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tan(x^2+e)^(3)
Trazado el gráfico de la función/ tan(x^2+e)^(3)

Gráfico de la función y = tan(x^2+e)^(3)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          3/ 2    \
f(x) = tan \x  + E/
f(x)=tan⁡3(x2+e)f{\left(x \right)} = \tan^{3}{\left(x^{2} + e \right)} 
f = tan(x^2 + E)^3
Gráfico de la función
02468-8-6-4-2-1010-500000500000
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⁡3(x2+e)=0\tan^{3}{\left(x^{2} + e \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−39.638613912455x_{1} = -39.638613912455
x2=83.9839200616833x_{2} = 83.9839200616833
x3=−44.4223306779322x_{3} = -44.4223306779322
x4=25.9974181054171x_{4} = 25.9974181054171
x5=−48.7387671818245x_{5} = -48.7387671818245
x6=64.2532184559972x_{6} = 64.2532184559972
x7=−27.7509188599651x_{7} = -27.7509188599651
x8=−65.7514619706497x_{8} = -65.7514619706497
x9=−72.0003553564783x_{9} = -72.0003553564783
x10=72.0003553564564x_{10} = 72.0003553564564
x11=24.2466738682761x_{11} = 24.2466738682761
x12=−95.7475366457913x_{12} = -95.7475366457913
x13=−75.7220563290132x_{13} = -75.7220563290132
x14=70.255714079848x_{14} = 70.255714079848
x15=16.2578295361657x_{15} = 16.2578295361657
x16=−13.7447773598198x_{16} = -13.7447773598198
x17=−18.0002042093335x_{17} = -18.0002042093335
x18=−45.7463015774115x_{18} = -45.7463015774115
x19=−99.9691431847019x_{19} = -99.9691431847019
x20=−22.0051925046565x_{20} = -22.0051925046565
x21=91.9993151720193x_{21} = 91.9993151720193
x22=−91.9993150234991x_{22} = -91.9993150234991
x23=22.0051925229838x_{23} = 22.0051925229838
x24=42.2474614545484x_{24} = 42.2474614545484
x25=28.2557560894112x_{25} = 28.2557560894112
x26=−73.746322642561x_{26} = -73.746322642561
x27=66.2512505794805x_{27} = 66.2512505794805
x28=−7.75329336151626x_{28} = -7.75329336151626
x29=−79.9990476079842x_{29} = -79.9990476079842
x30=78.2520753810225x_{30} = 78.2520753810225
x31=−77.7486182135598x_{31} = -77.7486182135598
x32=0.650608896182194x_{32} = 0.650608896182194
x33=53.9983460942051x_{33} = 53.9983460942051
x34=18.0002042087624x_{34} = 18.0002042087624
x35=58.2524104672049x_{35} = 58.2524104672049
x36=4.01636999743769x_{36} = 4.01636999743769
x37=−85.9984449022991x_{37} = -85.9984449022991
x38=98.0013875080094x_{38} = 98.0013875080094
x39=−4.01636996167528x_{39} = -4.01636996167528
x40=86.2901955543356x_{40} = 86.2901955543356
x41=−67.7516795752359x_{41} = -67.7516795752359
x42=−19.7479319966854x_{42} = -19.7479319966854
x43=−29.9819753853773x_{43} = -29.9819753853773
x44=32.2535331815425x_{44} = 32.2535331815425
x45=−53.9983458691648x_{45} = -53.9983458691648
x46=−1.88810131330552x_{46} = -1.88810131330552
x47=56.2495078859045x_{47} = 56.2495078859045
x48=−49.6646094899362x_{48} = -49.6646094899362
x49=−41.7989097836137x_{49} = -41.7989097836137
x50=60.2669426199727x_{50} = 60.2669426199727
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^2 + E)^3.
tan⁡3(02+e)\tan^{3}{\left(0^{2} + e \right)}
Resultado:
f(0)=tan⁡3(e)f{\left(0 \right)} = \tan^{3}{\left(e \right)}
Punto:
(0, tan(E)^3)
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
6x(tan⁡2(x2+e)+1)tan⁡2(x2+e)=06 x \left(\tan^{2}{\left(x^{2} + e \right)} + 1\right) \tan^{2}{\left(x^{2} + e \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=−−ilog⁡(−e−ei)x_{2} = - \sqrt{- i \log{\left(- e^{- e i} \right)}}
x3=−ilog⁡(−e−ei)x_{3} = \sqrt{- i \log{\left(- e^{- e i} \right)}}
Signos de extremos en los puntos:
       3    
(0, tan (E))

     ________________                          
    /       /  -E*I\      3/         /  -E*I\\ 
(-\/  -I*log\-e    /, tan \E - I*log\-e    //)

    ________________                          
   /       /  -E*I\      3/         /  -E*I\\ 
(\/  -I*log\-e    /, tan \E - I*log\-e    //)


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=0x_{1} = 0
La función no tiene puntos máximos
Decrece en los intervalos
[0,∞)\left[0, \infty\right)
Crece en los intervalos
(−∞,0]\left(-\infty, 0\right]
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
6(tan⁡2(x2+e)+1)(4x2(tan⁡2(x2+e)+1)+4x2tan⁡2(x2+e)+tan⁡(x2+e))tan⁡(x2+e)=06 \left(\tan^{2}{\left(x^{2} + e \right)} + 1\right) \left(4 x^{2} \left(\tan^{2}{\left(x^{2} + e \right)} + 1\right) + 4 x^{2} \tan^{2}{\left(x^{2} + e \right)} + \tan{\left(x^{2} + e \right)}\right) \tan{\left(x^{2} + e \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−75.7427983129756x_{1} = -75.7427983129756
x2=−34.0077845109936x_{2} = -34.0077845109936
x3=−84.0026214518797x_{3} = -84.0026214518797
x4=−74.086339338953x_{4} = -74.086339338953
x5=−29.981974747368x_{5} = -29.981974747368
x6=24.2466727005258x_{6} = 24.2466727005258
x7=−39.7573190190284x_{7} = -39.7573190190284
x8=−15.7673437984565x_{8} = -15.7673437984565
x9=89.996939791193x_{9} = 89.996939791193
x10=94.3094486949099x_{10} = 94.3094486949099
x11=42.2474607291862x_{11} = 42.2474607291862
x12=30.2427970435897x_{12} = 30.2427970435897
x13=52.2538737970178x_{13} = 52.2538737970178
x14=49.8225000091431x_{14} = 49.8225000091431
x15=−19.7479315339674x_{15} = -19.7479315339674
x16=0.268698338105772x_{16} = 0.268698338105772
x17=−93.9924563607082x_{17} = -93.9924563607082
x18=−61.7601553777445x_{18} = -61.7601553777445
x19=19.9851339207681x_{19} = 19.9851339207681
x20=−27.750919101079x_{20} = -27.750919101079
x21=46.2584901171642x_{21} = 46.2584901171642
x22=−53.9983452835034x_{22} = -53.9983452835034
x23=−43.8887184533301x_{23} = -43.8887184533301
x24=98.0013880149109x_{24} = 98.0013880149109
x25=92.0163871940733x_{25} = 92.0163871940733
x26=66.2512509156625x_{26} = 66.2512509156625
x27=60.0057371577005x_{27} = 60.0057371577005
x28=−79.9990471579977x_{28} = -79.9990471579977
x29=70.2557141420874x_{29} = 70.2557141420874
x30=100.173201238609x_{30} = 100.173201238609
x31=−35.7638297842412x_{31} = -35.7638297842412
x32=4.01637574102321x_{32} = 4.01637574102321
x33=−4.01637574102321x_{33} = -4.01637574102321
x34=32.3021991792655x_{34} = 32.3021991792655
x35=82.2451961758494x_{35} = 82.2451961758494
x36=78.2520759639745x_{36} = 78.2520759639745
x37=56.2495076688681x_{37} = 56.2495076688681
x38=64.2532182668084x_{38} = 64.2532182668084
x39=6.17433580785887x_{39} = 6.17433580785887
x40=7.95331150508558x_{40} = 7.95331150508558
x41=−56.0536882320416x_{41} = -56.0536882320416
x42=48.2529114886881x_{42} = 48.2529114886881
x43=−23.9861369635059x_{43} = -23.9861369635059
x44=−99.79615234419x_{44} = -99.79615234419
x45=−18.000204280643x_{45} = -18.000204280643
x46=34.2379521965629x_{46} = 34.2379521965629
x47=36.2437448399812x_{47} = 36.2437448399812
x48=−77.7486178419391x_{48} = -77.7486178419391
x49=−58.0092151543166x_{49} = -58.0092151543166
x50=−65.7514623576137x_{50} = -65.7514623576137
x51=−7.75329421880383x_{51} = -7.75329421880383
x52=80.0186799368818x_{52} = 80.0186799368818
x53=62.2415219343273x_{53} = 62.2415219343273
x54=−25.7545988268615x_{54} = -25.7545988268615
x55=−22.0051943749188x_{55} = -22.0051943749188
x56=53.9983452835034x_{56} = 53.9983452835034
x57=−31.7627856559757x_{57} = -31.7627856559757
x58=−69.9869006155553x_{58} = -69.9869006155553
x59=16.2578317658498x_{59} = 16.2578317658498
x60=74.2557635330431x_{60} = 74.2557635330431
x61=−95.7475361249581x_{61} = -95.7475361249581
x62=−51.7403155352152x_{62} = -51.7403155352152
x63=76.0119214385267x_{63} = 76.0119214385267
x64=−47.9917776906258x_{64} = -47.9917776906258
x65=−45.746300594867x_{65} = -45.746300594867
x66=10.2027382791873x_{66} = 10.2027382791873
x67=39.9936741435735x_{67} = 39.9936741435735
x68=−9.72990701049216x_{68} = -9.72990701049216
x69=1.88809519853225x_{69} = 1.88809519853225
x70=13.9714729126065x_{70} = 13.9714729126065
x71=18.000204280643x_{71} = 18.000204280643
x72=68.0062327798659x_{72} = 68.0062327798659
x73=43.8529134131597x_{73} = 43.8529134131597
x74=22.0051943749188x_{74} = 22.0051943749188
x75=−72.0003554800134x_{75} = -72.0003554800134
x76=28.2557577289468x_{76} = 28.2557577289468
x77=−89.752252755178x_{77} = -89.752252755178
x78=−67.7516802575391x_{78} = -67.7516802575391
x79=−85.9984442729955x_{79} = -85.9984442729955
x80=−41.7613126418156x_{80} = -41.7613126418156
x81=58.2524109089238x_{81} = 58.2524109089238
x82=72.0003554800134x_{82} = 72.0003554800134
x83=−64.0082817384299x_{83} = -64.0082817384299
x84=38.2675513640209x_{84} = 38.2675513640209
x85=−13.7447761000505x_{85} = -13.7447761000505
x86=−98.0013880149109x_{86} = -98.0013880149109
x87=−81.381205598052x_{87} = -81.381205598052
x88=−11.7751173065529x_{88} = -11.7751173065529
x89=−87.7523042704504x_{89} = -87.7523042704504
x90=12.2971443106699x_{90} = 12.2971443106699
x91=96.254766356069x_{91} = 96.254766356069
x92=−5.64262681390757x_{92} = -5.64262681390757
x93=25.9974177822902x_{93} = 25.9974177822902
x94=−91.9993147778614x_{94} = -91.9993147778614
x95=88.7313614271817x_{95} = 88.7313614271817
x96=−49.727826507905x_{96} = -49.727826507905
x97=84.2453625743546x_{97} = 84.2453625743546
x98=−59.7433893021072x_{98} = -59.7433893021072
x99=−1.88809519853225x_{99} = -1.88809519853225
x100=86.2537808709031x_{100} = 86.2537808709031

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
[100.173201238609,∞)\left[100.173201238609, \infty\right)
Convexa en los intervalos
[0.268698338105772,1.88809519853225]\left[0.268698338105772, 1.88809519853225\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⁡3(x2+e)y = \lim_{x \to -\infty} \tan^{3}{\left(x^{2} + e \right)}
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞tan⁡3(x2+e)y = \lim_{x \to \infty} \tan^{3}{\left(x^{2} + e \right)}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(x^2 + E)^3, 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⁡3(x2+e)x)y = x \lim_{x \to -\infty}\left(\frac{\tan^{3}{\left(x^{2} + e \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(tan⁡3(x2+e)x)y = x \lim_{x \to \infty}\left(\frac{\tan^{3}{\left(x^{2} + e \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⁡3(x2+e)=tan⁡3(x2+e)\tan^{3}{\left(x^{2} + e \right)} = \tan^{3}{\left(x^{2} + e \right)}
- Sí
tan⁡3(x2+e)=−tan⁡3(x2+e)\tan^{3}{\left(x^{2} + e \right)} = - \tan^{3}{\left(x^{2} + e \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = tan(x^2+e)^(3)
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