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Trazado el gráfico de la función/ tg(x)+arctg(x)

Gráfico de la función y = tg(x)+arctg(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = tan(x) + atan(x)
f(x)=tan⁡(x)+atan⁡(x)f{\left(x \right)} = \tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} 
f = tan(x) + atan(x)
Gráfico de la función
02468-8-6-4-2-1010-200200
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⁡(x)+atan⁡(x)=0\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=99.5299908589546x_{1} = 99.5299908589546
x2=−49.2675048394235x_{2} = -49.2675048394235
x3=42.9851917481598x_{3} = 42.9851917481598
x4=39.8441394477448x_{4} = 39.8441394477448
x5=−74.3982387986425x_{5} = -74.3982387986425
x6=80.6811187268223x_{6} = 80.6811187268223
x7=−46.1263183201969x_{7} = -46.1263183201969
x8=14.7242522567935x_{8} = 14.7242522567935
x9=68.1154155674723x_{9} = 68.1154155674723
x10=−8.45673516163459x_{10} = -8.45673516163459
x11=−86.9640430028565x_{11} = -86.9640430028565
x12=−61.83266643575x_{12} = -61.83266643575
x13=−33.562342086631x_{13} = -33.562342086631
x14=52.408740441672x_{14} = 52.408740441672
x15=77.5396725417462x_{15} = 77.5396725417462
x16=−64.9740303869064x_{16} = -64.9740303869064
x17=64.9740303869064x_{17} = 64.9740303869064
x18=71.2568191529722x_{18} = 71.2568191529722
x19=2.2831221774726x_{19} = 2.2831221774726
x20=−71.2568191529722x_{20} = -71.2568191529722
x21=11.5883131166121x_{21} = 11.5883131166121
x22=58.6913271474582x_{22} = 58.6913271474582
x23=−55.5500167379063x_{23} = -55.5500167379063
x24=−83.8225759478241x_{24} = -83.8225759478241
x25=55.5500167379063x_{25} = 55.5500167379063
x26=93.2470026209387x_{26} = 93.2470026209387
x27=21.0012872745219x_{27} = 21.0012872745219
x28=17.8622178450272x_{28} = 17.8622178450272
x29=−58.6913271474582x_{29} = -58.6913271474582
x30=90.1055188584242x_{30} = 90.1055188584242
x31=−90.1055188584242x_{31} = -90.1055188584242
x32=5.33753969228673x_{32} = 5.33753969228673
x33=96.3884935138203x_{33} = 96.3884935138203
x34=−96.3884935138203x_{34} = -96.3884935138203
x35=−42.9851917481598x_{35} = -42.9851917481598
x36=−5.33753969228673x_{36} = -5.33753969228673
x37=−39.8441394477448x_{37} = -39.8441394477448
x38=0x_{38} = 0
x39=8.45673516163459x_{39} = 8.45673516163459
x40=−2.2831221774726x_{40} = -2.2831221774726
x41=86.9640430028565x_{41} = 86.9640430028565
x42=−21.0012872745219x_{42} = -21.0012872745219
x43=−17.8622178450272x_{43} = -17.8622178450272
x44=−30.4216613078004x_{44} = -30.4216613078004
x45=49.2675048394235x_{45} = 49.2675048394235
x46=−93.2470026209387x_{46} = -93.2470026209387
x47=−99.5299908589546x_{47} = -99.5299908589546
x48=−80.6811187268223x_{48} = -80.6811187268223
x49=36.703180699745x_{49} = 36.703180699745
x50=46.1263183201969x_{50} = 46.1263183201969
x51=−11.5883131166121x_{51} = -11.5883131166121
x52=−52.408740441672x_{52} = -52.408740441672
x53=61.83266643575x_{53} = 61.83266643575
x54=−14.7242522567935x_{54} = -14.7242522567935
x55=33.562342086631x_{55} = 33.562342086631
x56=−77.5396725417462x_{56} = -77.5396725417462
x57=24.1410236736903x_{57} = 24.1410236736903
x58=−68.1154155674723x_{58} = -68.1154155674723
x59=30.4216613078004x_{59} = 30.4216613078004
x60=74.3982387986425x_{60} = 74.3982387986425
x61=27.2811936639259x_{61} = 27.2811936639259
x62=−24.1410236736903x_{62} = -24.1410236736903
x63=−27.2811936639259x_{63} = -27.2811936639259
x64=−36.703180699745x_{64} = -36.703180699745
x65=83.8225759478241x_{65} = 83.8225759478241
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(x) + atan(x).
tan⁡(0)+atan⁡(0)\tan{\left(0 \right)} + \operatorname{atan}{\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
tan⁡2(x)+1+1x2+1=0\tan^{2}{\left(x \right)} + 1 + \frac{1}{x^{2} + 1} = 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
2(−x(x2+1)2+(tan⁡2(x)+1)tan⁡(x))=02 \left(- \frac{x}{\left(x^{2} + 1\right)^{2}} + \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−15.7082191890374x_{1} = -15.7082191890374
x2=75.3982260183446x_{2} = 75.3982260183446
x3=−75.3982260183446x_{3} = -75.3982260183446
x4=−53.4070816709532x_{4} = -53.4070816709532
x5=53.4070816709532x_{5} = 53.4070816709532
x6=−47.1238993512507x_{6} = -47.1238993512507
x7=−40.8407191588716x_{7} = -40.8407191588716
x8=−94.2477808019261x_{8} = -94.2477808019261
x9=43.98230889158x_{9} = 43.98230889158
x10=3.16758836114823x_{10} = 3.16758836114823
x11=−62.8318571011953x_{11} = -62.8318571011953
x12=−9.42594558462241x_{12} = -9.42594558462241
x13=−18.8497043937593x_{13} = -18.8497043937593
x14=−91.1061882761657x_{14} = -91.1061882761657
x15=12.5668681635809x_{15} = 12.5668681635809
x16=−3.16758836114823x_{16} = -3.16758836114823
x17=21.9912422140771x_{17} = 21.9912422140771
x18=−21.9912422140771x_{18} = -21.9912422140771
x19=−81.6814108277603x_{19} = -81.6814108277603
x20=87.9645957693188x_{20} = 87.9645957693188
x21=6.28701317511834x_{21} = 6.28701317511834
x22=69.1150414065892x_{22} = 69.1150414065892
x23=84.8230032850167x_{23} = 84.8230032850167
x24=9.42594558462241x_{24} = 9.42594558462241
x25=−84.8230032850167x_{25} = -84.8230032850167
x26=−12.5668681635809x_{26} = -12.5668681635809
x27=−31.4159587220774x_{27} = -31.4159587220774
x28=−50.2654903251137x_{28} = -50.2654903251137
x29=−25.1328040205507x_{29} = -25.1328040205507
x30=−37.6991304808952x_{30} = -37.6991304808952
x31=62.8318571011953x_{31} = 62.8318571011953
x32=91.1061882761657x_{32} = 91.1061882761657
x33=97.3893733436485x_{33} = 97.3893733436485
x34=40.8407191588716x_{34} = 40.8407191588716
x35=31.4159587220774x_{35} = 31.4159587220774
x36=−56.548673291256x_{36} = -56.548673291256
x37=18.8497043937593x_{37} = 18.8497043937593
x38=56.548673291256x_{38} = 56.548673291256
x39=37.6991304808952x_{39} = 37.6991304808952
x40=−34.5575433799626x_{40} = -34.5575433799626
x41=−100.530965898917x_{41} = -100.530965898917
x42=−69.1150414065892x_{42} = -69.1150414065892
x43=−43.98230889158x_{43} = -43.98230889158
x44=0x_{44} = 0
x45=25.1328040205507x_{45} = 25.1328040205507
x46=−65.9734492062964x_{46} = -65.9734492062964
x47=47.1238993512507x_{47} = 47.1238993512507
x48=−6.28701317511834x_{48} = -6.28701317511834
x49=94.2477808019261x_{49} = 94.2477808019261
x50=78.5398184031738x_{50} = 78.5398184031738
x51=15.7082191890374x_{51} = 15.7082191890374
x52=−72.2566336822883x_{52} = -72.2566336822883
x53=−28.2743780124173x_{53} = -28.2743780124173
x54=−97.3893733436485x_{54} = -97.3893733436485
x55=72.2566336822883x_{55} = 72.2566336822883
x56=34.5575433799626x_{56} = 34.5575433799626
x57=28.2743780124173x_{57} = 28.2743780124173
x58=−87.9645957693188x_{58} = -87.9645957693188
x59=100.530965898917x_{59} = 100.530965898917
x60=50.2654903251137x_{60} = 50.2654903251137
x61=−78.5398184031738x_{61} = -78.5398184031738
x62=59.6902651176419x_{62} = 59.6902651176419
x63=−59.6902651176419x_{63} = -59.6902651176419
x64=65.9734492062964x_{64} = 65.9734492062964
x65=81.6814108277603x_{65} = 81.6814108277603

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
[100.530965898917,∞)\left[100.530965898917, \infty\right)
Convexa en los intervalos
(−∞,−100.530965898917]\left(-\infty, -100.530965898917\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⁡(x)+atan⁡(x))y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right)
True

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

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