Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ atan(6*x)*cot(2*x)

Gráfico de la función y = atan(6*x)*cot(2*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = atan(6*x)*cot(2*x)
f(x)=cot⁡(2x)atan⁡(6x)f{\left(x \right)} = \cot{\left(2 x \right)} \operatorname{atan}{\left(6 x \right)} 
f = cot(2*x)*atan(6*x)
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⁡(2x)atan⁡(6x)=0\cot{\left(2 x \right)} \operatorname{atan}{\left(6 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−3π4x_{1} = - \frac{3 \pi}{4}
x2=−π4x_{2} = - \frac{\pi}{4}
x3=π4x_{3} = \frac{\pi}{4}
x4=3π4x_{4} = \frac{3 \pi}{4}
Solución numérica
x1=−7.06858347057703x_{1} = -7.06858347057703
x2=32.2013246992954x_{2} = 32.2013246992954
x3=66.7588438887831x_{3} = 66.7588438887831
x4=−27.4889357189107x_{4} = -27.4889357189107
x5=10.2101761241668x_{5} = 10.2101761241668
x6=−32.2013246992954x_{6} = -32.2013246992954
x7=−41.6261026600648x_{7} = -41.6261026600648
x8=55.7632696012188x_{8} = 55.7632696012188
x9=−38.484510006475x_{9} = -38.484510006475
x10=44.7676953136546x_{10} = 44.7676953136546
x11=30.6305283725005x_{11} = 30.6305283725005
x12=−21.2057504117311x_{12} = -21.2057504117311
x13=27.4889357189107x_{13} = 27.4889357189107
x14=−19.6349540849362x_{14} = -19.6349540849362
x15=−52.621676947629x_{15} = -52.621676947629
x16=−3.92699081698724x_{16} = -3.92699081698724
x17=49.4800842940392x_{17} = 49.4800842940392
x18=82.4668071567321x_{18} = 82.4668071567321
x19=−69.9004365423729x_{19} = -69.9004365423729
x20=93.4623814442964x_{20} = 93.4623814442964
x21=52.621676947629x_{21} = 52.621676947629
x22=−77.7544181763474x_{22} = -77.7544181763474
x23=88.7499924639117x_{23} = 88.7499924639117
x24=16.4933614313464x_{24} = 16.4933614313464
x25=41.6261026600648x_{25} = 41.6261026600648
x26=96.6039740978861x_{26} = 96.6039740978861
x27=−91.8915851175014x_{27} = -91.8915851175014
x28=−74.6128255227576x_{28} = -74.6128255227576
x29=−46.3384916404494x_{29} = -46.3384916404494
x30=−5.49778714378214x_{30} = -5.49778714378214
x31=−84.037603483527x_{31} = -84.037603483527
x32=91.8915851175014x_{32} = 91.8915851175014
x33=−30.6305283725005x_{33} = -30.6305283725005
x34=77.7544181763474x_{34} = 77.7544181763474
x35=−55.7632696012188x_{35} = -55.7632696012188
x36=84.037603483527x_{36} = 84.037603483527
x37=−71.4712328691678x_{37} = -71.4712328691678
x38=8.63937979737193x_{38} = 8.63937979737193
x39=−76.1836218495525x_{39} = -76.1836218495525
x40=−65.1880475619882x_{40} = -65.1880475619882
x41=60.4756585816035x_{41} = 60.4756585816035
x42=3.92699081698724x_{42} = 3.92699081698724
x43=38.484510006475x_{43} = 38.484510006475
x44=90.3207887907066x_{44} = 90.3207887907066
x45=−90.3207887907066x_{45} = -90.3207887907066
x46=−79.3252145031423x_{46} = -79.3252145031423
x47=−16.4933614313464x_{47} = -16.4933614313464
x48=98.174770424681x_{48} = 98.174770424681
x49=−18.0641577581413x_{49} = -18.0641577581413
x50=76.1836218495525x_{50} = 76.1836218495525
x51=−13.3517687777566x_{51} = -13.3517687777566
x52=24.3473430653209x_{52} = 24.3473430653209
x53=−40.0553063332699x_{53} = -40.0553063332699
x54=−82.4668071567321x_{54} = -82.4668071567321
x55=80.8960108299372x_{55} = 80.8960108299372
x56=22.776546738526x_{56} = 22.776546738526
x57=−98.174770424681x_{57} = -98.174770424681
x58=−68.329640215578x_{58} = -68.329640215578
x59=46.3384916404494x_{59} = 46.3384916404494
x60=40.0553063332699x_{60} = 40.0553063332699
x61=−33.7721210260903x_{61} = -33.7721210260903
x62=−25.9181393921158x_{62} = -25.9181393921158
x63=−96.6039740978861x_{63} = -96.6039740978861
x64=−60.4756585816035x_{64} = -60.4756585816035
x65=−49.4800842940392x_{65} = -49.4800842940392
x66=−87.1791961371168x_{66} = -87.1791961371168
x67=−57.3340659280137x_{67} = -57.3340659280137
x68=25.9181393921158x_{68} = 25.9181393921158
x69=85.6083998103219x_{69} = 85.6083998103219
x70=−10.2101761241668x_{70} = -10.2101761241668
x71=−35.3429173528852x_{71} = -35.3429173528852
x72=−99.7455667514759x_{72} = -99.7455667514759
x73=58.9048622548086x_{73} = 58.9048622548086
x74=74.6128255227576x_{74} = 74.6128255227576
x75=62.0464549083984x_{75} = 62.0464549083984
x76=54.1924732744239x_{76} = 54.1924732744239
x77=−24.3473430653209x_{77} = -24.3473430653209
x78=68.329640215578x_{78} = 68.329640215578
x79=63.6172512351933x_{79} = 63.6172512351933
x80=19.6349540849362x_{80} = 19.6349540849362
x81=−85.6083998103219x_{81} = -85.6083998103219
x82=18.0641577581413x_{82} = 18.0641577581413
x83=36.9137136796801x_{83} = 36.9137136796801
x84=−93.4623814442964x_{84} = -93.4623814442964
x85=−54.1924732744239x_{85} = -54.1924732744239
x86=71.4712328691678x_{86} = 71.4712328691678
x87=−43.1968989868597x_{87} = -43.1968989868597
x88=2.35619449019234x_{88} = 2.35619449019234
x89=99.7455667514759x_{89} = 99.7455667514759
x90=−11.7809724509617x_{90} = -11.7809724509617
x91=11.7809724509617x_{91} = 11.7809724509617
x92=−62.0464549083984x_{92} = -62.0464549083984
x93=69.9004365423729x_{93} = 69.9004365423729
x94=−63.6172512351933x_{94} = -63.6172512351933
x95=−2.35619449019234x_{95} = -2.35619449019234
x96=33.7721210260903x_{96} = 33.7721210260903
x97=47.9092879672443x_{97} = 47.9092879672443
x98=5.49778714378214x_{98} = 5.49778714378214
x99=14.9225651045515x_{99} = 14.9225651045515
x100=−47.9092879672443x_{100} = -47.9092879672443
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 atan(6*x)*cot(2*x).
cot⁡(0⋅2)atan⁡(0⋅6)\cot{\left(0 \cdot 2 \right)} \operatorname{atan}{\left(0 \cdot 6 \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
(−2cot⁡2(2x)−2)atan⁡(6x)+6cot⁡(2x)36x2+1=0\left(- 2 \cot^{2}{\left(2 x \right)} - 2\right) \operatorname{atan}{\left(6 x \right)} + \frac{6 \cot{\left(2 x \right)}}{36 x^{2} + 1} = 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
8(−54xcot⁡(2x)(36x2+1)2+(cot⁡2(2x)+1)cot⁡(2x)atan⁡(6x)−3(cot⁡2(2x)+1)36x2+1)=08 \left(- \frac{54 x \cot{\left(2 x \right)}}{\left(36 x^{2} + 1\right)^{2}} + \left(\cot^{2}{\left(2 x \right)} + 1\right) \cot{\left(2 x \right)} \operatorname{atan}{\left(6 x \right)} - \frac{3 \left(\cot^{2}{\left(2 x \right)} + 1\right)}{36 x^{2} + 1}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−76.1836172728896x_{1} = -76.1836172728896
x2=−35.342896053799x_{2} = -35.342896053799
x3=40.0552897567363x_{3} = 40.0552897567363
x4=33.7720976963927x_{4} = 33.7720976963927
x5=−57.3340578436775x_{5} = -57.3340578436775
x6=3.92522378309214x_{6} = 3.92522378309214
x7=19.6348849121048x_{7} = 19.6348849121048
x8=−85.6083961864475x_{8} = -85.6083961864475
x9=99.7455640825119x_{9} = 99.7455640825119
x10=11.7807796230453x_{10} = 11.7807796230453
x11=−33.7720976963927x_{11} = -33.7720976963927
x12=−74.6128207512255x_{12} = -74.6128207512255
x13=66.7588379275038x_{13} = 66.7588379275038
x14=82.4668032513103x_{14} = 82.4668032513103
x15=63.617244670096x_{15} = 63.617244670096
x16=−32.2012990341245x_{16} = -32.2012990341245
x17=74.6128207512255x_{17} = 74.6128207512255
x18=24.3472981241373x_{18} = 24.3472981241373
x19=76.1836172728896x_{19} = 76.1836172728896
x20=−84.0375997228279x_{20} = -84.0375997228279
x21=5.49689266980637x_{21} = 5.49689266980637
x22=84.0375997228279x_{22} = 84.0375997228279
x23=30.6305000027684x_{23} = 30.6305000027684
x24=−41.626087312506x_{24} = -41.626087312506
x25=−13.3516188082001x_{25} = -13.3516188082001
x26=−65.1880413097189x_{26} = -65.1880413097189
x27=−90.3207855353206x_{27} = -90.3207855353206
x28=−3.92522378309214x_{28} = -3.92522378309214
x29=−49.4800734363937x_{29} = -49.4800734363937
x30=10.2099190514777x_{30} = 10.2099190514777
x31=44.7676820469119x_{31} = 44.7676820469119
x32=47.9092763851422x_{32} = 47.9092763851422
x33=−11.7807796230453x_{33} = -11.7807796230453
x34=38.4844920472151x_{34} = 38.4844920472151
x35=−87.1791926427337x_{35} = -87.1791926427337
x36=−18.0640759944023x_{36} = -18.0640759944023
x37=18.0640759944023x_{37} = 18.0640759944023
x38=49.4800734363937x_{38} = 49.4800734363937
x39=41.626087312506x_{39} = 41.626087312506
x40=25.9180997436115x_{40} = 25.9180997436115
x41=27.4889004803409x_{41} = 27.4889004803409
x42=−60.475651316076x_{42} = -60.475651316076
x43=85.6083961864475x_{43} = 85.6083961864475
x44=−68.3296345254368x_{44} = -68.3296345254368
x45=58.9048545962636x_{45} = 58.9048545962636
x46=98.1747676695802x_{46} = 98.1747676695802
x47=−46.3384792588889x_{47} = -46.3384792588889
x48=32.2012990341245x_{48} = 32.2012990341245
x49=−38.4844920472151x_{49} = -38.4844920472151
x50=−47.9092763851422x_{50} = -47.9092763851422
x51=22.776495369582x_{51} = 22.776495369582
x52=62.0464480063937x_{52} = 62.0464480063937
x53=55.7632610545696x_{53} = 55.7632610545696
x54=−98.1747676695802x_{54} = -98.1747676695802
x55=−93.4623784042012x_{55} = -93.4623784042012
x56=14.922445144379x_{56} = 14.922445144379
x57=−30.6305000027684x_{57} = -30.6305000027684
x58=−27.4889004803409x_{58} = -27.4889004803409
x59=90.3207855353206x_{59} = 90.3207855353206
x60=−25.9180997436115x_{60} = -25.9180997436115
x61=36.9136941571839x_{61} = 36.9136941571839
x62=93.4623784042012x_{62} = 93.4623784042012
x63=60.475651316076x_{63} = 60.475651316076
x64=71.4712276686177x_{64} = 71.4712276686177
x65=−16.4932632979721x_{65} = -16.4932632979721
x66=−55.7632610545696x_{66} = -55.7632610545696
x67=−10.2099190514777x_{67} = -10.2099190514777
x68=2.3511844143667x_{68} = 2.3511844143667
x69=68.3296345254368x_{69} = 68.3296345254368
x70=−19.6348849121048x_{70} = -19.6348849121048
x71=−5.49689266980637x_{71} = -5.49689266980637
x72=−8.63902007900938x_{72} = -8.63902007900938
x73=−2.3511844143667x_{73} = -2.3511844143667
x74=16.4932632979721x_{74} = 16.4932632979721
x75=−63.617244670096x_{75} = -63.617244670096
x76=46.3384792588889x_{76} = 46.3384792588889
x77=−62.0464480063937x_{77} = -62.0464480063937
x78=−21.205691130554x_{78} = -21.205691130554
x79=−77.7544137828553x_{79} = -77.7544137828553
x80=−99.7455640825119x_{80} = -99.7455640825119
x81=80.8960067712752x_{81} = 80.8960067712752
x82=−79.3252102820406x_{82} = -79.3252102820406
x83=−43.1968847365023x_{83} = -43.1968847365023
x84=52.6216673489391x_{84} = 52.6216673489391
x85=−24.3472981241373x_{85} = -24.3472981241373
x86=−91.8915819725224x_{86} = -91.8915819725224
x87=−54.1924642246406x_{87} = -54.1924642246406
x88=−71.4712276686177x_{88} = -71.4712276686177
x89=−96.6039712524103x_{89} = -96.6039712524103
x90=88.7499890922012x_{90} = 88.7499890922012
x91=8.63902007900938x_{91} = 8.63902007900938
x92=−52.6216673489391x_{92} = -52.6216673489391
x93=96.6039712524103x_{93} = 96.6039712524103
x94=−82.4668032513103x_{94} = -82.4668032513103
x95=−69.9004311052837x_{95} = -69.9004311052837
x96=69.9004311052837x_{96} = 69.9004311052837
x97=−40.0552897567363x_{97} = -40.0552897567363
x98=77.7544137828553x_{98} = 77.7544137828553
x99=54.1924642246406x_{99} = 54.1924642246406
x100=91.8915819725224x_{100} = 91.8915819725224

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
[−2.3511844143667,2.3511844143667]\left[-2.3511844143667, 2.3511844143667\right]
Convexa en los intervalos
(−∞,−99.7455640825119]\left(-\infty, -99.7455640825119\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→−∞(cot⁡(2x)atan⁡(6x))y = \lim_{x \to -\infty}\left(\cot{\left(2 x \right)} \operatorname{atan}{\left(6 x \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(cot⁡(2x)atan⁡(6x))y = \lim_{x \to \infty}\left(\cot{\left(2 x \right)} \operatorname{atan}{\left(6 x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función atan(6*x)*cot(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→−∞(cot⁡(2x)atan⁡(6x)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(2 x \right)} \operatorname{atan}{\left(6 x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(cot⁡(2x)atan⁡(6x)x)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(2 x \right)} \operatorname{atan}{\left(6 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:
cot⁡(2x)atan⁡(6x)=cot⁡(2x)atan⁡(6x)\cot{\left(2 x \right)} \operatorname{atan}{\left(6 x \right)} = \cot{\left(2 x \right)} \operatorname{atan}{\left(6 x \right)}
- Sí
cot⁡(2x)atan⁡(6x)=−cot⁡(2x)atan⁡(6x)\cot{\left(2 x \right)} \operatorname{atan}{\left(6 x \right)} = - \cot{\left(2 x \right)} \operatorname{atan}{\left(6 x \right)}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор