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Trazado el gráfico de la función/ tan(tan(2*x))

Gráfico de la función y = tan(tan(2*x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = tan(tan(2*x))
f(x)=tan⁡(tan⁡(2x))f{\left(x \right)} = \tan{\left(\tan{\left(2 x \right)} \right)} 
f = tan(tan(2*x))
Gráfico de la función
02468-8-6-4-2-1010-250250
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⁡(tan⁡(2x))=0\tan{\left(\tan{\left(2 x \right)} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=−40.0235182424307x_{1} = -40.0235182424307
x2=1.5707963267949x_{2} = 1.5707963267949
x3=−50.2654824574367x_{3} = -50.2654824574367
x4=14.1371669411541x_{4} = 14.1371669411541
x5=−3.14159265358979x_{5} = -3.14159265358979
x6=32.9867228626928x_{6} = 32.9867228626928
x7=−36.1283155162826x_{7} = -36.1283155162826
x8=7.85398163397448x_{8} = 7.85398163397448
x9=72.2566310325652x_{9} = 72.2566310325652
x10=42.4115008234622x_{10} = 42.4115008234622
x11=40.1081602314601x_{11} = 40.1081602314601
x12=−43.9822971502571x_{12} = -43.9822971502571
x13=26.0722239276738x_{13} = 26.0722239276738
x14=98.9601685880785x_{14} = 98.9601685880785
x15=83.2522053201295x_{15} = 83.2522053201295
x16=−37.6991118430775x_{16} = -37.6991118430775
x17=−89.5353906273091x_{17} = -89.5353906273091
x18=−25.7640548565578x_{18} = -25.7640548565578
x19=34.5575191894877x_{19} = 34.5575191894877
x20=−94.2477796076938x_{20} = -94.2477796076938
x21=−67.5442420521806x_{21} = -67.5442420521806
x22=−51.8362787842316x_{22} = -51.8362787842316
x23=−14.1371669411541x_{23} = -14.1371669411541
x24=−21.9911485751286x_{24} = -21.9911485751286
x25=92.6769832808989x_{25} = 92.6769832808989
x26=−15.707963267949x_{26} = -15.707963267949
x27=50.2654824574367x_{27} = 50.2654824574367
x28=64.4026493985908x_{28} = 64.4026493985908
x29=−77.7809191566922x_{29} = -77.7809191566922
x30=21.9911485751286x_{30} = 21.9911485751286
x31=−87.9645943005142x_{31} = -87.9645943005142
x32=76.9690200129499x_{32} = 76.9690200129499
x33=−72.2566310325652x_{33} = -72.2566310325652
x34=23.5619449019235x_{34} = 23.5619449019235
x35=−55.7433857219175x_{35} = -55.7433857219175
x36=59.6902604182061x_{36} = 59.6902604182061
x37=−63.6437522155381x_{37} = -63.6437522155381
x38=−85.687315405466x_{38} = -85.687315405466
x39=−105.243353895258x_{39} = -105.243353895258
x40=−80.1106126665397x_{40} = -80.1106126665397
x41=−9.42477796076938x_{41} = -9.42477796076938
x42=−29.845130209103x_{42} = -29.845130209103
x43=10.2630300223571x_{43} = 10.2630300223571
x44=−84.0058153926878x_{44} = -84.0058153926878
x45=20.4203522483337x_{45} = 20.4203522483337
x46=−65.9734457253857x_{46} = -65.9734457253857
x47=15.707963267949x_{47} = 15.707963267949
x48=28.2743338823081x_{48} = 28.2743338823081
x49=94.2477796076938x_{49} = 94.2477796076938
x50=37.6991118430775x_{50} = 37.6991118430775
x51=−1.5707963267949x_{51} = -1.5707963267949
x52=6.28318530717959x_{52} = 6.28318530717959
x53=46.1844071048915x_{53} = 46.1844071048915
x54=86.3937979737193x_{54} = 86.3937979737193
x55=48.0633725028023x_{55} = 48.0633725028023
x56=−53.4070751110265x_{56} = -53.4070751110265
x57=−19.5560384897921x_{57} = -19.5560384897921
x58=−69.7463520068149x_{58} = -69.7463520068149
x59=−6.28318530717959x_{59} = -6.28318530717959
x60=36.1283155162826x_{60} = 36.1283155162826
x61=−75.398223686155x_{61} = -75.398223686155
x62=51.8362787842316x_{62} = 51.8362787842316
x63=80.1106126665397x_{63} = 80.1106126665397
x64=3.95877890782638x_{64} = 3.95877890782638
x65=0x_{65} = 0
x66=58.1194640914112x_{66} = 58.1194640914112
x67=−62.0993088065887x_{67} = -62.0993088065887
x68=73.8274273593601x_{68} = 73.8274273593601
x69=−58.1194640914112x_{69} = -58.1194640914112
x70=62.2005394439564x_{70} = 62.2005394439564
x71=−97.3893722612836x_{71} = -97.3893722612836
x72=−95.8185759344887x_{72} = -95.8185759344887
x73=29.845130209103x_{73} = 29.845130209103
x74=−47.7552034316864x_{74} = -47.7552034316864
x75=−28.2743338823081x_{75} = -28.2743338823081
x76=81.6814089933346x_{76} = 81.6814089933346
x77=−91.7375005819435x_{77} = -91.7375005819435
x78=100.530964914873x_{78} = 100.530964914873
x79=−23.5619449019235x_{79} = -23.5619449019235
x80=18.2182422936993x_{80} = 18.2182422936993
x81=78.5398163397448x_{81} = 78.5398163397448
x82=70.6858347057703x_{82} = 70.6858347057703
x83=−11.8598880461058x_{83} = -11.8598880461058
x84=12.5663706143592x_{84} = 12.5663706143592
x85=87.9645943005142x_{85} = 87.9645943005142
x86=90.4748733262645x_{86} = 90.4748733262645
x87=43.9822971502571x_{87} = 43.9822971502571
x88=−31.4159265358979x_{88} = -31.4159265358979
x89=−81.6814089933346x_{89} = -81.6814089933346
x90=56.5486677646163x_{90} = 56.5486677646163
x91=68.2507246204339x_{91} = 68.2507246204339
x92=54.1135576792799x_{92} = 54.1135576792799
x93=−73.8274273593601x_{93} = -73.8274273593601
x94=−59.6902604182061x_{94} = -59.6902604182061
x95=−33.6180364905323x_{95} = -33.6180364905323
x96=65.9734457253857x_{96} = 65.9734457253857
x97=−45.553093477052x_{97} = -45.553093477052
x98=−7.85398163397448x_{98} = -7.85398163397448
x99=−17.2787595947439x_{99} = -17.2787595947439
x100=95.8185759344887x_{100} = 95.8185759344887
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(tan(2*x)).
tan⁡(tan⁡(0⋅2))\tan{\left(\tan{\left(0 \cdot 2 \right)} \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
(2tan⁡2(2x)+2)(tan⁡2(tan⁡(2x))+1)=0\left(2 \tan^{2}{\left(2 x \right)} + 2\right) \left(\tan^{2}{\left(\tan{\left(2 x \right)} \right)} + 1\right) = 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)tan⁡(tan⁡(2x))+tan⁡(2x))(tan⁡2(2x)+1)(tan⁡2(tan⁡(2x))+1)=08 \left(\left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(\tan{\left(2 x \right)} \right)} + \tan{\left(2 x \right)}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\tan^{2}{\left(\tan{\left(2 x \right)} \right)} + 1\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−69.7310392015724x_{1} = -69.7310392015724
x2=−99.9149640922764x_{2} = -99.9149640922764
x3=−25.7487420513153x_{3} = -25.7487420513153
x4=89.5353906273091x_{4} = 89.5353906273091
x5=−15.707963267949x_{5} = -15.707963267949
x6=−31.4159265358979x_{6} = -31.4159265358979
x7=42.4115008234622x_{7} = 42.4115008234622
x8=21.9911485751286x_{8} = 21.9911485751286
x9=−11.9503697917622x_{9} = -11.9503697917622
x10=0x_{10} = 0
x11=−42.4115008234622x_{11} = -42.4115008234622
x12=−29.845130209103x_{12} = -29.845130209103
x13=−3.75759347618678x_{13} = -3.75759347618678
x14=−21.9911485751286x_{14} = -21.9911485751286
x15=−36.1283155162826x_{15} = -36.1283155162826
x16=28.2743338823081x_{16} = 28.2743338823081
x17=86.3937979737193x_{17} = 86.3937979737193
x18=68.2487278775419x_{18} = 68.2487278775419
x19=72.2566310325652x_{19} = 72.2566310325652
x20=−47.7398906264439x_{20} = -47.7398906264439
x21=40.2247036740703x_{21} = 40.2247036740703
x22=84.2070008243274x_{22} = 84.2070008243274
x23=−94.2477796076938x_{23} = -94.2477796076938
x24=−61.261056745001x_{24} = -61.261056745001
x25=87.9645943005142x_{25} = 87.9645943005142
x26=−95.8185759344887x_{26} = -95.8185759344887
x27=−50.2654824574367x_{27} = -50.2654824574367
x28=23.5619449019235x_{28} = 23.5619449019235
x29=−43.9822971502571x_{29} = -43.9822971502571
x30=−97.3893722612836x_{30} = -97.3893722612836
x31=50.2654824574367x_{31} = 50.2654824574367
x32=−14.1371669411541x_{32} = -14.1371669411541
x33=59.6902604182061x_{33} = 59.6902604182061
x34=58.1194640914112x_{34} = 58.1194640914112
x35=−91.722187776701x_{35} = -91.722187776701
x36=−53.4070751110265x_{36} = -53.4070751110265
x37=−23.5619449019235x_{37} = -23.5619449019235
x38=98.0053730838806x_{38} = 98.0053730838806
x39=−86.3937979737193x_{39} = -86.3937979737193
x40=−17.2787595947439x_{40} = -17.2787595947439
x41=12.5663706143592x_{41} = 12.5663706143592
x42=−81.6814089933346x_{42} = -81.6814089933346
x43=94.2477796076938x_{43} = 94.2477796076938
x44=81.6814089933346x_{44} = 81.6814089933346
x45=−83.8682061427265x_{45} = -83.8682061427265
x46=−67.5442420521806x_{46} = -67.5442420521806
x47=−80.1106126665397x_{47} = -80.1106126665397
x48=−1.5707963267949x_{48} = -1.5707963267949
x49=26.0875367329163x_{49} = 26.0875367329163
x50=92.6769832808989x_{50} = 92.6769832808989
x51=−39.8859089924694x_{51} = -39.8859089924694
x52=36.1283155162826x_{52} = 36.1283155162826
x53=76.014224508752x_{53} = 76.014224508752
x54=−28.2743338823081x_{54} = -28.2743338823081
x55=4.71238898038469x_{55} = 4.71238898038469
x56=48.6946861306418x_{56} = 48.6946861306418
x57=54.0230759336235x_{57} = 54.0230759336235
x58=−72.2566310325652x_{58} = -72.2566310325652
x59=37.6991118430775x_{59} = 37.6991118430775
x60=70.6858347057703x_{60} = 70.6858347057703
x61=−45.553093477052x_{61} = -45.553093477052
x62=9.42477796076938x_{62} = 9.42477796076938
x63=−89.5353906273091x_{63} = -89.5353906273091
x64=65.9734457253857x_{64} = 65.9734457253857
x65=73.8274273593601x_{65} = 73.8274273593601
x66=20.4203522483337x_{66} = 20.4203522483337
x67=−87.9645943005142x_{67} = -87.9645943005142
x68=1.5707963267949x_{68} = 1.5707963267949
x69=32.0319273584949x_{69} = 32.0319273584949
x70=45.553093477052x_{70} = 45.553093477052
x71=78.5398163397448x_{71} = 78.5398163397448
x72=−6.28318530717959x_{72} = -6.28318530717959
x73=−55.9326669420193x_{73} = -55.9326669420193
x74=95.8185759344887x_{74} = 95.8185759344887
x75=−20.4203522483337x_{75} = -20.4203522483337
x76=15.707963267949x_{76} = 15.707963267949
x77=−58.1194640914112x_{77} = -58.1194640914112
x78=56.5486677646163x_{78} = 56.5486677646163
x79=80.1106126665397x_{79} = 80.1106126665397
x80=7.85398163397448x_{80} = 7.85398163397448
x81=29.845130209103x_{81} = 29.845130209103
x82=−65.9734457253857x_{82} = -65.9734457253857
x83=43.9822971502571x_{83} = 43.9822971502571
x84=14.1371669411541x_{84} = 14.1371669411541
x85=−37.6991118430775x_{85} = -37.6991118430775
x86=−59.6902604182061x_{86} = -59.6902604182061
x87=−33.9415183668907x_{87} = -33.9415183668907
x88=18.2335550989418x_{88} = 18.2335550989418
x89=−64.4026493985908x_{89} = -64.4026493985908
x90=−75.398223686155x_{90} = -75.398223686155
x91=51.8362787842316x_{91} = 51.8362787842316
x92=100.530964914873x_{92} = 100.530964914873
x93=−77.9238155171478x_{93} = -77.9238155171478
x94=64.4026493985908x_{94} = 64.4026493985908
x95=34.5575191894877x_{95} = 34.5575191894877
x96=−73.8274273593601x_{96} = -73.8274273593601
x97=6.28318530717959x_{97} = 6.28318530717959
x98=62.2158522491989x_{98} = 62.2158522491989
x99=−9.42477796076938x_{99} = -9.42477796076938
x100=−51.8362787842316x_{100} = -51.8362787842316
x101=−7.85398163397448x_{101} = -7.85398163397448

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
[100.530964914873,∞)\left[100.530964914873, \infty\right)
Convexa en los intervalos
(−∞,−99.9149640922764]\left(-\infty, -99.9149640922764\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→−∞tan⁡(tan⁡(2x))y = \lim_{x \to -\infty} \tan{\left(\tan{\left(2 x \right)} \right)}
True

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

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