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Trazado el gráfico de la función/ y=-2x*arcctg(x)

Gráfico de la función y = y=-2x*arcctg(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = -2*x*acot(x)
f(x)=−2xacot⁡(x)f{\left(x \right)} = - 2 x \operatorname{acot}{\left(x \right)} 
f = (-2*x)*acot(x)
Gráfico de la función
02468-8-6-4-2-10102-4
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:
−2xacot⁡(x)=0- 2 x \operatorname{acot}{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=0x_{1} = 0
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 (-2*x)*acot(x).
−0acot⁡(0)- 0 \operatorname{acot}{\left(0 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
2xx2+1−2acot⁡(x)=0\frac{2 x}{x^{2} + 1} - 2 \operatorname{acot}{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=18717.3825921039x_{1} = 18717.3825921039
x2=−41469.0097027937x_{2} = -41469.0097027937
x3=−27908.3228655288x_{3} = -27908.3228655288
x4=−21128.451639271x_{4} = -21128.451639271
x5=39057.5547154358x_{5} = 39057.5547154358
x6=28887.0585662606x_{6} = 28887.0585662606
x7=42447.8004663952x_{7} = 42447.8004663952
x8=29734.5801929714x_{8} = 29734.5801929714
x9=41600.2364126928x_{9} = 41600.2364126928
x10=23802.0523520614x_{10} = 23802.0523520614
x11=−28755.8386877142x_{11} = -28755.8386877142
x12=−40621.447587593x_{12} = -40621.447587593
x13=−19433.5887869564x_{13} = -19433.5887869564
x14=10245.0412006996x_{14} = 10245.0412006996
x15=22107.1135839482x_{15} = 22107.1135839482
x16=−29603.3595724168x_{16} = -29603.3595724168
x17=39905.113426091x_{17} = 39905.113426091
x18=−10960.8778764425x_{18} = -10960.8778764425
x19=−14349.4901108771x_{19} = -14349.4901108771
x20=19564.7930550097x_{20} = 19564.7930550097
x21=−32993.4859340089x_{21} = -32993.4859340089
x22=21259.6603274632x_{22} = 21259.6603274632
x23=−25365.810848258x_{23} = -25365.810848258
x24=−30450.8850969459x_{24} = -30450.8850969459
x25=36514.8912641685x_{25} = 36514.8912641685
x26=−11807.9338157736x_{26} = -11807.9338157736
x27=−33841.0266214391x_{27} = -33841.0266214391
x28=−16044.0821523617x_{28} = -16044.0821523617
x29=35667.3415095204x_{29} = 35667.3415095204
x30=−31298.4148843908x_{30} = -31298.4148843908
x31=−42316.5735052903x_{31} = -42316.5735052903
x32=17869.99050832x_{32} = 17869.99050832
x33=17022.6195482377x_{33} = 17022.6195482377
x34=−37231.2183043527x_{34} = -37231.2183043527
x35=33124.7089739066x_{35} = 33124.7089739066
x36=−12655.0631680385x_{36} = -12655.0631680385
x37=31429.6368125843x_{37} = 31429.6368125843
x38=27192.0307619732x_{38} = 27192.0307619732
x39=26344.5255783821x_{39} = 26344.5255783821
x40=−26213.3083727677x_{40} = -26213.3083727677
x41=12786.2285767811x_{41} = 12786.2285767811
x42=20412.2196081348x_{42} = 20412.2196081348
x43=−32145.9485975863x_{43} = -32145.9485975863
x44=−18586.1809993501x_{44} = -18586.1809993501
x45=28039.5419336107x_{45} = 28039.5419336107
x46=11939.0892257274x_{46} = 11939.0892257274
x47=22954.5779890773x_{47} = 22954.5779890773
x48=38209.9980244722x_{48} = 38209.9980244722
x49=16175.2730314267x_{49} = 16175.2730314267
x50=37362.4434906472x_{50} = 37362.4434906472
x51=33972.2501557217x_{51} = 33972.2501557217
x52=−35536.1170902848x_{52} = -35536.1170902848
x53=13633.4257117535x_{53} = 13633.4257117535
x54=−13502.2521178288x_{54} = -13502.2521178288
x55=−16891.4245591221x_{55} = -16891.4245591221
x56=−21975.9030580391x_{56} = -21975.9030580391
x57=30582.10639855x_{57} = 30582.10639855
x58=−36383.6664480288x_{58} = -36383.6664480288
x59=32277.1711035644x_{59} = 32277.1711035644
x60=−23670.8387235676x_{60} = -23670.8387235676
x61=−24518.3207010738x_{61} = -24518.3207010738
x62=40752.6740304247x_{62} = 40752.6740304247
x63=−27060.8125814987x_{63} = -27060.8125814987
x64=−17738.7919818582x_{64} = -17738.7919818582
x65=34819.7944072015x_{65} = 34819.7944072015
x66=25497.0269799384x_{66} = 25497.0269799384
x67=15327.9550113123x_{67} = 15327.9550113123
x68=−38926.3288599295x_{68} = -38926.3288599295
x69=−39773.8872675524x_{69} = -39773.8872675524
x70=−22823.365825763x_{70} = -22823.365825763
x71=−34688.5704142599x_{71} = -34688.5704142599
x72=−15196.7689456217x_{72} = -15196.7689456217
x73=−38078.7724924216x_{73} = -38078.7724924216
x74=24649.5356458528x_{74} = 24649.5356458528
x75=11092.020896022x_{75} = 11092.020896022
x76=−20281.012992025x_{76} = -20281.012992025
x77=14480.6704897809x_{77} = 14480.6704897809
Signos de extremos en los puntos:
(18717.38259210387, -1.99999999809709)

(-41469.00970279369, -1.99999999961233)

(-27908.322865528775, -1.99999999914406)

(-21128.451639271025, -1.99999999850661)

(39057.55471543576, -1.99999999956298)

(28887.058566260628, -1.99999999920108)

(42447.8004663952, -1.99999999963)

(29734.580192971367, -1.99999999924598)

(41600.23641269277, -1.99999999961477)

(23802.052352061375, -1.99999999882326)

(-28755.838687714244, -1.99999999919377)

(-40621.447587593044, -1.99999999959598)

(-19433.58878695641, -1.99999999823476)

(10245.041200699563, -1.99999999364843)

(22107.113583948245, -1.9999999986359)

(-29603.35957241684, -1.99999999923928)

(39905.113426091, -1.99999999958135)

(-10960.87787644252, -1.99999999445096)

(-14349.490110877094, -1.99999999676231)

(19564.79305500968, -1.99999999825836)

(-32993.48593400886, -1.99999999938758)

(21259.66032746322, -1.99999999852499)

(-25365.810848258036, -1.99999999896388)

(-30450.88509694595, -1.99999999928103)

(36514.89126416845, -1.9999999995)

(-11807.933815773578, -1.99999999521854)

(-33841.02662143908, -1.99999999941787)

(-16044.082152361707, -1.99999999741012)

(35667.3415095204, -1.99999999947596)

(-31298.41488439077, -1.99999999931944)

(-42316.573505290275, -1.9999999996277)

(17869.990508320032, -1.99999999791234)

(17022.619548237748, -1.99999999769932)

(-37231.21830435275, -1.99999999951906)

(33124.7089739066, -1.99999999939242)

(-12655.063168038458, -1.99999999583725)

(31429.636812584253, -1.99999999932511)

(27192.03076197315, -1.99999999909838)

(26344.525578382065, -1.99999999903943)

(-26213.308372767653, -1.99999999902979)

(12786.228576781117, -1.99999999592222)

(20412.219608134823, -1.99999999839997)

(-32145.948597586266, -1.99999999935486)

(-18586.180999350086, -1.99999999807013)

(28039.541933610733, -1.99999999915206)

(11939.089225727435, -1.99999999532301)

(22954.57798907731, -1.99999999873477)

(38209.99802447215, -1.99999999954338)

(16175.273031426721, -1.99999999745196)

(37362.44349064715, -1.99999999952243)

(33972.25015572168, -1.99999999942236)

(-35536.11709028476, -1.99999999947208)

(13633.425711753476, -1.99999999641327)

(-13502.25211782884, -1.99999999634324)

(-16891.424559122082, -1.99999999766344)

(-21975.903058039054, -1.99999999861957)

(30582.106398549957, -1.99999999928719)

(-36383.66644802879, -1.99999999949639)

(32277.171103564393, -1.99999999936009)

(-23670.838723567595, -1.99999999881018)

(-24518.320701073833, -1.99999999889101)

(40752.67403042468, -1.99999999959858)

(-27060.81258149867, -1.99999999908961)

(-17738.791981858223, -1.99999999788134)

(34819.79440720148, -1.99999999945013)

(25497.02697993837, -1.99999999897451)

(15327.955011312279, -1.99999999716247)

(-38926.32885992955, -1.99999999956003)

(-39773.887267552374, -1.99999999957858)

(-22823.365825763038, -1.99999999872018)

(-34688.570414259906, -1.99999999944597)

(-15196.768945621745, -1.99999999711327)

(-38078.77249242158, -1.99999999954023)

(24649.53564585282, -1.99999999890279)

(11092.020896021957, -1.9999999945814)

(-20281.012992024975, -1.9999999983792)

(14480.670489780925, -1.9999999968207)


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=18717.3825921039x_{1} = 18717.3825921039
x2=39057.5547154358x_{2} = 39057.5547154358
x3=10245.0412006996x_{3} = 10245.0412006996
x4=22107.1135839482x_{4} = 22107.1135839482
x5=−16044.0821523617x_{5} = -16044.0821523617
x6=−37231.2183043527x_{6} = -37231.2183043527
x7=−32145.9485975863x_{7} = -32145.9485975863
x8=11939.0892257274x_{8} = 11939.0892257274
x9=38209.9980244722x_{9} = 38209.9980244722
x10=37362.4434906472x_{10} = 37362.4434906472
x11=−16891.4245591221x_{11} = -16891.4245591221
x12=34819.7944072015x_{12} = 34819.7944072015
Puntos máximos de la función:
x12=−40621.447587593x_{12} = -40621.447587593
x12=−14349.4901108771x_{12} = -14349.4901108771
x12=−32993.4859340089x_{12} = -32993.4859340089
x12=−42316.5735052903x_{12} = -42316.5735052903
x12=17022.6195482377x_{12} = 17022.6195482377
x12=31429.6368125843x_{12} = 31429.6368125843
x12=−26213.3083727677x_{12} = -26213.3083727677
x12=32277.1711035644x_{12} = 32277.1711035644
x12=40752.6740304247x_{12} = 40752.6740304247
x12=25497.0269799384x_{12} = 25497.0269799384
x12=−22823.365825763x_{12} = -22823.365825763
x12=−38078.7724924216x_{12} = -38078.7724924216
x12=24649.5356458528x_{12} = 24649.5356458528
x12=14480.6704897809x_{12} = 14480.6704897809
Decrece en los intervalos
[39057.5547154358,∞)\left[39057.5547154358, \infty\right)
Crece en los intervalos
(−∞,−37231.2183043527]\left(-\infty, -37231.2183043527\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
4(−x2x2+1+1)x2+1=0\frac{4 \left(- \frac{x^{2}}{x^{2} + 1} + 1\right)}{x^{2} + 1} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−2xacot⁡(x))=−2\lim_{x \to -\infty}\left(- 2 x \operatorname{acot}{\left(x \right)}\right) = -2
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=−2y = -2
lim⁡x→∞(−2xacot⁡(x))=−2\lim_{x \to \infty}\left(- 2 x \operatorname{acot}{\left(x \right)}\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 (-2*x)*acot(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−2acot⁡(x))=0\lim_{x \to -\infty}\left(- 2 \operatorname{acot}{\left(x \right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−2acot⁡(x))=0\lim_{x \to \infty}\left(- 2 \operatorname{acot}{\left(x \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:
−2xacot⁡(x)=−2xacot⁡(x)- 2 x \operatorname{acot}{\left(x \right)} = - 2 x \operatorname{acot}{\left(x \right)}
- No
−2xacot⁡(x)=2xacot⁡(x)- 2 x \operatorname{acot}{\left(x \right)} = 2 x \operatorname{acot}{\left(x \right)}
- No
es decir, función
no es
par ni impar
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