138333577581x_{8} = 100.138333577581
x9=52.229129124363x_{9} = 52.229129124363
x10=10.2114930806688x_{10} = 10.2114930806688
x11=32.2016042033765x_{11} = 32.2016042033765
x12=54.1926177050632x_{12} = 54.1926177050632
x13=48.6948512929266x_{13} = 48.6948512929266
x14=78.1472090046011x_{14} = 78.1472090046011
x15=87.9646736530215x_{15} = 87.9646736530215
x16=86.0011804675551x_{16} = 86.0011804675551
x17=74.2202241982727x_{17} = 74.2202241982727
x18=84.0376874001288x_{18} = 84.0376874001288
x19=18.0647555011281x_{19} = 18.0647555011281
x20=16.1013607428328x_{20} = 16.1013607428328
x21=43.9824849270035x_{21} = 43.9824849270035
x22=36.12855664888x_{22} = 36.12855664888
x23=40.0555177446633x_{23} = 40.0555177446633
x24=76.1837165133704x_{24} = 76.1837165133704
x25=45.5532731128904x_{25} = 45.5532731128904
x26=23.9550547005005x_{26} = 23.9550547005005
x27=82.0741944600837x_{27} = 82.0741944600837
x28=56.1561068320796x_{28} = 56.1561068320796
x29=34.1650791332823x_{29} = 34.1650791332823
x30=96.211346143363x_{30} = 96.211346143363
x31=1.98604791452458x_{31} = 1.98604791452458
x32=58.1195964441983x_{32} = 58.1195964441983
x33=12.1746984486673x_{33} = 12.1746984486673
x34=21.9916083393454x_{34} = 21.9916083393454
x35=72.2567320751989x_{35} = 72.2567320751989
x36=64.0100677006671x_{36} = 64.0100677006671
x37=60.0830864888838x_{37} = 60.0830864888838
x38=6.28588866860407x_{38} = 6.28588866860407
x39=14.1380013572958x_{39} = 14.1380013572958
x40=28.2746645920275x_{40} = 28.2746645920275
x41=50.2656411617136x_{41} = 50.2656411617136
x42=65.973558794222x_{42} = 65.973558794222
x43=27.0965861761157x_{43} = 27.0965861761157
x44=42.0190006942842x_{44} = 42.0190006942842
x45=98.1748398222977x_{45} = 98.1748398222977
x46=38.0920363030603x_{46} = 38.0920363030603
x47=80.1107016577539x_{47} = 80.1107016577539

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
[100.138333577581,∞)\left[100.138333577581, \infty\right)
Convexa en los intervalos
(−∞,1.98604791452458]\left(-\infty, 1.98604791452458\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(x)sin⁡(8x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\log{\left(x \right)} \sin{\left(8 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⁡(8x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\log{\left(x \right)} \sin{\left(8 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)*sin(8*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(x)sin⁡(8x)x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(x \right)} \sin{\left(8 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⁡(8x)x)=0\lim_{x \to \infty}\left(\frac{\log{\left(x \right)} \sin{\left(8 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⁡(8x)=−log⁡(−x)sin⁡(8x)\log{\left(x \right)} \sin{\left(8 x \right)} = - \log{\left(- x \right)} \sin{\left(8 x \right)}
- No
log⁡(x)sin⁡(8x)=log⁡(−x)sin⁡(8x)\log{\left(x \right)} \sin{\left(8 x \right)} = \log{\left(- x \right)} \sin{\left(8 x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор