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Trazado el gráfico de la función/ tan(8*x)-cot(8*x)-2*cos(16*x)^2/sin(16*x)

Gráfico de la función y = tan(8*x)-cot(8*x)-2*cos(16*x)^2/sin(16*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                  2      
                             2*cos (16*x)
f(x) = tan(8*x) - cot(8*x) - ------------
                              sin(16*x)
f(x)=(tan⁡(8x)−cot⁡(8x))−2cos⁡2(16x)sin⁡(16x)f{\left(x \right)} = \left(\tan{\left(8 x \right)} - \cot{\left(8 x \right)}\right) - \frac{2 \cos^{2}{\left(16 x \right)}}{\sin{\left(16 x \right)}} 
f = tan(8*x) - cot(8*x) - 2*cos(16*x)^2/sin(16*x)
Gráfico de la función
0.000.250.500.751.001.251.501.752.002.252.502.753.00-50005000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
x2=0.196349540849362x_{2} = 0.196349540849362
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:
(tan⁡(8x)−cot⁡(8x))−2cos⁡2(16x)sin⁡(16x)=0\left(\tan{\left(8 x \right)} - \cot{\left(8 x \right)}\right) - \frac{2 \cos^{2}{\left(16 x \right)}}{\sin{\left(16 x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−π32x_{1} = - \frac{\pi}{32}
x2=π32x_{2} = \frac{\pi}{32}
Solución numérica
x1=−16.002487579223x_{1} = -16.002487579223
x2=94.1496048372691x_{2} = 94.1496048372691
x3=−33.9684705669396x_{3} = -33.9684705669396
x4=−89.7317401681585x_{4} = -89.7317401681585
x5=−36.0301407458579x_{5} = -36.0301407458579
x6=−5.6941366846315x_{6} = -5.6941366846315
x7=22.2856728864026x_{7} = 22.2856728864026
x8=−61.7519305971244x_{8} = -61.7519305971244
x9=78.2452920284708x_{9} = 78.2452920284708
x10=10.0138265833175x_{10} = 10.0138265833175
x11=−19.7331288553609x_{11} = -19.7331288553609
x12=82.2704576158827x_{12} = 82.2704576158827
x13=−27.9798095710341x_{13} = -27.9798095710341
x14=−47.8111131968197x_{14} = -47.8111131968197
x15=11.5846229101124x_{15} = 11.5846229101124
x16=−99.9419162923253x_{16} = -99.9419162923253
x17=86.1974484328699x_{17} = 86.1974484328699
x18=96.0149254753381x_{18} = 96.0149254753381
x19=2.25801971976766x_{19} = 2.25801971976766
x20=14.2353417115787x_{20} = 14.2353417115787
x21=−23.7582944427728x_{21} = -23.7582944427728
x22=−59.9847847294801x_{22} = -59.9847847294801
x23=−81.877758534184x_{23} = -81.877758534184
x24=69.9986113127976x_{24} = 69.9986113127976
x25=−80.012437896115x_{25} = -80.012437896115
x26=−91.6952355766521x_{26} = -91.6952355766521
x27=18.2605072989907x_{27} = 18.2605072989907
x28=4.1233403578366x_{28} = 4.1233403578366
x29=53.9961237335746x_{29} = 53.9961237335746
x30=32.004975158446x_{30} = 32.004975158446
x31=−3.73064127613788x_{31} = -3.73064127613788
x32=−67.7405915930299x_{32} = -67.7405915930299
x33=−7.7558068635498x_{33} = -7.7558068635498
x34=−34.2629948782137x_{34} = -34.2629948782137
x35=−96.0149254753381x_{35} = -96.0149254753381
x36=−87.4737204483908x_{36} = -87.4737204483908
x37=−21.7947990342792x_{37} = -21.7947990342792
x38=64.9916980211388x_{38} = 64.9916980211388
x39=−45.7494430179014x_{39} = -45.7494430179014
x40=49.9709581461627x_{40} = 49.9709581461627
x41=−41.7242774304894x_{41} = -41.7242774304894
x42=71.9621067212912x_{42} = 71.9621067212912
x43=36.2264902867073x_{43} = 36.2264902867073
x44=−49.7746086053133x_{44} = -49.7746086053133
x45=46.2403168700248x_{45} = 46.2403168700248
x46=−43.7859476094077x_{46} = -43.7859476094077
x47=37.8954613839269x_{47} = 37.8954613839269
x48=100.236440603599x_{48} = 100.236440603599
x49=91.9897598879261x_{49} = 91.9897598879261
x50=60.5738333520282x_{50} = 60.5738333520282
x51=−77.9507677171967x_{51} = -77.9507677171967
x52=−58.0212893209865x_{52} = -58.0212893209865
x53=20.2240027074843x_{53} = 20.2240027074843
x54=66.2679700366597x_{54} = 66.2679700366597
x55=−63.715426005618x_{55} = -63.715426005618
x56=62.2428044492478x_{56} = 62.2428044492478
x57=−29.7469554386784x_{57} = -29.7469554386784
x58=40.2516558741192x_{58} = 40.2516558741192
x59=80.3069622073891x_{59} = 80.3069622073891
x60=−65.7770961845363x_{60} = -65.7770961845363
x61=58.3158136322605x_{61} = 58.3158136322605
x62=56.3523182237669x_{62} = 56.3523182237669
x63=−83.7430791722529x_{63} = -83.7430791722529
x64=61.4574062858503x_{64} = 61.4574062858503
x65=26.0163141625405x_{65} = 26.0163141625405
x66=5.98866099590554x_{66} = 5.98866099590554
x67=−87.7682447596649x_{67} = -87.7682447596649
x68=90.1244392498572x_{68} = 90.1244392498572
x69=88.2591186117883x_{69} = 88.2591186117883
x70=84.2339530243763x_{70} = 84.2339530243763
x71=64.2062998577414x_{71} = 64.2062998577414
x72=−1.76714586764426x_{72} = -1.76714586764426
x73=−37.9936361543516x_{73} = -37.9936361543516
x74=−14.0389921707294x_{74} = -14.0389921707294
x75=42.2151512826128x_{75} = 42.2151512826128
x76=−71.7657571804418x_{76} = -71.7657571804418
x77=−32.004975158446x_{77} = -32.004975158446
x78=7.65763209312512x_{78} = 7.65763209312512
x79=74.0237769002095x_{79} = 74.0237769002095
x80=−11.9773219918111x_{80} = -11.9773219918111
x81=75.9872723087031x_{81} = 75.9872723087031
x82=27.9798095710341x_{82} = 27.9798095710341
x83=97.880246113407x_{83} = 97.880246113407
x84=−93.7569057555704x_{84} = -93.7569057555704
x85=−55.9596191420682x_{85} = -55.9596191420682
x86=44.2768214615311x_{86} = 44.2768214615311
x87=15.5116137270996x_{87} = 15.5116137270996
x88=−53.9961237335746x_{88} = -53.9961237335746
x89=−10.0138265833175x_{89} = -10.0138265833175
x90=30.0414797499524x_{90} = 30.0414797499524
x91=−75.9872723087031x_{91} = -75.9872723087031
x92=−18.5550316102647x_{92} = -18.5550316102647
x93=48.007462737669x_{93} = 48.007462737669
x94=−97.9784208838317x_{94} = -97.9784208838317
x95=−73.9256021297848x_{95} = -73.9256021297848
x96=52.0326283250809x_{96} = 52.0326283250809
x97=24.2491682948962x_{97} = 24.2491682948962
x98=−39.7607820219958x_{98} = -39.7607820219958
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
64cos⁡(16x)+8tan⁡2(8x)+8cot⁡2(8x)+16+32cos⁡3(16x)sin⁡2(16x)=064 \cos{\left(16 x \right)} + 8 \tan^{2}{\left(8 x \right)} + 8 \cot^{2}{\left(8 x \right)} + 16 + \frac{32 \cos^{3}{\left(16 x \right)}}{\sin^{2}{\left(16 x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−atan⁡(2+5)8x_{1} = - \frac{\operatorname{atan}{\left(\sqrt{2 + \sqrt{5}} \right)}}{8}
x2=atan⁡(2+5)8x_{2} = \frac{\operatorname{atan}{\left(\sqrt{2 + \sqrt{5}} \right)}}{8}
Signos de extremos en los puntos:
      /   ___________\                                           /      /   ___________\\ 
      |  /       ___ |                       ___________        2|      |  /       ___ || 
 -atan\\/  2 + \/ 5  /         1            /       ___    2*cos \2*atan\\/  2 + \/ 5  // 
(----------------------, -------------- - \/  2 + \/ 5   + ------------------------------)
           8                ___________                        /      /   ___________\\   
                           /       ___                         |      |  /       ___ ||   
                         \/  2 + \/ 5                       sin\2*atan\\/  2 + \/ 5  //   

     /   ___________\                                          /      /   ___________\\ 
     |  /       ___ |     ___________                         2|      |  /       ___ || 
 atan\\/  2 + \/ 5  /    /       ___          1          2*cos \2*atan\\/  2 + \/ 5  // 
(--------------------, \/  2 + \/ 5   - -------------- - ------------------------------)
          8                                ___________       /      /   ___________\\   
                                          /       ___        |      |  /       ___ ||   
                                        \/  2 + \/ 5      sin\2*atan\\/  2 + \/ 5  //


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=−atan⁡(2+5)8x_{1} = - \frac{\operatorname{atan}{\left(\sqrt{2 + \sqrt{5}} \right)}}{8}
Puntos máximos de la función:
x1=atan⁡(2+5)8x_{1} = \frac{\operatorname{atan}{\left(\sqrt{2 + \sqrt{5}} \right)}}{8}
Decrece en los intervalos
[−atan⁡(2+5)8,atan⁡(2+5)8]\left[- \frac{\operatorname{atan}{\left(\sqrt{2 + \sqrt{5}} \right)}}{8}, \frac{\operatorname{atan}{\left(\sqrt{2 + \sqrt{5}} \right)}}{8}\right]
Crece en los intervalos
(−∞,−atan⁡(2+5)8]∪[atan⁡(2+5)8,∞)\left(-\infty, - \frac{\operatorname{atan}{\left(\sqrt{2 + \sqrt{5}} \right)}}{8}\right] \cup \left[\frac{\operatorname{atan}{\left(\sqrt{2 + \sqrt{5}} \right)}}{8}, \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
128((tan⁡2(8x)+1)tan⁡(8x)−(cot⁡2(8x)+1)cot⁡(8x)−8sin⁡(16x)−12cos⁡2(16x)sin⁡(16x)−8cos⁡4(16x)sin⁡3(16x))=0128 \left(\left(\tan^{2}{\left(8 x \right)} + 1\right) \tan{\left(8 x \right)} - \left(\cot^{2}{\left(8 x \right)} + 1\right) \cot{\left(8 x \right)} - 8 \sin{\left(16 x \right)} - \frac{12 \cos^{2}{\left(16 x \right)}}{\sin{\left(16 x \right)}} - \frac{8 \cos^{4}{\left(16 x \right)}}{\sin^{3}{\left(16 x \right)}}\right) = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas verticales
Hay:
x1=0x_{1} = 0
x2=0.196349540849362x_{2} = 0.196349540849362
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞((tan⁡(8x)−cot⁡(8x))−2cos⁡2(16x)sin⁡(16x))y = \lim_{x \to -\infty}\left(\left(\tan{\left(8 x \right)} - \cot{\left(8 x \right)}\right) - \frac{2 \cos^{2}{\left(16 x \right)}}{\sin{\left(16 x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞((tan⁡(8x)−cot⁡(8x))−2cos⁡2(16x)sin⁡(16x))y = \lim_{x \to \infty}\left(\left(\tan{\left(8 x \right)} - \cot{\left(8 x \right)}\right) - \frac{2 \cos^{2}{\left(16 x \right)}}{\sin{\left(16 x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(8*x) - cot(8*x) - 2*cos(16*x)^2/sin(16*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→−∞((tan⁡(8x)−cot⁡(8x))−2cos⁡2(16x)sin⁡(16x)x)y = x \lim_{x \to -\infty}\left(\frac{\left(\tan{\left(8 x \right)} - \cot{\left(8 x \right)}\right) - \frac{2 \cos^{2}{\left(16 x \right)}}{\sin{\left(16 x \right)}}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((tan⁡(8x)−cot⁡(8x))−2cos⁡2(16x)sin⁡(16x)x)y = x \lim_{x \to \infty}\left(\frac{\left(\tan{\left(8 x \right)} - \cot{\left(8 x \right)}\right) - \frac{2 \cos^{2}{\left(16 x \right)}}{\sin{\left(16 x \right)}}}{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:
(tan⁡(8x)−cot⁡(8x))−2cos⁡2(16x)sin⁡(16x)=−tan⁡(8x)+cot⁡(8x)+2cos⁡2(16x)sin⁡(16x)\left(\tan{\left(8 x \right)} - \cot{\left(8 x \right)}\right) - \frac{2 \cos^{2}{\left(16 x \right)}}{\sin{\left(16 x \right)}} = - \tan{\left(8 x \right)} + \cot{\left(8 x \right)} + \frac{2 \cos^{2}{\left(16 x \right)}}{\sin{\left(16 x \right)}}
- No
(tan⁡(8x)−cot⁡(8x))−2cos⁡2(16x)sin⁡(16x)=tan⁡(8x)−cot⁡(8x)−2cos⁡2(16x)sin⁡(16x)\left(\tan{\left(8 x \right)} - \cot{\left(8 x \right)}\right) - \frac{2 \cos^{2}{\left(16 x \right)}}{\sin{\left(16 x \right)}} = \tan{\left(8 x \right)} - \cot{\left(8 x \right)} - \frac{2 \cos^{2}{\left(16 x \right)}}{\sin{\left(16 x \right)}}
- No
es decir, función
no es
par ni impar
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