Sr Examen

Otras calculadoras

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

Gráfico de la función y = sin(1/x)/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /1\
       sin|-|
          \x/
f(x) = ------
         x
f(x)=sin⁡(1x)xf{\left(x \right)} = \frac{\sin{\left(\frac{1}{x} \right)}}{x} 
f = sin(1/x)/x
Gráfico de la función
02468-8-6-4-2-1010-2020
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:
sin⁡(1x)x=0\frac{\sin{\left(\frac{1}{x} \right)}}{x} = 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 sin(1/x)/x.
sin⁡(10)0\frac{\sin{\left(\frac{1}{0} \right)}}{0}
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
−sin⁡(1x)x2−cos⁡(1x)x3=0- \frac{\sin{\left(\frac{1}{x} \right)}}{x^{2}} - \frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=31427.7538525251x_{1} = 31427.7538525251
x2=19561.7681653678x_{2} = 19561.7681653678
x3=−11802.9219858377x_{3} = -11802.9219858377
x4=−21973.2101350335x_{4} = -21973.2101350335
x5=32275.3375870094x_{5} = 32275.3375870094
x6=26342.2791561729x_{6} = 26342.2791561729
x7=−38077.2183563533x_{7} = -38077.2183563533
x8=14476.5835259143x_{8} = 14476.5835259143
x9=28885.0098692674x_{9} = 28885.0098692674
x10=−17735.4558269403x_{10} = -17735.4558269403
x11=−39772.3993665363x_{11} = -39772.3993665363
x12=40751.2218421629x_{12} = 40751.2218421629
x13=−22820.7728941576x_{13} = -22820.7728941576
x14=12781.5999770302x_{14} = 12781.5999770302
x15=36513.2705382744x_{15} = 36513.2705382744
x16=21256.8765949775x_{16} = 21256.8765949775
x17=−32991.6922603157x_{17} = -32991.6922603157
x18=34818.0947801175x_{18} = 34818.0947801175
x19=−19430.5435758771x_{19} = -19430.5435758771
x20=−14345.3659691272x_{20} = -14345.3659691272
x21=23799.5659677507x_{21} = 23799.5659677507
x22=−21125.6507050534x_{22} = -21125.6507050534
x23=22951.9998063081x_{23} = 22951.9998063081
x24=−41467.5826221759x_{24} = -41467.5826221759
x25=−25363.4778078947x_{25} = -25363.4778078947
x26=25494.7058871718x_{26} = 25494.7058871718
x27=−12650.3868310499x_{27} = -12650.3868310499
x28=20409.3203021652x_{28} = 20409.3203021652
x29=18714.2207495256x_{29} = 18714.2207495256
x30=13629.0847527609x_{30} = 13629.0847527609
x31=−24515.9070182535x_{31} = -24515.9070182535
x32=22104.4365655516x_{32} = 22104.4365655516
x33=27189.8543562071x_{33} = 27189.8543562071
x34=11934.1321841624x_{34} = 11934.1321841624
x35=−36382.0399055397x_{35} = -36382.0399055397
x36=−34686.8643893828x_{36} = -34686.8643893828
x37=15324.0939721606x_{37} = 15324.0939721606
x38=−31296.5240688142x_{38} = -31296.5240688142
x39=16171.6142548589x_{39} = 16171.6142548589
x40=−30448.9416552006x_{40} = -30448.9416552006
x41=29732.5898907212x_{41} = 29732.5898907212
x42=−27058.6256744435x_{42} = -27058.6256744435
x43=39056.0395006955x_{43} = 39056.0395006955
x44=−37229.6287891721x_{44} = -37229.6287891721
x45=33122.922370868x_{45} = 33122.922370868
x46=30580.1712546872x_{46} = 30580.1712546872
x47=−40619.9907312464x_{47} = -40619.9907312464
x48=−13497.8691947406x_{48} = -13497.8691947406
x49=17866.6787268417x_{49} = 17866.6787268417
x50=−20278.0950218298x_{50} = -20278.0950218298
x51=28037.4313121062x_{51} = 28037.4313121062
x52=−23668.3386246561x_{52} = -23668.3386246561
x53=42446.4062709612x_{53} = 42446.4062709612
x54=−33839.277869644x_{54} = -33839.277869644
x55=−18582.9969472475x_{55} = -18582.9969472475
x56=−16040.3936060323x_{56} = -16040.3936060323
x57=17019.1429035105x_{57} = 17019.1429035105
x58=−28753.7806881948x_{58} = -28753.7806881948
x59=−16887.9210447625x_{59} = -16887.9210447625
x60=24647.1347484752x_{60} = 24647.1347484752
x61=−42315.1750077004x_{61} = -42315.1750077004
x62=39903.6303938846x_{62} = 39903.6303938846
x63=−27906.2023693417x_{63} = -27906.2023693417
x64=35665.6822704316x_{64} = 35665.6822704316
x65=−35534.4517543759x_{65} = -35534.4517543759
x66=33970.5081255404x_{66} = 33970.5081255404
x67=−26211.0507610785x_{67} = -26211.0507610785
x68=−29601.3604914767x_{68} = -29601.3604914767
x69=37360.8595307203x_{69} = 37360.8595307203
x70=−32144.1076334276x_{70} = -32144.1076334276
x71=38208.4491995411x_{71} = 38208.4491995411
x72=−15192.8747405778x_{72} = -15192.8747405778
x73=41598.8138115962x_{73} = 41598.8138115962
x74=−38924.808562425x_{74} = -38924.808562425
Signos de extremos en los puntos:
(31427.75385252513, 1.01244936863584e-9)

(19561.768165367816, 2.61326700203962e-9)

(-11802.92198583774, 7.17828878815265e-9)

(-21973.210135033478, 2.07115681341263e-9)

(32275.33758700941, 9.59971661769784e-10)

(26342.27915617286, 1.44109729949528e-9)

(-38077.21835635332, 6.89714840869954e-10)

(14476.583525914342, 4.77164184811957e-9)

(28885.0098692674, 1.19854669695631e-9)

(-17735.455826940255, 3.17918126065533e-9)

(-39772.39936653633, 6.32173689106629e-10)

(40751.22184216292, 6.02169466601296e-10)

(-22820.772894157573, 1.92016832722739e-9)

(12781.599977030211, 6.12110115785768e-9)

(36513.27053827435, 7.50064359890521e-10)

(21256.876594977497, 2.21310030110204e-9)

(-32991.69226031572, 9.1873617026553e-10)

(34818.09478011748, 8.24878518050218e-10)

(-19430.54357587707, 2.6486837018706e-9)

(-14345.365969127153, 4.85933382770265e-9)

(23799.565967750717, 1.76547643881391e-9)

(-21125.650705053413, 2.24067985801215e-9)

(22951.999806308108, 1.89827416311931e-9)

(-41467.58262217587, 5.81543977719108e-10)

(-25363.47780789469, 1.55447017378804e-9)

(25494.705887171793, 1.53850880987833e-9)

(-12650.386831049884, 6.2487390379762e-9)

(20409.320302165244, 2.4007277737314e-9)

(18714.220749525568, 2.85533128463339e-9)

(13629.084752760904, 5.38352352507529e-9)

(-24515.90701825348, 1.66381129723624e-9)

(22104.43656555161, 2.04663831900024e-9)

(27189.854356207117, 1.35265249323197e-9)

(11934.132184162385, 7.02131264624273e-9)

(-36382.03990553967, 7.55485108297283e-10)

(-34686.864389382834, 8.31131830416913e-10)

(15324.09397216062, 4.25843856872904e-9)

(-31296.524068814153, 1.02095779385533e-9)

(16171.614254858863, 3.82378312981834e-9)

(-30448.941655200615, 1.07858802910931e-9)

(29732.58989072119, 1.13118729530794e-9)

(-27058.625674443527, 1.36580447228137e-9)

(39056.03950069551, 6.555768321039e-10)

(-37229.62878917209, 7.21477161603512e-10)

(33122.922370867964, 9.11470687336713e-10)

(30580.171254687153, 1.06935073800839e-9)

(-40619.99073124638, 6.06066612668086e-10)

(-13497.869194740559, 5.48870095304181e-9)

(17866.678726841652, 3.13265337670453e-9)

(-20278.09502182977, 2.43189988518305e-9)

(28037.43131210623, 1.27210674293587e-9)

(-23668.338624656142, 1.78510781631491e-9)

(42446.4062709612, 5.5503215853839e-10)

(-33839.277869644044, 8.73288672042809e-10)

(-18582.99694724754, 2.89579949932944e-9)

(-16040.393606032281, 3.88660099711682e-9)

(17019.142903510474, 3.45242799797342e-9)

(-28753.78068819479, 1.20951173961037e-9)

(-16887.92104476253, 3.50628827966937e-9)

(24647.13474847516, 1.64614133781089e-9)

(-42315.17500770036, 5.58480118576188e-10)

(39903.63039388465, 6.28022468513138e-10)

(-27906.202369341747, 1.28409903998383e-9)

(35665.68227043159, 7.8613824940945e-10)

(-35534.45175437587, 7.91955467372039e-10)

(33970.50812554042, 8.66554563844005e-10)

(-26211.05076107848, 1.45556343397359e-9)

(-29601.360491476687, 1.14123913560002e-9)

(37360.85953072034, 7.16417658076677e-10)

(-32144.107633427648, 9.67825928054685e-10)

(38208.44919954114, 6.84985184164745e-10)

(-15192.874740577829, 4.33231558626961e-9)

(41598.813811596214, 5.77880588151573e-10)

(-38924.80856242503, 6.60004701622595e-10)


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
Decrece 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
2sin⁡(1x)+2cos⁡(1x)−sin⁡(1x)xx+2cos⁡(1x)xx3=0\frac{2 \sin{\left(\frac{1}{x} \right)} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x} + \frac{2 \cos{\left(\frac{1}{x} \right)}}{x}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=10266.3810605679x_{1} = 10266.3810605679
x2=6777.35808590346x_{2} = 6777.35808590346
x3=−9142.28468120415x_{3} = -9142.28468120415
x4=6995.41907802694x_{4} = 6995.41907802694
x5=10702.5135499073x_{5} = 10702.5135499073
x6=5905.12094237548x_{6} = 5905.12094237548
x7=−2164.74632169628x_{7} = -2164.74632169628
x8=6341.238008277x_{8} = 6341.238008277
x9=−3036.76020548816x_{9} = -3036.76020548816
x10=−8924.21981899891x_{10} = -8924.21981899891
x11=−6525.53002008465x_{11} = -6525.53002008465
x12=6123.17905862427x_{12} = 6123.17905862427
x13=−4563.03036925716x_{13} = -4563.03036925716
x14=2416.48880150633x_{14} = 2416.48880150633
x15=8085.73151442808x_{15} = 8085.73151442808
x16=3288.54895555541x_{16} = 3288.54895555541
x17=−4344.98224128159x_{17} = -4344.98224128159
x18=6559.29770818451x_{18} = 6559.29770818451
x19=−4126.93645068434x_{19} = -4126.93645068434
x20=5687.06375543933x_{20} = 5687.06375543933
x21=5469.00760902946x_{21} = 5469.00760902946
x22=−8706.15522575402x_{22} = -8706.15522575402
x23=10048.3150754106x_{23} = 10048.3150754106
x24=−6089.4116072124x_{24} = -6089.4116072124
x25=1762.58748876663x_{25} = 1762.58748876663
x26=8303.79522195176x_{26} = 8303.79522195176
x27=4160.70174963449x_{27} = 4160.70174963449
x28=2198.50119034854x_{28} = 2198.50119034854
x29=3724.61784511459x_{29} = 3724.61784511459
x30=−5217.18584464859x_{30} = -5217.18584464859
x31=−2600.73083319832x_{31} = -2600.73083319832
x32=−7397.77464675724x_{32} = -7397.77464675724
x33=−8270.02693126976x_{33} = -8270.02693126976
x34=−3690.85354049825x_{34} = -3690.85354049825
x35=5032.89897907618x_{35} = 5032.89897907618
x36=−3254.78606949773x_{36} = -3254.78606949773
x37=−10668.7448619236x_{37} = -10668.7448619236
x38=4596.79639205497x_{38} = 4596.79639205497
x39=−5435.24062483556x_{39} = -5435.24062483556
x40=−10014.5464681664x_{40} = -10014.5464681664
x41=−6307.47043240041x_{41} = -6307.47043240041
x42=−8051.96327835772x_{42} = -8051.96327835772
x43=2852.50197826512x_{43} = 2852.50197826512
x44=10920.5800333585x_{44} = 10920.5800333585
x45=1980.53182026938x_{45} = 1980.53182026938
x46=−7179.71266310646x_{46} = -7179.71266310646
x47=3070.52214346087x_{47} = 3070.52214346087
x48=4814.84682750572x_{48} = 4814.84682750572
x49=−2382.73142636039x_{49} = -2382.73142636039
x50=7431.54268908389x_{50} = 7431.54268908389
x51=8739.92361362634x_{51} = 8739.92361362634
x52=4378.74792929501x_{52} = 4378.74792929501
x53=−10232.612424687x_{53} = -10232.612424687
x54=10484.4472223324x_{54} = 10484.4472223324
x55=8521.85926334984x_{55} = 8521.85926334984
x56=−9578.4151390019x_{56} = -9578.4151390019
x57=3942.65823222572x_{57} = 3942.65823222572
x58=9176.05315271381x_{58} = 9176.05315271381
x59=9830.24927861521x_{59} = 9830.24927861521
x60=7213.48062881799x_{60} = 7213.48062881799
x61=−7615.83710277534x_{61} = -7615.83710277534
x62=9394.11830258816x_{62} = 9394.11830258816
x63=9612.18368300356x_{63} = 9612.18368300356
x64=5250.95263284462x_{64} = 5250.95263284462
x65=−5871.3536295173x_{65} = -5871.3536295173
x66=−2818.74121582482x_{66} = -2818.74121582482
x67=−7833.89999170286x_{67} = -7833.89999170286
x68=3506.58117292141x_{68} = 3506.58117292141
x69=−6961.65119622668x_{69} = -6961.65119622668
x70=−10450.678559586x_{70} = -10450.678559586
x71=−6743.59029626745x_{71} = -6743.59029626745
x72=−4781.08051455468x_{72} = -4781.08051455468
x73=−4999.13241300762x_{73} = -4999.13241300762
x74=7649.60521524296x_{74} = 7649.60521524296
x75=7867.66816854676x_{75} = 7867.66816854676
x76=−9360.34979356815x_{76} = -9360.34979356815
x77=−5653.29659743359x_{77} = -5653.29659743359
x78=2634.49011359224x_{78} = 2634.49011359224
x79=8957.98825021979x_{79} = 8957.98825021979
x80=−8488.09092220372x_{80} = -8488.09092220372
x81=−1728.84075562599x_{81} = -1728.84075562599
x82=−3472.81751110804x_{82} = -3472.81751110804
x83=−10886.8113216367x_{83} = -10886.8113216367
x84=−1946.78034201181x_{84} = -1946.78034201181
x85=−3908.89338902105x_{85} = -3908.89338902105
x86=1544.67885484452x_{86} = 1544.67885484452
x87=−9796.48070193829x_{87} = -9796.48070193829
x88=−1510.93905386622x_{88} = -1510.93905386622
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→−∞(sin⁡(1x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(1x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x}\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 sin(1/x)/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(1x)x2)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(1x)x2)=0\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{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:
sin⁡(1x)x=sin⁡(1x)x\frac{\sin{\left(\frac{1}{x} \right)}}{x} = \frac{\sin{\left(\frac{1}{x} \right)}}{x}
- No
sin⁡(1x)x=−sin⁡(1x)x\frac{\sin{\left(\frac{1}{x} \right)}}{x} = - \frac{\sin{\left(\frac{1}{x} \right)}}{x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор