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Trazado el gráfico de la función/ sin(3*x)/((2*x))

Gráfico de la función y = sin(3*x)/((2*x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       sin(3*x)
f(x) = --------
         2*x
f(x)=sin⁡(3x)2xf{\left(x \right)} = \frac{\sin{\left(3 x \right)}}{2 x} 
f = sin(3*x)/((2*x))
Gráfico de la función
02468-8-6-4-2-10102-2
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⁡(3x)2x=0\frac{\sin{\left(3 x \right)}}{2 x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π3x_{1} = \frac{\pi}{3}
Solución numérica
x1=11.5191730631626x_{1} = 11.5191730631626
x2=−83.7758040957278x_{2} = -83.7758040957278
x3=−9.42477796076938x_{3} = -9.42477796076938
x4=−17.8023583703422x_{4} = -17.8023583703422
x5=46.0766922526503x_{5} = 46.0766922526503
x6=−26.1799387799149x_{6} = -26.1799387799149
x7=43.9822971502571x_{7} = 43.9822971502571
x8=−31.4159265358979x_{8} = -31.4159265358979
x9=78.5398163397448x_{9} = 78.5398163397448
x10=−46.0766922526503x_{10} = -46.0766922526503
x11=−35.6047167406843x_{11} = -35.6047167406843
x12=−15.707963267949x_{12} = -15.707963267949
x13=−21.9911485751286x_{13} = -21.9911485751286
x14=56.5486677646163x_{14} = 56.5486677646163
x15=98.4365698124802x_{15} = 98.4365698124802
x16=−13.6135681655558x_{16} = -13.6135681655558
x17=−72.2566310325652x_{17} = -72.2566310325652
x18=39.7935069454707x_{18} = 39.7935069454707
x19=4.18879020478639x_{19} = 4.18879020478639
x20=−55.5014702134197x_{20} = -55.5014702134197
x21=−50.2654824574367x_{21} = -50.2654824574367
x22=−85.870199198121x_{22} = -85.870199198121
x23=17.8023583703422x_{23} = 17.8023583703422
x24=63.8790506229925x_{24} = 63.8790506229925
x25=−93.2005820564972x_{25} = -93.2005820564972
x26=96.342174710087x_{26} = 96.342174710087
x27=−90.0589894029074x_{27} = -90.0589894029074
x28=70.162235930172x_{28} = 70.162235930172
x29=19.8967534727354x_{29} = 19.8967534727354
x30=85.870199198121x_{30} = 85.870199198121
x31=−94.2477796076938x_{31} = -94.2477796076938
x32=52.3598775598299x_{32} = 52.3598775598299
x33=−57.5958653158129x_{33} = -57.5958653158129
x34=−59.6902604182061x_{34} = -59.6902604182061
x35=10.471975511966x_{35} = 10.471975511966
x36=−4.18879020478639x_{36} = -4.18879020478639
x37=100.530964914873x_{37} = 100.530964914873
x38=54.4542726622231x_{38} = 54.4542726622231
x39=−68.0678408277789x_{39} = -68.0678408277789
x40=−70.162235930172x_{40} = -70.162235930172
x41=−2.0943951023932x_{41} = -2.0943951023932
x42=−92.1533845053006x_{42} = -92.1533845053006
x43=−99.4837673636768x_{43} = -99.4837673636768
x44=41.8879020478639x_{44} = 41.8879020478639
x45=−77.4926187885482x_{45} = -77.4926187885482
x46=21.9911485751286x_{46} = 21.9911485751286
x47=−54.4542726622231x_{47} = -54.4542726622231
x48=48.1710873550435x_{48} = 48.1710873550435
x49=−103.672557568463x_{49} = -103.672557568463
x50=83.7758040957278x_{50} = 83.7758040957278
x51=68.0678408277789x_{51} = 68.0678408277789
x52=34.5575191894877x_{52} = 34.5575191894877
x53=74.3510261349584x_{53} = 74.3510261349584
x54=−81.6814089933346x_{54} = -81.6814089933346
x55=50.2654824574367x_{55} = 50.2654824574367
x56=80.634211442138x_{56} = 80.634211442138
x57=−48.1710873550435x_{57} = -48.1710873550435
x58=−87.9645943005142x_{58} = -87.9645943005142
x59=24.0855436775217x_{59} = 24.0855436775217
x60=61.7846555205993x_{60} = 61.7846555205993
x61=37.6991118430775x_{61} = 37.6991118430775
x62=95.2949771588904x_{62} = 95.2949771588904
x63=32.4631240870945x_{63} = 32.4631240870945
x64=28.2743338823081x_{64} = 28.2743338823081
x65=−61.7846555205993x_{65} = -61.7846555205993
x66=−65.9734457253857x_{66} = -65.9734457253857
x67=92.1533845053006x_{67} = 92.1533845053006
x68=72.2566310325652x_{68} = 72.2566310325652
x69=−6.28318530717959x_{69} = -6.28318530717959
x70=−41.8879020478639x_{70} = -41.8879020478639
x71=30.3687289847013x_{71} = 30.3687289847013
x72=6.28318530717959x_{72} = 6.28318530717959
x73=−96.342174710087x_{73} = -96.342174710087
x74=94.2477796076938x_{74} = 94.2477796076938
x75=−33.5103216382911x_{75} = -33.5103216382911
x76=65.9734457253857x_{76} = 65.9734457253857
x77=−37.6991118430775x_{77} = -37.6991118430775
x78=−43.9822971502571x_{78} = -43.9822971502571
x79=87.9645943005142x_{79} = 87.9645943005142
x80=−11.5191730631626x_{80} = -11.5191730631626
x81=−24.0855436775217x_{81} = -24.0855436775217
x82=−323.584043319749x_{82} = -323.584043319749
x83=8.37758040957278x_{83} = 8.37758040957278
x84=−193.731546971371x_{84} = -193.731546971371
x85=−79.5870138909414x_{85} = -79.5870138909414
x86=76.4454212373516x_{86} = 76.4454212373516
x87=−19.8967534727354x_{87} = -19.8967534727354
x88=−39.7935069454707x_{88} = -39.7935069454707
x89=26.1799387799149x_{89} = 26.1799387799149
x90=−63.8790506229925x_{90} = -63.8790506229925
x91=2.0943951023932x_{91} = 2.0943951023932
x92=15.707963267949x_{92} = 15.707963267949
x93=−8.37758040957278x_{93} = -8.37758040957278
x94=−74.3510261349584x_{94} = -74.3510261349584
x95=90.0589894029074x_{95} = 90.0589894029074
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(3*x)/((2*x)).
sin⁡(0⋅3)0⋅2\frac{\sin{\left(0 \cdot 3 \right)}}{0 \cdot 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
312xcos⁡(3x)−sin⁡(3x)2x2=03 \frac{1}{2 x} \cos{\left(3 x \right)} - \frac{\sin{\left(3 x \right)}}{2 x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=88.4869374039331x_{1} = 88.4869374039331
x2=−65.448149267704x_{2} = -65.448149267704
x3=−6.79043431976252x_{3} = -6.79043431976252
x4=170.168949124421x_{4} = 170.168949124421
x5=−49.7396498613733x_{5} = -49.7396498613733
x6=−16.2247147439848x_{6} = -16.2247147439848
x7=−29.8414069768057x_{7} = -29.8414069768057
x8=49.7396498613733x_{8} = 49.7396498613733
x9=66.4953735549544x_{9} = 66.4953735549544
x10=51.8341352242202x_{10} = 51.8341352242202
x11=56.0230857030463x_{11} = 56.0230857030463
x12=36.1252398838916x_{12} = 36.1252398838916
x13=−50.7868934733971x_{13} = -50.7868934733971
x14=−23.5572285705398x_{14} = -23.5572285705398
x15=−1.49780315263635x_{15} = -1.49780315263635
x16=75.9203589492161x_{16} = 75.9203589492161
x17=80.1092256793491x_{17} = 80.1092256793491
x18=34.0306554883025x_{18} = 34.0306554883025
x19=−100.006255101775x_{19} = -100.006255101775
x20=62.3064710135101x_{20} = 62.3064710135101
x21=26.6993762096484x_{21} = 26.6993762096484
x22=−53.9286135759886x_{22} = -53.9286135759886
x23=−82.2036561197804x_{23} = -82.2036561197804
x24=−71.7314832814509x_{24} = -71.7314832814509
x25=−80.1092256793491x_{25} = -80.1092256793491
x26=−62.3064710135101x_{26} = -62.3064710135101
x27=−56.0230857030463x_{27} = -56.0230857030463
x28=82.2036561197804x_{28} = 82.2036561197804
x29=−84.2980848042281x_{29} = -84.2980848042281
x30=93.722995310418x_{30} = 93.722995310418
x31=−34.0306554883025x_{31} = -34.0306554883025
x32=86.3925118604052x_{32} = 86.3925118604052
x33=5.74025175731026x_{33} = 5.74025175731026
x34=−9.93719959696432x_{34} = -9.93719959696432
x35=−7.839817499563x_{35} = -7.839817499563
x36=58.1175522783976x_{36} = 58.1175522783976
x37=−75.9203589492161x_{37} = -75.9203589492161
x38=−67.5425970131389x_{38} = -67.5425970131389
x39=91.6285731099136x_{39} = 91.6285731099136
x40=−27.7467308235745x_{40} = -27.7467308235745
x41=73.8259223276238x_{41} = 73.8259223276238
x42=27.7467308235745x_{42} = 27.7467308235745
x43=−51.8341352242202x_{43} = -51.8341352242202
x44=−3.63470721980963x_{44} = -3.63470721980963
x45=60.212013881401x_{45} = 60.212013881401
x46=29.8414069768057x_{46} = 29.8414069768057
x47=−73.8259223276238x_{47} = -73.8259223276238
x48=−45.5506542337597x_{48} = -45.5506542337597
x49=95.8174163262761x_{49} = 95.8174163262761
x50=16.2247147439848x_{50} = 16.2247147439848
x51=38.2198035316744x_{51} = 38.2198035316744
x52=45.5506542337597x_{52} = 45.5506542337597
x53=−38.2198035316744x_{53} = -38.2198035316744
x54=2.5750839456459x_{54} = 2.5750839456459
x55=−25.6520087701104x_{55} = -25.6520087701104
x56=−93.722995310418x_{56} = -93.722995310418
x57=12.0335407481252x_{57} = 12.0335407481252
x58=−14.1293045227106x_{58} = -14.1293045227106
x59=−5.74025175731026x_{59} = -5.74025175731026
x60=−43.4561415684628x_{60} = -43.4561415684628
x61=7.839817499563x_{61} = 7.839817499563
x62=−69.6370415919254x_{62} = -69.6370415919254
x63=214.151380376346x_{63} = 214.151380376346
x64=−91.6285731099136x_{64} = -91.6285731099136
x65=71.7314832814509x_{65} = 71.7314832814509
x66=100.006255101775x_{66} = 100.006255101775
x67=−89.534149641627x_{67} = -89.534149641627
x68=−58.1175522783976x_{68} = -58.1175522783976
x69=53.9286135759886x_{69} = 53.9286135759886
x70=−78.014793341506x_{70} = -78.014793341506
x71=84.2980848042281x_{71} = 84.2980848042281
x72=40.3143496657172x_{72} = 40.3143496657172
x73=37.1725240820437x_{73} = 37.1725240820437
x74=−153.413716993446x_{74} = -153.413716993446
x75=−60.212013881401x_{75} = -60.212013881401
x76=−21.4623731968525x_{76} = -21.4623731968525
x77=78.014793341506x_{77} = 78.014793341506
x78=14.1293045227106x_{78} = 14.1293045227106
x79=−36.1252398838916x_{79} = -36.1252398838916
x80=20.4149100867915x_{80} = 20.4149100867915
x81=−12.0335407481252x_{81} = -12.0335407481252
x82=−31.9360462622872x_{82} = -31.9360462622872
x83=18.3198927626296x_{83} = 18.3198927626296
x84=31.9360462622872x_{84} = 31.9360462622872
x85=−95.8174163262761x_{85} = -95.8174163262761
x86=97.9118362335106x_{86} = 97.9118362335106
x87=44.5033992843595x_{87} = 44.5033992843595
x88=−47.6451565627964x_{88} = -47.6451565627964
x89=42.4088808811114x_{89} = 42.4088808811114
x90=−351.334462172035x_{90} = -351.334462172035
x91=−97.9118362335106x_{91} = -97.9118362335106
x92=69.6370415919254x_{92} = 69.6370415919254
x93=9.93719959696432x_{93} = 9.93719959696432
x94=64.4009241109425x_{94} = 64.4009241109425
x95=22.5098115923814x_{95} = 22.5098115923814
Signos de extremos en los puntos:
(88.4869374039331, 0.00565051144349552)

(-65.448149267704, 0.00763953634790889)

(-6.790434319762521, 0.0735444360211112)

(170.1689491244206, 0.00293825074030404)

(-49.73964986137327, -0.0100521168532409)

(-16.224714743984794, -0.0308106810626306)

(-29.84140697680573, 0.0167541969512603)

(49.73964986137327, -0.0100521168532409)

(66.49537355495444, -0.00751922564151797)

(51.83413522422022, -0.00964595356826895)

(56.02308570304626, -0.00892473421535613)

(36.12523988389156, 0.0138401493761729)

(-50.786893473397086, 0.00984484768974962)

(-23.557228570539834, 0.0212227830972996)

(-1.4978031526363547, -0.325850442316833)

(75.9203589492161, 0.00658578525869541)

(80.10922567934914, 0.00624142434739368)

(34.03065548830255, 0.0146919302201117)

(-100.00625510177518, -0.00499965949213873)

(62.306471013510084, -0.00802473381440714)

(26.69937620964837, -0.0187255699826685)

(-53.92861357598856, -0.00927133882917417)

(-82.20365611978043, 0.00608240451771827)

(-71.73148328145086, 0.00697036473602969)

(-80.10922567934914, 0.00624142434739368)

(-62.306471013510084, -0.00802473381440714)

(-56.02308570304626, -0.00892473421535613)

(82.20365611978043, 0.00608240451771827)

(-84.29808480422805, 0.00593128648461355)

(93.72299531041797, -0.00533483630200461)

(-34.03065548830255, 0.0146919302201117)

(86.39251186040515, 0.00578749555419506)

(5.740251757310256, -0.0869577035192308)

(-9.93719959696432, -0.0502877025320981)

(-7.839817499563003, -0.063719425466419)

(58.11755227839756, -0.00860311139416558)

(-75.9203589492161, 0.00658578525869541)

(-67.54259701313894, 0.00740264563812724)

(91.62857310991362, -0.00545677701319961)

(-27.746730823574467, 0.0180188407230791)

(73.82592232762377, 0.00677261980241303)

(27.746730823574467, 0.0180188407230791)

(-51.83413522422022, -0.00964595356826895)

(-3.6347072198096333, -0.136987804234587)

(60.212013881400964, -0.00830386340098503)

(29.84140697680573, 0.0167541969512603)

(-73.82592232762377, 0.00677261980241303)

(-45.55065423375966, -0.010976496851203)

(95.81741632627612, -0.00521822643124976)

(16.224714743984794, -0.0308106810626306)

(38.21980353167436, 0.0130817256715564)

(45.55065423375966, -0.010976496851203)

(-38.21980353167436, 0.0130817256715564)

(2.5750839456459023, 0.192561830288849)

(-25.652008770110395, 0.0194900054805641)

(-93.72299531041797, -0.00533483630200461)

(12.033540748125203, -0.0415345984517238)

(-14.12930452271064, -0.0353776023435246)

(-5.740251757310256, -0.0869577035192308)

(-43.456141568462805, -0.0115055150594557)

(7.839817499563003, -0.063719425466419)

(-69.63704159192544, 0.00718000449883474)

(214.15138037634594, 0.00233479416954943)

(-91.62857310991362, -0.00545677701319961)

(71.73148328145086, 0.00697036473602969)

(100.00625510177518, -0.00499965949213873)

(-89.53414964162702, -0.00558442266892595)

(-58.11755227839756, -0.00860311139416558)

(53.92861357598856, -0.00927133882917417)

(-78.01479334150599, 0.00640898238229569)

(84.29808480422805, 0.00593128648461355)

(40.31434966571717, 0.0124021077755364)

(37.17252408204367, -0.0134502542107399)

(-153.41371699344606, 0.00325915328541868)

(-60.212013881400964, -0.00830386340098503)

(-21.46237319685247, 0.023293775711192)

(78.01479334150599, 0.00640898238229569)

(14.12930452271064, -0.0353776023435246)

(-36.12523988389156, 0.0138401493761729)

(20.414910086791465, -0.0244886389814967)

(-12.033540748125203, -0.0415345984517238)

(-31.93604626228723, 0.0156554372018487)

(18.319892762629646, -0.0272882194827047)

(31.93604626228723, 0.0156554372018487)

(-95.81741632627612, -0.00521822643124976)

(97.91183623351056, -0.00510660530671998)

(44.50339928435946, 0.0112347816877682)

(-47.64515656279642, -0.010493989314784)

(42.40888088111144, 0.0117896191899248)

(-351.3344621720351, -0.00142314469201501)

(-97.91183623351056, -0.00510660530671998)

(69.63704159192544, 0.00718000449883474)

(9.93719959696432, -0.0502877025320981)

(64.40092411094253, -0.00776375975233391)

(22.50981159238137, -0.0222101009198238)


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=−49.7396498613733x_{1} = -49.7396498613733
x2=−16.2247147439848x_{2} = -16.2247147439848
x3=49.7396498613733x_{3} = 49.7396498613733
x4=66.4953735549544x_{4} = 66.4953735549544
x5=51.8341352242202x_{5} = 51.8341352242202
x6=56.0230857030463x_{6} = 56.0230857030463
x7=−1.49780315263635x_{7} = -1.49780315263635
x8=−100.006255101775x_{8} = -100.006255101775
x9=62.3064710135101x_{9} = 62.3064710135101
x10=26.6993762096484x_{10} = 26.6993762096484
x11=−53.9286135759886x_{11} = -53.9286135759886
x12=−62.3064710135101x_{12} = -62.3064710135101
x13=−56.0230857030463x_{13} = -56.0230857030463
x14=93.722995310418x_{14} = 93.722995310418
x15=5.74025175731026x_{15} = 5.74025175731026
x16=−9.93719959696432x_{16} = -9.93719959696432
x17=−7.839817499563x_{17} = -7.839817499563
x18=58.1175522783976x_{18} = 58.1175522783976
x19=91.6285731099136x_{19} = 91.6285731099136
x20=−51.8341352242202x_{20} = -51.8341352242202
x21=−3.63470721980963x_{21} = -3.63470721980963
x22=60.212013881401x_{22} = 60.212013881401
x23=−45.5506542337597x_{23} = -45.5506542337597
x24=95.8174163262761x_{24} = 95.8174163262761
x25=16.2247147439848x_{25} = 16.2247147439848
x26=45.5506542337597x_{26} = 45.5506542337597
x27=−93.722995310418x_{27} = -93.722995310418
x28=12.0335407481252x_{28} = 12.0335407481252
x29=−14.1293045227106x_{29} = -14.1293045227106
x30=−5.74025175731026x_{30} = -5.74025175731026
x31=−43.4561415684628x_{31} = -43.4561415684628
x32=7.839817499563x_{32} = 7.839817499563
x33=−91.6285731099136x_{33} = -91.6285731099136
x34=100.006255101775x_{34} = 100.006255101775
x35=−89.534149641627x_{35} = -89.534149641627
x36=−58.1175522783976x_{36} = -58.1175522783976
x37=53.9286135759886x_{37} = 53.9286135759886
x38=37.1725240820437x_{38} = 37.1725240820437
x39=−60.212013881401x_{39} = -60.212013881401
x40=14.1293045227106x_{40} = 14.1293045227106
x41=20.4149100867915x_{41} = 20.4149100867915
x42=−12.0335407481252x_{42} = -12.0335407481252
x43=18.3198927626296x_{43} = 18.3198927626296
x44=−95.8174163262761x_{44} = -95.8174163262761
x45=97.9118362335106x_{45} = 97.9118362335106
x46=−47.6451565627964x_{46} = -47.6451565627964
x47=−351.334462172035x_{47} = -351.334462172035
x48=−97.9118362335106x_{48} = -97.9118362335106
x49=9.93719959696432x_{49} = 9.93719959696432
x50=64.4009241109425x_{50} = 64.4009241109425
x51=22.5098115923814x_{51} = 22.5098115923814
Puntos máximos de la función:
x51=88.4869374039331x_{51} = 88.4869374039331
x51=−65.448149267704x_{51} = -65.448149267704
x51=−6.79043431976252x_{51} = -6.79043431976252
x51=170.168949124421x_{51} = 170.168949124421
x51=−29.8414069768057x_{51} = -29.8414069768057
x51=36.1252398838916x_{51} = 36.1252398838916
x51=−50.7868934733971x_{51} = -50.7868934733971
x51=−23.5572285705398x_{51} = -23.5572285705398
x51=75.9203589492161x_{51} = 75.9203589492161
x51=80.1092256793491x_{51} = 80.1092256793491
x51=34.0306554883025x_{51} = 34.0306554883025
x51=−82.2036561197804x_{51} = -82.2036561197804
x51=−71.7314832814509x_{51} = -71.7314832814509
x51=−80.1092256793491x_{51} = -80.1092256793491
x51=82.2036561197804x_{51} = 82.2036561197804
x51=−84.2980848042281x_{51} = -84.2980848042281
x51=−34.0306554883025x_{51} = -34.0306554883025
x51=86.3925118604052x_{51} = 86.3925118604052
x51=−75.9203589492161x_{51} = -75.9203589492161
x51=−67.5425970131389x_{51} = -67.5425970131389
x51=−27.7467308235745x_{51} = -27.7467308235745
x51=73.8259223276238x_{51} = 73.8259223276238
x51=27.7467308235745x_{51} = 27.7467308235745
x51=29.8414069768057x_{51} = 29.8414069768057
x51=−73.8259223276238x_{51} = -73.8259223276238
x51=38.2198035316744x_{51} = 38.2198035316744
x51=−38.2198035316744x_{51} = -38.2198035316744
x51=2.5750839456459x_{51} = 2.5750839456459
x51=−25.6520087701104x_{51} = -25.6520087701104
x51=−69.6370415919254x_{51} = -69.6370415919254
x51=214.151380376346x_{51} = 214.151380376346
x51=71.7314832814509x_{51} = 71.7314832814509
x51=−78.014793341506x_{51} = -78.014793341506
x51=84.2980848042281x_{51} = 84.2980848042281
x51=40.3143496657172x_{51} = 40.3143496657172
x51=−153.413716993446x_{51} = -153.413716993446
x51=−21.4623731968525x_{51} = -21.4623731968525
x51=78.014793341506x_{51} = 78.014793341506
x51=−36.1252398838916x_{51} = -36.1252398838916
x51=−31.9360462622872x_{51} = -31.9360462622872
x51=31.9360462622872x_{51} = 31.9360462622872
x51=44.5033992843595x_{51} = 44.5033992843595
x51=42.4088808811114x_{51} = 42.4088808811114
x51=69.6370415919254x_{51} = 69.6370415919254
Decrece en los intervalos
[100.006255101775,∞)\left[100.006255101775, \infty\right)
Crece en los intervalos
(−∞,−351.334462172035]\left(-\infty, -351.334462172035\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
−9sin⁡(3x)2−3cos⁡(3x)x+sin⁡(3x)x2x=0\frac{- \frac{9 \sin{\left(3 x \right)}}{2} - \frac{3 \cos{\left(3 x \right)}}{x} + \frac{\sin{\left(3 x \right)}}{x^{2}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=87.9620679348333x_{1} = 87.9620679348333
x2=120.425873075205x_{2} = 120.425873075205
x3=−39.7879215162522x_{3} = -39.7879215162522
x4=−72.253555400176x_{4} = -72.253555400176
x5=19.8855763345093x_{5} = 19.8855763345093
x6=50.2610609682413x_{6} = 50.2610609682413
x7=−94.2454216780301x_{7} = -94.2454216780301
x8=12.5486534410288x_{8} = 12.5486534410288
x9=−48.1664735791016x_{9} = -48.1664735791016
x10=−10.4506972421882x_{10} = -10.4506972421882
x11=−1.9801233301909x_{11} = -1.9801233301909
x12=−61.7810585195113x_{12} = -61.7810585195113
x13=98.4343122265222x_{13} = 98.4343122265222
x14=24.0763125873385x_{14} = 24.0763125873385
x15=65.9700771374731x_{15} = 65.9700771374731
x16=63.8755715737869x_{16} = 63.8755715737869
x17=−77.4897509904114x_{17} = -77.4897509904114
x18=−19.8855763345093x_{18} = -19.8855763345093
x19=−58.6392731370675x_{19} = -58.6392731370675
x20=78.5369867826151x_{20} = 78.5369867826151
x21=94.2454216780301x_{21} = 94.2454216780301
x22=−50.2610609682413x_{22} = -50.2610609682413
x23=54.4501913584021x_{23} = 54.4501913584021
x24=1.9801233301909x_{24} = 1.9801233301909
x25=−6.24754852825825x_{25} = -6.24754852825825
x26=−37.6932159861931x_{26} = -37.6932159861931
x27=−70.1590684769674x_{27} = -70.1590684769674
x28=−59.6865371859136x_{28} = -59.6865371859136
x29=−32.4562767876337x_{29} = -32.4562767876337
x30=−17.7898639401636x_{30} = -17.7898639401636
x31=32.4562767876337x_{31} = 32.4562767876337
x32=−26.1714468439769x_{32} = -26.1714468439769
x33=−68.0645759021945x_{33} = -68.0645759021945
x34=−63.8755715737869x_{34} = -63.8755715737869
x35=−138.228469107333x_{35} = -138.228469107333
x36=−90.0565217942609x_{36} = -90.0565217942609
x37=−13.597218410424x_{37} = -13.597218410424
x38=−15.6937991373847x_{38} = -15.6937991373847
x39=26.1714468439769x_{39} = 26.1714468439769
x40=28.2664714640675x_{40} = 28.2664714640675
x41=52.3556329692382x_{41} = 52.3556329692382
x42=−99.481533543212x_{42} = -99.481533543212
x43=85.8676112088714x_{43} = 85.8676112088714
x44=−92.1509729826281x_{44} = -92.1509729826281
x45=−57.5920066694884x_{45} = -57.5920066694884
x46=−35.5984739109814x_{46} = -35.5984739109814
x47=34.5510870905781x_{47} = 34.5510870905781
x48=21.9810373015517x_{48} = 21.9810373015517
x49=−85.8676112088714x_{49} = -85.8676112088714
x50=−24.0763125873385x_{50} = -24.0763125873385
x51=−83.7731514013501x_{51} = -83.7731514013501
x52=43.9772438382808x_{52} = 43.9772438382808
x53=6.24754852825825x_{53} = 6.24754852825825
x54=−43.9772438382808x_{54} = -43.9772438382808
x55=17.7898639401636x_{55} = 17.7898639401636
x56=−4.13481500730066x_{56} = -4.13481500730066
x57=−79.5842215683334x_{57} = -79.5842215683334
x58=−40.8352623351573x_{58} = -40.8352623351573
x59=70.1590684769674x_{59} = 70.1590684769674
x60=76.4425141503516x_{60} = 76.4425141503516
x61=90.0565217942609x_{61} = 90.0565217942609
x62=61.7810585195113x_{62} = 61.7810585195113
x63=68.0645759021945x_{63} = 68.0645759021945
x64=39.7879215162522x_{64} = 39.7879215162522
x65=−87.9620679348333x_{65} = -87.9620679348333
x66=83.7731514013501x_{66} = 83.7731514013501
x67=−21.9810373015517x_{67} = -21.9810373015517
x68=100.528754364743x_{68} = 100.528754364743
x69=74.3480371495245x_{69} = 74.3480371495245
x70=−11.4998383071223x_{70} = -11.4998383071223
x71=48.1664735791016x_{71} = 48.1664735791016
x72=−81.6786882751859x_{72} = -81.6786882751859
x73=37.6932159861931x_{73} = 37.6932159861931
x74=−28.2664714640675x_{74} = -28.2664714640675
x75=8.35094176033098x_{75} = 8.35094176033098
x76=96.3398680430731x_{76} = 96.3398680430731
x77=644.026148934294x_{77} = 644.026148934294
x78=10.4506972421882x_{78} = 10.4506972421882
x79=−33.5036884317571x_{79} = -33.5036884317571
x80=41.8825959869515x_{80} = 41.8825959869515
x81=−41.8825959869515x_{81} = -41.8825959869515
x82=92.1509729826281x_{82} = 92.1509729826281
x83=72.253555400176x_{83} = 72.253555400176
x84=−65.9700771374731x_{84} = -65.9700771374731
x85=−60.7337989411748x_{85} = -60.7337989411748
x86=30.3614091638229x_{86} = 30.3614091638229
x87=80.6314553867873x_{87} = 80.6314553867873
x88=−7.29989882649759x_{88} = -7.29989882649759
x89=56.5447376487538x_{89} = 56.5447376487538
x90=−55.4974659302913x_{90} = -55.4974659302913
x91=4.13481500730066x_{91} = 4.13481500730066
x92=−46.0718687023364x_{92} = -46.0718687023364
x93=−96.3398680430731x_{93} = -96.3398680430731
x94=46.0718687023364x_{94} = 46.0718687023364
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−(−9sin⁡(3x)2−3cos⁡(3x)x+sin⁡(3x)x2x)=−92\lim_{x \to 0^-}\left(\frac{- \frac{9 \sin{\left(3 x \right)}}{2} - \frac{3 \cos{\left(3 x \right)}}{x} + \frac{\sin{\left(3 x \right)}}{x^{2}}}{x}\right) = - \frac{9}{2}
lim⁡x→0+(−9sin⁡(3x)2−3cos⁡(3x)x+sin⁡(3x)x2x)=−92\lim_{x \to 0^+}\left(\frac{- \frac{9 \sin{\left(3 x \right)}}{2} - \frac{3 \cos{\left(3 x \right)}}{x} + \frac{\sin{\left(3 x \right)}}{x^{2}}}{x}\right) = - \frac{9}{2}
- 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
[644.026148934294,∞)\left[644.026148934294, \infty\right)
Convexa en los intervalos
(−∞,−138.228469107333]\left(-\infty, -138.228469107333\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⁡(3x)2x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(3 x \right)}}{2 x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(3x)2x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(3 x \right)}}{2 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(3*x)/((2*x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(12xsin⁡(3x)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{2 x} \sin{\left(3 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(12xsin⁡(3x)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{2 x} \sin{\left(3 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:
sin⁡(3x)2x=sin⁡(3x)2x\frac{\sin{\left(3 x \right)}}{2 x} = \frac{\sin{\left(3 x \right)}}{2 x}
- No
sin⁡(3x)2x=−sin⁡(3x)2x\frac{\sin{\left(3 x \right)}}{2 x} = - \frac{\sin{\left(3 x \right)}}{2 x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор