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

Gráfico de la función y = sin(2*x)/x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       sin(2*x)
f(x) = --------
          x
f(x)=sin⁡(2x)xf{\left(x \right)} = \frac{\sin{\left(2 x \right)}}{x} 
f = sin(2*x)/x
Gráfico de la función
02468-8-6-4-2-10102.5-2.5
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)x=0\frac{\sin{\left(2 x \right)}}{x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π2x_{1} = \frac{\pi}{2}
Solución numérica
x1=89.5353906273091x_{1} = 89.5353906273091
x2=−15.707963267949x_{2} = -15.707963267949
x3=−31.4159265358979x_{3} = -31.4159265358979
x4=42.4115008234622x_{4} = 42.4115008234622
x5=21.9911485751286x_{5} = 21.9911485751286
x6=−42.4115008234622x_{6} = -42.4115008234622
x7=−29.845130209103x_{7} = -29.845130209103
x8=−21.9911485751286x_{8} = -21.9911485751286
x9=−36.1283155162826x_{9} = -36.1283155162826
x10=28.2743338823081x_{10} = 28.2743338823081
x11=86.3937979737193x_{11} = 86.3937979737193
x12=72.2566310325652x_{12} = 72.2566310325652
x13=−94.2477796076938x_{13} = -94.2477796076938
x14=−61.261056745001x_{14} = -61.261056745001
x15=87.9645943005142x_{15} = 87.9645943005142
x16=−95.8185759344887x_{16} = -95.8185759344887
x17=−50.2654824574367x_{17} = -50.2654824574367
x18=23.5619449019235x_{18} = 23.5619449019235
x19=−43.9822971502571x_{19} = -43.9822971502571
x20=−97.3893722612836x_{20} = -97.3893722612836
x21=50.2654824574367x_{21} = 50.2654824574367
x22=−14.1371669411541x_{22} = -14.1371669411541
x23=59.6902604182061x_{23} = 59.6902604182061
x24=58.1194640914112x_{24} = 58.1194640914112
x25=−1668.18569905618x_{25} = -1668.18569905618
x26=−53.4070751110265x_{26} = -53.4070751110265
x27=−23.5619449019235x_{27} = -23.5619449019235
x28=−86.3937979737193x_{28} = -86.3937979737193
x29=−17.2787595947439x_{29} = -17.2787595947439
x30=12.5663706143592x_{30} = 12.5663706143592
x31=−81.6814089933346x_{31} = -81.6814089933346
x32=94.2477796076938x_{32} = 94.2477796076938
x33=81.6814089933346x_{33} = 81.6814089933346
x34=−67.5442420521806x_{34} = -67.5442420521806
x35=−80.1106126665397x_{35} = -80.1106126665397
x36=−1.5707963267949x_{36} = -1.5707963267949
x37=92.6769832808989x_{37} = 92.6769832808989
x38=36.1283155162826x_{38} = 36.1283155162826
x39=−39.2699081698724x_{39} = -39.2699081698724
x40=−28.2743338823081x_{40} = -28.2743338823081
x41=4.71238898038469x_{41} = 4.71238898038469
x42=−237.190245346029x_{42} = -237.190245346029
x43=48.6946861306418x_{43} = 48.6946861306418
x44=−72.2566310325652x_{44} = -72.2566310325652
x45=−271.747764535517x_{45} = -271.747764535517
x46=37.6991118430775x_{46} = 37.6991118430775
x47=70.6858347057703x_{47} = 70.6858347057703
x48=−45.553093477052x_{48} = -45.553093477052
x49=−89.5353906273091x_{49} = -89.5353906273091
x50=65.9734457253857x_{50} = 65.9734457253857
x51=73.8274273593601x_{51} = 73.8274273593601
x52=20.4203522483337x_{52} = 20.4203522483337
x53=−87.9645943005142x_{53} = -87.9645943005142
x54=1.5707963267949x_{54} = 1.5707963267949
x55=45.553093477052x_{55} = 45.553093477052
x56=78.5398163397448x_{56} = 78.5398163397448
x57=−6.28318530717959x_{57} = -6.28318530717959
x58=95.8185759344887x_{58} = 95.8185759344887
x59=15.707963267949x_{59} = 15.707963267949
x60=−20.4203522483337x_{60} = -20.4203522483337
x61=153.9380400259x_{61} = 153.9380400259
x62=−58.1194640914112x_{62} = -58.1194640914112
x63=56.5486677646163x_{63} = 56.5486677646163
x64=80.1106126665397x_{64} = 80.1106126665397
x65=7.85398163397448x_{65} = 7.85398163397448
x66=26.7035375555132x_{66} = 26.7035375555132
x67=29.845130209103x_{67} = 29.845130209103
x68=−65.9734457253857x_{68} = -65.9734457253857
x69=43.9822971502571x_{69} = 43.9822971502571
x70=14.1371669411541x_{70} = 14.1371669411541
x71=−317.300858012569x_{71} = -317.300858012569
x72=−37.6991118430775x_{72} = -37.6991118430775
x73=−59.6902604182061x_{73} = -59.6902604182061
x74=−64.4026493985908x_{74} = -64.4026493985908
x75=−83.2522053201295x_{75} = -83.2522053201295
x76=−75.398223686155x_{76} = -75.398223686155
x77=51.8362787842316x_{77} = 51.8362787842316
x78=100.530964914873x_{78} = 100.530964914873
x79=64.4026493985908x_{79} = 64.4026493985908
x80=34.5575191894877x_{80} = 34.5575191894877
x81=−73.8274273593601x_{81} = -73.8274273593601
x82=6.28318530717959x_{82} = 6.28318530717959
x83=−9.42477796076938x_{83} = -9.42477796076938
x84=−51.8362787842316x_{84} = -51.8362787842316
x85=67.5442420521806x_{85} = 67.5442420521806
x86=−7.85398163397448x_{86} = -7.85398163397448
x87=−370.707933123596x_{87} = -370.707933123596
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(2*x)/x.
sin⁡(0⋅2)0\frac{\sin{\left(0 \cdot 2 \right)}}{0}
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
2cos⁡(2x)x−sin⁡(2x)x2=0\frac{2 \cos{\left(2 x \right)}}{x} - \frac{\sin{\left(2 x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=2.24670472895453x_{1} = 2.24670472895453
x2=−10.1856514796438x_{2} = -10.1856514796438
x3=−24.3370721159772x_{3} = -24.3370721159772
x4=98.172223901556x_{4} = 98.172223901556
x5=−32.1935597952787x_{5} = -32.1935597952787
x6=69.8968599047927x_{6} = 69.8968599047927
x7=−46.3330961388114x_{7} = -46.3330961388114
x8=−98.172223901556x_{8} = -98.172223901556
x9=−77.7512028363303x_{9} = -77.7512028363303
x10=24.3370721159772x_{10} = 24.3370721159772
x11=−19.6222161805821x_{11} = -19.6222161805821
x12=41.6200962353617x_{12} = 41.6200962353617
x13=40.0490643144726x_{13} = 40.0490643144726
x14=−3.86262591846885x_{14} = -3.86262591846885
x15=46.3330961388114x_{15} = 46.3330961388114
x16=−16.4781945199112x_{16} = -16.4781945199112
x17=11.7597262493445x_{17} = 11.7597262493445
x18=85.6054794697228x_{18} = 85.6054794697228
x19=54.1878598258373x_{19} = 54.1878598258373
x20=66.7550989265392x_{20} = 66.7550989265392
x21=51.0459832324538x_{21} = 51.0459832324538
x22=−60.4715244985757x_{22} = -60.4715244985757
x23=84.0346285545694x_{23} = 84.0346285545694
x24=91.8888644664832x_{24} = 91.8888644664832
x25=−63.6133213216672x_{25} = -63.6133213216672
x26=52.6169257678188x_{26} = 52.6169257678188
x27=−54.1878598258373x_{27} = -54.1878598258373
x28=63.6133213216672x_{28} = 63.6133213216672
x29=99.7430603324317x_{29} = 99.7430603324317
x30=−76.1803402100956x_{30} = -76.1803402100956
x31=−35.3358428558098x_{31} = -35.3358428558098
x32=30.6223651301872x_{32} = 30.6223651301872
x33=−82.4637755597094x_{33} = -82.4637755597094
x34=13.3330271294063x_{34} = 13.3330271294063
x35=18.0503111221878x_{35} = 18.0503111221878
x36=−47.9040693934309x_{36} = -47.9040693934309
x37=−68.3259813506395x_{37} = -68.3259813506395
x38=32.1935597952787x_{38} = 32.1935597952787
x39=−99.7430603324317x_{39} = -99.7430603324317
x40=−41.6200962353617x_{40} = -41.6200962353617
x41=38.4780131551656x_{41} = 38.4780131551656
x42=76.1803402100956x_{42} = 76.1803402100956
x43=8.61037763596538x_{43} = 8.61037763596538
x44=−5.45206082971445x_{44} = -5.45206082971445
x45=55.7587861230655x_{45} = 55.7587861230655
x46=−27.4798391439445x_{46} = -27.4798391439445
x47=−85.6054794697228x_{47} = -85.6054794697228
x48=−58.9006179191122x_{48} = -58.9006179191122
x49=−55.7587861230655x_{49} = -55.7587861230655
x50=22.7655670069956x_{50} = 22.7655670069956
x51=−84.0346285545694x_{51} = -84.0346285545694
x52=−38.4780131551656x_{52} = -38.4780131551656
x53=47.9040693934309x_{53} = 47.9040693934309
x54=74.6094747920599x_{54} = 74.6094747920599
x55=204.987701063789x_{55} = 204.987701063789
x56=82.4637755597094x_{56} = 82.4637755597094
x57=−71.4677348441946x_{57} = -71.4677348441946
x58=−2.24670472895453x_{58} = -2.24670472895453
x59=33.7647173885721x_{59} = 33.7647173885721
x60=−49.4750314121659x_{60} = -49.4750314121659
x61=−62.0424254948814x_{61} = -62.0424254948814
x62=16.4781945199112x_{62} = 16.4781945199112
x63=−18.0503111221878x_{63} = -18.0503111221878
x64=77.7512028363303x_{64} = 77.7512028363303
x65=−91.8888644664832x_{65} = -91.8888644664832
x66=−33.7647173885721x_{66} = -33.7647173885721
x67=−40.0490643144726x_{67} = -40.0490643144726
x68=−69.8968599047927x_{68} = -69.8968599047927
x69=96.6013861664138x_{69} = 96.6013861664138
x70=60.4715244985757x_{70} = 60.4715244985757
x71=88.7471755026564x_{71} = 88.7471755026564
x72=10.1856514796438x_{72} = 10.1856514796438
x73=68.3259813506395x_{73} = 68.3259813506395
x74=44.7621104652086x_{74} = 44.7621104652086
x75=−13.3330271294063x_{75} = -13.3330271294063
x76=−79.3220628366317x_{76} = -79.3220628366317
x77=−90.3180208221014x_{77} = -90.3180208221014
x78=−57.3297052975115x_{78} = -57.3297052975115
x79=−25.9084912436398x_{79} = -25.9084912436398
x80=−11.7597262493445x_{80} = -11.7597262493445
x81=19.6222161805821x_{81} = 19.6222161805821
x82=25.9084912436398x_{82} = 25.9084912436398
x83=−93.4597065202651x_{83} = -93.4597065202651
x84=3.86262591846885x_{84} = 3.86262591846885
x85=62.0424254948814x_{85} = 62.0424254948814
x86=90.3180208221014x_{86} = 90.3180208221014
Signos de extremos en los puntos:
(2.246704728954532, -0.434467256422443)

(-10.18565147964378, 0.0980592480281483)

(-24.337072115977193, -0.0410809080835075)

(98.172223901556, 0.0101860484638785)

(-32.19355979527871, 0.0310583676149227)

(69.8968599047927, 0.0143064283116353)

(-46.33309613881142, -0.0215815876990685)

(-98.172223901556, 0.0101860484638785)

(-77.75120283633034, -0.0128612714243586)

(24.337072115977193, -0.0410809080835075)

(-19.622216180582097, 0.0509461061857615)

(41.6200962353617, 0.0240251209641055)

(40.04906431447256, -0.024967426643558)

(-3.8626259184688534, 0.256749107051798)

(46.33309613881142, -0.0215815876990685)

(-16.478194519911238, 0.0606583423726206)

(11.759726249344503, -0.0849592339552253)

(85.60547946972281, 0.0116812959808693)

(54.18785982583734, 0.0184535325015639)

(66.75509892653919, 0.0149797089170242)

(51.04598323245382, 0.0195892402934823)

(-60.47152449857575, 0.0165361437007152)

(84.0346285545694, -0.0118996456204748)

(91.88886446648316, 0.0108825503716759)

(-63.613321321667165, 0.015719492253233)

(52.6169257678188, -0.0190044332375671)

(-54.18785982583734, 0.0184535325015639)

(63.613321321667165, 0.015719492253233)

(99.74306033243167, -0.0100256341886906)

(-76.18034021009562, 0.0131264635863328)

(-35.33584285580975, 0.0282970441297328)

(30.6223651301872, -0.0326515186419956)

(-82.46377555970939, 0.0121263137918205)

(13.333027129406338, 0.0749490399878624)

(18.050311122187804, -0.0553794646022984)

(-47.90406939343085, 0.0208739162691316)

(-68.3259813506395, -0.0146353291349374)

(32.19355979527871, 0.0310583676149227)

(-99.74306033243167, -0.0100256341886906)

(-41.6200962353617, 0.0240251209641055)

(38.47801315516559, 0.0259866739740854)

(76.18034021009562, 0.0131264635863328)

(8.610377635965385, -0.115943604692308)

(-5.4520608297144495, -0.182650405646115)

(55.758786123065505, -0.0179336722809866)

(-27.479839143944467, -0.0363842926436063)

(-85.60547946972281, 0.0116812959808693)

(-58.90061791911219, -0.0169771388955304)

(-55.758786123065505, -0.0179336722809866)

(22.76556700699564, 0.0439153964569649)

(-84.0346285545694, -0.0118996456204748)

(-38.47801315516559, 0.0259866739740854)

(47.90406939343085, 0.0208739162691316)

(74.60947479205991, -0.0134028224709878)

(204.98770106378876, 0.00487832694374757)

(82.46377555970939, 0.0121263137918205)

(-71.46773484419464, -0.0139919857530453)

(-2.246704728954532, -0.434467256422443)

(33.76471738857206, -0.0296134678930985)

(-49.47503141216594, -0.0202111834730081)

(-62.04242549488138, -0.0161174796093628)

(16.478194519911238, 0.0606583423726206)

(-18.050311122187804, -0.0553794646022984)

(77.75120283633034, -0.0128612714243586)

(-91.88886446648316, 0.0108825503716759)

(-33.76471738857206, -0.0296134678930985)

(-40.04906431447256, -0.024967426643558)

(-69.8968599047927, 0.0143064283116353)

(96.60138616641379, -0.0103516796697785)

(60.47152449857575, 0.0165361437007152)

(88.7471755026564, 0.0112677854121748)

(10.18565147964378, 0.0980592480281483)

(68.3259813506395, -0.0146353291349374)

(44.76211046520859, 0.0223389292683471)

(-13.333027129406338, 0.0749490399878624)

(-79.32206283663172, 0.0126065825610424)

(-90.31802082210145, -0.01107181786798)

(-57.32970529751154, 0.0174423008954086)

(-25.908491243639833, 0.0385901989751759)

(-11.759726249344503, -0.0849592339552253)

(19.622216180582097, 0.0509461061857615)

(25.908491243639833, 0.0385901989751759)

(-93.45970652026512, -0.0106996450858762)

(3.8626259184688534, 0.256749107051798)

(62.04242549488138, -0.0161174796093628)

(90.31802082210145, -0.01107181786798)


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=2.24670472895453x_{1} = 2.24670472895453
x2=−24.3370721159772x_{2} = -24.3370721159772
x3=−46.3330961388114x_{3} = -46.3330961388114
x4=−77.7512028363303x_{4} = -77.7512028363303
x5=24.3370721159772x_{5} = 24.3370721159772
x6=40.0490643144726x_{6} = 40.0490643144726
x7=46.3330961388114x_{7} = 46.3330961388114
x8=11.7597262493445x_{8} = 11.7597262493445
x9=84.0346285545694x_{9} = 84.0346285545694
x10=52.6169257678188x_{10} = 52.6169257678188
x11=99.7430603324317x_{11} = 99.7430603324317
x12=30.6223651301872x_{12} = 30.6223651301872
x13=18.0503111221878x_{13} = 18.0503111221878
x14=−68.3259813506395x_{14} = -68.3259813506395
x15=−99.7430603324317x_{15} = -99.7430603324317
x16=8.61037763596538x_{16} = 8.61037763596538
x17=−5.45206082971445x_{17} = -5.45206082971445
x18=55.7587861230655x_{18} = 55.7587861230655
x19=−27.4798391439445x_{19} = -27.4798391439445
x20=−58.9006179191122x_{20} = -58.9006179191122
x21=−55.7587861230655x_{21} = -55.7587861230655
x22=−84.0346285545694x_{22} = -84.0346285545694
x23=74.6094747920599x_{23} = 74.6094747920599
x24=−71.4677348441946x_{24} = -71.4677348441946
x25=−2.24670472895453x_{25} = -2.24670472895453
x26=33.7647173885721x_{26} = 33.7647173885721
x27=−49.4750314121659x_{27} = -49.4750314121659
x28=−62.0424254948814x_{28} = -62.0424254948814
x29=−18.0503111221878x_{29} = -18.0503111221878
x30=77.7512028363303x_{30} = 77.7512028363303
x31=−33.7647173885721x_{31} = -33.7647173885721
x32=−40.0490643144726x_{32} = -40.0490643144726
x33=96.6013861664138x_{33} = 96.6013861664138
x34=68.3259813506395x_{34} = 68.3259813506395
x35=−90.3180208221014x_{35} = -90.3180208221014
x36=−11.7597262493445x_{36} = -11.7597262493445
x37=−93.4597065202651x_{37} = -93.4597065202651
x38=62.0424254948814x_{38} = 62.0424254948814
x39=90.3180208221014x_{39} = 90.3180208221014
Puntos máximos de la función:
x39=−10.1856514796438x_{39} = -10.1856514796438
x39=98.172223901556x_{39} = 98.172223901556
x39=−32.1935597952787x_{39} = -32.1935597952787
x39=69.8968599047927x_{39} = 69.8968599047927
x39=−98.172223901556x_{39} = -98.172223901556
x39=−19.6222161805821x_{39} = -19.6222161805821
x39=41.6200962353617x_{39} = 41.6200962353617
x39=−3.86262591846885x_{39} = -3.86262591846885
x39=−16.4781945199112x_{39} = -16.4781945199112
x39=85.6054794697228x_{39} = 85.6054794697228
x39=54.1878598258373x_{39} = 54.1878598258373
x39=66.7550989265392x_{39} = 66.7550989265392
x39=51.0459832324538x_{39} = 51.0459832324538
x39=−60.4715244985757x_{39} = -60.4715244985757
x39=91.8888644664832x_{39} = 91.8888644664832
x39=−63.6133213216672x_{39} = -63.6133213216672
x39=−54.1878598258373x_{39} = -54.1878598258373
x39=63.6133213216672x_{39} = 63.6133213216672
x39=−76.1803402100956x_{39} = -76.1803402100956
x39=−35.3358428558098x_{39} = -35.3358428558098
x39=−82.4637755597094x_{39} = -82.4637755597094
x39=13.3330271294063x_{39} = 13.3330271294063
x39=−47.9040693934309x_{39} = -47.9040693934309
x39=32.1935597952787x_{39} = 32.1935597952787
x39=−41.6200962353617x_{39} = -41.6200962353617
x39=38.4780131551656x_{39} = 38.4780131551656
x39=76.1803402100956x_{39} = 76.1803402100956
x39=−85.6054794697228x_{39} = -85.6054794697228
x39=22.7655670069956x_{39} = 22.7655670069956
x39=−38.4780131551656x_{39} = -38.4780131551656
x39=47.9040693934309x_{39} = 47.9040693934309
x39=204.987701063789x_{39} = 204.987701063789
x39=82.4637755597094x_{39} = 82.4637755597094
x39=16.4781945199112x_{39} = 16.4781945199112
x39=−91.8888644664832x_{39} = -91.8888644664832
x39=−69.8968599047927x_{39} = -69.8968599047927
x39=60.4715244985757x_{39} = 60.4715244985757
x39=88.7471755026564x_{39} = 88.7471755026564
x39=10.1856514796438x_{39} = 10.1856514796438
x39=44.7621104652086x_{39} = 44.7621104652086
x39=−13.3330271294063x_{39} = -13.3330271294063
x39=−79.3220628366317x_{39} = -79.3220628366317
x39=−57.3297052975115x_{39} = -57.3297052975115
x39=−25.9084912436398x_{39} = -25.9084912436398
x39=19.6222161805821x_{39} = 19.6222161805821
x39=25.9084912436398x_{39} = 25.9084912436398
x39=3.86262591846885x_{39} = 3.86262591846885
Decrece en los intervalos
[99.7430603324317,∞)\left[99.7430603324317, \infty\right)
Crece en los intervalos
(−∞,−99.7430603324317]\left(-\infty, -99.7430603324317\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(−2sin⁡(2x)−2cos⁡(2x)x+sin⁡(2x)x2)x=0\frac{2 \left(- 2 \sin{\left(2 x \right)} - \frac{2 \cos{\left(2 x \right)}}{x} + \frac{\sin{\left(2 x \right)}}{x^{2}}\right)}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−12.5264126404965x_{1} = -12.5264126404965
x2=−43.9709250198299x_{2} = -43.9709250198299
x3=−2.97018499528636x_{3} = -2.97018499528636
x4=15.6760458632822x_{4} = 15.6760458632822
x5=95.8133573606804x_{5} = 95.8133573606804
x6=−64.3948844946117x_{6} = -64.3948844946117
x7=−39.2571702659654x_{7} = -39.2571702659654
x8=58.1108594230163x_{8} = 58.1108594230163
x9=−59.6818822743783x_{9} = -59.6818822743783
x10=−20.3958276156359x_{10} = -20.3958276156359
x11=81.6752870376032x_{11} = 81.6752870376032
x12=−80.1043706477551x_{12} = -80.1043706477551
x13=31.4000002782599x_{13} = 31.4000002782599
x14=−17.2497574606835x_{14} = -17.2497574606835
x15=86.3880100042327x_{15} = 86.3880100042327
x16=−75.391591452362x_{16} = -75.391591452362
x17=−9.37132279238738x_{17} = -9.37132279238738
x18=29.828364501764x_{18} = 29.828364501764
x19=87.9589097056013x_{19} = 87.9589097056013
x20=14.1016805019762x_{20} = 14.1016805019762
x21=7.78961820519359x_{21} = 7.78961820519359
x22=2.97018499528636x_{22} = 2.97018499528636
x23=56.5398239792896x_{23} = 56.5398239792896
x24=23.5406987060771x_{24} = 23.5406987060771
x25=−73.8206539800394x_{25} = -73.8206539800394
x26=−86.3880100042327x_{26} = -86.3880100042327
x27=28.2566352310993x_{27} = 28.2566352310993
x28=51.8266306358671x_{28} = 51.8266306358671
x29=−67.536838414692x_{29} = -67.536838414692
x30=−4.60292007146833x_{30} = -4.60292007146833
x31=43.9709250198299x_{31} = 43.9709250198299
x32=50.2555326476356x_{32} = 50.2555326476356
x33=4.60292007146833x_{33} = 4.60292007146833
x34=−51.8266306358671x_{34} = -51.8266306358671
x35=−97.3842378699522x_{35} = -97.3842378699522
x36=92.671587779267x_{36} = 92.671587779267
x37=−29.828364501764x_{37} = -29.828364501764
x38=37.6858427046437x_{38} = 37.6858427046437
x39=−89.5298057788704x_{39} = -89.5298057788704
x40=45.5421137457344x_{40} = 45.5421137457344
x41=−21.9683807357099x_{41} = -21.9683807357099
x42=−15.6760458632822x_{42} = -15.6760458632822
x43=−95.8133573606804x_{43} = -95.8133573606804
x44=−28.2566352310993x_{44} = -28.2566352310993
x45=−249.754613990023x_{45} = -249.754613990023
x46=59.6818822743783x_{46} = 59.6818822743783
x47=−65.9658657574213x_{47} = -65.9658657574213
x48=−81.6752870376032x_{48} = -81.6752870376032
x49=−37.6858427046437x_{49} = -37.6858427046437
x50=36.1144688810077x_{50} = 36.1144688810077
x51=−6.20222251095099x_{51} = -6.20222251095099
x52=89.5298057788704x_{52} = 89.5298057788704
x53=80.1043706477551x_{53} = 80.1043706477551
x54=−72.2497103686524x_{54} = -72.2497103686524
x55=−100.525990994784x_{55} = -100.525990994784
x56=34.5430424733226x_{56} = 34.5430424733226
x57=−23.5406987060771x_{57} = -23.5406987060771
x58=−50.2555326476356x_{58} = -50.2555326476356
x59=48.6844151814505x_{59} = 48.6844151814505
x60=−45.5421137457344x_{60} = -45.5421137457344
x61=72.2497103686524x_{61} = 72.2497103686524
x62=−14.1016805019762x_{62} = -14.1016805019762
x63=6.20222251095099x_{63} = 6.20222251095099
x64=−58.1108594230163x_{64} = -58.1108594230163
x65=−42.3997071961013x_{65} = -42.3997071961013
x66=100.525990994784x_{66} = 100.525990994784
x67=70.6787602087186x_{67} = 70.6787602087186
x68=78.5334494538573x_{68} = 78.5334494538573
x69=67.536838414692x_{69} = 67.536838414692
x70=−87.9589097056013x_{70} = -87.9589097056013
x71=42.3997071961013x_{71} = 42.3997071961013
x72=−31.4000002782599x_{72} = -31.4000002782599
x73=−83.2461988954369x_{73} = -83.2461988954369
x74=−10.9498482397464x_{74} = -10.9498482397464
x75=94.2424740447043x_{75} = 94.2424740447043
x76=−94.2424740447043x_{76} = -94.2424740447043
x77=−53.3977108664721x_{77} = -53.3977108664721
x78=20.3958276156359x_{78} = 20.3958276156359
x79=12.5264126404965x_{79} = 12.5264126404965
x80=−61.252893502736x_{80} = -61.252893502736
x81=21.9683807357099x_{81} = 21.9683807357099
x82=64.3948844946117x_{82} = 64.3948844946117
x83=−36.1144688810077x_{83} = -36.1144688810077
x84=65.9658657574213x_{84} = 65.9658657574213
x85=26.6847959102454x_{85} = 26.6847959102454
x86=73.8206539800394x_{86} = 73.8206539800394
x87=−7.78961820519359x_{87} = -7.78961820519359
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(−2sin⁡(2x)−2cos⁡(2x)x+sin⁡(2x)x2)x)=−83\lim_{x \to 0^-}\left(\frac{2 \left(- 2 \sin{\left(2 x \right)} - \frac{2 \cos{\left(2 x \right)}}{x} + \frac{\sin{\left(2 x \right)}}{x^{2}}\right)}{x}\right) = - \frac{8}{3}
lim⁡x→0+(2(−2sin⁡(2x)−2cos⁡(2x)x+sin⁡(2x)x2)x)=−83\lim_{x \to 0^+}\left(\frac{2 \left(- 2 \sin{\left(2 x \right)} - \frac{2 \cos{\left(2 x \right)}}{x} + \frac{\sin{\left(2 x \right)}}{x^{2}}\right)}{x}\right) = - \frac{8}{3}
- los límites son iguales, es decir omitimos el punto correspondiente

Intervalos de convexidad y concavidad:
Hallemos los intervales donde la función es convexa o cóncava, para eso veamos cómo se comporta la función en los puntos de flexiones:
Cóncava en los intervalos
[95.8133573606804,∞)\left[95.8133573606804, \infty\right)
Convexa en los intervalos
(−∞,−100.525990994784]\left(-\infty, -100.525990994784\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)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡(2x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)}}{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)/x, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(2x)x2)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)}}{x^{2}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(2x)x2)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)}}{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⁡(2x)x=sin⁡(2x)x\frac{\sin{\left(2 x \right)}}{x} = \frac{\sin{\left(2 x \right)}}{x}
- No
sin⁡(2x)x=−sin⁡(2x)x\frac{\sin{\left(2 x \right)}}{x} = - \frac{\sin{\left(2 x \right)}}{x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = sin(2*x)/x
    © Sr Examen – Калькулятор