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Trazado el gráfico de la función/ (2*x)*sin(1/x)+cos(1/x)

Gráfico de la función y = (2*x)*sin(1/x)+cos(1/x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
              /1\      /1\
f(x) = 2*x*sin|-| + cos|-|
              \x/      \x/
f(x)=2xsin⁡(1x)+cos⁡(1x)f{\left(x \right)} = 2 x \sin{\left(\frac{1}{x} \right)} + \cos{\left(\frac{1}{x} \right)} 
f = (2*x)*sin(1/x) + cos(1/x)
Gráfico de la función
02468-8-6-4-2-10105-5
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:
2xsin⁡(1x)+cos⁡(1x)=02 x \sin{\left(\frac{1}{x} \right)} + \cos{\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 (2*x)*sin(1/x) + cos(1/x).
0⋅2sin⁡(10)+cos⁡(10)0 \cdot 2 \sin{\left(\frac{1}{0} \right)} + \cos{\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
2sin⁡(1x)−2cos⁡(1x)x+sin⁡(1x)x2=02 \sin{\left(\frac{1}{x} \right)} - \frac{2 \cos{\left(\frac{1}{x} \right)}}{x} + \frac{\sin{\left(\frac{1}{x} \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−23667.7807761837x_{1} = -23667.7807761837
x2=−24515.3684606503x_{2} = -24515.3684606503
x3=15323.23216915x_{3} = 15323.23216915
x4=−14344.4453659261x_{4} = -14344.4453659261
x5=−16039.5703456219x_{5} = -16039.5703456219
x6=21256.2554233482x_{6} = 21256.2554233482
x7=−36381.6770245262x_{7} = -36381.6770245262
x8=22103.839219867x_{8} = 22103.839219867
x9=−37229.2741705613x_{9} = -37229.2741705613
x10=−10954.2729524476x_{10} = -10954.2729524476
x11=25494.1879976264x_{11} = 25494.1879976264
x12=14475.6712322997x_{12} = 14475.6712322997
x13=−39772.0674218958x_{13} = -39772.0674218958
x14=−30448.5080552154x_{14} = -30448.5080552154
x15=41598.4964374195x_{15} = 41598.4964374195
x16=33970.119473448x_{16} = 33970.119473448
x17=−41467.2642484886x_{17} = -41467.2642484886
x18=27189.3687618399x_{18} = 27189.3687618399
x19=42446.0952348827x_{19} = 42446.0952348827
x20=−34686.483772001x_{20} = -34686.483772001
x21=11933.0253359143x_{21} = 11933.0253359143
x22=−32143.6969032752x_{22} = -32143.6969032752
x23=28036.9604004731x_{23} = 28036.9604004731
x24=13628.1156823495x_{24} = 13628.1156823495
x25=−26210.5470412737x_{25} = -26210.5470412737
x26=−21972.6092384625x_{26} = -21972.6092384625
x27=−38924.4693889889x_{27} = -38924.4693889889
x28=−33838.8877175908x_{28} = -33838.8877175908
x29=−31296.1022134924x_{29} = -31296.1022134924
x30=−29600.9144741046x_{30} = -29600.9144741046
x31=34817.7155904103x_{31} = 34817.7155904103
x32=40750.8978661988x_{32} = 40750.8978661988
x33=35665.3120932869x_{33} = 35665.3120932869
x34=−17734.7112890337x_{34} = -17734.7112890337
x35=22951.4245261489x_{35} = 22951.4245261489
x36=17018.3669835115x_{36} = 17018.3669835115
x37=−13496.8907426424x_{37} = -13496.8907426424
x38=−18582.2863816239x_{38} = -18582.2863816239
x39=30579.7395065666x_{39} = 30579.7395065666
x40=38208.1036605632x_{40} = 38208.1036605632
x41=−12649.3427697718x_{41} = -12649.3427697718
x42=−11801.8028806562x_{42} = -11801.8028806562
x43=28884.5527784603x_{43} = 28884.5527784603
x44=−32991.2920835956x_{44} = -32991.2920835956
x45=−27905.7292537638x_{45} = -27905.7292537638
x46=29732.1458325661x_{46} = 29732.1458325661
x47=37360.5061517164x_{47} = 37360.5061517164
x48=−20277.4438767813x_{48} = -20277.4438767813
x49=36512.9089552013x_{49} = 36512.9089552013
x50=12780.5665913212x_{50} = 12780.5665913212
x51=−19429.8640171832x_{51} = -19429.8640171832
x52=16170.7976456873x_{52} = 16170.7976456873
x53=−35534.0802167403x_{53} = -35534.0802167403
x54=−21125.025692681x_{54} = -21125.025692681
x55=31427.3337503038x_{55} = 31427.3337503038
x56=18713.51514406x_{56} = 18713.51514406
x57=24646.5990449056x_{57} = 24646.5990449056
x58=−40619.6657136875x_{58} = -40619.6657136875
x59=17865.9396329837x_{59} = 17865.9396329837
x60=−27058.1377362019x_{60} = -27058.1377362019
x61=39903.299535638x_{61} = 39903.299535638
x62=−16887.1391223858x_{62} = -16887.1391223858
x63=33122.5237720275x_{63} = 33122.5237720275
x64=39055.7014614063x_{64} = 39055.7014614063
x65=−28753.321521205x_{65} = -28753.321521205
x66=−22820.1943213495x_{66} = -22820.1943213495
x67=26341.7779339352x_{67} = 26341.7779339352
x68=20408.6733247176x_{68} = 20408.6733247176
x69=23799.0111808714x_{69} = 23799.0111808714
x70=11085.4937140326x_{70} = 11085.4937140326
x71=−38076.8716322562x_{71} = -38076.8716322562
x72=32274.9285188887x_{72} = 32274.9285188887
x73=−42314.8630116956x_{73} = -42314.8630116956
x74=−25362.9572514602x_{74} = -25362.9572514602
x75=−15192.0055262454x_{75} = -15192.0055262454
x76=19561.0931447107x_{76} = 19561.0931447107
Signos de extremos en los puntos:
(-23667.780776183714, 2.99999999851234)

(-24515.368460650297, 2.99999999861343)

(15323.23216915003, 2.9999999964509)

(-14344.44536592608, 2.99999999595004)

(-16039.57034562194, 2.99999999676083)

(21256.255423348193, 2.99999999815564)

(-36381.67702452621, 2.99999999937042)

(22103.839219867034, 2.99999999829438)

(-37229.27417056133, 2.99999999939876)

(-10954.272952447594, 2.99999999305533)

(25494.187997626384, 2.99999999871786)

(14475.671232299745, 2.99999999602313)

(-39772.06742189575, 2.99999999947318)

(-30448.508055215378, 2.99999999910115)

(41598.49643741948, 2.99999999951843)

(33970.11947344803, 2.99999999927785)

(-41467.26424848864, 2.99999999951537)

(27189.368761839905, 2.99999999887275)

(42446.09523488272, 2.99999999953747)

(-34686.48377200103, 2.99999999930737)

(11933.025335914264, 2.99999999414782)

(-32143.69690327515, 2.99999999919346)

(28036.96040047314, 2.99999999893988)

(13628.11568234953, 2.99999999551309)

(-26210.547041273716, 2.99999999878698)

(-21972.609238462534, 2.99999999827394)

(-38924.46938898889, 2.99999999944999)

(-33838.887717590784, 2.99999999927224)

(-31296.102213492435, 2.99999999914918)

(-29600.91447410458, 2.99999999904894)

(34817.71559041035, 2.99999999931259)

(40750.897866198764, 2.99999999949818)

(35665.312093286906, 2.99999999934487)

(-17734.71128903371, 2.99999999735046)

(22951.42452614892, 2.99999999841803)

(17018.36698351151, 2.99999999712271)

(-13496.890742642408, 2.99999999542542)

(-18582.286381623937, 2.99999999758665)

(30579.739506566602, 2.99999999910885)

(38208.10366056323, 2.99999999942917)

(-12649.342769771787, 2.99999999479186)

(-11801.80288065619, 2.99999999401696)

(28884.55277846027, 2.99999999900118)

(-32991.2920835956, 2.99999999923437)

(-27905.72925376384, 2.99999999892988)

(29732.14583256606, 2.99999999905732)

(37360.50615171638, 2.99999999940297)

(-20277.44387678132, 2.99999999797329)

(36512.90895520135, 2.99999999937493)

(12780.566591321194, 2.99999999489826)

(-19429.864017183212, 2.99999999779261)

(16170.797645687257, 2.99999999681319)

(-35534.08021674026, 2.99999999934002)

(-21125.025692681, 2.99999999813266)

(31427.33375030377, 2.99999999915627)

(18713.51514405997, 2.99999999762038)

(24646.59904490561, 2.99999999862816)

(-40619.66571368752, 2.99999999949494)

(17865.939632983664, 2.99999999738924)

(-27058.13773620189, 2.99999999886179)

(39903.29953563803, 2.99999999947664)

(-16887.139122385786, 2.99999999707782)

(33122.52377202749, 2.99999999924042)

(39055.701461406345, 2.99999999945368)

(-28753.321521205013, 2.99999999899204)

(-22820.194321349496, 2.99999999839978)

(26341.777933935235, 2.99999999879904)

(20408.673324717554, 2.99999999799927)

(23799.011180871425, 2.9999999985287)

(11085.493714032567, 2.99999999321877)

(-38076.87163225615, 2.99999999942523)

(32274.92851888869, 2.9999999992)

(-42314.86301169557, 2.99999999953459)

(-25362.95725146023, 2.99999999870456)

(-15192.005526245362, 2.99999999638932)

(19561.093144710703, 2.99999999782213)


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=−10954.2729524476x_{1} = -10954.2729524476
x2=−32143.6969032752x_{2} = -32143.6969032752
x3=−42314.8630116956x_{3} = -42314.8630116956
Puntos máximos de la función:
x3=−21972.6092384625x_{3} = -21972.6092384625
x3=30579.7395065666x_{3} = 30579.7395065666
x3=−20277.4438767813x_{3} = -20277.4438767813
x3=23799.0111808714x_{3} = 23799.0111808714
x3=−25362.9572514602x_{3} = -25362.9572514602
Decrece en los intervalos
[−32143.6969032752,−25362.9572514602]∪[−10954.2729524476,∞)\left[-32143.6969032752, -25362.9572514602\right] \cup \left[-10954.2729524476, \infty\right)
Crece en los intervalos
(−∞,−42314.8630116956]\left(-\infty, -42314.8630116956\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
−4sin⁡(1x)+cos⁡(1x)xx3=0- \frac{4 \sin{\left(\frac{1}{x} \right)} + \frac{\cos{\left(\frac{1}{x} \right)}}{x}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−6525.36713014802x_{1} = -6525.36713014802
x2=5904.94092902201x_{2} = 5904.94092902201
x3=−7397.63096784987x_{3} = -7397.63096784987
x4=−6961.49851379555x_{4} = -6961.49851379555
x5=3942.38854642211x_{5} = 3942.38854642211
x6=10702.414242073x_{6} = 10702.414242073
x7=2634.08627873514x_{7} = 2634.08627873514
x8=4596.56511220308x_{8} = 4596.56511220308
x9=7649.46626564119x_{9} = 7649.46626564119
x10=−8051.8312754052x_{10} = -8051.8312754052
x11=−2600.32176132822x_{11} = -2600.32176132822
x12=5686.87683680712x_{12} = 5686.87683680712
x13=−4998.91976261784x_{13} = -4998.91976261784
x14=7213.33327604652x_{14} = 7213.33327604652
x15=−9360.2362451289x_{15} = -9360.2362451289
x16=−8924.10072045444x_{16} = -8924.10072045444
x17=−3254.45932630835x_{17} = -3254.45932630835
x18=−3472.51130669693x_{18} = -3472.51130669693
x19=−5216.98208675838x_{19} = -5216.98208675838
x20=−3036.40996846126x_{20} = -3036.40996846126
x21=−8269.89840968735x_{21} = -8269.89840968735
x22=8303.66722154174x_{22} = 8303.66722154174
x23=4814.6260284385x_{23} = 4814.6260284385
x24=6341.07038000669x_{24} = 6341.07038000669
x25=−4780.85815987876x_{25} = -4780.85815987876
x26=6777.20124802895x_{26} = 6777.20124802895
x27=−10232.5085568502x_{27} = -10232.5085568502
x28=8957.86959935446x_{28} = 8957.86959935446
x29=−2382.28483739691x_{29} = -2382.28483739691
x30=−5435.0450454479x_{30} = -5435.0450454479
x31=2852.12906550084x_{31} = 2852.12906550084
x32=−7615.69753880319x_{32} = -7615.69753880319
x33=8085.60006120702x_{33} = 8085.60006120702
x34=6559.13565455569x_{34} = 6559.13565455569
x35=3724.33235560717x_{35} = 3724.33235560717
x36=−3908.62137862313x_{36} = -3908.62137862313
x37=−8487.96570306571x_{37} = -8487.96570306571
x38=9394.00516114469x_{38} = 9394.00516114469
x39=−2818.36384226923x_{39} = -2818.36384226923
x40=−8706.03314358394x_{40} = -8706.03314358394
x41=7867.53307105522x_{41} = 7867.53307105522
x42=−10668.6452406764x_{42} = -10668.6452406764
x43=−4344.73755322751x_{43} = -4344.73755322751
x44=10920.4827087942x_{44} = 10920.4827087942
x45=−1728.22455417254x_{45} = -1728.22455417254
x46=10484.3458487129x_{46} = 10484.3458487129
x47=−6743.43267518848x_{47} = -6743.43267518848
x48=5468.81323411963x_{48} = 5468.81323411963
x49=10266.277533389x_{49} = 10266.277533389
x50=4160.44620913561x_{50} = 4160.44620913561
x51=7431.39966122828x_{51} = 7431.39966122828
x52=5032.68775188277x_{52} = 5032.68775188277
x53=8521.73453899435x_{53} = 8521.73453899435
x54=−4562.79738200223x_{54} = -4562.79738200223
x55=−7179.56461921555x_{55} = -7179.56461921555
x56=9830.14115762299x_{56} = 9830.14115762299
x57=−10450.5768593569x_{57} = -10450.5768593569
x58=3070.17575169201x_{58} = 3070.17575169201
x59=−9796.37220932952x_{59} = -9796.37220932952
x60=3506.27791137237x_{60} = 3506.27791137237
x61=−5871.17258359613x_{61} = -5871.17258359613
x62=1543.98872010884x_{62} = 1543.98872010884
x63=−1510.23345998075x_{63} = -1510.23345998075
x64=1979.99420713499x_{64} = 1979.99420713499
x65=−3690.56544454926x_{65} = -3690.56544454926
x66=4378.50512379739x_{66} = 4378.50512379739
x67=−4126.6788240641x_{67} = -4126.6788240641
x68=9612.07310874602x_{68} = 9612.07310874602
x69=6995.26713059928x_{69} = 6995.26713059928
x70=5250.75018195528x_{70} = 5250.75018195528
x71=10048.2093011818x_{71} = 10048.2093011818
x72=6123.00545845857x_{72} = 6123.00545845857
x73=9175.9373220396x_{73} = 9175.9373220396
x74=1761.98310448831x_{74} = 1761.98310448831
x75=3288.22556128488x_{75} = 3288.22556128488
x76=−5653.10856523835x_{76} = -5653.10856523835
x77=−9142.16842391901x_{77} = -9142.16842391901
x78=−7833.76431350715x_{78} = -7833.76431350715
x79=8739.80200180385x_{79} = 8739.80200180385
x80=−2164.25463078275x_{80} = -2164.25463078275
x81=−10886.7136960724x_{81} = -10886.7136960724
x82=−1946.23340198059x_{82} = -1946.23340198059
x83=2198.01704835973x_{83} = 2198.01704835973
x84=−6089.2370469473x_{84} = -6089.2370469473
x85=2416.04844774274x_{85} = 2416.04844774274
x86=−6307.30190912842x_{86} = -6307.30190912842
x87=−10014.4403383052x_{87} = -10014.4403383052
x88=−9578.30417604128x_{88} = -9578.30417604128
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
[8739.80200180385,∞)\left[8739.80200180385, \infty\right)
Convexa en los intervalos
(−∞,8739.80200180385]\left(-\infty, 8739.80200180385\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→−∞(2xsin⁡(1x)+cos⁡(1x))=3\lim_{x \to -\infty}\left(2 x \sin{\left(\frac{1}{x} \right)} + \cos{\left(\frac{1}{x} \right)}\right) = 3
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=3y = 3
lim⁡x→∞(2xsin⁡(1x)+cos⁡(1x))=3\lim_{x \to \infty}\left(2 x \sin{\left(\frac{1}{x} \right)} + \cos{\left(\frac{1}{x} \right)}\right) = 3
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=3y = 3
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (2*x)*sin(1/x) + cos(1/x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(2xsin⁡(1x)+cos⁡(1x)x)=0\lim_{x \to -\infty}\left(\frac{2 x \sin{\left(\frac{1}{x} \right)} + \cos{\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→∞(2xsin⁡(1x)+cos⁡(1x)x)=0\lim_{x \to \infty}\left(\frac{2 x \sin{\left(\frac{1}{x} \right)} + \cos{\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:
2xsin⁡(1x)+cos⁡(1x)=2xsin⁡(1x)+cos⁡(1x)2 x \sin{\left(\frac{1}{x} \right)} + \cos{\left(\frac{1}{x} \right)} = 2 x \sin{\left(\frac{1}{x} \right)} + \cos{\left(\frac{1}{x} \right)}
- No
2xsin⁡(1x)+cos⁡(1x)=−2xsin⁡(1x)−cos⁡(1x)2 x \sin{\left(\frac{1}{x} \right)} + \cos{\left(\frac{1}{x} \right)} = - 2 x \sin{\left(\frac{1}{x} \right)} - \cos{\left(\frac{1}{x} \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор