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Gráfico de la función y = tan(sqrt(exp(x+2)))/sin(((7*x)/4)-(1/1000))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src]
          /   ________\
          |  /  x + 2 |
       tan\\/  e      /
f(x) = ----------------
          /7*x    1  \ 
       sin|--- - ----| 
          \ 4    1000/ 
$$f{\left(x \right)} = \frac{\tan{\left(\sqrt{e^{x + 2}} \right)}}{\sin{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}}$$
f = tan(sqrt(exp(x + 2)))/sin((7*x)/4 - 1/1000)
Gráfico de la función
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
$$x_{1} = 0.000571428571428571$$
$$x_{2} = 1.79576723062274$$
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:
$$\frac{\tan{\left(\sqrt{e^{x + 2}} \right)}}{\sin{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}} = 0$$
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
$$x_{1} = -689.647935743731$$
$$x_{2} = 38.2186529115769$$
$$x_{3} = -97.3767734998073$$
$$x_{4} = -79.574832601517$$
$$x_{5} = 4.18128006980943$$
$$x_{6} = 7.98975497511892$$
$$x_{7} = -188.146744907196$$
$$x_{8} = -149.020536692293$$
$$x_{9} = 22.3709467012917$$
$$x_{10} = -87.6690346846249$$
$$x_{11} = -86.8047897798513$$
$$x_{12} = 26.0361455466237$$
$$x_{13} = -96.3818983398437$$
$$x_{14} = -87.3103780775632$$
$$x_{15} = -146.942658621189$$
$$x_{16} = -73.4063204564011$$
$$x_{17} = -123.55484047147$$
$$x_{18} = -65.2188422418512$$
$$x_{19} = 2.48668434903502$$
$$x_{20} = 20.2456291531846$$
$$x_{21} = -66.6387283863113$$
$$x_{22} = 13.8538438837124$$
$$x_{23} = -76.0628029826387$$
$$x_{24} = 40.460018911466$$
$$x_{25} = -84.9951044538107$$
$$x_{26} = -91.9167961947003$$
$$x_{27} = -75.2781061685407$$
$$x_{28} = 18.2356555642637$$
$$x_{29} = -175.426059018885$$
$$x_{30} = -86.7831771219654$$
$$x_{31} = 30.1117740114503$$
$$x_{32} = -100.901041369809$$
$$x_{33} = 15.6143954020993$$
$$x_{34} = -103.419473214707$$
$$x_{35} = -121.248098219823$$
$$x_{36} = 44.2452539636124$$
$$x_{37} = -107.820918186212$$
$$x_{38} = -79.8436194528914$$
$$x_{39} = 24.2389511745147$$
$$x_{40} = -79.8840409881522$$
$$x_{41} = 12.0053260806657$$
$$x_{42} = 6.17833773003168$$
$$x_{43} = -81.2327533684886$$
$$x_{44} = -128.426733929944$$
$$x_{45} = 10.0398544181011$$
$$x_{46} = -96.6965883006325$$
$$x_{47} = 35.9734038227516$$
$$x_{48} = -80.2860247663765$$
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(sqrt(exp(x + 2)))/sin((7*x)/4 - 1/1000).
$$\frac{\tan{\left(\sqrt{e^{2}} \right)}}{\sin{\left(- \frac{1}{1000} + \frac{0 \cdot 7}{4} \right)}}$$
Resultado:
$$f{\left(0 \right)} = - \frac{\tan{\left(e \right)}}{\sin{\left(\frac{1}{1000} \right)}}$$
Punto:
(0, -tan(E)/sin(1/1000))
Extremos de la función
Para hallar los extremos hay que resolver la ecuación
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
$$\frac{d}{d x} f{\left(x \right)} = $$
primera derivada
$$\frac{\left(\tan^{2}{\left(\sqrt{e^{x + 2}} \right)} + 1\right) e^{- x - 2} e^{\frac{x}{2} + 1} e^{x + 2}}{2 \sin{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}} - \frac{7 \cos{\left(\frac{7 x}{4} - \frac{1}{1000} \right)} \tan{\left(\sqrt{e^{x + 2}} \right)}}{4 \sin^{2}{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}} = 0$$
Resolvermos esta ecuación
Raíces de esta ecuación
$$x_{1} = -65.6831037228758$$
$$x_{2} = -81.8398659413375$$
$$x_{3} = -0.370825100569674$$
$$x_{4} = -42.3455582962087$$
$$x_{5} = -56.7071247126192$$
$$x_{6} = -8.23703501208389$$
$$x_{7} = -15.4176214152433$$
$$x_{8} = -27.9839918797988$$
$$x_{9} = -22.5984044737583$$
$$x_{10} = -96.201432357748$$
$$x_{11} = -49.526341504414$$
$$x_{12} = -74.6590827331323$$
$$x_{13} = -99.7918239618507$$
$$x_{14} = -35.1647750880035$$
$$x_{15} = -31.5743834839009$$
$$x_{16} = -69.2734953269784$$
$$x_{17} = -72.863886931081$$
$$x_{18} = -11.8272350915717$$
$$x_{19} = -83.6350617433889$$
$$x_{20} = -80.0446701392862$$
$$x_{21} = -85.4302575454402$$
$$x_{22} = -4.65374211427633$$
$$x_{23} = -2.90016061870246$$
$$x_{24} = -94.4062365556967$$
$$x_{25} = -44.14075409826$$
$$x_{26} = -47.7311457023626$$
$$x_{27} = -97.9966281597993$$
$$x_{28} = -13.6224263653095$$
$$x_{29} = -90.8158449515941$$
$$x_{30} = -51.3215373064653$$
$$x_{31} = -62.0927121187731$$
$$x_{32} = -67.4782995249271$$
$$x_{33} = -6.44283173147076$$
$$x_{34} = -17.2128170923721$$
$$x_{35} = -78.2494743372349$$
$$x_{36} = -29.7791876818496$$
$$x_{37} = -24.3936002757145$$
$$x_{38} = -63.8879079208244$$
$$x_{39} = -38.7551666921061$$
$$x_{40} = -10.0320665556463$$
$$x_{41} = -76.4542785351836$$
$$x_{42} = -92.6110407536454$$
$$x_{43} = -60.2975163167218$$
$$x_{44} = -40.5503624941574$$
$$x_{45} = -26.1887960777501$$
$$x_{46} = -87.2254533474915$$
$$x_{47} = -58.5023205146705$$
$$x_{48} = -20.8032086722794$$
$$x_{49} = -45.9359499003113$$
$$x_{50} = -33.3695792859522$$
$$x_{51} = -54.9119289105679$$
Signos de extremos en los puntos:
(-65.68310372287576, -1.5432263358434e-14)

(-81.83986594133755, 4.78666779628583e-18)

(-0.37082510056967405, 2.01266358109232)

(-42.345558296208715, 1.80348787186641e-9)

(-56.7071247126192, 1.37258327613389e-12)

(-8.237035012083892, -0.0460268214591987)

(-15.417621415243326, -0.00126893878679156)

(-27.983991879798754, 2.36966933848724e-6)

(-22.59840447375831, -3.50068630440781e-5)

(-96.20143235774803, 3.64299658904347e-21)

(-49.526341504413956, 4.97537666075055e-11)

(-74.65908273313231, 1.7350841767129e-16)

(-99.79182396185065, 6.05083254103891e-22)

(-35.16477508800347, 6.53733119268715e-8)

(-31.574383483900867, 3.9359005682082e-7)

(-69.27349532697838, -2.56322066268017e-15)

(-72.863886931081, -4.25737949968296e-16)

(-11.82723509157167, -0.00763997433255879)

(-83.63506174338886, -1.95079427453795e-18)

(-80.04467013928624, -1.17450563040157e-17)

(-85.43025754544017, 7.9504124028055e-19)

(-4.653742114276328, -0.283641696748437)

(-2.9001606187024604, 0.792690797093598)

(-94.40623655569672, -8.93882798527462e-21)

(-44.140754098260025, -7.35006890882542e-10)

(-47.731145702362646, -1.22080916205515e-10)

(-97.99662815979934, -1.48469398557002e-21)

(-13.622426365309462, 0.00311360538532328)

(-90.8158449515941, -5.38175856620394e-20)

(-51.32153730646527, -2.02770209183792e-11)

(-62.09271211877313, -9.29123098262817e-14)

(-67.47829952492707, 6.28937964446894e-15)

(-6.442831731470764, 0.113307341311121)

(-17.212817092372088, 0.00051715255810779)

(-78.24947433723493, 2.88188680425113e-17)

(-29.77918768184963, -9.65752706224852e-7)

(-24.393600275714547, 1.42669579090917e-5)

(-63.88790792082444, 3.7866175334718e-14)

(-38.755166692106094, 1.08581755006931e-8)

(-10.032066555646251, 0.0187479133661977)

(-76.45427853518362, -7.07129139063991e-17)

(-92.61104075364541, 2.19332200284352e-20)

(-60.29751631672182, 2.27979119648242e-13)

(-40.550362494157405, -4.42522178270163e-9)

(-26.188796077750066, -5.81446237490517e-6)

(-87.22545334749148, -3.24017033470406e-19)

(-58.50232051467051, -5.59392819883223e-13)

(-20.803208672279414, 8.58964097275147e-5)

(-45.935949900311336, 2.99550187207933e-10)

(-33.36957928595216, -1.6040662566073e-7)

(-54.91192891056789, -3.36791031804043e-12)


Intervalos de crecimiento y decrecimiento de la función:
Hallemos los intervalos donde la función crece y decrece y también los puntos mínimos y máximos de la función, para lo cual miramos cómo se comporta la función en los extremos con desviación mínima del extremo:
Puntos mínimos de la función:
$$x_{1} = -81.8398659413375$$
$$x_{2} = -0.370825100569674$$
$$x_{3} = -42.3455582962087$$
$$x_{4} = -56.7071247126192$$
$$x_{5} = -27.9839918797988$$
$$x_{6} = -96.201432357748$$
$$x_{7} = -49.526341504414$$
$$x_{8} = -74.6590827331323$$
$$x_{9} = -99.7918239618507$$
$$x_{10} = -35.1647750880035$$
$$x_{11} = -31.5743834839009$$
$$x_{12} = -85.4302575454402$$
$$x_{13} = -2.90016061870246$$
$$x_{14} = -13.6224263653095$$
$$x_{15} = -67.4782995249271$$
$$x_{16} = -6.44283173147076$$
$$x_{17} = -17.2128170923721$$
$$x_{18} = -78.2494743372349$$
$$x_{19} = -24.3936002757145$$
$$x_{20} = -63.8879079208244$$
$$x_{21} = -38.7551666921061$$
$$x_{22} = -10.0320665556463$$
$$x_{23} = -92.6110407536454$$
$$x_{24} = -60.2975163167218$$
$$x_{25} = -20.8032086722794$$
$$x_{26} = -45.9359499003113$$
Puntos máximos de la función:
$$x_{26} = -65.6831037228758$$
$$x_{26} = -8.23703501208389$$
$$x_{26} = -15.4176214152433$$
$$x_{26} = -22.5984044737583$$
$$x_{26} = -69.2734953269784$$
$$x_{26} = -72.863886931081$$
$$x_{26} = -11.8272350915717$$
$$x_{26} = -83.6350617433889$$
$$x_{26} = -80.0446701392862$$
$$x_{26} = -4.65374211427633$$
$$x_{26} = -94.4062365556967$$
$$x_{26} = -44.14075409826$$
$$x_{26} = -47.7311457023626$$
$$x_{26} = -97.9966281597993$$
$$x_{26} = -90.8158449515941$$
$$x_{26} = -51.3215373064653$$
$$x_{26} = -62.0927121187731$$
$$x_{26} = -29.7791876818496$$
$$x_{26} = -76.4542785351836$$
$$x_{26} = -40.5503624941574$$
$$x_{26} = -26.1887960777501$$
$$x_{26} = -87.2254533474915$$
$$x_{26} = -58.5023205146705$$
$$x_{26} = -33.3695792859522$$
$$x_{26} = -54.9119289105679$$
Decrece en los intervalos
$$\left[-0.370825100569674, \infty\right)$$
Crece en los intervalos
$$\left(-\infty, -99.7918239618507\right]$$
Asíntotas verticales
Hay:
$$x_{1} = 0.000571428571428571$$
$$x_{2} = 1.79576723062274$$
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(\sqrt{e^{x + 2}} \right)}}{\sin{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}}\right) = 0$$
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
$$y = 0$$
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
$$y = \lim_{x \to \infty}\left(\frac{\tan{\left(\sqrt{e^{x + 2}} \right)}}{\sin{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}}\right)$$
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(sqrt(exp(x + 2)))/sin((7*x)/4 - 1/1000), dividida por x con x->+oo y x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\tan{\left(\sqrt{e^{x + 2}} \right)}}{x \sin{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}}\right) = 0$$
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\sqrt{e^{x + 2}} \right)}}{x \sin{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}}\right)$$
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:
$$\frac{\tan{\left(\sqrt{e^{x + 2}} \right)}}{\sin{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}} = - \frac{\tan{\left(e^{1 - \frac{x}{2}} \right)}}{\sin{\left(\frac{7 x}{4} + \frac{1}{1000} \right)}}$$
- No
$$\frac{\tan{\left(\sqrt{e^{x + 2}} \right)}}{\sin{\left(\frac{7 x}{4} - \frac{1}{1000} \right)}} = \frac{\tan{\left(e^{1 - \frac{x}{2}} \right)}}{\sin{\left(\frac{7 x}{4} + \frac{1}{1000} \right)}}$$
- No
es decir, función
no es
par ni impar