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Trazado el gráfico de la función/ tan(2*x)+2/3*tan(2*x)^(3)+1/5*tan(2*x)^(5)

Gráfico de la función y = tan(2*x)+2/3*tan(2*x)^(3)+1/5*tan(2*x)^(5)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                       3           5     
                  2*tan (2*x)   tan (2*x)
f(x) = tan(2*x) + ----------- + ---------
                       3            5
f(x)=(2tan⁡3(2x)3+tan⁡(2x))+tan⁡5(2x)5f{\left(x \right)} = \left(\frac{2 \tan^{3}{\left(2 x \right)}}{3} + \tan{\left(2 x \right)}\right) + \frac{\tan^{5}{\left(2 x \right)}}{5} 
f = 2*tan(2*x)^3/3 + tan(2*x) + tan(2*x)^5/5
Gráfico de la función
02468-8-6-4-2-1010-25000000002500000000
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:
(2tan⁡3(2x)3+tan⁡(2x))+tan⁡5(2x)5=0\left(\frac{2 \tan^{3}{\left(2 x \right)}}{3} + \tan{\left(2 x \right)}\right) + \frac{\tan^{5}{\left(2 x \right)}}{5} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=48.6946861306418x_{1} = 48.6946861306418
x2=81.6814089933346x_{2} = 81.6814089933346
x3=−14.1371669411541x_{3} = -14.1371669411541
x4=−56.5486677646163x_{4} = -56.5486677646163
x5=−86.3937979737193x_{5} = -86.3937979737193
x6=−1.5707963267949x_{6} = -1.5707963267949
x7=23.5619449019235x_{7} = 23.5619449019235
x8=59.6902604182061x_{8} = 59.6902604182061
x9=73.8274273593601x_{9} = 73.8274273593601
x10=4.71238898038469x_{10} = 4.71238898038469
x11=34.5575191894877x_{11} = 34.5575191894877
x12=−21.9911485751286x_{12} = -21.9911485751286
x13=−20.4203522483337x_{13} = -20.4203522483337
x14=15.707963267949x_{14} = 15.707963267949
x15=−91.106186954104x_{15} = -91.106186954104
x16=−95.8185759344887x_{16} = -95.8185759344887
x17=26.7035375555132x_{17} = 26.7035375555132
x18=−81.6814089933346x_{18} = -81.6814089933346
x19=31.4159265358979x_{19} = 31.4159265358979
x20=20.4203522483337x_{20} = 20.4203522483337
x21=−12.5663706143592x_{21} = -12.5663706143592
x22=−94.2477796076938x_{22} = -94.2477796076938
x23=67.5442420521806x_{23} = 67.5442420521806
x24=−59.6902604182061x_{24} = -59.6902604182061
x25=36.1283155162826x_{25} = 36.1283155162826
x26=−43.9822971502571x_{26} = -43.9822971502571
x27=58.1194640914112x_{27} = 58.1194640914112
x28=−29.845130209103x_{28} = -29.845130209103
x29=−31.4159265358979x_{29} = -31.4159265358979
x30=12.5663706143592x_{30} = 12.5663706143592
x31=43.9822971502571x_{31} = 43.9822971502571
x32=−3.14159265358979x_{32} = -3.14159265358979
x33=7.85398163397448x_{33} = 7.85398163397448
x34=−15.707963267949x_{34} = -15.707963267949
x35=9.42477796076938x_{35} = 9.42477796076938
x36=0x_{36} = 0
x37=89.5353906273091x_{37} = 89.5353906273091
x38=−65.9734457253857x_{38} = -65.9734457253857
x39=75.398223686155x_{39} = 75.398223686155
x40=−28.2743338823081x_{40} = -28.2743338823081
x41=51.8362787842316x_{41} = 51.8362787842316
x42=−69.1150383789755x_{42} = -69.1150383789755
x43=70.6858347057703x_{43} = 70.6858347057703
x44=−50.2654824574367x_{44} = -50.2654824574367
x45=80.1106126665397x_{45} = 80.1106126665397
x46=62.8318530717959x_{46} = 62.8318530717959
x47=−75.398223686155x_{47} = -75.398223686155
x48=45.553093477052x_{48} = 45.553093477052
x49=−100.530964914873x_{49} = -100.530964914873
x50=14.1371669411541x_{50} = 14.1371669411541
x51=28.2743338823081x_{51} = 28.2743338823081
x52=65.9734457253857x_{52} = 65.9734457253857
x53=−67.5442420521806x_{53} = -67.5442420521806
x54=40.8407044966673x_{54} = 40.8407044966673
x55=42.4115008234622x_{55} = 42.4115008234622
x56=−45.553093477052x_{56} = -45.553093477052
x57=−58.1194640914112x_{57} = -58.1194640914112
x58=−47.1238898038469x_{58} = -47.1238898038469
x59=−87.9645943005142x_{59} = -87.9645943005142
x60=−6.28318530717959x_{60} = -6.28318530717959
x61=−83.2522053201295x_{61} = -83.2522053201295
x62=−97.3893722612836x_{62} = -97.3893722612836
x63=94.2477796076938x_{63} = 94.2477796076938
x64=84.8230016469244x_{64} = 84.8230016469244
x65=−17.2787595947439x_{65} = -17.2787595947439
x66=95.8185759344887x_{66} = 95.8185759344887
x67=−39.2699081698724x_{67} = -39.2699081698724
x68=72.2566310325652x_{68} = 72.2566310325652
x69=−36.1283155162826x_{69} = -36.1283155162826
x70=1.5707963267949x_{70} = 1.5707963267949
x71=−9.42477796076938x_{71} = -9.42477796076938
x72=−64.4026493985908x_{72} = -64.4026493985908
x73=56.5486677646163x_{73} = 56.5486677646163
x74=92.6769832808989x_{74} = 92.6769832808989
x75=−51.8362787842316x_{75} = -51.8362787842316
x76=100.530964914873x_{76} = 100.530964914873
x77=−89.5353906273091x_{77} = -89.5353906273091
x78=6.28318530717959x_{78} = 6.28318530717959
x79=−61.261056745001x_{79} = -61.261056745001
x80=−53.4070751110265x_{80} = -53.4070751110265
x81=−73.8274273593601x_{81} = -73.8274273593601
x82=21.9911485751286x_{82} = 21.9911485751286
x83=29.845130209103x_{83} = 29.845130209103
x84=−25.1327412287183x_{84} = -25.1327412287183
x85=18.8495559215388x_{85} = 18.8495559215388
x86=−72.2566310325652x_{86} = -72.2566310325652
x87=87.9645943005142x_{87} = 87.9645943005142
x88=37.6991118430775x_{88} = 37.6991118430775
x89=50.2654824574367x_{89} = 50.2654824574367
x90=86.3937979737193x_{90} = 86.3937979737193
x91=64.4026493985908x_{91} = 64.4026493985908
x92=53.4070751110265x_{92} = 53.4070751110265
x93=97.3893722612836x_{93} = 97.3893722612836
x94=−37.6991118430775x_{94} = -37.6991118430775
x95=−23.5619449019235x_{95} = -23.5619449019235
x96=−7.85398163397448x_{96} = -7.85398163397448
x97=78.5398163397448x_{97} = 78.5398163397448
x98=−42.4115008234622x_{98} = -42.4115008234622
x99=−78.5398163397448x_{99} = -78.5398163397448
x100=−80.1106126665397x_{100} = -80.1106126665397
x101=−34.5575191894877x_{101} = -34.5575191894877
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 tan(2*x) + 2*tan(2*x)^3/3 + tan(2*x)^5/5.
(tan⁡(0⋅2)+2tan⁡3(0⋅2)3)+tan⁡5(0⋅2)5\left(\tan{\left(0 \cdot 2 \right)} + \frac{2 \tan^{3}{\left(0 \cdot 2 \right)}}{3}\right) + \frac{\tan^{5}{\left(0 \cdot 2 \right)}}{5}
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
2(6tan⁡2(2x)+6)tan⁡2(2x)3+(10tan⁡2(2x)+10)tan⁡4(2x)5+2tan⁡2(2x)+2=0\frac{2 \left(6 \tan^{2}{\left(2 x \right)} + 6\right) \tan^{2}{\left(2 x \right)}}{3} + \frac{\left(10 \tan^{2}{\left(2 x \right)} + 10\right) \tan^{4}{\left(2 x \right)}}{5} + 2 \tan^{2}{\left(2 x \right)} + 2 = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
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
8(tan⁡2(2x)+1)(2(tan⁡2(2x)+1)tan⁡2(2x)+tan⁡4(2x)+4tan⁡2(2x)+3)tan⁡(2x)=08 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \tan^{2}{\left(2 x \right)} + \tan^{4}{\left(2 x \right)} + 4 \tan^{2}{\left(2 x \right)} + 3\right) \tan{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0

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
[0,∞)\left[0, \infty\right)
Convexa en los intervalos
(−∞,0]\left(-\infty, 0\right]
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→−∞((2tan⁡3(2x)3+tan⁡(2x))+tan⁡5(2x)5)y = \lim_{x \to -\infty}\left(\left(\frac{2 \tan^{3}{\left(2 x \right)}}{3} + \tan{\left(2 x \right)}\right) + \frac{\tan^{5}{\left(2 x \right)}}{5}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞((2tan⁡3(2x)3+tan⁡(2x))+tan⁡5(2x)5)y = \lim_{x \to \infty}\left(\left(\frac{2 \tan^{3}{\left(2 x \right)}}{3} + \tan{\left(2 x \right)}\right) + \frac{\tan^{5}{\left(2 x \right)}}{5}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(2*x) + 2*tan(2*x)^3/3 + tan(2*x)^5/5, 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→−∞((2tan⁡3(2x)3+tan⁡(2x))+tan⁡5(2x)5x)y = x \lim_{x \to -\infty}\left(\frac{\left(\frac{2 \tan^{3}{\left(2 x \right)}}{3} + \tan{\left(2 x \right)}\right) + \frac{\tan^{5}{\left(2 x \right)}}{5}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((2tan⁡3(2x)3+tan⁡(2x))+tan⁡5(2x)5x)y = x \lim_{x \to \infty}\left(\frac{\left(\frac{2 \tan^{3}{\left(2 x \right)}}{3} + \tan{\left(2 x \right)}\right) + \frac{\tan^{5}{\left(2 x \right)}}{5}}{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:
(2tan⁡3(2x)3+tan⁡(2x))+tan⁡5(2x)5=−tan⁡5(2x)5−2tan⁡3(2x)3−tan⁡(2x)\left(\frac{2 \tan^{3}{\left(2 x \right)}}{3} + \tan{\left(2 x \right)}\right) + \frac{\tan^{5}{\left(2 x \right)}}{5} = - \frac{\tan^{5}{\left(2 x \right)}}{5} - \frac{2 \tan^{3}{\left(2 x \right)}}{3} - \tan{\left(2 x \right)}
- No
(2tan⁡3(2x)3+tan⁡(2x))+tan⁡5(2x)5=tan⁡5(2x)5+2tan⁡3(2x)3+tan⁡(2x)\left(\frac{2 \tan^{3}{\left(2 x \right)}}{3} + \tan{\left(2 x \right)}\right) + \frac{\tan^{5}{\left(2 x \right)}}{5} = \frac{\tan^{5}{\left(2 x \right)}}{5} + \frac{2 \tan^{3}{\left(2 x \right)}}{3} + \tan{\left(2 x \right)}
- No
es decir, función
no es
par ni impar
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