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

Gráfico de la función y = tan(x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          / 2\
f(x) = tan\x /
f(x)=tan⁡(x2)f{\left(x \right)} = \tan{\left(x^{2} \right)} 
f = tan(x^2)
Gráfico de la función
02468-8-6-4-2-1010-200200
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⁡(x2)=0\tan{\left(x^{2} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=67.9569069394495x_{1} = 67.9569069394495
x2=−39.751994978311x_{2} = -39.751994978311
x3=−3.96332729760601x_{3} = -3.96332729760601
x4=−80.5248237316253x_{4} = -80.5248237316253
x5=24.1079787811859x_{5} = 24.1079787811859
x6=8.1224039375905x_{6} = 8.1224039375905
x7=80.2708323895589x_{7} = 80.2708323895589
x8=−75.9264283325462x_{8} = -75.9264283325462
x9=−1.77245385090552x_{9} = -1.77245385090552
x10=100.249463334393x_{10} = 100.249463334393
x11=60.002209801089x_{11} = 60.002209801089
x12=−27.7432910832421x_{12} = -27.7432910832421
x13=−17.9009064202391x_{13} = -17.9009064202391
x14=10.026513098524x_{14} = 10.026513098524
x15=−5.87856438167413x_{15} = -5.87856438167413
x16=14.2899797929646x_{16} = 14.2899797929646
x17=−51.4927587106166x_{17} = -51.4927587106166
x18=−29.7645616180101x_{18} = -29.7645616180101
x19=−58.0326401661019x_{19} = -58.0326401661019
x20=−67.7717379115238x_{20} = -67.7717379115238
x21=78.2493711274826x_{21} = 78.2493711274826
x22=−11.7571287633483x_{22} = -11.7571287633483
x23=−87.8572316310395x_{23} = -87.8572316310395
x24=42.2424505354389x_{24} = 42.2424505354389
x25=−23.7138163312566x_{25} = -23.7138163312566
x26=74.1682461464185x_{26} = 74.1682461464185
x27=184.820282768104x_{27} = 184.820282768104
x28=0x_{28} = 0
x29=58.248777376671x_{29} = 58.248777376671
x30=62.0612003855959x_{30} = 62.0612003855959
x31=−84.4291047624884x_{31} = -84.4291047624884
x32=64.2499240996983x_{32} = 64.2499240996983
x33=72.1064202529227x_{33} = 72.1064202529227
x34=−43.883895641232x_{34} = -43.883895641232
x35=32.2956461615465x_{35} = 32.2956461615465
x36=28.2482660354898x_{36} = 28.2482660354898
x37=−13.7293684929565x_{37} = -13.7293684929565
x38=81.6676264107824x_{38} = 81.6676264107824
x39=−80.0160348181288x_{39} = -80.0160348181288
x40=25.8681123241458x_{40} = 25.8681123241458
x41=−38.4667406968773x_{41} = -38.4667406968773
x42=−32.6343363586107x_{42} = -32.6343363586107
x43=3.96332729760601x_{43} = 3.96332729760601
x44=−64.7613016588022x_{44} = -64.7613016588022
x45=33.9553310080679x_{45} = 33.9553310080679
x46=−47.6589077565298x_{46} = -47.6589077565298
x47=−74.1258763936626x_{47} = -74.1258763936626
x48=40.1451993554841x_{48} = 40.1451993554841
x49=−91.9799381175988x_{49} = -91.9799381175988
x50=−34.0015602071588x_{50} = -34.0015602071588
x51=30.2877046810784x_{51} = 30.2877046810784
x52=84.0001017854304x_{52} = 84.0001017854304
x53=−19.7372107716651x_{53} = -19.7372107716651
x54=−98.0473024615403x_{54} = -98.0473024615403
x55=22.0668724858422x_{55} = 22.0668724858422
x56=−69.9614276176015x_{56} = -69.9614276176015
x57=−103.183610631804x_{57} = -103.183610631804
x58=93.1845593138809x_{58} = 93.1845593138809
x59=−45.7760013538141x_{59} = -45.7760013538141
x60=−95.5975576533873x_{60} = -95.5975576533873
x61=92.2187145500715x_{61} = 92.2187145500715
x62=−35.7579111527931x_{62} = -35.7579111527931
x63=−41.7562441168065x_{63} = -41.7562441168065
x64=54.4290535682477x_{64} = 54.4290535682477
x65=57.4614120860502x_{65} = 57.4614120860502
x66=−53.8487698955452x_{66} = -53.8487698955452
x67=96.2688855658457x_{67} = 96.2688855658457
x68=−21.9955738423172x_{68} = -21.9955738423172
x69=52.2498231190263x_{69} = 52.2498231190263
x70=−59.9760251130426x_{70} = -59.9760251130426
x71=36.7543853307821x_{71} = 36.7543853307821
x72=−61.7312877624015x_{72} = -61.7312877624015
x73=76.3596488013313x_{73} = 76.3596488013313
x74=86.3787158522038x_{74} = 86.3787158522038
x75=46.2539143975367x_{75} = 46.2539143975367
x76=−56.8291936234389x_{76} = -56.8291936234389
x77=1.77245385090552x_{77} = 1.77245385090552
x78=18.2485292908913x_{78} = 18.2485292908913
x79=6.63191504395654x_{79} = 6.63191504395654
x80=49.2155253544827x_{80} = 49.2155253544827
x81=−93.6553612390807x_{81} = -93.6553612390807
x82=97.9992282725941x_{82} = 97.9992282725941
x83=19.9745404704565x_{83} = 19.9745404704565
x84=−77.7660970999597x_{84} = -77.7660970999597
x85=70.2750571073632x_{85} = 70.2750571073632
x86=36.2379045917467x_{86} = 36.2379045917467
x87=16.244807875181x_{87} = 16.244807875181
x88=−86.8321505469921x_{88} = -86.8321505469921
x89=90.1863813452118x_{89} = 90.1863813452118
x90=−25.5011701553691x_{90} = -25.5011701553691
x91=−4.34160752734961x_{91} = -4.34160752734961
x92=−72.0410373587883x_{92} = -72.0410373587883
x93=−65.7482432566799x_{93} = -65.7482432566799
x94=88.249692284161x_{94} = 88.249692284161
x95=48.2485249133073x_{95} = 48.2485249133073
x96=−89.7498945058111x_{96} = -89.7498945058111
x97=43.4883751991386x_{97} = 43.4883751991386
x98=66.2480561002732x_{98} = 66.2480561002732
x99=−52.12943166154x_{99} = -52.12943166154
x100=−15.7539144225679x_{100} = -15.7539144225679
x101=−7.72594721818665x_{101} = -7.72594721818665
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).
tan⁡(02)\tan{\left(0^{2} \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
2x(tan⁡2(x2)+1)=02 x \left(\tan^{2}{\left(x^{2} \right)} + 1\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
Signos de extremos en los puntos:
(0, 0)


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
2(4x2(tan⁡2(x2)+1)tan⁡(x2)+tan⁡2(x2)+1)=02 \left(4 x^{2} \left(\tan^{2}{\left(x^{2} \right)} + 1\right) \tan{\left(x^{2} \right)} + \tan^{2}{\left(x^{2} \right)} + 1\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−45.084479855911x_{1} = -45.084479855911
x2=70.2974053113422x_{2} = 70.2974053113422
x3=−15.7538824522744x_{3} = -15.7538824522744
x4=98.0152554894128x_{4} = 98.0152554894128
x5=−23.9773021993105x_{5} = -23.9773021993105
x6=−58.0055657475439x_{6} = -58.0055657475439
x7=67.9337880141555x_{7} = 67.9337880141555
x8=−38.2620177957948x_{8} = -38.2620177957948
x9=−65.7482428168775x_{9} = -65.7482428168775
x10=−21.9955620959205x_{10} = -21.9955620959205
x11=−53.9944246823664x_{11} = -53.9944246823664
x12=30.2357931718758x_{12} = 30.2357931718758
x13=−3.96131707443208x_{13} = -3.96131707443208
x14=10.0263890839373x_{14} = 10.0263890839373
x15=21.9955620959205x_{15} = 21.9955620959205
x16=−59.9760245336446x_{16} = -59.9760245336446
x17=46.2539131343586x_{17} = 46.2539131343586
x18=−19.7371945141746x_{18} = -19.7371945141746
x19=100.374735941116x_{19} = 100.374735941116
x20=−9.70799294173983x_{20} = -9.70799294173983
x21=84.0001015745334x_{21} = 84.0001015745334
x22=−51.9785493213814x_{22} = -51.9785493213814
x23=36.2379019649839x_{23} = 36.2379019649839
x24=26.1699605167475x_{24} = 26.1699605167475
x25=−77.7458952237758x_{25} = -77.7458952237758
x26=66.2480556703503x_{26} = 66.2480556703503
x27=24.2379330940786x_{27} = 24.2379330940786
x28=93.7894418451826x_{28} = 93.7894418451826
x29=28.2482604900649x_{29} = 28.2482604900649
x30=18.2485087212107x_{30} = 18.2485087212107
x31=42.2424488771403x_{31} = 42.2424488771403
x32=−25.6852891421007x_{32} = -25.6852891421007
x33=90.1166853515224x_{33} = 90.1166853515224
x34=−49.5019410506056x_{34} = -49.5019410506056
x35=64.249923628404x_{35} = 64.249923628404
x36=−93.6721316773806x_{36} = -93.6721316773806
x37=3.96131707443208x_{37} = 3.96131707443208
x38=8.1221706533684x_{38} = 8.1221706533684
x39=−11.757051848034x_{39} = -11.757051848034
x40=−97.9992281397806x_{40} = -97.9992281397806
x41=43.812246930533x_{41} = 43.812246930533
x42=88.249692102287x_{42} = 88.249692102287
x43=50.0384776453561x_{43} = 50.0384776453561
x44=−5.87794891821309x_{44} = -5.87794891821309
x45=−105.412585554328x_{45} = -105.412585554328
x46=−89.7498943329059x_{46} = -89.7498943329059
x47=56.2736644965707x_{47} = 56.2736644965707
x48=−41.7562423999034x_{48} = -41.7562423999034
x49=34.001557027258x_{49} = 34.001557027258
x50=62.0105583185608x_{50} = 62.0105583185608
x51=−47.9218543142169x_{51} = -47.9218543142169
x52=−27.7432852294635x_{52} = -27.7432852294635
x53=−87.9108519779118x_{53} = -87.9108519779118
x54=86.2148953763774x_{54} = 86.2148953763774
x55=−56.0779294085062x_{55} = -56.0779294085062
x56=6.39019423112011x_{56} = 6.39019423112011
x57=−95.7453253997488x_{57} = -95.7453253997488
x58=−81.3399907221167x_{58} = -81.3399907221167
x59=32.2956424506555x_{59} = 32.2956424506555
x60=72.0192295354316x_{60} = 72.0192295354316
x61=58.2218034714853x_{61} = 58.2218034714853
x62=54.255619353996x_{62} = 54.255619353996
x63=78.2493708665866x_{63} = 78.2493708665866
x64=1.74930740111531x_{64} = 1.74930740111531
x65=11.757051848034x_{65} = 11.757051848034
x66=92.1335081311262x_{66} = 92.1335081311262
x67=48.2485238004028x_{67} = 48.2485238004028
x68=−17.7245160605986x_{68} = -17.7245160605986
x69=−75.9884679014017x_{69} = -75.9884679014017
x70=−31.7066144592746x_{70} = -31.7066144592746
x71=−29.764556877646x_{71} = -29.764556877646
x72=−85.9959828872556x_{72} = -85.9959828872556
x73=−84.0001015745334x_{73} = -84.0001015745334
x74=14.1795869624156x_{74} = 14.1795869624156
x75=40.1060504564238x_{75} = 40.1060504564238
x76=−34.001557027258x_{76} = -34.001557027258
x77=−73.8711472326801x_{77} = -73.8711472326801
x78=−35.7579084188236x_{78} = -35.7579084188236
x79=76.2567240235449x_{79} = 76.2567240235449
x80=−39.7519929884018x_{80} = -39.7519929884018
x81=−67.8180771660422x_{81} = -67.8180771660422
x82=52.2498222427207x_{82} = 52.2498222427207
x83=16.2447787164853x_{83} = 16.2447787164853
x84=−45.7416723328422x_{84} = -45.7416723328422
x85=−70.0287518186291x_{85} = -70.0287518186291
x86=74.2529128163118x_{86} = 74.2529128163118
x87=81.2627080522657x_{87} = 81.2627080522657
x88=−13.7293201913147x_{88} = -13.7293201913147
x89=96.2525672823938x_{89} = 96.2525672823938
x90=−79.996401145904x_{90} = -79.996401145904
x91=−61.7312872310352x_{91} = -61.7312872310352
x92=−91.9970139691073x_{92} = -91.9970139691073
x93=−64.029511570492x_{93} = -64.029511570492
x94=−7.72567614263021x_{94} = -7.72567614263021
x95=60.0022092224492x_{95} = 60.0022092224492
x96=−1.74930740111531x_{96} = -1.74930740111531
x97=38.2620177957948x_{97} = 38.2620177957948
x98=80.2512610557666x_{98} = 80.2512610557666
x99=19.9745247856047x_{99} = 19.9745247856047
x100=−47.7576816532727x_{100} = -47.7576816532727

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
[100.374735941116,∞)\left[100.374735941116, \infty\right)
Convexa en los intervalos
[−1.74930740111531,1.74930740111531]\left[-1.74930740111531, 1.74930740111531\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞tan⁡(x2)=⟨−∞,∞⟩\lim_{x \to -\infty} \tan{\left(x^{2} \right)} = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞tan⁡(x2)=⟨−∞,∞⟩\lim_{x \to \infty} \tan{\left(x^{2} \right)} = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(x^2), 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⁡(x2)x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x^{2} \right)}}{x}\right)
True

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