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Trazado el gráfico de la función/ tan(sqrt(exp(x+2)))/sin(((7*x)/4)-(1/1000))

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(x)=tan⁡(ex+2)sin⁡(7x4−11000)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
02468-8-6-4-2-1010-10001000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0.000571428571428571x_{1} = 0.000571428571428571
x2=1.79576723062274x_{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:
tan⁡(ex+2)sin⁡(7x4−11000)=0\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
x1=−689.647935743731x_{1} = -689.647935743731
x2=38.2186529115769x_{2} = 38.2186529115769
x3=−97.3767734998073x_{3} = -97.3767734998073
x4=−79.574832601517x_{4} = -79.574832601517
x5=4.18128006980943x_{5} = 4.18128006980943
x6=7.98975497511892x_{6} = 7.98975497511892
x7=−188.146744907196x_{7} = -188.146744907196
x8=−149.020536692293x_{8} = -149.020536692293
x9=22.3709467012917x_{9} = 22.3709467012917
x10=−87.6690346846249x_{10} = -87.6690346846249
x11=−86.8047897798513x_{11} = -86.8047897798513
x12=26.0361455466237x_{12} = 26.0361455466237
x13=−96.3818983398437x_{13} = -96.3818983398437
x14=−87.3103780775632x_{14} = -87.3103780775632
x15=−146.942658621189x_{15} = -146.942658621189
x16=−73.4063204564011x_{16} = -73.4063204564011
x17=−123.55484047147x_{17} = -123.55484047147
x18=−65.2188422418512x_{18} = -65.2188422418512
x19=2.48668434903502x_{19} = 2.48668434903502
x20=20.2456291531846x_{20} = 20.2456291531846
x21=−66.6387283863113x_{21} = -66.6387283863113
x22=13.8538438837124x_{22} = 13.8538438837124
x23=−76.0628029826387x_{23} = -76.0628029826387
x24=40.460018911466x_{24} = 40.460018911466
x25=−84.9951044538107x_{25} = -84.9951044538107
x26=−91.9167961947003x_{26} = -91.9167961947003
x27=−75.2781061685407x_{27} = -75.2781061685407
x28=18.2356555642637x_{28} = 18.2356555642637
x29=−175.426059018885x_{29} = -175.426059018885
x30=−86.7831771219654x_{30} = -86.7831771219654
x31=30.1117740114503x_{31} = 30.1117740114503
x32=−100.901041369809x_{32} = -100.901041369809
x33=15.6143954020993x_{33} = 15.6143954020993
x34=−103.419473214707x_{34} = -103.419473214707
x35=−121.248098219823x_{35} = -121.248098219823
x36=44.2452539636124x_{36} = 44.2452539636124
x37=−107.820918186212x_{37} = -107.820918186212
x38=−79.8436194528914x_{38} = -79.8436194528914
x39=24.2389511745147x_{39} = 24.2389511745147
x40=−79.8840409881522x_{40} = -79.8840409881522
x41=12.0053260806657x_{41} = 12.0053260806657
x42=6.17833773003168x_{42} = 6.17833773003168
x43=−81.2327533684886x_{43} = -81.2327533684886
x44=−128.426733929944x_{44} = -128.426733929944
x45=10.0398544181011x_{45} = 10.0398544181011
x46=−96.6965883006325x_{46} = -96.6965883006325
x47=35.9734038227516x_{47} = 35.9734038227516
x48=−80.2860247663765x_{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).
tan⁡(e2)sin⁡(−11000+0⋅74)\frac{\tan{\left(\sqrt{e^{2}} \right)}}{\sin{\left(- \frac{1}{1000} + \frac{0 \cdot 7}{4} \right)}}
Resultado:
f(0)=−tan⁡(e)sin⁡(11000)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
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(ex+2)+1)e−x−2ex2+1ex+22sin⁡(7x4−11000)−7cos⁡(7x4−11000)tan⁡(ex+2)4sin⁡2(7x4−11000)=0\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
x1=−65.6831037228758x_{1} = -65.6831037228758
x2=−81.8398659413375x_{2} = -81.8398659413375
x3=−0.370825100569674x_{3} = -0.370825100569674
x4=−42.3455582962087x_{4} = -42.3455582962087
x5=−56.7071247126192x_{5} = -56.7071247126192
x6=−8.23703501208389x_{6} = -8.23703501208389
x7=−15.4176214152433x_{7} = -15.4176214152433
x8=−27.9839918797988x_{8} = -27.9839918797988
x9=−22.5984044737583x_{9} = -22.5984044737583
x10=−96.201432357748x_{10} = -96.201432357748
x11=−49.526341504414x_{11} = -49.526341504414
x12=−74.6590827331323x_{12} = -74.6590827331323
x13=−99.7918239618507x_{13} = -99.7918239618507
x14=−35.1647750880035x_{14} = -35.1647750880035
x15=−31.5743834839009x_{15} = -31.5743834839009
x16=−69.2734953269784x_{16} = -69.2734953269784
x17=−72.863886931081x_{17} = -72.863886931081
x18=−11.8272350915717x_{18} = -11.8272350915717
x19=−83.6350617433889x_{19} = -83.6350617433889
x20=−80.0446701392862x_{20} = -80.0446701392862
x21=−85.4302575454402x_{21} = -85.4302575454402
x22=−4.65374211427633x_{22} = -4.65374211427633
x23=−2.90016061870246x_{23} = -2.90016061870246
x24=−94.4062365556967x_{24} = -94.4062365556967
x25=−44.14075409826x_{25} = -44.14075409826
x26=−47.7311457023626x_{26} = -47.7311457023626
x27=−97.9966281597993x_{27} = -97.9966281597993
x28=−13.6224263653095x_{28} = -13.6224263653095
x29=−90.8158449515941x_{29} = -90.8158449515941
x30=−51.3215373064653x_{30} = -51.3215373064653
x31=−62.0927121187731x_{31} = -62.0927121187731
x32=−67.4782995249271x_{32} = -67.4782995249271
x33=−6.44283173147076x_{33} = -6.44283173147076
x34=−17.2128170923721x_{34} = -17.2128170923721
x35=−78.2494743372349x_{35} = -78.2494743372349
x36=−29.7791876818496x_{36} = -29.7791876818496
x37=−24.3936002757145x_{37} = -24.3936002757145
x38=−63.8879079208244x_{38} = -63.8879079208244
x39=−38.7551666921061x_{39} = -38.7551666921061
x40=−10.0320665556463x_{40} = -10.0320665556463
x41=−76.4542785351836x_{41} = -76.4542785351836
x42=−92.6110407536454x_{42} = -92.6110407536454
x43=−60.2975163167218x_{43} = -60.2975163167218
x44=−40.5503624941574x_{44} = -40.5503624941574
x45=−26.1887960777501x_{45} = -26.1887960777501
x46=−87.2254533474915x_{46} = -87.2254533474915
x47=−58.5023205146705x_{47} = -58.5023205146705
x48=−20.8032086722794x_{48} = -20.8032086722794
x49=−45.9359499003113x_{49} = -45.9359499003113
x50=−33.3695792859522x_{50} = -33.3695792859522
x51=−54.9119289105679x_{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:
x1=−81.8398659413375x_{1} = -81.8398659413375
x2=−0.370825100569674x_{2} = -0.370825100569674
x3=−42.3455582962087x_{3} = -42.3455582962087
x4=−56.7071247126192x_{4} = -56.7071247126192
x5=−27.9839918797988x_{5} = -27.9839918797988
x6=−96.201432357748x_{6} = -96.201432357748
x7=−49.526341504414x_{7} = -49.526341504414
x8=−74.6590827331323x_{8} = -74.6590827331323
x9=−99.7918239618507x_{9} = -99.7918239618507
x10=−35.1647750880035x_{10} = -35.1647750880035
x11=−31.5743834839009x_{11} = -31.5743834839009
x12=−85.4302575454402x_{12} = -85.4302575454402
x13=−2.90016061870246x_{13} = -2.90016061870246
x14=−13.6224263653095x_{14} = -13.6224263653095
x15=−67.4782995249271x_{15} = -67.4782995249271
x16=−6.44283173147076x_{16} = -6.44283173147076
x17=−17.2128170923721x_{17} = -17.2128170923721
x18=−78.2494743372349x_{18} = -78.2494743372349
x19=−24.3936002757145x_{19} = -24.3936002757145
x20=−63.8879079208244x_{20} = -63.8879079208244
x21=−38.7551666921061x_{21} = -38.7551666921061
x22=−10.0320665556463x_{22} = -10.0320665556463
x23=−92.6110407536454x_{23} = -92.6110407536454
x24=−60.2975163167218x_{24} = -60.2975163167218
x25=−20.8032086722794x_{25} = -20.8032086722794
x26=−45.9359499003113x_{26} = -45.9359499003113
Puntos máximos de la función:
x26=−65.6831037228758x_{26} = -65.6831037228758
x26=−8.23703501208389x_{26} = -8.23703501208389
x26=−15.4176214152433x_{26} = -15.4176214152433
x26=−22.5984044737583x_{26} = -22.5984044737583
x26=−69.2734953269784x_{26} = -69.2734953269784
x26=−72.863886931081x_{26} = -72.863886931081
x26=−11.8272350915717x_{26} = -11.8272350915717
x26=−83.6350617433889x_{26} = -83.6350617433889
x26=−80.0446701392862x_{26} = -80.0446701392862
x26=−4.65374211427633x_{26} = -4.65374211427633
x26=−94.4062365556967x_{26} = -94.4062365556967
x26=−44.14075409826x_{26} = -44.14075409826
x26=−47.7311457023626x_{26} = -47.7311457023626
x26=−97.9966281597993x_{26} = -97.9966281597993
x26=−90.8158449515941x_{26} = -90.8158449515941
x26=−51.3215373064653x_{26} = -51.3215373064653
x26=−62.0927121187731x_{26} = -62.0927121187731
x26=−29.7791876818496x_{26} = -29.7791876818496
x26=−76.4542785351836x_{26} = -76.4542785351836
x26=−40.5503624941574x_{26} = -40.5503624941574
x26=−26.1887960777501x_{26} = -26.1887960777501
x26=−87.2254533474915x_{26} = -87.2254533474915
x26=−58.5023205146705x_{26} = -58.5023205146705
x26=−33.3695792859522x_{26} = -33.3695792859522
x26=−54.9119289105679x_{26} = -54.9119289105679
Decrece en los intervalos
[−0.370825100569674,∞)\left[-0.370825100569674, \infty\right)
Crece en los intervalos
(−∞,−99.7918239618507]\left(-\infty, -99.7918239618507\right]
Asíntotas verticales
Hay:
x1=0.000571428571428571x_{1} = 0.000571428571428571
x2=1.79576723062274x_{2} = 1.79576723062274
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(tan⁡(ex+2)sin⁡(7x4−11000))=0\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=0y = 0
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(ex+2)sin⁡(7x4−11000))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→−∞(tan⁡(ex+2)xsin⁡(7x4−11000))=0\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=xlim⁡x→∞(tan⁡(ex+2)xsin⁡(7x4−11000))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:
tan⁡(ex+2)sin⁡(7x4−11000)=−tan⁡(e1−x2)sin⁡(7x4+11000)\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
tan⁡(ex+2)sin⁡(7x4−11000)=tan⁡(e1−x2)sin⁡(7x4+11000)\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
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