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Trazado el gráfico de la función/ tan(2*(x^3)-5)

Gráfico de la función y = tan(2*(x^3)-5)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          /   3    \
f(x) = tan\2*x  - 5/
f(x)=tan⁡(2x3−5)f{\left(x \right)} = \tan{\left(2 x^{3} - 5 \right)} 
f = tan(2*x^3 - 5)
Gráfico de la función
02468-8-6-4-2-1010-10001000
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⁡(2x3−5)=0\tan{\left(2 x^{3} - 5 \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=223532x_{1} = \frac{2^{\frac{2}{3}} \sqrt[3]{5}}{2}
Solución numérica
x1=76.6482099081789x_{1} = 76.6482099081789
x2=81.934678860784x_{2} = 81.934678860784
x3=−6.11274822093914x_{3} = -6.11274822093914
x4=55.7179107000844x_{4} = 55.7179107000844
x5=−59.7945904792717x_{5} = -59.7945904792717
x6=−97.2725649134396x_{6} = -97.2725649134396
x7=40.0594467910745x_{7} = 40.0594467910745
x8=−67.936504221419x_{8} = -67.936504221419
x9=−99.7672760640937x_{9} = -99.7672760640937
x10=25.9296058535865x_{10} = 25.9296058535865
x11=34.0736189975448x_{11} = 34.0736189975448
x12=8.2968114248492x_{12} = 8.2968114248492
x13=10.2100833163083x_{13} = 10.2100833163083
x14=28.2512003785704x_{14} = 28.2512003785704
x15=36.2752279039579x_{15} = 36.2752279039579
x16=−79.0955293045537x_{16} = -79.0955293045537
x17=−49.5845773073551x_{17} = -49.5845773073551
x18=−13.8365414658988x_{18} = -13.8365414658988
x19=98.3216485037071x_{19} = 98.3216485037071
x20=74.0179698096328x_{20} = 74.0179698096328
x21=−32.2383656106496x_{21} = -32.2383656106496
x22=−21.7921525731502x_{22} = -21.7921525731502
x23=−64.3476455003615x_{23} = -64.3476455003615
x24=−15.7397455053722x_{24} = -15.7397455053722
x25=14.1289257644055x_{25} = 14.1289257644055
x26=−1.74941276993431x_{26} = -1.74941276993431
x27=−53.5600413066043x_{27} = -53.5600413066043
x28=69.4102180268385x_{28} = 69.4102180268385
x29=42.0642253887896x_{29} = 42.0642253887896
x30=−55.959530520481x_{30} = -55.959530520481
x31=29.9673021057691x_{31} = 29.9673021057691
x32=−38.7351182286188x_{32} = -38.7351182286188
x33=58.3392590391614x_{33} = 58.3392590391614
x34=52.0214382998985x_{34} = 52.0214382998985
x35=84.5800990049548x_{35} = 84.5800990049548
x36=−39.7638872129735x_{36} = -39.7638872129735
x37=95.9956910604597x_{37} = 95.9956910604597
x38=−10.0250108539974x_{38} = -10.0250108539974
x39=59.1421404288583x_{39} = 59.1421404288583
x40=80.2060667634553x_{40} = 80.2060667634553
x41=−19.7648336257452x_{41} = -19.7648336257452
x42=4.02755666065725x_{42} = 4.02755666065725
x43=−66.2626484127125x_{43} = -66.2626484127125
x44=−39.7201276567449x_{44} = -39.7201276567449
x45=−35.8512254823998x_{45} = -35.8512254823998
x46=92.2354872027367x_{46} = 92.2354872027367
x47=−74.3590939179168x_{47} = -74.3590939179168
x48=−7.74190071134124x_{48} = -7.74190071134124
x49=48.4090960189642x_{49} = 48.4090960189642
x50=2.28463469115738x_{50} = 2.28463469115738
x51=62.1071869927831x_{51} = 62.1071869927831
x52=68.1394417321101x_{52} = 68.1394417321101
x53=24.3060225589568x_{53} = 24.3060225589568
x54=79.5036201706066x_{54} = 79.5036201706066
x55=−95.9477581380849x_{55} = -95.9477581380849
x56=94.4359112909735x_{56} = 94.4359112909735
x57=−74.1899617033973x_{57} = -74.1899617033973
x58=−3.50483938659206x_{58} = -3.50483938659206
x59=12.2441189337843x_{59} = 12.2441189337843
x60=5.97196838828103x_{60} = 5.97196838828103
x61=−75.8476378307427x_{61} = -75.8476378307427
x62=−51.8205709534001x_{62} = -51.8205709534001
x63=−11.8651525983693x_{63} = -11.8651525983693
x64=22.2754972000303x_{64} = 22.2754972000303
x65=−91.9490118738006x_{65} = -91.9490118738006
x66=86.2471253997886x_{66} = 86.2471253997886
x67=45.4002126268166x_{67} = 45.4002126268166
x68=−71.7527404062393x_{68} = -71.7527404062393
x69=−45.4143872976867x_{69} = -45.4143872976867
x70=−58.203842925346x_{70} = -58.203842925346
x71=−88.1476133902684x_{71} = -88.1476133902684
x72=−17.8083098718863x_{72} = -17.8083098718863
x73=−85.7278240618925x_{73} = -85.7278240618925
x74=32.1535978473001x_{74} = 32.1535978473001
x75=−69.3779049652637x_{75} = -69.3779049652637
x76=49.9859887603547x_{76} = 49.9859887603547
x77=38.110270507257x_{77} = 38.110270507257
x78=−82.5233382353162x_{78} = -82.5233382353162
x79=−99.8705368729717x_{79} = -99.8705368729717
x80=89.9538076502397x_{80} = 89.9538076502397
x81=56.6807467682476x_{81} = 56.6807467682476
x82=44.2377330376607x_{82} = 44.2377330376607
x83=70.8828901912564x_{83} = 70.8828901912564
x84=19.7731175568073x_{84} = 19.7731175568073
x85=100.231252520829x_{85} = 100.231252520829
x86=−34.253430842836x_{86} = -34.253430842836
x87=−1.90605436779673x_{87} = -1.90605436779673
x88=66.0704775425801x_{88} = 66.0704775425801
x89=−23.7508740369822x_{89} = -23.7508740369822
x90=−89.8316549869289x_{90} = -89.8316549869289
x91=−25.8129027136937x_{91} = -25.8129027136937
x92=18.2468048193849x_{92} = 18.2468048193849
x93=−47.8159800718115x_{93} = -47.8159800718115
x94=88.2469772083323x_{94} = 88.2469772083323
x95=−29.6840816155032x_{95} = -29.6840816155032
x96=16.1479293411635x_{96} = 16.1479293411635
x97=64.2359422413301x_{97} = 64.2359422413301
x98=−60.9787187763012x_{98} = -60.9787187763012
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(2*x^3 - 5).
tan⁡(−5+2⋅03)\tan{\left(-5 + 2 \cdot 0^{3} \right)}
Resultado:
f(0)=−tan⁡(5)f{\left(0 \right)} = - \tan{\left(5 \right)}
Punto:
(0, -tan(5))
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
6x2(tan⁡2(2x3−5)+1)=06 x^{2} \left(\tan^{2}{\left(2 x^{3} - 5 \right)} + 1\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
Signos de extremos en los puntos:
(0, -tan(5))


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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
No cambia el valor en todo el eje numérico
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
12x(6x3tan⁡(2x3−5)+1)(tan⁡2(2x3−5)+1)=012 x \left(6 x^{3} \tan{\left(2 x^{3} - 5 \right)} + 1\right) \left(\tan^{2}{\left(2 x^{3} - 5 \right)} + 1\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=70.295281566701x_{1} = 70.295281566701
x2=12.2510999666511x_{2} = 12.2510999666511
x3=−49.7484430752804x_{3} = -49.7484430752804
x4=2.28418808472882x_{4} = 2.28418808472882
x5=−65.5532911338708x_{5} = -65.5532911338708
x6=−34.0636118133651x_{6} = -34.0636118133651
x7=−11.8688705360588x_{7} = -11.8688705360588
x8=−31.7498784374608x_{8} = -31.7498784374608
x9=−15.750305913016x_{9} = -15.750305913016
x10=10.1849073739188x_{10} = 10.1849073739188
x11=48.2512862549167x_{11} = 48.2512862549167
x12=46.2499395428594x_{12} = 46.2499395428594
x13=16.189987796377x_{13} = 16.189987796377
x14=28.2498882527283x_{14} = 28.2498882527283
x15=−25.6968658382215x_{15} = -25.6968658382215
x16=24.2295622248834x_{16} = 24.2295622248834
x17=−59.7509179512753x_{17} = -59.7509179512753
x18=−57.9995721874007x_{18} = -57.9995721874007
x19=22.1588126268466x_{19} = 22.1588126268466
x20=−39.7877155220288x_{20} = -39.7877155220288
x21=−51.9215695698998x_{21} = -51.9215695698998
x22=20.2503194195012x_{22} = 20.2503194195012
x23=59.6779661243175x_{23} = 59.6779661243175
x24=41.6328046645448x_{24} = 41.6328046645448
x25=0x_{25} = 0
x26=92.2501329160581x_{26} = 92.2501329160581
x27=55.5763841130453x_{27} = 55.5763841130453
x28=32.252557824129x_{28} = 32.252557824129
x29=44.1785241952741x_{29} = 44.1785241952741
x30=−13.767827910863x_{30} = -13.767827910863
x31=−9.90372147552056x_{31} = -9.90372147552056
x32=72.2500755059285x_{32} = 72.2500755059285
x33=52.0001468930514x_{33} = 52.0001468930514
x34=25.999506155863x_{34} = 25.999506155863
x35=38.333199717978x_{35} = 38.333199717978
x36=74.0046831121765x_{36} = 74.0046831121765
x37=14.0045582989738x_{37} = 14.0045582989738
x38=32.7482507002735x_{38} = 32.7482507002735
x39=−5.99852096262923x_{39} = -5.99852096262923
x40=8.25091924366563x_{40} = 8.25091924366563
x41=58.2606935214268x_{41} = 58.2606935214268
x42=−21.5963497145096x_{42} = -21.5963497145096
x43=6.23880872868215x_{43} = 6.23880872868215
x44=54.0901496842349x_{44} = 54.0901496842349
x45=−38.2180517042139x_{45} = -38.2180517042139
x46=40.0614043673613x_{46} = 40.0614043673613
x47=29.9970080289849x_{47} = 29.9970080289849
x48=4.24187764600161x_{48} = 4.24187764600161
x49=−63.7262106861335x_{49} = -63.7262106861335
x50=−54.0046165783125x_{50} = -54.0046165783125
x51=−24.7026114363995x_{51} = -24.7026114363995
x52=36.2561184458008x_{52} = 36.2561184458008
x53=−23.7136880251628x_{53} = -23.7136880251628
x54=−41.7499300930881x_{54} = -41.7499300930881
x55=−7.75062569435054x_{55} = -7.75062569435054
x56=−43.7324087637508x_{56} = -43.7324087637508
x57=61.5553937376623x_{57} = 61.5553937376623
x58=49.9998156931923x_{58} = 49.9998156931923
x59=−1.74771145350859x_{59} = -1.74771145350859
x60=−17.7719130049327x_{60} = -17.7719130049327
x61=−47.7170727558083x_{61} = -47.7170727558083
x62=18.2499495121719x_{62} = 18.2499495121719
x63=−35.8658848599295x_{63} = -35.8658848599295
x64=−19.7500789557821x_{64} = -19.7500789557821
x65=64.2495170721608x_{65} = 64.2495170721608
x66=−93.7737568301729x_{66} = -93.7737568301729
x67=−60.2480515937067x_{67} = -60.2480515937067
x68=−3.74387226611335x_{68} = -3.74387226611335
x69=−55.9986292773793x_{69} = -55.9986292773793
x70=−45.7568085612358x_{70} = -45.7568085612358
x71=−77.9165361566427x_{71} = -77.9165361566427
x72=−29.7481197334334x_{72} = -29.7481197334334

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
[92.2501329160581,∞)\left[92.2501329160581, \infty\right)
Convexa en los intervalos
(−∞,−93.7737568301729]\left(-\infty, -93.7737568301729\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⁡(2x3−5)y = \lim_{x \to -\infty} \tan{\left(2 x^{3} - 5 \right)}
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞tan⁡(2x3−5)y = \lim_{x \to \infty} \tan{\left(2 x^{3} - 5 \right)}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(2*x^3 - 5), 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⁡(2x3−5)x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(2 x^{3} - 5 \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(tan⁡(2x3−5)x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(2 x^{3} - 5 \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⁡(2x3−5)=−tan⁡(2x3+5)\tan{\left(2 x^{3} - 5 \right)} = - \tan{\left(2 x^{3} + 5 \right)}
- No
tan⁡(2x3−5)=tan⁡(2x3+5)\tan{\left(2 x^{3} - 5 \right)} = \tan{\left(2 x^{3} + 5 \right)}
- No
es decir, función
no es
par ni impar
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