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

Gráfico de la función y = sin(x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          / 2\
f(x) = sin\x /
f(x)=sin⁡(x2)f{\left(x \right)} = \sin{\left(x^{2} \right)} 
f = sin(x^2)
Gráfico de la función
02468-8-6-4-2-10102-2
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)=0\sin{\left(x^{2} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−πx_{2} = - \sqrt{\pi}
x3=πx_{3} = \sqrt{\pi}
Solución numérica
x1=−39.7914902637393x_{1} = -39.7914902637393
x2=−22.2793994368607x_{2} = -22.2793994368607
x3=70.4759390524558x_{3} = 70.4759390524558
x4=6.13996024767893x_{4} = 6.13996024767893
x5=30.3395228621317x_{5} = 30.3395228621317
x6=−67.446468865301x_{6} = -67.446468865301
x7=−25.6240680728152x_{7} = -25.6240680728152
x8=−102.358250765251x_{8} = -102.358250765251
x9=−7.72594721818665x_{9} = -7.72594721818665
x10=86.2148955714351x_{10} = 86.2148955714351
x11=3.96332729760601x_{11} = 3.96332729760601
x12=−86.2331131935235x_{12} = -86.2331131935235
x13=−95.8600986425016x_{13} = -95.8600986425016
x14=−50.3514177699791x_{14} = -50.3514177699791
x15=23.7138163312566x_{15} = 23.7138163312566
x16=−79.405141242584x_{16} = -79.405141242584
x17=−53.9070793548543x_{17} = -53.9070793548543
x18=83.568896540101x_{18} = 83.568896540101
x19=−33.862683274665x_{19} = -33.862683274665
x20=8.1224039375905x_{20} = 8.1224039375905
x21=16.244807875181x_{21} = 16.244807875181
x22=−55.8533929588406x_{22} = -55.8533929588406
x23=−19.8166364880301x_{23} = -19.8166364880301
x24=5.60499121639793x_{24} = 5.60499121639793
x25=−3.96332729760601x_{25} = -3.96332729760601
x26=−65.7004437278195x_{26} = -65.7004437278195
x27=−509.434811845833x_{27} = -509.434811845833
x28=97.6620463450291x_{28} = 97.6620463450291
x29=64.2499240996983x_{29} = 64.2499240996983
x30=1523.05464995283x_{30} = 1523.05464995283
x31=0x_{31} = 0
x32=84.7262604713124x_{32} = 84.7262604713124
x33=−15.7539144225679x_{33} = -15.7539144225679
x34=−93.8731452190634x_{34} = -93.8731452190634
x35=−30.2357976940353x_{35} = -30.2357976940353
x36=53.8779325133859x_{36} = 53.8779325133859
x37=−1.77245385090552x_{37} = -1.77245385090552
x38=25.8681123241458x_{38} = 25.8681123241458
x39=−64.1275664870346x_{39} = -64.1275664870346
x40=52.2498231190263x_{40} = 52.2498231190263
x41=−89.7498945058111x_{41} = -89.7498945058111
x42=−11.6227571644753x_{42} = -11.6227571644753
x43=94.1238082693386x_{43} = 94.1238082693386
x44=−30.9037888727466x_{44} = -30.9037888727466
x45=−60.9888981852461x_{45} = -60.9888981852461
x46=18.2485292908913x_{46} = 18.2485292908913
x47=−47.3281688005095x_{47} = -47.3281688005095
x48=82.1853040708499x_{48} = 82.1853040708499
x49=96.2035963853511x_{49} = 96.2035963853511
x50=−43.8480866628973x_{50} = -43.8480866628973
x51=105.367871942952x_{51} = 105.367871942952
x52=28.2482660354898x_{52} = 28.2482660354898
x53=−13.6144763601762x_{53} = -13.6144763601762
x54=78.2292943160867x_{54} = 78.2292943160867
x55=66.2243410266297x_{55} = 66.2243410266297
x56=−41.8314129339366x_{56} = -41.8314129339366
x57=−9.86860538583257x_{57} = -9.86860538583257
x58=42.2424505354389x_{58} = 42.2424505354389
x59=−5.87856438167413x_{59} = -5.87856438167413
x60=−219.061242084455x_{60} = -219.061242084455
x61=9.86860538583257x_{61} = 9.86860538583257
x62=80.212104788192x_{62} = 80.212104788192
x63=−43.5966002736661x_{63} = -43.5966002736661
x64=22.137941502317x_{64} = 22.137941502317
x65=−59.4499094640902x_{65} = -59.4499094640902
x66=46.1519210773927x_{66} = 46.1519210773927
x67=60.0806953935677x_{67} = 60.0806953935677
x68=−44.5587988737213x_{68} = -44.5587988737213
x69=56.6908218728778x_{69} = 56.6908218728778
x70=32.6343363586107x_{70} = 32.6343363586107
x71=56.3015718028556x_{71} = 56.3015718028556
x72=−67.5860615683408x_{72} = -67.5860615683408
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).
sin⁡(02)\sin{\left(0^{2} \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
2xcos⁡(x2)=02 x \cos{\left(x^{2} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=−2π2x_{2} = - \frac{\sqrt{2} \sqrt{\pi}}{2}
x3=2π2x_{3} = \frac{\sqrt{2} \sqrt{\pi}}{2}
Signos de extremos en los puntos:
(0, 0)

    ___   ____     
 -\/ 2 *\/ pi      
(--------------, 1)
       2           

   ___   ____    
 \/ 2 *\/ pi     
(------------, 1)
      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:
Puntos mínimos de la función:
x1=0x_{1} = 0
Puntos máximos de la función:
x1=−2π2x_{1} = - \frac{\sqrt{2} \sqrt{\pi}}{2}
x1=2π2x_{1} = \frac{\sqrt{2} \sqrt{\pi}}{2}
Decrece en los intervalos
(−∞,−2π2]∪[0,∞)\left(-\infty, - \frac{\sqrt{2} \sqrt{\pi}}{2}\right] \cup \left[0, \infty\right)
Crece en los intervalos
(−∞,0]∪[2π2,∞)\left(-\infty, 0\right] \cup \left[\frac{\sqrt{2} \sqrt{\pi}}{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
2(−2x2sin⁡(x2)+cos⁡(x2))=02 \left(- 2 x^{2} \sin{\left(x^{2} \right)} + \cos{\left(x^{2} \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=3.96733151576786x_{1} = 3.96733151576786
x2=55.486583629928x_{2} = 55.486583629928
x3=−3.07855253413366x_{3} = -3.07855253413366
x4=58.4103566874062x_{4} = 58.4103566874062
x5=80.2316859157274x_{5} = 80.2316859157274
x6=70.0287529105774x_{6} = 70.0287529105774
x7=−26.049659306849x_{7} = -26.049659306849
x8=−6.63277181383793x_{8} = -6.63277181383793
x9=−41.8314163492637x_{9} = -41.8314163492637
x10=56.2457461521626x_{10} = 56.2457461521626
x11=−95.8600989263108x_{11} = -95.8600989263108
x12=−0.808251932935767x_{12} = -0.808251932935767
x13=36.839766486297x_{13} = 36.839766486297
x14=66.979059569236x_{14} = 66.979059569236
x15=−91.9799384388618x_{15} = -91.9799384388618
x16=22.1379645446803x_{16} = 22.1379645446803
x17=9.86886548575814x_{17} = 9.86886548575814
x18=−11.6229163837568x_{18} = -11.6229163837568
x19=15.6539537308685x_{19} = 15.6539537308685
x20=−51.3400080449723x_{20} = -51.3400080449723
x21=4.34465629687618x_{21} = 4.34465629687618
x22=−14.1797184947322x_{22} = -14.1797184947322
x23=−43.8480896283294x_{23} = -43.8480896283294
x24=−8.50079702379371x_{24} = -8.50079702379371
x25=−33.81627068654x_{25} = -33.81627068654
x26=94.1238085691455x_{26} = 94.1238085691455
x27=59.3176526349719x_{27} = 59.3176526349719
x28=33.0171638614708x_{28} = 33.0171638614708
x29=−19.8166686134035x_{29} = -19.8166686134035
x30=96.1709354546847x_{30} = 96.1709354546847
x31=28.2482771263222x_{31} = 28.2482771263222
x32=−64.7127740607003x_{32} = -64.7127740607003
x33=−59.8186774525393x_{33} = -59.8186774525393
x34=64.2499250422868x_{34} = 64.2499250422868
x35=33.3485269383943x_{35} = 33.3485269383943
x36=−24.3672290918915x_{36} = -24.3672290918915
x37=−48.0200918842023x_{37} = -48.0200918842023
x38=86.2148959615504x_{38} = 86.2148959615504
x39=−39.7914942317174x_{39} = -39.7914942317174
x40=78.2292948382806x_{40} = 78.2292948382806
x41=−5.87979427872852x_{41} = -5.87979427872852
x42=22.698505802003x_{42} = 22.698505802003
x43=45.3969715219982x_{43} = 45.3969715219982
x44=16.244866191735x_{44} = 16.244866191735
x45=42.3167589941069x_{45} = 42.3167589941069
x46=18.2485704298816x_{46} = 18.2485704298816
x47=5.319022925319x_{47} = 5.319022925319
x48=−53.8779341118659x_{48} = -53.8779341118659
x49=1.81447238096425x_{49} = 1.81447238096425
x50=6.14103975852348x_{50} = 6.14103975852348
x51=86.7959633259416x_{51} = 86.7959633259416
x52=−9.86886548575814x_{52} = -9.86886548575814
x53=−89.7498948516216x_{53} = -89.7498948516216
x54=−16.4371171083255x_{54} = -16.4371171083255
x55=−100.312119502767x_{55} = -100.312119502767
x56=−4.34465629687618x_{56} = -4.34465629687618
x57=52.2498248716371x_{57} = 52.2498248716371
x58=48.3460975487031x_{58} = 48.3460975487031
x59=−48.3785772894901x_{59} = -48.3785772894901
x60=−81.5521411296421x_{60} = -81.5521411296421
x61=−33.8626897130311x_{61} = -33.8626897130311
x62=−22.0668957515546x_{62} = -22.0668957515546
x63=−17.7245834055135x_{63} = -17.7245834055135
x64=−66.6970412378293x_{64} = -66.6970412378293
x65=−104.859784155019x_{65} = -104.859784155019
x66=−32.8263117664724x_{66} = -32.8263117664724
x67=−74.0622769317944x_{67} = -74.0622769317944
x68=3.07855253413366x_{68} = 3.07855253413366
x69=12.1514707300601x_{69} = 12.1514707300601
x70=0.808251932935767x_{70} = 0.808251932935767
x71=−7.72648921717798x_{71} = -7.72648921717798
x72=−5.319022925319x_{72} = -5.319022925319
x73=25.3157200470873x_{73} = 25.3157200470873
x74=82.1853045212075x_{74} = 82.1853045212075
x75=−83.9065504458045x_{75} = -83.9065504458045
x76=−77.3407557244222x_{76} = -77.3407557244222
x77=−3.96733151576786x_{77} = -3.96733151576786
x78=8.12287039886387x_{78} = 8.12287039886387
x79=−15.7539783621169x_{79} = -15.7539783621169
x80=−87.803578724131x_{80} = -87.803578724131
x81=−1.81447238096425x_{81} = -1.81447238096425
x82=−55.8533943936404x_{82} = -55.8533943936404
x83=48.9274341336244x_{83} = 48.9274341336244

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
[80.2316859157274,∞)\left[80.2316859157274, \infty\right)
Convexa en los intervalos
(−∞,−104.859784155019]\left(-\infty, -104.859784155019\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞sin⁡(x2)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(x^{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)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(x^{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), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x2)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x^{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)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x^{2} \right)}}{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:
sin⁡(x2)=sin⁡(x2)\sin{\left(x^{2} \right)} = \sin{\left(x^{2} \right)}
- Sí
sin⁡(x2)=−sin⁡(x2)\sin{\left(x^{2} \right)} = - \sin{\left(x^{2} \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = sin(x^2)
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