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Trazado el gráfico de la función/ -cos(1/x)*exp(sin(1/x))/x^2

Gráfico de la función y = -cos(1/x)*exp(sin(1/x))/x^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                   /1\
                sin|-|
           /1\     \x/
       -cos|-|*e      
           \x/        
f(x) = ---------------
               2      
              x
f(x)=esin⁡(1x)(−cos⁡(1x))x2f{\left(x \right)} = \frac{e^{\sin{\left(\frac{1}{x} \right)}} \left(- \cos{\left(\frac{1}{x} \right)}\right)}{x^{2}} 
f = (exp(sin(1/x))*(-cos(1/x)))/x^2
Gráfico de la función
02468-8-6-4-2-1010-500500
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:
esin⁡(1x)(−cos⁡(1x))x2=0\frac{e^{\sin{\left(\frac{1}{x} \right)}} \left(- \cos{\left(\frac{1}{x} \right)}\right)}{x^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−2πx_{1} = - \frac{2}{\pi}
x2=2πx_{2} = \frac{2}{\pi}
Solución numérica
x1=−0.636619772367581x_{1} = -0.636619772367581
x2=0.636619772367581x_{2} = 0.636619772367581
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 ((-cos(1/x))*exp(sin(1/x)))/x^2.
esin⁡(10)(−cos⁡(10))02\frac{e^{\sin{\left(\frac{1}{0} \right)}} \left(- \cos{\left(\frac{1}{0} \right)}\right)}{0^{2}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
−esin⁡(1x)sin⁡(1x)x2+esin⁡(1x)cos⁡2(1x)x2x2+2esin⁡(1x)cos⁡(1x)x3=0\frac{- \frac{e^{\sin{\left(\frac{1}{x} \right)}} \sin{\left(\frac{1}{x} \right)}}{x^{2}} + \frac{e^{\sin{\left(\frac{1}{x} \right)}} \cos^{2}{\left(\frac{1}{x} \right)}}{x^{2}}}{x^{2}} + \frac{2 e^{\sin{\left(\frac{1}{x} \right)}} \cos{\left(\frac{1}{x} \right)}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=11729.2827331713x_{1} = 11729.2827331713
x2=−23035.702228767x_{2} = -23035.702228767
x3=25286.4015309975x_{3} = 25286.4015309975
x4=−25578.049958303x_{4} = -25578.049958303
x5=12576.3233313138x_{5} = 12576.3233313138
x6=26981.3585313262x_{6} = 26981.3585313262
x7=−26425.5193986275x_{7} = -26425.5193986275
x8=−13715.2816460207x_{8} = -13715.2816460207
x9=−34053.0499622728x_{9} = -34053.0499622728
x10=18507.0250901888x_{10} = 18507.0250901888
x11=−37443.1835415977x_{11} = -37443.1835415977
x12=34608.9665898629x_{12} = 34608.9665898629
x13=−12868.210294926x_{13} = -12868.210294926
x14=19354.3923520808x_{14} = 19354.3923520808
x15=38846.6661221215x_{15} = 38846.6661221215
x16=−29815.4748584263x_{16} = -29815.4748584263
x17=−18798.7417899514x_{17} = -18798.7417899514
x18=−32358.0057216508x_{18} = -32358.0057216508
x19=−15409.6051398523x_{19} = -15409.6051398523
x20=−33205.5257200765x_{20} = -33205.5257200765
x21=−40833.364644882x_{21} = -40833.364644882
x22=−34900.5781354018x_{22} = -34900.5781354018
x23=32913.9097045018x_{23} = 32913.9097045018
x24=−30662.9798827596x_{24} = -30662.9798827596
x25=−16256.8379240003x_{25} = -16256.8379240003
x26=−19646.0964830336x_{26} = -19646.0964830336
x27=21896.6023719434x_{27} = 21896.6023719434
x28=37151.5775927233x_{28} = 37151.5775927233
x29=24438.9359148168x_{29} = 24438.9359148168
x30=−38290.7248604382x_{30} = -38290.7248604382
x31=−14562.416874509x_{31} = -14562.416874509
x32=−42528.4694320437x_{32} = -42528.4694320437
x33=21049.1831346725x_{33} = 21049.1831346725
x34=40541.7645884147x_{34} = 40541.7645884147
x35=−36595.6451696578x_{35} = -36595.6451696578
x36=−22188.2771135289x_{36} = -22188.2771135289
x37=15117.8154386715x_{37} = 15117.8154386715
x38=41389.3171698112x_{38} = 41389.3171698112
x39=31218.869083731x_{39} = 31218.869083731
x40=26133.8760111575x_{40} = 26133.8760111575
x41=−35748.1099566389x_{41} = -35748.1099566389
x42=16812.3607049125x_{42} = 16812.3607049125
x43=−41680.9159739694x_{43} = -41680.9159739694
x44=−31510.4903139375x_{44} = -31510.4903139375
x45=−17104.1084408519x_{45} = -17104.1084408519
x46=−17951.4112172345x_{46} = -17951.4112172345
x47=−39985.8155815512x_{47} = -39985.8155815512
x48=17659.6801083926x_{48} = 17659.6801083926
x49=15965.0708630119x_{49} = 15965.0708630119
x50=−39138.2689327188x_{50} = -39138.2689327188
x51=23591.4801031795x_{51} = 23591.4801031795
x52=−20493.4722414666x_{52} = -20493.4722414666
x53=−27272.9973442816x_{53} = -27272.9973442816
x54=30371.3557132254x_{54} = 30371.3557132254
x55=−28967.9757223352x_{55} = -28967.9757223352
x56=39694.2141919723x_{56} = 39694.2141919723
x57=36304.0374848652x_{57} = 36304.0374848652
x58=37999.1205330129x_{58} = 37999.1205330129
x59=−24730.5899126083x_{59} = -24730.5899126083
x60=32066.3872016969x_{60} = 32066.3872016969
x61=22744.0351746262x_{61} = 22744.0351746262
x62=35456.500410424x_{62} = 35456.500410424
x63=42236.8718057933x_{63} = 42236.8718057933
x64=−12021.2168342714x_{64} = -12021.2168342714
x65=28676.344876692x_{65} = 28676.344876692
x66=−21340.8665053599x_{66} = -21340.8665053599
x67=20201.7791396619x_{67} = 20201.7791396619
x68=13423.4332415118x_{68} = 13423.4332415118
x69=33761.4362655411x_{69} = 33761.4362655411
x70=27828.8483666012x_{70} = 27828.8483666012
x71=−23883.1402793202x_{71} = -23883.1402793202
x72=−28120.4830147873x_{72} = -28120.4830147873
x73=29523.8474939237x_{73} = 29523.8474939237
x74=14270.6004135672x_{74} = 14270.6004135672
Signos de extremos en los puntos:
(11729.28273317127, -7.26932548470924e-9)

(-23035.702228767022, -1.88442229801962e-9)

(25286.401530997504, -1.56402292581562e-9)

(-25578.04995830298, -1.52843916611491e-9)

(12576.323331313766, -6.32305766698647e-9)

(26981.35853132615, -1.37368915605624e-9)

(-26425.519398627475, -1.4319785104796e-9)

(-13715.281646020736, -5.31568054194769e-9)

(-34053.049962272795, -8.62333415088094e-10)

(18507.025090188792, -2.9197807324725e-9)

(-37443.18354159769, -7.13251785560454e-10)

(34608.96658986287, -8.34901584325048e-10)

(-12868.210294926044, -6.03851222746354e-9)

(19354.392352080762, -2.66970550397916e-9)

(38846.66612212154, -6.62679710236189e-10)

(-29815.474858426292, -1.12486906276991e-9)

(-18798.741789951386, -2.82956276426244e-9)

(-32358.00572165081, -9.55043346466261e-10)

(-15409.605139852336, -4.21103445606305e-9)

(-33205.52572007646, -9.06914195578946e-10)

(-40833.36464488198, -5.99734496607604e-10)

(-34900.5781354018, -8.20960598011055e-10)

(32913.90970450184, -9.23111683946838e-10)

(-30662.979882759606, -1.06354806096677e-9)

(-16256.837924000338, -3.78356441910748e-9)

(-19646.09648303359, -2.59074905507852e-9)

(21896.60237194337, -2.08576977387785e-9)

(37151.57759272331, -7.24531328859198e-10)

(24438.935914816822, -1.67437675102237e-9)

(-38290.72486043821, -6.82026865579266e-10)

(-14562.416874508972, -4.7152340790446e-9)

(-42528.46943204374, -5.52879238645454e-10)

(21049.183134672516, -2.25709655868495e-9)

(40541.76458841468, -6.0842271129898e-10)

(-36595.64516965781, -7.46671055563229e-10)

(-22188.277113528922, -2.03110915620415e-9)

(15117.815438671483, -4.37573132758386e-9)

(41389.31716981125, -5.83759511232649e-10)

(31218.869083731006, -1.02607611453962e-9)

(26133.876011157452, -1.46422886862711e-9)

(-35748.109956638946, -7.82495197701497e-10)

(16812.360704912455, -3.53808638941307e-9)

(-41680.91597396938, -5.7559242732677e-10)

(-31510.490313937473, -1.00710765085038e-9)

(-17104.108440851887, -3.4180131196585e-9)

(-17951.411217234523, -3.10297742160075e-9)

(-39985.81558155118, -6.25427857332484e-10)

(17659.68010839262, -3.20670439437443e-9)

(15965.070863011882, -3.92360700144872e-9)

(-39138.2689327188, -6.52808314582343e-10)

(23591.480103179503, -1.7968343061238e-9)

(-20493.47224146656, -2.38093594691601e-9)

(-27272.997344281626, -1.34436852295342e-9)

(30371.355713225414, -1.08414143360717e-9)

(-28967.975722335188, -1.19164998496853e-9)

(39694.21419197232, -6.34682499184232e-10)

(36304.03748486518, -7.5875594803806e-10)

(37999.1205330129, -6.92571057451402e-10)

(-24730.5899126083, -1.63498392362752e-9)

(32066.38720169688, -9.72553449684463e-10)

(22744.03517462623, -1.93323236747701e-9)

(35456.50041042397, -7.95463967998435e-10)

(42236.87180579333, -5.60566049350814e-10)

(-12021.21683427144, -6.9193772516006e-9)

(28676.34487669204, -1.21609514551747e-9)

(-21340.86650535988, -2.1956117731444e-9)

(20201.779139661903, -2.45042976840474e-9)

(13423.43324151179, -5.55015532977707e-9)

(33761.436265541124, -8.77346273108155e-10)

(27828.848366601185, -1.29129401102774e-9)

(-23883.14027932021, -1.75306878363521e-9)

(-28120.483014787296, -1.26455872795516e-9)

(29523.84749392373, -1.14727836294168e-9)

(14270.600413567196, -4.91072868424304e-9)


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
La función no tiene puntos máximos
Crece en todo el eje numérico
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
(−6cos⁡(1x)+4(sin⁡(1x)−cos⁡2(1x))x+−(2cos⁡(1x)−sin⁡(1x)x+cos⁡2(1x)x)cos⁡(1x)+2sin⁡(1x)+2sin⁡(1x)cos⁡(1x)x+cos⁡(1x)xx)esin⁡(1x)x4=0\frac{\left(- 6 \cos{\left(\frac{1}{x} \right)} + \frac{4 \left(\sin{\left(\frac{1}{x} \right)} - \cos^{2}{\left(\frac{1}{x} \right)}\right)}{x} + \frac{- \left(2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x} + \frac{\cos^{2}{\left(\frac{1}{x} \right)}}{x}\right) \cos{\left(\frac{1}{x} \right)} + 2 \sin{\left(\frac{1}{x} \right)} + \frac{2 \sin{\left(\frac{1}{x} \right)} \cos{\left(\frac{1}{x} \right)}}{x} + \frac{\cos{\left(\frac{1}{x} \right)}}{x}}{x}\right) e^{\sin{\left(\frac{1}{x} \right)}}}{x^{4}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−4400.50562719818x_{1} = -4400.50562719818
x2=−5926.47958466645x_{2} = -5926.47958466645
x3=9994.93059879699x_{3} = 9994.93059879699
x4=−9415.05874545408x_{4} = -9415.05874545408
x5=−6798.57101244811x_{5} = -6798.57101244811
x6=−8542.87165026114x_{6} = -8542.87165026114
x7=9558.82769201256x_{7} = 9558.82769201256
x8=10649.0906656941x_{8} = 10649.0906656941
x9=5198.13791550948x_{9} = 5198.13791550948
x10=−1569.47439156728x_{10} = -1569.47439156728
x11=−9633.1082081595x_{11} = -9633.1082081595
x12=3018.54075715693x_{12} = 3018.54075715693
x13=−9851.15860177388x_{13} = -9851.15860177388
x14=−10287.2619420795x_{14} = -10287.2619420795
x15=9122.72833188829x_{15} = 9122.72833188829
x16=−10069.2098650711x_{16} = -10069.2098650711
x17=−3964.58868909644x_{17} = -3964.58868909644
x18=8904.68014265024x_{18} = 8904.68014265024
x19=8468.58712470397x_{19} = 8468.58712470397
x20=−7016.6013625404x_{20} = -7016.6013625404
x21=−3310.82813490797x_{21} = -3310.82813490797
x22=−8978.96288782897x_{22} = -8978.96288782897
x23=1929.63134921358x_{23} = 1929.63134921358
x24=2582.8364842131x_{24} = 2582.8364842131
x25=3672.28600540643x_{25} = 3672.28600540643
x26=3454.35191893277x_{26} = 3454.35191893277
x27=2365.04216107262x_{27} = 2365.04216107262
x28=−6580.5433558822x_{28} = -6580.5433558822
x29=−1786.77240913177x_{29} = -1786.77240913177
x30=2147.30135394966x_{30} = 2147.30135394966
x31=−4182.54114920052x_{31} = -4182.54114920052
x32=1712.05780372662x_{32} = 1712.05780372662
x33=9340.77753793679x_{33} = 9340.77753793679
x34=6506.24591943596x_{34} = 6506.24591943596
x35=4980.13557690145x_{35} = 4980.13557690145
x36=−2657.30062559394x_{36} = -2657.30062559394
x37=7160.34221099565x_{37} = 7160.34221099565
x38=−8760.91664780018x_{38} = -8760.91664780018
x39=−6362.51867533298x_{39} = -6362.51867533298
x40=3236.43559524588x_{40} = 3236.43559524588
x41=−7234.6341583316x_{41} = -7234.6341583316
x42=8250.54247126404x_{42} = 8250.54247126404
x43=9776.87873140012x_{43} = 9776.87873140012
x44=7596.4171929777x_{44} = 7596.4171929777
x45=2800.67224732124x_{45} = 2800.67224732124
x46=−3528.72907516106x_{46} = -3528.72907516106
x47=5416.14561212382x_{47} = 5416.14561212382
x48=−3092.95210796945x_{48} = -3092.95210796945
x49=−5054.45520876147x_{49} = -5054.45520876147
x50=−5272.45315210189x_{50} = -5272.45315210189
x51=7814.4573859636x_{51} = 7814.4573859636
x52=−6144.49729423505x_{52} = -6144.49729423505
x53=−8324.82799431772x_{53} = -8324.82799431772
x54=10431.0366121143x_{54} = 10431.0366121143
x55=3890.2349454905x_{55} = 3890.2349454905
x56=−7670.70623895426x_{56} = -7670.70623895426
x57=−4836.46395714845x_{57} = -4836.46395714845
x58=5852.17472716282x_{58} = 5852.17472716282
x59=10867.1453617746x_{59} = 10867.1453617746
x60=4108.19643275641x_{60} = 4108.19643275641
x61=5634.15805635517x_{61} = 5634.15805635517
x62=1494.6205339573x_{62} = 1494.6205339573
x63=−2439.54420407903x_{63} = -2439.54420407903
x64=−9197.01028076965x_{64} = -9197.01028076965
x65=−2221.85333012333x_{65} = -2221.85333012333
x66=−2004.25099565857x_{66} = -2004.25099565857
x67=6724.27558375039x_{67} = 6724.27558375039
x68=−10941.4225633499x_{68} = -10941.4225633499
x69=6288.21901966254x_{69} = 6288.21901966254
x70=4762.139316056x_{70} = 4762.139316056
x71=−5708.46597685361x_{71} = -5708.46597685361
x72=−7452.66918149776x_{72} = -7452.66918149776
x73=−8106.78578990542x_{73} = -8106.78578990542
x74=8686.63304582561x_{74} = 8686.63304582561
x75=−7888.74515931039x_{75} = -7888.74515931039
x76=−10723.3683363724x_{76} = -10723.3683363724
x77=−5490.45697099633x_{77} = -5490.45697099633
x78=−10505.3147815301x_{78} = -10505.3147815301
x79=4544.14998775219x_{79} = 4544.14998775219
x80=6070.19517707899x_{80} = 6070.19517707899
x81=−3746.65042047421x_{81} = -3746.65042047421
x82=4326.16861502922x_{82} = 4326.16861502922
x83=−4618.48037201593x_{83} = -4618.48037201593
x84=7378.37874887203x_{84} = 7378.37874887203
x85=−2875.10693799594x_{85} = -2875.10693799594
x86=10212.9832417419x_{86} = 10212.9832417419
x87=8032.49918733725x_{87} = 8032.49918733725
x88=6942.30775616108x_{88} = 6942.30775616108
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=0x_{1} = 0

True

True

- los límites no son iguales, signo
x1=0x_{1} = 0
- 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:
No tiene corvaduras en todo el eje numérico
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→−∞(esin⁡(1x)(−cos⁡(1x))x2)=0\lim_{x \to -\infty}\left(\frac{e^{\sin{\left(\frac{1}{x} \right)}} \left(- \cos{\left(\frac{1}{x} \right)}\right)}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(esin⁡(1x)(−cos⁡(1x))x2)=0\lim_{x \to \infty}\left(\frac{e^{\sin{\left(\frac{1}{x} \right)}} \left(- \cos{\left(\frac{1}{x} \right)}\right)}{x^{2}}\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 ((-cos(1/x))*exp(sin(1/x)))/x^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−esin⁡(1x)cos⁡(1x)xx2)=0\lim_{x \to -\infty}\left(- \frac{e^{\sin{\left(\frac{1}{x} \right)}} \cos{\left(\frac{1}{x} \right)}}{x x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−esin⁡(1x)cos⁡(1x)xx2)=0\lim_{x \to \infty}\left(- \frac{e^{\sin{\left(\frac{1}{x} \right)}} \cos{\left(\frac{1}{x} \right)}}{x x^{2}}\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:
esin⁡(1x)(−cos⁡(1x))x2=−e−sin⁡(1x)cos⁡(1x)x2\frac{e^{\sin{\left(\frac{1}{x} \right)}} \left(- \cos{\left(\frac{1}{x} \right)}\right)}{x^{2}} = - \frac{e^{- \sin{\left(\frac{1}{x} \right)}} \cos{\left(\frac{1}{x} \right)}}{x^{2}}
- No
esin⁡(1x)(−cos⁡(1x))x2=e−sin⁡(1x)cos⁡(1x)x2\frac{e^{\sin{\left(\frac{1}{x} \right)}} \left(- \cos{\left(\frac{1}{x} \right)}\right)}{x^{2}} = \frac{e^{- \sin{\left(\frac{1}{x} \right)}} \cos{\left(\frac{1}{x} \right)}}{x^{2}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор