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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       -1 + cos(2*x)
f(x) = -------------
              2     
             x
f(x)=cos⁡(2x)−1x2f{\left(x \right)} = \frac{\cos{\left(2 x \right)} - 1}{x^{2}} 
f = (cos(2*x) - 1)/x^2
Gráfico de la función
02468-8-6-4-2-10102-4
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:
cos⁡(2x)−1x2=0\frac{\cos{\left(2 x \right)} - 1}{x^{2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=πx_{1} = \pi
Solución numérica
x1=81.6814091740375x_{1} = 81.6814091740375
x2=91.1061867208972x_{2} = 91.1061867208972
x3=−40.8407042430283x_{3} = -40.8407042430283
x4=9.42477816834986x_{4} = 9.42477816834986
x5=50.2654824463366x_{5} = 50.2654824463366
x6=47.1238899954189x_{6} = 47.1238899954189
x7=−56.5486675120423x_{7} = -56.5486675120423
x8=31.4159268994904x_{8} = 31.4159268994904
x9=69.1150381457816x_{9} = 69.1150381457816
x10=−6.28318511325755x_{10} = -6.28318511325755
x11=40.8407042462337x_{11} = 40.8407042462337
x12=62.8318524971054x_{12} = 62.8318524971054
x13=84.8230011943097x_{13} = 84.8230011943097
x14=53.4070753544334x_{14} = 53.4070753544334
x15=−182.212330270911x_{15} = -182.212330270911
x16=−62.831852823464x_{16} = -62.831852823464
x17=−69.1150386188422x_{17} = -69.1150386188422
x18=97.3893726288047x_{18} = 97.3893726288047
x19=−21.991148586426x_{19} = -21.991148586426
x20=−84.8230013997108x_{20} = -84.8230013997108
x21=72.2566310277176x_{21} = 72.2566310277176
x22=−75.3982253232481x_{22} = -75.3982253232481
x23=3.14159230647624x_{23} = 3.14159230647624
x24=−100.530964668746x_{24} = -100.530964668746
x25=−9.42477811279807x_{25} = -9.42477811279807
x26=3.14159202258703x_{26} = 3.14159202258703
x27=87.9645943356948x_{27} = 87.9645943356948
x28=−87.9645943586046x_{28} = -87.9645943586046
x29=−25.1327414564238x_{29} = -25.1327414564238
x30=−18.8495560547772x_{30} = -18.8495560547772
x31=−3.1415918086506x_{31} = -3.1415918086506
x32=25.1327409761656x_{32} = 25.1327409761656
x33=333.008669396267x_{33} = 333.008669396267
x34=−18.8495563021403x_{34} = -18.8495563021403
x35=−28.274336411954x_{35} = -28.274336411954
x36=69.1150385723079x_{36} = 69.1150385723079
x37=−72.2566308724232x_{37} = -72.2566308724232
x38=−91.1061871962484x_{38} = -91.1061871962484
x39=−75.398223859952x_{39} = -75.398223859952
x40=−47.1238900096216x_{40} = -47.1238900096216
x41=−12.5663703305055x_{41} = -12.5663703305055
x42=25.1327414061616x_{42} = 25.1327414061616
x43=−25.1327413659795x_{43} = -25.1327413659795
x44=40.8407039198138x_{44} = 40.8407039198138
x45=43.98229716939x_{45} = 43.98229716939
x46=−50.2654822927432x_{46} = -50.2654822927432
x47=59.690260594979x_{47} = 59.690260594979
x48=−91.1061871357964x_{48} = -91.1061871357964
x49=18.8495556571753x_{49} = 18.8495556571753
x50=−18.8495556424861x_{50} = -18.8495556424861
x51=−40.8407046587706x_{51} = -40.8407046587706
x52=47.1238895673742x_{52} = 47.1238895673742
x53=−43.98229717452x_{53} = -43.98229717452
x54=−1709.02630102645x_{54} = -1709.02630102645
x55=78.5398161863985x_{55} = 78.5398161863985
x56=18.849559744074x_{56} = 18.849559744074
x57=31.4159267729052x_{57} = 31.4159267729052
x58=21.9911485851759x_{58} = 21.9911485851759
x59=15.7079634314657x_{59} = 15.7079634314657
x60=56.5486676070174x_{60} = 56.5486676070174
x61=53.4070755245567x_{61} = 53.4070755245567
x62=−62.8318532373291x_{62} = -62.8318532373291
x63=12.5663704410235x_{63} = 12.5663704410235
x64=65.9734457528465x_{64} = 65.9734457528465
x65=−81.6814090378975x_{65} = -81.6814090378975
x66=−37.6991118770152x_{66} = -37.6991118770152
x67=−25.1327415660596x_{67} = -25.1327415660596
x68=−3.14159278291973x_{68} = -3.14159278291973
x69=−69.1150385823665x_{69} = -69.1150385823665
x70=−34.5575189305341x_{70} = -34.5575189305341
x71=91.1061871454997x_{71} = 91.1061871454997
x72=−59.6902604575246x_{72} = -59.6902604575246
x73=94.2477796093523x_{73} = 94.2477796093523
x74=−65.9734457649134x_{74} = -65.9734457649134
x75=56.5486679526703x_{75} = 56.5486679526703
x76=−15.7079632962971x_{76} = -15.7079632962971
x77=6.28318528388037x_{77} = 6.28318528388037
x78=75.3982240865665x_{78} = 75.3982240865665
x79=−87.964591359568x_{79} = -87.964591359568
x80=−28.2743337117191x_{80} = -28.2743337117191
x81=−84.82300181x_{81} = -84.82300181
x82=−97.3893724386893x_{82} = -97.3893724386893
x83=−78.5398160908277x_{83} = -78.5398160908277
x84=37.6991120151215x_{84} = 37.6991120151215
x85=−81.6814095337443x_{85} = -81.6814095337443
x86=−37.6991119106453x_{86} = -37.6991119106453
x87=−53.4070752808355x_{87} = -53.4070752808355
x88=9.42477789820696x_{88} = 9.42477789820696
x89=84.823001404869x_{89} = 84.823001404869
x90=28.2743338651556x_{90} = 28.2743338651556
x91=97.3893725100508x_{91} = 97.3893725100508
x92=15.7079650823801x_{92} = 15.7079650823801
x93=−31.4159267006674x_{93} = -31.4159267006674
x94=75.3982239327941x_{94} = 75.3982239327941
x95=−72.2566351619525x_{95} = -72.2566351619525
x96=−47.1238900400185x_{96} = -47.1238900400185
x97=34.5575190268142x_{97} = 34.5575190268142
x98=−94.2477794517506x_{98} = -94.2477794517506
x99=62.8318528264673x_{99} = 62.8318528264673
x100=100.530964765533x_{100} = 100.530964765533
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 (-1 + cos(2*x))/x^2.
−1+cos⁡(0⋅2)02\frac{-1 + \cos{\left(0 \cdot 2 \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
−2sin⁡(2x)x2−2(cos⁡(2x)−1)x3=0- \frac{2 \sin{\left(2 x \right)}}{x^{2}} - \frac{2 \left(\cos{\left(2 x \right)} - 1\right)}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=48.6741442319544x_{1} = 48.6741442319544
x2=34.5575191894877x_{2} = 34.5575191894877
x3=−939.336203423348x_{3} = -939.336203423348
x4=−20.3713029592876x_{4} = -20.3713029592876
x5=−86.3822220347287x_{5} = -86.3822220347287
x6=67.5294347771441x_{6} = 67.5294347771441
x7=−43.9822971502571x_{7} = -43.9822971502571
x8=28.2743338823081x_{8} = 28.2743338823081
x9=−39.2444323611642x_{9} = -39.2444323611642
x10=4.49340945790906x_{10} = 4.49340945790906
x11=86.3822220347287x_{11} = 86.3822220347287
x12=58.1022547544956x_{12} = 58.1022547544956
x13=−80.0981286289451x_{13} = -80.0981286289451
x14=73.8138806006806x_{14} = 73.8138806006806
x15=−12.5663706143592x_{15} = -12.5663706143592
x16=59.6902604182061x_{16} = 59.6902604182061
x17=56.5486677646163x_{17} = 56.5486677646163
x18=100.530964914873x_{18} = 100.530964914873
x19=−54.9596782878889x_{19} = -54.9596782878889
x20=15.707963267949x_{20} = 15.707963267949
x21=−58.1022547544956x_{21} = -58.1022547544956
x22=−15.707963267949x_{22} = -15.707963267949
x23=−37.6991118430775x_{23} = -37.6991118430775
x24=−29.811598790893x_{24} = -29.811598790893
x25=−51.8169824872797x_{25} = -51.8169824872797
x26=14.0661939128315x_{26} = 14.0661939128315
x27=−72.2566310325652x_{27} = -72.2566310325652
x28=−36.1006222443756x_{28} = -36.1006222443756
x29=−11909.7777497589x_{29} = -11909.7777497589
x30=47.1238898038469x_{30} = 47.1238898038469
x31=−89.5242209304172x_{31} = -89.5242209304172
x32=65.9734457253857x_{32} = 65.9734457253857
x33=12.5663706143592x_{33} = 12.5663706143592
x34=21.9911485751286x_{34} = 21.9911485751286
x35=−3.14159265358979x_{35} = -3.14159265358979
x36=−6.28318530717959x_{36} = -6.28318530717959
x37=−65.9734457253857x_{37} = -65.9734457253857
x38=36.1006222443756x_{38} = 36.1006222443756
x39=7.72525183693771x_{39} = 7.72525183693771
x40=81.6814089933346x_{40} = 81.6814089933346
x41=78.5398163397448x_{41} = 78.5398163397448
x42=−73.8138806006806x_{42} = -73.8138806006806
x43=23.519452498689x_{43} = 23.519452498689
x44=−67.5294347771441x_{44} = -67.5294347771441
x45=−83.2401924707234x_{45} = -83.2401924707234
x46=42.3879135681319x_{46} = 42.3879135681319
x47=51.8169824872797x_{47} = 51.8169824872797
x48=−56.5486677646163x_{48} = -56.5486677646163
x49=−97.3893722612836x_{49} = -97.3893722612836
x50=37.6991118430775x_{50} = 37.6991118430775
x51=−21.9911485751286x_{51} = -21.9911485751286
x52=3.14159265358979x_{52} = 3.14159265358979
x53=69.1150383789755x_{53} = 69.1150383789755
x54=29.811598790893x_{54} = 29.811598790893
x55=−50.2654824574367x_{55} = -50.2654824574367
x56=−94.2477796076938x_{56} = -94.2477796076938
x57=80.0981286289451x_{57} = 80.0981286289451
x58=−7.72525183693771x_{58} = -7.72525183693771
x59=−53.4070751110265x_{59} = -53.4070751110265
x60=−61.2447302603744x_{60} = -61.2447302603744
x61=92.6661922776228x_{61} = 92.6661922776228
x62=89.5242209304172x_{62} = 89.5242209304172
x63=−42.3879135681319x_{63} = -42.3879135681319
x64=−59.6902604182061x_{64} = -59.6902604182061
x65=−23.519452498689x_{65} = -23.519452498689
x66=−207.345115136926x_{66} = -207.345115136926
x67=26.6660542588127x_{67} = 26.6660542588127
x68=20.3713029592876x_{68} = 20.3713029592876
x69=−87.9645943005142x_{69} = -87.9645943005142
x70=−28.2743338823081x_{70} = -28.2743338823081
x71=−95.8081387868617x_{71} = -95.8081387868617
x72=64.3871195905574x_{72} = 64.3871195905574
x73=6.28318530717959x_{73} = 6.28318530717959
x74=70.6716857116195x_{74} = 70.6716857116195
x75=45.5311340139913x_{75} = 45.5311340139913
x76=87.9645943005142x_{76} = 87.9645943005142
x77=43.9822971502571x_{77} = 43.9822971502571
x78=185.353966561798x_{78} = 185.353966561798
x79=−4.49340945790906x_{79} = -4.49340945790906
x80=−64.3871195905574x_{80} = -64.3871195905574
x81=−75.398223686155x_{81} = -75.398223686155
x82=−9.42477796076938x_{82} = -9.42477796076938
x83=−14.0661939128315x_{83} = -14.0661939128315
x84=−31.4159265358979x_{84} = -31.4159265358979
x85=−45.5311340139913x_{85} = -45.5311340139913
x86=72.2566310325652x_{86} = 72.2566310325652
x87=94.2477796076938x_{87} = 94.2477796076938
x88=−100.530964914873x_{88} = -100.530964914873
x89=95.8081387868617x_{89} = 95.8081387868617
x90=−81.6814089933346x_{90} = -81.6814089933346
x91=50.2654824574367x_{91} = 50.2654824574367
Signos de extremos en los puntos:
(48.674144231954386, -0.000843820504482794)

(34.55751918948773, 0)

(-939.3362034233481, 0)

(-20.37130295928756, -0.00480780806192296)

(-86.38222203472871, -0.000267992756153105)

(67.52943477714412, -0.000438478740327786)

(-43.982297150257104, 0)

(28.274333882308138, 0)

(-39.24443236116419, -0.00129775286774544)

(4.493409457909064, -0.0943808984516225)

(86.38222203472871, -0.000267992756153105)

(58.10225475449559, -0.000592264082122352)

(-80.09812862894512, -0.000311686196600725)

(73.81388060068065, -0.00036700689023421)

(-12.566370614359172, 0)

(59.69026041820607, 0)

(56.548667764616276, 0)

(100.53096491487338, 0)

(-54.959678287888934, -0.000661908375587791)

(15.707963267948966, 0)

(-58.10225475449559, -0.000592264082122352)

(-15.707963267948966, 0)

(-37.69911184307752, 0)

(-29.81159879089296, -0.00224786935640603)

(-51.81698248727967, -0.000744601728471834)

(14.066193912831473, -0.0100574374624647)

(-72.25663103256524, 0)

(-36.10062224437561, -0.00153344254981861)

(-11909.777749758907, 0)

(47.1238898038469, 0)

(-89.52422093041719, -0.000249513880428108)

(65.97344572538566, 0)

(12.566370614359172, 0)

(21.991148575128552, 0)

(-3.141592653589793, 0)

(-6.283185307179586, 0)

(-65.97344572538566, 0)

(36.10062224437561, -0.00153344254981861)

(7.725251836937707, -0.0329600519859479)

(81.68140899333463, 0)

(78.53981633974483, 0)

(-73.81388060068065, -0.00036700689023421)

(23.519452498689006, -0.00360903571712935)

(-67.52943477714412, -0.000438478740327786)

(-83.2401924707234, -0.00028860321866995)

(42.38791356813192, -0.00111251088673472)

(51.81698248727967, -0.000744601728471834)

(-56.548667764616276, 0)

(-97.3893722612836, 0)

(37.69911184307752, 0)

(-21.991148575128552, 0)

(3.141592653589793, 0)

(69.11503837897546, 0)

(29.81159879089296, -0.00224786935640603)

(-50.26548245743669, 0)

(-94.2477796076938, 0)

(80.09812862894512, -0.000311686196600725)

(-7.725251836937707, -0.0329600519859479)

(-53.40707511102649, 0)

(-61.2447302603744, -0.000533060834814295)

(92.66619227762284, -0.000232882463806292)

(89.52422093041719, -0.000249513880428108)

(-42.38791356813192, -0.00111251088673472)

(-59.69026041820607, 0)

(-23.519452498689006, -0.00360903571712935)

(-207.34511513692635, 0)

(26.666054258812675, -0.0028086792975511)

(20.37130295928756, -0.00480780806192296)

(-87.96459430051421, 0)

(-28.274333882308138, 0)

(-95.8081387868617, -0.000217860190205359)

(64.38711959055742, -0.000482311099451838)

(6.283185307179586, 0)

(70.6716857116195, -0.000400361353240022)

(45.53113401399128, -0.000964281022986201)

(87.96459430051421, 0)

(43.982297150257104, 0)

(185.3539665617978, 0)

(-4.493409457909064, -0.0943808984516225)

(-64.38711959055742, -0.000482311099451838)

(-75.39822368615503, 0)

(-9.42477796076938, 0)

(-14.066193912831473, -0.0100574374624647)

(-31.41592653589793, 0)

(-45.53113401399128, -0.000964281022986201)

(72.25663103256524, 0)

(94.2477796076938, 0)

(-100.53096491487338, 0)

(95.8081387868617, -0.000217860190205359)

(-81.68140899333463, 0)

(50.26548245743669, 0)


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=48.6741442319544x_{1} = 48.6741442319544
x2=−20.3713029592876x_{2} = -20.3713029592876
x3=−86.3822220347287x_{3} = -86.3822220347287
x4=67.5294347771441x_{4} = 67.5294347771441
x5=−39.2444323611642x_{5} = -39.2444323611642
x6=4.49340945790906x_{6} = 4.49340945790906
x7=86.3822220347287x_{7} = 86.3822220347287
x8=58.1022547544956x_{8} = 58.1022547544956
x9=−80.0981286289451x_{9} = -80.0981286289451
x10=73.8138806006806x_{10} = 73.8138806006806
x11=−54.9596782878889x_{11} = -54.9596782878889
x12=−58.1022547544956x_{12} = -58.1022547544956
x13=−29.811598790893x_{13} = -29.811598790893
x14=−51.8169824872797x_{14} = -51.8169824872797
x15=14.0661939128315x_{15} = 14.0661939128315
x16=−36.1006222443756x_{16} = -36.1006222443756
x17=−89.5242209304172x_{17} = -89.5242209304172
x18=36.1006222443756x_{18} = 36.1006222443756
x19=7.72525183693771x_{19} = 7.72525183693771
x20=−73.8138806006806x_{20} = -73.8138806006806
x21=23.519452498689x_{21} = 23.519452498689
x22=−67.5294347771441x_{22} = -67.5294347771441
x23=−83.2401924707234x_{23} = -83.2401924707234
x24=42.3879135681319x_{24} = 42.3879135681319
x25=51.8169824872797x_{25} = 51.8169824872797
x26=29.811598790893x_{26} = 29.811598790893
x27=80.0981286289451x_{27} = 80.0981286289451
x28=−7.72525183693771x_{28} = -7.72525183693771
x29=−61.2447302603744x_{29} = -61.2447302603744
x30=92.6661922776228x_{30} = 92.6661922776228
x31=89.5242209304172x_{31} = 89.5242209304172
x32=−42.3879135681319x_{32} = -42.3879135681319
x33=−23.519452498689x_{33} = -23.519452498689
x34=26.6660542588127x_{34} = 26.6660542588127
x35=20.3713029592876x_{35} = 20.3713029592876
x36=−95.8081387868617x_{36} = -95.8081387868617
x37=64.3871195905574x_{37} = 64.3871195905574
x38=70.6716857116195x_{38} = 70.6716857116195
x39=45.5311340139913x_{39} = 45.5311340139913
x40=−4.49340945790906x_{40} = -4.49340945790906
x41=−64.3871195905574x_{41} = -64.3871195905574
x42=−14.0661939128315x_{42} = -14.0661939128315
x43=−45.5311340139913x_{43} = -45.5311340139913
x44=95.8081387868617x_{44} = 95.8081387868617
Puntos máximos de la función:
x44=34.5575191894877x_{44} = 34.5575191894877
x44=−939.336203423348x_{44} = -939.336203423348
x44=−43.9822971502571x_{44} = -43.9822971502571
x44=28.2743338823081x_{44} = 28.2743338823081
x44=−12.5663706143592x_{44} = -12.5663706143592
x44=59.6902604182061x_{44} = 59.6902604182061
x44=56.5486677646163x_{44} = 56.5486677646163
x44=100.530964914873x_{44} = 100.530964914873
x44=15.707963267949x_{44} = 15.707963267949
x44=−15.707963267949x_{44} = -15.707963267949
x44=−37.6991118430775x_{44} = -37.6991118430775
x44=−72.2566310325652x_{44} = -72.2566310325652
x44=−11909.7777497589x_{44} = -11909.7777497589
x44=47.1238898038469x_{44} = 47.1238898038469
x44=65.9734457253857x_{44} = 65.9734457253857
x44=12.5663706143592x_{44} = 12.5663706143592
x44=21.9911485751286x_{44} = 21.9911485751286
x44=−3.14159265358979x_{44} = -3.14159265358979
x44=−6.28318530717959x_{44} = -6.28318530717959
x44=−65.9734457253857x_{44} = -65.9734457253857
x44=81.6814089933346x_{44} = 81.6814089933346
x44=78.5398163397448x_{44} = 78.5398163397448
x44=−56.5486677646163x_{44} = -56.5486677646163
x44=−97.3893722612836x_{44} = -97.3893722612836
x44=37.6991118430775x_{44} = 37.6991118430775
x44=−21.9911485751286x_{44} = -21.9911485751286
x44=3.14159265358979x_{44} = 3.14159265358979
x44=69.1150383789755x_{44} = 69.1150383789755
x44=−50.2654824574367x_{44} = -50.2654824574367
x44=−94.2477796076938x_{44} = -94.2477796076938
x44=−53.4070751110265x_{44} = -53.4070751110265
x44=−59.6902604182061x_{44} = -59.6902604182061
x44=−207.345115136926x_{44} = -207.345115136926
x44=−87.9645943005142x_{44} = -87.9645943005142
x44=−28.2743338823081x_{44} = -28.2743338823081
x44=6.28318530717959x_{44} = 6.28318530717959
x44=87.9645943005142x_{44} = 87.9645943005142
x44=43.9822971502571x_{44} = 43.9822971502571
x44=185.353966561798x_{44} = 185.353966561798
x44=−75.398223686155x_{44} = -75.398223686155
x44=−9.42477796076938x_{44} = -9.42477796076938
x44=−31.4159265358979x_{44} = -31.4159265358979
x44=72.2566310325652x_{44} = 72.2566310325652
x44=94.2477796076938x_{44} = 94.2477796076938
x44=−100.530964914873x_{44} = -100.530964914873
x44=−81.6814089933346x_{44} = -81.6814089933346
x44=50.2654824574367x_{44} = 50.2654824574367
Decrece en los intervalos
[95.8081387868617,∞)\left[95.8081387868617, \infty\right)
Crece en los intervalos
(−∞,−95.8081387868617]\left(-\infty, -95.8081387868617\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
2(−2cos⁡(2x)+4sin⁡(2x)x+3(cos⁡(2x)−1)x2)x2=0\frac{2 \left(- 2 \cos{\left(2 x \right)} + \frac{4 \sin{\left(2 x \right)}}{x} + \frac{3 \left(\cos{\left(2 x \right)} - 1\right)}{x^{2}}\right)}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=1335.96152783466x_{1} = 1335.96152783466
x2=−71.4570911320099x_{2} = -71.4570911320099
x3=24.3049190457346x_{3} = 24.3049190457346
x4=−77.7414305824544x_{4} = -77.7414305824544
x5=40.0298544804573x_{5} = 40.0298544804573
x6=22.7339953909907x_{6} = 22.7339953909907
x7=8.51135078767434x_{7} = 8.51135078767434
x8=77.7414305824544x_{8} = 77.7414305824544
x9=30.5970389725585x_{9} = 30.5970389725585
x10=−38.4590122703912x_{10} = -38.4590122703912
x11=−47.888731676718x_{11} = -47.888731676718
x12=−99.7354646652444x_{12} = -99.7354646652444
x13=33.7418214227106x_{13} = 33.7418214227106
x14=25.8806097494708x_{14} = 25.8806097494708
x15=−27.4515052410928x_{15} = -27.4515052410928
x16=84.025595846992x_{16} = 84.025595846992
x17=−32.1709600835167x_{17} = -32.1709600835167
x18=−57.3168464266514x_{18} = -57.3168464266514
x19=−5.28103240630265x_{19} = -5.28103240630265
x20=−91.880790070283x_{20} = -91.880790070283
x21=−63.6017131240042x_{21} = -63.6017131240042
x22=66.7440290551475x_{22} = 66.7440290551475
x23=85.5968192243094x_{23} = 85.5968192243094
x24=−11.6898582204548x_{24} = -11.6898582204548
x25=90.3096235943467x_{25} = 90.3096235943467
x26=−35.3151982819233x_{26} = -35.3151982819233
x27=29.0261632399594x_{27} = 29.0261632399594
x28=−69.8862806383665x_{28} = -69.8862806383665
x29=−25.8806097494708x_{29} = -25.8806097494708
x30=62.0301381205536x_{30} = 62.0301381205536
x31=−79.3127250697764x_{31} = -79.3127250697764
x32=−54.1742687855514x_{32} = -54.1742687855514
x33=44.7457194481397x_{33} = 44.7457194481397
x34=−49.4595578500579x_{34} = -49.4595578500579
x35=−24.3049190457346x_{35} = -24.3049190457346
x36=41.6024965392658x_{36} = 41.6024965392658
x37=55.745088528446x_{37} = 55.745088528446
x38=−93.4515946389277x_{38} = -93.4515946389277
x39=−3.70722846405825x_{39} = -3.70722846405825
x40=−41.6024965392658x_{40} = -41.6024965392658
x41=38.4590122703912x_{41} = 38.4590122703912
x42=60.4593229336188x_{42} = 60.4593229336188
x43=742.199918425809x_{43} = 742.199918425809
x44=−76.1706223070459x_{44} = -76.1706223070459
x45=−68.3148408970371x_{45} = -68.3148408970371
x46=96.5935408750673x_{46} = 96.5935408750673
x47=−46.3165498784734x_{47} = -46.3165498784734
x48=−82.4547893068791x_{48} = -82.4547893068791
x49=−55.745088528446x_{49} = -55.745088528446
x50=98.1646611098724x_{50} = 98.1646611098724
x51=−8.51135078767434x_{51} = -8.51135078767434
x52=11.6898582204548x_{52} = 11.6898582204548
x53=−19.5858273496712x_{53} = -19.5858273496712
x54=1.30308171092781x_{54} = 1.30308171092781
x55=46.3165498784734x_{55} = 46.3165498784734
x56=52.6023942608824x_{56} = 52.6023942608824
x57=18.0062837080869x_{57} = 18.0062837080869
x58=−33.7418214227106x_{58} = -33.7418214227106
x59=32.1709600835167x_{59} = 32.1709600835167
x60=54.1742687855514x_{60} = 54.1742687855514
x61=91.880790070283x_{61} = 91.880790070283
x62=74.5992854099041x_{62} = 74.5992854099041
x63=−85.5968192243094x_{63} = -85.5968192243094
x64=−84.025595846992x_{64} = -84.025595846992
x65=68.3148408970371x_{65} = 68.3148408970371
x66=99.7354646652444x_{66} = 99.7354646652444
x67=82.4547893068791x_{67} = 82.4547893068791
x68=−62.0301381205536x_{68} = -62.0301381205536
x69=−102.877368081182x_{69} = -102.877368081182
x70=88.7388184372412x_{70} = 88.7388184372412
x71=47.888731676718x_{71} = 47.888731676718
x72=63.6017131240042x_{72} = 63.6017131240042
x73=−40.0298544804573x_{73} = -40.0298544804573
x74=69.8862806383665x_{74} = 69.8862806383665
x75=−90.3096235943467x_{75} = -90.3096235943467
x76=−374.632259996929x_{76} = -374.632259996929
x77=10.118480698715x_{77} = 10.118480698715
x78=−16.4352509360813x_{78} = -16.4352509360813
x79=−1.30308171092781x_{79} = -1.30308171092781
x80=−10.118480698715x_{80} = -10.118480698715
x81=19.5858273496712x_{81} = 19.5858273496712
x82=3.70722846405825x_{82} = 3.70722846405825
x83=−18.0062837080869x_{83} = -18.0062837080869
x84=76.1706223070459x_{84} = 76.1706223070459
x85=−13.2806415733888x_{85} = -13.2806415733888
x86=−60.4593229336188x_{86} = -60.4593229336188
x87=16.4352509360813x_{87} = 16.4352509360813
x88=−98.1646611098724x_{88} = -98.1646611098724
x89=−110.732171281162x_{89} = -110.732171281162
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

lim⁡x→0−(2(−2cos⁡(2x)+4sin⁡(2x)x+3(cos⁡(2x)−1)x2)x2)=43\lim_{x \to 0^-}\left(\frac{2 \left(- 2 \cos{\left(2 x \right)} + \frac{4 \sin{\left(2 x \right)}}{x} + \frac{3 \left(\cos{\left(2 x \right)} - 1\right)}{x^{2}}\right)}{x^{2}}\right) = \frac{4}{3}
lim⁡x→0+(2(−2cos⁡(2x)+4sin⁡(2x)x+3(cos⁡(2x)−1)x2)x2)=43\lim_{x \to 0^+}\left(\frac{2 \left(- 2 \cos{\left(2 x \right)} + \frac{4 \sin{\left(2 x \right)}}{x} + \frac{3 \left(\cos{\left(2 x \right)} - 1\right)}{x^{2}}\right)}{x^{2}}\right) = \frac{4}{3}
- los límites son iguales, es decir omitimos el punto correspondiente

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
[1335.96152783466,∞)\left[1335.96152783466, \infty\right)
Convexa en los intervalos
(−∞,−102.877368081182]\left(-\infty, -102.877368081182\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→−∞(cos⁡(2x)−1x2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(2 x \right)} - 1}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(cos⁡(2x)−1x2)=0\lim_{x \to \infty}\left(\frac{\cos{\left(2 x \right)} - 1}{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 (-1 + cos(2*x))/x^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(2x)−1xx2)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(2 x \right)} - 1}{x x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(2x)−1xx2)=0\lim_{x \to \infty}\left(\frac{\cos{\left(2 x \right)} - 1}{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:
cos⁡(2x)−1x2=cos⁡(2x)−1x2\frac{\cos{\left(2 x \right)} - 1}{x^{2}} = \frac{\cos{\left(2 x \right)} - 1}{x^{2}}
- Sí
cos⁡(2x)−1x2=−cos⁡(2x)−1x2\frac{\cos{\left(2 x \right)} - 1}{x^{2}} = - \frac{\cos{\left(2 x \right)} - 1}{x^{2}}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор