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

Gráfico de la función y = x/tan(2*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          x    
f(x) = --------
       tan(2*x)
f(x)=xtan⁡(2x)f{\left(x \right)} = \frac{x}{\tan{\left(2 x \right)}} 
f = x/tan(2*x)
Gráfico de la función
02468-8-6-4-2-1010-20002000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
xtan⁡(2x)=0\frac{x}{\tan{\left(2 x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=58.9048622548086x_{1} = 58.9048622548086
x2=−74.6128255227576x_{2} = -74.6128255227576
x3=55.7632696012188x_{3} = 55.7632696012188
x4=−33.7721210260903x_{4} = -33.7721210260903
x5=3.92699081698724x_{5} = 3.92699081698724
x6=−96.6039740978861x_{6} = -96.6039740978861
x7=−43.1968989868597x_{7} = -43.1968989868597
x8=−47.9092879672443x_{8} = -47.9092879672443
x9=−16.4933614313464x_{9} = -16.4933614313464
x10=71.4712328691678x_{10} = 71.4712328691678
x11=46.3384916404494x_{11} = 46.3384916404494
x12=16.4933614313464x_{12} = 16.4933614313464
x13=−76.1836218495525x_{13} = -76.1836218495525
x14=90.3207887907066x_{14} = 90.3207887907066
x15=60.4756585816035x_{15} = 60.4756585816035
x16=−99.7455667514759x_{16} = -99.7455667514759
x17=−98.174770424681x_{17} = -98.174770424681
x18=−10.2101761241668x_{18} = -10.2101761241668
x19=−7.06858347057703x_{19} = -7.06858347057703
x20=−85.6083998103219x_{20} = -85.6083998103219
x21=−2.35619449019234x_{21} = -2.35619449019234
x22=−55.7632696012188x_{22} = -55.7632696012188
x23=63.6172512351933x_{23} = 63.6172512351933
x24=32.2013246992954x_{24} = 32.2013246992954
x25=18.0641577581413x_{25} = 18.0641577581413
x26=−82.4668071567321x_{26} = -82.4668071567321
x27=−91.8915851175014x_{27} = -91.8915851175014
x28=77.7544181763474x_{28} = 77.7544181763474
x29=−90.3207887907066x_{29} = -90.3207887907066
x30=−60.4756585816035x_{30} = -60.4756585816035
x31=−13.3517687777566x_{31} = -13.3517687777566
x32=91.8915851175014x_{32} = 91.8915851175014
x33=−3.92699081698724x_{33} = -3.92699081698724
x34=−71.4712328691678x_{34} = -71.4712328691678
x35=40.0553063332699x_{35} = 40.0553063332699
x36=−25.9181393921158x_{36} = -25.9181393921158
x37=49.4800842940392x_{37} = 49.4800842940392
x38=33.7721210260903x_{38} = 33.7721210260903
x39=2.35619449019234x_{39} = 2.35619449019234
x40=47.9092879672443x_{40} = 47.9092879672443
x41=99.7455667514759x_{41} = 99.7455667514759
x42=80.8960108299372x_{42} = 80.8960108299372
x43=96.6039740978861x_{43} = 96.6039740978861
x44=−11.7809724509617x_{44} = -11.7809724509617
x45=14.9225651045515x_{45} = 14.9225651045515
x46=−62.0464549083984x_{46} = -62.0464549083984
x47=−18.0641577581413x_{47} = -18.0641577581413
x48=82.4668071567321x_{48} = 82.4668071567321
x49=54.1924732744239x_{49} = 54.1924732744239
x50=5.49778714378214x_{50} = 5.49778714378214
x51=−49.4800842940392x_{51} = -49.4800842940392
x52=84.037603483527x_{52} = 84.037603483527
x53=88.7499924639117x_{53} = 88.7499924639117
x54=−77.7544181763474x_{54} = -77.7544181763474
x55=−46.3384916404494x_{55} = -46.3384916404494
x56=24.3473430653209x_{56} = 24.3473430653209
x57=−87.1791961371168x_{57} = -87.1791961371168
x58=−38.484510006475x_{58} = -38.484510006475
x59=93.4623814442964x_{59} = 93.4623814442964
x60=22.776546738526x_{60} = 22.776546738526
x61=19.6349540849362x_{61} = 19.6349540849362
x62=44.7676953136546x_{62} = 44.7676953136546
x63=85.6083998103219x_{63} = 85.6083998103219
x64=62.0464549083984x_{64} = 62.0464549083984
x65=−57.3340659280137x_{65} = -57.3340659280137
x66=76.1836218495525x_{66} = 76.1836218495525
x67=69.9004365423729x_{67} = 69.9004365423729
x68=8.63937979737193x_{68} = 8.63937979737193
x69=−69.9004365423729x_{69} = -69.9004365423729
x70=68.329640215578x_{70} = 68.329640215578
x71=−63.6172512351933x_{71} = -63.6172512351933
x72=98.174770424681x_{72} = 98.174770424681
x73=41.6261026600648x_{73} = 41.6261026600648
x74=−19.6349540849362x_{74} = -19.6349540849362
x75=−24.3473430653209x_{75} = -24.3473430653209
x76=−93.4623814442964x_{76} = -93.4623814442964
x77=−41.6261026600648x_{77} = -41.6261026600648
x78=−27.4889357189107x_{78} = -27.4889357189107
x79=−52.621676947629x_{79} = -52.621676947629
x80=−30.6305283725005x_{80} = -30.6305283725005
x81=30.6305283725005x_{81} = 30.6305283725005
x82=−84.037603483527x_{82} = -84.037603483527
x83=10.2101761241668x_{83} = 10.2101761241668
x84=36.9137136796801x_{84} = 36.9137136796801
x85=25.9181393921158x_{85} = 25.9181393921158
x86=74.6128255227576x_{86} = 74.6128255227576
x87=−68.329640215578x_{87} = -68.329640215578
x88=−65.1880475619882x_{88} = -65.1880475619882
x89=−79.3252145031423x_{89} = -79.3252145031423
x90=−40.0553063332699x_{90} = -40.0553063332699
x91=52.621676947629x_{91} = 52.621676947629
x92=11.7809724509617x_{92} = 11.7809724509617
x93=−54.1924732744239x_{93} = -54.1924732744239
x94=−32.2013246992954x_{94} = -32.2013246992954
x95=27.4889357189107x_{95} = 27.4889357189107
x96=38.484510006475x_{96} = 38.484510006475
x97=−5.49778714378214x_{97} = -5.49778714378214
x98=−35.3429173528852x_{98} = -35.3429173528852
x99=−21.2057504117311x_{99} = -21.2057504117311
x100=66.7588438887831x_{100} = 66.7588438887831
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 x/tan(2*x).
0tan⁡(0⋅2)\frac{0}{\tan{\left(0 \cdot 2 \right)}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
x(−2tan⁡2(2x)−2)tan⁡2(2x)+1tan⁡(2x)=0\frac{x \left(- 2 \tan^{2}{\left(2 x \right)} - 2\right)}{\tan^{2}{\left(2 x \right)}} + \frac{1}{\tan{\left(2 x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=1.1380893669753⋅10−18x_{1} = 1.1380893669753 \cdot 10^{-18}
x2=−5.90893574501939⋅10−17x_{2} = -5.90893574501939 \cdot 10^{-17}
x3=−2.41399978188354⋅10−18x_{3} = -2.41399978188354 \cdot 10^{-18}
x4=7.03587341264306⋅10−15x_{4} = 7.03587341264306 \cdot 10^{-15}
x5=−1.4225078896882⋅10−17x_{5} = -1.4225078896882 \cdot 10^{-17}
x6=2.49026233707955⋅10−17x_{6} = 2.49026233707955 \cdot 10^{-17}
x7=−7.50209094402789⋅10−17x_{7} = -7.50209094402789 \cdot 10^{-17}
x8=3.92739216229596⋅10−16x_{8} = 3.92739216229596 \cdot 10^{-16}
x9=−1.05781253595098⋅10−17x_{9} = -1.05781253595098 \cdot 10^{-17}
x10=−1.10884376018502⋅10−14x_{10} = -1.10884376018502 \cdot 10^{-14}
x11=−1.67282620640646⋅10−17x_{11} = -1.67282620640646 \cdot 10^{-17}
x12=7.80848009522299⋅10−15x_{12} = 7.80848009522299 \cdot 10^{-15}
x13=−4.00425592300622⋅10−16x_{13} = -4.00425592300622 \cdot 10^{-16}
x14=−1.79708817793883⋅10−15x_{14} = -1.79708817793883 \cdot 10^{-15}
x15=3.20594351989922⋅10−18x_{15} = 3.20594351989922 \cdot 10^{-18}
x16=−2.69375147402637⋅10−17x_{16} = -2.69375147402637 \cdot 10^{-17}
x17=2.01833444948684⋅10−19x_{17} = 2.01833444948684 \cdot 10^{-19}
x18=−6.36166437288271⋅10−17x_{18} = -6.36166437288271 \cdot 10^{-17}
x19=3.55301334149869⋅10−15x_{19} = 3.55301334149869 \cdot 10^{-15}
x20=5.34450584611906⋅10−19x_{20} = 5.34450584611906 \cdot 10^{-19}
x21=−5.56269649471253⋅10−19x_{21} = -5.56269649471253 \cdot 10^{-19}
x22=−7.3010844051703⋅10−16x_{22} = -7.3010844051703 \cdot 10^{-16}
x23=−4.63291123159765⋅10−17x_{23} = -4.63291123159765 \cdot 10^{-17}
x24=1.32232491390115⋅10−17x_{24} = 1.32232491390115 \cdot 10^{-17}
x25=−2.66190717917483⋅10−16x_{25} = -2.66190717917483 \cdot 10^{-16}
x26=4.84872040258331⋅10−17x_{26} = 4.84872040258331 \cdot 10^{-17}
x27=−6.25806367615583⋅10−16x_{27} = -6.25806367615583 \cdot 10^{-16}
x28=−5.4239813684212⋅10−16x_{28} = -5.4239813684212 \cdot 10^{-16}
x29=−3.04227761475379⋅10−16x_{29} = -3.04227761475379 \cdot 10^{-16}
x30=4.40573155138887⋅10−15x_{30} = 4.40573155138887 \cdot 10^{-15}
x31=−3.30688802212003⋅10−15x_{31} = -3.30688802212003 \cdot 10^{-15}
x32=−6.06605785541262⋅10−16x_{32} = -6.06605785541262 \cdot 10^{-16}
x33=−1.3634662391789⋅10−14x_{33} = -1.3634662391789 \cdot 10^{-14}
x34=2.11199163050304⋅10−15x_{34} = 2.11199163050304 \cdot 10^{-15}
x35=1.27445010097268⋅10−14x_{35} = 1.27445010097268 \cdot 10^{-14}
x36=−3.65765477836298⋅10−14x_{36} = -3.65765477836298 \cdot 10^{-14}
x37=−4.06820352760193⋅10−17x_{37} = -4.06820352760193 \cdot 10^{-17}
x38=−2.43157266583309⋅10−15x_{38} = -2.43157266583309 \cdot 10^{-15}
x39=2.50156589687295⋅10−17x_{39} = 2.50156589687295 \cdot 10^{-17}
x40=8.65013654714942⋅10−19x_{40} = 8.65013654714942 \cdot 10^{-19}
x41=−7.33393749295904⋅10−19x_{41} = -7.33393749295904 \cdot 10^{-19}
x42=1.14622047659147⋅10−14x_{42} = 1.14622047659147 \cdot 10^{-14}
x43=−5.17930852382622⋅10−19x_{43} = -5.17930852382622 \cdot 10^{-19}
x44=8.11798499309856⋅10−16x_{44} = 8.11798499309856 \cdot 10^{-16}
x45=5.35011389775258⋅10−18x_{45} = 5.35011389775258 \cdot 10^{-18}
x46=−1.62959612501489⋅10−18x_{46} = -1.62959612501489 \cdot 10^{-18}
x47=2.42268441917384⋅10−14x_{47} = 2.42268441917384 \cdot 10^{-14}
x48=2.55984081486131⋅10−16x_{48} = 2.55984081486131 \cdot 10^{-16}
x49=−2.43661477614666⋅10−18x_{49} = -2.43661477614666 \cdot 10^{-18}
x50=9.22678841222072⋅10−17x_{50} = 9.22678841222072 \cdot 10^{-17}
x51=−1.97262339979227⋅10−16x_{51} = -1.97262339979227 \cdot 10^{-16}
Signos de extremos en los puntos:
(1.138089366975301e-18, 0.5)

(-5.908935745019393e-17, 0.5)

(-2.4139997818835416e-18, 0.5)

(7.035873412643059e-15, 0.5)

(-1.4225078896881972e-17, 0.5)

(2.490262337079547e-17, 0.5)

(-7.502090944027891e-17, 0.5)

(3.927392162295961e-16, 0.5)

(-1.0578125359509778e-17, 0.5)

(-1.1088437601850204e-14, 0.5)

(-1.672826206406457e-17, 0.5)

(7.80848009522299e-15, 0.5)

(-4.0042559230062224e-16, 0.5)

(-1.797088177938827e-15, 0.5)

(3.2059435198992216e-18, 0.5)

(-2.6937514740263702e-17, 0.5)

(2.0183344494868357e-19, 0.5)

(-6.36166437288271e-17, 0.5)

(3.553013341498688e-15, 0.5)

(5.344505846119064e-19, 0.5)

(-5.562696494712527e-19, 0.5)

(-7.301084405170298e-16, 0.5)

(-4.6329112315976487e-17, 0.5)

(1.3223249139011519e-17, 0.5)

(-2.661907179174829e-16, 0.5)

(4.848720402583308e-17, 0.5)

(-6.258063676155835e-16, 0.5)

(-5.423981368421204e-16, 0.5)

(-3.0422776147537936e-16, 0.5)

(4.405731551388869e-15, 0.5)

(-3.3068880221200325e-15, 0.5)

(-6.066057855412624e-16, 0.5)

(-1.3634662391788959e-14, 0.5)

(2.1119916305030396e-15, 0.5)

(1.274450100972679e-14, 0.5)

(-3.6576547783629764e-14, 0.5)

(-4.068203527601934e-17, 0.5)

(-2.4315726658330897e-15, 0.5)

(2.5015658968729537e-17, 0.5)

(8.650136547149423e-19, 0.5)

(-7.333937492959042e-19, 0.5)

(1.1462204765914748e-14, 0.5)

(-5.179308523826221e-19, 0.5)

(8.117984993098563e-16, 0.5)

(5.35011389775258e-18, 0.5)

(-1.6295961250148949e-18, 0.5)

(2.4226844191738444e-14, 0.5)

(2.5598408148613075e-16, 0.5)

(-2.4366147761466602e-18, 0.5)

(9.226788412220725e-17, 0.5)

(-1.972623399792274e-16, 0.5)


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
Puntos máximos de la función:
x51=1.1380893669753⋅10−18x_{51} = 1.1380893669753 \cdot 10^{-18}
x51=−5.90893574501939⋅10−17x_{51} = -5.90893574501939 \cdot 10^{-17}
x51=−2.41399978188354⋅10−18x_{51} = -2.41399978188354 \cdot 10^{-18}
x51=7.03587341264306⋅10−15x_{51} = 7.03587341264306 \cdot 10^{-15}
x51=−1.4225078896882⋅10−17x_{51} = -1.4225078896882 \cdot 10^{-17}
x51=2.49026233707955⋅10−17x_{51} = 2.49026233707955 \cdot 10^{-17}
x51=−7.50209094402789⋅10−17x_{51} = -7.50209094402789 \cdot 10^{-17}
x51=3.92739216229596⋅10−16x_{51} = 3.92739216229596 \cdot 10^{-16}
x51=−1.05781253595098⋅10−17x_{51} = -1.05781253595098 \cdot 10^{-17}
x51=−1.10884376018502⋅10−14x_{51} = -1.10884376018502 \cdot 10^{-14}
x51=−1.67282620640646⋅10−17x_{51} = -1.67282620640646 \cdot 10^{-17}
x51=7.80848009522299⋅10−15x_{51} = 7.80848009522299 \cdot 10^{-15}
x51=−4.00425592300622⋅10−16x_{51} = -4.00425592300622 \cdot 10^{-16}
x51=−1.79708817793883⋅10−15x_{51} = -1.79708817793883 \cdot 10^{-15}
x51=3.20594351989922⋅10−18x_{51} = 3.20594351989922 \cdot 10^{-18}
x51=−2.69375147402637⋅10−17x_{51} = -2.69375147402637 \cdot 10^{-17}
x51=2.01833444948684⋅10−19x_{51} = 2.01833444948684 \cdot 10^{-19}
x51=−6.36166437288271⋅10−17x_{51} = -6.36166437288271 \cdot 10^{-17}
x51=3.55301334149869⋅10−15x_{51} = 3.55301334149869 \cdot 10^{-15}
x51=5.34450584611906⋅10−19x_{51} = 5.34450584611906 \cdot 10^{-19}
x51=−5.56269649471253⋅10−19x_{51} = -5.56269649471253 \cdot 10^{-19}
x51=−7.3010844051703⋅10−16x_{51} = -7.3010844051703 \cdot 10^{-16}
x51=−4.63291123159765⋅10−17x_{51} = -4.63291123159765 \cdot 10^{-17}
x51=1.32232491390115⋅10−17x_{51} = 1.32232491390115 \cdot 10^{-17}
x51=−2.66190717917483⋅10−16x_{51} = -2.66190717917483 \cdot 10^{-16}
x51=4.84872040258331⋅10−17x_{51} = 4.84872040258331 \cdot 10^{-17}
x51=−6.25806367615583⋅10−16x_{51} = -6.25806367615583 \cdot 10^{-16}
x51=−5.4239813684212⋅10−16x_{51} = -5.4239813684212 \cdot 10^{-16}
x51=−3.04227761475379⋅10−16x_{51} = -3.04227761475379 \cdot 10^{-16}
x51=4.40573155138887⋅10−15x_{51} = 4.40573155138887 \cdot 10^{-15}
x51=−3.30688802212003⋅10−15x_{51} = -3.30688802212003 \cdot 10^{-15}
x51=−6.06605785541262⋅10−16x_{51} = -6.06605785541262 \cdot 10^{-16}
x51=−1.3634662391789⋅10−14x_{51} = -1.3634662391789 \cdot 10^{-14}
x51=2.11199163050304⋅10−15x_{51} = 2.11199163050304 \cdot 10^{-15}
x51=1.27445010097268⋅10−14x_{51} = 1.27445010097268 \cdot 10^{-14}
x51=−3.65765477836298⋅10−14x_{51} = -3.65765477836298 \cdot 10^{-14}
x51=−4.06820352760193⋅10−17x_{51} = -4.06820352760193 \cdot 10^{-17}
x51=−2.43157266583309⋅10−15x_{51} = -2.43157266583309 \cdot 10^{-15}
x51=2.50156589687295⋅10−17x_{51} = 2.50156589687295 \cdot 10^{-17}
x51=8.65013654714942⋅10−19x_{51} = 8.65013654714942 \cdot 10^{-19}
x51=−7.33393749295904⋅10−19x_{51} = -7.33393749295904 \cdot 10^{-19}
x51=1.14622047659147⋅10−14x_{51} = 1.14622047659147 \cdot 10^{-14}
x51=−5.17930852382622⋅10−19x_{51} = -5.17930852382622 \cdot 10^{-19}
x51=8.11798499309856⋅10−16x_{51} = 8.11798499309856 \cdot 10^{-16}
x51=5.35011389775258⋅10−18x_{51} = 5.35011389775258 \cdot 10^{-18}
x51=−1.62959612501489⋅10−18x_{51} = -1.62959612501489 \cdot 10^{-18}
x51=2.42268441917384⋅10−14x_{51} = 2.42268441917384 \cdot 10^{-14}
x51=2.55984081486131⋅10−16x_{51} = 2.55984081486131 \cdot 10^{-16}
x51=−2.43661477614666⋅10−18x_{51} = -2.43661477614666 \cdot 10^{-18}
x51=9.22678841222072⋅10−17x_{51} = 9.22678841222072 \cdot 10^{-17}
x51=−1.97262339979227⋅10−16x_{51} = -1.97262339979227 \cdot 10^{-16}
Decrece en los intervalos
(−∞,−3.65765477836298⋅10−14]\left(-\infty, -3.65765477836298 \cdot 10^{-14}\right]
Crece en los intervalos
[2.42268441917384⋅10−14,∞)\left[2.42268441917384 \cdot 10^{-14}, \infty\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
4(2x(tan⁡2(2x)+1tan⁡2(2x)−1)−1tan⁡(2x))(tan⁡2(2x)+1)tan⁡(2x)=0\frac{4 \left(2 x \left(\frac{\tan^{2}{\left(2 x \right)} + 1}{\tan^{2}{\left(2 x \right)}} - 1\right) - \frac{1}{\tan{\left(2 x \right)}}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right)}{\tan{\left(2 x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−30.6223651301872x_{1} = -30.6223651301872
x2=−96.6013861664138x_{2} = -96.6013861664138
x3=41.6200962353617x_{3} = 41.6200962353617
x4=−82.4637755597094x_{4} = -82.4637755597094
x5=66.7550989265392x_{5} = 66.7550989265392
x6=11.7597262493445x_{6} = 11.7597262493445
x7=−41.6200962353617x_{7} = -41.6200962353617
x8=−54.1878598258373x_{8} = -54.1878598258373
x9=−10.1856514796438x_{9} = -10.1856514796438
x10=−46.3330961388114x_{10} = -46.3330961388114
x11=44.7621104652086x_{11} = 44.7621104652086
x12=84.0346285545694x_{12} = 84.0346285545694
x13=80.8929203639828x_{13} = 80.8929203639828
x14=−52.6169257678188x_{14} = -52.6169257678188
x15=−27.4798391439445x_{15} = -27.4798391439445
x16=3.86262591846885x_{16} = 3.86262591846885
x17=19.6222161805821x_{17} = 19.6222161805821
x18=77.7512028363303x_{18} = 77.7512028363303
x19=−40.0490643144726x_{19} = -40.0490643144726
x20=18.0503111221878x_{20} = 18.0503111221878
x21=33.7647173885721x_{21} = 33.7647173885721
x22=2.24670472895453x_{22} = 2.24670472895453
x23=−16.4781945199112x_{23} = -16.4781945199112
x24=−55.7587861230655x_{24} = -55.7587861230655
x25=−98.172223901556x_{25} = -98.172223901556
x26=−11.7597262493445x_{26} = -11.7597262493445
x27=−60.4715244985757x_{27} = -60.4715244985757
x28=88.7471755026564x_{28} = 88.7471755026564
x29=−90.3180208221014x_{29} = -90.3180208221014
x30=−32.1935597952787x_{30} = -32.1935597952787
x31=−57.3297052975115x_{31} = -57.3297052975115
x32=68.3259813506395x_{32} = 68.3259813506395
x33=−69.8968599047927x_{33} = -69.8968599047927
x34=60.4715244985757x_{34} = 60.4715244985757
x35=−68.3259813506395x_{35} = -68.3259813506395
x36=98.172223901556x_{36} = 98.172223901556
x37=−77.7512028363303x_{37} = -77.7512028363303
x38=−43.1911110173644x_{38} = -43.1911110173644
x39=−63.6133213216672x_{39} = -63.6133213216672
x40=24.3370721159772x_{40} = 24.3370721159772
x41=71.4677348441946x_{41} = 71.4677348441946
x42=−8.61037763596538x_{42} = -8.61037763596538
x43=16.4781945199112x_{43} = 16.4781945199112
x44=46.3330961388114x_{44} = 46.3330961388114
x45=−47.9040693934309x_{45} = -47.9040693934309
x46=−85.6054794697228x_{46} = -85.6054794697228
x47=62.0424254948814x_{47} = 62.0424254948814
x48=−18.0503111221878x_{48} = -18.0503111221878
x49=−93.4597065202651x_{49} = -93.4597065202651
x50=82.4637755597094x_{50} = 82.4637755597094
x51=−65.1842123526942x_{51} = -65.1842123526942
x52=8.61037763596538x_{52} = 8.61037763596538
x53=−79.3220628366317x_{53} = -79.3220628366317
x54=74.6094747920599x_{54} = 74.6094747920599
x55=58.9006179191122x_{55} = 58.9006179191122
x56=63.6133213216672x_{56} = 63.6133213216672
x57=47.9040693934309x_{57} = 47.9040693934309
x58=−99.7430603324317x_{58} = -99.7430603324317
x59=25.9084912436398x_{59} = 25.9084912436398
x60=−25.9084912436398x_{60} = -25.9084912436398
x61=40.0490643144726x_{61} = 40.0490643144726
x62=−5.45206082971445x_{62} = -5.45206082971445
x63=49.4750314121659x_{63} = 49.4750314121659
x64=−76.1803402100956x_{64} = -76.1803402100956
x65=−19.6222161805821x_{65} = -19.6222161805821
x66=−71.4677348441946x_{66} = -71.4677348441946
x67=−84.0346285545694x_{67} = -84.0346285545694
x68=−91.8888644664832x_{68} = -91.8888644664832
x69=55.7587861230655x_{69} = 55.7587861230655
x70=−35.3358428558098x_{70} = -35.3358428558098
x71=−13.3330271294063x_{71} = -13.3330271294063
x72=−33.7647173885721x_{72} = -33.7647173885721
x73=38.4780131551656x_{73} = 38.4780131551656
x74=−3.86262591846885x_{74} = -3.86262591846885
x75=36.9069403003403x_{75} = 36.9069403003403
x76=91.8888644664832x_{76} = 91.8888644664832
x77=54.1878598258373x_{77} = 54.1878598258373
x78=14.9057993954465x_{78} = 14.9057993954465
x79=32.1935597952787x_{79} = 32.1935597952787
x80=5.45206082971445x_{80} = 5.45206082971445
x81=99.7430603324317x_{81} = 99.7430603324317
x82=85.6054794697228x_{82} = 85.6054794697228
x83=−2.24670472895453x_{83} = -2.24670472895453
x84=−49.4750314121659x_{84} = -49.4750314121659
x85=−24.3370721159772x_{85} = -24.3370721159772
x86=−21.193956784066x_{86} = -21.193956784066
x87=−87.1763284175963x_{87} = -87.1763284175963
x88=10.1856514796438x_{88} = 10.1856514796438
x89=−74.6094747920599x_{89} = -74.6094747920599
x90=22.7655670069956x_{90} = 22.7655670069956
x91=52.6169257678188x_{91} = 52.6169257678188
x92=93.4597065202651x_{92} = 93.4597065202651
x93=30.6223651301872x_{93} = 30.6223651301872
x94=−62.0424254948814x_{94} = -62.0424254948814
x95=90.3180208221014x_{95} = 90.3180208221014
x96=−38.4780131551656x_{96} = -38.4780131551656
x97=69.8968599047927x_{97} = 69.8968599047927
x98=96.6013861664138x_{98} = 96.6013861664138
x99=76.1803402100956x_{99} = 76.1803402100956
x100=27.4798391439445x_{100} = 27.4798391439445
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(4(2x(tan⁡2(2x)+1tan⁡2(2x)−1)−1tan⁡(2x))(tan⁡2(2x)+1)tan⁡(2x))=−43\lim_{x \to 0^-}\left(\frac{4 \left(2 x \left(\frac{\tan^{2}{\left(2 x \right)} + 1}{\tan^{2}{\left(2 x \right)}} - 1\right) - \frac{1}{\tan{\left(2 x \right)}}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right)}{\tan{\left(2 x \right)}}\right) = - \frac{4}{3}
lim⁡x→0+(4(2x(tan⁡2(2x)+1tan⁡2(2x)−1)−1tan⁡(2x))(tan⁡2(2x)+1)tan⁡(2x))=−43\lim_{x \to 0^+}\left(\frac{4 \left(2 x \left(\frac{\tan^{2}{\left(2 x \right)} + 1}{\tan^{2}{\left(2 x \right)}} - 1\right) - \frac{1}{\tan{\left(2 x \right)}}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right)}{\tan{\left(2 x \right)}}\right) = - \frac{4}{3}
- los límites son iguales, es decir omitimos el punto correspondiente

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
[−2.24670472895453,2.24670472895453]\left[-2.24670472895453, 2.24670472895453\right]
Convexa en los intervalos
(−∞,−99.7430603324317]\left(-\infty, -99.7430603324317\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
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→−∞(xtan⁡(2x))y = \lim_{x \to -\infty}\left(\frac{x}{\tan{\left(2 x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(xtan⁡(2x))y = \lim_{x \to \infty}\left(\frac{x}{\tan{\left(2 x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x/tan(2*x), 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→−∞1tan⁡(2x)y = x \lim_{x \to -\infty} \frac{1}{\tan{\left(2 x \right)}}
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞1tan⁡(2x)y = x \lim_{x \to \infty} \frac{1}{\tan{\left(2 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:
xtan⁡(2x)=xtan⁡(2x)\frac{x}{\tan{\left(2 x \right)}} = \frac{x}{\tan{\left(2 x \right)}}
- Sí
xtan⁡(2x)=−xtan⁡(2x)\frac{x}{\tan{\left(2 x \right)}} = - \frac{x}{\tan{\left(2 x \right)}}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор