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Trazado el gráfico de la función/ arcctgx/(x-3)

Gráfico de la función y = arcctgx/(x-3)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       acot(x)
f(x) = -------
        x - 3
f(x)=acot⁡(x)x−3f{\left(x \right)} = \frac{\operatorname{acot}{\left(x \right)}}{x - 3} 
f = acot(x)/(x - 3)
Gráfico de la función
02468-8-6-4-2-1010-2020
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=3x_{1} = 3
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:
acot⁡(x)x−3=0\frac{\operatorname{acot}{\left(x \right)}}{x - 3} = 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 acot(x)/(x - 3).
acot⁡(0)−3\frac{\operatorname{acot}{\left(0 \right)}}{-3}
Resultado:
f(0)=−π6f{\left(0 \right)} = - \frac{\pi}{6}
Punto:
(0, -pi/6)
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
−1(x−3)(x2+1)−acot⁡(x)(x−3)2=0- \frac{1}{\left(x - 3\right) \left(x^{2} + 1\right)} - \frac{\operatorname{acot}{\left(x \right)}}{\left(x - 3\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−35313.0668342955x_{1} = -35313.0668342955
x2=15518.8336328721x_{2} = 15518.8336328721
x3=−38704.1170790913x_{3} = -38704.1170790913
x4=41805.3607928303x_{4} = 41805.3607928303
x5=34175.5123312146x_{5} = 34175.5123312146
x6=−42942.7903957194x_{6} = -42942.7903957194
x7=−16657.783408065x_{7} = -16657.783408065
x8=−27682.6438443875x_{8} = -27682.6438443875
x9=29936.3941106863x_{9} = 29936.3941106863
x10=26544.8446743911x_{10} = 26544.8446743911
x11=−14960.8860813988x_{11} = -14960.8860813988
x12=−25986.8382213314x_{12} = -25986.8382213314
x13=−33617.4943084304x_{13} = -33617.4943084304
x14=−34465.2850023106x_{14} = -34465.2850023106
x15=−20050.7379682616x_{15} = -20050.7379682616
x16=35871.0864641394x_{16} = 35871.0864641394
x17=23152.9596471974x_{17} = 23152.9596471974
x18=−21746.9237165814x_{18} = -21746.9237165814
x19=40957.6259784778x_{19} = 40957.6259784778
x20=12972.6101323084x_{20} = 12972.6101323084
x21=40109.8856055808x_{21} = 40109.8856055808
x22=−39551.8628248821x_{22} = -39551.8628248821
x23=27392.7585806149x_{23} = 27392.7585806149
x24=−29378.3813978389x_{24} = -29378.3813978389
x25=−13263.5701308315x_{25} = -13263.5701308315
x26=−18354.3754214744x_{26} = -18354.3754214744
x27=−12414.7034718108x_{27} = -12414.7034718108
x28=−42095.0661510489x_{28} = -42095.0661510489
x29=−23442.9703742917x_{29} = -23442.9703742917
x30=−15809.3788845165x_{30} = -15809.3788845165
x31=−37856.3650546609x_{31} = -37856.3650546609
x32=−31074.0618885764x_{32} = -31074.0618885764
x33=36718.860771076x_{33} = 36718.860771076
x34=−37008.6063243624x_{34} = -37008.6063243624
x35=20608.7247596784x_{35} = 20608.7247596784
x36=−22594.9625089587x_{36} = -22594.9625089587
x37=18912.3527533172x_{37} = 18912.3527533172
x38=21456.8409729567x_{38} = 21456.8409729567
x39=−41247.3370154355x_{39} = -41247.3370154355
x40=28240.6537163389x_{40} = 28240.6537163389
x41=30784.242139886x_{41} = 30784.242139886
x42=−24290.9504985193x_{42} = -24290.9504985193
x43=−31921.8835434698x_{43} = -31921.8835434698
x44=31632.0769879872x_{44} = 31632.0769879872
x45=37566.6273450369x_{45} = 37566.6273450369
x46=22304.9178115051x_{46} = 22304.9178115051
x47=−17506.1121672217x_{47} = -17506.1121672217
x48=19760.5640406041x_{48} = 19760.5640406041
x49=−30226.22818334x_{49} = -30226.22818334
x50=14670.2260130038x_{50} = 14670.2260130038
x51=24000.9702280612x_{51} = 24000.9702280612
x52=−20898.8503044486x_{52} = -20898.8503044486
x53=32479.8996955549x_{53} = 32479.8996955549
x54=−14112.2895735892x_{54} = -14112.2895735892
x55=35023.3038583412x_{55} = 35023.3038583412
x56=−19202.5816602195x_{56} = -19202.5816602195
x57=25696.9101211369x_{57} = 25696.9101211369
x58=24848.9527856772x_{58} = 24848.9527856772
x59=39262.1393115611x_{59} = 39262.1393115611
x60=18064.0836828544x_{60} = 18064.0836828544
x61=16367.3357554312x_{61} = 16367.3357554312
x62=33327.7111964611x_{62} = 33327.7111964611
x63=−28530.5203814017x_{63} = -28530.5203814017
x64=29088.5317374502x_{64} = 29088.5317374502
x65=17215.7481743741x_{65} = 17215.7481743741
x66=−25138.9056447972x_{66} = -25138.9056447972
x67=−32769.6940726206x_{67} = -32769.6940726206
x68=12123.544841056x_{68} = 12123.544841056
x69=38414.3867016121x_{69} = 38414.3867016121
x70=1.26241821842168x_{70} = 1.26241821842168
x71=−36160.8404213184x_{71} = -36160.8404213184
x72=−26834.7503365653x_{72} = -26834.7503365653
x73=13821.4931681849x_{73} = 13821.4931681849
x74=−40399.6026837339x_{74} = -40399.6026837339
Signos de extremos en los puntos:
(-35313.06683429549, 8.01848338181294e-10)

(15518.833632872089, 4.15303707114861e-9)

(-38704.11707909125, 6.67501135843943e-10)

(41805.36079283031, 5.72225509807911e-10)

(34175.512331214624, 8.56264733025152e-10)

(-42942.790395719414, 5.42236987377855e-10)

(-16657.78340806499, 3.60319171329807e-9)

(-27682.64384438751, 1.30478154940659e-9)

(29936.394110686302, 1.11594951668977e-9)

(26544.844674391126, 1.41934740490877e-9)

(-14960.886081398789, 4.46681833383236e-9)

(-25986.8382213314, 1.48061785005386e-9)

(-33617.49430843041, 8.84770360411383e-10)

(-34465.285002310586, 8.41779704251999e-10)

(-20050.737968261594, 2.48699150728489e-9)

(35871.08646413939, 7.77225893994018e-10)

(23152.959647197393, 1.86570616473751e-9)

(-21746.923716581445, 2.11419202623933e-9)

(40957.625978477816, 5.96159213298478e-10)

(12972.610132308384, 5.94354713795122e-9)

(40109.885605580756, 6.21626667528365e-10)

(-39551.86282488213, 6.39194713316016e-10)

(27392.758580614878, 1.33283388544353e-9)

(-29378.38139783892, 1.15851036688784e-9)

(-13263.570130831458, 5.68304250099865e-9)

(-18354.375421474422, 2.96790366013197e-9)

(-12414.703471810783, 6.48667834896136e-9)

(-42095.066151048944, 5.64295591459207e-10)

(-23442.970374291708, 1.81936212368839e-9)

(-15809.37888451648, 4.00025778966488e-9)

(-37856.365054660906, 6.97730588555291e-10)

(-31074.061888576398, 1.03552841581763e-9)

(36718.86077107599, 7.41749200934459e-10)

(-37008.606324362416, 7.30061312882698e-10)

(20608.7247596784, 2.35483773297575e-9)

(-22594.962508958713, 1.95847977440683e-9)

(18912.352753317246, 2.79626145482265e-9)

(21456.840972956685, 2.17234680811585e-9)

(-41247.33701543549, 5.87728273655331e-10)

(28240.65371633886, 1.25399742921036e-9)

(30784.242139885995, 1.05532297275285e-9)

(-24290.95049851926, 1.69456155240711e-9)

(-31921.883543469776, 9.81255647453413e-10)

(31632.076987987162, 9.99506844645914e-10)

(37566.62734503691, 7.0864751696245e-10)

(22304.91781150513, 2.01028282114719e-9)

(-17506.11216722175, 3.26246730310394e-9)

(19760.564040604117, 2.56134019256726e-9)

(-30226.22818334005, 1.09443250786901e-9)

(14670.226013003838, 4.64745518220953e-9)

(24000.970228061182, 1.73618776467237e-9)

(-20898.85030444859, 2.2892481474175e-9)

(32479.89969555486, 9.48005280641194e-10)

(-14112.289573589214, 5.02010415642315e-9)

(35023.303858341176, 8.15310391916053e-10)

(-19202.581660219523, 2.7115206374669e-9)

(25696.91012113692, 1.51456840774484e-9)

(24848.95278567718, 1.61970623227391e-9)

(39262.1393115611, 6.48761793868724e-10)

(18064.083682854387, 3.06506901790584e-9)

(16367.335755431184, 3.73356422411508e-9)

(33327.71119646112, 9.00384720241422e-10)

(-28530.520381401748, 1.22838625950304e-9)

(29088.531737450157, 1.18195567830066e-9)

(17215.748174374123, 3.37461228493546e-9)

(-25138.905644797193, 1.58217835799154e-9)

(-32769.694072620616, 9.31141040812685e-10)

(12123.544841055986, 6.80531501472155e-9)

(38414.38670161214, 6.77713458004886e-10)

(1.262418218421678, -0.385549636898932)

(-36160.84042131838, 7.64692692897442e-10)

(-26834.750336565336, 1.38853340464679e-9)

(13821.493168184918, 5.23581565927066e-9)

(-40399.602683733865, 6.12651587531458e-10)


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
Puntos máximos de la función:
x74=1.26241821842168x_{74} = 1.26241821842168
Decrece en los intervalos
(−∞,1.26241821842168]\left(-\infty, 1.26241821842168\right]
Crece en los intervalos
[1.26241821842168,∞)\left[1.26241821842168, \infty\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
2(x(x2+1)2+1(x−3)(x2+1)+acot⁡(x)(x−3)2)x−3=0\frac{2 \left(\frac{x}{\left(x^{2} + 1\right)^{2}} + \frac{1}{\left(x - 3\right) \left(x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=5739.03095087926x_{1} = 5739.03095087926
x2=1805.43955191568x_{2} = 1805.43955191568
x3=−4502.04100865049x_{3} = -4502.04100865049
x4=8138.9045667575x_{4} = 8138.9045667575
x5=−9956.2508846458x_{5} = -9956.2508846458
x6=0.17935688002721x_{6} = 0.17935688002721
x7=−6466.02475936076x_{7} = -6466.02475936076
x8=9665.80990054675x_{8} = 9665.80990054675
x9=−5156.81392122052x_{9} = -5156.81392122052
x10=6393.62489744838x_{10} = 6393.62489744838
x11=−3847.08306824419x_{11} = -3847.08306824419
x12=6175.4360524004x_{12} = 6175.4360524004
x13=2024.84149664708x_{13} = 2024.84149664708
x14=−2754.73556955859x_{14} = -2754.73556955859
x15=9883.92871280749x_{15} = 9883.92871280749
x16=−8647.51899910909x_{16} = -8647.51899910909
x17=−9738.13463310465x_{17} = -9738.13463310465
x18=9447.68882539942x_{18} = 9447.68882539942
x19=−6902.36298248242x_{19} = -6902.36298248242
x20=−1660.00043820182x_{20} = -1660.00043820182
x21=5520.81250035928x_{21} = 5520.81250035928
x22=−2317.3026441142x_{22} = -2317.3026441142
x23=4647.7973577865x_{23} = 4647.7973577865
x24=2681.72337804868x_{24} = 2681.72337804868
x25=5957.23840880638x_{25} = 5957.23840880638
x26=−3628.70599204772x_{26} = -3628.70599204772
x27=−2536.07142944323x_{27} = -2536.07142944323
x28=6829.97959979156x_{28} = 6829.97959979156
x29=−2973.31679944955x_{29} = -2973.31679944955
x30=7266.30829889645x_{30} = 7266.30829889645
x31=10320.1601270208x_{31} = 10320.1601270208
x32=−10828.6975404165x_{32} = -10828.6975404165
x33=−2098.39928129367x_{33} = -2098.39928129367
x34=−3191.8312612551x_{34} = -3191.8312612551
x35=−7556.82816022102x_{35} = -7556.82816022102
x36=7484.46433232432x_{36} = 7484.46433232432
x37=−4720.31551331578x_{37} = -4720.31551331578
x38=3774.43639658305x_{38} = 3774.43639658305
x39=5084.33688450853x_{39} = 5084.33688450853
x40=2462.92383062021x_{40} = 2462.92383062021
x41=8575.17841051419x_{41} = 8575.17841051419
x42=−8865.64726589311x_{42} = -8865.64726589311
x43=−7993.11546158443x_{43} = -7993.11546158443
x44=8357.04324622462x_{44} = 8357.04324622462
x45=−6247.8454847632x_{45} = -6247.8454847632
x46=6611.80582431964x_{46} = 6611.80582431964
x47=3556.01176544918x_{47} = 3556.01176544918
x48=7048.14689478809x_{48} = 7048.14689478809
x49=−6029.65847446908x_{49} = -6029.65847446908
x50=−8429.38776883049x_{50} = -8429.38776883049
x51=9229.56532555109x_{51} = 9229.56532555109
x52=1585.66222122544x_{52} = 1585.66222122544
x53=−7774.97384050646x_{53} = -7774.97384050646
x54=2243.97915038475x_{54} = 2243.97915038475
x55=3992.82165401903x_{55} = 3992.82165401903
x56=−3410.29118322484x_{56} = -3410.29118322484
x57=2900.4110627802x_{57} = 2900.4110627802
x58=−10174.3651844545x_{58} = -10174.3651844545
x59=−1879.31904952349x_{59} = -1879.31904952349
x60=−9301.8957324112x_{60} = -9301.8957324112
x61=4866.07619926592x_{61} = 4866.07619926592
x62=3337.53990547011x_{62} = 3337.53990547011
x63=9011.43922338841x_{63} = 9011.43922338841
x64=3119.01072872497x_{64} = 3119.01072872497
x65=8793.31032352348x_{65} = 8793.31032352348
x66=−8211.2533422616x_{66} = -8211.2533422616
x67=−9083.7727799182x_{67} = -9083.7727799182
x68=−4938.5724258037x_{68} = -4938.5724258037
x69=−4065.42827072866x_{69} = -4065.42827072866
x70=−7338.6780646605x_{70} = -7338.6780646605
x71=10756.3841066263x_{71} = 10756.3841066263
x72=4211.17374472313x_{72} = 4211.17374472313
x73=−9520.01629729564x_{73} = -9520.01629729564
x74=7702.6154558089x_{74} = 7702.6154558089
x75=−6684.19704156285x_{75} = -6684.19704156285
x76=7920.76207887546x_{76} = 7920.76207887546
x77=−7120.52315489378x_{77} = -7120.52315489378
x78=−5375.04183086147x_{78} = -5375.04183086147
x79=−10392.4776540664x_{79} = -10392.4776540664
x80=−10610.5884051352x_{80} = -10610.5884051352
x81=5302.58168270876x_{81} = 5302.58168270876
x82=10538.2729880046x_{82} = 10538.2729880046
x83=−4283.7463084719x_{83} = -4283.7463084719
x84=4429.49763773049x_{84} = 4429.49763773049
x85=−5593.25770745173x_{85} = -5593.25770745173
x86=10102.0454099151x_{86} = 10102.0454099151
x87=−11046.8051546367x_{87} = -11046.8051546367
x88=−5811.46287609826x_{88} = -5811.46287609826
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=3x_{1} = 3

lim⁡x→3−(2(x(x2+1)2+1(x−3)(x2+1)+acot⁡(x)(x−3)2)x−3)=−∞\lim_{x \to 3^-}\left(\frac{2 \left(\frac{x}{\left(x^{2} + 1\right)^{2}} + \frac{1}{\left(x - 3\right) \left(x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3}\right) = -\infty
lim⁡x→3+(2(x(x2+1)2+1(x−3)(x2+1)+acot⁡(x)(x−3)2)x−3)=∞\lim_{x \to 3^+}\left(\frac{2 \left(\frac{x}{\left(x^{2} + 1\right)^{2}} + \frac{1}{\left(x - 3\right) \left(x^{2} + 1\right)} + \frac{\operatorname{acot}{\left(x \right)}}{\left(x - 3\right)^{2}}\right)}{x - 3}\right) = \infty
- los límites no son iguales, signo
x1=3x_{1} = 3
- es el punto de flexión

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
(−∞,0.17935688002721]\left(-\infty, 0.17935688002721\right]
Convexa en los intervalos
[0.17935688002721,∞)\left[0.17935688002721, \infty\right)
Asíntotas verticales
Hay:
x1=3x_{1} = 3
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(acot⁡(x)x−3)=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x - 3}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(acot⁡(x)x−3)=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x - 3}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función acot(x)/(x - 3), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(acot⁡(x)x(x−3))=0\lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x \left(x - 3\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(acot⁡(x)x(x−3))=0\lim_{x \to \infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{x \left(x - 3\right)}\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:
acot⁡(x)x−3=−acot⁡(x)−x−3\frac{\operatorname{acot}{\left(x \right)}}{x - 3} = - \frac{\operatorname{acot}{\left(x \right)}}{- x - 3}
- No
acot⁡(x)x−3=acot⁡(x)−x−3\frac{\operatorname{acot}{\left(x \right)}}{x - 3} = \frac{\operatorname{acot}{\left(x \right)}}{- x - 3}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор