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y=x*arctgx
Trazado el gráfico de la función/ y=x*arctgx

Gráfico de la función y = y=x*arctgx

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = x*acot(x)
f(x)=xacot⁡(x)f{\left(x \right)} = x \operatorname{acot}{\left(x \right)} 
f = x*acot(x)
Gráfico de la función
02468-8-6-4-2-101002
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:
xacot⁡(x)=0x \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 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
−xx2+1+acot⁡(x)=0- \frac{x}{x^{2} + 1} + \operatorname{acot}{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=30582.10639855x_{1} = 30582.10639855
x2=8551.45303022722x_{2} = 8551.45303022722
x3=−8420.37201352502x_{3} = -8420.37201352502
x4=35667.3415095204x_{4} = 35667.3415095204
x5=−27060.8125814987x_{5} = -27060.8125814987
x6=31429.6368125843x_{6} = 31429.6368125843
x7=−22823.365825763x_{7} = -22823.365825763
x8=22954.5779890773x_{8} = 22954.5779890773
x9=15327.9550113123x_{9} = 15327.9550113123
x10=32277.1711035644x_{10} = 32277.1711035644
x11=10245.0412006996x_{11} = 10245.0412006996
x12=25497.0269799384x_{12} = 25497.0269799384
x13=−38078.7724924216x_{13} = -38078.7724924216
x14=−19433.5887869564x_{14} = -19433.5887869564
x15=−15196.7689456217x_{15} = -15196.7689456217
x16=13633.4257117535x_{16} = 13633.4257117535
x17=33972.2501557217x_{17} = 33972.2501557217
x18=−13502.2521178288x_{18} = -13502.2521178288
x19=−37231.2183043527x_{19} = -37231.2183043527
x20=−30450.8850969459x_{20} = -30450.8850969459
x21=−18586.1809993501x_{21} = -18586.1809993501
x22=−11807.9338157736x_{22} = -11807.9338157736
x23=41600.2364126928x_{23} = 41600.2364126928
x24=39057.5547154358x_{24} = 39057.5547154358
x25=36514.8912641685x_{25} = 36514.8912641685
x26=−39773.8872675524x_{26} = -39773.8872675524
x27=−32993.4859340089x_{27} = -32993.4859340089
x28=17869.99050832x_{28} = 17869.99050832
x29=−21975.9030580391x_{29} = -21975.9030580391
x30=−29603.3595724168x_{30} = -29603.3595724168
x31=26344.5255783821x_{31} = 26344.5255783821
x32=22107.1135839482x_{32} = 22107.1135839482
x33=−24518.3207010738x_{33} = -24518.3207010738
x34=27192.0307619732x_{34} = 27192.0307619732
x35=−42316.5735052903x_{35} = -42316.5735052903
x36=−21128.451639271x_{36} = -21128.451639271
x37=20412.2196081348x_{37} = 20412.2196081348
x38=−31298.4148843908x_{38} = -31298.4148843908
x39=17022.6195482377x_{39} = 17022.6195482377
x40=−35536.1170902848x_{40} = -35536.1170902848
x41=12786.2285767811x_{41} = 12786.2285767811
x42=33124.7089739066x_{42} = 33124.7089739066
x43=−25365.810848258x_{43} = -25365.810848258
x44=−26213.3083727677x_{44} = -26213.3083727677
x45=−40621.447587593x_{45} = -40621.447587593
x46=−10960.8778764425x_{46} = -10960.8778764425
x47=−27908.3228655288x_{47} = -27908.3228655288
x48=−41469.0097027937x_{48} = -41469.0097027937
x49=16175.2730314267x_{49} = 16175.2730314267
x50=−34688.5704142599x_{50} = -34688.5704142599
x51=42447.8004663952x_{51} = 42447.8004663952
x52=24649.5356458528x_{52} = 24649.5356458528
x53=11092.020896022x_{53} = 11092.020896022
x54=29734.5801929714x_{54} = 29734.5801929714
x55=−10113.913790961x_{55} = -10113.913790961
x56=38209.9980244722x_{56} = 38209.9980244722
x57=11939.0892257274x_{57} = 11939.0892257274
x58=−9267.06673722176x_{58} = -9267.06673722176
x59=−32145.9485975863x_{59} = -32145.9485975863
x60=14480.6704897809x_{60} = 14480.6704897809
x61=28039.5419336107x_{61} = 28039.5419336107
x62=28887.0585662606x_{62} = 28887.0585662606
x63=−20281.012992025x_{63} = -20281.012992025
x64=19564.7930550097x_{64} = 19564.7930550097
x65=37362.4434906472x_{65} = 37362.4434906472
x66=39905.113426091x_{66} = 39905.113426091
x67=−38926.3288599295x_{67} = -38926.3288599295
x68=−28755.8386877142x_{68} = -28755.8386877142
x69=23802.0523520614x_{69} = 23802.0523520614
x70=−33841.0266214391x_{70} = -33841.0266214391
x71=−23670.8387235676x_{71} = -23670.8387235676
x72=−12655.0631680385x_{72} = -12655.0631680385
x73=18717.3825921039x_{73} = 18717.3825921039
x74=−16891.4245591221x_{74} = -16891.4245591221
x75=21259.6603274632x_{75} = 21259.6603274632
x76=−14349.4901108771x_{76} = -14349.4901108771
x77=40752.6740304247x_{77} = 40752.6740304247
x78=−17738.7919818582x_{78} = -17738.7919818582
x79=−36383.6664480288x_{79} = -36383.6664480288
x80=9398.17409812154x_{80} = 9398.17409812154
x81=−16044.0821523617x_{81} = -16044.0821523617
x82=34819.7944072015x_{82} = 34819.7944072015
Signos de extremos en los puntos:
(30582.106398549957, 0.999999999643595)

(8551.453030227216, 0.999999995441742)

(-8420.37201352502, 0.999999995298719)

(35667.3415095204, 0.999999999737978)

(-27060.81258149867, 0.999999999544805)

(31429.636812584253, 0.999999999662557)

(-22823.365825763038, 0.999999999360089)

(22954.57798907731, 0.999999999367384)

(15327.955011312279, 0.999999998581235)

(32277.171103564393, 0.999999999680046)

(10245.041200699563, 0.999999996824213)

(25497.02697993837, 0.999999999487257)

(-38078.77249242158, 0.999999999770114)

(-19433.58878695641, 0.999999999117382)

(-15196.768945621745, 0.999999998556635)

(13633.425711753476, 0.999999998206635)

(33972.25015572168, 0.999999999711178)

(-13502.25211782884, 0.999999998171621)

(-37231.21830435275, 0.999999999759528)

(-30450.88509694595, 0.999999999640517)

(-18586.180999350086, 0.999999999035064)

(-11807.933815773578, 0.999999997609268)

(41600.23641269277, 0.999999999807386)

(39057.55471543576, 0.999999999781491)

(36514.89126416845, 0.999999999750001)

(-39773.887267552374, 0.999999999789291)

(-32993.48593400886, 0.999999999693788)

(17869.990508320032, 0.999999998956169)

(-21975.903058039054, 0.999999999309784)

(-29603.35957241684, 0.999999999619638)

(26344.525578382065, 0.999999999519716)

(22107.113583948245, 0.999999999317952)

(-24518.320701073833, 0.999999999445505)

(27192.03076197315, 0.999999999549188)

(-42316.573505290275, 0.999999999813852)

(-21128.451639271025, 0.999999999253305)

(20412.219608134823, 0.999999999199985)

(-31298.41488439077, 0.999999999659722)

(17022.619548237748, 0.999999998849661)

(-35536.11709028476, 0.99999999973604)

(12786.228576781117, 0.99999999796111)

(33124.7089739066, 0.999999999696209)

(-25365.810848258036, 0.999999999481939)

(-26213.308372767653, 0.999999999514896)

(-40621.447587593044, 0.999999999797992)

(-10960.87787644252, 0.999999997225479)

(-27908.322865528775, 0.999999999572032)

(-41469.00970279369, 0.999999999806165)

(16175.273031426721, 0.999999998725982)

(-34688.570414259906, 0.999999999722983)

(42447.8004663952, 0.999999999815001)

(24649.53564585282, 0.999999999451393)

(11092.020896021957, 0.999999997290698)

(29734.580192971367, 0.999999999622988)

(-10113.913790961042, 0.999999996741331)

(38209.99802447215, 0.99999999977169)

(11939.089225727435, 0.999999997661506)

(-9267.06673722176, 0.999999996118549)

(-32145.948597586266, 0.999999999677428)

(14480.670489780925, 0.99999999841035)

(28039.541933610733, 0.999999999576028)

(28887.058566260628, 0.999999999600541)

(-20281.012992024975, 0.9999999991896)

(19564.79305500968, 0.99999999912918)

(37362.44349064715, 0.999999999761214)

(39905.113426091, 0.999999999790675)

(-38926.32885992955, 0.999999999780016)

(-28755.838687714244, 0.999999999596887)

(23802.052352061375, 0.999999999411631)

(-33841.02662143908, 0.999999999708934)

(-23670.838723567595, 0.99999999940509)

(-12655.063168038458, 0.999999997918626)

(18717.38259210387, 0.999999999048545)

(-16891.424559122082, 0.999999998831722)

(21259.66032746322, 0.999999999262493)

(-14349.490110877094, 0.999999998381153)

(40752.67403042468, 0.999999999799291)

(-17738.791981858223, 0.999999998940671)

(-36383.66644802879, 0.999999999748194)

(9398.174098121537, 0.999999996226088)

(-16044.082152361707, 0.999999998705062)

(34819.79440720148, 0.999999999725067)


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=31429.6368125843x_{1} = 31429.6368125843
x2=−22823.365825763x_{2} = -22823.365825763
x3=32277.1711035644x_{3} = 32277.1711035644
x4=25497.0269799384x_{4} = 25497.0269799384
x5=−38078.7724924216x_{5} = -38078.7724924216
x6=−32993.4859340089x_{6} = -32993.4859340089
x7=−42316.5735052903x_{7} = -42316.5735052903
x8=17022.6195482377x_{8} = 17022.6195482377
x9=−26213.3083727677x_{9} = -26213.3083727677
x10=−40621.447587593x_{10} = -40621.447587593
x11=24649.5356458528x_{11} = 24649.5356458528
x12=14480.6704897809x_{12} = 14480.6704897809
x13=−14349.4901108771x_{13} = -14349.4901108771
x14=40752.6740304247x_{14} = 40752.6740304247
Puntos máximos de la función:
x14=10245.0412006996x_{14} = 10245.0412006996
x14=−37231.2183043527x_{14} = -37231.2183043527
x14=39057.5547154358x_{14} = 39057.5547154358
x14=22107.1135839482x_{14} = 22107.1135839482
x14=38209.9980244722x_{14} = 38209.9980244722
x14=11939.0892257274x_{14} = 11939.0892257274
x14=−32145.9485975863x_{14} = -32145.9485975863
x14=37362.4434906472x_{14} = 37362.4434906472
x14=18717.3825921039x_{14} = 18717.3825921039
x14=−16891.4245591221x_{14} = -16891.4245591221
x14=−16044.0821523617x_{14} = -16044.0821523617
x14=34819.7944072015x_{14} = 34819.7944072015
Decrece en los intervalos
[40752.6740304247,∞)\left[40752.6740304247, \infty\right)
Crece en los intervalos
(−∞,−42316.5735052903]\left(-\infty, -42316.5735052903\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(x2x2+1−1)x2+1=0\frac{2 \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→−∞(xacot⁡(x))=1\lim_{x \to -\infty}\left(x \operatorname{acot}{\left(x \right)}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=1y = 1
lim⁡x→∞(xacot⁡(x))=1\lim_{x \to \infty}\left(x \operatorname{acot}{\left(x \right)}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=1y = 1
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x*acot(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞acot⁡(x)=0\lim_{x \to -\infty} \operatorname{acot}{\left(x \right)} = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞acot⁡(x)=0\lim_{x \to \infty} \operatorname{acot}{\left(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:
xacot⁡(x)=xacot⁡(x)x \operatorname{acot}{\left(x \right)} = x \operatorname{acot}{\left(x \right)}
- Sí
xacot⁡(x)=−xacot⁡(x)x \operatorname{acot}{\left(x \right)} = - x \operatorname{acot}{\left(x \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = y=x*arctgx
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