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Trazado el gráfico de la función/ (sin(9*x)*tan(3*x))/(sqrt(2-sin(4*x)^2)-sqrt(2+sin(4*x)^2))

Gráfico de la función y = (sin(9*x)*tan(3*x))/(sqrt(2-sin(4*x)^2)-sqrt(2+sin(4*x)^2))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                  sin(9*x)*tan(3*x)           
f(x) = ---------------------------------------
          _______________      _______________
         /        2           /        2      
       \/  2 - sin (4*x)  - \/  2 + sin (4*x)
f(x)=sin⁡(9x)tan⁡(3x)2−sin⁡2(4x)−sin⁡2(4x)+2f{\left(x \right)} = \frac{\sin{\left(9 x \right)} \tan{\left(3 x \right)}}{\sqrt{2 - \sin^{2}{\left(4 x \right)}} - \sqrt{\sin^{2}{\left(4 x \right)} + 2}} 
f = (sin(9*x)*tan(3*x))/(sqrt(2 - sin(4*x)^2) - sqrt(sin(4*x)^2 + 2))
Gráfico de la función
02468-8-6-4-2-1010-2000020000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=0.785398163397448x_{2} = 0.785398163397448
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⁡(9x)tan⁡(3x)2−sin⁡2(4x)−sin⁡2(4x)+2=0\frac{\sin{\left(9 x \right)} \tan{\left(3 x \right)}}{\sqrt{2 - \sin^{2}{\left(4 x \right)}} - \sqrt{\sin^{2}{\left(4 x \right)} + 2}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−79.587013957071x_{1} = -79.587013957071
x2=58.2939970166106x_{2} = 58.2939970166106
x3=−91.8043186549017x_{3} = -91.8043186549017
x4=−8.02851455917392x_{4} = -8.02851455917392
x5=41.8879019551629x_{5} = 41.8879019551629
x6=66.3225115757845x_{6} = 66.3225115757845
x7=−31.7649923862968x_{7} = -31.7649923862968
x8=98.4365698456188x_{8} = 98.4365698456188
x9=32.1140582366957x_{9} = 32.1140582366957
x10=60.0393262686049x_{10} = 60.0393262686049
x11=−9.77384381116824x_{11} = -9.77384381116824
x12=−16.0570291183478x_{12} = -16.0570291183478
x13=16.0570291183478x_{13} = 16.0570291183478
x14=−93.8987137572949x_{14} = -93.8987137572949
x15=76.0963553869528x_{15} = 76.0963553869528
x16=−83.7758040786232x_{16} = -83.7758040786232
x17=12.2173047639603x_{17} = 12.2173047639603
x18=70.1622359788179x_{18} = 70.1622359788179
x19=−71.9075651821664x_{19} = -71.9075651821664
x20=92.1533845581785x_{20} = 92.1533845581785
x21=18.151424220741x_{21} = 18.151424220741
x22=62.1337213709981x_{22} = 62.1337213709981
x23=−97.7384381116825x_{23} = -97.7384381116825
x24=−60.0393262686049x_{24} = -60.0393262686049
x25=94.5968454580927x_{25} = 94.5968454580927
x26=−38.0481776934764x_{26} = -38.0481776934764
x27=−74.0019602845596x_{27} = -74.0019602845596
x28=82.0304748437335x_{28} = 82.0304748437335
x29=72.6056968829641x_{29} = 72.6056968829641
x30=−13.613568229432x_{30} = -13.613568229432
x31=−55.8505360638185x_{31} = -55.8505360638185
x32=80.2851455917392x_{32} = 80.2851455917392
x33=−5.93411945678072x_{33} = -5.93411945678072
x34=63.8790507190579x_{34} = 63.8790507190579
x35=−99.8328332140756x_{35} = -99.8328332140756
x36=84.1248699461267x_{36} = 84.1248699461267
x37=−17.8023583387693x_{37} = -17.8023583387693
x38=90.0589893442465x_{38} = 90.0589893442465
x39=−69.8131700797732x_{39} = -69.8131700797732
x40=−77.8416846389471x_{40} = -77.8416846389471
x41=38.0481776934764x_{41} = 38.0481776934764
x42=28.623399732707x_{42} = 28.623399732707
x43=−19.8967535778114x_{43} = -19.8967535778114
x44=−33.85938748869x_{44} = -33.85938748869
x45=2.09439500680622x_{45} = 2.09439500680622
x46=48.1710873994927x_{46} = 48.1710873994927
x47=−65.6243798749868x_{47} = -65.6243798749868
x48=−27.9252680319093x_{48} = -27.9252680319093
x49=−95.9931088596881x_{49} = -95.9931088596881
x50=44.331363000656x_{50} = 44.331363000656
x51=8.3775803464205x_{51} = 8.3775803464205
x52=78.190750489346x_{52} = 78.190750489346
x53=68.0678407607945x_{53} = 68.0678407607945
x54=26.1799388201561x_{54} = 26.1799388201561
x55=85.8701993037672x_{55} = 85.8701993037672
x56=4.18879024075387x_{56} = 4.18879024075387
x57=−87.6155284501153x_{57} = -87.6155284501153
x58=−68.0678408119881x_{58} = -68.0678408119881
x59=74.3510260903306x_{59} = 74.3510260903306
x60=−46.0766922667605x_{60} = -46.0766922667605
x61=−24.4346095279206x_{61} = -24.4346095279206
x62=52.3598774972308x_{62} = 52.3598774972308
x63=−39.7935069184688x_{63} = -39.7935069184688
x64=−41.8879021208032x_{64} = -41.8879021208032
x65=100.181899064475x_{65} = 100.181899064475
x66=−35.6047168053919x_{66} = -35.6047168053919
x67=36.3028484414821x_{67} = 36.3028484414821
x68=−53.7561409614254x_{68} = -53.7561409614254
x69=22.3402144255274x_{69} = 22.3402144255274
x70=−47.8220215046446x_{70} = -47.8220215046446
x71=−90.0589893805018x_{71} = -90.0589893805018
x72=−82.0304748437335x_{72} = -82.0304748437335
x73=54.1052068118242x_{73} = 54.1052068118242
x74=−43.6332312998582x_{74} = -43.6332312998582
x75=10.1229096615671x_{75} = 10.1229096615671
x76=−57.5958653813132x_{76} = -57.5958653813132
x77=−11.8682389135614x_{77} = -11.8682389135614
x78=34.2084533390889x_{78} = 34.2084533390889
x79=24.0855435922613x_{79} = 24.0855435922613
x80=−3.83972435438753x_{80} = -3.83972435438753
x81=30.3687289219779x_{81} = 30.3687289219779
x82=−49.9164166070378x_{82} = -49.9164166070378
x83=−148.003920569119x_{83} = -148.003920569119
x84=56.1996019142174x_{84} = 56.1996019142174
x85=−52.010811709431x_{85} = -52.010811709431
x86=20.2458193231342x_{86} = 20.2458193231342
x87=14.3116998663535x_{87} = 14.3116998663535
x88=88.3136601509131x_{88} = 88.3136601509131
x89=46.0766921768866x_{89} = 46.0766921768866
x90=−75.7472895365539x_{90} = -75.7472895365539
x91=−61.7846554984007x_{91} = -61.7846554984007
x92=40.1425727958696x_{92} = 40.1425727958696
x93=−30.0196631343025x_{93} = -30.0196631343025
x94=−25.8308729295161x_{94} = -25.8308729295161
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(9*x)*tan(3*x))/(sqrt(2 - sin(4*x)^2) - sqrt(2 + sin(4*x)^2)).
sin⁡(0⋅9)tan⁡(0⋅3)−sin⁡2(0⋅4)+2+2−sin⁡2(0⋅4)\frac{\sin{\left(0 \cdot 9 \right)} \tan{\left(0 \cdot 3 \right)}}{- \sqrt{\sin^{2}{\left(0 \cdot 4 \right)} + 2} + \sqrt{2 - \sin^{2}{\left(0 \cdot 4 \right)}}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=0.785398163397448x_{2} = 0.785398163397448
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
No se ha logrado calcular el límite a la izquierda
lim⁡x→−∞(sin⁡(9x)tan⁡(3x)2−sin⁡2(4x)−sin⁡2(4x)+2)\lim_{x \to -\infty}\left(\frac{\sin{\left(9 x \right)} \tan{\left(3 x \right)}}{\sqrt{2 - \sin^{2}{\left(4 x \right)}} - \sqrt{\sin^{2}{\left(4 x \right)} + 2}}\right)
No se ha logrado calcular el límite a la derecha
lim⁡x→∞(sin⁡(9x)tan⁡(3x)2−sin⁡2(4x)−sin⁡2(4x)+2)\lim_{x \to \infty}\left(\frac{\sin{\left(9 x \right)} \tan{\left(3 x \right)}}{\sqrt{2 - \sin^{2}{\left(4 x \right)}} - \sqrt{\sin^{2}{\left(4 x \right)} + 2}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (sin(9*x)*tan(3*x))/(sqrt(2 - sin(4*x)^2) - sqrt(2 + sin(4*x)^2)), dividida por x con x->+oo y x ->-oo
No se ha logrado calcular el límite a la izquierda
lim⁡x→−∞(sin⁡(9x)tan⁡(3x)x(2−sin⁡2(4x)−sin⁡2(4x)+2))\lim_{x \to -\infty}\left(\frac{\sin{\left(9 x \right)} \tan{\left(3 x \right)}}{x \left(\sqrt{2 - \sin^{2}{\left(4 x \right)}} - \sqrt{\sin^{2}{\left(4 x \right)} + 2}\right)}\right)
No se ha logrado calcular el límite a la derecha
lim⁡x→∞(sin⁡(9x)tan⁡(3x)x(2−sin⁡2(4x)−sin⁡2(4x)+2))\lim_{x \to \infty}\left(\frac{\sin{\left(9 x \right)} \tan{\left(3 x \right)}}{x \left(\sqrt{2 - \sin^{2}{\left(4 x \right)}} - \sqrt{\sin^{2}{\left(4 x \right)} + 2}\right)}\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:
sin⁡(9x)tan⁡(3x)2−sin⁡2(4x)−sin⁡2(4x)+2=sin⁡(9x)tan⁡(3x)2−sin⁡2(4x)−sin⁡2(4x)+2\frac{\sin{\left(9 x \right)} \tan{\left(3 x \right)}}{\sqrt{2 - \sin^{2}{\left(4 x \right)}} - \sqrt{\sin^{2}{\left(4 x \right)} + 2}} = \frac{\sin{\left(9 x \right)} \tan{\left(3 x \right)}}{\sqrt{2 - \sin^{2}{\left(4 x \right)}} - \sqrt{\sin^{2}{\left(4 x \right)} + 2}}
- Sí
sin⁡(9x)tan⁡(3x)2−sin⁡2(4x)−sin⁡2(4x)+2=−sin⁡(9x)tan⁡(3x)2−sin⁡2(4x)−sin⁡2(4x)+2\frac{\sin{\left(9 x \right)} \tan{\left(3 x \right)}}{\sqrt{2 - \sin^{2}{\left(4 x \right)}} - \sqrt{\sin^{2}{\left(4 x \right)} + 2}} = - \frac{\sin{\left(9 x \right)} \tan{\left(3 x \right)}}{\sqrt{2 - \sin^{2}{\left(4 x \right)}} - \sqrt{\sin^{2}{\left(4 x \right)} + 2}}
- No
es decir, función
es
par
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