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xarctg(1/x)
Trazado el gráfico de la función/ xarctg(1/x)

Gráfico de la función y = xarctg(1/x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
             /1\
f(x) = x*atan|-|
             \x/
f(x)=xatan⁡(1x)f{\left(x \right)} = x \operatorname{atan}{\left(\frac{1}{x} \right)} 
f = x*atan(1/x)
Gráfico de la función
02468-8-6-4-2-101002
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
xatan⁡(1x)=0x \operatorname{atan}{\left(\frac{1}{x} \right)} = 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 x*atan(1/x).
0atan⁡(10)0 \operatorname{atan}{\left(\frac{1}{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
atan⁡(1x)−1x(1+1x2)=0\operatorname{atan}{\left(\frac{1}{x} \right)} - \frac{1}{x \left(1 + \frac{1}{x^{2}}\right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=14480.6704897809x_{1} = 14480.6704897809
x2=11092.020896022x_{2} = 11092.020896022
x3=21259.6603274632x_{3} = 21259.6603274632
x4=−38078.7724924216x_{4} = -38078.7724924216
x5=18717.3825921039x_{5} = 18717.3825921039
x6=−26213.3083727677x_{6} = -26213.3083727677
x7=−27060.8125814987x_{7} = -27060.8125814987
x8=−29603.3595724168x_{8} = -29603.3595724168
x9=28887.0585662606x_{9} = 28887.0585662606
x10=32277.1711035644x_{10} = 32277.1711035644
x11=42447.8004663952x_{11} = 42447.8004663952
x12=16175.2730314267x_{12} = 16175.2730314267
x13=−28755.8386877142x_{13} = -28755.8386877142
x14=−33841.0266214391x_{14} = -33841.0266214391
x15=23802.0523520614x_{15} = 23802.0523520614
x16=−25365.810848258x_{16} = -25365.810848258
x17=19564.7930550097x_{17} = 19564.7930550097
x18=29734.5801929714x_{18} = 29734.5801929714
x19=27192.0307619732x_{19} = 27192.0307619732
x20=−21128.451639271x_{20} = -21128.451639271
x21=−16044.0821523617x_{21} = -16044.0821523617
x22=22954.5779890773x_{22} = 22954.5779890773
x23=−24518.3207010738x_{23} = -24518.3207010738
x24=−37231.2183043527x_{24} = -37231.2183043527
x25=26344.5255783821x_{25} = 26344.5255783821
x26=−16891.4245591221x_{26} = -16891.4245591221
x27=−21975.9030580391x_{27} = -21975.9030580391
x28=22107.1135839482x_{28} = 22107.1135839482
x29=−8420.37201352502x_{29} = -8420.37201352502
x30=−22823.365825763x_{30} = -22823.365825763
x31=28039.5419336107x_{31} = 28039.5419336107
x32=24649.5356458528x_{32} = 24649.5356458528
x33=40752.6740304247x_{33} = 40752.6740304247
x34=11939.0892257274x_{34} = 11939.0892257274
x35=−10113.913790961x_{35} = -10113.913790961
x36=39057.5547154358x_{36} = 39057.5547154358
x37=36514.8912641685x_{37} = 36514.8912641685
x38=−14349.4901108771x_{38} = -14349.4901108771
x39=−11807.9338157736x_{39} = -11807.9338157736
x40=−18586.1809993501x_{40} = -18586.1809993501
x41=37362.4434906472x_{41} = 37362.4434906472
x42=15327.9550113123x_{42} = 15327.9550113123
x43=−31298.4148843908x_{43} = -31298.4148843908
x44=33972.2501557217x_{44} = 33972.2501557217
x45=−9267.06673722176x_{45} = -9267.06673722176
x46=−35536.1170902848x_{46} = -35536.1170902848
x47=−15196.7689456217x_{47} = -15196.7689456217
x48=−30450.8850969459x_{48} = -30450.8850969459
x49=34819.7944072015x_{49} = 34819.7944072015
x50=35667.3415095204x_{50} = 35667.3415095204
x51=12786.2285767811x_{51} = 12786.2285767811
x52=−12655.0631680385x_{52} = -12655.0631680385
x53=−19433.5887869564x_{53} = -19433.5887869564
x54=10245.0412006996x_{54} = 10245.0412006996
x55=−34688.5704142599x_{55} = -34688.5704142599
x56=−13502.2521178288x_{56} = -13502.2521178288
x57=−32993.4859340089x_{57} = -32993.4859340089
x58=13633.4257117535x_{58} = 13633.4257117535
x59=−27908.3228655288x_{59} = -27908.3228655288
x60=−20281.012992025x_{60} = -20281.012992025
x61=8551.45303022722x_{61} = 8551.45303022722
x62=39905.113426091x_{62} = 39905.113426091
x63=−42316.5735052903x_{63} = -42316.5735052903
x64=17022.6195482377x_{64} = 17022.6195482377
x65=−36383.6664480288x_{65} = -36383.6664480288
x66=−23670.8387235676x_{66} = -23670.8387235676
x67=31429.6368125843x_{67} = 31429.6368125843
x68=−32145.9485975863x_{68} = -32145.9485975863
x69=−40621.447587593x_{69} = -40621.447587593
x70=33124.7089739066x_{70} = 33124.7089739066
x71=25497.0269799384x_{71} = 25497.0269799384
x72=30582.10639855x_{72} = 30582.10639855
x73=38209.9980244722x_{73} = 38209.9980244722
x74=20412.2196081348x_{74} = 20412.2196081348
x75=17869.99050832x_{75} = 17869.99050832
x76=−17738.7919818582x_{76} = -17738.7919818582
x77=−38926.3288599295x_{77} = -38926.3288599295
x78=9398.17409812154x_{78} = 9398.17409812154
x79=−39773.8872675524x_{79} = -39773.8872675524
x80=−10960.8778764425x_{80} = -10960.8778764425
x81=−41469.0097027937x_{81} = -41469.0097027937
x82=41600.2364126928x_{82} = 41600.2364126928
Signos de extremos en los puntos:
(14480.670489780925, 0.99999999841035)

(11092.020896021957, 0.999999997290698)

(21259.66032746322, 0.999999999262493)

(-38078.77249242158, 0.999999999770114)

(18717.38259210387, 0.999999999048545)

(-26213.308372767653, 0.999999999514896)

(-27060.81258149867, 0.999999999544805)

(-29603.35957241684, 0.999999999619638)

(28887.058566260628, 0.999999999600541)

(32277.171103564393, 0.999999999680046)

(42447.8004663952, 0.999999999815001)

(16175.273031426723, 0.999999998725982)

(-28755.838687714244, 0.999999999596887)

(-33841.02662143908, 0.999999999708934)

(23802.052352061375, 0.999999999411631)

(-25365.810848258036, 0.999999999481939)

(19564.79305500968, 0.99999999912918)

(29734.580192971367, 0.999999999622988)

(27192.03076197315, 0.999999999549188)

(-21128.451639271025, 0.999999999253305)

(-16044.082152361707, 0.999999998705062)

(22954.57798907731, 0.999999999367384)

(-24518.320701073833, 0.999999999445505)

(-37231.21830435275, 0.999999999759528)

(26344.525578382065, 0.999999999519716)

(-16891.424559122082, 0.999999998831722)

(-21975.903058039054, 0.999999999309784)

(22107.113583948245, 0.999999999317952)

(-8420.37201352502, 0.999999995298719)

(-22823.365825763038, 0.999999999360089)

(28039.541933610733, 0.999999999576028)

(24649.53564585282, 0.999999999451393)

(40752.67403042468, 0.999999999799291)

(11939.089225727435, 0.999999997661505)

(-10113.913790961042, 0.999999996741331)

(39057.55471543576, 0.999999999781491)

(36514.89126416845, 0.999999999750001)

(-14349.490110877094, 0.999999998381153)

(-11807.933815773578, 0.999999997609268)

(-18586.180999350086, 0.999999999035064)

(37362.44349064715, 0.999999999761214)

(15327.955011312279, 0.999999998581235)

(-31298.41488439077, 0.999999999659722)

(33972.25015572168, 0.999999999711178)

(-9267.06673722176, 0.999999996118549)

(-35536.11709028476, 0.99999999973604)

(-15196.768945621745, 0.999999998556635)

(-30450.88509694595, 0.999999999640517)

(34819.79440720148, 0.999999999725067)

(35667.3415095204, 0.999999999737978)

(12786.228576781117, 0.99999999796111)

(-12655.063168038458, 0.999999997918626)

(-19433.58878695641, 0.999999999117382)

(10245.041200699563, 0.999999996824213)

(-34688.570414259906, 0.999999999722983)

(-13502.25211782884, 0.999999998171621)

(-32993.48593400886, 0.999999999693788)

(13633.425711753476, 0.999999998206635)

(-27908.322865528775, 0.999999999572032)

(-20281.012992024975, 0.9999999991896)

(8551.453030227216, 0.999999995441742)

(39905.113426091, 0.999999999790675)

(-42316.573505290275, 0.999999999813852)

(17022.619548237748, 0.999999998849661)

(-36383.66644802879, 0.999999999748194)

(-23670.838723567595, 0.99999999940509)

(31429.636812584253, 0.999999999662557)

(-32145.948597586266, 0.999999999677428)

(-40621.447587593044, 0.999999999797992)

(33124.7089739066, 0.999999999696209)

(25497.02697993837, 0.999999999487257)

(30582.106398549957, 0.999999999643595)

(38209.99802447215, 0.99999999977169)

(20412.219608134823, 0.999999999199985)

(17869.990508320032, 0.999999998956169)

(-17738.791981858223, 0.999999998940671)

(-38926.32885992955, 0.999999999780016)

(9398.174098121537, 0.999999996226088)

(-39773.887267552374, 0.999999999789291)

(-10960.87787644252, 0.999999997225479)

(-41469.00970279369, 0.999999999806165)

(41600.23641269277, 0.999999999807386)


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=−33841.0266214391x_{1} = -33841.0266214391
x2=22107.1135839482x_{2} = 22107.1135839482
x3=11939.0892257274x_{3} = 11939.0892257274
x4=39057.5547154358x_{4} = 39057.5547154358
x5=−15196.7689456217x_{5} = -15196.7689456217
x6=35667.3415095204x_{6} = 35667.3415095204
x7=17022.6195482377x_{7} = 17022.6195482377
x8=31429.6368125843x_{8} = 31429.6368125843
x9=25497.0269799384x_{9} = 25497.0269799384
Puntos máximos de la función:
x9=−27060.8125814987x_{9} = -27060.8125814987
x9=−37231.2183043527x_{9} = -37231.2183043527
x9=−22823.365825763x_{9} = -22823.365825763
x9=36514.8912641685x_{9} = 36514.8912641685
x9=−14349.4901108771x_{9} = -14349.4901108771
x9=−30450.8850969459x_{9} = -30450.8850969459
x9=12786.2285767811x_{9} = 12786.2285767811
x9=39905.113426091x_{9} = 39905.113426091
x9=−42316.5735052903x_{9} = -42316.5735052903
Decrece en los intervalos
[39057.5547154358,∞)\left[39057.5547154358, \infty\right)
Crece en los intervalos
(−∞,−33841.0266214391]\left(-\infty, -33841.0266214391\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
−2x4(1+1x2)2=0- \frac{2}{x^{4} \left(1 + \frac{1}{x^{2}}\right)^{2}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xatan⁡(1x))=1\lim_{x \to -\infty}\left(x \operatorname{atan}{\left(\frac{1}{x} \right)}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=1y = 1
lim⁡x→∞(xatan⁡(1x))=1\lim_{x \to \infty}\left(x \operatorname{atan}{\left(\frac{1}{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*atan(1/x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞atan⁡(1x)=0\lim_{x \to -\infty} \operatorname{atan}{\left(\frac{1}{x} \right)} = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞atan⁡(1x)=0\lim_{x \to \infty} \operatorname{atan}{\left(\frac{1}{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:
xatan⁡(1x)=xatan⁡(1x)x \operatorname{atan}{\left(\frac{1}{x} \right)} = x \operatorname{atan}{\left(\frac{1}{x} \right)}
- Sí
xatan⁡(1x)=−xatan⁡(1x)x \operatorname{atan}{\left(\frac{1}{x} \right)} = - x \operatorname{atan}{\left(\frac{1}{x} \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = xarctg(1/x)
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