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

Gráfico de la función y = ctg(x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          / 2\
f(x) = cot\x /
f(x)=cot⁡(x2)f{\left(x \right)} = \cot{\left(x^{2} \right)} 
f = cot(x^2)
Gráfico de la función
02468-8-6-4-2-1010-500500
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:
cot⁡(x2)=0\cot{\left(x^{2} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−2π2x_{1} = - \frac{\sqrt{2} \sqrt{\pi}}{2}
x2=2π2x_{2} = \frac{\sqrt{2} \sqrt{\pi}}{2}
Solución numérica
x1=86.0963870678969x_{1} = 86.0963870678969
x2=−90.0033147016335x_{2} = -90.0033147016335
x3=30.1056386838189x_{3} = 30.1056386838189
x4=−65.4489234683786x_{4} = -65.4489234683786
x5=−19.7769635023157x_{5} = -19.7769635023157
x6=64.0172446814754x_{6} = 64.0172446814754
x7=−15.5026397106951x_{7} = -15.5026397106951
x8=46.1008397987878x_{8} = 46.1008397987878
x9=36.3461099821495x_{9} = 36.3461099821495
x10=44.6116458567618x_{10} = 44.6116458567618
x11=74.2000075836661x_{11} = 74.2000075836661
x12=51.9936580595558x_{12} = 51.9936580595558
x13=−51.7514032388041x_{13} = -51.7514032388041
x14=58.1002694166063x_{14} = 58.1002694166063
x15=−31.5825215509269x_{15} = -31.5825215509269
x16=−97.7504683154682x_{16} = -97.7504683154682
x17=−50.4293491241057x_{17} = -50.4293491241057
x18=−75.9160834277591x_{18} = -75.9160834277591
x19=88.1695583594924x_{19} = 88.1695583594924
x20=−55.5007351018416x_{20} = -55.5007351018416
x21=96.0483567129934x_{21} = 96.0483567129934
x22=17.9447275541579x_{22} = 17.9447275541579
x23=−53.7757941608435x_{23} = -53.7757941608435
x24=8.40748682459689x_{24} = 8.40748682459689
x25=10.2588183479024x_{25} = 10.2588183479024
x26=98.2313688976238x_{26} = 98.2313688976238
x27=38.200389887964x_{27} = 38.200389887964
x28=48.3298472783022x_{28} = 48.3298472783022
x29=22.2441192889355x_{29} = 22.2441192889355
x30=−5.74340690380656x_{30} = -5.74340690380656
x31=−17.6801716346925x_{31} = -17.6801716346925
x32=42.0000508927152x_{32} = 42.0000508927152
x33=23.6142477333534x_{33} = 23.6142477333534
x34=6.2665706865775x_{34} = 6.2665706865775
x35=−41.4352553411563x_{35} = -41.4352553411563
x36=−73.5621733352707x_{36} = -73.5621733352707
x37=−12.0865238340859x_{37} = -12.0865238340859
x38=34.2547056054462x_{38} = 34.2547056054462
x39=51.9332002971783x_{39} = 51.9332002971783
x40=−79.7505757169974x_{40} = -79.7505757169974
x41=−71.7460769861801x_{41} = -71.7460769861801
x42=−77.8569394794164x_{42} = -77.8569394794164
x43=90.1254004476896x_{43} = 90.1254004476896
x44=94.1655205728572x_{44} = 94.1655205728572
x45=−62.2255007657586x_{45} = -62.2255007657586
x46=−42.0000508927152x_{46} = -42.0000508927152
x47=56.0919338458764x_{47} = 56.0919338458764
x48=−67.922226198768x_{48} = -67.922226198768
x49=−3.7599424119465x_{49} = -3.7599424119465
x50=14.1241330177449x_{50} = 14.1241330177449
x51=−28.1089052148573x_{51} = -28.1089052148573
x52=−91.7491008963786x_{52} = -91.7491008963786
x53=−85.7673550856064x_{53} = -85.7673550856064
x54=−83.7472721892434x_{54} = -83.7472721892434
x55=−9.78869633477761x_{55} = -9.78869633477761
x56=−33.7465159227779x_{56} = -33.7465159227779
x57=40.6699954771946x_{57} = 40.6699954771946
x58=12.3437127194011x_{58} = 12.3437127194011
x59=92.1590784403794x_{59} = 92.1590784403794
x60=72.2478802037779x_{60} = 72.2478802037779
x61=−23.8129686089945x_{61} = -23.8129686089945
x62=−21.7441875995693x_{62} = -21.7441875995693
x63=80.0650973249266x_{63} = 80.0650973249266
x64=−25.8377328511584x_{64} = -25.8377328511584
x65=32.1738213359532x_{65} = 32.1738213359532
x66=54.1251803643413x_{66} = 54.1251803643413
x67=2.1708037636748x_{67} = 2.1708037636748
x68=75.9574545934608x_{68} = 75.9574545934608
x69=62.2507392582852x_{69} = 62.2507392582852
x70=−8.02512612976212x_{70} = -8.02512612976212
x71=27.9969170993996x_{71} = 27.9969170993996
x72=−16.0012437412711x_{72} = -16.0012437412711
x73=−45.7588407154027x_{73} = -45.7588407154027
x74=−29.8436176978044x_{74} = -29.8436176978044
x75=60.2503967723284x_{75} = 60.2503967723284
x76=−89.74114311439x_{76} = -89.74114311439
x77=78.1388848285651x_{77} = 78.1388848285651
x78=−93.8313029414246x_{78} = -93.8313029414246
x79=26.0797777885892x_{79} = 26.0797777885892
x80=−39.9687360864765x_{80} = -39.9687360864765
x81=20.2479095536667x_{81} = 20.2479095536667
x82=−59.7792743637658x_{82} = -59.7792743637658
x83=−63.7960287378891x_{83} = -63.7960287378891
x84=102.166244873501x_{84} = 102.166244873501
x85=−69.9726528768182x_{85} = -69.9726528768182
x86=−82.9746967475147x_{86} = -82.9746967475147
x87=−47.9382417919236x_{87} = -47.9382417919236
x88=−35.9113322545768x_{88} = -35.9113322545768
x89=16.0012437412711x_{89} = 16.0012437412711
x90=−87.9019176781946x_{90} = -87.9019176781946
x91=69.9950979945973x_{91} = 69.9950979945973
x92=−57.6933047803503x_{92} = -57.6933047803503
x93=84.2521726769473x_{93} = 84.2521726769473
x94=−13.0849837455246x_{94} = -13.0849837455246
x95=−2.1708037636748x_{95} = -2.1708037636748
x96=4.15677273792348x_{96} = 4.15677273792348
x97=68.3831899290199x_{97} = 68.3831899290199
x98=−100.037709879876x_{98} = -100.037709879876
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 cot(x^2).
cot⁡(02)\cot{\left(0^{2} \right)}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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(−cot⁡2(x2)−1)=02 x \left(- \cot^{2}{\left(x^{2} \right)} - 1\right) = 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
2(4x2(cot⁡2(x2)+1)cot⁡(x2)−cot⁡2(x2)−1)=02 \left(4 x^{2} \left(\cot^{2}{\left(x^{2} \right)} + 1\right) \cot{\left(x^{2} \right)} - \cot^{2}{\left(x^{2} \right)} - 1\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=45.9985057804882x_{1} = 45.9985057804882
x2=−9.78856305960857x_{2} = -9.78856305960857
x3=72.2478798723152x_{3} = 72.2478798723152
x4=−67.9453482609181x_{4} = -67.9453482609181
x5=−38.893144607161x_{5} = -38.893144607161
x6=2.15842031656434x_{6} = 2.15842031656434
x7=34.254702495525x_{7} = 34.254702495525
x8=−87.9376499827146x_{8} = -87.9376499827146
x9=−97.7504681816382x_{9} = -97.7504681816382
x10=80.0062184622719x_{10} = 80.0062184622719
x11=20.2478944955663x_{11} = 20.2478944955663
x12=12.3436462566901x_{12} = 12.3436462566901
x13=−75.7503725285458x_{13} = -75.7503725285458
x14=−91.7491007345316x_{14} = -91.7491007345316
x15=−3.75758735559995x_{15} = -3.75758735559995
x16=−51.7514023369347x_{16} = -51.7514023369347
x17=−57.7477317893834x_{17} = -57.7477317893834
x18=−14.0124326083036x_{18} = -14.0124326083036
x19=90.1428276001121x_{19} = 90.1428276001121
x20=88.2229888812924x_{20} = 88.2229888812924
x21=96.0320009554325x_{21} = 96.0320009554325
x22=62.2507387401102x_{22} = 62.2507387401102
x23=−85.7490383137135x_{23} = -85.7490383137135
x24=40.1256287139733x_{24} = 40.1256287139733
x25=−77.977897765382x_{25} = -77.977897765382
x26=16.0012132306729x_{26} = 16.0012132306729
x27=42.0000492055384x_{27} = 42.0000492055384
x28=−45.7588394107798x_{28} = -45.7588394107798
x29=32.173817582749x_{29} = 32.173817582749
x30=−5.74274694367216x_{30} = -5.74274694367216
x31=74.2211740185213x_{31} = 74.2211740185213
x32=58.019104216233x_{32} = 58.019104216233
x33=54.1251795760043x_{33} = 54.1251795760043
x34=4.15503066662393x_{34} = 4.15503066662393
x35=37.7869512658615x_{35} = 37.7869512658615
x36=−95.7535280406693x_{36} = -95.7535280406693
x37=−71.7460766477137x_{37} = -71.7460766477137
x38=−55.754872883948x_{38} = -55.754872883948
x39=−40.0080154571131x_{39} = -40.0080154571131
x40=82.4619746424392x_{40} = 82.4619746424392
x41=48.0037300937962x_{41} = 48.0037300937962
x42=51.9936571702341x_{42} = 51.9936571702341
x43=30.0011002714247x_{43} = 30.0011002714247
x44=−42.0000492055384x_{44} = -42.0000492055384
x45=68.2452273442545x_{45} = 68.2452273442545
x46=65.9986223856813x_{46} = 65.9986223856813
x47=−8.02488425811924x_{47} = -8.02488425811924
x48=−23.7469037558692x_{48} = -23.7469037558692
x49=−42.8516133870942x_{49} = -42.8516133870942
x50=−47.83983840802x_{50} = -47.83983840802
x51=−65.9986223856813x_{51} = -65.9986223856813
x52=−21.7441754410407x_{52} = -21.7441754410407
x53=99.9905925327219x_{53} = 99.9905925327219
x54=51.0177363026731x_{54} = 51.0177363026731
x55=−35.9987052085305x_{55} = -35.9987052085305
x56=24.0100357823293x_{56} = 24.0100357823293
x57=35.955043923882x_{57} = 35.955043923882
x58=−17.768772323048x_{58} = -17.768772323048
x59=84.2521724679374x_{59} = 84.2521724679374
x60=26.259840534989x_{60} = 26.259840534989
x61=−53.7757933570406x_{61} = -53.7757933570406
x62=94.2488896480008x_{62} = 94.2488896480008
x63=−20.0138062145804x_{63} = -20.0138062145804
x64=−79.7505754705589x_{64} = -79.7505754705589
x65=−90.0033145301847x_{65} = -90.0033145301847
x66=56.2038373604716x_{66} = 56.2038373604716
x67=−81.8693342414289x_{67} = -81.8693342414289
x68=92.1249831707776x_{68} = 92.1249831707776
x69=27.9969114032736x_{69} = 27.9969114032736
x70=8.2183052138805x_{70} = 8.2183052138805
x71=−69.770321522697x_{71} = -69.770321522697
x72=−83.7472719764304x_{72} = -83.7472719764304
x73=6.26606264138994x_{73} = 6.26606264138994
x74=−73.7541024874687x_{74} = -73.7541024874687
x75=−26.0194705476381x_{75} = -26.0194705476381
x76=63.9927024191448x_{76} = 63.9927024191448
x77=76.2464239323044x_{77} = 76.2464239323044
x78=86.0051153657684x_{78} = 86.0051153657684
x79=69.8603187519718x_{79} = 69.8603187519718
x80=10.2587025689995x_{80} = 10.2587025689995
x81=−94.3987687094098x_{81} = -94.3987687094098
x82=−2.15842031656434x_{82} = -2.15842031656434
x83=−63.7467649053452x_{83} = -63.7467649053452
x84=60.25039620081x_{84} = 60.25039620081
x85=−33.7465126702331x_{85} = -33.7465126702331
x86=−62.2255002469528x_{86} = -62.2255002469528
x87=77.9980392867871x_{87} = 77.9980392867871
x88=−31.7313755955179x_{88} = -31.7313755955179
x89=−50.1795417731822x_{89} = -50.1795417731822
x90=−59.752991396122x_{90} = -59.752991396122
x91=−27.9969114032736x_{91} = -27.9969114032736
x92=98.2473582456787x_{92} = 98.2473582456787
x93=−12.0864530375985x_{93} = -12.0864530375985
x94=−16.0012132306729x_{94} = -16.0012132306729
x95=−99.8176391013562x_{95} = -99.8176391013562
x96=−29.896200933167x_{96} = -29.896200933167
x97=18.0320290256093x_{97} = 18.0320290256093
x98=14.1240886540409x_{98} = 14.1240886540409
x99=22.2441079319026x_{99} = 22.2441079319026
x100=45.032187761776x_{100} = 45.032187761776

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
[−2.15842031656434,2.15842031656434]\left[-2.15842031656434, 2.15842031656434\right]
Convexa en los intervalos
(−∞,−99.8176391013562]\left(-\infty, -99.8176391013562\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞cot⁡(x2)=cot⁡(∞)\lim_{x \to -\infty} \cot{\left(x^{2} \right)} = \cot{\left(\infty \right)}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=cot⁡(∞)y = \cot{\left(\infty \right)}
lim⁡x→∞cot⁡(x2)=cot⁡(∞)\lim_{x \to \infty} \cot{\left(x^{2} \right)} = \cot{\left(\infty \right)}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=cot⁡(∞)y = \cot{\left(\infty \right)}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cot(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→−∞(cot⁡(x2)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(x^{2} \right)}}{x}\right)
True

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