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Gráfico de la función y = sin(4*a)-cos(4*a)+cos(2*a)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src]
f(a) = sin(4*a) - cos(4*a) + cos(2*a)
f(a)=(sin(4a)cos(4a))+cos(2a)f{\left(a \right)} = \left(\sin{\left(4 a \right)} - \cos{\left(4 a \right)}\right) + \cos{\left(2 a \right)}
f = sin(4*a) - cos(4*a) + cos(2*a)
Gráfico de la función
02468-8-6-4-2-10105-5
Puntos de cruce con el eje de coordenadas X
El gráfico de la función cruce el eje A con f = 0
o sea hay que resolver la ecuación:
(sin(4a)cos(4a))+cos(2a)=0\left(\sin{\left(4 a \right)} - \cos{\left(4 a \right)}\right) + \cos{\left(2 a \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje A:

Solución analítica
a1=0a_{1} = 0
a2=πa_{2} = \pi
Solución numérica
a1=34.0706943074751a_{1} = 34.0706943074751
a2=64.0611580490107a_{2} = -64.0611580490107
a3=13.795675591574a_{3} = -13.795675591574
a4=39.9099728008373a_{4} = -39.9099728008373
a5=51.4947874346515a_{5} = -51.4947874346515
a6=96.1600672840687a_{6} = 96.1600672840687
a7=72.2566310325652a_{7} = -72.2566310325652
a8=84.3361767649118a_{8} = 84.3361767649118
a9=19.7802876173688a_{9} = 19.7802876173688
a10=59.6902604182061a_{10} = -59.6902604182061
a11=76.3289553819851a_{11} = 76.3289553819851
a12=80.4521040161198a_{12} = 80.4521040161198
a13=72.2566310325652a_{13} = 72.2566310325652
a14=2.21086095775973a_{14} = -2.21086095775973
a15=56.0618428826036a_{15} = 56.0618428826036
a16=10.3555096565994a_{16} = 10.3555096565994
a17=94.2477796076938a_{17} = 94.2477796076938
a18=57.7779727418311a_{18} = -57.7779727418311
a19=47.6107146858595a_{19} = -47.6107146858595
a20=4.07232434941985a_{20} = 4.07232434941985
a21=37.2122869610649a_{21} = 37.2122869610649
a22=87.9645943005142a_{22} = -87.9645943005142
a23=45.894584826632a_{23} = 45.894584826632
a24=68.1843066831454a_{24} = -68.1843066831454
a25=3.62841753560244a_{25} = -3.62841753560244
a26=0a_{26} = 0
a27=17.9188242257087a_{27} = -17.9188242257087
a28=12.0795457323465a_{28} = 12.0795457323465
a29=100.044140032861a_{29} = 100.044140032861
a30=50.2654824574367a_{30} = 50.2654824574367
a31=31.9027514179106a_{31} = -31.9027514179106
a32=48.054621499677a_{32} = 48.054621499677
a33=35.7868241667026a_{33} = -35.7868241667026
a34=70.0457700748055a_{34} = 70.0457700748055
a35=14.4786582907341a_{35} = 14.4786582907341
a36=79.7691213169597a_{36} = -79.7691213169597
a37=30.1866215586831a_{37} = 30.1866215586831
a38=85.7537333427545a_{38} = 85.7537333427545
a39=21.9911485751286a_{39} = -21.9911485751286
a40=7.51249028439444a_{40} = -7.51249028439444
a41=36.4698068658627a_{41} = 36.4698068658627
a42=23.9034362515035a_{42} = 23.9034362515035
a43=78.0529914577322a_{43} = 78.0529914577322
a44=59.2034355361934a_{44} = 59.2034355361934
a45=83.8922699510944a_{45} = -83.8922699510944
a46=87.9645943005142a_{46} = 87.9645943005142
a47=52.1777701338116a_{47} = 52.1777701338116
a48=61.9011213759658a_{48} = -61.9011213759658
a49=29.503638859523a_{49} = -29.503638859523
a50=77.6090846439148a_{50} = -77.6090846439148
a51=32.346658231728a_{51} = 32.346658231728
a52=46.1931581080168a_{52} = -46.1931581080168
a53=15.2211383859363a_{53} = 15.2211383859363
a54=43.9822971502571a_{54} = -43.9822971502571
a55=37.6991118430775a_{55} = -37.6991118430775
a56=65.9734457253857a_{56} = -65.9734457253857
a57=41.7714361924974a_{57} = 41.7714361924974
a58=63.7625847676259a_{58} = 63.7625847676259
a59=6.28318530717959a_{59} = 6.28318530717959
a60=89.8768819768892a_{60} = 89.8768819768892
a61=28.2743338823081a_{61} = 28.2743338823081
a62=91.5930118361167a_{62} = -91.5930118361167
a63=8.19547298355452a_{63} = 8.19547298355452
a64=6.77001018919224a_{64} = -6.77001018919224
a65=94.2477796076938a_{65} = -94.2477796076938
a66=99.6002332190433a_{66} = -99.6002332190433
a67=24.2020095328883a_{67} = -24.2020095328883
a68=33.6267874936577a_{68} = -33.6267874936577
a69=25.619566110731a_{69} = -25.619566110731
a70=43.9822971502571a_{70} = 43.9822971502571
a71=40.3538796146547a_{71} = 40.3538796146547
a72=54.3378068068565a_{72} = 54.3378068068565
a73=65.9734457253857a_{73} = 65.9734457253857
a74=18.3627310395261a_{74} = 18.3627310395261
a75=11.6356389185291a_{75} = -11.6356389185291
a76=15.707963267949a_{76} = -15.707963267949
a77=50.2654824574367a_{77} = -50.2654824574367
a78=90.1754552582739a_{78} = -90.1754552582739
a79=26.0634729245484a_{79} = 26.0634729245484
a80=62.3450281897832a_{80} = 62.3450281897832
a81=20.0788608987536a_{81} = -20.0788608987536
a82=58.4609554409912a_{82} = 58.4609554409912
a83=55.6179360687862a_{83} = -55.6179360687862
a84=95.4770845849087a_{84} = -95.4770845849087
a85=9.91160284278203a_{85} = -9.91160284278203
a86=98.3201039571136a_{86} = 98.3201039571136
a87=74.1689187089402a_{87} = 74.1689187089402
a88=21.9911485751286a_{88} = 21.9911485751286
a89=86.0523066241393a_{89} = -86.0523066241393
a90=42.0700094738822a_{90} = -42.0700094738822
a91=73.4859360097801a_{91} = -73.4859360097801
a92=1.91228767637494a_{92} = 1.91228767637494
a93=28.2743338823081a_{93} = -28.2743338823081
a94=67.8857334017606a_{94} = 67.8857334017606
a95=97.8761971432962a_{95} = -97.8761971432962
a96=92.0369186499341a_{96} = 92.0369186499341
a97=69.6018632609881a_{97} = -69.6018632609881
a98=53.8938999930391a_{98} = -53.8938999930391
a99=81.6814089933346a_{99} = -81.6814089933346
a100=75.8850485681677a_{100} = -75.8850485681677
Puntos de cruce con el eje de coordenadas Y
El gráfico cruce el eje Y cuando a es igual a 0:
sustituimos a = 0 en sin(4*a) - cos(4*a) + cos(2*a).
(cos(04)+sin(04))+cos(02)\left(- \cos{\left(0 \cdot 4 \right)} + \sin{\left(0 \cdot 4 \right)}\right) + \cos{\left(0 \cdot 2 \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
ddaf(a)=0\frac{d}{d a} f{\left(a \right)} = 0
(la derivada es igual a cero),
y las raíces de esta ecuación serán los extremos de esta función:
ddaf(a)=\frac{d}{d a} f{\left(a \right)} =
primera derivada
2sin(2a)+4sin(4a)+4cos(4a)=0- 2 \sin{\left(2 a \right)} + 4 \sin{\left(4 a \right)} + 4 \cos{\left(4 a \right)} = 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
d2da2f(a)=0\frac{d^{2}}{d a^{2}} f{\left(a \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:
d2da2f(a)=\frac{d^{2}}{d a^{2}} f{\left(a \right)} =
segunda derivada
4(4sin(4a)cos(2a)+4cos(4a))=04 \left(- 4 \sin{\left(4 a \right)} - \cos{\left(2 a \right)} + 4 \cos{\left(4 a \right)}\right) = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con a->+oo y a->-oo
lima((sin(4a)cos(4a))+cos(2a))=3,3\lim_{a \to -\infty}\left(\left(\sin{\left(4 a \right)} - \cos{\left(4 a \right)}\right) + \cos{\left(2 a \right)}\right) = \left\langle -3, 3\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=3,3y = \left\langle -3, 3\right\rangle
lima((sin(4a)cos(4a))+cos(2a))=3,3\lim_{a \to \infty}\left(\left(\sin{\left(4 a \right)} - \cos{\left(4 a \right)}\right) + \cos{\left(2 a \right)}\right) = \left\langle -3, 3\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=3,3y = \left\langle -3, 3\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(4*a) - cos(4*a) + cos(2*a), dividida por a con a->+oo y a ->-oo
lima((sin(4a)cos(4a))+cos(2a)a)=0\lim_{a \to -\infty}\left(\frac{\left(\sin{\left(4 a \right)} - \cos{\left(4 a \right)}\right) + \cos{\left(2 a \right)}}{a}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lima((sin(4a)cos(4a))+cos(2a)a)=0\lim_{a \to \infty}\left(\frac{\left(\sin{\left(4 a \right)} - \cos{\left(4 a \right)}\right) + \cos{\left(2 a \right)}}{a}\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(-a) и f = -f(-a).
Pues, comprobamos:
(sin(4a)cos(4a))+cos(2a)=sin(4a)+cos(2a)cos(4a)\left(\sin{\left(4 a \right)} - \cos{\left(4 a \right)}\right) + \cos{\left(2 a \right)} = - \sin{\left(4 a \right)} + \cos{\left(2 a \right)} - \cos{\left(4 a \right)}
- No
(sin(4a)cos(4a))+cos(2a)=sin(4a)cos(2a)+cos(4a)\left(\sin{\left(4 a \right)} - \cos{\left(4 a \right)}\right) + \cos{\left(2 a \right)} = \sin{\left(4 a \right)} - \cos{\left(2 a \right)} + \cos{\left(4 a \right)}
- No
es decir, función
no es
par ni impar