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

Gráfico de la función y = sqrt(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{\sqrt{\sin{\left(2 x \right)}}}{x} 
f = sqrt(sin(2*x))/x
Gráfico de la función
02468-8-6-4-2-1010-510
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{\sqrt{\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=1.5707963267949x_{1} = 1.5707963267949
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 sqrt(sin(2*x))/x.
sin⁡(0⋅2)0\frac{\sqrt{\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
cos⁡(2x)xsin⁡(2x)−sin⁡(2x)x2=0\frac{\cos{\left(2 x \right)}}{x \sqrt{\sin{\left(2 x \right)}}} - \frac{\sqrt{\sin{\left(2 x \right)}}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−63.609391408151x_{1} = -63.609391408151
x2=66.7513539643031x_{2} = 66.7513539643031
x3=−99.7405539133885x_{3} = -99.7405539133885
x4=−62.0383960813757x_{4} = -62.0383960813757
x5=33.7573137512912x_{5} = 33.7573137512912
x6=36.9001669211527x_{6} = 36.9001669211527
x7=−46.3277006372222x_{7} = -46.3277006372222
x8=−98.1696773784321x_{8} = -98.1696773784321
x9=74.6061240613667x_{9} = 74.6061240613667
x10=11.7384800939942x_{10} = 11.7384800939942
x11=58.8963735834305x_{11} = 58.8963735834305
x12=−13.3142855057572x_{12} = -13.3142855057572
x13=22.7545872771683x_{13} = 22.7545872771683
x14=63.609391408151x_{14} = 63.609391408151
x15=−65.1803771434091x_{15} = -65.1803771434091
x16=−30.6142018882607x_{16} = -30.6142018882607
x17=−19.6094782798075x_{17} = -19.6094782798075
x18=76.1770585706428x_{18} = 76.1770585706428
x19=−18.0364644916681x_{19} = -18.0364644916681
x20=−27.4707425696429x_{20} = -27.4707425696429
x21=41.614089810742x_{21} = 41.614089810742
x22=−47.8988508196586x_{22} = -47.8988508196586
x23=−43.1853230479384x_{23} = -43.1853230479384
x24=−41.614089810742x_{24} = -41.614089810742
x25=47.8988508196586x_{25} = 47.8988508196586
x26=−57.3253446670262x_{26} = -57.3253446670262
x27=85.602559129126x_{27} = 85.602559129126
x28=18.0364644916681x_{28} = 18.0364644916681
x29=44.7565256168206x_{29} = 44.7565256168206
x30=−85.602559129126x_{30} = -85.602559129126
x31=−90.3152528534981x_{31} = -90.3152528534981
x32=−52.6121745880344x_{32} = -52.6121745880344
x33=40.0428222957763x_{33} = 40.0428222957763
x34=−5.40633666694364x_{34} = -5.40633666694364
x35=98.1696773784321x_{35} = 98.1696773784321
x36=−79.3189111701245x_{36} = -79.3189111701245
x37=99.7405539133885x_{37} = 99.7405539133885
x38=−40.0428222957763x_{38} = -40.0428222957763
x39=16.4630276170453x_{39} = 16.4630276170453
x40=−54.1832463772731x_{40} = -54.1832463772731
x41=27.4707425696429x_{41} = 27.4707425696429
x42=3.79827300987529x_{42} = 3.79827300987529
x43=14.8890337004883x_{43} = 14.8890337004883
x44=24.3268011678533x_{44} = 24.3268011678533
x45=55.7543026449315x_{45} = 55.7543026449315
x46=88.744358541403x_{46} = 88.744358541403
x47=−93.4570315962354x_{47} = -93.4570315962354
x48=60.4673904155608x_{48} = 60.4673904155608
x49=−69.8932832672187x_{49} = -69.8932832672187
x50=71.4642368192271x_{50} = 71.4642368192271
x51=2.13739113572906x_{51} = 2.13739113572906
x52=77.747987496317x_{52} = 77.747987496317
x53=−76.1770585706428x_{53} = -76.1770585706428
x54=49.4699785303278x_{54} = 49.4699785303278
x55=8.58137569421011x_{55} = 8.58137569421011
x56=19.6094782798075x_{56} = 19.6094782798075
x57=−35.3287683589234x_{57} = -35.3287683589234
x58=−55.7543026449315x_{58} = -55.7543026449315
x59=−84.0316536256143x_{59} = -84.0316536256143
x60=−32.1857948915632x_{60} = -32.1857948915632
x61=−38.4715163039797x_{61} = -38.4715163039797
x62=−60.4673904155608x_{62} = -60.4673904155608
x63=−49.4699785303278x_{63} = -49.4699785303278
x64=32.1857948915632x_{64} = 32.1857948915632
x65=46.3277006372222x_{65} = 46.3277006372222
x66=62.0383960813757x_{66} = 62.0383960813757
x67=96.5987982349427x_{67} = 96.5987982349427
x68=−71.4642368192271x_{68} = -71.4642368192271
x69=−16.4630276170453x_{69} = -16.4630276170453
x70=−96.5987982349427x_{70} = -96.5987982349427
x71=10.1611269299962x_{71} = 10.1611269299962
x72=82.4607439626894x_{72} = 82.4607439626894
x73=−2.13739113572906x_{73} = -2.13739113572906
x74=68.322322485708x_{74} = 68.322322485708
x75=−87.173460698078x_{75} = -87.173460698078
x76=30.6142018882607x_{76} = 30.6142018882607
x77=−91.8861438154664x_{77} = -91.8861438154664
x78=84.0316536256143x_{78} = 84.0316536256143
x79=−21.182163158836x_{79} = -21.182163158836
x80=−24.3268011678533x_{80} = -24.3268011678533
x81=−10.1611269299962x_{81} = -10.1611269299962
x82=−11.7384800939942x_{82} = -11.7384800939942
x83=54.1832463772731x_{83} = 54.1832463772731
x84=−25.898843096056x_{84} = -25.898843096056
x85=−74.6061240613667x_{85} = -74.6061240613667
x86=90.3152528534981x_{86} = 90.3152528534981
x87=−3.79827300987529x_{87} = -3.79827300987529
x88=−33.7573137512912x_{88} = -33.7573137512912
x89=−82.4607439626894x_{89} = -82.4607439626894
x90=80.8898298980315x_{90} = 80.8898298980315
x91=38.4715163039797x_{91} = 38.4715163039797
x92=69.8932832672187x_{92} = 69.8932832672187
x93=5.40633666694364x_{93} = 5.40633666694364
x94=−8.58137569421011x_{94} = -8.58137569421011
x95=25.898843096056x_{95} = 25.898843096056
x96=−68.322322485708x_{96} = -68.322322485708
x97=91.8861438154664x_{97} = 91.8861438154664
x98=52.6121745880344x_{98} = 52.6121745880344
x99=93.4570315962354x_{99} = 93.4570315962354
x100=−77.747987496317x_{100} = -77.747987496317
Signos de extremos en los puntos:
(-63.60939140815101, -0.0157199778263343*I)

(66.75135396430314, 0.0149801291078152)

(-99.74055391338845, -0.0100257601558681)

(-62.03839608137567, -0.0161180030072405)

(33.75731375129117, 0.0296167148833163*I)

(36.900166921152675, 0.0270951749854015*I)

(-46.327700637222165, -0.0215828443509178)

(-98.16967737843208, -0.0101861805750159*I)

(74.60612406136674, 0.0134031234387214*I)

(11.73848009399415, 0.0850360393335331*I)

(58.89637358343045, 0.016977750594038*I)

(-13.314285505757224, -0.0750017471309216*I)

(22.754587277168262, 0.0439259886853852)

(63.60939140815101, 0.0157199778263343)

(-65.18037714340906, -0.0153411380529724)

(-30.6142018882607, -0.0326558712102137)

(-19.60947827980746, -0.0509626466768668*I)

(76.17705857064283, 0.0131267463173753)

(-18.036464491668116, -0.0554007125582391)

(-27.470742569642862, -0.0363903155700504)

(41.614089810742044, 0.02402685466694)

(-47.89885081965864, -0.0208750533021526*I)

(-43.185323047938354, -0.0231529122413857)

(-41.614089810742044, -0.02402685466694*I)

(47.89885081965864, 0.0208750533021526)

(-57.32534466702617, -0.0174429642673855*I)

(85.60255912912602, 0.0116814952312444)

(18.036464491668116, 0.0554007125582391*I)

(44.756525616820596, 0.0223403229249365)

(-85.60255912912602, -0.0116814952312444*I)

(-90.31525285349807, -0.0110719875286708)

(-52.612174588034385, -0.0190052912967891)

(40.04282229577635, 0.0249693724726356*I)

(-5.406336666943637, -0.18341903022942)

(98.16967737843208, 0.0101861805750159)

(-79.31891117012448, -0.0126068330109199*I)

(99.74055391338845, 0.0100257601558681*I)

(-40.04282229577635, -0.0249693724726356)

(16.463027617045263, 0.0606862687517582)

(-54.18324637727305, -0.018454318078295*I)

(27.470742569642862, 0.0363903155700504*I)

(3.798273009875294, 0.258903189710353)

(14.889033700488254, 0.067087996040512*I)

(24.326801167853258, 0.0410895782745605*I)

(55.75430264493151, 0.0179343933146333*I)

(88.74435854140305, 0.0112679642422872)

(-93.45703159623535, -0.0106997982057615)

(60.46739041556085, 0.0165367089563368)

(-69.89328326721875, -0.0143067943509695*I)

(71.46423681922705, 0.0139923281824918*I)

(2.137391135729064, 0.445271188713631*I)

(77.74798749631695, 0.0128615373628376*I)

(-76.17705857064283, -0.0131267463173753*I)

(49.46997853032777, 0.0202122156003825*I)

(8.581375694210113, 0.116139144217135*I)

(19.60947827980746, 0.0509626466768668)

(-35.328768358923384, -0.028299877004799*I)

(-55.75430264493151, -0.0179343933146333)

(-84.03165362561431, -0.0118998562546094)

(-32.18579489156321, -0.0310621135424149*I)

(-38.471516303979705, -0.0259888679989479*I)

(-60.46739041556085, -0.0165367089563368*I)

(-49.46997853032777, -0.0202122156003825)

(32.18579489156321, 0.0310621135424149)

(46.327700637222165, 0.0215828443509178*I)

(62.03839608137567, 0.0161180030072405*I)

(96.59879823494268, 0.010351818331022*I)

(-71.46423681922705, -0.0139923281824918)

(-16.463027617045263, -0.0606862687517582*I)

(-96.59879823494268, -0.010351818331022)

(10.16112692999623, 0.0981774182935068)

(82.46074396268945, 0.0121265366936926)

(-2.137391135729064, -0.445271188713631)

(68.32232248570797, 0.0146357210055436*I)

(-87.17346069807797, -0.0114710038646054)

(30.6142018882607, 0.0326558712102137*I)

(-91.88614381546644, -0.0108827114792659*I)

(84.03165362561431, 0.0118998562546094*I)

(-21.18216315883595, -0.0471832636912932)

(-24.326801167853258, -0.0410895782745605)

(-10.16112692999623, -0.0981774182935068*I)

(-11.73848009399415, -0.0850360393335331)

(54.18324637727305, 0.018454318078295)

(-25.89884309605599, -0.0385973854559027*I)

(-74.60612406136674, -0.0134031234387214)

(90.31525285349807, 0.0110719875286708*I)

(-3.798273009875294, -0.258903189710353*I)

(-33.75731375129117, -0.0296167148833163)

(-82.46074396268945, -0.0121265366936926*I)

(80.88982989803151, 0.0123620212466517*I)

(38.471516303979705, 0.0259888679989479)

(69.89328326721875, 0.0143067943509695)

(5.406336666943637, 0.18341903022942*I)

(-8.581375694210113, -0.116139144217135)

(25.89884309605599, 0.0385973854559027)

(-68.32232248570797, -0.0146357210055436)

(91.88614381546644, 0.0108827114792659)

(52.612174588034385, 0.0190052912967891*I)

(93.45703159623535, 0.0106997982057615*I)

(-77.74798749631695, -0.0128615373628376)


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=−99.7405539133885x_{1} = -99.7405539133885
x2=−62.0383960813757x_{2} = -62.0383960813757
x3=−46.3277006372222x_{3} = -46.3277006372222
x4=−65.1803771434091x_{4} = -65.1803771434091
x5=−30.6142018882607x_{5} = -30.6142018882607
x6=−18.0364644916681x_{6} = -18.0364644916681
x7=−27.4707425696429x_{7} = -27.4707425696429
x8=−43.1853230479384x_{8} = -43.1853230479384
x9=−90.3152528534981x_{9} = -90.3152528534981
x10=−52.6121745880344x_{10} = -52.6121745880344
x11=−5.40633666694364x_{11} = -5.40633666694364
x12=−40.0428222957763x_{12} = -40.0428222957763
x13=−93.4570315962354x_{13} = -93.4570315962354
x14=−55.7543026449315x_{14} = -55.7543026449315
x15=−84.0316536256143x_{15} = -84.0316536256143
x16=−49.4699785303278x_{16} = -49.4699785303278
x17=−71.4642368192271x_{17} = -71.4642368192271
x18=−96.5987982349427x_{18} = -96.5987982349427
x19=−2.13739113572906x_{19} = -2.13739113572906
x20=−87.173460698078x_{20} = -87.173460698078
x21=−21.182163158836x_{21} = -21.182163158836
x22=−24.3268011678533x_{22} = -24.3268011678533
x23=−11.7384800939942x_{23} = -11.7384800939942
x24=−74.6061240613667x_{24} = -74.6061240613667
x25=−33.7573137512912x_{25} = -33.7573137512912
x26=−8.58137569421011x_{26} = -8.58137569421011
x27=−68.322322485708x_{27} = -68.322322485708
x28=−77.747987496317x_{28} = -77.747987496317
Puntos máximos de la función:
x28=66.7513539643031x_{28} = 66.7513539643031
x28=22.7545872771683x_{28} = 22.7545872771683
x28=63.609391408151x_{28} = 63.609391408151
x28=76.1770585706428x_{28} = 76.1770585706428
x28=41.614089810742x_{28} = 41.614089810742
x28=47.8988508196586x_{28} = 47.8988508196586
x28=85.602559129126x_{28} = 85.602559129126
x28=44.7565256168206x_{28} = 44.7565256168206
x28=98.1696773784321x_{28} = 98.1696773784321
x28=16.4630276170453x_{28} = 16.4630276170453
x28=3.79827300987529x_{28} = 3.79827300987529
x28=88.744358541403x_{28} = 88.744358541403
x28=60.4673904155608x_{28} = 60.4673904155608
x28=19.6094782798075x_{28} = 19.6094782798075
x28=32.1857948915632x_{28} = 32.1857948915632
x28=10.1611269299962x_{28} = 10.1611269299962
x28=82.4607439626894x_{28} = 82.4607439626894
x28=54.1832463772731x_{28} = 54.1832463772731
x28=38.4715163039797x_{28} = 38.4715163039797
x28=69.8932832672187x_{28} = 69.8932832672187
x28=25.898843096056x_{28} = 25.898843096056
x28=91.8861438154664x_{28} = 91.8861438154664
Decrece en los intervalos
[−2.13739113572906,3.79827300987529]\left[-2.13739113572906, 3.79827300987529\right]
Crece en los intervalos
(−∞,−99.7405539133885]\left(-\infty, -99.7405539133885\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
−2sin⁡(2x)−cos⁡2(2x)sin⁡32(2x)−2cos⁡(2x)xsin⁡(2x)+2sin⁡(2x)x2x=0\frac{- 2 \sqrt{\sin{\left(2 x \right)}} - \frac{\cos^{2}{\left(2 x \right)}}{\sin^{\frac{3}{2}}{\left(2 x \right)}} - \frac{2 \cos{\left(2 x \right)}}{x \sqrt{\sin{\left(2 x \right)}}} + \frac{2 \sqrt{\sin{\left(2 x \right)}}}{x^{2}}}{x} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
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{\sqrt{\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{\sqrt{\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 sqrt(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{\sqrt{\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{\sqrt{\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{\sqrt{\sin{\left(2 x \right)}}}{x} = - \frac{\sqrt{- \sin{\left(2 x \right)}}}{x}
- No
sin⁡(2x)x=−sin⁡(2x)x\frac{\sqrt{\sin{\left(2 x \right)}}}{x} = \frac{\sqrt{- \sin{\left(2 x \right)}}}{x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор