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Trazado el gráfico de la función/ ln(x/(x-2))/x-2

Gráfico de la función y = ln(x/(x-2))/x-2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /  x  \    
       log|-----|    
          \x - 2/    
f(x) = ---------- - 2
           x
f(x)=−2+log⁡(xx−2)xf{\left(x \right)} = -2 + \frac{\log{\left(\frac{x}{x - 2} \right)}}{x} 
f = -2 + log(x/(x - 2))/x
Gráfico de la función
02468-8-6-4-2-1010-100100
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=2x_{2} = 2
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:
−2+log⁡(xx−2)x=0-2 + \frac{\log{\left(\frac{x}{x - 2} \right)}}{x} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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/(x - 2))/x - 2.
log⁡(0−2)0−2\frac{\log{\left(\frac{0}{-2} \right)}}{0} - 2
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
(x−2)(−x(x−2)2+1x−2)x2−log⁡(xx−2)x2=0\frac{\left(x - 2\right) \left(- \frac{x}{\left(x - 2\right)^{2}} + \frac{1}{x - 2}\right)}{x^{2}} - \frac{\log{\left(\frac{x}{x - 2} \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−17093.4045100989x_{1} = -17093.4045100989
x2=22736.0503030179x_{2} = 22736.0503030179
x3=27822.319868693x_{3} = 27822.319868693
x4=26126.9249240027x_{4} = 26126.9249240027
x5=−19636.785209234x_{5} = -19636.785209234
x6=−40828.8963066525x_{6} = -40828.8963066525
x7=−23875.4880583312x_{7} = -23875.4880583312
x8=−24723.2009749401x_{8} = -24723.2009749401
x9=−30657.0250198402x_{9} = -30657.0250198402
x10=−32352.3637029924x_{10} = -32352.3637029924
x11=−41676.5386909395x_{11} = -41676.5386909395
x12=28670.0089688525x_{12} = 28670.0089688525
x13=−29809.3501890986x_{13} = -29809.3501890986
x14=33756.053127688x_{14} = 33756.053127688
x15=30365.3727892908x_{15} = 30365.3727892908
x16=31213.0482994943x_{16} = 31213.0482994943
x17=−34895.3482422306x_{17} = -34895.3482422306
x18=−31504.6960937631x_{18} = -31504.6960937631
x19=18497.2165176913x_{19} = 18497.2165176913
x20=25279.2179406976x_{20} = 25279.2179406976
x21=−35743.0043749846x_{21} = -35743.0043749846
x22=−39133.6066245065x_{22} = -39133.6066245065
x23=23583.7814819518x_{23} = 23583.7814819518
x24=37146.6849254491x_{24} = 37146.6849254491
x25=−20484.5479648147x_{25} = -20484.5479648147
x26=−28961.6712756215x_{26} = -28961.6712756215
x27=32908.3881861732x_{27} = 32908.3881861732
x28=39689.6344692787x_{28} = 39689.6344692787
x29=−17941.2156237514x_{29} = -17941.2156237514
x30=−16245.5722638748x_{30} = -16245.5722638748
x31=−23027.7672609082x_{31} = -23027.7672609082
x32=16801.5657645544x_{32} = 16801.5657645544
x33=−21332.2983264744x_{33} = -21332.2983264744
x34=38841.986630203x_{34} = 38841.986630203
x35=−33200.0281101306x_{35} = -33200.0281101306
x36=−42524.1795668896x_{36} = -42524.1795668896
x37=−38285.9591110553x_{37} = -38285.9591110553
x38=−34047.6895518949x_{38} = -34047.6895518949
x39=20192.791690122x_{39} = 20192.791690122
x40=19345.0122864752x_{40} = 19345.0122864752
x41=−25570.9067838829x_{41} = -25570.9067838829
x42=−36590.6581262118x_{42} = -36590.6581262118
x43=−15397.7154722609x_{43} = -15397.7154722609
x44=35451.3742149795x_{44} = 35451.3742149795
x45=24431.5037239393x_{45} = 24431.5037239393
x46=15953.7043452856x_{46} = 15953.7043452856
x47=−37438.3096561705x_{47} = -37438.3096561705
x48=−39981.2523189299x_{48} = -39981.2523189299
x49=21888.3091318714x_{49} = 21888.3091318714
x50=40537.2804689772x_{50} = 40537.2804689772
x51=29517.6931364115x_{51} = 29517.6931364115
x52=17649.4019777359x_{52} = 17649.4019777359
x53=−18789.0084129199x_{53} = -18789.0084129199
x54=37994.3368274949x_{54} = 37994.3368274949
x55=32060.7199992469x_{55} = 32060.7199992469
x56=15105.8133749906x_{56} = 15105.8133749906
x57=21040.5567402861x_{57} = 21040.5567402861
x58=34603.7150647342x_{58} = 34603.7150647342
x59=−28113.9879146686x_{59} = -28113.9879146686
x60=−27266.2996967173x_{60} = -27266.2996967173
x61=−22180.037693162x_{61} = -22180.037693162
x62=36299.0307755608x_{62} = 36299.0307755608
x63=42232.567396967x_{63} = 42232.567396967
x64=41384.9247432867x_{64} = 41384.9247432867
x65=26974.6253647857x_{65} = 26974.6253647857
x66=−26418.6061603723x_{66} = -26418.6061603723
Signos de extremos en los puntos:
(-17093.404510098877, -1.99999999315541)

(22736.050303017877, -1.99999999613082)

(27822.31986869305, -1.9999999974162)

(26126.92492400273, -1.99999999706998)

(-19636.785209233993, -1.99999999481359)

(-40828.896306652525, -1.99999999880027)

(-23875.488058331208, -1.99999999649161)

(-24723.20097494007, -1.99999999672808)

(-30657.025019840166, -1.99999999787208)

(-32352.363702992432, -1.99999999808925)

(-41676.538690939495, -1.99999999884857)

(28670.00896885247, -1.99999999756673)

(-29809.350189098583, -1.99999999774934)

(33756.05312768803, -1.99999999824475)

(30365.372789290846, -1.99999999783086)

(31213.04829949433, -1.99999999794708)

(-34895.3482422306, -1.99999999835759)

(-31504.69609376308, -1.99999999798504)

(18497.216517691293, -1.99999999415424)

(25279.217940697632, -1.99999999687018)

(-35743.0043749846, -1.99999999843456)

(-39133.606624506516, -1.99999999869407)

(23583.781481951846, -1.99999999640398)

(37146.68492544914, -1.99999999855056)

(-20484.54796481471, -1.99999999523398)

(-28961.671275621513, -1.99999999761566)

(32908.388186173244, -1.99999999815316)

(39689.63446927872, -1.99999999873034)

(-17941.215623751443, -1.99999999378699)

(-16245.57226387476, -1.99999999242237)

(-23027.767260908204, -1.99999999622856)

(16801.5657645544, -1.99999999291473)

(-21332.298326474356, -1.99999999560525)

(38841.986630203035, -1.99999999867432)

(-33200.028110130595, -1.99999999818557)

(-42524.17956688957, -1.99999999889402)

(-38285.95911105529, -1.99999999863561)

(-34047.68955189494, -1.99999999827479)

(20192.791690121972, -1.99999999509478)

(19345.01228647517, -1.99999999465541)

(-25570.906783882878, -1.99999999694141)

(-36590.65812621183, -1.99999999850625)

(-15397.71547226093, -1.99999999156492)

(35451.37421497947, -1.99999999840861)

(24431.503723939313, -1.99999999664921)

(15953.704345285605, -1.9999999921416)

(-37438.30965617054, -1.99999999857312)

(-39981.25231892988, -1.99999999874886)

(21888.309131871432, -1.9999999958253)

(40537.28046897723, -1.99999999878289)

(29517.693136411537, -1.99999999770449)

(17649.40197773589, -1.99999999357912)

(-18789.008412919877, -1.99999999433501)

(37994.336827494895, -1.99999999861451)

(32060.719999246903, -1.99999999805421)

(15105.813374990552, -1.99999999123462)

(21040.556740286076, -1.9999999954821)

(34603.71506473418, -1.99999999832969)

(-28113.987914668593, -1.99999999746971)

(-27266.29969671734, -1.99999999730994)

(-22180.03769316204, -1.99999999593476)

(36299.03077556081, -1.99999999848207)

(42232.567396967, -1.99999999887864)

(41384.92474328668, -1.99999999883223)

(26974.62536478566, -1.99999999725125)

(-26418.606160372317, -1.99999999713454)


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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
No cambia el valor en todo el eje numérico
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
xx−2−1x−2+3(xx−2−1)x+2log⁡(xx−2)xx2=0\frac{\frac{\frac{x}{x - 2} - 1}{x - 2} + \frac{3 \left(\frac{x}{x - 2} - 1\right)}{x} + \frac{2 \log{\left(\frac{x}{x - 2} \right)}}{x}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−8104.97188497642x_{1} = -8104.97188497642
x2=−10285.8335605286x_{2} = -10285.8335605286
x3=2358.87276629222x_{3} = 2358.87276629222
x4=7812.5859751793x_{4} = 7812.5859751793
x5=−3088.16472429732x_{5} = -3088.16472429732
x6=−9195.41206291057x_{6} = -9195.41206291057
x7=9557.29722522818x_{7} = 9557.29722522818
x8=−9849.66681034432x_{8} = -9849.66681034432
x9=−9413.4976514751x_{9} = -9413.4976514751
x10=−5269.6584495685x_{10} = -5269.6584495685
x11=7158.30019830726x_{11} = 7158.30019830726
x12=9775.38234056315x_{12} = 9775.38234056315
x13=−2215.15126693443x_{13} = -2215.15126693443
x14=−7014.50444655045x_{14} = -7014.50444655045
x15=−6142.10127943092x_{15} = -6142.10127943092
x16=−7886.88093508863x_{16} = -7886.88093508863
x17=3668.31033365313x_{17} = 3668.31033365313
x18=−3960.86402875143x_{18} = -3960.86402875143
x19=3013.70585597089x_{19} = 3013.70585597089
x20=−8541.15064998913x_{20} = -8541.15064998913
x21=7376.39695876702x_{21} = 7376.39695876702
x22=8248.76978986642x_{22} = 8248.76978986642
x23=−5487.7738899372x_{23} = -5487.7738899372
x24=−10721.9981930631x_{24} = -10721.9981930631
x25=8684.94919622346x_{25} = 8684.94919622346
x26=8030.67847905738x_{26} = 8030.67847905738
x27=−7232.60069823897x_{27} = -7232.60069823897
x28=2140.50616315703x_{28} = 2140.50616315703
x29=−6360.20516267939x_{29} = -6360.20516267939
x30=−10503.9161251815x_{30} = -10503.9161251815
x31=10865.7988664952x_{31} = 10865.7988664952
x32=−9631.58255222162x_{32} = -9631.58255222162
x33=−4397.15291495801x_{33} = -4397.15291495801
x34=7594.49217399665x_{34} = 7594.49217399665
x35=−3524.53949687421x_{35} = -3524.53949687421
x36=−3306.35959425606x_{36} = -3306.35959425606
x37=8466.86000096627x_{37} = 8466.86000096627
x38=−5051.53918701948x_{38} = -5051.53918701948
x39=4540.93557365292x_{39} = 4540.93557365292
x40=6940.20174468926x_{40} = 6940.20174468926
x41=−5705.88593829684x_{41} = -5705.88593829684
x42=8903.03745118803x_{42} = 8903.03745118803
x43=−4179.01224745344x_{43} = -4179.01224745344
x44=3450.12595958138x_{44} = 3450.12595958138
x45=10211.5506438931x_{45} = 10211.5506438931
x46=−5923.99496259865x_{46} = -5923.99496259865
x47=−6578.30685049238x_{47} = -6578.30685049238
x48=−4833.41559616591x_{48} = -4833.41559616591
x49=1922.06842960452x_{49} = 1922.06842960452
x50=−10067.7504671628x_{50} = -10067.7504671628
x51=−1996.80304313164x_{51} = -1996.80304313164
x52=3886.48153939024x_{52} = 3886.48153939024
x53=−2651.71588722832x_{53} = -2651.71588722832
x54=−2433.45177650721x_{54} = -2433.45177650721
x55=6503.99906541776x_{55} = 6503.99906541776
x56=5631.56424007234x_{56} = 5631.56424007234
x57=4322.7926186106x_{57} = 4322.7926186106
x58=−6796.40655092572x_{58} = -6796.40655092572
x59=9993.46679928136x_{59} = 9993.46679928136
x60=10429.6339133166x_{60} = 10429.6339133166
x61=4977.20194889188x_{61} = 4977.20194889188
x62=2577.18707074862x_{62} = 2577.18707074862
x63=−8759.23861931871x_{63} = -8759.23861931871
x64=6067.78728248457x_{64} = 6067.78728248457
x65=5413.44762908865x_{65} = 5413.44762908865
x66=−7668.78882306102x_{66} = -7668.78882306102
x67=3231.92566125734x_{67} = 3231.92566125734
x68=6722.10143048014x_{68} = 6722.10143048014
x69=−8323.06176291277x_{69} = -8323.06176291277
x70=−3742.70697568559x_{70} = -3742.70697568559
x71=2795.46179898202x_{71} = 2795.46179898202
x72=−1778.39055536145x_{72} = -1778.39055536145
x73=9339.21140674983x_{73} = 9339.21140674983
x74=−2869.95159818425x_{74} = -2869.95159818425
x75=−7450.69544829572x_{75} = -7450.69544829572
x76=10647.7166432494x_{76} = 10647.7166432494
x77=−4615.28707739227x_{77} = -4615.28707739227
x78=9121.12483409941x_{78} = 9121.12483409941
x79=5195.32704331287x_{79} = 5195.32704331287
x80=5849.67732903244x_{80} = 5849.67732903244
x81=4104.64173142376x_{81} = 4104.64173142376
x82=−8977.32573699418x_{82} = -8977.32573699418
x83=6285.89443238248x_{83} = 6285.89443238248
x84=−10940.0797935925x_{84} = -10940.0797935925
x85=4759.07171188489x_{85} = 4759.07171188489
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0
x2=2x_{2} = 2

lim⁡x→0−(xx−2−1x−2+3(xx−2−1)x+2log⁡(xx−2)xx2)=∞\lim_{x \to 0^-}\left(\frac{\frac{\frac{x}{x - 2} - 1}{x - 2} + \frac{3 \left(\frac{x}{x - 2} - 1\right)}{x} + \frac{2 \log{\left(\frac{x}{x - 2} \right)}}{x}}{x^{2}}\right) = \infty
lim⁡x→0+(xx−2−1x−2+3(xx−2−1)x+2log⁡(xx−2)xx2)=−∞\lim_{x \to 0^+}\left(\frac{\frac{\frac{x}{x - 2} - 1}{x - 2} + \frac{3 \left(\frac{x}{x - 2} - 1\right)}{x} + \frac{2 \log{\left(\frac{x}{x - 2} \right)}}{x}}{x^{2}}\right) = -\infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión
lim⁡x→2−(xx−2−1x−2+3(xx−2−1)x+2log⁡(xx−2)xx2)=∞\lim_{x \to 2^-}\left(\frac{\frac{\frac{x}{x - 2} - 1}{x - 2} + \frac{3 \left(\frac{x}{x - 2} - 1\right)}{x} + \frac{2 \log{\left(\frac{x}{x - 2} \right)}}{x}}{x^{2}}\right) = \infty
lim⁡x→2+(xx−2−1x−2+3(xx−2−1)x+2log⁡(xx−2)xx2)=∞\lim_{x \to 2^+}\left(\frac{\frac{\frac{x}{x - 2} - 1}{x - 2} + \frac{3 \left(\frac{x}{x - 2} - 1\right)}{x} + \frac{2 \log{\left(\frac{x}{x - 2} \right)}}{x}}{x^{2}}\right) = \infty
- los límites son iguales, es decir omitimos el punto correspondiente

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:
No tiene corvaduras en todo el eje numérico
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=2x_{2} = 2
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−2+log⁡(xx−2)x)=−2\lim_{x \to -\infty}\left(-2 + \frac{\log{\left(\frac{x}{x - 2} \right)}}{x}\right) = -2
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=−2y = -2
lim⁡x→∞(−2+log⁡(xx−2)x)=−2\lim_{x \to \infty}\left(-2 + \frac{\log{\left(\frac{x}{x - 2} \right)}}{x}\right) = -2
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=−2y = -2
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función log(x/(x - 2))/x - 2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−2+log⁡(xx−2)xx)=0\lim_{x \to -\infty}\left(\frac{-2 + \frac{\log{\left(\frac{x}{x - 2} \right)}}{x}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−2+log⁡(xx−2)xx)=0\lim_{x \to \infty}\left(\frac{-2 + \frac{\log{\left(\frac{x}{x - 2} \right)}}{x}}{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:
−2+log⁡(xx−2)x=−2−log⁡(−x−x−2)x-2 + \frac{\log{\left(\frac{x}{x - 2} \right)}}{x} = -2 - \frac{\log{\left(- \frac{x}{- x - 2} \right)}}{x}
- No
−2+log⁡(xx−2)x=2+log⁡(−x−x−2)x-2 + \frac{\log{\left(\frac{x}{x - 2} \right)}}{x} = 2 + \frac{\log{\left(- \frac{x}{- x - 2} \right)}}{x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор