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

Gráfico de la función y = cos(x)*sin(2*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = cos(x)*sin(2*x)
f(x)=sin⁡(2x)cos⁡(x)f{\left(x \right)} = \sin{\left(2 x \right)} \cos{\left(x \right)} 
f = sin(2*x)*cos(x)
Gráfico de la función
0.00.51.01.52.02.53.03.54.04.55.05.56.02-2
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:
sin⁡(2x)cos⁡(x)=0\sin{\left(2 x \right)} \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Solución numérica
x1=−83.2522054524035x_{1} = -83.2522054524035
x2=70.6858345559153x_{2} = 70.6858345559153
x3=81.6814089933346x_{3} = 81.6814089933346
x4=−36.128315423197x_{4} = -36.128315423197
x5=20.4203521537986x_{5} = 20.4203521537986
x6=−80.1106125824842x_{6} = -80.1106125824842
x7=59.6902604182061x_{7} = 59.6902604182061
x8=73.8274274722061x_{8} = 73.8274274722061
x9=48.6946859820148x_{9} = 48.6946859820148
x10=34.5575191894877x_{10} = 34.5575191894877
x11=−21.9911485751286x_{11} = -21.9911485751286
x12=15.707963267949x_{12} = 15.707963267949
x13=−81.6814089933346x_{13} = -81.6814089933346
x14=80.1106131546315x_{14} = 80.1106131546315
x15=95.8185760508519x_{15} = 95.8185760508519
x16=−20.4203520921076x_{16} = -20.4203520921076
x17=−94.2477796076938x_{17} = -94.2477796076938
x18=−95.818575585294x_{18} = -95.818575585294
x19=−59.6902604182061x_{19} = -59.6902604182061
x20=−43.9822971502571x_{20} = -43.9822971502571
x21=−61.261056881309x_{21} = -61.261056881309
x22=−31.4159265358979x_{22} = -31.4159265358979
x23=42.4115007327518x_{23} = 42.4115007327518
x24=12.5663706143592x_{24} = 12.5663706143592
x25=4.71238883532779x_{25} = 4.71238883532779
x26=43.9822971502571x_{26} = 43.9822971502571
x27=−29.8451300981866x_{27} = -29.8451300981866
x28=−67.5442421609972x_{28} = -67.5442421609972
x29=−15.707963267949x_{29} = -15.707963267949
x30=9.42477796076938x_{30} = 9.42477796076938
x31=23.5619450555027x_{31} = 23.5619450555027
x32=−1.57079642505341x_{32} = -1.57079642505341
x33=92.6769831301454x_{33} = 92.6769831301454
x34=0x_{34} = 0
x35=−65.9734457253857x_{35} = -65.9734457253857
x36=36.1283160593477x_{36} = 36.1283160593477
x37=−28.2743338823081x_{37} = -28.2743338823081
x38=−39.2699083096144x_{38} = -39.2699083096144
x39=−50.2654824574367x_{39} = -50.2654824574367
x40=−42.4115006663339x_{40} = -42.4115006663339
x41=−17.278759737384x_{41} = -17.278759737384
x42=29.8451303144929x_{42} = 29.8451303144929
x43=−75.398223686155x_{43} = -75.398223686155
x44=−95.8185758682892x_{44} = -95.8185758682892
x45=28.2743338823081x_{45} = 28.2743338823081
x46=−89.5353907394375x_{46} = -89.5353907394375
x47=65.9734457253857x_{47} = 65.9734457253857
x48=67.5442422018325x_{48} = 67.5442422018325
x49=−92.6769836764771x_{49} = -92.6769836764771
x50=64.4026493118058x_{50} = 64.4026493118058
x51=−51.8362786906154x_{51} = -51.8362786906154
x52=−7.85398150264842x_{52} = -7.85398150264842
x53=−87.9645943005142x_{53} = -87.9645943005142
x54=14.1371670924752x_{54} = 14.1371670924752
x55=−6.28318530717959x_{55} = -6.28318530717959
x56=−23.561945003804x_{56} = -23.561945003804
x57=−70.6858349962623x_{57} = -70.6858349962623
x58=7.85398164444075x_{58} = 7.85398164444075
x59=26.7035374084741x_{59} = 26.7035374084741
x60=−73.8274272804402x_{60} = -73.8274272804402
x61=−97.3893722612836x_{61} = -97.3893722612836
x62=94.2477796076938x_{62} = 94.2477796076938
x63=72.2566310325652x_{63} = 72.2566310325652
x64=45.5530936288414x_{64} = 45.5530936288414
x65=−9.42477796076938x_{65} = -9.42477796076938
x66=56.5486677646163x_{66} = 56.5486677646163
x67=−86.3937978155375x_{67} = -86.3937978155375
x68=100.530964914873x_{68} = 100.530964914873
x69=1.57079648184495x_{69} = 1.57079648184495
x70=6.28318530717959x_{70} = 6.28318530717959
x71=86.3937978909611x_{71} = 86.3937978909611
x72=−53.4070751110265x_{72} = -53.4070751110265
x73=89.5353907744432x_{73} = 89.5353907744432
x74=21.9911485751286x_{74} = 21.9911485751286
x75=87.9645943005142x_{75} = 87.9645943005142
x76=−72.2566310325652x_{76} = -72.2566310325652
x77=37.6991118430775x_{77} = 37.6991118430775
x78=−14.13716684381x_{78} = -14.13716684381
x79=50.2654824574367x_{79} = 50.2654824574367
x80=−64.4026492408158x_{80} = -64.4026492408158
x81=7.85398173541774x_{81} = 7.85398173541774
x82=−37.6991118430775x_{82} = -37.6991118430775
x83=78.5398163397448x_{83} = 78.5398163397448
x84=51.8362788934209x_{84} = 51.8362788934209
x85=−58.1194640027517x_{85} = -58.1194640027517
x86=−45.5530935824522x_{86} = -45.5530935824522
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 cos(x)*sin(2*x).
sin⁡(0⋅2)cos⁡(0)\sin{\left(0 \cdot 2 \right)} \cos{\left(0 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
−(4sin⁡(x)cos⁡(2x)+5sin⁡(2x)cos⁡(x))=0- (4 \sin{\left(x \right)} \cos{\left(2 x \right)} + 5 \sin{\left(2 x \right)} \cos{\left(x \right)}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=πx_{2} = \pi
x3=i(log⁡(9)−log⁡(−5−214i))2x_{3} = \frac{i \left(\log{\left(9 \right)} - \log{\left(-5 - 2 \sqrt{14} i \right)}\right)}{2}
x4=i(log⁡(9)−log⁡(−5+214i))2x_{4} = \frac{i \left(\log{\left(9 \right)} - \log{\left(-5 + 2 \sqrt{14} i \right)}\right)}{2}
x5=−ilog⁡(−−5−214i3)x_{5} = - i \log{\left(- \frac{\sqrt{-5 - 2 \sqrt{14} i}}{3} \right)}
x6=−ilog⁡(−−5+214i3)x_{6} = - i \log{\left(- \frac{\sqrt{-5 + 2 \sqrt{14} i}}{3} \right)}

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
[π,∞)\left[\pi, \infty\right)
Convexa en los intervalos
(−∞,−π2+atan⁡(2145)2]\left(-\infty, - \frac{\pi}{2} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{14}}{5} \right)}}{2}\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(2x)cos⁡(x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞(sin⁡(2x)cos⁡(x))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin{\left(2 x \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x)*sin(2*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(2x)cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(2x)cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la izquierda
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:
sin⁡(2x)cos⁡(x)=−sin⁡(2x)cos⁡(x)\sin{\left(2 x \right)} \cos{\left(x \right)} = - \sin{\left(2 x \right)} \cos{\left(x \right)}
- No
sin⁡(2x)cos⁡(x)=sin⁡(2x)cos⁡(x)\sin{\left(2 x \right)} \cos{\left(x \right)} = \sin{\left(2 x \right)} \cos{\left(x \right)}
- Sí
es decir, función
es
impar
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