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Trazado el gráfico de la función/ 3*x-4*log(x)-5*sin(x)

Gráfico de la función y = 3*x-4*log(x)-5*sin(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = 3*x - 4*log(x) - 5*sin(x)
f(x)=(3x−4log⁡(x))−5sin⁡(x)f{\left(x \right)} = \left(3 x - 4 \log{\left(x \right)}\right) - 5 \sin{\left(x \right)} 
f = 3*x - 4*log(x) - 5*sin(x)
Gráfico de la función
02468-8-6-4-2-1010-2525
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:
(3x−4log⁡(x))−5sin⁡(x)=0\left(3 x - 4 \log{\left(x \right)}\right) - 5 \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=2.33430786236425x_{1} = 2.33430786236425
x2=0.748755401410195x_{2} = 0.748755401410195
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 3*x - 4*log(x) - 5*sin(x).
(−4log⁡(0)+0⋅3)−5sin⁡(0)\left(- 4 \log{\left(0 \right)} + 0 \cdot 3\right) - 5 \sin{\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
−5cos⁡(x)+3−4x=0- 5 \cos{\left(x \right)} + 3 - \frac{4}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=8659.15676399116x_{1} = 8659.15676399116
x2=99.593666256374x_{2} = 99.593666256374
x3=−74.4844227888752x_{3} = -74.4844227888752
x4=−101.448366030563x_{4} = -101.448366030563
x5=19.8263891583539x_{5} = 19.8263891583539
x6=−43.0784232507229x_{6} = -43.0784232507229
x7=−63.7433663282863x_{7} = -63.7433663282863
x8=32.3737658845994x_{8} = 32.3737658845994
x9=−36.7992772881958x_{9} = -36.7992772881958
x10=38.6520354336436x_{10} = 38.6520354336436
x11=−55.6394691921915x_{11} = -55.6394691921915
x12=5.17398805428147x_{12} = 5.17398805428147
x13=−19.7251283309285x_{13} = -19.7251283309285
x14=−44.8871230552348x_{14} = -44.8871230552348
x15=26.0978273148013x_{15} = 26.0978273148013
x16=68.1731539284796x_{16} = 68.1731539284796
x17=−87.0488370640837x_{17} = -87.0488370640837
x18=13.5655092601862x_{18} = 13.5655092601862
x19=76.3385550865966x_{19} = 76.3385550865966
x20=−24.2473585373772x_{20} = -24.2473585373772
x21=−38.600240762574x_{21} = -38.600240762574
x22=57.4932451903028x_{22} = 57.4932451903028
x23=61.8884958151393x_{23} = 61.8884958151393
x24=−68.2024874381788x_{24} = -68.2024874381788
x25=82.6207535550359x_{25} = 82.6207535550359
x26=87.0258570893134x_{26} = 87.0258570893134
x27=43.0319604346148x_{27} = 43.0319604346148
x28=−88.8805903515249x_{28} = -88.8805903515249
x29=−70.0279757928347x_{29} = -70.0279757928347
x30=−49.358604874442x_{30} = -49.358604874442
x31=95.1855397465583x_{31} = 95.1855397465583
x32=−95.164524774114x_{32} = -95.164524774114
x33=−82.5965413923539x_{33} = -82.5965413923539
x34=1.50300976346137x_{34} = 1.50300976346137
x35=−30.5218137237599x_{35} = -30.5218137237599
x36=88.9030909044499x_{36} = 88.9030909044499
x37=−7.05976001620841x_{37} = -7.05976001620841
x38=−17.9791235997861x_{38} = -17.9791235997861
x39=−80.7665534881257x_{39} = -80.7665534881257
x40=−32.311900428637x_{40} = -32.311900428637
x41=17.8673939120046x_{41} = 17.8673939120046
x42=−26.0210254679911x_{42} = -26.0210254679911
x43=51.2121645607065x_{43} = 51.2121645607065
x44=80.7417852981937x_{44} = 80.7417852981937
x45=30.4561862941834x_{45} = 30.4561862941834
x46=−13.4168451851825x_{46} = -13.4168451851825
x47=70.056532656695x_{47} = 70.056532656695
x48=−51.1730895286278x_{48} = -51.1730895286278
x49=55.603506810047x_{49} = 55.603506810047
x50=−93.331242524074x_{50} = -93.331242524074
x51=49.3180612245419x_{51} = 49.3180612245419
x52=44.931667445655x_{52} = 44.931667445655
x53=−57.4584431018328x_{53} = -57.4584431018328
x54=24.1646752551122x_{54} = 24.1646752551122
x55=36.7448709415722x_{55} = 36.7448709415722
x56=93.3098099288294x_{56} = 93.3098099288294
x57=63.7747379736411x_{57} = 63.7747379736411
x58=−5.55158029993605x_{58} = -5.55158029993605
x59=−76.3123494474998x_{59} = -76.3123494474998
x60=−61.9208072429477x_{60} = -61.9208072429477
x61=7.34072265652191x_{61} = 7.34072265652191
x62=−11.7273830842277x_{62} = -11.7273830842277
x63=−99.6137467221083x_{63} = -99.6137467221083
x64=11.5550788546483x_{64} = 11.5550788546483
x65=74.4575645731811x_{65} = 74.4575645731811
Signos de extremos en los puntos:
(8659.156763991157, 25937.2044550593)

(99.59366625637401, 284.406414171067)

(-74.48442278887525, -244.654782498991 - 4*pi*I)

(-101.44836603056332, -318.853175539216 - 4*pi*I)

(19.826389158353894, 43.3874658772698)

(-43.07842325072291, -148.2160041589 - 4*pi*I)

(-63.7433663282863, -203.897401518402 - 4*pi*I)

(32.373765884599386, 79.1221516522013)

(-36.799277288195846, -128.735864971748 - 4*pi*I)

(38.65203543364358, 97.262145407066)

(-55.63946919219154, -186.93903695161 - 4*pi*I)

(5.173988054281468, 13.4240962948854)

(-19.725128330928538, -67.2634070381814 - 4*pi*I)

(-44.88712305523475, -145.946384978343 - 4*pi*I)

(26.097827314801343, 61.1355840183632)

(68.17315392847964, 191.674598846685)

(-87.04883706408371, -282.977508995339 - 4*pi*I)

(13.565509260186223, 26.0613794247474)

(76.33855508659659, 207.636185220272)

(-24.247358537377245, -89.3660927770684 - 4*pi*I)

(-38.60024076257399, -126.493615553358 - 4*pi*I)

(57.49324519030283, 152.221818999426)

(61.88849581513934, 173.211818277258)

(-68.20248743817879, -225.452720453556 - 4*pi*I)

(82.62075355503595, 226.169360267915)

(87.02585708931342, 247.246813533313)

(43.03196043461475, 118.116165484073)

(-88.88059035152487, -280.625098323609 - 4*pi*I)

(-70.0279757928347, -223.122990862607 - 4*pi*I)

(-49.358604874441994, -167.610180703007 - 4*pi*I)

(95.18553974655828, 263.302131929885)

(-95.16452477411396, -299.747875428763 - 4*pi*I)

(-82.59654139235393, -261.482278830599 - 4*pi*I)

(1.503009763461372, -2.10936598923885)

(-30.521813723759905, -109.137476806866 - 4*pi*I)

(88.9030909044499, 244.72573255676)

(-7.059760016208413, -25.4927232212381 - 4*pi*I)

(-17.979123599786057, -69.3172541292041 - 4*pi*I)

(-80.76655348812574, -263.828284550978 - 4*pi*I)

(-32.311900428637024, -106.933354176166 - 4*pi*I)

(17.867393912004644, 46.2287703886753)

(-26.021025467991123, -87.2187425170716 - 4*pi*I)

(51.212164560706455, 133.835179977029)

(80.74178529819368, 228.69701150275)

(30.456186294183357, 81.7986151594691)

(-13.416845185182497, -46.8786119925272 - 4*pi*I)

(70.05653265669497, 189.130195340137)

(-51.17308952862778, -165.319959626718 - 4*pi*I)

(55.60350681004703, 154.790491350987)

(-93.33124252407403, -302.105841969844 - 4*pi*I)

(49.318061224541914, 136.420586087109)

(44.93166744565501, 115.509185723959)

(-57.458443101832806, -184.632747967188 - 4*pi*I)

(24.164675255112222, 63.8734126345027)

(36.744870941572245, 99.8979934426889)

(93.30980992882942, 265.817523689837)

(63.77473797364106, 170.656504208801)

(-5.551580299936049, -26.8513953329272 - 4*pi*I)

(-76.31234944749977, -242.316241550732 - 4*pi*I)

(-61.920807242947724, -206.216572697207 - 4*pi*I)

(7.340722656521914, 9.69267421445369)

(-11.727383084227696, -48.749690182 - 4*pi*I)

(-99.61374672210832, -321.216007208559 - 4*pi*I)

(11.555078854648341, 29.1143271627237)

(74.45756457318113, 210.171509588833)


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=−74.4844227888752x_{1} = -74.4844227888752
x2=19.8263891583539x_{2} = 19.8263891583539
x3=−43.0784232507229x_{3} = -43.0784232507229
x4=32.3737658845994x_{4} = 32.3737658845994
x5=−36.7992772881958x_{5} = -36.7992772881958
x6=38.6520354336436x_{6} = 38.6520354336436
x7=−55.6394691921915x_{7} = -55.6394691921915
x8=26.0978273148013x_{8} = 26.0978273148013
x9=−87.0488370640837x_{9} = -87.0488370640837
x10=13.5655092601862x_{10} = 13.5655092601862
x11=76.3385550865966x_{11} = 76.3385550865966
x12=−24.2473585373772x_{12} = -24.2473585373772
x13=57.4932451903028x_{13} = 57.4932451903028
x14=−68.2024874381788x_{14} = -68.2024874381788
x15=82.6207535550359x_{15} = 82.6207535550359
x16=−49.358604874442x_{16} = -49.358604874442
x17=95.1855397465583x_{17} = 95.1855397465583
x18=1.50300976346137x_{18} = 1.50300976346137
x19=−30.5218137237599x_{19} = -30.5218137237599
x20=88.9030909044499x_{20} = 88.9030909044499
x21=−17.9791235997861x_{21} = -17.9791235997861
x22=−80.7665534881257x_{22} = -80.7665534881257
x23=51.2121645607065x_{23} = 51.2121645607065
x24=70.056532656695x_{24} = 70.056532656695
x25=−93.331242524074x_{25} = -93.331242524074
x26=44.931667445655x_{26} = 44.931667445655
x27=63.7747379736411x_{27} = 63.7747379736411
x28=−5.55158029993605x_{28} = -5.55158029993605
x29=−61.9208072429477x_{29} = -61.9208072429477
x30=7.34072265652191x_{30} = 7.34072265652191
x31=−11.7273830842277x_{31} = -11.7273830842277
x32=−99.6137467221083x_{32} = -99.6137467221083
Puntos máximos de la función:
x32=99.593666256374x_{32} = 99.593666256374
x32=−101.448366030563x_{32} = -101.448366030563
x32=−63.7433663282863x_{32} = -63.7433663282863
x32=5.17398805428147x_{32} = 5.17398805428147
x32=−19.7251283309285x_{32} = -19.7251283309285
x32=−44.8871230552348x_{32} = -44.8871230552348
x32=68.1731539284796x_{32} = 68.1731539284796
x32=−38.600240762574x_{32} = -38.600240762574
x32=61.8884958151393x_{32} = 61.8884958151393
x32=87.0258570893134x_{32} = 87.0258570893134
x32=43.0319604346148x_{32} = 43.0319604346148
x32=−88.8805903515249x_{32} = -88.8805903515249
x32=−70.0279757928347x_{32} = -70.0279757928347
x32=−95.164524774114x_{32} = -95.164524774114
x32=−82.5965413923539x_{32} = -82.5965413923539
x32=−7.05976001620841x_{32} = -7.05976001620841
x32=−32.311900428637x_{32} = -32.311900428637
x32=17.8673939120046x_{32} = 17.8673939120046
x32=−26.0210254679911x_{32} = -26.0210254679911
x32=80.7417852981937x_{32} = 80.7417852981937
x32=30.4561862941834x_{32} = 30.4561862941834
x32=−13.4168451851825x_{32} = -13.4168451851825
x32=−51.1730895286278x_{32} = -51.1730895286278
x32=55.603506810047x_{32} = 55.603506810047
x32=49.3180612245419x_{32} = 49.3180612245419
x32=−57.4584431018328x_{32} = -57.4584431018328
x32=24.1646752551122x_{32} = 24.1646752551122
x32=36.7448709415722x_{32} = 36.7448709415722
x32=93.3098099288294x_{32} = 93.3098099288294
x32=−76.3123494474998x_{32} = -76.3123494474998
x32=11.5550788546483x_{32} = 11.5550788546483
x32=74.4575645731811x_{32} = 74.4575645731811
Decrece en los intervalos
[95.1855397465583,∞)\left[95.1855397465583, \infty\right)
Crece en los intervalos
(−∞,−99.6137467221083]\left(-\infty, -99.6137467221083\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
5sin⁡(x)+4x2=05 \sin{\left(x \right)} + \frac{4}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−84.8228904574051x_{1} = -84.8228904574051
x2=−50.265799082152x_{2} = -50.265799082152
x3=100.530885757574x_{3} = 100.530885757574
x4=−12.5714326161909x_{4} = -12.5714326161909
x5=53.4073555819491x_{5} = 53.4073555819491
x6=25.1314745859145x_{6} = 25.1314745859145
x7=62.8316504281201x_{7} = 62.8316504281201
x8=12.5613004429586x_{8} = 12.5613004429586
x9=−9.41575423991801x_{9} = -9.41575423991801
x10=−72.256477805172x_{10} = -72.256477805172
x11=−69.1152058511954x_{11} = -69.1152058511954
x12=18.8473037996473x_{12} = 18.8473037996473
x13=−15.7047196449385x_{13} = -15.7047196449385
x14=37.6985489307727x_{14} = 37.6985489307727
x15=−204.203503298257x_{15} = -204.203503298257
x16=−43.9827106983436x_{16} = -43.9827106983436
x17=−40.8402248584783x_{17} = -40.8402248584783
x18=31.4151159245089x_{18} = 31.4151159245089
x19=−59.6900358820359x_{19} = -59.6900358820359
x20=−75.3983644094964x_{20} = -75.3983644094964
x21=−87.9646976892343x_{21} = -87.9646976892343
x22=94.2476895442472x_{22} = 94.2476895442472
x23=56.548417586638x_{23} = 56.548417586638
x24=−100.531044071924x_{24} = -100.531044071924
x25=−34.5568492713411x_{25} = -34.5568492713411
x26=47.1242500514441x_{26} = 47.1242500514441
x27=34.5581890556913x_{27} = 34.5581890556913
x28=87.9644909113081x_{28} = 87.9644909113081
x29=−94.2478696707961x_{29} = -94.2478696707961
x30=−78.5396866482011x_{30} = -78.5396866482011
x31=59.6904849509977x_{31} = 59.6904849509977
x32=28.2753345146972x_{32} = 28.2753345146972
x33=−25.1340076162278x_{33} = -25.1340076162278
x34=43.9818835866163x_{34} = 43.9818835866163
x35=−31.4167370636322x_{35} = -31.4167370636322
x36=−97.3892879146786x_{36} = -97.3892879146786
x37=−56.5489179381674x_{37} = -56.5489179381674
x38=65.9736295269633x_{38} = 65.9736295269633
x39=−138.230118626183x_{39} = -138.230118626183
x40=9.43376725346464x_{40} = 9.43376725346464
x41=−37.6996747217633x_{41} = -37.6996747217633
x42=50.2651658247435x_{42} = 50.2651658247435
x43=78.5399460304319x_{43} = 78.5399460304319
x44=97.3894566075964x_{44} = 97.3894566075964
x45=−18.8518069676194x_{45} = -18.8518069676194
x46=−47.1235295452334x_{46} = -47.1235295452334
x47=15.711204213987x_{47} = 15.711204213987
x48=3.21888076015164x_{48} = 3.21888076015164
x49=81.6812890862561x_{49} = 81.6812890862561
x50=84.8231128358607x_{50} = 84.8231128358607
x51=−3.05581617184158x_{51} = -3.05581617184158
x52=−62.8320557128574x_{52} = -62.8320557128574
x53=−113.097398073104x_{53} = -113.097398073104
x54=40.8411841123256x_{54} = 40.8411841123256
x55=−6.30332164086276x_{55} = -6.30332164086276
x56=−65.9732619217597x_{56} = -65.9732619217597
x57=−81.6815288997091x_{57} = -81.6815288997091
x58=75.3980829617631x_{58} = 75.3980829617631
x59=6.26278743998271x_{59} = 6.26278743998271
x60=−53.4067946342121x_{60} = -53.4067946342121
x61=69.1148709051322x_{61} = 69.1148709051322
x62=−28.273333108249x_{62} = -28.273333108249
x63=21.9928025504857x_{63} = 21.9928025504857
x64=72.2567842586587x_{64} = 72.2567842586587
x65=−91.1060905722746x_{65} = -91.1060905722746
x66=−21.9894941020329x_{66} = -21.9894941020329
x67=91.1062833355256x_{67} = 91.1062833355256

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
[100.530885757574,∞)\left[100.530885757574, \infty\right)
Convexa en los intervalos
(−∞,−138.230118626183]\left(-\infty, -138.230118626183\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((3x−4log⁡(x))−5sin⁡(x))=−∞\lim_{x \to -\infty}\left(\left(3 x - 4 \log{\left(x \right)}\right) - 5 \sin{\left(x \right)}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞((3x−4log⁡(x))−5sin⁡(x))y = \lim_{x \to \infty}\left(\left(3 x - 4 \log{\left(x \right)}\right) - 5 \sin{\left(x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 3*x - 4*log(x) - 5*sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((3x−4log⁡(x))−5sin⁡(x)x)=3\lim_{x \to -\infty}\left(\frac{\left(3 x - 4 \log{\left(x \right)}\right) - 5 \sin{\left(x \right)}}{x}\right) = 3
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=3xy = 3 x
lim⁡x→∞((3x−4log⁡(x))−5sin⁡(x)x)=3\lim_{x \to \infty}\left(\frac{\left(3 x - 4 \log{\left(x \right)}\right) - 5 \sin{\left(x \right)}}{x}\right) = 3
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=3xy = 3 x
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:
(3x−4log⁡(x))−5sin⁡(x)=−3x−4log⁡(−x)+5sin⁡(x)\left(3 x - 4 \log{\left(x \right)}\right) - 5 \sin{\left(x \right)} = - 3 x - 4 \log{\left(- x \right)} + 5 \sin{\left(x \right)}
- No
(3x−4log⁡(x))−5sin⁡(x)=3x+4log⁡(−x)−5sin⁡(x)\left(3 x - 4 \log{\left(x \right)}\right) - 5 \sin{\left(x \right)} = 3 x + 4 \log{\left(- x \right)} - 5 \sin{\left(x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор