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

Gráfico de la función y = x^2/cos(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          2  
         x   
f(x) = ------
       cos(x)
f(x)=x2cos⁡(x)f{\left(x \right)} = \frac{x^{2}}{\cos{\left(x \right)}} 
f = x^2/cos(x)
Gráfico de la función
02468-8-6-4-2-1010-25002500
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:
x2cos⁡(x)=0\frac{x^{2}}{\cos{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=0x_{1} = 0
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^2/cos(x).
02cos⁡(0)\frac{0^{2}}{\cos{\left(0 \right)}}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
x2sin⁡(x)cos⁡2(x)+2xcos⁡(x)=0\frac{x^{2} \sin{\left(x \right)}}{\cos^{2}{\left(x \right)}} + \frac{2 x}{\cos{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=56.5132926241755x_{1} = 56.5132926241755
x2=18.7432530945386x_{2} = 18.7432530945386
x3=34.4996123350132x_{3} = 34.4996123350132
x4=−53.3696181339615x_{4} = -53.3696181339615
x5=−97.3688346960149x_{5} = -97.3688346960149
x6=−50.2256832197934x_{6} = -50.2256832197934
x7=40.7917141624847x_{7} = 40.7917141624847
x8=−9.21096438740149x_{8} = -9.21096438740149
x9=100.511069234565x_{9} = 100.511069234565
x10=28.2035393053095x_{10} = 28.2035393053095
x11=59.6567478435559x_{11} = 59.6567478435559
x12=21.9000773156394x_{12} = 21.9000773156394
x13=91.0842327848165x_{13} = 91.0842327848165
x14=−94.2265573558031x_{14} = -94.2265573558031
x15=37.6460352959305x_{15} = 37.6460352959305
x16=5.95939190757933x_{16} = 5.95939190757933
x17=−31.3522215217643x_{17} = -31.3522215217643
x18=−34.4996123350132x_{18} = -34.4996123350132
x19=69.0860970774096x_{19} = 69.0860970774096
x20=−2.45871417599962x_{20} = -2.45871417599962
x21=−47.0814357397523x_{21} = -47.0814357397523
x22=25.053079662454x_{22} = 25.053079662454
x23=65.9431258539286x_{23} = 65.9431258539286
x24=75.3716947511882x_{24} = 75.3716947511882
x25=−100.511069234565x_{25} = -100.511069234565
x26=31.3522215217643x_{26} = 31.3522215217643
x27=−81.6569211705466x_{27} = -81.6569211705466
x28=−56.5132926241755x_{28} = -56.5132926241755
x29=0x_{29} = 0
x30=−75.3716947511882x_{30} = -75.3716947511882
x31=53.3696181339615x_{31} = 53.3696181339615
x32=50.2256832197934x_{32} = 50.2256832197934
x33=81.6569211705466x_{33} = 81.6569211705466
x34=−59.6567478435559x_{34} = -59.6567478435559
x35=−78.5143487963623x_{35} = -78.5143487963623
x36=87.9418559209576x_{36} = 87.9418559209576
x37=−65.9431258539286x_{37} = -65.9431258539286
x38=−72.2289483771681x_{38} = -72.2289483771681
x39=72.2289483771681x_{39} = 72.2289483771681
x40=−5.95939190757933x_{40} = -5.95939190757933
x41=−87.9418559209576x_{41} = -87.9418559209576
x42=−62.8000167068325x_{42} = -62.8000167068325
x43=−84.7994209518635x_{43} = -84.7994209518635
x44=−15.5802941824244x_{44} = -15.5802941824244
x45=9.21096438740149x_{45} = 9.21096438740149
x46=97.3688346960149x_{46} = 97.3688346960149
x47=78.5143487963623x_{47} = 78.5143487963623
x48=62.8000167068325x_{48} = 62.8000167068325
x49=47.0814357397523x_{49} = 47.0814357397523
x50=−28.2035393053095x_{50} = -28.2035393053095
x51=84.7994209518635x_{51} = 84.7994209518635
x52=−69.0860970774096x_{52} = -69.0860970774096
x53=−21.9000773156394x_{53} = -21.9000773156394
x54=−18.7432530945386x_{54} = -18.7432530945386
x55=−40.7917141624847x_{55} = -40.7917141624847
x56=2.45871417599962x_{56} = 2.45871417599962
x57=−12.4065403639626x_{57} = -12.4065403639626
x58=−91.0842327848165x_{58} = -91.0842327848165
x59=12.4065403639626x_{59} = 12.4065403639626
x60=−43.9368086315937x_{60} = -43.9368086315937
x61=94.2265573558031x_{61} = 94.2265573558031
x62=−25.053079662454x_{62} = -25.053079662454
x63=15.5802941824244x_{63} = 15.5802941824244
x64=43.9368086315937x_{64} = 43.9368086315937
x65=−37.6460352959305x_{65} = -37.6460352959305
Signos de extremos en los puntos:
(56.513292624175506, 3195.75161739489)

(18.74325309453857, 353.303875761973)

(34.4996123350132, -1192.22157372685)

(-53.36961813396146, -2850.31543808823)

(-97.36883469601494, -9482.68975914927)

(-50.2256832197934, 2524.61846269625)

(40.79171416248471, -1665.96274380725)

(-9.210964387401486, -86.8188315245924)

(100.51106923456473, 10104.4748407434)

(28.20353930530947, -797.437121315344)

(59.656747843555884, -3560.92700161815)

(21.90007731563936, -481.609233704586)

(91.08423278481655, -8298.33722098652)

(-94.22655735580307, 8880.64388591755)

(37.64603529593052, 1419.22256428091)

(5.9593919075793265, 37.4610010422361)

(-31.352221521764292, 984.959763812075)

(-34.4996123350132, -1192.22157372685)

(69.08609707740959, 4774.88839053132)

(-2.4587141759996247, -7.79271815542963)

(-47.081435739752315, -2218.66068987187)

(25.053079662453992, 629.653624231737)

(65.94312585392862, -4350.49538766933)

(75.37169475118824, 5682.89201773282)

(-100.51106923456473, 10104.4748407434)

(31.352221521764292, 984.959763812075)

(-81.65692117054658, 6669.85247519616)

(-56.513292624175506, 3195.75161739489)

(0, 0)

(-75.37169475118824, 5682.89201773282)

(53.36961813396146, -2850.31543808823)

(50.2256832197934, 2524.61846269625)

(81.65692117054658, 6669.85247519616)

(-59.656747843555884, -3560.92700161815)

(-78.51434879636227, -6166.50264258389)

(87.94185592095755, 7735.76976428322)

(-65.94312585392862, -4350.49538766933)

(-72.22894837716808, -5219.02060045796)

(72.22894837716808, -5219.02060045796)

(-5.9593919075793265, 37.4610010422361)

(-87.94185592095755, 7735.76976428322)

(-62.80001670683253, 3945.84159151576)

(-84.79942095186354, -7192.941515721)

(-15.580294182424433, -244.737394922768)

(9.210964387401486, -86.8188315245924)

(97.36883469601494, -9482.68975914927)

(78.51434879636227, -6166.50264258389)

(62.80001670683253, 3945.84159151576)

(47.081435739752315, -2218.66068987187)

(-28.20353930530947, -797.437121315344)

(84.79942095186354, -7192.941515721)

(-69.08609707740959, 4774.88839053132)

(-21.90007731563936, -481.609233704586)

(-18.74325309453857, 353.303875761973)

(-40.79171416248471, -1665.96274380725)

(2.4587141759996247, -7.79271815542963)

(-12.406540363962565, 155.909416368761)

(-91.08423278481655, -8298.33722098652)

(12.406540363962565, 155.909416368761)

(-43.936808631593706, 1932.44211776972)

(94.22655735580307, 8880.64388591755)

(-25.053079662453992, 629.653624231737)

(15.580294182424433, -244.737394922768)

(43.936808631593706, 1932.44211776972)

(-37.64603529593052, 1419.22256428091)


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=56.5132926241755x_{1} = 56.5132926241755
x2=18.7432530945386x_{2} = 18.7432530945386
x3=−50.2256832197934x_{3} = -50.2256832197934
x4=100.511069234565x_{4} = 100.511069234565
x5=−94.2265573558031x_{5} = -94.2265573558031
x6=37.6460352959305x_{6} = 37.6460352959305
x7=5.95939190757933x_{7} = 5.95939190757933
x8=−31.3522215217643x_{8} = -31.3522215217643
x9=69.0860970774096x_{9} = 69.0860970774096
x10=25.053079662454x_{10} = 25.053079662454
x11=75.3716947511882x_{11} = 75.3716947511882
x12=−100.511069234565x_{12} = -100.511069234565
x13=31.3522215217643x_{13} = 31.3522215217643
x14=−81.6569211705466x_{14} = -81.6569211705466
x15=−56.5132926241755x_{15} = -56.5132926241755
x16=0x_{16} = 0
x17=−75.3716947511882x_{17} = -75.3716947511882
x18=50.2256832197934x_{18} = 50.2256832197934
x19=81.6569211705466x_{19} = 81.6569211705466
x20=87.9418559209576x_{20} = 87.9418559209576
x21=−5.95939190757933x_{21} = -5.95939190757933
x22=−87.9418559209576x_{22} = -87.9418559209576
x23=−62.8000167068325x_{23} = -62.8000167068325
x24=62.8000167068325x_{24} = 62.8000167068325
x25=−69.0860970774096x_{25} = -69.0860970774096
x26=−18.7432530945386x_{26} = -18.7432530945386
x27=−12.4065403639626x_{27} = -12.4065403639626
x28=12.4065403639626x_{28} = 12.4065403639626
x29=−43.9368086315937x_{29} = -43.9368086315937
x30=94.2265573558031x_{30} = 94.2265573558031
x31=−25.053079662454x_{31} = -25.053079662454
x32=43.9368086315937x_{32} = 43.9368086315937
x33=−37.6460352959305x_{33} = -37.6460352959305
Puntos máximos de la función:
x33=34.4996123350132x_{33} = 34.4996123350132
x33=−53.3696181339615x_{33} = -53.3696181339615
x33=−97.3688346960149x_{33} = -97.3688346960149
x33=40.7917141624847x_{33} = 40.7917141624847
x33=−9.21096438740149x_{33} = -9.21096438740149
x33=28.2035393053095x_{33} = 28.2035393053095
x33=59.6567478435559x_{33} = 59.6567478435559
x33=21.9000773156394x_{33} = 21.9000773156394
x33=91.0842327848165x_{33} = 91.0842327848165
x33=−34.4996123350132x_{33} = -34.4996123350132
x33=−2.45871417599962x_{33} = -2.45871417599962
x33=−47.0814357397523x_{33} = -47.0814357397523
x33=65.9431258539286x_{33} = 65.9431258539286
x33=53.3696181339615x_{33} = 53.3696181339615
x33=−59.6567478435559x_{33} = -59.6567478435559
x33=−78.5143487963623x_{33} = -78.5143487963623
x33=−65.9431258539286x_{33} = -65.9431258539286
x33=−72.2289483771681x_{33} = -72.2289483771681
x33=72.2289483771681x_{33} = 72.2289483771681
x33=−84.7994209518635x_{33} = -84.7994209518635
x33=−15.5802941824244x_{33} = -15.5802941824244
x33=9.21096438740149x_{33} = 9.21096438740149
x33=97.3688346960149x_{33} = 97.3688346960149
x33=78.5143487963623x_{33} = 78.5143487963623
x33=47.0814357397523x_{33} = 47.0814357397523
x33=−28.2035393053095x_{33} = -28.2035393053095
x33=84.7994209518635x_{33} = 84.7994209518635
x33=−21.9000773156394x_{33} = -21.9000773156394
x33=−40.7917141624847x_{33} = -40.7917141624847
x33=2.45871417599962x_{33} = 2.45871417599962
x33=−91.0842327848165x_{33} = -91.0842327848165
x33=15.5802941824244x_{33} = 15.5802941824244
Decrece en los intervalos
[100.511069234565,∞)\left[100.511069234565, \infty\right)
Crece en los intervalos
(−∞,−100.511069234565]\left(-\infty, -100.511069234565\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
x2(2sin⁡2(x)cos⁡2(x)+1)+4xsin⁡(x)cos⁡(x)+2cos⁡(x)=0\frac{x^{2} \left(\frac{2 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{4 x \sin{\left(x \right)}}{\cos{\left(x \right)}} + 2}{\cos{\left(x \right)}} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
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→−∞(x2cos⁡(x))y = \lim_{x \to -\infty}\left(\frac{x^{2}}{\cos{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(x2cos⁡(x))y = \lim_{x \to \infty}\left(\frac{x^{2}}{\cos{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x^2/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→−∞(xcos⁡(x))y = x \lim_{x \to -\infty}\left(\frac{x}{\cos{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(xcos⁡(x))y = x \lim_{x \to \infty}\left(\frac{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:
x2cos⁡(x)=x2cos⁡(x)\frac{x^{2}}{\cos{\left(x \right)}} = \frac{x^{2}}{\cos{\left(x \right)}}
- Sí
x2cos⁡(x)=−x2cos⁡(x)\frac{x^{2}}{\cos{\left(x \right)}} = - \frac{x^{2}}{\cos{\left(x \right)}}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = x^2/cos(x)
    © Sr Examen – Калькулятор