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

Gráfico de la función y = x/cos(x)^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          x   
f(x) = -------
          2   
       cos (x)
f(x)=xcos⁡2(x)f{\left(x \right)} = \frac{x}{\cos^{2}{\left(x \right)}} 
f = x/cos(x)^2
Gráfico de la función
0.000.100.200.300.400.500.600.700.800.901.0005
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:
xcos⁡2(x)=0\frac{x}{\cos^{2}{\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/cos(x)^2.
0cos⁡2(0)\frac{0}{\cos^{2}{\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
2xsin⁡(x)cos⁡3(x)+1cos⁡2(x)=0\frac{2 x \sin{\left(x \right)}}{\cos^{3}{\left(x \right)}} + \frac{1}{\cos^{2}{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−69.1078034322536x_{1} = -69.1078034322536
x2=15.6760783451944x_{2} = 15.6760783451944
x3=50.2555336325565x_{3} = 50.2555336325565
x4=18.8229989180076x_{4} = 18.8229989180076
x5=12.5264763376692x_{5} = 12.5264763376692
x6=−75.3915917440781x_{6} = -75.3915917440781
x7=−100.525991117835x_{7} = -100.525991117835
x8=−47.1132774827275x_{8} = -47.1132774827275
x9=59.6818828624266x_{9} = 59.6818828624266
x10=94.2424741940464x_{10} = 94.2424741940464
x11=31.4000043168626x_{11} = 31.4000043168626
x12=2.97508632168828x_{12} = 2.97508632168828
x13=62.8238944845809x_{13} = 62.8238944845809
x14=72.2497107001058x_{14} = 72.2497107001058
x15=−28.2566407733299x_{15} = -28.2566407733299
x16=−59.6818828624266x_{16} = -59.6818828624266
x17=−81.6752872670354x_{17} = -81.6752872670354
x18=84.817106677999x_{18} = 84.817106677999
x19=43.9709264903445x_{19} = 43.9709264903445
x20=21.9683925318703x_{20} = 21.9683925318703
x21=−6.20274981679304x_{21} = -6.20274981679304
x22=100.525991117835x_{22} = 100.525991117835
x23=81.6752872670354x_{23} = 81.6752872670354
x24=56.5398246709304x_{24} = 56.5398246709304
x25=6.20274981679304x_{25} = 6.20274981679304
x26=−87.9589098892909x_{26} = -87.9589098892909
x27=47.1132774827275x_{27} = 47.1132774827275
x28=34.5430455066495x_{28} = 34.5430455066495
x29=−15.6760783451944x_{29} = -15.6760783451944
x30=−43.9709264903445x_{30} = -43.9709264903445
x31=−84.817106677999x_{31} = -84.817106677999
x32=69.1078034322536x_{32} = 69.1078034322536
x33=9.37147510585595x_{33} = 9.37147510585595
x34=37.6858450405302x_{34} = 37.6858450405302
x35=91.1006985770946x_{35} = 91.1006985770946
x36=−40.8284587489214x_{36} = -40.8284587489214
x37=78.5334497119428x_{37} = 78.5334497119428
x38=87.9589098892909x_{38} = 87.9589098892909
x39=−21.9683925318703x_{39} = -21.9683925318703
x40=−12.5264763376692x_{40} = -12.5264763376692
x41=−62.8238944845809x_{41} = -62.8238944845809
x42=−50.2555336325565x_{42} = -50.2555336325565
x43=28.2566407733299x_{43} = 28.2566407733299
x44=−94.2424741940464x_{44} = -94.2424741940464
x45=−78.5334497119428x_{45} = -78.5334497119428
x46=−53.397711687542x_{46} = -53.397711687542
x47=97.3842380053013x_{47} = 97.3842380053013
x48=25.1128337203766x_{48} = 25.1128337203766
x49=−2.97508632168828x_{49} = -2.97508632168828
x50=65.9658661929102x_{50} = 65.9658661929102
x51=−37.6858450405302x_{51} = -37.6858450405302
x52=−18.8229989180076x_{52} = -18.8229989180076
x53=75.3915917440781x_{53} = 75.3915917440781
x54=−97.3842380053013x_{54} = -97.3842380053013
x55=−65.9658661929102x_{55} = -65.9658661929102
x56=−72.2497107001058x_{56} = -72.2497107001058
x57=−9.37147510585595x_{57} = -9.37147510585595
x58=−34.5430455066495x_{58} = -34.5430455066495
x59=−25.1128337203766x_{59} = -25.1128337203766
x60=40.8284587489214x_{60} = 40.8284587489214
x61=−56.5398246709304x_{61} = -56.5398246709304
x62=−31.4000043168626x_{62} = -31.4000043168626
x63=53.397711687542x_{63} = 53.397711687542
x64=−91.1006985770946x_{64} = -91.1006985770946
Signos de extremos en los puntos:
(-69.10780343225363, -69.1114209687341)

(15.676078345194368, 15.692026211395)

(50.255533632556485, 50.2605082091241)

(18.822998918007553, 18.8362805423167)

(12.5264763376692, 12.5464340650668)

(-75.39159174407808, -75.3949077637325)

(-100.52599111783519, -100.528478036862)

(-47.11327748272753, -47.1185838424919)

(59.681882862426576, 59.6860717383134)

(94.24247419404638, 94.2451269257593)

(31.400004316862624, 31.4079660992075)

(2.9750863216882792, 3.05911749691083)

(62.82389448458093, 62.8278738622055)

(72.24971070010584, 72.2531709215735)

(-28.256640773329945, -28.2654882510611)

(-59.681882862426576, -59.6860717383134)

(-81.67528726703536, -81.6783481684214)

(84.817106677999, 84.8200541966045)

(43.97092649034452, 43.9766120653359)

(21.968392531870297, 21.9797725178951)

(-6.202749816793043, -6.2430545215424)

(100.52599111783519, 100.528478036862)

(81.67528726703536, 81.6783481684214)

(56.53982467093041, 56.5442463330324)

(6.202749816793043, 6.2430545215424)

(-87.95890988929088, -87.9617521255159)

(47.11327748272753, 47.1185838424919)

(34.54304550664949, 34.5502828534536)

(-15.676078345194368, -15.692026211395)

(-43.97092649034452, -43.9766120653359)

(-84.817106677999, -84.8200541966045)

(69.10780343225363, 69.1114209687341)

(9.371475105855954, 9.3981518026594)

(37.68584504053022, 37.6924788310086)

(91.10069857709462, 91.1034427931534)

(-40.8284587489214, -40.8345819288714)

(78.53344971194282, 78.5366330688553)

(87.95890988929088, 87.9617521255159)

(-21.968392531870297, -21.9797725178951)

(-12.5264763376692, -12.5464340650668)

(-62.82389448458093, -62.8278738622055)

(-50.255533632556485, -50.2605082091241)

(28.256640773329945, 28.2654882510611)

(-94.24247419404638, -94.2451269257593)

(-78.53344971194282, -78.5366330688553)

(-53.39771168754203, -53.40239353611)

(97.38423800530128, 97.3868051558497)

(25.112833720376596, 25.1227887896764)

(-2.9750863216882792, -3.05911749691083)

(65.96586619291024, 65.9696560317228)

(-37.68584504053022, -37.6924788310086)

(-18.822998918007553, -18.8362805423167)

(75.39159174407808, 75.3949077637325)

(-97.38423800530128, -97.3868051558497)

(-65.96586619291024, -65.9696560317228)

(-72.24971070010584, -72.2531709215735)

(-9.371475105855954, -9.3981518026594)

(-34.54304550664949, -34.5502828534536)

(-25.112833720376596, -25.1227887896764)

(40.8284587489214, 40.8345819288714)

(-56.53982467093041, -56.5442463330324)

(-31.400004316862624, -31.4079660992075)

(53.39771168754203, 53.40239353611)

(-91.10069857709462, -91.1034427931534)


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=15.6760783451944x_{1} = 15.6760783451944
x2=50.2555336325565x_{2} = 50.2555336325565
x3=18.8229989180076x_{3} = 18.8229989180076
x4=12.5264763376692x_{4} = 12.5264763376692
x5=59.6818828624266x_{5} = 59.6818828624266
x6=94.2424741940464x_{6} = 94.2424741940464
x7=31.4000043168626x_{7} = 31.4000043168626
x8=2.97508632168828x_{8} = 2.97508632168828
x9=62.8238944845809x_{9} = 62.8238944845809
x10=72.2497107001058x_{10} = 72.2497107001058
x11=84.817106677999x_{11} = 84.817106677999
x12=43.9709264903445x_{12} = 43.9709264903445
x13=21.9683925318703x_{13} = 21.9683925318703
x14=100.525991117835x_{14} = 100.525991117835
x15=81.6752872670354x_{15} = 81.6752872670354
x16=56.5398246709304x_{16} = 56.5398246709304
x17=6.20274981679304x_{17} = 6.20274981679304
x18=47.1132774827275x_{18} = 47.1132774827275
x19=34.5430455066495x_{19} = 34.5430455066495
x20=69.1078034322536x_{20} = 69.1078034322536
x21=9.37147510585595x_{21} = 9.37147510585595
x22=37.6858450405302x_{22} = 37.6858450405302
x23=91.1006985770946x_{23} = 91.1006985770946
x24=78.5334497119428x_{24} = 78.5334497119428
x25=87.9589098892909x_{25} = 87.9589098892909
x26=28.2566407733299x_{26} = 28.2566407733299
x27=97.3842380053013x_{27} = 97.3842380053013
x28=25.1128337203766x_{28} = 25.1128337203766
x29=65.9658661929102x_{29} = 65.9658661929102
x30=75.3915917440781x_{30} = 75.3915917440781
x31=40.8284587489214x_{31} = 40.8284587489214
x32=53.397711687542x_{32} = 53.397711687542
Puntos máximos de la función:
x32=−69.1078034322536x_{32} = -69.1078034322536
x32=−75.3915917440781x_{32} = -75.3915917440781
x32=−100.525991117835x_{32} = -100.525991117835
x32=−47.1132774827275x_{32} = -47.1132774827275
x32=−28.2566407733299x_{32} = -28.2566407733299
x32=−59.6818828624266x_{32} = -59.6818828624266
x32=−81.6752872670354x_{32} = -81.6752872670354
x32=−6.20274981679304x_{32} = -6.20274981679304
x32=−87.9589098892909x_{32} = -87.9589098892909
x32=−15.6760783451944x_{32} = -15.6760783451944
x32=−43.9709264903445x_{32} = -43.9709264903445
x32=−84.817106677999x_{32} = -84.817106677999
x32=−40.8284587489214x_{32} = -40.8284587489214
x32=−21.9683925318703x_{32} = -21.9683925318703
x32=−12.5264763376692x_{32} = -12.5264763376692
x32=−62.8238944845809x_{32} = -62.8238944845809
x32=−50.2555336325565x_{32} = -50.2555336325565
x32=−94.2424741940464x_{32} = -94.2424741940464
x32=−78.5334497119428x_{32} = -78.5334497119428
x32=−53.397711687542x_{32} = -53.397711687542
x32=−2.97508632168828x_{32} = -2.97508632168828
x32=−37.6858450405302x_{32} = -37.6858450405302
x32=−18.8229989180076x_{32} = -18.8229989180076
x32=−97.3842380053013x_{32} = -97.3842380053013
x32=−65.9658661929102x_{32} = -65.9658661929102
x32=−72.2497107001058x_{32} = -72.2497107001058
x32=−9.37147510585595x_{32} = -9.37147510585595
x32=−34.5430455066495x_{32} = -34.5430455066495
x32=−25.1128337203766x_{32} = -25.1128337203766
x32=−56.5398246709304x_{32} = -56.5398246709304
x32=−31.4000043168626x_{32} = -31.4000043168626
x32=−91.1006985770946x_{32} = -91.1006985770946
Decrece en los intervalos
[100.525991117835,∞)\left[100.525991117835, \infty\right)
Crece en los intervalos
[−2.97508632168828,2.97508632168828]\left[-2.97508632168828, 2.97508632168828\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
2(x(3sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))cos⁡2(x)=0\frac{2 \left(x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{\cos^{2}{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
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=1.5707963267949x_{1} = 1.5707963267949
x2=4.71238898038469x_{2} = 4.71238898038469

lim⁡x→1.5707963267949−(2(x(3sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))cos⁡2(x))=6.70421144615668⋅1065\lim_{x \to 1.5707963267949^-}\left(\frac{2 \left(x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{\cos^{2}{\left(x \right)}}\right) = 6.70421144615668 \cdot 10^{65}
lim⁡x→1.5707963267949+(2(x(3sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))cos⁡2(x))=6.70421144615668⋅1065\lim_{x \to 1.5707963267949^+}\left(\frac{2 \left(x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{\cos^{2}{\left(x \right)}}\right) = 6.70421144615668 \cdot 10^{65}
- los límites son iguales, es decir omitimos el punto correspondiente
lim⁡x→4.71238898038469−(2(x(3sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))cos⁡2(x))=2.48304127635433⋅1064\lim_{x \to 4.71238898038469^-}\left(\frac{2 \left(x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{\cos^{2}{\left(x \right)}}\right) = 2.48304127635433 \cdot 10^{64}
lim⁡x→4.71238898038469+(2(x(3sin⁡2(x)cos⁡2(x)+1)+2sin⁡(x)cos⁡(x))cos⁡2(x))=2.48304127635433⋅1064\lim_{x \to 4.71238898038469^+}\left(\frac{2 \left(x \left(\frac{3 \sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1\right) + \frac{2 \sin{\left(x \right)}}{\cos{\left(x \right)}}\right)}{\cos^{2}{\left(x \right)}}\right) = 2.48304127635433 \cdot 10^{64}
- 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
[0,∞)\left[0, \infty\right)
Convexa en los intervalos
(−∞,0]\left(-\infty, 0\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→−∞(xcos⁡2(x))y = \lim_{x \to -\infty}\left(\frac{x}{\cos^{2}{\left(x \right)}}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(xcos⁡2(x))y = \lim_{x \to \infty}\left(\frac{x}{\cos^{2}{\left(x \right)}}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función x/cos(x)^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞1cos⁡2(x)=⟨0,∞⟩\lim_{x \to -\infty} \frac{1}{\cos^{2}{\left(x \right)}} = \left\langle 0, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨0,∞⟩xy = \left\langle 0, \infty\right\rangle x
lim⁡x→∞1cos⁡2(x)=⟨0,∞⟩\lim_{x \to \infty} \frac{1}{\cos^{2}{\left(x \right)}} = \left\langle 0, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨0,∞⟩xy = \left\langle 0, \infty\right\rangle x
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:
xcos⁡2(x)=−xcos⁡2(x)\frac{x}{\cos^{2}{\left(x \right)}} = - \frac{x}{\cos^{2}{\left(x \right)}}
- No
xcos⁡2(x)=xcos⁡2(x)\frac{x}{\cos^{2}{\left(x \right)}} = \frac{x}{\cos^{2}{\left(x \right)}}
- No
es decir, función
no es
par ni impar
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