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

Gráfico de la función y = sqrt(x)sin(2x)+2x

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
         ___               
f(x) = \/ x *sin(2*x) + 2*x
f(x)=xsin⁡(2x)+2xf{\left(x \right)} = \sqrt{x} \sin{\left(2 x \right)} + 2 x 
f = sqrt(x)*sin(2*x) + 2*x
Gráfico de la función
02468-8-6-4-2-1010025
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:
xsin⁡(2x)+2x=0\sqrt{x} \sin{\left(2 x \right)} + 2 x = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=0x_{1} = 0
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)*sin(2*x) + 2*x.
0sin⁡(0⋅2)+0⋅2\sqrt{0} \sin{\left(0 \cdot 2 \right)} + 0 \cdot 2
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
2xcos⁡(2x)+2+sin⁡(2x)2x=02 \sqrt{x} \cos{\left(2 x \right)} + 2 + \frac{\sin{\left(2 x \right)}}{2 \sqrt{x}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=96.5542962825001x_{1} = 96.5542962825001
x2=8.47891764709577x_{2} = 8.47891764709577
x3=1.37356019503363x_{3} = 1.37356019503363
x4=98.2265781614379x_{4} = 98.2265781614379
x5=91.9451835523682x_{5} = 91.9451835523682
x6=74.5564656873091x_{6} = 74.5564656873091
x7=39.9790233888753x_{7} = 39.9790233888753
x8=38.5686118468664x_{8} = 38.5686118468664
x9=85.6639864504356x_{9} = 85.6639864504356
x10=24.2502578026685x_{10} = 24.2502578026685
x11=76.2426495504873x_{11} = 76.2426495504873
x12=16.6247581295506x_{12} = 16.6247581295506
x13=26.0215946120854x_{13} = 26.0215946120854
x14=52.5548650379479x_{14} = 52.5548650379479
x15=55.6983163751646x_{15} = 55.6983163751646
x16=69.9621437559624x_{16} = 69.9621437559624
x17=148.482173833143x_{17} = 148.482173833143
x18=30.5436517489923x_{18} = 30.5436517489923
x19=4.21066772273144x_{19} = 4.21066772273144
x20=41.706833463119x_{20} = 41.706833463119
x21=99.6966606084695x_{21} = 99.6966606084695
x22=17.9519995096557x_{22} = 17.9519995096557
x23=47.9843256304734x_{23} = 47.9843256304734
x24=63.6820346035394x_{24} = 63.6820346035394
x25=77.6991815134592x_{25} = 77.6991815134592
x26=11.6430255466494x_{26} = 11.6430255466494
x27=60.5421609718334x_{27} = 60.5421609718334
x28=10.3799683707267x_{28} = 10.3799683707267
x29=90.2694501727238x_{29} = 90.2694501727238
x30=46.2674193617919x_{30} = 46.2674193617919
x31=2.03234002532634x_{31} = 2.03234002532634
x32=32.293638795604x_{32} = 32.293638795604
x33=61.9847921754224x_{33} = 61.9847921754224
x34=54.2628627375957x_{34} = 54.2628627375957
x35=33.6892579325605x_{35} = 33.6892579325605
x36=83.9844237047965x_{36} = 83.9844237047965
x37=87.126961486212x_{37} = 87.126961486212
x38=68.2708093195111x_{38} = 68.2708093195111
x39=82.5234737485926x_{39} = 82.5234737485926
x40=19.7547387878667x_{40} = 19.7547387878667
Signos de extremos en los puntos:
(96.55429628250008, 183.330848001607)

(8.478917647095766, 14.1946442597821)

(1.3735601950336325, 3.19754074153171)

(98.22657816143793, 206.31093345149)

(91.94518355236825, 193.424131668234)

(74.55646568730914, 140.533119195779)

(39.979023388875326, 73.7085943762326)

(38.56861184686642, 83.2599406091253)

(85.66398645043564, 180.526319345167)

(24.2502578026685, 43.6686006103094)

(76.24264955048727, 161.156226222749)

(16.624758129550557, 37.186880060478)

(26.02159461208536, 57.0355200865998)

(52.55486503794787, 97.9248812529022)

(55.69831637516463, 103.99638649791)

(69.96214375596243, 148.225006819255)

(148.48217383314267, 309.106870758076)

(30.54365174899228, 55.6438877095042)

(4.210667722731443, 10.1518329305587)

(41.70683346311898, 89.7877518238841)

(99.69666060846947, 189.456225245806)

(17.951999509655668, 31.7731707360955)

(47.984325630473386, 102.817861652733)

(63.682034603539364, 135.277282138302)

(77.69918151345915, 146.637383375041)

(11.64302554664937, 20.0029050677465)

(60.542160971833354, 128.796484694271)

(10.37996837072672, 23.7977459613798)

(90.26945017272377, 171.087915538578)

(46.267419361791895, 85.8014248876055)

(2.0323400253263353, 2.92780803339502)

(32.293638795604025, 70.1734460836237)

(61.98479217542244, 116.156337609873)

(54.262862737595675, 115.819183416222)

(33.689257932560544, 61.6537960633879)

(83.98442370479654, 158.856331888086)

(87.12696148621202, 164.97063010536)

(68.27080931951107, 128.336132572278)

(82.52347374859264, 174.072912200107)

(19.754738787866657, 43.827170323947)


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=96.5542962825001x_{1} = 96.5542962825001
x2=8.47891764709577x_{2} = 8.47891764709577
x3=74.5564656873091x_{3} = 74.5564656873091
x4=39.9790233888753x_{4} = 39.9790233888753
x5=24.2502578026685x_{5} = 24.2502578026685
x6=52.5548650379479x_{6} = 52.5548650379479
x7=55.6983163751646x_{7} = 55.6983163751646
x8=30.5436517489923x_{8} = 30.5436517489923
x9=99.6966606084695x_{9} = 99.6966606084695
x10=17.9519995096557x_{10} = 17.9519995096557
x11=77.6991815134592x_{11} = 77.6991815134592
x12=11.6430255466494x_{12} = 11.6430255466494
x13=90.2694501727238x_{13} = 90.2694501727238
x14=46.2674193617919x_{14} = 46.2674193617919
x15=2.03234002532634x_{15} = 2.03234002532634
x16=61.9847921754224x_{16} = 61.9847921754224
x17=33.6892579325605x_{17} = 33.6892579325605
x18=83.9844237047965x_{18} = 83.9844237047965
x19=87.126961486212x_{19} = 87.126961486212
x20=68.2708093195111x_{20} = 68.2708093195111
Puntos máximos de la función:
x20=1.37356019503363x_{20} = 1.37356019503363
x20=98.2265781614379x_{20} = 98.2265781614379
x20=91.9451835523682x_{20} = 91.9451835523682
x20=38.5686118468664x_{20} = 38.5686118468664
x20=85.6639864504356x_{20} = 85.6639864504356
x20=76.2426495504873x_{20} = 76.2426495504873
x20=16.6247581295506x_{20} = 16.6247581295506
x20=26.0215946120854x_{20} = 26.0215946120854
x20=69.9621437559624x_{20} = 69.9621437559624
x20=148.482173833143x_{20} = 148.482173833143
x20=4.21066772273144x_{20} = 4.21066772273144
x20=41.706833463119x_{20} = 41.706833463119
x20=47.9843256304734x_{20} = 47.9843256304734
x20=63.6820346035394x_{20} = 63.6820346035394
x20=60.5421609718334x_{20} = 60.5421609718334
x20=10.3799683707267x_{20} = 10.3799683707267
x20=32.293638795604x_{20} = 32.293638795604
x20=54.2628627375957x_{20} = 54.2628627375957
x20=82.5234737485926x_{20} = 82.5234737485926
x20=19.7547387878667x_{20} = 19.7547387878667
Decrece en los intervalos
[99.6966606084695,∞)\left[99.6966606084695, \infty\right)
Crece en los intervalos
(−∞,2.03234002532634]\left(-\infty, 2.03234002532634\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
−4xsin⁡(2x)+2cos⁡(2x)x−sin⁡(2x)4x32=0- 4 \sqrt{x} \sin{\left(2 x \right)} + \frac{2 \cos{\left(2 x \right)}}{\sqrt{x}} - \frac{\sin{\left(2 x \right)}}{4 x^{\frac{3}{2}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−108.387253064509x_{1} = -108.387253064509
x2=−20.4325833360263x_{2} = -20.4325833360263
x3=119.38261492224x_{3} = 119.38261492224
x4=−58.1237650724076x_{4} = -58.1237650724076
x5=−17.2932090920213x_{5} = -17.2932090920213
x6=−6.32258264927498x_{6} = -6.32258264927498
x7=89.5381826814368x_{7} = 89.5381826814368
x8=50.2704552704552x_{8} = 50.2704552704552
x9=87.9674362076958x_{9} = 87.9674362076958
x10=−51.8411009510188x_{10} = -51.8411009510188
x11=−1.71019274472844x_{11} = -1.71019274472844
x12=−9.45118658622095x_{12} = -9.45118658622095
x13=−73.8308133888566x_{13} = -73.8308133888566
x14=−87.9674362076958x_{14} = -87.9674362076958
x15=22.0025074952079x_{15} = 22.0025074952079
x16=−25.1426821866891x_{16} = -25.1426821866891
x17=−23.5725476766968x_{17} = -23.5725476766968
x18=95.8211849194386x_{18} = 95.8211849194386
x19=42.417394154352x_{19} = 42.417394154352
x20=51.8411009510188x_{20} = 51.8411009510188
x21=1.71019274472844x_{21} = 1.71019274472844
x22=−43.9879800927231x_{22} = -43.9879800927231
x23=−36.135233199145x_{23} = -36.135233199145
x24=80.1137331592428x_{24} = 80.1137331592428
x25=−97.3919391693718x_{25} = -97.3919391693718
x26=67.5479430088459x_{26} = 67.5479430088459
x27=70.6893712021347x_{27} = 70.6893712021347
x28=−75.4015391832964x_{28} = -75.4015391832964
x29=−59.6944482409725x_{29} = -59.6944482409725
x30=36.135233199145x_{30} = 36.135233199145
x31=15.7238533086656x_{31} = 15.7238533086656
x32=100.533451613468x_{32} = 100.533451613468
x33=−61.265137210639x_{33} = -61.265137210639
x34=−29.8535030657731x_{34} = -29.8535030657731
x35=7.88561085819129x_{35} = 7.88561085819129
x36=−39.2762727343286x_{36} = -39.2762727343286
x37=−94.250432071883x_{37} = -94.250432071883
x38=−37.7057414530669x_{38} = -37.7057414530669
x39=59.6944482409725x_{39} = 59.6944482409725
x40=12.5862153543327x_{40} = 12.5862153543327
x41=81.6844694837518x_{41} = 81.6844694837518
x42=26.7128944196388x_{42} = 26.7128944196388
x43=−78.5429992343723x_{43} = -78.5429992343723
x44=73.8308133888566x_{44} = 73.8308133888566
x45=29.8535030657731x_{45} = 29.8535030657731
x46=48.6998193042876x_{46} = 48.6998193042876
x47=58.1237650724076x_{47} = 58.1237650724076
x48=−45.5585805320393x_{48} = -45.5585805320393
x49=−64.4065308560795x_{49} = -64.4065308560795
x50=94.250432071883x_{50} = 94.250432071883
x51=78.5429992343723x_{51} = 78.5429992343723
x52=−80.1137331592428x_{52} = -80.1137331592428
x53=14.1548159332138x_{53} = 14.1548159332138
x54=92.6796806979995x_{54} = 92.6796806979995
x55=−4.76452623548112x_{55} = -4.76452623548112
x56=86.3966915465097x_{56} = 86.3966915465097
x57=−89.5381826814368x_{57} = -89.5381826814368
x58=−31.4238810939663x_{58} = -31.4238810939663
x59=−95.8211849194386x_{59} = -95.8211849194386
x60=−7.88561085819129x_{60} = -7.88561085819129
x61=−22.0025074952079x_{61} = -22.0025074952079
x62=56.5530881881393x_{62} = 56.5530881881393
x63=−65.9772347842297x_{63} = -65.9772347842297
x64=6.32258264927498x_{64} = 6.32258264927498
x65=−53.4117554893238x_{65} = -53.4117554893238
x66=−67.5479430088459x_{66} = -67.5479430088459
x67=−15.7238533086656x_{67} = -15.7238533086656
x68=28.2831714497815x_{68} = 28.2831714497815
x69=−14.1548159332138x_{69} = -14.1548159332138
x70=−72.2600906603582x_{70} = -72.2600906603582
x71=20.4325833360263x_{71} = 20.4325833360263
x72=37.7057414530669x_{72} = 37.7057414530669
x73=45.5585805320393x_{73} = 45.5585805320393
x74=34.5647511087531x_{74} = 34.5647511087531
x75=72.2600906603582x_{75} = 72.2600906603582
x76=−86.3966915465097x_{76} = -86.3966915465097
x77=65.9772347842297x_{77} = 65.9772347842297
x78=−42.417394154352x_{78} = -42.417394154352
x79=64.4065308560795x_{79} = 64.4065308560795
x80=−28.2831714497815x_{80} = -28.2831714497815
x81=−83.2552080721026x_{81} = -83.2552080721026
x82=−81.6844694837518x_{82} = -81.6844694837518
x83=43.9879800927231x_{83} = 43.9879800927231
x84=−50.2704552704552x_{84} = -50.2704552704552
x85=23.5725476766968x_{85} = 23.5725476766968

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
[95.8211849194386,∞)\left[95.8211849194386, \infty\right)
Convexa en los intervalos
(−∞,1.71019274472844]\left(-\infty, 1.71019274472844\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xsin⁡(2x)+2x)=−∞\lim_{x \to -\infty}\left(\sqrt{x} \sin{\left(2 x \right)} + 2 x\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(xsin⁡(2x)+2x)=∞\lim_{x \to \infty}\left(\sqrt{x} \sin{\left(2 x \right)} + 2 x\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sqrt(x)*sin(2*x) + 2*x, dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(xsin⁡(2x)+2xx)y = x \lim_{x \to -\infty}\left(\frac{\sqrt{x} \sin{\left(2 x \right)} + 2 x}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(xsin⁡(2x)+2xx)y = x \lim_{x \to \infty}\left(\frac{\sqrt{x} \sin{\left(2 x \right)} + 2 x}{x}\right)
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:
xsin⁡(2x)+2x=−2x−−xsin⁡(2x)\sqrt{x} \sin{\left(2 x \right)} + 2 x = - 2 x - \sqrt{- x} \sin{\left(2 x \right)}
- No
xsin⁡(2x)+2x=2x+−xsin⁡(2x)\sqrt{x} \sin{\left(2 x \right)} + 2 x = 2 x + \sqrt{- x} \sin{\left(2 x \right)}
- No
es decir, función
no es
par ni impar
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