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sin(x)+sin(2*x)+sin(3*x)
Trazado el gráfico de la función/ sin(x)+sin(2*x)+sin(3*x)

Gráfico de la función y = sin(x)+sin(2*x)+sin(3*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = sin(x) + sin(2*x) + sin(3*x)
f(x)=(sin⁡(x)+sin⁡(2x))+sin⁡(3x)f{\left(x \right)} = \left(\sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \sin{\left(3 x \right)} 
f = sin(x) + sin(2*x) + sin(3*x)
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 X con f = 0
o sea hay que resolver la ecuación:
(sin⁡(x)+sin⁡(2x))+sin⁡(3x)=0\left(\sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \sin{\left(3 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−2π3x_{2} = - \frac{2 \pi}{3}
x3=−π2x_{3} = - \frac{\pi}{2}
x4=π2x_{4} = \frac{\pi}{2}
x5=2π3x_{5} = \frac{2 \pi}{3}
x6=πx_{6} = \pi
Solución numérica
x1=−39.7935069454707x_{1} = -39.7935069454707
x2=73.8274273593601x_{2} = 73.8274273593601
x3=48.1710873550435x_{3} = 48.1710873550435
x4=70.6858347057703x_{4} = 70.6858347057703
x5=−64.4026493985908x_{5} = -64.4026493985908
x6=14.1371669411541x_{6} = 14.1371669411541
x7=−41.8879020478639x_{7} = -41.8879020478639
x8=90.0589894029074x_{8} = 90.0589894029074
x9=4.18879020478639x_{9} = 4.18879020478639
x10=14.6607657167524x_{10} = 14.6607657167524
x11=−53.4070751110265x_{11} = -53.4070751110265
x12=−67.5442420521806x_{12} = -67.5442420521806
x13=52.3598775598299x_{13} = 52.3598775598299
x14=−114.668131856027x_{14} = -114.668131856027
x15=−20.4203522483337x_{15} = -20.4203522483337
x16=28.2743338823081x_{16} = 28.2743338823081
x17=50.2654824574367x_{17} = 50.2654824574367
x18=36.1283155162826x_{18} = 36.1283155162826
x19=−2.0943951023932x_{19} = -2.0943951023932
x20=−90.0589894029074x_{20} = -90.0589894029074
x21=−33.5103216382911x_{21} = -33.5103216382911
x22=34.5575191894877x_{22} = 34.5575191894877
x23=−29.845130209103x_{23} = -29.845130209103
x24=65.9734457253857x_{24} = 65.9734457253857
x25=0x_{25} = 0
x26=23.5619449019235x_{26} = 23.5619449019235
x27=20.4203522483337x_{27} = 20.4203522483337
x28=78.5398163397448x_{28} = 78.5398163397448
x29=98.4365698124802x_{29} = 98.4365698124802
x30=59.6902604182061x_{30} = 59.6902604182061
x31=100.530964914873x_{31} = 100.530964914873
x32=−85.870199198121x_{32} = -85.870199198121
x33=−72.2566310325652x_{33} = -72.2566310325652
x34=21.9911485751286x_{34} = 21.9911485751286
x35=−17.2787595947439x_{35} = -17.2787595947439
x36=84.8230016469244x_{36} = 84.8230016469244
x37=−79.5870138909414x_{37} = -79.5870138909414
x38=79.5870138909414x_{38} = 79.5870138909414
x39=−37.6991118430775x_{39} = -37.6991118430775
x40=−81.6814089933346x_{40} = -81.6814089933346
x41=−21.9911485751286x_{41} = -21.9911485751286
x42=26.7035375555132x_{42} = 26.7035375555132
x43=−46.0766922526503x_{43} = -46.0766922526503
x44=−58.1194640914112x_{44} = -58.1194640914112
x45=−4.18879020478639x_{45} = -4.18879020478639
x46=64.4026493985908x_{46} = 64.4026493985908
x47=12.5663706143592x_{47} = 12.5663706143592
x48=−87.9645943005142x_{48} = -87.9645943005142
x49=−3.14159265358979x_{49} = -3.14159265358979
x50=−77.4926187885482x_{50} = -77.4926187885482
x51=41.8879020478639x_{51} = 41.8879020478639
x52=−14.1371669411541x_{52} = -14.1371669411541
x53=−94.2477796076938x_{53} = -94.2477796076938
x54=−51.8362787842316x_{54} = -51.8362787842316
x55=15.707963267949x_{55} = 15.707963267949
x56=−43.9822971502571x_{56} = -43.9822971502571
x57=39.7935069454707x_{57} = 39.7935069454707
x58=−6.28318530717959x_{58} = -6.28318530717959
x59=58.1194640914112x_{59} = 58.1194640914112
x60=−83.7758040957278x_{60} = -83.7758040957278
x61=−28.2743338823081x_{61} = -28.2743338823081
x62=−34.5575191894877x_{62} = -34.5575191894877
x63=−95.8185759344887x_{63} = -95.8185759344887
x64=81.6814089933346x_{64} = 81.6814089933346
x65=−75.398223686155x_{65} = -75.398223686155
x66=94.2477796076938x_{66} = 94.2477796076938
x67=−48.1710873550435x_{67} = -48.1710873550435
x68=46.0766922526503x_{68} = 46.0766922526503
x69=−59.6902604182061x_{69} = -59.6902604182061
x70=87.9645943005142x_{70} = 87.9645943005142
x71=−92.1533845053006x_{71} = -92.1533845053006
x72=2.0943951023932x_{72} = 2.0943951023932
x73=−15.707963267949x_{73} = -15.707963267949
x74=−23.5619449019235x_{74} = -23.5619449019235
x75=83.7758040957278x_{75} = 83.7758040957278
x76=−61.261056745001x_{76} = -61.261056745001
x77=−7.85398163397448x_{77} = -7.85398163397448
x78=67.5442420521806x_{78} = 67.5442420521806
x79=80.1106126665397x_{79} = 80.1106126665397
x80=96.342174710087x_{80} = 96.342174710087
x81=6.28318530717959x_{81} = 6.28318530717959
x82=29.845130209103x_{82} = 29.845130209103
x83=−50.2654824574367x_{83} = -50.2654824574367
x84=54.4542726622231x_{84} = 54.4542726622231
x85=−73.8274273593601x_{85} = -73.8274273593601
x86=37.6991118430775x_{86} = 37.6991118430775
x87=−86.3937979737193x_{87} = -86.3937979737193
x88=43.9822971502571x_{88} = 43.9822971502571
x89=56.5486677646163x_{89} = 56.5486677646163
x90=−65.9734457253857x_{90} = -65.9734457253857
x91=58.6430628670095x_{91} = 58.6430628670095
x92=92.1533845053006x_{92} = 92.1533845053006
x93=−31.4159265358979x_{93} = -31.4159265358979
x94=35.6047167406843x_{94} = 35.6047167406843
x95=72.2566310325652x_{95} = 72.2566310325652
x96=10.471975511966x_{96} = 10.471975511966
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 sin(x) + sin(2*x) + sin(3*x).
(sin⁡(0)+sin⁡(0⋅2))+sin⁡(0⋅3)\left(\sin{\left(0 \right)} + \sin{\left(0 \cdot 2 \right)}\right) + \sin{\left(0 \cdot 3 \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
−(sin⁡(x)+4sin⁡(2x)+9sin⁡(3x))=0- (\sin{\left(x \right)} + 4 \sin{\left(2 x \right)} + 9 \sin{\left(3 x \right)}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=πx_{2} = \pi
x3=−ilog⁡(−19+199−i219+619)x_{3} = - i \log{\left(- \frac{1}{9} + \frac{\sqrt{19}}{9} - \frac{i \sqrt{2 \sqrt{19} + 61}}{9} \right)}
x4=−ilog⁡(−19+199+i219+619)x_{4} = - i \log{\left(- \frac{1}{9} + \frac{\sqrt{19}}{9} + \frac{i \sqrt{2 \sqrt{19} + 61}}{9} \right)}
x5=−ilog⁡(−199−19−i61−2199)x_{5} = - i \log{\left(- \frac{\sqrt{19}}{9} - \frac{1}{9} - \frac{i \sqrt{61 - 2 \sqrt{19}}}{9} \right)}
x6=−ilog⁡(−199−19+i61−2199)x_{6} = - i \log{\left(- \frac{\sqrt{19}}{9} - \frac{1}{9} + \frac{i \sqrt{61 - 2 \sqrt{19}}}{9} \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
(−∞,−atan⁡(219+61−1+19)]\left(-\infty, - \operatorname{atan}{\left(\frac{\sqrt{2 \sqrt{19} + 61}}{-1 + \sqrt{19}} \right)}\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((sin⁡(x)+sin⁡(2x))+sin⁡(3x))=⟨−3,3⟩\lim_{x \to -\infty}\left(\left(\sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \sin{\left(3 x \right)}\right) = \left\langle -3, 3\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
lim⁡x→∞((sin⁡(x)+sin⁡(2x))+sin⁡(3x))=⟨−3,3⟩\lim_{x \to \infty}\left(\left(\sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \sin{\left(3 x \right)}\right) = \left\langle -3, 3\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x) + sin(2*x) + sin(3*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((sin⁡(x)+sin⁡(2x))+sin⁡(3x)x)=0\lim_{x \to -\infty}\left(\frac{\left(\sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \sin{\left(3 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((sin⁡(x)+sin⁡(2x))+sin⁡(3x)x)=0\lim_{x \to \infty}\left(\frac{\left(\sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \sin{\left(3 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⁡(x)+sin⁡(2x))+sin⁡(3x)=−sin⁡(x)−sin⁡(2x)−sin⁡(3x)\left(\sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \sin{\left(3 x \right)} = - \sin{\left(x \right)} - \sin{\left(2 x \right)} - \sin{\left(3 x \right)}
- No
(sin⁡(x)+sin⁡(2x))+sin⁡(3x)=sin⁡(x)+sin⁡(2x)+sin⁡(3x)\left(\sin{\left(x \right)} + \sin{\left(2 x \right)}\right) + \sin{\left(3 x \right)} = \sin{\left(x \right)} + \sin{\left(2 x \right)} + \sin{\left(3 x \right)}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = sin(x)+sin(2*x)+sin(3*x)
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