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tan(32*x)/((4*x))
Trazado el gráfico de la función/ tan(32*x)/((4*x))

Gráfico de la función y = tan(32*x)/((4*x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       tan(32*x)
f(x) = ---------
          4*x
f(x)=tan⁡(32x)4xf{\left(x \right)} = \frac{\tan{\left(32 x \right)}}{4 x} 
f = tan(32*x)/((4*x))
Gráfico de la función
02468-8-6-4-2-1010-200100
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:
tan⁡(32x)4x=0\frac{\tan{\left(32 x \right)}}{4 x} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−65.875270954961x_{1} = -65.875270954961
x2=−55.7632696012188x_{2} = -55.7632696012188
x3=−27.7834600301847x_{3} = -27.7834600301847
x4=−53.9961237335746x_{4} = -53.9961237335746
x5=−17.7696334468673x_{5} = -17.7696334468673
x6=−89.2408663160351x_{6} = -89.2408663160351
x7=62.2428044492478x_{7} = 62.2428044492478
x8=−45.7494430179014x_{8} = -45.7494430179014
x9=−21.9911485751286x_{9} = -21.9911485751286
x10=74.1219516706342x_{10} = 74.1219516706342
x11=−23.7582944427728x_{11} = -23.7582944427728
x12=−13.7444678594553x_{12} = -13.7444678594553
x13=−42.3133260530375x_{13} = -42.3133260530375
x14=25.7217898512664x_{14} = 25.7217898512664
x15=52.0326283250809x_{15} = 52.0326283250809
x16=−69.9986113127976x_{16} = -69.9986113127976
x17=34.2629948782137x_{17} = 34.2629948782137
x18=50.167307687012x_{18} = 50.167307687012
x19=43.393248527709x_{19} = 43.393248527709
x20=−67.7405915930299x_{20} = -67.7405915930299
x21=−1.76714586764426x_{21} = -1.76714586764426
x22=48.007462737669x_{22} = 48.007462737669
x23=−33.5757714852409x_{23} = -33.5757714852409
x24=68.2314654451533x_{24} = 68.2314654451533
x25=−81.877758534184x_{25} = -81.877758534184
x26=−35.8337912050086x_{26} = -35.8337912050086
x27=66.4643195775091x_{27} = 66.4643195775091
x28=22.0893233455532x_{28} = 22.0893233455532
x29=−11.8791472213864x_{29} = -11.8791472213864
x30=−99.7455667514759x_{30} = -99.7455667514759
x31=100.236440603599x_{31} = 100.236440603599
x32=−73.8274273593601x_{32} = -73.8274273593601
x33=80.012437896115x_{33} = 80.012437896115
x34=−51.7381040138069x_{34} = -51.7381040138069
x35=72.2566310325652x_{35} = 72.2566310325652
x36=−86.0010988920206x_{36} = -86.0010988920206
x37=64.009950316892x_{37} = 64.009950316892
x38=88.2591186117883x_{38} = 88.2591186117883
x39=10.0138265833175x_{39} = 10.0138265833175
x40=−19.8313036257856x_{40} = -19.8313036257856
x41=30.2378292908018x_{41} = 30.2378292908018
x42=40.2516558741192x_{42} = 40.2516558741192
x43=−93.7569057555704x_{43} = -93.7569057555704
x44=59.9847847294801x_{44} = 59.9847847294801
x45=−61.7519305971244x_{45} = -61.7519305971244
x46=−41.8224522009141x_{46} = -41.8224522009141
x47=24.2491682948962x_{47} = 24.2491682948962
x48=−5.98866099590554x_{48} = -5.98866099590554
x49=−60.0829594999048x_{49} = -60.0829594999048
x50=28.2743338823081x_{50} = 28.2743338823081
x51=20.1258279370596x_{51} = 20.1258279370596
x52=90.1244392498572x_{52} = 90.1244392498572
x53=8.24668071567321x_{53} = 8.24668071567321
x54=−75.9872723087031x_{54} = -75.9872723087031
x55=32.004975158446x_{55} = 32.004975158446
x56=−49.9709581461627x_{56} = -49.9709581461627
x57=46.2403168700248x_{57} = 46.2403168700248
x58=−87.7682447596649x_{58} = -87.7682447596649
x59=12.2718463030851x_{59} = 12.2718463030851
x60=81.1905351412112x_{60} = 81.1905351412112
x61=38.0918109247762x_{61} = 38.0918109247762
x62=−47.9092879672443x_{62} = -47.9092879672443
x63=94.2477796076938x_{63} = 94.2477796076938
x64=−95.9167507049134x_{64} = -95.9167507049134
x65=92.3824589696249x_{65} = 92.3824589696249
x66=44.0804719206818x_{66} = 44.0804719206818
x67=69.9986113127976x_{67} = 69.9986113127976
x68=4.1233403578366x_{68} = 4.1233403578366
x69=−39.7607820219958x_{69} = -39.7607820219958
x70=−71.7657571804418x_{70} = -71.7657571804418
x71=−25.9181393921158x_{71} = -25.9181393921158
x72=−7.7558068635498x_{72} = -7.7558068635498
x73=42.3133260530375x_{73} = 42.3133260530375
x74=18.2605072989907x_{74} = 18.2605072989907
x75=6.38136007760427x_{75} = 6.38136007760427
x76=16.002487579223x_{76} = 16.002487579223
x77=58.3158136322605x_{77} = 58.3158136322605
x78=56.2541434533422x_{78} = 56.2541434533422
x79=−91.9897598879261x_{79} = -91.9897598879261
x80=−32.004975158446x_{80} = -32.004975158446
x81=−16.002487579223x_{81} = -16.002487579223
x82=−29.7469554386784x_{82} = -29.7469554386784
x83=84.2339530243763x_{83} = 84.2339530243763
x84=86.0010988920206x_{84} = 86.0010988920206
x85=−10.1120013537421x_{85} = -10.1120013537421
x86=78.2452920284708x_{86} = 78.2452920284708
x87=−77.7544181763474x_{87} = -77.7544181763474
x88=14.2353417115787x_{88} = 14.2353417115787
x89=−78.6379911101695x_{89} = -78.6379911101695
x90=−37.9936361543516x_{90} = -37.9936361543516
x91=−83.7430791722529x_{91} = -83.7430791722529
x92=98.174770424681x_{92} = 98.174770424681
x93=−63.8136007760427x_{93} = -63.8136007760427
x94=53.9961237335746x_{94} = 53.9961237335746
x95=95.42587685279x_{95} = 95.42587685279
x96=80.6996612890878x_{96} = 80.6996612890878
x97=2.25801971976766x_{97} = 2.25801971976766
x98=−57.9231145505618x_{98} = -57.9231145505618
x99=−56.4504929941916x_{99} = -56.4504929941916
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(32*x)/((4*x)).
tan⁡(0⋅32)0⋅4\frac{\tan{\left(0 \cdot 32 \right)}}{0 \cdot 4}
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
14x(32tan⁡2(32x)+32)−tan⁡(32x)4x2=0\frac{1}{4 x} \left(32 \tan^{2}{\left(32 x \right)} + 32\right) - \frac{\tan{\left(32 x \right)}}{4 x^{2}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
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
512(tan⁡2(32x)+1)tan⁡(32x)−16(tan⁡2(32x)+1)x+tan⁡(32x)2x2x=0\frac{512 \left(\tan^{2}{\left(32 x \right)} + 1\right) \tan{\left(32 x \right)} - \frac{16 \left(\tan^{2}{\left(32 x \right)} + 1\right)}{x} + \frac{\tan{\left(32 x \right)}}{2 x^{2}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−69.9986252639607x_{1} = -69.9986252639607
x2=16.0025486043484x_{2} = 16.0025486043484
x3=42.0188249828175x_{3} = 42.0188249828175
x4=−17.7696884032881x_{4} = -17.7696884032881
x5=−41.9206502668213x_{5} = -41.9206502668213
x6=53.9961418193494x_{6} = 53.9961418193494
x7=−55.7632871138538x_{7} = -55.7632871138538
x8=64.0099655733006x_{8} = 64.0099655733006
x9=60.2793252413693x_{9} = 60.2793252413693
x10=−64.0099655733006x_{10} = -64.0099655733006
x11=−67.7406060092314x_{11} = -67.7406060092314
x12=8.24679913056223x_{12} = 8.24679913056223
x13=−91.9897705039147x_{13} = -91.9897705039147
x14=56.2541608131621x_{14} = 56.2541608131621
x15=28.0780191217217x_{15} = 28.0780191217217
x16=46.2403379892888x_{16} = 46.2403379892888
x17=21.9911929819905x_{17} = 21.9911929819905
x18=−23.7583355466728x_{18} = -23.7583355466728
x19=−39.7608065829084x_{19} = -39.7608065829084
x20=78.245304509249x_{20} = 78.245304509249
x21=40.2516801355094x_{21} = 40.2516801355094
x22=86.0011102472499x_{22} = 86.0011102472499
x23=−80.0124501012444x_{23} = -80.0124501012444
x24=24.2492085667399x_{24} = 24.2492085667399
x25=66.1698100246605x_{25} = 66.1698100246605
x26=94.2477899693411x_{26} = 94.2477899693411
x27=−11.6828812689226x_{27} = -11.6828812689226
x28=80.0124501012444x_{28} = 80.0124501012444
x29=79.7179258350631x_{29} = 79.7179258350631
x30=26.0163516989573x_{30} = 26.0163516989573
x31=−71.7657707880746x_{31} = -71.7657707880746
x32=58.0213061520816x_{32} = 58.0213061520816
x33=−51.7381228889027x_{33} = -51.7381228889027
x34=−48.0074830795384x_{34} = -48.0074830795384
x35=−32.0050056712123x_{35} = -32.0050056712123
x36=−93.7569161714671x_{36} = -93.7569161714671
x37=34.2630233801181x_{37} = 34.2630233801181
x38=74.2201395987041x_{38} = 74.2201395987041
x39=−57.9231314101842x_{39} = -57.9231314101842
x40=32.0050056712123x_{40} = 32.0050056712123
x41=−59.9848010096399x_{41} = -59.9848010096399
x42=−75.9872851603569x_{42} = -75.9872851603569
x43=2.25845201290999x_{43} = 2.25845201290999
x44=90.1244500855665x_{44} = 90.1244500855665
x45=44.3750182389745x_{45} = 44.3750182389745
x46=100.236450346187x_{46} = 100.236450346187
x47=69.9986252639607x_{47} = 69.9986252639607
x48=−77.7544307359183x_{48} = -77.7544307359183
x49=6.08699619491471x_{49} = 6.08699619491471
x50=88.2591296765058x_{50} = 88.2591296765058
x51=−49.8728029567908x_{51} = -49.8728029567908
x52=−3.82907106292848x_{52} = -3.82907106292848
x53=72.2566445477549x_{53} = 72.2566445477549
x54=−10.0139241025127x_{54} = -10.0139241025127
x55=37.9936618576287x_{55} = 37.9936618576287
x56=20.1258764596351x_{56} = 20.1258764596351
x57=−81.7795957051525x_{57} = -81.7795957051525
x58=98.2729551323501x_{58} = 98.2729551323501
x59=18.2605607781088x_{59} = 18.2605607781088
x60=−37.9936618576287x_{60} = -37.9936618576287
x61=−29.7469882675837x_{61} = -29.7469882675837
x62=−86.0011102472499x_{62} = -86.0011102472499
x63=−7.7559327725012x_{63} = -7.7559327725012
x64=−19.7331783435486x_{64} = -19.7331783435486
x65=−16.0025486043484x_{65} = -16.0025486043484
x66=−27.9798444733323x_{66} = -27.9798444733323
x67=−95.7204113663014x_{67} = -95.7204113663014
x68=68.2314797576412x_{68} = 68.2314797576412
x69=14.235410312076x_{69} = 14.235410312076
x70=−83.7430908336606x_{70} = -83.7430908336606
x71=12.2719258793527x_{71} = 12.2719258793527
x72=−45.7494643637664x_{72} = -45.7494643637664
x73=48.0074830795384x_{73} = 48.0074830795384
x74=−89.8299258098194x_{74} = -89.8299258098194
x75=−99.7455765420091x_{75} = -99.7455765420091
x76=−43.9823193537665x_{76} = -43.9823193537665
x77=10.0139241025127x_{77} = 10.0139241025127
x78=−5.98882405413538x_{78} = -5.98882405413538
x79=−33.7721499422649x_{79} = -33.7721499422649
x80=91.9897705039147x_{80} = 91.9897705039147
x81=52.2289965636289x_{81} = 52.2289965636289
x82=−61.7519464113978x_{82} = -61.7519464113978
x83=−21.9911929819905x_{83} = -21.9911929819905
x84=4.12357716381613x_{84} = 4.12357716381613
x85=−97.7820813301124x_{85} = -97.7820813301124
x86=36.2265172437986x_{86} = 36.2265172437986
x87=96.0149356462803x_{87} = 96.0149356462803
x88=75.9872851603569x_{88} = 75.9872851603569
x89=−87.7682558862655x_{89} = -87.7682558862655
x90=84.2339646178271x_{90} = 84.2339646178271
x91=62.2428201388031x_{91} = 62.2428201388031
x92=−13.7445389099122x_{92} = -13.7445389099122
x93=−53.9961418193494x_{93} = -53.9961418193494
x94=−1.76769808630633x_{94} = -1.76769808630633
x95=−65.8752857793704x_{95} = -65.8752857793704
x96=−72.9438678133976x_{96} = -72.9438678133976
x97=30.237861586773x_{97} = 30.237861586773
x98=−25.820002443554x_{98} = -25.820002443554
x99=−35.7356437619636x_{99} = -35.7356437619636
x100=50.2655018855128x_{100} = 50.2655018855128
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−(512(tan⁡2(32x)+1)tan⁡(32x)−16(tan⁡2(32x)+1)x+tan⁡(32x)2x2x)=163843\lim_{x \to 0^-}\left(\frac{512 \left(\tan^{2}{\left(32 x \right)} + 1\right) \tan{\left(32 x \right)} - \frac{16 \left(\tan^{2}{\left(32 x \right)} + 1\right)}{x} + \frac{\tan{\left(32 x \right)}}{2 x^{2}}}{x}\right) = \frac{16384}{3}
lim⁡x→0+(512(tan⁡2(32x)+1)tan⁡(32x)−16(tan⁡2(32x)+1)x+tan⁡(32x)2x2x)=163843\lim_{x \to 0^+}\left(\frac{512 \left(\tan^{2}{\left(32 x \right)} + 1\right) \tan{\left(32 x \right)} - \frac{16 \left(\tan^{2}{\left(32 x \right)} + 1\right)}{x} + \frac{\tan{\left(32 x \right)}}{2 x^{2}}}{x}\right) = \frac{16384}{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
[100.236450346187,∞)\left[100.236450346187, \infty\right)
Convexa en los intervalos
[−1.76769808630633,2.25845201290999]\left[-1.76769808630633, 2.25845201290999\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→−∞(tan⁡(32x)4x)y = \lim_{x \to -\infty}\left(\frac{\tan{\left(32 x \right)}}{4 x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(tan⁡(32x)4x)y = \lim_{x \to \infty}\left(\frac{\tan{\left(32 x \right)}}{4 x}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(32*x)/((4*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→−∞(14xtan⁡(32x)x)y = x \lim_{x \to -\infty}\left(\frac{\frac{1}{4 x} \tan{\left(32 x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(14xtan⁡(32x)x)y = x \lim_{x \to \infty}\left(\frac{\frac{1}{4 x} \tan{\left(32 x \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⁡(32x)4x=tan⁡(32x)4x\frac{\tan{\left(32 x \right)}}{4 x} = \frac{\tan{\left(32 x \right)}}{4 x}
- No
tan⁡(32x)4x=−tan⁡(32x)4x\frac{\tan{\left(32 x \right)}}{4 x} = - \frac{\tan{\left(32 x \right)}}{4 x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = tan(32*x)/((4*x))
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