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Trazado el gráfico de la función/ tan(x)-1/(4*x)-1

Gráfico de la función y = tan(x)-1/(4*x)-1

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                 1     
f(x) = tan(x) - --- - 1
                4*x
f(x)=(tan⁡(x)−14x)−1f{\left(x \right)} = \left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1 
f = tan(x) - 1/(4*x) - 1
Gráfico de la función
02468-8-6-4-2-1010-200200
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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)−14x)−1=0\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1 = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=38.4877472680578x_{1} = 38.4877472680578
x2=63.6192121950275x_{2} = 63.6192121950275
x3=19.6412779024164x_{3} = 19.6412779024164
x4=−80.8975583839916x_{4} = -80.8975583839916
x5=−30.6346254172489x_{5} = -30.6346254172489
x6=85.609857792695x_{6} = 85.609857792695
x7=−84.039093101098x_{7} = -84.039093101098
x8=−27.4935029811728x_{8} = -27.4935029811728
x9=60.4777211923292x_{9} = 60.4777211923292
x10=−11.7916863086823x_{10} = -11.7916863086823
x11=−77.7560283558673x_{11} = -77.7560283558673
x12=44.7704795511379x_{12} = 44.7704795511379
x13=25.9229381995883x_{13} = 25.9229381995883
x14=88.7513989106074x_{14} = 88.7513989106074
x15=82.4683205951513x_{15} = 82.4683205951513
x16=3.95759900591204x_{16} = 3.95759900591204
x17=−40.0584365319855x_{17} = -40.0584365319855
x18=−71.4729848439766x_{18} = -71.4729848439766
x19=−52.6240579382353x_{19} = -52.6240579382353
x20=−36.9171111340582x_{20} = -36.9171111340582
x21=−87.1806319995788x_{21} = -87.1806319995788
x22=7.08591656277011x_{22} = 7.08591656277011
x23=98.1760420280135x_{23} = 98.1760420280135
x24=−33.775835627013x_{24} = -33.775835627013
x25=66.7607127461121x_{25} = 66.7607127461121
x26=−55.7655161617889x_{26} = -55.7655161617889
x27=−74.6145036099593x_{27} = -74.6145036099593
x28=−21.2116782555737x_{28} = -21.2116782555737
x29=−5.52094852985428x_{29} = -5.52094852985428
x30=−8.65403456872203x_{30} = -8.65403456872203
x31=79.3267877829935x_{31} = 79.3267877829935
x32=16.5008796888555x_{32} = 16.5008796888555
x33=69.9022215606051x_{33} = 69.9022215606051
x34=−68.3314728841616x_{34} = -68.3314728841616
x35=10.2222560339859x_{35} = 10.2222560339859
x36=47.9118901281572x_{36} = 47.9118901281572
x37=−99.746821495994x_{37} = -99.746821495994
x38=95.0344913544903x_{38} = 95.0344913544903
x39=35.3464412999816x_{39} = 35.3464412999816
x40=51.0533230563024x_{40} = 51.0533230563024
x41=−58.9069887533201x_{41} = -58.9069887533201
x42=−43.1998009073906x_{42} = -43.1998009073906
x43=22.7820035307747x_{43} = 22.7820035307747
x44=−90.3221746425414x_{44} = -90.3221746425414
x45=−18.0711229382539x_{45} = -18.0711229382539
x46=54.19477445812x_{46} = 54.19477445812
x47=−2.41082493019699x_{47} = -2.41082493019699
x48=41.6290963693637x_{48} = 41.6290963693637
x49=57.3362413039713x_{49} = 57.3362413039713
x50=101.317595301642x_{50} = 101.317595301642
x51=−65.1899687163157x_{51} = -65.1899687163157
x52=32.205191035161x_{52} = 32.205191035161
x53=76.1852598978093x_{53} = 76.1852598978093
x54=−24.3525024452919x_{54} = -24.3525024452919
x55=−62.0484735262681x_{55} = -62.0484735262681
x56=−14.9310074230454x_{56} = -14.9310074230454
x57=−96.6052696988463x_{57} = -96.6052696988463
x58=13.3610373580831x_{58} = 13.3610373580831
x59=29.0640144525706x_{59} = 29.0640144525706
x60=73.0437375741928x_{60} = 73.0437375741928
x61=−46.3411963133816x_{61} = -46.3411963133816
x62=−49.4826168258104x_{62} = -49.4826168258104
x63=91.8929435473717x_{63} = 91.8929435473717
x64=−93.4637206519092x_{64} = -93.4637206519092
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) - 1/(4*x) - 1.
−1+(tan⁡(0)−10⋅4)-1 + \left(\tan{\left(0 \right)} - \frac{1}{0 \cdot 4}\right)
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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+14x2=0\tan^{2}{\left(x \right)} + 1 + \frac{1}{4 x^{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
2(tan⁡2(x)+1)tan⁡(x)−12x3=02 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{1}{2 x^{3}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=81.6814094520786x_{1} = 81.6814094520786
x2=43.9823000886252x_{2} = 43.9823000886252
x3=−72.2566316952498x_{3} = -72.2566316952498
x4=75.3982242694076x_{4} = 75.3982242694076
x5=−3.14959356380938x_{5} = -3.14959356380938
x6=−59.6902615937248x_{6} = -59.6902615937248
x7=−31.4159345987753x_{7} = -31.4159345987753
x8=−94.2477799063191x_{8} = -94.2477799063191
x9=−37.6991165090964x_{9} = -37.6991165090964
x10=−47.1238921928491x_{10} = -47.1238921928491
x11=−65.9734465960134x_{11} = -65.9734465960134
x12=−100.530965160933x_{12} = -100.530965160933
x13=18.8495932494818x_{13} = 18.8495932494818
x14=37.6991165090964x_{14} = 37.6991165090964
x15=6.28419268165633x_{15} = 6.28419268165633
x16=−78.5398168557694x_{16} = -78.5398168557694
x17=47.1238921928491x_{17} = 47.1238921928491
x18=−43.9823000886252x_{18} = -43.9823000886252
x19=3.14959356380938x_{19} = 3.14959356380938
x20=−69.1150391361959x_{20} = -69.1150391361959
x21=−50.2654844259139x_{21} = -50.2654844259139
x22=50.2654844259139x_{22} = 50.2654844259139
x23=9.42507655767126x_{23} = 9.42507655767126
x24=−87.9645946678103x_{24} = -87.9645946678103
x25=−97.3893725319319x_{25} = -97.3893725319319
x26=−34.5575252472483x_{26} = -34.5575252472483
x27=−18.8495932494818x_{27} = -18.8495932494818
x28=−6.28419268165633x_{28} = -6.28419268165633
x29=91.1061872846991x_{29} = 91.1061872846991
x30=15.7080277702229x_{30} = 15.7080277702229
x31=62.8318540796563x_{31} = 62.8318540796563
x32=40.8407081666179x_{32} = 40.8407081666179
x33=−91.1061872846991x_{33} = -91.1061872846991
x34=−21.9911720820024x_{34} = -21.9911720820024
x35=69.1150391361959x_{35} = 69.1150391361959
x36=28.2743449424922x_{36} = 28.2743449424922
x37=−75.3982242694076x_{37} = -75.3982242694076
x38=56.5486691471408x_{38} = 56.5486691471408
x39=−40.8407081666179x_{39} = -40.8407081666179
x40=53.4070767521588x_{40} = 53.4070767521588
x41=84.8230020565613x_{41} = 84.8230020565613
x42=31.4159345987753x_{42} = 31.4159345987753
x43=59.6902615937248x_{43} = 59.6902615937248
x44=−9.42507655767126x_{44} = -9.42507655767126
x45=94.2477799063191x_{45} = 94.2477799063191
x46=34.5575252472483x_{46} = 34.5575252472483
x47=87.9645946678103x_{47} = 87.9645946678103
x48=21.9911720820024x_{48} = 21.9911720820024
x49=−12.566496593124x_{49} = -12.566496593124
x50=72.2566316952498x_{50} = 72.2566316952498
x51=12.566496593124x_{51} = 12.566496593124
x52=100.530965160933x_{52} = 100.530965160933
x53=65.9734465960134x_{53} = 65.9734465960134
x54=−28.2743449424922x_{54} = -28.2743449424922
x55=−25.1327569765083x_{55} = -25.1327569765083
x56=−84.8230020565613x_{56} = -84.8230020565613
x57=−53.4070767521588x_{57} = -53.4070767521588
x58=−15.7080277702229x_{58} = -15.7080277702229
x59=78.5398168557694x_{59} = 78.5398168557694
x60=−56.5486691471408x_{60} = -56.5486691471408
x61=−62.8318540796563x_{61} = -62.8318540796563
x62=97.3893725319319x_{62} = 97.3893725319319
x63=25.1327569765083x_{63} = 25.1327569765083
x64=−81.6814094520786x_{64} = -81.6814094520786
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=0x_{1} = 0

lim⁡x→0−(2(tan⁡2(x)+1)tan⁡(x)−12x3)=∞\lim_{x \to 0^-}\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{1}{2 x^{3}}\right) = \infty
lim⁡x→0+(2(tan⁡2(x)+1)tan⁡(x)−12x3)=−∞\lim_{x \to 0^+}\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{1}{2 x^{3}}\right) = -\infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

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
[100.530965160933,∞)\left[100.530965160933, \infty\right)
Convexa en los intervalos
(−∞,−100.530965160933]\left(-\infty, -100.530965160933\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
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)−14x)−1)y = \lim_{x \to -\infty}\left(\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞((tan⁡(x)−14x)−1)y = \lim_{x \to \infty}\left(\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(x) - 1/(4*x) - 1, 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)−14x)−1x)y = x \lim_{x \to -\infty}\left(\frac{\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((tan⁡(x)−14x)−1x)y = x \lim_{x \to \infty}\left(\frac{\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1}{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)−14x)−1=−tan⁡(x)−1+14x\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1 = - \tan{\left(x \right)} - 1 + \frac{1}{4 x}
- No
(tan⁡(x)−14x)−1=tan⁡(x)+1−14x\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1 = \tan{\left(x \right)} + 1 - \frac{1}{4 x}
- No
es decir, función
no es
par ni impar
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