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Trazado el gráfico de la función/ arctan(1/x)/(x+sqrt(x^2+5))

Gráfico de la función y = arctan(1/x)/(x+sqrt(x^2+5))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
               /1\    
           atan|-|    
               \x/    
f(x) = ---------------
              ________
             /  2     
       x + \/  x  + 5
f(x)=atan⁡(1x)x+x2+5f{\left(x \right)} = \frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + \sqrt{x^{2} + 5}} 
f = atan(1/x)/(x + sqrt(x^2 + 5))
Gráfico de la función
55010152025303540450.000.25
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:
atan⁡(1x)x+x2+5=0\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + \sqrt{x^{2} + 5}} = 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 atan(1/x)/(x + sqrt(x^2 + 5)).
atan⁡(10)02+5\frac{\operatorname{atan}{\left(\frac{1}{0} \right)}}{\sqrt{0^{2} + 5}}
Resultado:
f(0)=5⟨−π10,π10⟩f{\left(0 \right)} = \sqrt{5} \left\langle - \frac{\pi}{10}, \frac{\pi}{10}\right\rangle
Punto:
(0, AccumBounds(-pi/10, pi/10)*sqrt(5))
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+5−1)atan⁡(1x)(x+x2+5)2−1x2(1+1x2)(x+x2+5)=0\frac{\left(- \frac{x}{\sqrt{x^{2} + 5}} - 1\right) \operatorname{atan}{\left(\frac{1}{x} \right)}}{\left(x + \sqrt{x^{2} + 5}\right)^{2}} - \frac{1}{x^{2} \left(1 + \frac{1}{x^{2}}\right) \left(x + \sqrt{x^{2} + 5}\right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=30586.4960317016x_{1} = 30586.4960317016
x2=−41473.4250678483x_{2} = -41473.4250678483
x3=18724.5503307565x_{3} = 18724.5503307565
x4=29739.0948477105x_{4} = 29739.0948477105
x5=−21984.2320113325x_{5} = -21984.2320113325
x6=−20290.0372329335x_{6} = -20290.0372329335
x7=14489.9291942501x_{7} = 14489.9291942501
x8=42450.9635985586x_{8} = 42450.9635985586
x9=−9286.77399607226x_{9} = -9286.77399607226
x10=39908.4780233899x_{10} = 39908.4780233899
x11=−11823.4164644708x_{11} = -11823.4164644708
x12=−38931.0325474025x_{12} = -38931.0325474025
x13=−38083.5808415457x_{13} = -38083.5808415457
x14=40755.9686812106x_{14} = 40755.9686812106
x15=−16902.2569581704x_{15} = -16902.2569581704
x16=−18596.0270685x_{16} = -18596.0270685
x17=37366.0369535824x_{17} = 37366.0369535824
x18=34823.650131951x_{18} = 34823.650131951
x19=33128.7618861189x_{19} = 33128.7618861189
x20=−39778.49075349x_{20} = -39778.49075349
x21=−17749.107716313x_{21} = -17749.107716313
x22=−32999.0350923798x_{22} = -32999.0350923798
x23=−21137.1143172439x_{23} = -21137.1143172439
x24=−26220.2919562211x_{24} = -26220.2919562211
x25=−8442.04857510117x_{25} = -8442.04857510117
x26=−12669.5125313416x_{26} = -12669.5125313416
x27=25502.291220155x_{27} = 25502.291220155
x28=24654.9806766047x_{28} = 24654.9806766047
x29=41603.4639652552x_{29} = 41603.4639652552
x30=39060.99229278x_{30} = 39060.99229278
x31=17030.4992728625x_{31} = 17030.4992728625
x32=27196.9671750743x_{32} = 27196.9671750743
x33=−13515.7972632899x_{33} = -13515.7972632899
x34=−37236.1360768888x_{34} = -37236.1360768888
x35=17877.4974187058x_{35} = 17877.4974187058
x36=23807.6910229565x_{36} = 23807.6910229565
x37=−36388.6987394278x_{37} = -36388.6987394278
x38=−40625.9550520886x_{38} = -40625.9550520886
x39=33976.2020168722x_{39} = 33976.2020168722
x40=−10977.552558557x_{40} = -10977.552558557
x41=20418.793240764x_{41} = 20418.793240764
x42=−34693.8485179822x_{42} = -34693.8485179822
x43=19571.6509121846x_{43} = 19571.6509121846
x44=−10131.9787605055x_{44} = -10131.9787605055
x45=−22831.3857984867x_{45} = -22831.3857984867
x46=15336.7034773974x_{46} = 15336.7034773974
x47=32281.330368093x_{47} = 32281.330368093
x48=31433.908158897x_{48} = 31433.908158897
x49=−35541.2693589076x_{49} = -35541.2693589076
x50=−27067.5775930096x_{50} = -27067.5775930096
x51=36518.5680929011x_{51} = 36518.5680929011
x52=28044.3292740089x_{52} = 28044.3292740089
x53=−24525.7867030123x_{53} = -24525.7867030123
x54=21265.9723356637x_{54} = 21265.9723356637
x55=−42320.9004555853x_{55} = -42320.9004555853
x56=−30456.8973738324x_{56} = -30456.8973738324
x57=12796.7093513341x_{57} = 12796.7093513341
x58=−31304.2644344512x_{58} = -31304.2644344512
x59=16183.5644707302x_{59} = 16183.5644707302
x60=11104.0940915695x_{60} = 11104.0940915695
x61=−15208.8069857835x_{61} = -15208.8069857835
x62=13643.2577105972x_{62} = 13643.2577105972
x63=−32151.6439945354x_{63} = -32151.6439945354
x64=−19443.0060610182x_{64} = -19443.0060610182
x65=−25373.0275918757x_{65} = -25373.0275918757
x66=−33846.4368615773x_{66} = -33846.4368615773
x67=35671.1056626215x_{67} = 35671.1056626215
x68=22960.4245689479x_{68} = 22960.4245689479
x69=26349.6206405067x_{69} = 26349.6206405067
x70=22113.1839776482x_{70} = 22113.1839776482
x71=−23678.5718081203x_{71} = -23678.5718081203
x72=11950.3101659441x_{72} = 11950.3101659441
x73=28891.705569053x_{73} = 28891.705569053
x74=10258.1061741605x_{74} = 10258.1061741605
x75=38213.5118170516x_{75} = 38213.5118170516
x76=−28762.2051678377x_{76} = -28762.2051678377
x77=9412.40750297377x_{77} = 9412.40750297377
x78=−14362.2373797794x_{78} = -14362.2373797794
x79=−27914.8825661469x_{79} = -27914.8825661469
x80=−29609.5438845902x_{80} = -29609.5438845902
x81=−16055.4856324891x_{81} = -16055.4856324891
Signos de extremos en los puntos:
(30586.496031701594, 5.34454267167801e-10)

(-41473.42506784832, -0.400000012279845)

(18724.550330756523, 1.42609089961922e-9)

(29739.094847710516, 5.65346244465958e-10)

(-21984.23201133254, -0.400000007576938)

(-20290.037232933544, -0.399999998420807)

(14489.929194250144, 2.38142810367477e-9)

(42450.96359855857, 2.77456496714578e-10)

(-9286.773996072257, -0.400000004294441)

(39908.47802338989, 3.13934953560287e-10)

(-11823.416464470785, -0.400000003821089)

(-38931.03254740248, -0.399999992247514)

(-38083.58084154573, -0.400000006054859)

(40755.968681210645, 3.01014602577858e-10)

(-16902.256958170354, -0.399999995778915)

(-18596.027068500047, -0.399999994416919)

(37366.03695358239, 3.58109569013238e-10)

(34823.65013195104, 4.12307677711122e-10)

(33128.76188611892, 4.55574694822403e-10)

(-39778.49075349002, -0.400000010580786)

(-17749.107716313, -0.400000004254328)

(-32999.03509237983, -0.399999976167207)

(-21137.114317243886, -0.399999998873498)

(-26220.291956221063, -0.400000005418882)

(-8442.048575101167, -0.400000005681411)

(-12669.512531341603, -0.40000000415797)

(25502.29122015501, 7.68796861392899e-10)

(24654.980676604722, 8.22546900543479e-10)

(41603.4639652552, 2.88875705867705e-10)

(39060.992292779985, 3.27705296254185e-10)

(17030.49927286251, 1.72391259148653e-9)

(27196.96717507431, 6.75972532570977e-10)

(-13515.797263289915, -0.40000000168503)

(-37236.13607688878, -0.400000008200881)

(17877.49741870577, 1.56443150805898e-9)

(23807.691022956493, 8.82135800707402e-10)

(-36388.69873942783, -0.400000007965551)

(-40625.95505208863, -0.399999996278548)

(33976.20201687218, 4.33132072279703e-10)

(-10977.552558556952, -0.400000004788246)

(20418.793240763964, 1.19925036588736e-9)

(-34693.848517982166, -0.400000023398145)

(19571.65091218462, 1.30531425461478e-9)

(-10131.978760505515, -0.400000004657371)

(-22831.38579848668, -0.400000005334315)

(15336.703477397365, 2.12571951496136e-9)

(32281.33036809296, 4.7980763540737e-10)

(31433.908158897022, 5.06026479996294e-10)

(-35541.26935890759, -0.399999993067072)

(-27067.577593009584, -0.399999996708123)

(36518.56809290106, 3.7492337958797e-10)

(28044.32927400888, 6.35740513324481e-10)

(-24525.786703012323, -0.400000002768438)

(21265.972335663708, 1.10560377693177e-9)

(-42320.90045558528, -0.40000000277653)

(-30456.897373832446, -0.399999996926122)

(12796.709351334095, 3.05332749182599e-9)

(-31304.26443445121, -0.400000009825044)

(16183.564470730218, 1.90906905136312e-9)

(11104.094091569514, 4.05512020506996e-9)

(-15208.806985783538, -0.400000000644032)

(13643.257710597203, 2.68617210785303e-9)

(-32151.643994535425, -0.399999998043644)

(-19443.0060610182, -0.399999996480221)

(-25373.027591875747, -0.400000005855039)

(-33846.4368615773, -0.399999999692163)

(35671.10566262152, 3.92949609835289e-10)

(22960.424568947943, 9.48440681818139e-10)

(26349.620640506673, 7.20147188252385e-10)

(22113.183977648223, 1.02250971869938e-9)

(-23678.571808120276, -0.400000008922719)

(11950.310165944054, 3.50115747384076e-9)

(28891.70556905302, 5.98995614500285e-10)

(10258.10617416046, 4.75155344102293e-9)

(38213.51181705159, 3.42401849284767e-10)

(-28762.205167837703, -0.399999995778117)

(9412.407502973765, 5.64376021685755e-9)

(-14362.237379779373, -0.400000002638577)

(-27914.882566146887, -0.400000003445607)

(-29609.543884590195, -0.400000005657061)

(-16055.4856324891, -0.400000000203136)


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=−21984.2320113325x_{1} = -21984.2320113325
x2=−11823.4164644708x_{2} = -11823.4164644708
x3=−26220.2919562211x_{3} = -26220.2919562211
x4=−10977.552558557x_{4} = -10977.552558557
x5=−34693.8485179822x_{5} = -34693.8485179822
x6=−10131.9787605055x_{6} = -10131.9787605055
x7=−31304.2644344512x_{7} = -31304.2644344512
x8=−23678.5718081203x_{8} = -23678.5718081203
Puntos máximos de la función:
x8=−18596.0270685x_{8} = -18596.0270685
Decrece en los intervalos
[−10131.9787605055,∞)\left[-10131.9787605055, \infty\right)
Crece en los intervalos
(−∞,−34693.8485179822]\left(-\infty, -34693.8485179822\right]
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→−∞(atan⁡(1x)x+x2+5)=−25\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + \sqrt{x^{2} + 5}}\right) = - \frac{2}{5}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=−25y = - \frac{2}{5}
lim⁡x→∞(atan⁡(1x)x+x2+5)=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + \sqrt{x^{2} + 5}}\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 atan(1/x)/(x + sqrt(x^2 + 5)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(atan⁡(1x)x(x+x2+5))=0\lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x \left(x + \sqrt{x^{2} + 5}\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(atan⁡(1x)x(x+x2+5))=0\lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x \left(x + \sqrt{x^{2} + 5}\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:
atan⁡(1x)x+x2+5=−atan⁡(1x)−x+x2+5\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + \sqrt{x^{2} + 5}} = - \frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{- x + \sqrt{x^{2} + 5}}
- No
atan⁡(1x)x+x2+5=atan⁡(1x)−x+x2+5\frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{x + \sqrt{x^{2} + 5}} = \frac{\operatorname{atan}{\left(\frac{1}{x} \right)}}{- x + \sqrt{x^{2} + 5}}
- No
es decir, función
no es
par ni impar
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