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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          / 2        \
f(x) = sin\x  + x - 2/
f(x)=sin⁡((x2+x)−2)f{\left(x \right)} = \sin{\left(\left(x^{2} + x\right) - 2 \right)} 
f = sin(x^2 + x - 2)
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⁡((x2+x)−2)=0\sin{\left(\left(x^{2} + x\right) - 2 \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−2x_{1} = -2
x2=1x_{2} = 1
x3=−12+9+4π2x_{3} = - \frac{1}{2} + \frac{\sqrt{9 + 4 \pi}}{2}
x4=−9+4π2−12x_{4} = - \frac{\sqrt{9 + 4 \pi}}{2} - \frac{1}{2}
Solución numérica
x1=−116.939779715697x_{1} = -116.939779715697
x2=−21.748476403341x_{2} = -21.748476403341
x3=−7.74675668540325x_{3} = -7.74675668540325
x4=−29.8776511498725x_{4} = -29.8776511498725
x5=−17.7499191139391x_{5} = -17.7499191139391
x6=8.13172236767178x_{6} = 8.13172236767178
x7=58.3047780282247x_{7} = 58.3047780282247
x8=28.1741291513262x_{8} = 28.1741291513262
x9=−66.8361116966145x_{9} = -66.8361116966145
x10=−43.8333683688807x_{10} = -43.8333683688807
x11=30.4908980080424x_{11} = 30.4908980080424
x12=−98.9743871723917x_{12} = -98.9743871723917
x13=−89.8414536490093x_{13} = -89.8414536490093
x14=−45.084039263718x_{14} = -45.084039263718
x15=−41.9813353390849x_{15} = -41.9813353390849
x16=22.2479932123413x_{16} = 22.2479932123413
x17=26.1880111606559x_{17} = 26.1880111606559
x18=−57.8167891895676x_{18} = -57.8167891895676
x19=−29.7705176205755x_{19} = -29.7705176205755
x20=76.0592600470052x_{20} = 76.0592600470052
x21=92.0878758747221x_{21} = 92.0878758747221
x22=14.5100992136919x_{22} = 14.5100992136919
x23=−83.8565315802369x_{23} = -83.8565315802369
x24=−16.4241141759026x_{24} = -16.4241141759026
x25=86.850082787116x_{25} = 86.850082787116
x26=72.2724784606021x_{26} = 72.2724784606021
x27=96.2038248317161x_{27} = 96.2038248317161
x28=−19.812118814972x_{28} = -19.812118814972
x29=30.287487535207x_{29} = 30.287487535207
x30=64.6654945120484x_{30} = 64.6654945120484
x31=11.7435638079246x_{31} = 11.7435638079246
x32=82.2133435841218x_{32} = 82.2133435841218
x33=−83.5922924219616x_{33} = -83.5922924219616
x34=−50.1829974243502x_{34} = -50.1829974243502
x35=74.1057007842398x_{35} = 74.1057007842398
x36=44.4349806231698x_{36} = 44.4349806231698
x37=−2x_{37} = -2
x38=−9.99813635933461x_{38} = -9.99813635933461
x39=72.1644731490627x_{39} = 72.1644731490627
x40=10.2399876875736x_{40} = 10.2399876875736
x41=−95.8131384517795x_{41} = -95.8131384517795
x42=−45.9563128004075x_{42} = -45.9563128004075
x43=20.148607900646x_{43} = 20.148607900646
x44=34.0840311179036x_{44} = 34.0840311179036
x45=−14.4243366760854x_{45} = -14.4243366760854
x46=−31.7432988433174x_{46} = -31.7432988433174
x47=−68.5658326283559x_{47} = -68.5658326283559
x48=−37.6673132594477x_{48} = -37.6673132594477
x49=−32.1429528306313x_{49} = -32.1429528306313
x50=34.0385815532504x_{50} = 34.0385815532504
x51=85.8917389122117x_{51} = 85.8917389122117
x52=30.5921042171814x_{52} = 30.5921042171814
x53=52.2499715490631x_{53} = 52.2499715490631
x54=−3.91683742088636x_{54} = -3.91683742088636
x55=78.1840995854429x_{55} = 78.1840995854429
x56=−83.9695208624596x_{56} = -83.9695208624596
x57=−47.8519330335048x_{57} = -47.8519330335048
x58=66.1667984245229x_{58} = 66.1667984245229
x59=42.2494483208342x_{59} = 42.2494483208342
x60=−38.9142806067433x_{60} = -38.9142806067433
x61=94.2511488653623x_{61} = 94.2511488653623
x62=4.09342529290929x_{62} = 4.09342529290929
x63=69.4101321435751x_{63} = 69.4101321435751
x64=54.2087397608233x_{64} = 54.2087397608233
x65=−50.0880575855439x_{65} = -50.0880575855439
x66=−18.8100743111686x_{66} = -18.8100743111686
x67=−64.7429852525815x_{67} = -64.7429852525815
x68=−65.641385327196x_{68} = -65.641385327196
x69=3.73768371494959x_{69} = 3.73768371494959
x70=−23.928340140194x_{70} = -23.928340140194
x71=46.2510095532161x_{71} = 46.2510095532161
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(x^2 + x - 2).
sin⁡(−2+02)\sin{\left(-2 + 0^{2} \right)}
Resultado:
f(0)=−sin⁡(2)f{\left(0 \right)} = - \sin{\left(2 \right)}
Punto:
(0, -sin(2))
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
(2x+1)cos⁡(x2+x−2)=0\left(2 x + 1\right) \cos{\left(x^{2} + x - 2 \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−12x_{1} = - \frac{1}{2}
x2=−12+2π+92x_{2} = - \frac{1}{2} + \frac{\sqrt{2 \pi + 9}}{2}
x3=−2π+92−12x_{3} = - \frac{\sqrt{2 \pi + 9}}{2} - \frac{1}{2}
Signos de extremos en los puntos:
(-1/2, -sin(9/4))

                        /                          2               \ 
         __________     |      /        __________\      __________| 
   1   \/ 9 + 2*pi      |  5   |  1   \/ 9 + 2*pi |    \/ 9 + 2*pi | 
(- - + ------------, sin|- - + |- - + ------------|  + ------------|)
   2        2           \  2   \  2        2      /         2      / 

                         /                                       2\ 
         __________      |      __________   /        __________\ | 
   1   \/ 9 + 2*pi       |5   \/ 9 + 2*pi    |  1   \/ 9 + 2*pi | | 
(- - - ------------, -sin|- + ------------ - |- - - ------------| |)
   2        2            \2        2         \  2        2      / /


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:
La función no tiene puntos mínimos
Puntos máximos de la función:
x3=−12x_{3} = - \frac{1}{2}
x3=−12+2π+92x_{3} = - \frac{1}{2} + \frac{\sqrt{2 \pi + 9}}{2}
x3=−2π+92−12x_{3} = - \frac{\sqrt{2 \pi + 9}}{2} - \frac{1}{2}
Decrece en los intervalos
(−∞,−2π+92−12]\left(-\infty, - \frac{\sqrt{2 \pi + 9}}{2} - \frac{1}{2}\right]
Crece en los intervalos
[−12+2π+92,∞)\left[- \frac{1}{2} + \frac{\sqrt{2 \pi + 9}}{2}, \infty\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
−(2x+1)2sin⁡(x2+x−2)+2cos⁡(x2+x−2)=0- \left(2 x + 1\right)^{2} \sin{\left(x^{2} + x - 2 \right)} + 2 \cos{\left(x^{2} + x - 2 \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−40.7513419056834x_{1} = -40.7513419056834
x2=−2.0656371303885x_{2} = -2.0656371303885
x3=−89.8414539995843x_{3} = -89.8414539995843
x4=−0.0133406737356569x_{4} = -0.0133406737356569
x5=66.0724849647371x_{5} = 66.0724849647371
x6=−57.8167905172507x_{6} = -57.8167905172507
x7=−5.09600066724405x_{7} = -5.09600066724405
x8=1.84148236799324x_{8} = 1.84148236799324
x9=83.7562037768776x_{9} = 83.7562037768776
x10=47.9990878002422x_{10} = 47.9990878002422
x11=−70.4101328752524x_{11} = -70.4101328752524
x12=82.2133440259091x_{12} = 82.2133440259091
x13=−6.56803901820778x_{13} = -6.56803901820778
x14=−65.4965433294761x_{14} = -65.4965433294761
x15=−85.3877024313226x_{15} = -85.3877024313226
x16=−17.7499678191407x_{16} = -17.7499678191407
x17=22.2480144501444x_{17} = 22.2480144501444
x18=15.7174008816694x_{18} = 15.7174008816694
x19=−73.8315424852899x_{19} = -73.8315424852899
x20=−3.43109502092136x_{20} = -3.43109502092136
x21=13.5367831850899x_{21} = 13.5367831850899
x22=−29.8776610101329x_{22} = -29.8776610101329
x23=6.52737605310492x_{23} = 6.52737605310492
x24=8.1321110476879x_{24} = 8.1321110476879
x25=67.7502045415264x_{25} = 67.7502045415264
x26=−43.833371441244x_{26} = -43.833371441244
x27=−60.2323848675122x_{27} = -60.2323848675122
x28=−62.2749432414173x_{28} = -62.2749432414173
x29=75.9771470036566x_{29} = 75.9771470036566
x30=−31.6425922627417x_{30} = -31.6425922627417
x31=54.151287160648x_{31} = 54.151287160648
x32=−30.876589495331x_{32} = -30.876589495331
x33=−10.3235920492209x_{33} = -10.3235920492209
x34=11.9976494277443x_{34} = 11.9976494277443
x35=34.0840371617435x_{35} = 34.0840371617435
x36=−62.0711854045399x_{36} = -62.0711854045399
x37=−90.192403554272x_{37} = -90.192403554272
x38=55.5419594601788x_{38} = 55.5419594601788
x39=78.602219015687x_{39} = 78.602219015687
x40=28.1741397553048x_{40} = 28.1741397553048
x41=−19.8121535244453x_{41} = -19.8121535244453
x42=94.2179871076815x_{42} = 94.2179871076815
x43=−9.99842809331267x_{43} = -9.99842809331267
x44=−95.7966569399566x_{44} = -95.7966569399566
x45=−77.9161951649827x_{45} = -77.9161951649827
x46=−50.6863120187489x_{46} = -50.6863120187489
x47=−39.843537062414x_{47} = -39.843537062414
x48=10.2401894803142x_{48} = 10.2401894803142
x49=46.2510119998386x_{49} = 46.2510119998386
x50=−67.6129822284992x_{50} = -67.6129822284992
x51=−5.73459518223488x_{51} = -5.73459518223488
x52=22.658620066539x_{52} = 22.658620066539
x53=−21.7485024621832x_{53} = -21.7485024621832
x54=−3.92307224376124x_{54} = -3.92307224376124
x55=90.9784744967844x_{55} = 90.9784744967844
x56=91.2356810937167x_{56} = 91.2356810937167
x57=3.74096167002634x_{57} = 3.74096167002634
x58=−99.2133672310587x_{58} = -99.2133672310587
x59=96.1550827046514x_{59} = 96.1550827046514
x60=−0.986659326264343x_{60} = -0.986659326264343
x61=74.084643772935x_{61} = 74.084643772935
x62=−45.7832049878908x_{62} = -45.7832049878908
x63=52.249973252293x_{63} = 52.249973252293
x64=4.09600066724405x_{64} = 4.09600066724405
x65=20.1486362972225x_{65} = 20.1486362972225
x66=−23.9283595810071x_{66} = -23.9283595810071
x67=50.462787643336x_{67} = 50.462787643336
x68=77.4218011154984x_{68} = 77.4218011154984
x69=−13.3691816760991x_{69} = -13.3691816760991
x70=1.0656371303885x_{70} = 1.0656371303885
x71=−7.74741343143817x_{71} = -7.74741343143817
x72=26.1880243126247x_{72} = 26.1880243126247
x73=10.6703232113286x_{73} = 10.6703232113286
x74=42.2494515208223x_{74} = 42.2494515208223
x75=72.2724791092941x_{75} = 72.2724791092941
x76=−2.84148236799324x_{76} = -2.84148236799324
x77=−35.4005267500721x_{77} = -35.4005267500721
x78=58.3314852444104x_{78} = 58.3314852444104
x79=−83.8565320118763x_{79} = -83.8565320118763
x80=111.833815640882x_{80} = 111.833815640882
x81=−52.7412921597752x_{81} = -52.7412921597752
x82=−33.8822445029248x_{82} = -33.8822445029248

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
[96.1550827046514,∞)\left[96.1550827046514, \infty\right)
Convexa en los intervalos
(−∞,−95.7966569399566]\left(-\infty, -95.7966569399566\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞sin⁡((x2+x)−2)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(\left(x^{2} + x\right) - 2 \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞sin⁡((x2+x)−2)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(\left(x^{2} + x\right) - 2 \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x^2 + x - 2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡((x2+x)−2)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(\left(x^{2} + x\right) - 2 \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡((x2+x)−2)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(\left(x^{2} + x\right) - 2 \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
Gráfico
Gráfico de la función y = sin(x^2+x-2)
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