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Trazado el gráfico de la función/ sin(3*x)^2/(-1+sqrt(1-3*x^2))

Gráfico de la función y = sin(3*x)^2/(-1+sqrt(1-3*x^2))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
              2          
           sin (3*x)     
f(x) = ------------------
               __________
              /        2 
       -1 + \/  1 - 3*x
f(x)=sin⁡2(3x)1−3x2−1f{\left(x \right)} = \frac{\sin^{2}{\left(3 x \right)}}{\sqrt{1 - 3 x^{2}} - 1} 
f = sin(3*x)^2/(sqrt(1 - 3*x^2) - 1)
Gráfico de la función
02468-8-6-4-2-10100-10
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⁡2(3x)1−3x2−1=0\frac{\sin^{2}{\left(3 x \right)}}{\sqrt{1 - 3 x^{2}} - 1} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π3x_{1} = \frac{\pi}{3}
Solución numérica
x1=83.7758041738755x_{1} = 83.7758041738755
x2=−68.0678407177314x_{2} = -68.0678407177314
x3=−2.09439495341714x_{3} = -2.09439495341714
x4=32.4631239373878x_{4} = 32.4631239373878
x5=−53.407075547284x_{5} = -53.407075547284
x6=−11.5191731251854x_{6} = -11.5191731251854
x7=−17.8023582939866x_{7} = -17.8023582939866
x8=−43.9822971850546x_{8} = -43.9822971850546
x9=−4.18879001288644x_{9} = -4.18879001288644
x10=61.7846556072272x_{10} = 61.7846556072272
x11=76.4454210731484x_{11} = 76.4454210731484
x12=−72.2566309870438x_{12} = -72.2566309870438
x13=−28.2743338947898x_{13} = -28.2743338947898
x14=59.6902603644233x_{14} = 59.6902603644233
x15=−41.8879020073281x_{15} = -41.8879020073281
x16=−31.4159264946804x_{16} = -31.4159264946804
x17=−99.4837673648131x_{17} = -99.4837673648131
x18=98.436569643698x_{18} = 98.436569643698
x19=−50.2654824494464x_{19} = -50.2654824494464
x20=−75.3982236304748x_{20} = -75.3982236304748
x21=8.37758024407212x_{21} = 8.37758024407212
x22=−90.0589893044654x_{22} = -90.0589893044654
x23=−35.6047168432264x_{23} = -35.6047168432264
x24=−70.1622357540224x_{24} = -70.1622357540224
x25=−87.9645943668825x_{25} = -87.9645943668825
x26=2.09439524233262x_{26} = 2.09439524233262
x27=74.3510259933659x_{27} = 74.3510259933659
x28=−9.42477797919813x_{28} = -9.42477797919813
x29=−21.9911485925066x_{29} = -21.9911485925066
x30=56.5486678712747x_{30} = 56.5486678712747
x31=−85.8701982254335x_{31} = -85.8701982254335
x32=34.557519296064x_{32} = 34.557519296064
x33=−53.4070750921797x_{33} = -53.4070750921797
x34=41.887902141974x_{34} = 41.887902141974
x35=15.7079632015201x_{35} = 15.7079632015201
x36=43.9822971706907x_{36} = 43.9822971706907
x37=81.6814089893432x_{37} = 81.6814089893432
x38=−19.8967533927881x_{38} = -19.8967533927881
x39=−33.5103216930609x_{39} = -33.5103216930609
x40=24.0855436989385x_{40} = 24.0855436989385
x41=−55.501470253734x_{41} = -55.501470253734
x42=70.1622361499298x_{42} = 70.1622361499298
x43=63.879050721003x_{43} = 63.879050721003
x44=26.1799389485711x_{44} = 26.1799389485711
x45=−59.6902604772027x_{45} = -59.6902604772027
x46=62.8318525722289x_{46} = 62.8318525722289
x47=39.7935070383395x_{47} = 39.7935070383395
x48=72.2566310199357x_{48} = 72.2566310199357
x49=−79.5870139959988x_{49} = -79.5870139959988
x50=83.7758040068571x_{50} = 83.7758040068571
x51=65.97344575532x_{51} = 65.97344575532
x52=−57.595865420076x_{52} = -57.595865420076
x53=−24.085543546371x_{53} = -24.085543546371
x54=21.9911485855667x_{54} = 21.9911485855667
x55=−64.9262478936832x_{55} = -64.9262478936832
x56=−39.7935068843454x_{56} = -39.7935068843454
x57=−37.6991118947743x_{57} = -37.6991118947743
x58=87.9645943394306x_{58} = 87.9645943394306
x59=48.171087549349x_{59} = 48.171087549349
x60=92.1533847504801x_{60} = 92.1533847504801
x61=17.8023584664455x_{61} = 17.8023584664455
x62=−26.1799386039895x_{62} = -26.1799386039895
x63=4.18879034564693x_{63} = 4.18879034564693
x64=28.2743338384102x_{64} = 28.2743338384102
x65=−15.7079633117906x_{65} = -15.7079633117906
x66=−46.0766921317752x_{66} = -46.0766921317752
x67=6.28318524850749x_{67} = 6.28318524850749
x68=54.4542725045089x_{68} = 54.4542725045089
x69=46.0766922309492x_{69} = 46.0766922309492
x70=52.3598774106518x_{70} = 52.3598774106518
x71=94.247779612042x_{71} = 94.247779612042
x72=−13.6135682647737x_{72} = -13.6135682647737
x73=96.3421745768098x_{73} = 96.3421745768098
x74=−92.1533843304102x_{74} = -92.1533843304102
x75=−81.6814090591358x_{75} = -81.6814090591358
x76=12.5663707133305x_{76} = 12.5663707133305
x77=12.5663690482218x_{77} = 12.5663690482218
x78=37.6991117734004x_{78} = 37.6991117734004
x79=−6.28318532046115x_{79} = -6.28318532046115
x80=90.0589893483166x_{80} = 90.0589893483166
x81=−96.3421749004023x_{81} = -96.3421749004023
x82=−83.775804061806x_{82} = -83.775804061806
x83=30.3687288284625x_{83} = 30.3687288284625
x84=−77.4926188107758x_{84} = -77.4926188107758
x85=−9.42477805140116x_{85} = -9.42477805140116
x86=85.8701992993614x_{86} = 85.8701992993614
x87=100.530964970124x_{87} = 100.530964970124
x88=10.4719753649727x_{88} = 10.4719753649727
x89=−31.415926551159x_{89} = -31.415926551159
x90=−94.2477795332638x_{90} = -94.2477795332638
x91=50.2654824287478x_{91} = 50.2654824287478
x92=19.8967535620831x_{92} = 19.8967535620831
x93=−65.9734457769287x_{93} = -65.9734457769287
x94=−74.3510261761185x_{94} = -74.3510261761185
x95=−48.1710871787495x_{95} = -48.1710871787495
x96=68.0678407822198x_{96} = 68.0678407822198
x97=78.5398164271464x_{97} = 78.5398164271464
x98=−61.7846554735345x_{98} = -61.7846554735345
x99=−96.3421746111554x_{99} = -96.3421746111554
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(3*x)^2/(-1 + sqrt(1 - 3*x^2)).
sin⁡2(0⋅3)−1+1−3⋅02\frac{\sin^{2}{\left(0 \cdot 3 \right)}}{-1 + \sqrt{1 - 3 \cdot 0^{2}}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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⁡2(3x)1−3x2−1)=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(3 x \right)}}{\sqrt{1 - 3 x^{2}} - 1}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(sin⁡2(3x)1−3x2−1)=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(3 x \right)}}{\sqrt{1 - 3 x^{2}} - 1}\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 sin(3*x)^2/(-1 + sqrt(1 - 3*x^2)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡2(3x)x(1−3x2−1))=0\lim_{x \to -\infty}\left(\frac{\sin^{2}{\left(3 x \right)}}{x \left(\sqrt{1 - 3 x^{2}} - 1\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡2(3x)x(1−3x2−1))=0\lim_{x \to \infty}\left(\frac{\sin^{2}{\left(3 x \right)}}{x \left(\sqrt{1 - 3 x^{2}} - 1\right)}\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⁡2(3x)1−3x2−1=sin⁡2(3x)1−3x2−1\frac{\sin^{2}{\left(3 x \right)}}{\sqrt{1 - 3 x^{2}} - 1} = \frac{\sin^{2}{\left(3 x \right)}}{\sqrt{1 - 3 x^{2}} - 1}
- Sí
sin⁡2(3x)1−3x2−1=−sin⁡2(3x)1−3x2−1\frac{\sin^{2}{\left(3 x \right)}}{\sqrt{1 - 3 x^{2}} - 1} = - \frac{\sin^{2}{\left(3 x \right)}}{\sqrt{1 - 3 x^{2}} - 1}
- No
es decir, función
es
par
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