Sr Examen

Otras calculadoras

sin(x^2)/x^2
Trazado el gráfico de la función/ sin(x^2)/x^2

Gráfico de la función y = sin(x^2)/x^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

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

Solución analítica
x1=−πx_{1} = - \sqrt{\pi}
x2=πx_{2} = \sqrt{\pi}
Solución numérica
x1=46.1519210773927x_{1} = 46.1519210773927
x2=66.2243410266297x_{2} = 66.2243410266297
x3=95.0041877696109x_{3} = 95.0041877696109
x4=3.96332729760601x_{4} = 3.96332729760601
x5=64.2499240996983x_{5} = 64.2499240996983
x6=−81.378604766654x_{6} = -81.378604766654
x7=56.2178104297146x_{7} = 56.2178104297146
x8=−18.2485292908913x_{8} = -18.2485292908913
x9=48.572997466143x_{9} = 48.572997466143
x10=22.137941502317x_{10} = 22.137941502317
x11=−86.9225536433638x_{11} = -86.9225536433638
x12=−56.7738853926973x_{12} = -56.7738853926973
x13=28.2482660354898x_{13} = 28.2482660354898
x14=−91.8261111135335x_{14} = -91.8261111135335
x15=−89.7498945058111x_{15} = -89.7498945058111
x16=16.244807875181x_{16} = 16.244807875181
x17=25.0035385108662x_{17} = 25.0035385108662
x18=−55.8533929588406x_{18} = -55.8533929588406
x19=8.1224039375905x_{19} = 8.1224039375905
x20=9.7081295627785x_{20} = 9.7081295627785
x21=−31.2072703486357x_{21} = -31.2072703486357
x22=86.2148955714351x_{22} = 86.2148955714351
x23=42.2424505354389x_{23} = 42.2424505354389
x24=156.939754059142x_{24} = 156.939754059142
x25=43.1620718174968x_{25} = 43.1620718174968
x26=82.1853040708499x_{26} = 82.1853040708499
x27=−71.735129236519x_{27} = -71.735129236519
x28=−83.4748615192751x_{28} = -83.4748615192751
x29=−53.9070793548543x_{29} = -53.9070793548543
x30=−35.4490770181103x_{30} = -35.4490770181103
x31=−31.8548964385305x_{31} = -31.8548964385305
x32=−66.8616956564689x_{32} = -66.8616956564689
x33=78.2092123508435x_{33} = 78.2092123508435
x34=−5.87856438167413x_{34} = -5.87856438167413
x35=−70.7872889894487x_{35} = -70.7872889894487
x36=−15.7539144225679x_{36} = -15.7539144225679
x37=−41.8314129339366x_{37} = -41.8314129339366
x38=−85.2622155494847x_{38} = -85.2622155494847
x39=−39.7914902637393x_{39} = -39.7914902637393
x40=96.1546004074416x_{40} = 96.1546004074416
x41=120.487757271179x_{41} = 120.487757271179
x42=−11.6227571644753x_{42} = -11.6227571644753
x43=−83.8691000988299x_{43} = -83.8691000988299
x44=15.039769647786x_{44} = 15.039769647786
x45=−46.3556833091199x_{45} = -46.3556833091199
x46=−30.2877046810784x_{46} = -30.2877046810784
x47=29.6588257185807x_{47} = 29.6588257185807
x48=−62.9907256783402x_{48} = -62.9907256783402
x49=−9.86860538583257x_{49} = -9.86860538583257
x50=52.2498231190263x_{50} = 52.2498231190263
x51=18.2485292908913x_{51} = 18.2485292908913
x52=83.9065500225979x_{52} = 83.9065500225979
x53=50.6934195989623x_{53} = 50.6934195989623
x54=94.1238082693386x_{54} = 94.1238082693386
x55=118.846962815798x_{55} = 118.846962815798
x56=−5.60499121639793x_{56} = -5.60499121639793
x57=100.17108832176x_{57} = 100.17108832176
x58=−1.77245385090552x_{58} = -1.77245385090552
x59=−48.1833681990255x_{59} = -48.1833681990255
x60=60.0806953935677x_{60} = 60.0806953935677
x61=−87.7498922731452x_{61} = -87.7498922731452
x62=−64.2499240996983x_{62} = -64.2499240996983
x63=29.2320577752649x_{63} = 29.2320577752649
x64=6.13996024767893x_{64} = 6.13996024767893
x65=−90.6034304197701x_{65} = -90.6034304197701
x66=69.057495235125x_{66} = 69.057495235125
x67=−7.72594721818665x_{67} = -7.72594721818665
x68=−43.8480866628973x_{68} = -43.8480866628973
x69=112.658929164549x_{69} = 112.658929164549
x70=−19.8166364880301x_{70} = -19.8166364880301
x71=−65.5568360288295x_{71} = -65.5568360288295
x72=−13.3817331184947x_{72} = -13.3817331184947
x73=98.0953530904779x_{73} = 98.0953530904779
x74=−95.8600986425016x_{74} = -95.8600986425016
x75=−64.201009038097x_{75} = -64.201009038097
x76=−33.862683274665x_{76} = -33.862683274665
x77=−53.8487698955452x_{77} = -53.8487698955452
x78=−3.96332729760601x_{78} = -3.96332729760601
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(x^2)/x^2.
sin⁡(02)02\frac{\sin{\left(0^{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
2xcos⁡(x2)x2−2sin⁡(x2)x3=0\frac{2 x \cos{\left(x^{2} \right)}}{x^{2}} - \frac{2 \sin{\left(x^{2} \right)}}{x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−27.487300879733x_{1} = -27.487300879733
x2=−81.8117534286295x_{2} = -81.8117534286295
x3=56.1199279909343x_{3} = 56.1199279909343
x4=−51.294088034684x_{4} = -51.294088034684
x5=7.19840138414632x_{5} = 7.19840138414632
x6=46.0326381269472x_{6} = 46.0326381269472
x7=70.1296163961195x_{7} = 70.1296163961195
x8=−77.5739683255625x_{8} = -77.5739683255625
x9=82.2139674644798x_{9} = 82.2139674644798
x10=−97.7504677801482x_{10} = -97.7504677801482
x11=30.4686650830675x_{11} = 30.4686650830675
x12=−16.0011216977146x_{12} = -16.0011216977146
x13=−63.4751315873709x_{13} = -63.4751315873709
x14=−57.7477298421119x_{14} = -57.7477298421119
x15=−53.6295415482154x_{15} = -53.6295415482154
x16=−12.8424122001834x_{16} = -12.8424122001834
x17=9.29420367942992x_{17} = 9.29420367942992
x18=−111.755990954751x_{18} = -111.755990954751
x19=92.1079304031337x_{19} = 92.1079304031337
x20=68.5896113576236x_{20} = 68.5896113576236
x21=67.6209131009439x_{21} = 67.6209131009439
x22=6.26453768135879x_{22} = 6.26453768135879
x23=−17.856889938083x_{23} = -17.856889938083
x24=−47.8726582137852x_{24} = -47.8726582137852
x25=−38.5686848262369x_{25} = -38.5686848262369
x26=−21.7441389653189x_{26} = -21.7441389653189
x27=−84.4756034159993x_{27} = -84.4756034159993
x28=−33.7930178276353x_{28} = -33.7930178276353
x29=−7.8259012426924x_{29} = -7.8259012426924
x30=−85.7673542930979x_{30} = -85.7673542930979
x31=147.556117595223x_{31} = 147.556117595223
x32=27.9968943148725x_{32} = 27.9968943148725
x33=86.3878070945521x_{33} = 86.3878070945521
x34=26.0194492595522x_{34} = 26.0194492595522
x35=−3.75049248937143x_{35} = -3.75049248937143
x36=−90.2995216474216x_{36} = -90.2995216474216
x37=98.4230706903319x_{37} = 98.4230706903319
x38=−91.8859626824559x_{38} = -91.8859626824559
x39=−93.7978150743204x_{39} = -93.7978150743204
x40=22.3145788067872x_{40} = 22.3145788067872
x41=78.6997443140484x_{41} = 78.6997443140484
x42=−52.3549350628188x_{42} = -52.3549350628188
x43=−23.812931580963x_{43} = -23.812931580963
x44=23.6806356632024x_{44} = 23.6806356632024
x45=68.1300442553852x_{45} = 68.1300442553852
x46=58.1813182114996x_{46} = 58.1813182114996
x47=32.1249469684196x_{47} = 32.1249469684196
x48=16.0989917566665x_{48} = 16.0989917566665
x49=54.1251772109928x_{49} = 54.1251772109928
x50=−87.848290956425x_{50} = -87.848290956425
x51=−47.7083162370527x_{51} = -47.7083162370527
x52=4.14978978647482x_{52} = 4.14978978647482
x53=−225.808811928315x_{53} = -225.808811928315
x54=84.2148756127912x_{54} = 84.2148756127912
x55=339.12512890907x_{55} = 339.12512890907
x56=80.1827238875859x_{56} = 80.1827238875859
x57=18.1188682387635x_{57} = 18.1188682387635
x58=34.2088060145132x_{58} = 34.2088060145132
x59=94.332184745798x_{59} = 94.332184745798
x60=−65.8556635962687x_{60} = -65.8556635962687
x61=−2.11976636870884x_{61} = -2.11976636870884
x62=261.64865047635x_{62} = 261.64865047635
x63=51.5994127711782x_{63} = 51.5994127711782
x64=20.2478493210409x_{64} = 20.2478493210409
x65=62.2254986905353x_{65} = 62.2254986905353
x66=−68.0146672875878x_{66} = -68.0146672875878
x67=−42.0000441440065x_{67} = -42.0000441440065
x68=42.1120943325309x_{68} = 42.1120943325309
x69=−98.1833843061319x_{69} = -98.1833843061319
x70=37.828491311387x_{70} = 37.828491311387
x71=10.2583552061545x_{71} = 10.2583552061545
x72=40.164750811318x_{72} = 40.164750811318
x73=−29.8961868990962x_{73} = -29.8961868990962
x74=60.2503944862544x_{74} = 60.2503944862544
x75=−76.902847810038x_{75} = -76.902847810038
x76=36.0423055930793x_{76} = 36.0423055930793
x77=2.11976636870884x_{77} = 2.11976636870884
x78=−69.7703204185698x_{78} = -69.7703204185698
Signos de extremos en los puntos:
(-27.487300879732977, 0.001323534989533)

(-81.81175342862946, 0.000149406190911279)

(56.11992799093429, 0.000317516111949359)

(-51.29408803468397, -0.000380071533342589)

(7.198401384146321, 0.019295099487588)

(46.032638126947205, 0.000471919824505443)

(70.12961639611953, -0.00020332794172061)

(-77.57396832556249, -0.000166175876052604)

(82.21396746447982, -0.000147947892013696)

(-97.75046778014821, -0.000104655560719047)

(30.468665083067464, -0.00107719144114341)

(-16.001121697714638, -0.00390567256417978)

(-63.475131587370946, 0.000248194850672783)

(-57.74772984211192, -0.000299868017423585)

(-53.62954154821543, -0.000347689683696048)

(-12.842412200183375, 0.00606315689591026)

(9.294203679429923, -0.0115756804584678)

(-111.75599095475077, -8.00678876126555e-5)

(92.10793040313365, 0.000117870723345681)

(68.58961135762361, -0.000212560863689315)

(67.62091310094392, -0.000218694533696834)

(6.264537681358792, 0.0254730530928808)

(-17.856889938083004, -0.00313607341346806)

(-47.8726582137852, -0.000436339844394369)

(-38.568684826236904, -0.000672249119554206)

(-21.74413896531894, 0.00211502058560655)

(-84.4756034159993, -0.000140132022588208)

(-33.79301782763527, -0.000875680902961545)

(-7.825901242692397, -0.0163257593209978)

(-85.76735429309788, -0.000135942724375418)

(147.5561175952228, 4.59288487871757e-5)

(27.996894314872534, -0.00127579216525722)

(86.3878070945521, -0.000133997006542391)

(26.01944925955225, -0.00147707764928763)

(-3.7504924893714255, 0.0709134594504622)

(-90.29952164742163, -0.000122639140272602)

(98.42307069033188, -0.000103230059309167)

(-91.88596268245591, -0.0001184408887131)

(-93.79781507432041, 0.000113661806191721)

(22.314578806787196, 0.00200826831595226)

(78.69974431404836, -0.000161455688730729)

(-52.35493506281884, 0.000364825108729964)

(-23.81293158096305, 0.00176349241629226)

(23.680635663202445, 0.00178325149744762)

(68.13004425538523, -0.000215438168237391)

(58.18131821149962, -0.000295415220485631)

(32.124946968419636, 0.000968980321516034)

(16.098991756666486, 0.00385833036946338)

(54.125177210992774, 0.000341351104266613)

(-87.848290956425, 0.000129578623592958)

(-47.70831623705273, 0.000439351162050216)

(4.149789786474824, -0.0579718023461539)

(-225.80881192831538, 1.9611834893873e-5)

(84.21487561279123, -0.000141001058404429)

(339.1251289090697, -8.69520962086523e-6)

(80.18272388758591, 0.000155538670918207)

(18.118868238763504, 0.00304604175008073)

(34.20880601451324, 0.000854523496376895)

(94.33218474579803, 0.000112377718691516)

(-65.85566359626866, 0.000230575801988624)

(-2.1197663687088406, -0.217233628211222)

(261.64865047634953, -1.46070663431957e-5)

(51.5994127711782, -0.000375586912843737)

(20.247849321040928, 0.00243916347187379)

(62.22549869053529, 0.000258263607950864)

(-68.01466728758777, 0.000216169707043122)

(-42.00004414400653, -0.000566892141283866)

(42.112094332530894, 0.000563879427437908)

(-98.18338430613193, 0.000103734687272752)

(37.828491311387026, -0.000698814410331)

(10.258355206154523, -0.00950221661878354)

(40.16475081131796, -0.000619883052268501)

(-29.896186899096215, 0.00111884037052573)

(60.25039448625436, -0.000275473732809509)

(-76.902847810038, 0.000169088919380717)

(36.04230559307926, -0.000769794390559548)

(2.1197663687088406, -0.217233628211222)

(-69.77032041856978, -0.0002054274881576)


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=−51.294088034684x_{1} = -51.294088034684
x2=70.1296163961195x_{2} = 70.1296163961195
x3=−77.5739683255625x_{3} = -77.5739683255625
x4=82.2139674644798x_{4} = 82.2139674644798
x5=−97.7504677801482x_{5} = -97.7504677801482
x6=30.4686650830675x_{6} = 30.4686650830675
x7=−16.0011216977146x_{7} = -16.0011216977146
x8=−57.7477298421119x_{8} = -57.7477298421119
x9=−53.6295415482154x_{9} = -53.6295415482154
x10=9.29420367942992x_{10} = 9.29420367942992
x11=−111.755990954751x_{11} = -111.755990954751
x12=68.5896113576236x_{12} = 68.5896113576236
x13=67.6209131009439x_{13} = 67.6209131009439
x14=−17.856889938083x_{14} = -17.856889938083
x15=−47.8726582137852x_{15} = -47.8726582137852
x16=−38.5686848262369x_{16} = -38.5686848262369
x17=−84.4756034159993x_{17} = -84.4756034159993
x18=−33.7930178276353x_{18} = -33.7930178276353
x19=−7.8259012426924x_{19} = -7.8259012426924
x20=−85.7673542930979x_{20} = -85.7673542930979
x21=27.9968943148725x_{21} = 27.9968943148725
x22=86.3878070945521x_{22} = 86.3878070945521
x23=26.0194492595522x_{23} = 26.0194492595522
x24=−90.2995216474216x_{24} = -90.2995216474216
x25=98.4230706903319x_{25} = 98.4230706903319
x26=−91.8859626824559x_{26} = -91.8859626824559
x27=78.6997443140484x_{27} = 78.6997443140484
x28=68.1300442553852x_{28} = 68.1300442553852
x29=58.1813182114996x_{29} = 58.1813182114996
x30=4.14978978647482x_{30} = 4.14978978647482
x31=84.2148756127912x_{31} = 84.2148756127912
x32=339.12512890907x_{32} = 339.12512890907
x33=−2.11976636870884x_{33} = -2.11976636870884
x34=261.64865047635x_{34} = 261.64865047635
x35=51.5994127711782x_{35} = 51.5994127711782
x36=−42.0000441440065x_{36} = -42.0000441440065
x37=37.828491311387x_{37} = 37.828491311387
x38=10.2583552061545x_{38} = 10.2583552061545
x39=40.164750811318x_{39} = 40.164750811318
x40=60.2503944862544x_{40} = 60.2503944862544
x41=36.0423055930793x_{41} = 36.0423055930793
x42=2.11976636870884x_{42} = 2.11976636870884
x43=−69.7703204185698x_{43} = -69.7703204185698
Puntos máximos de la función:
x43=−27.487300879733x_{43} = -27.487300879733
x43=−81.8117534286295x_{43} = -81.8117534286295
x43=56.1199279909343x_{43} = 56.1199279909343
x43=7.19840138414632x_{43} = 7.19840138414632
x43=46.0326381269472x_{43} = 46.0326381269472
x43=−63.4751315873709x_{43} = -63.4751315873709
x43=−12.8424122001834x_{43} = -12.8424122001834
x43=92.1079304031337x_{43} = 92.1079304031337
x43=6.26453768135879x_{43} = 6.26453768135879
x43=−21.7441389653189x_{43} = -21.7441389653189
x43=147.556117595223x_{43} = 147.556117595223
x43=−3.75049248937143x_{43} = -3.75049248937143
x43=−93.7978150743204x_{43} = -93.7978150743204
x43=22.3145788067872x_{43} = 22.3145788067872
x43=−52.3549350628188x_{43} = -52.3549350628188
x43=−23.812931580963x_{43} = -23.812931580963
x43=23.6806356632024x_{43} = 23.6806356632024
x43=32.1249469684196x_{43} = 32.1249469684196
x43=16.0989917566665x_{43} = 16.0989917566665
x43=54.1251772109928x_{43} = 54.1251772109928
x43=−87.848290956425x_{43} = -87.848290956425
x43=−47.7083162370527x_{43} = -47.7083162370527
x43=−225.808811928315x_{43} = -225.808811928315
x43=80.1827238875859x_{43} = 80.1827238875859
x43=18.1188682387635x_{43} = 18.1188682387635
x43=34.2088060145132x_{43} = 34.2088060145132
x43=94.332184745798x_{43} = 94.332184745798
x43=−65.8556635962687x_{43} = -65.8556635962687
x43=20.2478493210409x_{43} = 20.2478493210409
x43=62.2254986905353x_{43} = 62.2254986905353
x43=−68.0146672875878x_{43} = -68.0146672875878
x43=42.1120943325309x_{43} = 42.1120943325309
x43=−98.1833843061319x_{43} = -98.1833843061319
x43=−29.8961868990962x_{43} = -29.8961868990962
x43=−76.902847810038x_{43} = -76.902847810038
Decrece en los intervalos
[339.12512890907,∞)\left[339.12512890907, \infty\right)
Crece en los intervalos
(−∞,−111.755990954751]\left(-\infty, -111.755990954751\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(−2x2sin⁡(x2)−3cos⁡(x2)+3sin⁡(x2)x2)x2=0\frac{2 \left(- 2 x^{2} \sin{\left(x^{2} \right)} - 3 \cos{\left(x^{2} \right)} + \frac{3 \sin{\left(x^{2} \right)}}{x^{2}}\right)}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−22.0668026877238x_{1} = -22.0668026877238
x2=9.86782481171723x_{2} = 9.86782481171723
x3=−63.7590819002399x_{3} = -63.7590819002399
x4=66.2243384443197x_{4} = 66.2243384443197
x5=−47.2284883472124x_{5} = -47.2284883472124
x6=−15.7537225935399x_{6} = -15.7537225935399
x7=38.8325054432974x_{7} = 38.8325054432974
x8=−33.8626639595177x_{8} = -33.8626639595177
x9=−60.1851801981387x_{9} = -60.1851801981387
x10=−55.8533886544398x_{10} = -55.8533886544398
x11=−81.4557760892944x_{11} = -81.4557760892944
x12=3.52778397201587x_{12} = 3.52778397201587
x13=−89.7498934683796x_{13} = -89.7498934683796
x14=60.0806919353168x_{14} = 60.0806919353168
x15=−89.8897998629067x_{15} = -89.8897998629067
x16=82.185302719777x_{16} = 82.185302719777
x17=−31.8055239171221x_{17} = -31.8055239171221
x18=−56.9672252171924x_{18} = -56.9672252171924
x19=−7.72431969840891x_{19} = -7.72431969840891
x20=−43.848077766593x_{20} = -43.848077766593
x21=−9.86782481171723x_{21} = -9.86782481171723
x22=−101.309343365408x_{22} = -101.309343365408
x23=−19.8165401098264x_{23} = -19.8165401098264
x24=94.1238073699181x_{24} = 94.1238073699181
x25=29.9748866464009x_{25} = 29.9748866464009
x26=74.0834805642647x_{26} = 74.0834805642647
x27=96.2035955430112x_{27} = 96.2035955430112
x28=−47.261736163224x_{28} = -47.261736163224
x29=−55.9938299267056x_{29} = -55.9938299267056
x30=−91.997013166397x_{30} = -91.997013166397
x31=78.229292749505x_{31} = 78.229292749505
x32=78.2694412248249x_{32} = 78.2694412248249
x33=−72.8648838744917x_{33} = -72.8648838744917
x34=51.9483159842114x_{34} = 51.9483159842114
x35=52.2498178611914x_{35} = 52.2498178611914
x36=6.13671409961045x_{36} = 6.13671409961045
x37=86.2148944010888x_{37} = 86.2148944010888
x38=−41.8314026879441x_{38} = -41.8314026879441
x39=−83.8690988275086x_{39} = -83.8690988275086
x40=54.0816257858468x_{40} = 54.0816257858468
x41=28.2482327628185x_{41} = 28.2482327628185
x42=12.1509133337361x_{42} = 12.1509133337361
x43=−78.6498295765939x_{43} = -78.6498295765939
x44=58.1948145254578x_{44} = 58.1948145254578
x45=21.7079643197458x_{45} = 21.7079643197458
x46=64.2499212719322x_{46} = 64.2499212719322
x47=−25.5011249298425x_{47} = -25.5011249298425
x48=56.3294603860904x_{48} = 56.3294603860904
x49=42.2424405856399x_{49} = 42.2424405856399
x50=16.2446329171445x_{50} = 16.2446329171445
x51=68.6925894679026x_{51} = 68.6925894679026
x52=−18.0754265473486x_{52} = -18.0754265473486
x53=98.0953522959375x_{53} = 98.0953522959375
x54=25.8072739815158x_{54} = 25.8072739815158
x55=34.958236723676x_{55} = 34.958236723676
x56=−12.4067842296203x_{56} = -12.4067842296203
x57=−53.9070745671867x_{57} = -53.9070745671867
x58=44.5587903963516x_{58} = 44.5587903963516
x59=−36.8823605035375x_{59} = -36.8823605035375
x60=−5.87486435966933x_{60} = -5.87486435966933
x61=−53.0553153296769x_{61} = -53.0553153296769
x62=−42.1307370479931x_{62} = -42.1307370479931
x63=8.12100348114634x_{63} = 8.12100348114634
x64=−95.8600977910742x_{64} = -95.8600977910742
x65=−1.56860943902247x_{65} = -1.56860943902247
x66=22.1378723742674x_{66} = 22.1378723742674
x67=83.5688952550296x_{67} = 83.5688952550296
x68=−93.8731443124186x_{68} = -93.8731443124186
x69=3.95114986952551x_{69} = 3.95114986952551
x70=−13.8430243840388x_{70} = -13.8430243840388
x71=−39.791478359789x_{71} = -39.791478359789
x72=91.2082077797512x_{72} = 91.2082077797512
x73=−3.95114986952551x_{73} = -3.95114986952551
x74=23.8458724385199x_{74} = 23.8458724385199
x75=−87.6603412922761x_{75} = -87.6603412922761
x76=18.2484058702107x_{76} = 18.2484058702107
x77=−14.0681469260875x_{77} = -14.0681469260875
x78=88.3386436351664x_{78} = 88.3386436351664
x79=46.1178655318556x_{79} = 46.1178655318556
x80=−98.1433794006371x_{80} = -98.1433794006371
x81=−74.0410623025856x_{81} = -74.0410623025856
x82=22.6984202929001x_{82} = 22.6984202929001
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(−2x2sin⁡(x2)−3cos⁡(x2)+3sin⁡(x2)x2)x2)=0\lim_{x \to 0^-}\left(\frac{2 \left(- 2 x^{2} \sin{\left(x^{2} \right)} - 3 \cos{\left(x^{2} \right)} + \frac{3 \sin{\left(x^{2} \right)}}{x^{2}}\right)}{x^{2}}\right) = 0
lim⁡x→0+(2(−2x2sin⁡(x2)−3cos⁡(x2)+3sin⁡(x2)x2)x2)=0\lim_{x \to 0^+}\left(\frac{2 \left(- 2 x^{2} \sin{\left(x^{2} \right)} - 3 \cos{\left(x^{2} \right)} + \frac{3 \sin{\left(x^{2} \right)}}{x^{2}}\right)}{x^{2}}\right) = 0
- 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
[98.0953522959375,∞)\left[98.0953522959375, \infty\right)
Convexa en los intervalos
(−∞,−98.1433794006371]\left(-\infty, -98.1433794006371\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→−∞(sin⁡(x2)x2)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x^{2} \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(x2)x2)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x^{2} \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 sin(x^2)/x^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x2)xx2)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x^{2} \right)}}{x x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(x2)xx2)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x^{2} \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:
sin⁡(x2)x2=sin⁡(x2)x2\frac{\sin{\left(x^{2} \right)}}{x^{2}} = \frac{\sin{\left(x^{2} \right)}}{x^{2}}
- Sí
sin⁡(x2)x2=−sin⁡(x2)x2\frac{\sin{\left(x^{2} \right)}}{x^{2}} = - \frac{\sin{\left(x^{2} \right)}}{x^{2}}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = sin(x^2)/x^2
    © Sr Examen – Калькулятор