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

Gráfico de la función y = sqrt(x)*cos(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         ___       
f(x) = \/ x *cos(x)
f(x)=xcos⁡(x)f{\left(x \right)} = \sqrt{x} \cos{\left(x \right)} 
f = sqrt(x)*cos(x)
Gráfico de la función
02468-8-6-4-2-10105-5
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:
xcos⁡(x)=0\sqrt{x} \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
x3=3π2x_{3} = \frac{3 \pi}{2}
Solución numérica
x1=48.6946861306418x_{1} = 48.6946861306418
x2=−48.6946861306418x_{2} = -48.6946861306418
x3=92.6769832808989x_{3} = 92.6769832808989
x4=86.3937979737193x_{4} = 86.3937979737193
x5=−7.85398163397448x_{5} = -7.85398163397448
x6=−86.3937979737193x_{6} = -86.3937979737193
x7=1.5707963267949x_{7} = 1.5707963267949
x8=−64.4026493985908x_{8} = -64.4026493985908
x9=−58.1194640914112x_{9} = -58.1194640914112
x10=−83.2522053201295x_{10} = -83.2522053201295
x11=−54.9778714378214x_{11} = -54.9778714378214
x12=54.9778714378214x_{12} = 54.9778714378214
x13=89.5353906273091x_{13} = 89.5353906273091
x14=−20.4203522483337x_{14} = -20.4203522483337
x15=32.9867228626928x_{15} = 32.9867228626928
x16=−17.2787595947439x_{16} = -17.2787595947439
x17=23.5619449019235x_{17} = 23.5619449019235
x18=−45.553093477052x_{18} = -45.553093477052
x19=64.4026493985908x_{19} = 64.4026493985908
x20=45.553093477052x_{20} = 45.553093477052
x21=83.2522053201295x_{21} = 83.2522053201295
x22=−29.845130209103x_{22} = -29.845130209103
x23=−51.8362787842316x_{23} = -51.8362787842316
x24=80.1106126665397x_{24} = 80.1106126665397
x25=−39.2699081698724x_{25} = -39.2699081698724
x26=−92.6769832808989x_{26} = -92.6769832808989
x27=4.71238898038469x_{27} = 4.71238898038469
x28=70.6858347057703x_{28} = 70.6858347057703
x29=36.1283155162826x_{29} = 36.1283155162826
x30=−70.6858347057703x_{30} = -70.6858347057703
x31=42.4115008234622x_{31} = 42.4115008234622
x32=−42.4115008234622x_{32} = -42.4115008234622
x33=−67.5442420521806x_{33} = -67.5442420521806
x34=10.9955742875643x_{34} = 10.9955742875643
x35=98.9601685880785x_{35} = 98.9601685880785
x36=−23.5619449019235x_{36} = -23.5619449019235
x37=20.4203522483337x_{37} = 20.4203522483337
x38=−61.261056745001x_{38} = -61.261056745001
x39=−10.9955742875643x_{39} = -10.9955742875643
x40=17.2787595947439x_{40} = 17.2787595947439
x41=−95.8185759344887x_{41} = -95.8185759344887
x42=−36.1283155162826x_{42} = -36.1283155162826
x43=61.261056745001x_{43} = 61.261056745001
x44=73.8274273593601x_{44} = 73.8274273593601
x45=14.1371669411541x_{45} = 14.1371669411541
x46=−26.7035375555132x_{46} = -26.7035375555132
x47=51.8362787842316x_{47} = 51.8362787842316
x48=−89.5353906273091x_{48} = -89.5353906273091
x49=39.2699081698724x_{49} = 39.2699081698724
x50=−32.9867228626928x_{50} = -32.9867228626928
x51=−14.1371669411541x_{51} = -14.1371669411541
x52=−4.71238898038469x_{52} = -4.71238898038469
x53=−76.9690200129499x_{53} = -76.9690200129499
x54=95.8185759344887x_{54} = 95.8185759344887
x55=76.9690200129499x_{55} = 76.9690200129499
x56=58.1194640914112x_{56} = 58.1194640914112
x57=−80.1106126665397x_{57} = -80.1106126665397
x58=−73.8274273593601x_{58} = -73.8274273593601
x59=7.85398163397448x_{59} = 7.85398163397448
x60=−1.5707963267949x_{60} = -1.5707963267949
x61=29.845130209103x_{61} = 29.845130209103
x62=0x_{62} = 0
x63=67.5442420521806x_{63} = 67.5442420521806
x64=26.7035375555132x_{64} = 26.7035375555132
x65=−98.9601685880785x_{65} = -98.9601685880785
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(x)*cos(x).
0cos⁡(0)\sqrt{0} \cos{\left(0 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
−xsin⁡(x)+cos⁡(x)2x=0- \sqrt{x} \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{2 \sqrt{x}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−12.6060134442754x_{1} = -12.6060134442754
x2=84.8288957966139x_{2} = 84.8288957966139
x3=72.26355003974x_{3} = 72.26355003974
x4=78.5461819355535x_{4} = 78.5461819355535
x5=94.253084424113x_{5} = 94.253084424113
x6=87.970277977177x_{6} = 87.970277977177
x7=−100.535938219808x_{7} = -100.535938219808
x8=15.7397193560049x_{8} = 15.7397193560049
x9=91.1116746699497x_{9} = 91.1116746699497
x10=−28.2920048800691x_{10} = -28.2920048800691
x11=−81.6875298021918x_{11} = -81.6875298021918
x12=−56.5575080935408x_{12} = -56.5575080935408
x13=−43.9936619344429x_{13} = -43.9936619344429
x14=12.6060134442754x_{14} = 12.6060134442754
x15=−65.9810235167388x_{15} = -65.9810235167388
x16=−84.8288957966139x_{16} = -84.8288957966139
x17=28.2920048800691x_{17} = 28.2920048800691
x18=−47.1344973476771x_{18} = -47.1344973476771
x19=100.535938219808x_{19} = 100.535938219808
x20=−91.1116746699497x_{20} = -91.1116746699497
x21=−18.8760383379859x_{21} = -18.8760383379859
x22=22.013857636623x_{22} = 22.013857636623
x23=25.1526172579356x_{23} = 25.1526172579356
x24=34.5719807601687x_{24} = 34.5719807601687
x25=3.29231002128209x_{25} = 3.29231002128209
x26=−75.4048544617952x_{26} = -75.4048544617952
x27=47.1344973476771x_{27} = 47.1344973476771
x28=−22.013857636623x_{28} = -22.013857636623
x29=−25.1526172579356x_{29} = -25.1526172579356
x30=97.3945059759883x_{30} = 97.3945059759883
x31=−59.6986356231676x_{31} = -59.6986356231676
x32=−78.5461819355535x_{32} = -78.5461819355535
x33=−50.2754273458806x_{33} = -50.2754273458806
x34=−97.3945059759883x_{34} = -97.3945059759883
x35=−40.8529429059734x_{35} = -40.8529429059734
x36=53.4164352526291x_{36} = 53.4164352526291
x37=−31.43183263459x_{37} = -31.43183263459
x38=81.6875298021918x_{38} = 81.6875298021918
x39=59.6986356231676x_{39} = 59.6986356231676
x40=69.1222718113619x_{40} = 69.1222718113619
x41=−15.7397193560049x_{41} = -15.7397193560049
x42=−53.4164352526291x_{42} = -53.4164352526291
x43=−6.36162039206566x_{43} = -6.36162039206566
x44=−9.4774857054208x_{44} = -9.4774857054208
x45=65.9810235167388x_{45} = 65.9810235167388
x46=−72.26355003974x_{46} = -72.26355003974
x47=18.8760383379859x_{47} = 18.8760383379859
x48=6.36162039206566x_{48} = 6.36162039206566
x49=−62.8398096434599x_{49} = -62.8398096434599
x50=56.5575080935408x_{50} = 56.5575080935408
x51=62.8398096434599x_{51} = 62.8398096434599
x52=9.4774857054208x_{52} = 9.4774857054208
x53=−87.970277977177x_{53} = -87.970277977177
x54=43.9936619344429x_{54} = 43.9936619344429
x55=31.43183263459x_{55} = 31.43183263459
x56=−34.5719807601687x_{56} = -34.5719807601687
x57=−3.29231002128209x_{57} = -3.29231002128209
x58=50.2754273458806x_{58} = 50.2754273458806
x59=−94.253084424113x_{59} = -94.253084424113
x60=−37.7123693157661x_{60} = -37.7123693157661
x61=0.653271187094403x_{61} = 0.653271187094403
x62=37.7123693157661x_{62} = 37.7123693157661
x63=75.4048544617952x_{63} = 75.4048544617952
x64=40.8529429059734x_{64} = 40.8529429059734
x65=−69.1222718113619x_{65} = -69.1222718113619
Signos de extremos en los puntos:
(-12.606013444275414, 3.54770528507369*I)

(84.8288957966139, -9.21010036807552)

(72.26355003974, -8.50059354672143)

(78.54618193555346, -8.86244882770153)

(94.25308442411298, 9.70826617196213)

(87.970277977177, 9.3790957026809)

(-100.53593821980844, 10.0266371036526*I)

(15.73971935600487, -3.96533125786786)

(91.11167466994975, -9.54509983536653)

(-28.292004880069126, -5.31819247681142*I)

(-81.6875298021918, 9.03794608714833*I)

(-56.55750809354077, 7.52017873187663*I)

(-43.993661934442905, 6.63234347961736*I)

(12.606013444275414, 3.54770528507369)

(-65.9810235167388, -8.12263718050406*I)

(-84.8288957966139, -9.21010036807552*I)

(28.292004880069126, -5.31819247681142)

(-47.13449734767706, -6.86507057309731*I)

(100.53593821980844, 10.0266371036526)

(-91.11167466994975, -9.54509983536653*I)

(-18.876038337985854, 4.34313289225214*I)

(22.013857636622962, -4.69068300028599)

(25.152617257935617, 5.01424788582548)

(34.57198076016866, -5.87917944809784)

(3.2923100212820864, -1.79390283516354)

(-75.40485446179518, 8.68340596604541*I)

(47.13449734767706, -6.86507057309731)

(-22.013857636622962, -4.69068300028599*I)

(-25.152617257935617, 5.01424788582548*I)

(97.39450597598831, -9.86873543893722)

(-59.698635623167625, -7.72621823510751*I)

(-78.54618193555346, -8.86244882770153*I)

(-50.27542734588058, 7.0901660932241*I)

(-97.39450597598831, -9.86873543893722*I)

(-40.85294290597337, -6.39115203326596*I)

(53.41643525262913, -7.3083346567585)

(-31.431832634590037, 5.60570075250289*I)

(81.6875298021918, 9.03794608714833)

(59.698635623167625, -7.72621823510751)

(69.1222718113619, 8.3137630000695)

(-15.73971935600487, -3.96533125786786*I)

(-53.41643525262913, -7.3083346567585*I)

(-6.361620392065665, 2.51447081861791*I)

(-9.477485705420795, -3.07427725087097*I)

(65.9810235167388, -8.12263718050406)

(-72.26355003974, -8.50059354672143*I)

(18.876038337985854, 4.34313289225214)

(6.361620392065665, 2.51447081861791)

(-62.839809643459915, 7.92690554538958*I)

(56.55750809354077, 7.52017873187663)

(62.839809643459915, 7.92690554538958)

(9.477485705420795, -3.07427725087097)

(-87.970277977177, 9.3790957026809*I)

(43.993661934442905, 6.63234347961736)

(31.431832634590037, 5.60570075250289)

(-34.57198076016866, -5.87917944809784*I)

(-3.2923100212820864, -1.79390283516354*I)

(50.27542734588058, 7.0901660932241)

(-94.25308442411298, 9.70826617196213*I)

(-37.712369315766125, 6.14050009006662*I)

(0.6532711870944031, 0.641832750676974)

(37.712369315766125, 6.14050009006662)

(75.40485446179518, 8.68340596604541)

(40.85294290597337, -6.39115203326596)

(-69.1222718113619, 8.3137630000695*I)


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=84.8288957966139x_{1} = 84.8288957966139
x2=72.26355003974x_{2} = 72.26355003974
x3=78.5461819355535x_{3} = 78.5461819355535
x4=15.7397193560049x_{4} = 15.7397193560049
x5=91.1116746699497x_{5} = 91.1116746699497
x6=28.2920048800691x_{6} = 28.2920048800691
x7=22.013857636623x_{7} = 22.013857636623
x8=34.5719807601687x_{8} = 34.5719807601687
x9=3.29231002128209x_{9} = 3.29231002128209
x10=47.1344973476771x_{10} = 47.1344973476771
x11=97.3945059759883x_{11} = 97.3945059759883
x12=53.4164352526291x_{12} = 53.4164352526291
x13=59.6986356231676x_{13} = 59.6986356231676
x14=65.9810235167388x_{14} = 65.9810235167388
x15=9.4774857054208x_{15} = 9.4774857054208
x16=40.8529429059734x_{16} = 40.8529429059734
Puntos máximos de la función:
x16=94.253084424113x_{16} = 94.253084424113
x16=87.970277977177x_{16} = 87.970277977177
x16=12.6060134442754x_{16} = 12.6060134442754
x16=100.535938219808x_{16} = 100.535938219808
x16=25.1526172579356x_{16} = 25.1526172579356
x16=81.6875298021918x_{16} = 81.6875298021918
x16=69.1222718113619x_{16} = 69.1222718113619
x16=18.8760383379859x_{16} = 18.8760383379859
x16=6.36162039206566x_{16} = 6.36162039206566
x16=56.5575080935408x_{16} = 56.5575080935408
x16=62.8398096434599x_{16} = 62.8398096434599
x16=43.9936619344429x_{16} = 43.9936619344429
x16=31.43183263459x_{16} = 31.43183263459
x16=50.2754273458806x_{16} = 50.2754273458806
x16=0.653271187094403x_{16} = 0.653271187094403
x16=37.7123693157661x_{16} = 37.7123693157661
x16=75.4048544617952x_{16} = 75.4048544617952
Decrece en los intervalos
[97.3945059759883,∞)\left[97.3945059759883, \infty\right)
Crece en los intervalos
(−∞,3.29231002128209]\left(-\infty, 3.29231002128209\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
−(xcos⁡(x)+sin⁡(x)x+cos⁡(x)4x32)=0- (\sqrt{x} \cos{\left(x \right)} + \frac{\sin{\left(x \right)}}{\sqrt{x}} + \frac{\cos{\left(x \right)}}{4 x^{\frac{3}{2}}}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−36.1559611393004x_{1} = -36.1559611393004
x2=26.7409029817025x_{2} = 26.7409029817025
x3=80.1230923289863x_{3} = 80.1230923289863
x4=86.4053704242642x_{4} = 86.4053704242642
x5=42.4350586138523x_{5} = 42.4350586138523
x6=−76.9820087826371x_{6} = -76.9820087826371
x7=−7.97819025123437x_{7} = -7.97819025123437
x8=−70.6999773315004x_{8} = -70.6999773315004
x9=−33.0169941017832x_{9} = -33.0169941017832
x10=67.559042028453x_{10} = 67.559042028453
x11=−67.559042028453x_{11} = -67.559042028453
x12=−11.0853581860961x_{12} = -11.0853581860961
x13=51.8555589377593x_{13} = 51.8555589377593
x14=58.136661973445x_{14} = 58.136661973445
x15=33.0169941017832x_{15} = 33.0169941017832
x16=−48.7152085571549x_{16} = -48.7152085571549
x17=−26.7409029817025x_{17} = -26.7409029817025
x18=7.97819025123437x_{18} = 7.97819025123437
x19=−14.2073505099925x_{19} = -14.2073505099925
x20=−73.8409685283396x_{20} = -73.8409685283396
x21=−23.6042658400483x_{21} = -23.6042658400483
x22=4.91125081295869x_{22} = 4.91125081295869
x23=89.5465571901753x_{23} = 89.5465571901753
x24=36.1559611393004x_{24} = 36.1559611393004
x25=64.4181707871237x_{25} = 64.4181707871237
x26=−54.9960510556604x_{26} = -54.9960510556604
x27=76.9820087826371x_{27} = 76.9820087826371
x28=2.0090972384408x_{28} = 2.0090972384408
x29=17.3363302997334x_{29} = 17.3363302997334
x30=−89.5465571901753x_{30} = -89.5465571901753
x31=11.0853581860961x_{31} = 11.0853581860961
x32=−124.100967466518x_{32} = -124.100967466518
x33=−80.1230923289863x_{33} = -80.1230923289863
x34=−45.5750291575042x_{34} = -45.5750291575042
x35=−64.4181707871237x_{35} = -64.4181707871237
x36=54.9960510556604x_{36} = 54.9960510556604
x37=−92.6877714581404x_{37} = -92.6877714581404
x38=92.6877714581404x_{38} = 92.6877714581404
x39=29.8785771570692x_{39} = 29.8785771570692
x40=20.4691384083001x_{40} = 20.4691384083001
x41=14.2073505099925x_{41} = 14.2073505099925
x42=83.2642142711524x_{42} = 83.2642142711524
x43=−58.136661973445x_{43} = -58.136661973445
x44=70.6999773315004x_{44} = 70.6999773315004
x45=−51.8555589377593x_{45} = -51.8555589377593
x46=−83.2642142711524x_{46} = -83.2642142711524
x47=−95.8290105250036x_{47} = -95.8290105250036
x48=−42.4350586138523x_{48} = -42.4350586138523
x49=39.2953468672842x_{49} = 39.2953468672842
x50=23.6042658400483x_{50} = 23.6042658400483
x51=−133.525176756856x_{51} = -133.525176756856
x52=45.5750291575042x_{52} = 45.5750291575042
x53=−61.2773734476957x_{53} = -61.2773734476957
x54=95.8290105250036x_{54} = 95.8290105250036
x55=−39.2953468672842x_{55} = -39.2953468672842
x56=−86.4053704242642x_{56} = -86.4053704242642
x57=−2.0090972384408x_{57} = -2.0090972384408
x58=−98.9702720305701x_{58} = -98.9702720305701
x59=−4.91125081295869x_{59} = -4.91125081295869
x60=−20.4691384083001x_{60} = -20.4691384083001
x61=61.2773734476957x_{61} = 61.2773734476957
x62=98.9702720305701x_{62} = 98.9702720305701
x63=−29.8785771570692x_{63} = -29.8785771570692
x64=−17.3363302997334x_{64} = -17.3363302997334
x65=73.8409685283396x_{65} = 73.8409685283396
x66=48.7152085571549x_{66} = 48.7152085571549

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
[95.8290105250036,∞)\left[95.8290105250036, \infty\right)
Convexa en los intervalos
(−∞,2.0090972384408]\left(-\infty, 2.0090972384408\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xcos⁡(x))=⟨−∞,∞⟩i\lim_{x \to -\infty}\left(\sqrt{x} \cos{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle i
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩iy = \left\langle -\infty, \infty\right\rangle i
lim⁡x→∞(xcos⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\sqrt{x} \cos{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sqrt(x)*cos(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{\sqrt{x}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{\sqrt{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:
xcos⁡(x)=−xcos⁡(x)\sqrt{x} \cos{\left(x \right)} = \sqrt{- x} \cos{\left(x \right)}
- No
xcos⁡(x)=−−xcos⁡(x)\sqrt{x} \cos{\left(x \right)} = - \sqrt{- x} \cos{\left(x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = sqrt(x)*cos(x)
    © Sr Examen – Калькулятор