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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       sin(2*x)
f(x) = --------
         3*x
f(x)=sin⁡(2x)3xf{\left(x \right)} = \frac{\sin{\left(2 x \right)}}{3 x} 
f = sin(2*x)/((3*x))
Gráfico de la función
02468-8-6-4-2-10101.0-1.0
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⁡(2x)3x=0\frac{\sin{\left(2 x \right)}}{3 x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π2x_{1} = \frac{\pi}{2}
Solución numérica
x1=73.8274273593601x_{1} = 73.8274273593601
x2=70.6858347057703x_{2} = 70.6858347057703
x3=−89.5353906273091x_{3} = -89.5353906273091
x4=42.4115008234622x_{4} = 42.4115008234622
x5=−271.747764535517x_{5} = -271.747764535517
x6=−64.4026493985908x_{6} = -64.4026493985908
x7=14.1371669411541x_{7} = 14.1371669411541
x8=−1.5707963267949x_{8} = -1.5707963267949
x9=−53.4070751110265x_{9} = -53.4070751110265
x10=−67.5442420521806x_{10} = -67.5442420521806
x11=89.5353906273091x_{11} = 89.5353906273091
x12=−9.42477796076938x_{12} = -9.42477796076938
x13=−20.4203522483337x_{13} = -20.4203522483337
x14=28.2743338823081x_{14} = 28.2743338823081
x15=50.2654824574367x_{15} = 50.2654824574367
x16=−80.1106126665397x_{16} = -80.1106126665397
x17=36.1283155162826x_{17} = 36.1283155162826
x18=−97.3893722612836x_{18} = -97.3893722612836
x19=−370.707933123596x_{19} = -370.707933123596
x20=95.8185759344887x_{20} = 95.8185759344887
x21=45.553093477052x_{21} = 45.553093477052
x22=−237.190245346029x_{22} = -237.190245346029
x23=92.6769832808989x_{23} = 92.6769832808989
x24=34.5575191894877x_{24} = 34.5575191894877
x25=1.5707963267949x_{25} = 1.5707963267949
x26=−29.845130209103x_{26} = -29.845130209103
x27=65.9734457253857x_{27} = 65.9734457253857
x28=23.5619449019235x_{28} = 23.5619449019235
x29=20.4203522483337x_{29} = 20.4203522483337
x30=−1669.75649538298x_{30} = -1669.75649538298
x31=78.5398163397448x_{31} = 78.5398163397448
x32=59.6902604182061x_{32} = 59.6902604182061
x33=100.530964914873x_{33} = 100.530964914873
x34=−72.2566310325652x_{34} = -72.2566310325652
x35=21.9911485751286x_{35} = 21.9911485751286
x36=7.85398163397448x_{36} = 7.85398163397448
x37=−17.2787595947439x_{37} = -17.2787595947439
x38=4.71238898038469x_{38} = 4.71238898038469
x39=−37.6991118430775x_{39} = -37.6991118430775
x40=−81.6814089933346x_{40} = -81.6814089933346
x41=153.9380400259x_{41} = 153.9380400259
x42=−21.9911485751286x_{42} = -21.9911485751286
x43=26.7035375555132x_{43} = 26.7035375555132
x44=−58.1194640914112x_{44} = -58.1194640914112
x45=12.5663706143592x_{45} = 12.5663706143592
x46=−87.9645943005142x_{46} = -87.9645943005142
x47=−42.4115008234622x_{47} = -42.4115008234622
x48=−14.1371669411541x_{48} = -14.1371669411541
x49=−94.2477796076938x_{49} = -94.2477796076938
x50=−51.8362787842316x_{50} = -51.8362787842316
x51=15.707963267949x_{51} = 15.707963267949
x52=−43.9822971502571x_{52} = -43.9822971502571
x53=−6.28318530717959x_{53} = -6.28318530717959
x54=58.1194640914112x_{54} = 58.1194640914112
x55=−28.2743338823081x_{55} = -28.2743338823081
x56=48.6946861306418x_{56} = 48.6946861306418
x57=−83.2522053201295x_{57} = -83.2522053201295
x58=−95.8185759344887x_{58} = -95.8185759344887
x59=81.6814089933346x_{59} = 81.6814089933346
x60=−75.398223686155x_{60} = -75.398223686155
x61=−36.1283155162826x_{61} = -36.1283155162826
x62=94.2477796076938x_{62} = 94.2477796076938
x63=86.3937979737193x_{63} = 86.3937979737193
x64=−59.6902604182061x_{64} = -59.6902604182061
x65=87.9645943005142x_{65} = 87.9645943005142
x66=−317.300858012569x_{66} = -317.300858012569
x67=−15.707963267949x_{67} = -15.707963267949
x68=−23.5619449019235x_{68} = -23.5619449019235
x69=175.929188601028x_{69} = 175.929188601028
x70=−61.261056745001x_{70} = -61.261056745001
x71=−7.85398163397448x_{71} = -7.85398163397448
x72=67.5442420521806x_{72} = 67.5442420521806
x73=80.1106126665397x_{73} = 80.1106126665397
x74=6.28318530717959x_{74} = 6.28318530717959
x75=29.845130209103x_{75} = 29.845130209103
x76=−50.2654824574367x_{76} = -50.2654824574367
x77=−73.8274273593601x_{77} = -73.8274273593601
x78=37.6991118430775x_{78} = 37.6991118430775
x79=−86.3937979737193x_{79} = -86.3937979737193
x80=51.8362787842316x_{80} = 51.8362787842316
x81=43.9822971502571x_{81} = 43.9822971502571
x82=56.5486677646163x_{82} = 56.5486677646163
x83=−45.553093477052x_{83} = -45.553093477052
x84=−65.9734457253857x_{84} = -65.9734457253857
x85=−39.2699081698724x_{85} = -39.2699081698724
x86=−31.4159265358979x_{86} = -31.4159265358979
x87=72.2566310325652x_{87} = 72.2566310325652
x88=64.4026493985908x_{88} = 64.4026493985908
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
213xcos⁡(2x)−sin⁡(2x)3x2=02 \frac{1}{3 x} \cos{\left(2 x \right)} - \frac{\sin{\left(2 x \right)}}{3 x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=74.6094747920599x_{1} = 74.6094747920599
x2=96.6013861664138x_{2} = 96.6013861664138
x3=68.3259813506395x_{3} = 68.3259813506395
x4=−85.6054794697228x_{4} = -85.6054794697228
x5=−57.3297052975115x_{5} = -57.3297052975115
x6=−69.8968599047927x_{6} = -69.8968599047927
x7=204.987701063789x_{7} = 204.987701063789
x8=13.3330271294063x_{8} = 13.3330271294063
x9=−38.4780131551656x_{9} = -38.4780131551656
x10=30.6223651301872x_{10} = 30.6223651301872
x11=−19.6222161805821x_{11} = -19.6222161805821
x12=32.1935597952787x_{12} = 32.1935597952787
x13=38.4780131551656x_{13} = 38.4780131551656
x14=82.4637755597094x_{14} = 82.4637755597094
x15=−2.24670472895453x_{15} = -2.24670472895453
x16=−55.7587861230655x_{16} = -55.7587861230655
x17=−33.7647173885721x_{17} = -33.7647173885721
x18=−91.8888644664832x_{18} = -91.8888644664832
x19=18.0503111221878x_{19} = 18.0503111221878
x20=−41.6200962353617x_{20} = -41.6200962353617
x21=99.7430603324317x_{21} = 99.7430603324317
x22=−40.0490643144726x_{22} = -40.0490643144726
x23=−93.4597065202651x_{23} = -93.4597065202651
x24=98.172223901556x_{24} = 98.172223901556
x25=2.24670472895453x_{25} = 2.24670472895453
x26=−76.1803402100956x_{26} = -76.1803402100956
x27=22.7655670069956x_{27} = 22.7655670069956
x28=−47.9040693934309x_{28} = -47.9040693934309
x29=3.86262591846885x_{29} = 3.86262591846885
x30=46.3330961388114x_{30} = 46.3330961388114
x31=16.4781945199112x_{31} = 16.4781945199112
x32=88.7471755026564x_{32} = 88.7471755026564
x33=−16.4781945199112x_{33} = -16.4781945199112
x34=−24.3370721159772x_{34} = -24.3370721159772
x35=−63.6133213216672x_{35} = -63.6133213216672
x36=−11.7597262493445x_{36} = -11.7597262493445
x37=−68.3259813506395x_{37} = -68.3259813506395
x38=−630.67432880713x_{38} = -630.67432880713
x39=−5.45206082971445x_{39} = -5.45206082971445
x40=−58.9006179191122x_{40} = -58.9006179191122
x41=84.0346285545694x_{41} = 84.0346285545694
x42=−82.4637755597094x_{42} = -82.4637755597094
x43=−49.4750314121659x_{43} = -49.4750314121659
x44=40.0490643144726x_{44} = 40.0490643144726
x45=25.9084912436398x_{45} = 25.9084912436398
x46=91.8888644664832x_{46} = 91.8888644664832
x47=−46.3330961388114x_{47} = -46.3330961388114
x48=90.3180208221014x_{48} = 90.3180208221014
x49=19.6222161805821x_{49} = 19.6222161805821
x50=−3.86262591846885x_{50} = -3.86262591846885
x51=−60.4715244985757x_{51} = -60.4715244985757
x52=−99.7430603324317x_{52} = -99.7430603324317
x53=−71.4677348441946x_{53} = -71.4677348441946
x54=−54.1878598258373x_{54} = -54.1878598258373
x55=69.8968599047927x_{55} = 69.8968599047927
x56=−90.3180208221014x_{56} = -90.3180208221014
x57=11.7597262493445x_{57} = 11.7597262493445
x58=62.0424254948814x_{58} = 62.0424254948814
x59=10.1856514796438x_{59} = 10.1856514796438
x60=76.1803402100956x_{60} = 76.1803402100956
x61=−25.9084912436398x_{61} = -25.9084912436398
x62=−84.0346285545694x_{62} = -84.0346285545694
x63=85.6054794697228x_{63} = 85.6054794697228
x64=−77.7512028363303x_{64} = -77.7512028363303
x65=60.4715244985757x_{65} = 60.4715244985757
x66=−98.172223901556x_{66} = -98.172223901556
x67=33.7647173885721x_{67} = 33.7647173885721
x68=−62.0424254948814x_{68} = -62.0424254948814
x69=−32.1935597952787x_{69} = -32.1935597952787
x70=−18.0503111221878x_{70} = -18.0503111221878
x71=51.0459832324538x_{71} = 51.0459832324538
x72=8.61037763596538x_{72} = 8.61037763596538
x73=44.7621104652086x_{73} = 44.7621104652086
x74=63.6133213216672x_{74} = 63.6133213216672
x75=77.7512028363303x_{75} = 77.7512028363303
x76=−79.3220628366317x_{76} = -79.3220628366317
x77=−10.1856514796438x_{77} = -10.1856514796438
x78=55.7587861230655x_{78} = 55.7587861230655
x79=54.1878598258373x_{79} = 54.1878598258373
x80=66.7550989265392x_{80} = 66.7550989265392
x81=47.9040693934309x_{81} = 47.9040693934309
x82=−35.3358428558098x_{82} = -35.3358428558098
x83=24.3370721159772x_{83} = 24.3370721159772
x84=−27.4798391439445x_{84} = -27.4798391439445
x85=41.6200962353617x_{85} = 41.6200962353617
x86=−13.3330271294063x_{86} = -13.3330271294063
x87=52.6169257678188x_{87} = 52.6169257678188
Signos de extremos en los puntos:
(74.60947479205991, -0.00446760749032927)

(96.60138616641379, -0.00345055988992618)

(68.3259813506395, -0.00487844304497913)

(-85.60547946972281, 0.00389376532695642)

(-57.32970529751154, 0.00581410029846953)

(-69.8968599047927, 0.00476880943721178)

(204.98770106378876, 0.00162610898124919)

(13.333027129406338, 0.0249830133292875)

(-38.47801315516559, 0.00866222465802848)

(30.6223651301872, -0.0108838395473319)

(-19.622216180582097, 0.0169820353952538)

(32.19355979527871, 0.0103527892049742)

(38.47801315516559, 0.00866222465802848)

(82.46377555970939, 0.0040421045972735)

(-2.246704728954532, -0.144822418807481)

(-55.758786123065505, -0.00597789076032886)

(-33.76471738857206, -0.00987115596436615)

(-91.88886446648316, 0.00362751679055862)

(18.050311122187804, -0.0184598215340995)

(-41.6200962353617, 0.00800837365470182)

(99.74306033243167, -0.00334187806289688)

(-40.04906431447256, -0.00832247554785266)

(-93.45970652026512, -0.00356654836195873)

(98.172223901556, 0.00339534948795951)

(2.246704728954532, -0.144822418807481)

(-76.18034021009562, 0.00437548786211094)

(22.76556700699564, 0.014638465485655)

(-47.90406939343085, 0.00695797208971055)

(3.8626259184688534, 0.0855830356839328)

(46.33309613881142, -0.00719386256635616)

(16.478194519911238, 0.0202194474575402)

(88.7471755026564, 0.00375592847072495)

(-16.478194519911238, 0.0202194474575402)

(-24.337072115977193, -0.0136936360278358)

(-63.613321321667165, 0.00523983075107767)

(-11.759726249344503, -0.0283197446517418)

(-68.3259813506395, -0.00487844304497913)

(-630.6743288071303, -0.00052853463880156)

(-5.4520608297144495, -0.0608834685487051)

(-58.90061791911219, -0.00565904629851015)

(84.0346285545694, -0.00396654854015828)

(-82.46377555970939, 0.0040421045972735)

(-49.47503141216594, -0.00673706115766935)

(40.04906431447256, -0.00832247554785266)

(25.908491243639833, 0.012863399658392)

(91.88886446648316, 0.00362751679055862)

(-46.33309613881142, -0.00719386256635616)

(90.31802082210145, -0.00369060595599335)

(19.622216180582097, 0.0169820353952538)

(-3.8626259184688534, 0.0855830356839328)

(-60.47152449857575, 0.00551204790023838)

(-99.74306033243167, -0.00334187806289688)

(-71.46773484419464, -0.0046639952510151)

(-54.18785982583734, 0.00615117750052131)

(69.8968599047927, 0.00476880943721178)

(-90.31802082210145, -0.00369060595599335)

(11.759726249344503, -0.0283197446517418)

(62.04242549488138, -0.00537249320312092)

(10.18565147964378, 0.0326864160093828)

(76.18034021009562, 0.00437548786211094)

(-25.908491243639833, 0.012863399658392)

(-84.0346285545694, -0.00396654854015828)

(85.60547946972281, 0.00389376532695642)

(-77.75120283633034, -0.0042870904747862)

(60.47152449857575, 0.00551204790023838)

(-98.172223901556, 0.00339534948795951)

(33.76471738857206, -0.00987115596436615)

(-62.04242549488138, -0.00537249320312092)

(-32.19355979527871, 0.0103527892049742)

(-18.050311122187804, -0.0184598215340995)

(51.04598323245382, 0.00652974676449409)

(8.610377635965385, -0.0386478682307693)

(44.76211046520859, 0.0074463097561157)

(63.613321321667165, 0.00523983075107767)

(77.75120283633034, -0.0042870904747862)

(-79.32206283663172, 0.00420219418701412)

(-10.18565147964378, 0.0326864160093828)

(55.758786123065505, -0.00597789076032886)

(54.18785982583734, 0.00615117750052131)

(66.75509892653919, 0.00499323630567473)

(47.90406939343085, 0.00695797208971055)

(-35.33584285580975, 0.00943234804324425)

(24.337072115977193, -0.0136936360278358)

(-27.479839143944467, -0.0121280975478688)

(41.6200962353617, 0.00800837365470182)

(-13.333027129406338, 0.0249830133292875)

(52.6169257678188, -0.00633481107918903)


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=74.6094747920599x_{1} = 74.6094747920599
x2=96.6013861664138x_{2} = 96.6013861664138
x3=68.3259813506395x_{3} = 68.3259813506395
x4=30.6223651301872x_{4} = 30.6223651301872
x5=−2.24670472895453x_{5} = -2.24670472895453
x6=−55.7587861230655x_{6} = -55.7587861230655
x7=−33.7647173885721x_{7} = -33.7647173885721
x8=18.0503111221878x_{8} = 18.0503111221878
x9=99.7430603324317x_{9} = 99.7430603324317
x10=−40.0490643144726x_{10} = -40.0490643144726
x11=−93.4597065202651x_{11} = -93.4597065202651
x12=2.24670472895453x_{12} = 2.24670472895453
x13=46.3330961388114x_{13} = 46.3330961388114
x14=−24.3370721159772x_{14} = -24.3370721159772
x15=−11.7597262493445x_{15} = -11.7597262493445
x16=−68.3259813506395x_{16} = -68.3259813506395
x17=−630.67432880713x_{17} = -630.67432880713
x18=−5.45206082971445x_{18} = -5.45206082971445
x19=−58.9006179191122x_{19} = -58.9006179191122
x20=84.0346285545694x_{20} = 84.0346285545694
x21=−49.4750314121659x_{21} = -49.4750314121659
x22=40.0490643144726x_{22} = 40.0490643144726
x23=−46.3330961388114x_{23} = -46.3330961388114
x24=90.3180208221014x_{24} = 90.3180208221014
x25=−99.7430603324317x_{25} = -99.7430603324317
x26=−71.4677348441946x_{26} = -71.4677348441946
x27=−90.3180208221014x_{27} = -90.3180208221014
x28=11.7597262493445x_{28} = 11.7597262493445
x29=62.0424254948814x_{29} = 62.0424254948814
x30=−84.0346285545694x_{30} = -84.0346285545694
x31=−77.7512028363303x_{31} = -77.7512028363303
x32=33.7647173885721x_{32} = 33.7647173885721
x33=−62.0424254948814x_{33} = -62.0424254948814
x34=−18.0503111221878x_{34} = -18.0503111221878
x35=8.61037763596538x_{35} = 8.61037763596538
x36=77.7512028363303x_{36} = 77.7512028363303
x37=55.7587861230655x_{37} = 55.7587861230655
x38=24.3370721159772x_{38} = 24.3370721159772
x39=−27.4798391439445x_{39} = -27.4798391439445
x40=52.6169257678188x_{40} = 52.6169257678188
Puntos máximos de la función:
x40=−85.6054794697228x_{40} = -85.6054794697228
x40=−57.3297052975115x_{40} = -57.3297052975115
x40=−69.8968599047927x_{40} = -69.8968599047927
x40=204.987701063789x_{40} = 204.987701063789
x40=13.3330271294063x_{40} = 13.3330271294063
x40=−38.4780131551656x_{40} = -38.4780131551656
x40=−19.6222161805821x_{40} = -19.6222161805821
x40=32.1935597952787x_{40} = 32.1935597952787
x40=38.4780131551656x_{40} = 38.4780131551656
x40=82.4637755597094x_{40} = 82.4637755597094
x40=−91.8888644664832x_{40} = -91.8888644664832
x40=−41.6200962353617x_{40} = -41.6200962353617
x40=98.172223901556x_{40} = 98.172223901556
x40=−76.1803402100956x_{40} = -76.1803402100956
x40=22.7655670069956x_{40} = 22.7655670069956
x40=−47.9040693934309x_{40} = -47.9040693934309
x40=3.86262591846885x_{40} = 3.86262591846885
x40=16.4781945199112x_{40} = 16.4781945199112
x40=88.7471755026564x_{40} = 88.7471755026564
x40=−16.4781945199112x_{40} = -16.4781945199112
x40=−63.6133213216672x_{40} = -63.6133213216672
x40=−82.4637755597094x_{40} = -82.4637755597094
x40=25.9084912436398x_{40} = 25.9084912436398
x40=91.8888644664832x_{40} = 91.8888644664832
x40=19.6222161805821x_{40} = 19.6222161805821
x40=−3.86262591846885x_{40} = -3.86262591846885
x40=−60.4715244985757x_{40} = -60.4715244985757
x40=−54.1878598258373x_{40} = -54.1878598258373
x40=69.8968599047927x_{40} = 69.8968599047927
x40=10.1856514796438x_{40} = 10.1856514796438
x40=76.1803402100956x_{40} = 76.1803402100956
x40=−25.9084912436398x_{40} = -25.9084912436398
x40=85.6054794697228x_{40} = 85.6054794697228
x40=60.4715244985757x_{40} = 60.4715244985757
x40=−98.172223901556x_{40} = -98.172223901556
x40=−32.1935597952787x_{40} = -32.1935597952787
x40=51.0459832324538x_{40} = 51.0459832324538
x40=44.7621104652086x_{40} = 44.7621104652086
x40=63.6133213216672x_{40} = 63.6133213216672
x40=−79.3220628366317x_{40} = -79.3220628366317
x40=−10.1856514796438x_{40} = -10.1856514796438
x40=54.1878598258373x_{40} = 54.1878598258373
x40=66.7550989265392x_{40} = 66.7550989265392
x40=47.9040693934309x_{40} = 47.9040693934309
x40=−35.3358428558098x_{40} = -35.3358428558098
x40=41.6200962353617x_{40} = 41.6200962353617
x40=−13.3330271294063x_{40} = -13.3330271294063
Decrece en los intervalos
[99.7430603324317,∞)\left[99.7430603324317, \infty\right)
Crece en los intervalos
(−∞,−630.67432880713]\left(-\infty, -630.67432880713\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⁡(2x)3x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)}}{3 x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(2x)3x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)}}{3 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(2*x)/((3*x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(13xsin⁡(2x)x)=0\lim_{x \to -\infty}\left(\frac{\frac{1}{3 x} \sin{\left(2 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(13xsin⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\frac{1}{3 x} \sin{\left(2 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⁡(2x)3x=sin⁡(2x)3x\frac{\sin{\left(2 x \right)}}{3 x} = \frac{\sin{\left(2 x \right)}}{3 x}
- No
sin⁡(2x)3x=−sin⁡(2x)3x\frac{\sin{\left(2 x \right)}}{3 x} = - \frac{\sin{\left(2 x \right)}}{3 x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = sin(2*x)/((3*x))
    © Sr Examen – Калькулятор