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Trazado el gráfico de la función/ atan(sqrt(2*tan(x)))/(1-tan(x))

Gráfico de la función y = atan(sqrt(2*tan(x)))/(1-tan(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           /  __________\
       atan\\/ 2*tan(x) /
f(x) = ------------------
           1 - tan(x)
f(x)=atan⁡(2tan⁡(x))1−tan⁡(x)f{\left(x \right)} = \frac{\operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{1 - \tan{\left(x \right)}} 
f = atan(sqrt(2*tan(x)))/(1 - tan(x))
Gráfico de la función
02468-8-6-4-2-1010-100100
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0.785398163397448x_{1} = 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:
atan⁡(2tan⁡(x))1−tan⁡(x)=0\frac{\operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{1 - \tan{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=−95.8185759344887x_{1} = -95.8185759344887
x2=92.6769832808989x_{2} = 92.6769832808989
x3=20.4203522483337x_{3} = 20.4203522483337
x4=−86.3937979737193x_{4} = -86.3937979737193
x5=−51.8362787842316x_{5} = -51.8362787842316
x6=29.845130209103x_{6} = 29.845130209103
x7=−39.2699081698724x_{7} = -39.2699081698724
x8=67.5442420521806x_{8} = 67.5442420521806
x9=−1.57079632679489x_{9} = -1.57079632679489
x10=−64.4026493985908x_{10} = -64.4026493985908
x11=42.4115008234622x_{11} = 42.4115008234622
x12=83.2522053201295x_{12} = 83.2522053201295
x13=51.8362787842316x_{13} = 51.8362787842316
x14=0x_{14} = 0
x15=64.4026493985908x_{15} = 64.4026493985908
x16=−92.6769832808989x_{16} = -92.6769832808989
x17=−42.4115008234622x_{17} = -42.4115008234622
x18=−4.71238898038469x_{18} = -4.71238898038469
x19=86.3937979737193x_{19} = 86.3937979737193
x20=4.71238898038469x_{20} = 4.71238898038469
x21=89.5353906273091x_{21} = 89.5353906273091
x22=70.6858347057703x_{22} = 70.6858347057703
x23=17.2787595947439x_{23} = 17.2787595947439
x24=−36.1283155162826x_{24} = -36.1283155162826
x25=−80.1106126665397x_{25} = -80.1106126665397
x26=−58.1194640914112x_{26} = -58.1194640914112
x27=7.85398163397448x_{27} = 7.85398163397448
x28=−83.2522053201295x_{28} = -83.2522053201295
x29=−73.8274273593601x_{29} = -73.8274273593601
x30=−70.6858347057703x_{30} = -70.6858347057703
x31=−23.5619449019235x_{31} = -23.5619449019235
x32=32.9867228626928x_{32} = 32.9867228626928
x33=−14.1371669411541x_{33} = -14.1371669411541
x34=−89.5353906273091x_{34} = -89.5353906273091
x35=45.553093477052x_{35} = 45.553093477052
x36=80.1106126665397x_{36} = 80.1106126665397
x37=26.7035375555132x_{37} = 26.7035375555132
x38=61.261056745001x_{38} = 61.261056745001
x39=−48.6946861306418x_{39} = -48.6946861306418
x40=−7.85398163397448x_{40} = -7.85398163397448
x41=−61.261056745001x_{41} = -61.261056745001
x42=−26.7035375555132x_{42} = -26.7035375555132
x43=−67.5442420521806x_{43} = -67.5442420521806
x44=14.1371669411541x_{44} = 14.1371669411541
x45=95.8185759344887x_{45} = 95.8185759344887
x46=48.6946861306418x_{46} = 48.6946861306418
x47=−29.845130209103x_{47} = -29.845130209103
x48=1.5707963267949x_{48} = 1.5707963267949
x49=23.5619449019235x_{49} = 23.5619449019235
x50=39.2699081698724x_{50} = 39.2699081698724
x51=−17.2787595947439x_{51} = -17.2787595947439
x52=36.1283155162826x_{52} = 36.1283155162826
x53=−20.4203522483337x_{53} = -20.4203522483337
x54=73.8274273593601x_{54} = 73.8274273593601
x55=−45.553093477052x_{55} = -45.553093477052
x56=58.1194640914112x_{56} = 58.1194640914112
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 atan(sqrt(2*tan(x)))/(1 - tan(x)).
atan⁡(2tan⁡(0))1−tan⁡(0)\frac{\operatorname{atan}{\left(\sqrt{2 \tan{\left(0 \right)}} \right)}}{1 - \tan{\left(0 \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
2(tan⁡2(x)+1)2(1−tan⁡(x))(2tan⁡(x)+1)tan⁡(x)+(tan⁡2(x)+1)atan⁡(2tan⁡(x))(1−tan⁡(x))2=0\frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + 1\right)}{2 \left(1 - \tan{\left(x \right)}\right) \left(2 \tan{\left(x \right)} + 1\right) \sqrt{\tan{\left(x \right)}}} + \frac{\left(\tan^{2}{\left(x \right)} + 1\right) \operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{\left(1 - \tan{\left(x \right)}\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
(tan⁡2(x)+1)(2(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x)+4(tan⁡2(x)+1)(2tan⁡(x)+1)tan⁡(x))4(2tan⁡(x)+1)+2(tan⁡(x)−tan⁡2(x)+1tan⁡(x)−1)atan⁡(2tan⁡(x))tan⁡(x)−1+2(tan⁡2(x)+1)(tan⁡(x)−1)(2tan⁡(x)+1)tan⁡(x))tan⁡(x)−1=0\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{\sqrt{2} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(2 \tan{\left(x \right)} + 1\right) \sqrt{\tan{\left(x \right)}}}\right)}{4 \left(2 \tan{\left(x \right)} + 1\right)} + \frac{2 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} - 1}\right) \operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{\tan{\left(x \right)} - 1} + \frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} - 1\right) \left(2 \tan{\left(x \right)} + 1\right) \sqrt{\tan{\left(x \right)}}}\right)}{\tan{\left(x \right)} - 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−37.9018316439784x_{1} = -37.9018316439784
x2=81.8584815483035x_{2} = 81.8584815483035
x3=86.3647767666273x_{3} = 86.3647767666273
x4=15.8850358229178x_{4} = 15.8850358229178
x5=−31.2388539809291x_{5} = -31.2388539809291
x6=92.6479620738068x_{6} = 92.6479620738068
x7=−87.7875217455454x_{7} = -87.7875217455454
x8=−81.8841287942355x_{8} = -81.8841287942355
x9=−6.10611275221076x_{9} = -6.10611275221076
x10=31.5929990908668x_{10} = 31.5929990908668
x11=28.0716140814072x_{11} = 28.0716140814072
x12=64.3736281914987x_{12} = 64.3736281914987
x13=22.1682211300974x_{13} = 22.1682211300974
x14=−23.5909661090155x_{14} = -23.5909661090155
x15=88.141666855483x_{15} = 88.141666855483
x16=−94.070707052725x_{16} = -94.070707052725
x17=−34.3804466345189x_{17} = -34.3804466345189
x18=42.3824796163701x_{18} = 42.3824796163701
x19=−50.0884099024679x_{19} = -50.0884099024679
x20=−89.5644118344012x_{20} = -89.5644118344012
x21=−83.2812265272216x_{21} = -83.2812265272216
x22=−56.3715952096474x_{22} = -56.3715952096474
x23=44.1593697052259x_{23} = 44.1593697052259
x24=50.0627626565358x_{24} = 50.0627626565358
x25=−100.353892359905x_{25} = -100.353892359905
x26=−12.3892980593903x_{26} = -12.3892980593903
x27=−15.9106830688499x_{27} = -15.9106830688499
x28=−43.8052245952883x_{28} = -43.8052245952883
x29=−45.5821146841441x_{29} = -45.5821146841441
x30=−9.62749776167028x_{30} = -9.62749776167028
x31=75.5752962411239x_{31} = 75.5752962411239
x32=−28.0972613273393x_{32} = -28.0972613273393
x33=−76.998041220042x_{33} = -76.998041220042
x34=56.3459479637154x_{34} = 56.3459479637154
x35=53.5841476659953x_{35} = 53.5841476659953
x36=−21.8140760201597x_{36} = -21.8140760201597
x37=−72.0795584775964x_{37} = -72.0795584775964
x38=−61.290077952093x_{38} = -61.290077952093
x39=6.08046550627869x_{39} = 6.08046550627869
x40=59.8673329731749x_{40} = 59.8673329731749
x41=−59.892980219107x_{41} = -59.892980219107
x42=100.328245113972x_{42} = 100.328245113972
x43=72.0539112316643x_{43} = 72.0539112316643
x44=−65.7963731704168x_{44} = -65.7963731704168
x45=94.0450598067929x_{45} = 94.0450598067929
x46=9.60185051573821x_{46} = 9.60185051573821
x47=97.5664448162524x_{47} = 97.5664448162524
x48=78.3370965388439x_{48} = 78.3370965388439
x49=66.1505182803545x_{49} = 66.1505182803545
x50=26.6745163484212x_{50} = 26.6745163484212
x51=37.8761843980463x_{51} = 37.8761843980463
x52=−39.2989293769645x_{52} = -39.2989293769645
x53=34.3547993885868x_{53} = 34.3547993885868
x54=70.6568134986783x_{54} = 70.6568134986783
x55=4.68336777329263x_{55} = 4.68336777329263
x56=−78.362743784776x_{56} = -78.362743784776
x57=12.3636508134583x_{57} = 12.3636508134583
x58=−1.59981753388696x_{58} = -1.59981753388696
x59=20.3913310412416x_{59} = 20.3913310412416
x60=−17.3077808018359x_{60} = -17.3077808018359
x61=−67.5732632592726x_{61} = -67.5732632592726
x62=48.6656649235497x_{62} = 48.6656649235497
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0.785398163397448x_{1} = 0.785398163397448

lim⁡x→0.785398163397448−((tan⁡2(x)+1)(2(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x)+4(tan⁡2(x)+1)(2tan⁡(x)+1)tan⁡(x))4(2tan⁡(x)+1)+2(tan⁡(x)−tan⁡2(x)+1tan⁡(x)−1)atan⁡(2tan⁡(x))tan⁡(x)−1+2(tan⁡2(x)+1)(tan⁡(x)−1)(2tan⁡(x)+1)tan⁡(x))tan⁡(x)−1)=1.08172851219476⋅10322+5.84600654932361⋅1048atan⁡(12)\lim_{x \to 0.785398163397448^-}\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{\sqrt{2} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(2 \tan{\left(x \right)} + 1\right) \sqrt{\tan{\left(x \right)}}}\right)}{4 \left(2 \tan{\left(x \right)} + 1\right)} + \frac{2 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} - 1}\right) \operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{\tan{\left(x \right)} - 1} + \frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} - 1\right) \left(2 \tan{\left(x \right)} + 1\right) \sqrt{\tan{\left(x \right)}}}\right)}{\tan{\left(x \right)} - 1}\right) = 1.08172851219476 \cdot 10^{32} \sqrt{2} + 5.84600654932361 \cdot 10^{48} \operatorname{atan}{\left(1 \sqrt{2} \right)}
lim⁡x→0.785398163397448+((tan⁡2(x)+1)(2(tan⁡2(x)+1tan⁡32(x)−4tan⁡(x)+4(tan⁡2(x)+1)(2tan⁡(x)+1)tan⁡(x))4(2tan⁡(x)+1)+2(tan⁡(x)−tan⁡2(x)+1tan⁡(x)−1)atan⁡(2tan⁡(x))tan⁡(x)−1+2(tan⁡2(x)+1)(tan⁡(x)−1)(2tan⁡(x)+1)tan⁡(x))tan⁡(x)−1)=1.08172851219476⋅10322+5.84600654932361⋅1048atan⁡(12)\lim_{x \to 0.785398163397448^+}\left(\frac{\left(\tan^{2}{\left(x \right)} + 1\right) \left(\frac{\sqrt{2} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{\frac{3}{2}}{\left(x \right)}} - 4 \sqrt{\tan{\left(x \right)}} + \frac{4 \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(2 \tan{\left(x \right)} + 1\right) \sqrt{\tan{\left(x \right)}}}\right)}{4 \left(2 \tan{\left(x \right)} + 1\right)} + \frac{2 \left(\tan{\left(x \right)} - \frac{\tan^{2}{\left(x \right)} + 1}{\tan{\left(x \right)} - 1}\right) \operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{\tan{\left(x \right)} - 1} + \frac{\sqrt{2} \left(\tan^{2}{\left(x \right)} + 1\right)}{\left(\tan{\left(x \right)} - 1\right) \left(2 \tan{\left(x \right)} + 1\right) \sqrt{\tan{\left(x \right)}}}\right)}{\tan{\left(x \right)} - 1}\right) = 1.08172851219476 \cdot 10^{32} \sqrt{2} + 5.84600654932361 \cdot 10^{48} \operatorname{atan}{\left(1 \sqrt{2} \right)}
- los límites son iguales, es decir omitimos el punto correspondiente

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
[97.5664448162524,∞)\left[97.5664448162524, \infty\right)
Convexa en los intervalos
(−∞,−100.353892359905]\left(-\infty, -100.353892359905\right]
Asíntotas verticales
Hay:
x1=0.785398163397448x_{1} = 0.785398163397448
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→−∞(atan⁡(2tan⁡(x))1−tan⁡(x))y = \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{1 - \tan{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(atan⁡(2tan⁡(x))1−tan⁡(x))y = \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{1 - \tan{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función atan(sqrt(2*tan(x)))/(1 - tan(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→−∞(atan⁡(2tan⁡(x))x(1−tan⁡(x)))y = x \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{x \left(1 - \tan{\left(x \right)}\right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(atan⁡(2tan⁡(x))x(1−tan⁡(x)))y = x \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{x \left(1 - \tan{\left(x \right)}\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:
atan⁡(2tan⁡(x))1−tan⁡(x)=atan⁡(2−tan⁡(x))tan⁡(x)+1\frac{\operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{1 - \tan{\left(x \right)}} = \frac{\operatorname{atan}{\left(\sqrt{2} \sqrt{- \tan{\left(x \right)}} \right)}}{\tan{\left(x \right)} + 1}
- No
atan⁡(2tan⁡(x))1−tan⁡(x)=−atan⁡(2−tan⁡(x))tan⁡(x)+1\frac{\operatorname{atan}{\left(\sqrt{2 \tan{\left(x \right)}} \right)}}{1 - \tan{\left(x \right)}} = - \frac{\operatorname{atan}{\left(\sqrt{2} \sqrt{- \tan{\left(x \right)}} \right)}}{\tan{\left(x \right)} + 1}
- No
es decir, función
no es
par ni impar
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