Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ cbrt(x*sin(1/x))

Gráfico de la función y = cbrt(x*sin(1/x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           __________
          /      /1\ 
f(x) = 3 /  x*sin|-| 
       \/        \x/
f(x)=xsin⁡(1x)3f{\left(x \right)} = \sqrt[3]{x \sin{\left(\frac{1}{x} \right)}} 
f = (x*sin(1/x))^(1/3)
Gráfico de la función
02468-8-6-4-2-10100.01.5
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:
xsin⁡(1x)3=0\sqrt[3]{x \sin{\left(\frac{1}{x} \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=1πx_{1} = \frac{1}{\pi}
Solución numérica
x1=0.318309886183791x_{1} = 0.318309886183791
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*sin(1/x))^(1/3).
0sin⁡(10)3\sqrt[3]{0 \sin{\left(\frac{1}{0} \right)}}
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
xsin⁡(1x)3(sin⁡(1x)3−cos⁡(1x)3x)xsin⁡(1x)=0\frac{\sqrt[3]{x \sin{\left(\frac{1}{x} \right)}} \left(\frac{\sin{\left(\frac{1}{x} \right)}}{3} - \frac{\cos{\left(\frac{1}{x} \right)}}{3 x}\right)}{x \sin{\left(\frac{1}{x} \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=34817.4192223057x_{1} = 34817.4192223057
x2=23798.5775858423x_{2} = 23798.5775858423
x3=41598.2483807415x_{3} = 41598.2483807415
x4=−40619.4116833074x_{4} = -40619.4116833074
x5=−38924.2042951088x_{5} = -38924.2042951088
x6=−28752.9626499176x_{6} = -28752.9626499176
x7=9389.37064503813x_{7} = 9389.37064503813
x8=18712.9637011367x_{8} = 18712.9637011367
x9=−20276.9349867333x_{9} = -20276.9349867333
x10=−6715.3678731502x_{10} = -6715.3678731502
x11=−32990.9793136108x_{11} = -32990.9793136108
x12=19560.5655990897x_{12} = 19560.5655990897
x13=8541.7770616821x_{13} = 8541.7770616821
x14=−22819.7421392385x_{14} = -22819.7421392385
x15=−37228.9970056193x_{15} = -37228.9970056193
x16=30579.4020631111x_{16} = 30579.4020631111
x17=42445.8521317744x_{17} = 42445.8521317744
x18=29731.7987687807x_{18} = 29731.7987687807
x19=38207.8335917311x_{19} = 38207.8335917311
x20=39903.0409404711x_{20} = 39903.0409404711
x21=−27905.3594818086x_{21} = -27905.3594818086
x22=22950.9749170045x_{22} = 22950.9749170045
x23=−7562.95524847852x_{23} = -7562.95524847852
x24=11084.5626918887x_{24} = 11084.5626918887
x25=−31295.7725013399x_{25} = -31295.7725013399
x26=11932.1604631924x_{26} = 11932.1604631924
x27=−5867.7853612278x_{27} = -5867.7853612278
x28=6846.59778638007x_{28} = 6846.59778638007
x29=−12648.5269316607x_{29} = -12648.5269316607
x30=14474.9583152524x_{30} = 14474.9583152524
x31=−16038.92698059x_{31} = -16038.92698059
x32=7694.18581988464x_{32} = 7694.18581988464
x33=−27057.75638061x_{33} = -27057.75638061
x34=12779.7590910511x_{34} = 12779.7590910511
x35=15322.5586940399x_{35} = 15322.5586940399
x36=35665.0227687838x_{36} = 35665.0227687838
x37=−42314.6191583508x_{37} = -42314.6191583508
x38=26341.3861972523x_{38} = 26341.3861972523
x39=27188.9892379387x_{39} = 27188.9892379387
x40=−36381.3934022026x_{40} = -36381.3934022026
x41=−18581.7310631525x_{41} = -18581.7310631525
x42=33122.2122351905x_{42} = 33122.2122351905
x43=−29600.565879189x_{43} = -29600.565879189
x44=10236.9659895336x_{44} = 10236.9659895336
x45=16170.1594768851x_{45} = 16170.1594768851
x46=32274.6088001109x_{46} = 32274.6088001109
x47=−24514.9475478824x_{47} = -24514.9475478824
x48=−38076.6006372014x_{48} = -38076.6006372014
x49=13627.3584158347x_{49} = 13627.3584158347
x50=−21124.537222287x_{50} = -21124.537222287
x51=−30448.1691645146x_{51} = -30448.1691645146
x52=24646.1803624273x_{52} = 24646.1803624273
x53=−16886.5280528128x_{53} = -16886.5280528128
x54=−35533.789828967x_{54} = -35533.789828967
x55=31427.005408221x_{55} = 31427.005408221
x56=−32143.3758855904x_{56} = -32143.3758855904
x57=−23667.3447887024x_{57} = -23667.3447887024
x58=−11800.928436082x_{58} = -11800.928436082
x59=−10105.7343329223x_{59} = -10105.7343329223
x60=−41467.0154106392x_{60} = -41467.0154106392
x61=−8410.54601916266x_{61} = -8410.54601916266
x62=40750.6446498788x_{62} = 40750.6446498788
x63=−19429.3329257269x_{63} = -19429.3329257269
x64=−39771.8079776584x_{64} = -39771.8079776584
x65=33969.815710228x_{65} = 33969.815710228
x66=21255.7699541006x_{66} = 21255.7699541006
x67=5151.43764695395x_{67} = 5151.43764695395
x68=5999.0143153995x_{68} = 5999.0143153995
x69=28036.5923508734x_{69} = 28036.5923508734
x70=−17734.1294274606x_{70} = -17734.1294274606
x71=−13496.1261481671x_{71} = -13496.1261481671
x72=−10953.3308287156x_{72} = -10953.3308287156
x73=20408.1676911466x_{73} = 20408.1676911466
x74=−9258.13925377442x_{74} = -9258.13925377442
x75=−33838.5827821116x_{75} = -33838.5827821116
x76=17017.7606034652x_{76} = 17017.7606034652
x77=−21972.1396121883x_{77} = -21972.1396121883
x78=−26210.1533528135x_{78} = -26210.1533528135
x79=−25362.5504057785x_{79} = -25362.5504057785
x80=17865.3620248945x_{80} = 17865.3620248945
x81=36512.626347267x_{81} = 36512.626347267
x82=37360.2299555776x_{82} = 37360.2299555776
x83=25493.7832360173x_{83} = 25493.7832360173
x84=−15191.3262614066x_{84} = -15191.3262614066
x85=−14343.7259578365x_{85} = -14343.7259578365
x86=39055.437253915x_{86} = 39055.437253915
x87=22103.3723683027x_{87} = 22103.3723683027
x88=−34686.1862881253x_{88} = -34686.1862881253
x89=28884.1955296985x_{89} = 28884.1955296985
x90=−5020.21016824119x_{90} = -5020.21016824119
Signos de extremos en los puntos:
(34817.41922230575, 0.999999999954172)

(23798.577585842322, 0.99999999990191)

(41598.24838074155, 0.999999999967895)

(-40619.4116833074, 0.999999999966329)

(-38924.20429510883, 0.999999999963332)

(-28752.962649917623, 0.999999999932801)

(9389.370645038134, 0.999999999369835)

(18712.96370113669, 0.999999999841349)

(-20276.934986733297, 0.999999999864879)

(-6715.3678731502, 0.999999998768065)

(-32990.97931361078, 0.999999999948957)

(19560.56559908972, 0.999999999854801)

(8541.777061682104, 0.999999999238568)

(-22819.74213923852, 0.999999999893314)

(-37228.99700561929, 0.999999999959917)

(30579.402063111127, 0.999999999940589)

(42445.85213177438, 0.999999999969164)

(29731.798768780714, 0.999999999937153)

(38207.83359173108, 0.999999999961944)

(39903.04094047107, 0.999999999965109)

(-27905.35948180857, 0.999999999928657)

(22950.974917004532, 0.999999999894531)

(-7562.955248478524, 0.99999999902872)

(11084.562691888697, 0.999999999547842)

(-31295.772501339878, 0.999999999943277)

(11932.160463192366, 0.999999999609798)

(-5867.785361227798, 0.999999998386463)

(6846.597786380071, 0.999999998814838)

(-12648.526931660737, 0.999999999652746)

(14474.958315252354, 0.999999999734849)

(-16038.926980590011, 0.999999999784038)

(7694.185819884641, 0.999999999061569)

(-27057.75638060996, 0.999999999924117)

(12779.75909105108, 0.999999999659841)

(15322.558694039883, 0.999999999763373)

(35665.02276878378, 0.999999999956324)

(-42314.61915835083, 0.999999999968972)

(26341.38619725229, 0.999999999919934)

(27188.989237938695, 0.999999999924848)

(-36381.39340220262, 0.999999999958027)

(-18581.731063152518, 0.9999999998391)

(33122.212235190535, 0.999999999949361)

(-29600.565879189005, 0.999999999936594)

(10236.965989533557, 0.999999999469867)

(16170.159476885148, 0.999999999787529)

(32274.608800110884, 0.999999999946666)

(-24514.947547882402, 0.999999999907559)

(-38076.60063720139, 0.999999999961681)

(13627.358415834657, 0.99999999970084)

(-21124.537222287006, 0.999999999875505)

(-30448.16916451465, 0.999999999940075)

(24646.180362427283, 0.999999999908541)

(-16886.528052812755, 0.999999999805174)

(-35533.78982896705, 0.999999999956001)

(31427.005408221008, 0.99999999994375)

(-32143.375885590434, 0.999999999946229)

(-23667.34478870235, 0.999999999900819)

(-11800.928436081998, 0.999999999601071)

(-10105.734332922339, 0.999999999456009)

(-41467.01541063924, 0.999999999967691)

(-8410.546019162664, 0.999999999214621)

(40750.64464987877, 0.999999999966545)

(-19429.332925726867, 0.999999999852833)

(-39771.80797765841, 0.999999999964878)

(33969.81571022795, 0.999999999951856)

(21255.76995410064, 0.999999999877037)

(5151.437646953951, 0.999999997906511)

(5999.014315399501, 0.999999998456283)

(28036.592350873423, 0.999999999929323)

(-17734.129427460648, 0.999999999823352)

(-13496.12614816708, 0.999999999694993)

(-10953.33082871563, 0.999999999536942)

(20408.16769114656, 0.999999999866611)

(-9258.139253774418, 0.999999999351843)

(-33838.58278211164, 0.999999999951482)

(17017.76060346523, 0.999999999808167)

(-21972.13961218834, 0.999999999884924)

(-26210.153352813464, 0.99999999991913)

(-25362.55040577853, 0.999999999913634)

(17865.362024894453, 0.999999999825938)

(36512.626347266996, 0.999999999958328)

(37360.229955577626, 0.999999999960198)

(25493.783236017323, 0.999999999914521)

(-15191.326261406602, 0.999999999759267)

(-14343.725957836505, 0.999999999729975)

(39055.43725391503, 0.999999999963578)

(22103.372368302742, 0.999999999886287)

(-34686.18628812525, 0.999999999953824)

(28884.19552969851, 0.99999999993341)

(-5020.210168241192, 0.999999997795634)


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=−40619.4116833074x_{1} = -40619.4116833074
x2=38207.8335917311x_{2} = 38207.8335917311
x3=−7562.95524847852x_{3} = -7562.95524847852
x4=−24514.9475478824x_{4} = -24514.9475478824
x5=−32143.3758855904x_{5} = -32143.3758855904
Puntos máximos de la función:
x5=31427.005408221x_{5} = 31427.005408221
x5=−19429.3329257269x_{5} = -19429.3329257269
x5=−34686.1862881253x_{5} = -34686.1862881253
x5=28884.1955296985x_{5} = 28884.1955296985
Decrece en los intervalos
[38207.8335917311,∞)\left[38207.8335917311, \infty\right)
Crece en los intervalos
(−∞,−40619.4116833074]\left(-\infty, -40619.4116833074\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
xsin⁡(1x)3((sin⁡(1x)−cos⁡(1x)x)2sin⁡2(1x)−3(sin⁡(1x)−cos⁡(1x)x)sin⁡(1x)+3(sin⁡(1x)−cos⁡(1x)x)cos⁡(1x)xsin⁡2(1x)−3x2)9x2=0\frac{\sqrt[3]{x \sin{\left(\frac{1}{x} \right)}} \left(\frac{\left(\sin{\left(\frac{1}{x} \right)} - \frac{\cos{\left(\frac{1}{x} \right)}}{x}\right)^{2}}{\sin^{2}{\left(\frac{1}{x} \right)}} - \frac{3 \left(\sin{\left(\frac{1}{x} \right)} - \frac{\cos{\left(\frac{1}{x} \right)}}{x}\right)}{\sin{\left(\frac{1}{x} \right)}} + \frac{3 \left(\sin{\left(\frac{1}{x} \right)} - \frac{\cos{\left(\frac{1}{x} \right)}}{x}\right) \cos{\left(\frac{1}{x} \right)}}{x \sin^{2}{\left(\frac{1}{x} \right)}} - \frac{3}{x^{2}}\right)}{9 x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=2633.77082319224x_{1} = 2633.77082319224
x2=−8269.7979620211x_{2} = -8269.7979620211
x3=−9796.28741410893x_{3} = -9796.28741410893
x4=7649.35766891786x_{4} = 7649.35766891786
x5=−4344.5463440775x_{5} = -4344.5463440775
x6=9830.05665282476x_{6} = 9830.05665282476
x7=−2818.06903497193x_{7} = -2818.06903497193
x8=−2381.93603007282x_{8} = -2381.93603007282
x9=4596.38437661394x_{9} = 4596.38437661394
x10=889.293504947498x_{10} = 889.293504947498
x11=−1727.74358315394x_{11} = -1727.74358315394
x12=8303.56718115672x_{12} = 8303.56718115672
x13=1107.33872136477x_{13} = 1107.33872136477
x14=−6743.30948814206x_{14} = -6743.30948814206
x15=7431.28787758986x_{15} = 7431.28787758986
x16=10484.2666170969x_{16} = 10484.2666170969
x17=6122.86978572131x_{17} = 6122.86978572131
x18=2197.63895799782x_{18} = 2197.63895799782
x19=−4780.6843964014x_{19} = -4780.6843964014
x20=2415.7045036474x_{20} = 2415.7045036474
x21=9611.98668670116x_{21} = 9611.98668670116
x22=−4998.7535802187x_{22} = -4998.7535802187
x23=−4126.47750891055x_{23} = -4126.47750891055
x24=−6525.23982605818x_{24} = -6525.23982605818
x25=−7179.44891589531x_{25} = -7179.44891589531
x26=−5434.89219988536x_{26} = -5434.89219988536
x27=4378.31538513339x_{27} = 4378.31538513339
x28=−3690.34033347607x_{28} = -3690.34033347607
x29=−9578.21745022251x_{29} = -9578.21745022251
x30=−7615.58846198222x_{30} = -7615.58846198222
x31=−9142.07756058549x_{31} = -9142.07756058549
x32=5468.66132960416x_{32} = 5468.66132960416
x33=1761.51133313114x_{33} = 1761.51133313114
x34=3724.10927995637x_{34} = 3724.10927995637
x35=4814.453480199x_{35} = 4814.453480199
x36=−1073.57335953873x_{36} = -1073.57335953873
x37=8085.49732256461x_{37} = 8085.49732256461
x38=5686.73075789502x_{38} = 5686.73075789502
x39=−7397.5186754556x_{39} = -7397.5186754556
x40=−3908.40882950243x_{40} = -3908.40882950243
x41=−3036.13634098551x_{41} = -3036.13634098551
x42=10702.3366249177x_{42} = 10702.3366249177
x43=−4562.61531266235x_{43} = -4562.61531266235
x44=−2163.87065514191x_{44} = -2163.87065514191
x45=5250.59196786852x_{45} = 5250.59196786852
x46=−10668.5673785808x_{46} = -10668.5673785808
x47=−5652.96161628327x_{47} = -5652.96161628327
x48=−1945.80636882279x_{48} = -1945.80636882279
x49=6995.14837698266x_{49} = 6995.14837698266
x50=−8924.00763669924x_{50} = -8924.00763669924
x51=3069.90512579814x_{51} = 3069.90512579814
x52=−9360.14749875178x_{52} = -9360.14749875178
x53=−6089.10062403147x_{53} = -6089.10062403147
x54=−1509.6829479089x_{54} = -1509.6829479089
x55=1979.57444104256x_{55} = 1979.57444104256
x56=−10014.3573895997x_{56} = -10014.3573895997
x57=−3472.27205539527x_{57} = -3472.27205539527
x58=9175.84679210093x_{58} = 9175.84679210093
x59=3506.0409579492x_{59} = 3506.0409579492
x60=−5216.82285155971x_{60} = -5216.82285155971
x61=4160.24652336267x_{61} = 4160.24652336267
x62=10266.1966187223x_{62} = 10266.1966187223
x63=8957.77686545636x_{63} = 8957.77686545636
x64=3942.17781279665x_{64} = 3942.17781279665
x65=7213.21811278518x_{65} = 7213.21811278518
x66=−5871.03109316551x_{66} = -5871.03109316551
x67=5032.5226813311x_{67} = 5032.5226813311
x68=−8051.72810716762x_{68} = -8051.72810716762
x69=5904.80024537367x_{69} = 5904.80024537367
x70=−7833.65827322243x_{70} = -7833.65827322243
x71=10048.1266304085x_{71} = 10048.1266304085
x72=1325.39210843559x_{72} = 1325.39210843559
x73=−1291.62555068893x_{73} = -1291.62555068893
x74=−8487.86783617086x_{74} = -8487.86783617086
x75=8739.7069539558x_{75} = 8739.7069539558
x76=−6961.37918583642x_{76} = -6961.37918583642
x77=6777.07867298x_{77} = 6777.07867298
x78=−855.530336490659x_{78} = -855.530336490659
x79=1543.4502282718x_{79} = 1543.4502282718
x80=6340.93937348765x_{80} = 6340.93937348765
x81=−8705.93772816645x_{81} = -8705.93772816645
x82=−6307.17020327982x_{82} = -6307.17020327982
x83=−2600.00221972999x_{83} = -2600.00221972999
x84=9393.91673283538x_{84} = 9393.91673283538
x85=7867.42748456496x_{85} = 7867.42748456496
x86=8521.63705876126x_{86} = 8521.63705876126
x87=6559.00900394663x_{87} = 6559.00900394663
x88=2851.83773954251x_{88} = 2851.83773954251
x89=10920.4066416189x_{89} = 10920.4066416189
x90=−10450.497372488x_{90} = -10450.497372488
x91=−10886.6373936567x_{91} = -10886.6373936567
x92=−3254.20403909026x_{92} = -3254.20403909026
x93=−10232.4273759528x_{93} = -10232.4273759528
x94=3287.97288864819x_{94} = 3287.97288864819
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:
Cóncava en los intervalos
[9393.91673283538,∞)\left[9393.91673283538, \infty\right)
Convexa en los intervalos
(−∞,−10450.497372488]\left(-\infty, -10450.497372488\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→−∞xsin⁡(1x)3=1\lim_{x \to -\infty} \sqrt[3]{x \sin{\left(\frac{1}{x} \right)}} = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=1y = 1
lim⁡x→∞xsin⁡(1x)3=1\lim_{x \to \infty} \sqrt[3]{x \sin{\left(\frac{1}{x} \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*sin(1/x))^(1/3), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(xsin⁡(1x)3x)=0\lim_{x \to -\infty}\left(\frac{\sqrt[3]{x \sin{\left(\frac{1}{x} \right)}}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(xsin⁡(1x)3x)=0\lim_{x \to \infty}\left(\frac{\sqrt[3]{x \sin{\left(\frac{1}{x} \right)}}}{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:
xsin⁡(1x)3=xsin⁡(1x)3\sqrt[3]{x \sin{\left(\frac{1}{x} \right)}} = \sqrt[3]{x \sin{\left(\frac{1}{x} \right)}}
- Sí
xsin⁡(1x)3=−xsin⁡(1x)3\sqrt[3]{x \sin{\left(\frac{1}{x} \right)}} = - \sqrt[3]{x \sin{\left(\frac{1}{x} \right)}}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор