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Trazado el gráfico de la función/ acot(x)/cos(x)

Gráfico de la función y = acot(x)/cos(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       acot(x)
f(x) = -------
        cos(x)
f(x)=acot⁡(x)cos⁡(x)f{\left(x \right)} = \frac{\operatorname{acot}{\left(x \right)}}{\cos{\left(x \right)}} 
f = acot(x)/cos(x)
Gráfico de la función
02468-8-6-4-2-1010-100100
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469
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:
acot⁡(x)cos⁡(x)=0\frac{\operatorname{acot}{\left(x \right)}}{\cos{\left(x \right)}} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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 acot(x)/cos(x).
acot⁡(0)cos⁡(0)\frac{\operatorname{acot}{\left(0 \right)}}{\cos{\left(0 \right)}}
Resultado:
f(0)=π2f{\left(0 \right)} = \frac{\pi}{2}
Punto:
(0, pi/2)
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
sin⁡(x)acot⁡(x)cos⁡2(x)−1(x2+1)cos⁡(x)=0\frac{\sin{\left(x \right)} \operatorname{acot}{\left(x \right)}}{\cos^{2}{\left(x \right)}} - \frac{1}{\left(x^{2} + 1\right) \cos{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−56.5663405998409x_{1} = -56.5663405998409
x2=31.4476932635934x_{2} = 31.4476932635934
x3=44.0050101088746x_{3} = 44.0050101088746
x4=−72.2704652951335x_{4} = -72.2704652951335
x5=100.540910131064x_{5} = 100.540910131064
x6=62.8477605108147x_{6} = 62.8477605108147
x7=59.7070041757719x_{7} = 59.7070041757719
x8=−40.8651605782548x_{8} = -40.8651605782548
x9=84.8347876265671x_{9} = 84.8347876265671
x10=3.41222718253932x_{10} = 3.41222718253932
x11=50.2853611006615x_{11} = 50.2853611006615
x12=−28.3096135754975x_{12} = -28.3096135754975
x13=−44.0050101088746x_{13} = -44.0050101088746
x14=−65.9885963799416x_{14} = -65.9885963799416
x15=−9.52858748961069x_{15} = -9.52858748961069
x16=15.7711168638648x_{16} = 15.7711168638648
x17=78.552544609485x_{17} = 78.552544609485
x18=−37.7256004362115x_{18} = -37.7256004362115
x19=69.1295009571176x_{19} = 69.1295009571176
x20=6.43495952623824x_{20} = 6.43495952623824
x21=−97.3996381577911x_{21} = -97.3996381577911
x22=−78.552544609485x_{22} = -78.552544609485
x23=56.5663405998409x_{23} = 56.5663405998409
x24=−25.1724047189968x_{24} = -25.1724047189968
x25=34.5864081402316x_{25} = 34.5864081402316
x26=−94.2583875492313x_{26} = -94.2583875492313
x27=28.3096135754975x_{27} = 28.3096135754975
x28=−69.1295009571176x_{28} = -69.1295009571176
x29=−91.1171605135005x_{29} = -91.1171605135005
x30=−18.9023120326678x_{30} = -18.9023120326678
x31=−87.9759595737791x_{31} = -87.9759595737791
x32=−47.1450913828748x_{32} = -47.1450913828748
x33=25.1724047189968x_{33} = 25.1724047189968
x34=94.2583875492313x_{34} = 94.2583875492313
x35=91.1171605135005x_{35} = 91.1171605135005
x36=53.4257861100249x_{36} = 53.4257861100249
x37=−50.2853611006615x_{37} = -50.2853611006615
x38=−75.4114819351114x_{38} = -75.4114819351114
x39=75.4114819351114x_{39} = 75.4114819351114
x40=−3.41222718253932x_{40} = -3.41222718253932
x41=−81.6936480133593x_{41} = -81.6936480133593
x42=22.0364347942926x_{42} = 22.0364347942926
x43=−15.7711168638648x_{43} = -15.7711168638648
x44=−22.0364347942926x_{44} = -22.0364347942926
x45=−100.540910131064x_{45} = -100.540910131064
x46=87.9759595737791x_{46} = 87.9759595737791
x47=37.7256004362115x_{47} = 37.7256004362115
x48=81.6936480133593x_{48} = 81.6936480133593
x49=−12.6449633157096x_{49} = -12.6449633157096
x50=97.3996381577911x_{50} = 97.3996381577911
x51=65.9885963799416x_{51} = 65.9885963799416
x52=72.2704652951335x_{52} = 72.2704652951335
x53=9.52858748961069x_{53} = 9.52858748961069
x54=18.9023120326678x_{54} = 18.9023120326678
x55=−6.43495952623824x_{55} = -6.43495952623824
x56=40.8651605782548x_{56} = 40.8651605782548
x57=0.618285031575678x_{57} = 0.618285031575678
x58=−34.5864081402316x_{58} = -34.5864081402316
x59=12.6449633157096x_{59} = 12.6449633157096
x60=−84.8347876265671x_{60} = -84.8347876265671
x61=−53.4257861100249x_{61} = -53.4257861100249
x62=−59.7070041757719x_{62} = -59.7070041757719
x63=−31.4476932635934x_{63} = -31.4476932635934
x64=47.1450913828748x_{64} = 47.1450913828748
x65=−62.8477605108147x_{65} = -62.8477605108147
Signos de extremos en los puntos:
(-56.56634059984088, -0.0176792771595112)

(31.44769326359337, 0.0318041690805221)

(44.005010108874586, 0.0227266364452673)

(-72.27046529513353, 0.013837352203916)

(100.54091013106385, 0.00994636391387385)

(62.847760510814744, 0.0159121365515481)

(59.707004175771914, -0.0167492358049121)

(-40.86516057825484, 0.0244731579892852)

(84.83478762656715, -0.0117878899838998)

(3.4122271825393202, -0.295849648991141)

(50.285361100661476, 0.0198878117162592)

(-28.3096135754975, 0.0353309938310325)

(-44.005010108874586, -0.0227266364452673)

(-65.98859637994163, 0.0151547129278715)

(-9.528587489610688, 0.105130537633322)

(15.771116863864762, -0.0634487663771082)

(78.55254460948498, -0.0127306759480261)

(-37.72560043621149, -0.0265102932706506)

(69.12950095711761, 0.0144661082255695)

(6.434959526238242, 0.155960863934427)

(-97.3996381577911, 0.0102671588823597)

(-78.55254460948498, 0.0127306759480261)

(56.56634059984088, 0.0176792771595112)

(-25.17240471899681, -0.03973641590568)

(34.586408140231576, -0.028917105084519)

(-94.25838754923129, -0.0106093343558632)

(28.3096135754975, -0.0353309938310325)

(-69.12950095711761, -0.0144661082255695)

(-91.11716051350052, 0.0109751012656029)

(-18.902312032667847, -0.052927946264819)

(-87.97595957377905, -0.0113669862311388)

(-47.14509138287476, 0.0212127032217431)

(25.17240471899681, 0.03973641590568)

(94.25838754923129, 0.0106093343558632)

(91.11716051350052, -0.0109751012656029)

(53.42578611002485, -0.018718644514749)

(-50.285361100661476, -0.0198878117162592)

(-75.41148193511141, -0.0132609684634528)

(75.41148193511141, 0.0132609684634528)

(-3.4122271825393202, 0.295849648991141)

(-81.69364801335925, -0.0122411592624557)

(22.036434794292596, -0.0453948209266728)

(-15.771116863864762, 0.0634487663771082)

(-22.036434794292596, 0.0453948209266728)

(-100.54091013106385, -0.00994636391387385)

(87.97595957377905, 0.0113669862311388)

(37.72560043621149, 0.0265102932706506)

(81.69364801335925, 0.0122411592624557)

(-12.6449633157096, -0.0791629842530595)

(97.3996381577911, -0.0102671588823597)

(65.98859637994163, -0.0151547129278715)

(72.27046529513353, -0.013837352203916)

(9.528587489610688, -0.105130537633322)

(18.902312032667847, 0.052927946264819)

(-6.434959526238242, -0.155960863934427)

(40.86516057825484, -0.0244731579892852)

(0.6182850315756779, 1.24809563946823)

(-34.586408140231576, 0.028917105084519)

(12.6449633157096, 0.0791629842530595)

(-84.83478762656715, 0.0117878899838998)

(-53.42578611002485, 0.018718644514749)

(-59.707004175771914, 0.0167492358049121)

(-31.44769326359337, -0.0318041690805221)

(47.14509138287476, -0.0212127032217431)

(-62.847760510814744, -0.0159121365515481)


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=31.4476932635934x_{1} = 31.4476932635934
x2=44.0050101088746x_{2} = 44.0050101088746
x3=−72.2704652951335x_{3} = -72.2704652951335
x4=100.540910131064x_{4} = 100.540910131064
x5=62.8477605108147x_{5} = 62.8477605108147
x6=−40.8651605782548x_{6} = -40.8651605782548
x7=50.2853611006615x_{7} = 50.2853611006615
x8=−28.3096135754975x_{8} = -28.3096135754975
x9=−65.9885963799416x_{9} = -65.9885963799416
x10=−9.52858748961069x_{10} = -9.52858748961069
x11=69.1295009571176x_{11} = 69.1295009571176
x12=6.43495952623824x_{12} = 6.43495952623824
x13=−97.3996381577911x_{13} = -97.3996381577911
x14=−78.552544609485x_{14} = -78.552544609485
x15=56.5663405998409x_{15} = 56.5663405998409
x16=−91.1171605135005x_{16} = -91.1171605135005
x17=−47.1450913828748x_{17} = -47.1450913828748
x18=25.1724047189968x_{18} = 25.1724047189968
x19=94.2583875492313x_{19} = 94.2583875492313
x20=75.4114819351114x_{20} = 75.4114819351114
x21=−3.41222718253932x_{21} = -3.41222718253932
x22=−15.7711168638648x_{22} = -15.7711168638648
x23=−22.0364347942926x_{23} = -22.0364347942926
x24=87.9759595737791x_{24} = 87.9759595737791
x25=37.7256004362115x_{25} = 37.7256004362115
x26=81.6936480133593x_{26} = 81.6936480133593
x27=18.9023120326678x_{27} = 18.9023120326678
x28=0.618285031575678x_{28} = 0.618285031575678
x29=−34.5864081402316x_{29} = -34.5864081402316
x30=12.6449633157096x_{30} = 12.6449633157096
x31=−84.8347876265671x_{31} = -84.8347876265671
x32=−53.4257861100249x_{32} = -53.4257861100249
x33=−59.7070041757719x_{33} = -59.7070041757719
Puntos máximos de la función:
x33=−56.5663405998409x_{33} = -56.5663405998409
x33=59.7070041757719x_{33} = 59.7070041757719
x33=84.8347876265671x_{33} = 84.8347876265671
x33=3.41222718253932x_{33} = 3.41222718253932
x33=−44.0050101088746x_{33} = -44.0050101088746
x33=15.7711168638648x_{33} = 15.7711168638648
x33=78.552544609485x_{33} = 78.552544609485
x33=−37.7256004362115x_{33} = -37.7256004362115
x33=−25.1724047189968x_{33} = -25.1724047189968
x33=34.5864081402316x_{33} = 34.5864081402316
x33=−94.2583875492313x_{33} = -94.2583875492313
x33=28.3096135754975x_{33} = 28.3096135754975
x33=−69.1295009571176x_{33} = -69.1295009571176
x33=−18.9023120326678x_{33} = -18.9023120326678
x33=−87.9759595737791x_{33} = -87.9759595737791
x33=91.1171605135005x_{33} = 91.1171605135005
x33=53.4257861100249x_{33} = 53.4257861100249
x33=−50.2853611006615x_{33} = -50.2853611006615
x33=−75.4114819351114x_{33} = -75.4114819351114
x33=−81.6936480133593x_{33} = -81.6936480133593
x33=22.0364347942926x_{33} = 22.0364347942926
x33=−100.540910131064x_{33} = -100.540910131064
x33=−12.6449633157096x_{33} = -12.6449633157096
x33=97.3996381577911x_{33} = 97.3996381577911
x33=65.9885963799416x_{33} = 65.9885963799416
x33=72.2704652951335x_{33} = 72.2704652951335
x33=9.52858748961069x_{33} = 9.52858748961069
x33=−6.43495952623824x_{33} = -6.43495952623824
x33=40.8651605782548x_{33} = 40.8651605782548
x33=−31.4476932635934x_{33} = -31.4476932635934
x33=47.1450913828748x_{33} = 47.1450913828748
x33=−62.8477605108147x_{33} = -62.8477605108147
Decrece en los intervalos
[100.540910131064,∞)\left[100.540910131064, \infty\right)
Crece en los intervalos
(−∞,−97.3996381577911]\left(-\infty, -97.3996381577911\right]
Asíntotas verticales
Hay:
x1=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469
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→−∞(acot⁡(x)cos⁡(x))y = \lim_{x \to -\infty}\left(\frac{\operatorname{acot}{\left(x \right)}}{\cos{\left(x \right)}}\right)
True

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

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