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sin(4*x)/((5*x))
Trazado el gráfico de la función/ sin(4*x)/((5*x))

Gráfico de la función y = sin(4*x)/((5*x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       sin(4*x)
f(x) = --------
         5*x
f(x)=sin⁡(4x)5xf{\left(x \right)} = \frac{\sin{\left(4 x \right)}}{5 x} 
f = sin(4*x)/((5*x))
Gráfico de la función
02468-8-6-4-2-10101-1
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⁡(4x)5x=0\frac{\sin{\left(4 x \right)}}{5 x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π4x_{1} = \frac{\pi}{4}
Solución numérica
x1=10.2101761241668x_{1} = 10.2101761241668
x2=51.8362787842316x_{2} = 51.8362787842316
x3=−36.1283155162826x_{3} = -36.1283155162826
x4=68.329640215578x_{4} = 68.329640215578
x5=−91.8915851175014x_{5} = -91.8915851175014
x6=3.92699081698724x_{6} = 3.92699081698724
x7=36.1283155162826x_{7} = 36.1283155162826
x8=−3.92699081698724x_{8} = -3.92699081698724
x9=−11.7809724509617x_{9} = -11.7809724509617
x10=−59.6902604182061x_{10} = -59.6902604182061
x11=−54.1924732744239x_{11} = -54.1924732744239
x12=21.9911485751286x_{12} = 21.9911485751286
x13=43.9822971502571x_{13} = 43.9822971502571
x14=69.9004365423729x_{14} = 69.9004365423729
x15=−10.2101761241668x_{15} = -10.2101761241668
x16=98.174770424681x_{16} = 98.174770424681
x17=−76.1836218495525x_{17} = -76.1836218495525
x18=−25.9181393921158x_{18} = -25.9181393921158
x19=−80.1106126665397x_{19} = -80.1106126665397
x20=7.85398163397448x_{20} = 7.85398163397448
x21=−29.845130209103x_{21} = -29.845130209103
x22=−278.030949842697x_{22} = -278.030949842697
x23=−33.7721210260903x_{23} = -33.7721210260903
x24=−23.5619449019235x_{24} = -23.5619449019235
x25=−57.3340659280137x_{25} = -57.3340659280137
x26=−251.327412287183x_{26} = -251.327412287183
x27=14.1371669411541x_{27} = 14.1371669411541
x28=−18.0641577581413x_{28} = -18.0641577581413
x29=58.1194640914112x_{29} = 58.1194640914112
x30=94.2477796076938x_{30} = 94.2477796076938
x31=−51.8362787842316x_{31} = -51.8362787842316
x32=−69.9004365423729x_{32} = -69.9004365423729
x33=80.1106126665397x_{33} = 80.1106126665397
x34=2.35619449019234x_{34} = 2.35619449019234
x35=−19.6349540849362x_{35} = -19.6349540849362
x36=24.3473430653209x_{36} = 24.3473430653209
x37=64.4026493985908x_{37} = 64.4026493985908
x38=−85.6083998103219x_{38} = -85.6083998103219
x39=−77.7544181763474x_{39} = -77.7544181763474
x40=90.3207887907066x_{40} = 90.3207887907066
x41=75.398223686155x_{41} = 75.398223686155
x42=−58.1194640914112x_{42} = -58.1194640914112
x43=−81.6814089933346x_{43} = -81.6814089933346
x44=86.3937979737193x_{44} = 86.3937979737193
x45=40.0553063332699x_{45} = 40.0553063332699
x46=−14.1371669411541x_{46} = -14.1371669411541
x47=−87.9645943005142x_{47} = -87.9645943005142
x48=6.28318530717959x_{48} = 6.28318530717959
x49=−98.174770424681x_{49} = -98.174770424681
x50=−62.0464549083984x_{50} = -62.0464549083984
x51=−37.6991118430775x_{51} = -37.6991118430775
x52=−7.85398163397448x_{52} = -7.85398163397448
x53=84.037603483527x_{53} = 84.037603483527
x54=−63.6172512351933x_{54} = -63.6172512351933
x55=54.1924732744239x_{55} = 54.1924732744239
x56=65.9734457253857x_{56} = 65.9734457253857
x57=−99.7455667514759x_{57} = -99.7455667514759
x58=20.4203522483337x_{58} = 20.4203522483337
x59=46.3384916404494x_{59} = 46.3384916404494
x60=−32.2013246992954x_{60} = -32.2013246992954
x61=72.2566310325652x_{61} = 72.2566310325652
x62=−43.9822971502571x_{62} = -43.9822971502571
x63=−41.6261026600648x_{63} = -41.6261026600648
x64=73.8274273593601x_{64} = 73.8274273593601
x65=−88.7499924639117x_{65} = -88.7499924639117
x66=47.9092879672443x_{66} = 47.9092879672443
x67=18.0641577581413x_{67} = 18.0641577581413
x68=−55.7632696012188x_{68} = -55.7632696012188
x69=−47.9092879672443x_{69} = -47.9092879672443
x70=−95.8185759344887x_{70} = -95.8185759344887
x71=−97.3893722612836x_{71} = -97.3893722612836
x72=87.9645943005142x_{72} = 87.9645943005142
x73=42.4115008234622x_{73} = 42.4115008234622
x74=95.8185759344887x_{74} = 95.8185759344887
x75=62.0464549083984x_{75} = 62.0464549083984
x76=32.2013246992954x_{76} = 32.2013246992954
x77=29.845130209103x_{77} = 29.845130209103
x78=−73.8274273593601x_{78} = -73.8274273593601
x79=28.2743338823081x_{79} = 28.2743338823081
x80=83.2522053201295x_{80} = 83.2522053201295
x81=1535.45340944201x_{81} = 1535.45340944201
x82=−84.037603483527x_{82} = -84.037603483527
x83=−21.9911485751286x_{83} = -21.9911485751286
x84=−132.732289614169x_{84} = -132.732289614169
x85=−65.9734457253857x_{85} = -65.9734457253857
x86=76.1836218495525x_{86} = 76.1836218495525
x87=−15.707963267949x_{87} = -15.707963267949
x88=−45.553093477052x_{88} = -45.553093477052
x89=−1.5707963267949x_{89} = -1.5707963267949
x90=25.9181393921158x_{90} = 25.9181393921158
x91=−40.0553063332699x_{91} = -40.0553063332699
x92=50.2654824574367x_{92} = 50.2654824574367
x93=91.8915851175014x_{93} = 91.8915851175014
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(4*x)/((5*x)).
sin⁡(0⋅4)0⋅5\frac{\sin{\left(0 \cdot 4 \right)}}{0 \cdot 5}
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
415xcos⁡(4x)−sin⁡(4x)5x2=04 \frac{1}{5 x} \cos{\left(4 x \right)} - \frac{\sin{\left(4 x \right)}}{5 x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−13.7399195719722x_{1} = -13.7399195719722
x2=−43.5881642087982x_{2} = -43.5881642087982
x3=40.4464601819914x_{3} = 40.4464601819914
x4=71.8630622447071x_{4} = 71.8630622447071
x5=44.3735877513282x_{5} = 44.3735877513282
x6=70.2922464828135x_{6} = 70.2922464828135
x7=18.4534701501702x_{7} = 18.4534701501702
x8=74.2192843450433x_{8} = 74.2192843450433
x9=82.0733465634182x_{9} = 82.0733465634182
x10=93.8544146025812x_{10} = 93.8544146025812
x11=−12.1685360579886x_{11} = -12.1685360579886
x12=−60.0819192595118x_{12} = -60.0819192595118
x13=394.662418744048x_{13} = 394.662418744048
x14=−16.0967798976394x_{14} = -16.0967798976394
x15=45.9444322332416x_{15} = 45.9444322332416
x16=−42.0173142772847x_{16} = -42.0173142772847
x17=23.9520346967154x_{17} = 23.9520346967154
x18=−71.8630622447071x_{18} = -71.8630622447071
x19=−27.8793930615328x_{19} = -27.8793930615328
x20=12.1685360579886x_{20} = 12.1685360579886
x21=16.0967798976394x_{21} = 16.0967798976394
x22=−9.81110809029105x_{22} = -9.81110809029105
x23=75.0046913263314x_{23} = 75.0046913263314
x24=−100.13764169355x_{24} = -100.13764169355
x25=31.8066606608336x_{25} = 31.8066606608336
x26=−83.6441571910727x_{26} = -83.6441571910727
x27=−78.146317479006x_{27} = -78.146317479006
x28=1.93131295923443x_{28} = 1.93131295923443
x29=8.23909725995562x_{29} = 8.23909725995562
x30=−59.2965073177715x_{30} = -59.2965073177715
x31=−144.905529831764x_{31} = -144.905529831764
x32=−79.7171295652025x_{32} = -79.7171295652025
x33=−20.0245321572363x_{33} = -20.0245321572363
x34=64.0089738962685x_{34} = 64.0089738962685
x35=−56.1548556965816x_{35} = -56.1548556965816
x36=56.1548556965816x_{36} = 56.1548556965816
x37=62.4381530049142x_{37} = 62.4381530049142
x38=67.9360211548046x_{38} = 67.9360211548046
x39=53.7986124610881x_{39} = 53.7986124610881
x40=−87.5711815156414x_{40} = -87.5711815156414
x41=4.30518881798269x_{41} = 4.30518881798269
x42=49.8715301662158x_{42} = 49.8715301662158
x43=−75.7900981248612x_{43} = -75.7900981248612
x44=−67.9360211548046x_{44} = -67.9360211548046
x45=−68.7214298324352x_{45} = -68.7214298324352
x46=−35.7338674220973x_{46} = -35.7338674220973
x47=−5.87986312467225x_{47} = -5.87986312467225
x48=84.4295623053495x_{48} = 84.4295623053495
x49=−61.6527420898353x_{49} = -61.6527420898353
x50=38.0901701050478x_{50} = 38.0901701050478
x51=−45.9444322332416x_{51} = -45.9444322332416
x52=60.0819192595118x_{52} = 60.0819192595118
x53=42.0173142772847x_{53} = 42.0173142772847
x54=−23.9520346967154x_{54} = -23.9520346967154
x55=−64.0089738962685x_{55} = -64.0089738962685
x56=−86.0003721530265x_{56} = -86.0003721530265
x57=−17.6679214279049x_{57} = -17.6679214279049
x58=5.87986312467225x_{58} = 5.87986312467225
x59=96.2106254012507x_{59} = 96.2106254012507
x60=−89.9273947056755x_{60} = -89.9273947056755
x61=−53.7986124610881x_{61} = -53.7986124610881
x62=−66.3652030529498x_{62} = -66.3652030529498
x63=−49.8715301662158x_{63} = -49.8715301662158
x64=9.81110809029105x_{64} = 9.81110809029105
x65=−39.6610314183159x_{65} = -39.6610314183159
x66=89.9273947056755x_{66} = 89.9273947056755
x67=34.1629906753197x_{67} = 34.1629906753197
x68=22.3810552326043x_{68} = 22.3810552326043
x69=20.0245321572363x_{69} = 20.0245321572363
x70=−82.0733465634182x_{70} = -82.0733465634182
x71=30.2357622492879x_{71} = 30.2357622492879
x72=78.146317479006x_{72} = 78.146317479006
x73=48.3006930832069x_{73} = 48.3006930832069
x74=52.2277811939441x_{74} = 52.2277811939441
x75=−65.5797936108425x_{75} = -65.5797936108425
x76=−57.725682309487x_{76} = -57.725682309487
x77=−75.0046913263314x_{77} = -75.0046913263314
x78=96.9960288247349x_{78} = 96.9960288247349
x79=26.3084628839094x_{79} = 26.3084628839094
x80=88.3565860231351x_{80} = 88.3565860231351
x81=−30.2357622492879x_{81} = -30.2357622492879
x82=−97.7814321637186x_{82} = -97.7814321637186
x83=−31.8066606608336x_{83} = -31.8066606608336
x84=66.3652030529498x_{84} = 66.3652030529498
x85=100.13764169355x_{85} = 100.13764169355
x86=−1.93131295923443x_{86} = -1.93131295923443
x87=27.8793930615328x_{87} = 27.8793930615328
x88=−21.5955555086822x_{88} = -21.5955555086822
x89=−38.0901701050478x_{89} = -38.0901701050478
x90=86.0003721530265x_{90} = 86.0003721530265
x91=92.2836069408097x_{91} = 92.2836069408097
x92=−93.8544146025812x_{92} = -93.8544146025812
x93=−34.1629906753197x_{93} = -34.1629906753197
Signos de extremos en los puntos:
(-13.739919571972234, -0.0145537170574425)

(-43.58816420879817, -0.00458832607688831)

(40.44646018199142, -0.00494471404222622)

(71.86306224470708, -0.00278305409666654)

(44.3735877513282, 0.00450711416486993)

(70.29224648281348, -0.00284524602773579)

(18.453470150170162, -0.010837075476112)

(74.2192843450433, 0.00269470215409468)

(82.07334656341823, 0.00243683340006458)

(93.85441460258117, -0.00213095240453964)

(-12.168536057988597, -0.016432363233403)

(-60.08191925951185, 0.0033287596519433)

(394.66241874404795, 0.000506762109527025)

(-16.096779897639355, 0.0124233470459691)

(45.94443223324158, 0.00435302014867034)

(-42.0173142772847, -0.00475985824818994)

(23.952034696715426, 0.00834956650765266)

(-71.86306224470708, -0.00278305409666654)

(-27.879393061532753, -0.00717346891239463)

(12.168536057988597, -0.016432363233403)

(16.096779897639355, 0.0124233470459691)

(-9.811108090291048, 0.0203784424743046)

(75.00469132633138, -0.00266648506247399)

(-100.13764169354951, -0.00199724472573406)

(31.806660660833582, 0.00628779690129321)

(-83.6441571910727, 0.00239107085777116)

(-78.14631747900596, -0.00255928856302387)

(1.9313129592344267, 0.102699642820719)

(8.239097259955619, 0.0242633369490482)

(-59.296507317771535, -0.00337284996234339)

(-144.90552983176443, 0.00138020752265271)

(-79.7171295652025, -0.00250885873077445)

(-20.02453215723628, -0.00998697065742319)

(64.00897389626854, -0.00312453805134383)

(-56.15485569658162, -0.0035615445100986)

(56.15485569658162, -0.0035615445100986)

(62.438153004914184, -0.0032031440268943)

(67.9360211548046, 0.00294392639462058)

(53.79861246108815, 0.00371752785921583)

(-87.57118151564137, -0.00228384705497296)

(4.3051888179826925, -0.0463774418769231)

(49.871530166215834, -0.00401025367547625)

(-75.79009812486125, 0.00263885279068767)

(-67.9360211548046, 0.00294392639462058)

(-68.72142983243522, -0.00291028107370646)

(-35.73386742209732, -0.00559679430121812)

(-5.8798631246722515, -0.0339836935820901)

(84.4295623053495, -0.00236882814223086)

(-61.65274208983533, 0.00324394907611641)

(38.09017010504781, 0.00525058543453313)

(-45.94443223324158, 0.00435302014867034)

(60.08191925951185, 0.0033287596519433)

(42.0173142772847, -0.00475985824818994)

(-23.952034696715426, 0.00834956650765266)

(-64.00897389626854, -0.00312453805134383)

(-86.0003721530265, -0.00232556150577369)

(-17.667921427904876, 0.0113188176518931)

(5.8798631246722515, -0.0339836935820901)

(96.21062540125068, 0.00207876545825337)

(-89.92739470567548, 0.00222400779882111)

(-53.79861246108815, 0.00371752785921583)

(-66.36520305294982, 0.00301360610319761)

(-49.871530166215834, -0.00401025367547625)

(9.811108090291048, 0.0203784424743046)

(-39.66103141831586, 0.00504263302441695)

(89.92739470567548, 0.00222400779882111)

(34.16299067531975, -0.00585413165397496)

(22.381055232604297, 0.00893557170733885)

(20.02453215723628, -0.00998697065742319)

(-82.07334656341823, 0.00243683340006458)

(30.235762249287873, 0.00661445748028606)

(78.14631747900596, -0.00255928856302387)

(48.300693083206895, -0.00414067186791142)

(52.227781193944075, 0.00382933573271186)

(-65.5797936108425, -0.00304969771564102)

(-57.72568230948696, -0.00346462989119866)

(-75.00469132633138, -0.00266648506247399)

(96.99602882473488, -0.00206193323700578)

(26.3084628839094, -0.00760177329502683)

(88.35658602313505, 0.00226354602901002)

(-30.235762249287873, 0.00661445748028606)

(-97.78143216371859, 0.00204537141555956)

(-31.806660660833582, 0.00628779690129321)

(66.36520305294982, 0.00301360610319761)

(100.13764169354951, -0.00199724472573406)

(-1.9313129592344267, 0.102699642820719)

(27.879393061532753, -0.00717346891239463)

(-21.595555508682178, -0.00926054436677421)

(-38.09017010504781, 0.00525058543453313)

(86.0003721530265, -0.00232556150577369)

(92.28360694080966, -0.00216722419879452)

(-93.85441460258117, -0.00213095240453964)

(-34.16299067531975, -0.00585413165397496)


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=−13.7399195719722x_{1} = -13.7399195719722
x2=−43.5881642087982x_{2} = -43.5881642087982
x3=40.4464601819914x_{3} = 40.4464601819914
x4=71.8630622447071x_{4} = 71.8630622447071
x5=70.2922464828135x_{5} = 70.2922464828135
x6=18.4534701501702x_{6} = 18.4534701501702
x7=93.8544146025812x_{7} = 93.8544146025812
x8=−12.1685360579886x_{8} = -12.1685360579886
x9=−42.0173142772847x_{9} = -42.0173142772847
x10=−71.8630622447071x_{10} = -71.8630622447071
x11=−27.8793930615328x_{11} = -27.8793930615328
x12=12.1685360579886x_{12} = 12.1685360579886
x13=75.0046913263314x_{13} = 75.0046913263314
x14=−100.13764169355x_{14} = -100.13764169355
x15=−78.146317479006x_{15} = -78.146317479006
x16=−59.2965073177715x_{16} = -59.2965073177715
x17=−79.7171295652025x_{17} = -79.7171295652025
x18=−20.0245321572363x_{18} = -20.0245321572363
x19=64.0089738962685x_{19} = 64.0089738962685
x20=−56.1548556965816x_{20} = -56.1548556965816
x21=56.1548556965816x_{21} = 56.1548556965816
x22=62.4381530049142x_{22} = 62.4381530049142
x23=−87.5711815156414x_{23} = -87.5711815156414
x24=4.30518881798269x_{24} = 4.30518881798269
x25=49.8715301662158x_{25} = 49.8715301662158
x26=−68.7214298324352x_{26} = -68.7214298324352
x27=−35.7338674220973x_{27} = -35.7338674220973
x28=−5.87986312467225x_{28} = -5.87986312467225
x29=84.4295623053495x_{29} = 84.4295623053495
x30=42.0173142772847x_{30} = 42.0173142772847
x31=−64.0089738962685x_{31} = -64.0089738962685
x32=−86.0003721530265x_{32} = -86.0003721530265
x33=5.87986312467225x_{33} = 5.87986312467225
x34=−49.8715301662158x_{34} = -49.8715301662158
x35=34.1629906753197x_{35} = 34.1629906753197
x36=20.0245321572363x_{36} = 20.0245321572363
x37=78.146317479006x_{37} = 78.146317479006
x38=48.3006930832069x_{38} = 48.3006930832069
x39=−65.5797936108425x_{39} = -65.5797936108425
x40=−57.725682309487x_{40} = -57.725682309487
x41=−75.0046913263314x_{41} = -75.0046913263314
x42=96.9960288247349x_{42} = 96.9960288247349
x43=26.3084628839094x_{43} = 26.3084628839094
x44=100.13764169355x_{44} = 100.13764169355
x45=27.8793930615328x_{45} = 27.8793930615328
x46=−21.5955555086822x_{46} = -21.5955555086822
x47=86.0003721530265x_{47} = 86.0003721530265
x48=92.2836069408097x_{48} = 92.2836069408097
x49=−93.8544146025812x_{49} = -93.8544146025812
x50=−34.1629906753197x_{50} = -34.1629906753197
Puntos máximos de la función:
x50=44.3735877513282x_{50} = 44.3735877513282
x50=74.2192843450433x_{50} = 74.2192843450433
x50=82.0733465634182x_{50} = 82.0733465634182
x50=−60.0819192595118x_{50} = -60.0819192595118
x50=394.662418744048x_{50} = 394.662418744048
x50=−16.0967798976394x_{50} = -16.0967798976394
x50=45.9444322332416x_{50} = 45.9444322332416
x50=23.9520346967154x_{50} = 23.9520346967154
x50=16.0967798976394x_{50} = 16.0967798976394
x50=−9.81110809029105x_{50} = -9.81110809029105
x50=31.8066606608336x_{50} = 31.8066606608336
x50=−83.6441571910727x_{50} = -83.6441571910727
x50=1.93131295923443x_{50} = 1.93131295923443
x50=8.23909725995562x_{50} = 8.23909725995562
x50=−144.905529831764x_{50} = -144.905529831764
x50=67.9360211548046x_{50} = 67.9360211548046
x50=53.7986124610881x_{50} = 53.7986124610881
x50=−75.7900981248612x_{50} = -75.7900981248612
x50=−67.9360211548046x_{50} = -67.9360211548046
x50=−61.6527420898353x_{50} = -61.6527420898353
x50=38.0901701050478x_{50} = 38.0901701050478
x50=−45.9444322332416x_{50} = -45.9444322332416
x50=60.0819192595118x_{50} = 60.0819192595118
x50=−23.9520346967154x_{50} = -23.9520346967154
x50=−17.6679214279049x_{50} = -17.6679214279049
x50=96.2106254012507x_{50} = 96.2106254012507
x50=−89.9273947056755x_{50} = -89.9273947056755
x50=−53.7986124610881x_{50} = -53.7986124610881
x50=−66.3652030529498x_{50} = -66.3652030529498
x50=9.81110809029105x_{50} = 9.81110809029105
x50=−39.6610314183159x_{50} = -39.6610314183159
x50=89.9273947056755x_{50} = 89.9273947056755
x50=22.3810552326043x_{50} = 22.3810552326043
x50=−82.0733465634182x_{50} = -82.0733465634182
x50=30.2357622492879x_{50} = 30.2357622492879
x50=52.2277811939441x_{50} = 52.2277811939441
x50=88.3565860231351x_{50} = 88.3565860231351
x50=−30.2357622492879x_{50} = -30.2357622492879
x50=−97.7814321637186x_{50} = -97.7814321637186
x50=−31.8066606608336x_{50} = -31.8066606608336
x50=66.3652030529498x_{50} = 66.3652030529498
x50=−1.93131295923443x_{50} = -1.93131295923443
x50=−38.0901701050478x_{50} = -38.0901701050478
Decrece en los intervalos
[100.13764169355,∞)\left[100.13764169355, \infty\right)
Crece en los intervalos
(−∞,−100.13764169355]\left(-\infty, -100.13764169355\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(−8sin⁡(4x)−4cos⁡(4x)x+sin⁡(4x)x2)5x=0\frac{2 \left(- 8 \sin{\left(4 x \right)} - \frac{4 \cos{\left(4 x \right)}}{x} + \frac{\sin{\left(4 x \right)}}{x^{2}}\right)}{5 x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=14.1283176155496x_{1} = 14.1283176155496
x2=28.2699119896448x_{2} = 28.2699119896448
x3=−43.9794548528007x_{3} = -43.9794548528007
x4=−69.8986482235832x_{4} = -69.8986482235832
x5=−83.2507038222954x_{5} = -83.2507038222954
x6=−81.679878619113x_{6} = -81.679878619113
x7=−54.190166550132x_{7} = -54.190166550132
x8=3.8948091025968x_{8} = 3.8948091025968
x9=91.8902247919924x_{9} = 91.8902247919924
x10=−33.768419207346x_{10} = -33.768419207346
x11=73.8257341698916x_{11} = 73.8257341698916
x12=−32.1974422473059x_{12} = -32.1974422473059
x13=7.83802293164112x_{13} = 7.83802293164112
x14=−76.1819810298243x_{14} = -76.1819810298243
x15=−7.83802293164112x_{15} = -7.83802293164112
x16=29.8409411371892x_{16} = 29.8409411371892
x17=25.9133153179335x_{17} = 25.9133153179335
x18=−88.7485839832841x_{18} = -88.7485839832841
x19=47.9066786803402x_{19} = 47.9066786803402
x20=176.713879405348x_{20} = 176.713879405348
x21=−3.8948091025968x_{21} = -3.8948091025968
x22=80.1090522834368x_{22} = 80.1090522834368
x23=64.4007084066535x_{23} = 64.4007084066535
x24=−73.8257341698916x_{24} = -73.8257341698916
x25=55.7610278621434x_{25} = 55.7610278621434
x26=18.0572344405039x_{26} = 18.0572344405039
x27=−85.6069396400225x_{27} = -85.6069396400225
x28=65.971550951125x_{28} = 65.971550951125
x29=−95.8172713620925x_{29} = -95.8172713620925
x30=−91.8902247919924x_{30} = -91.8902247919924
x31=2.30146003573417x_{31} = 2.30146003573417
x32=−36.1248551843262x_{32} = -36.1248551843262
x33=−15.7000001391299x_{33} = -15.7000001391299
x34=−23.5566381436421x_{34} = -23.5566381436421
x35=−25.9133153179335x_{35} = -25.9133153179335
x36=−65.971550951125x_{36} = -65.971550951125
x37=−58.1173132428085x_{37} = -58.1173132428085
x38=21.9854625099149x_{38} = 21.9854625099149
x39=54.190166550132x_{39} = 54.190166550132
x40=43.9794548528007x_{40} = 43.9794548528007
x41=−77.7528105063391x_{41} = -77.7528105063391
x42=−99.7443135419539x_{42} = -99.7443135419539
x43=−19.6285851329827x_{43} = -19.6285851329827
x44=−51.8338671961091x_{44} = -51.8338671961091
x45=−98.1734971631186x_{45} = -98.1734971631186
x46=−87.9631732436274x_{46} = -87.9631732436274
x47=69.8986482235832x_{47} = 69.8986482235832
x48=−10.197913807818x_{48} = -10.197913807818
x49=32.1974422473059x_{49} = 32.1974422473059
x50=6.26320632024824x_{50} = 6.26320632024824
x51=95.8172713620925x_{51} = 95.8172713620925
x52=76.1819810298243x_{52} = 76.1819810298243
x53=−75.3965657735576x_{53} = -75.3965657735576
x54=−137.443769129712x_{54} = -137.443769129712
x55=46.3357938896335x_{55} = 46.3357938896335
x56=541.924502084771x_{56} = 541.924502084771
x57=42.4085532365654x_{57} = 42.4085532365654
x58=−40.0521853238775x_{58} = -40.0521853238775
x59=10.197913807818x_{59} = 10.197913807818
x60=−11.7703493530385x_{60} = -11.7703493530385
x61=−29.8409411371892x_{61} = -29.8409411371892
x62=−47.9066786803402x_{62} = -47.9066786803402
x63=49.4775578531048x_{63} = 49.4775578531048
x64=24.3422075907252x_{64} = 24.3422075907252
x65=36.1248551843262x_{65} = 36.1248551843262
x66=90.3194048064041x_{66} = 90.3194048064041
x67=40.0521853238775x_{67} = 40.0521853238775
x68=83.2507038222954x_{68} = 83.2507038222954
x69=72.2549010323048x_{69} = 72.2549010323048
x70=58.1173132428085x_{70} = 58.1173132428085
x71=68.3278107831092x_{71} = 68.3278107831092
x72=84.0361160190483x_{72} = 84.0361160190483
x73=98.1734971631186x_{73} = 98.1734971631186
x74=−84.0361160190483x_{74} = -84.0361160190483
x75=−80.1090522834368x_{75} = -80.1090522834368
x76=62.0444402016406x_{76} = 62.0444402016406
x77=−18.0572344405039x_{77} = -18.0572344405039
x78=−59.68816617625x_{78} = -59.68816617625
x79=50.262995497392x_{79} = 50.262995497392
x80=−45.5503492058811x_{80} = -45.5503492058811
x81=−63.6152862784309x_{81} = -63.6152862784309
x82=−14.1283176155496x_{82} = -14.1283176155496
x83=−21.9854625099149x_{83} = -21.9854625099149
x84=94.2464532916151x_{84} = 94.2464532916151
x85=87.9631732436274x_{85} = 87.9631732436274
x86=−41.6230994477184x_{86} = -41.6230994477184
x87=51.8338671961091x_{87} = 51.8338671961091
x88=20.4142284560092x_{88} = 20.4142284560092
x89=−55.7610278621434x_{89} = -55.7610278621434
x90=86.392351078291x_{90} = 86.392351078291
x91=−62.0444402016406x_{91} = -62.0444402016406
x92=−37.695795726181x_{92} = -37.695795726181
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(−8sin⁡(4x)−4cos⁡(4x)x+sin⁡(4x)x2)5x)=−6415\lim_{x \to 0^-}\left(\frac{2 \left(- 8 \sin{\left(4 x \right)} - \frac{4 \cos{\left(4 x \right)}}{x} + \frac{\sin{\left(4 x \right)}}{x^{2}}\right)}{5 x}\right) = - \frac{64}{15}
lim⁡x→0+(2(−8sin⁡(4x)−4cos⁡(4x)x+sin⁡(4x)x2)5x)=−6415\lim_{x \to 0^+}\left(\frac{2 \left(- 8 \sin{\left(4 x \right)} - \frac{4 \cos{\left(4 x \right)}}{x} + \frac{\sin{\left(4 x \right)}}{x^{2}}\right)}{5 x}\right) = - \frac{64}{15}
- 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
[176.713879405348,∞)\left[176.713879405348, \infty\right)
Convexa en los intervalos
(−∞,−95.8172713620925]\left(-\infty, -95.8172713620925\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⁡(4x)5x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(4 x \right)}}{5 x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(4x)5x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(4 x \right)}}{5 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(4*x)/((5*x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(15xsin⁡(4x)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{5 x} \sin{\left(4 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(15xsin⁡(4x)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{5 x} \sin{\left(4 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⁡(4x)5x=sin⁡(4x)5x\frac{\sin{\left(4 x \right)}}{5 x} = \frac{\sin{\left(4 x \right)}}{5 x}
- No
sin⁡(4x)5x=−sin⁡(4x)5x\frac{\sin{\left(4 x \right)}}{5 x} = - \frac{\sin{\left(4 x \right)}}{5 x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = sin(4*x)/((5*x))
    © Sr Examen – Калькулятор