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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = sin(x)*sin(3*x)
f(x)=sin⁡(x)sin⁡(3x)f{\left(x \right)} = \sin{\left(x \right)} \sin{\left(3 x \right)} 
f = sin(x)*sin(3*x)
Gráfico de la función
0-11-10-9-8-7-6-5-4-3-2-12-2
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⁡(x)sin⁡(3x)=0\sin{\left(x \right)} \sin{\left(3 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=6.2831852928604x_{1} = 6.2831852928604
x2=−4.18879020478639x_{2} = -4.18879020478639
x3=28.2743338653253x_{3} = 28.2743338653253
x4=33.5103216382911x_{4} = 33.5103216382911
x5=39.7935069454707x_{5} = 39.7935069454707
x6=8.37758040957278x_{6} = 8.37758040957278
x7=−46.0766922526503x_{7} = -46.0766922526503
x8=−31.4159264982662x_{8} = -31.4159264982662
x9=63.8790506229925x_{9} = 63.8790506229925
x10=59.6902605103337x_{10} = 59.6902605103337
x11=41.8879020478639x_{11} = 41.8879020478639
x12=−63.8790506229925x_{12} = -63.8790506229925
x13=48.1710873550435x_{13} = 48.1710873550435
x14=−41.8879020478639x_{14} = -41.8879020478639
x15=81.6814090805726x_{15} = 81.6814090805726
x16=−81.6814090367173x_{16} = -81.6814090367173
x17=30.3687289847013x_{17} = 30.3687289847013
x18=78.539816242374x_{18} = 78.539816242374
x19=−43.9822971748057x_{19} = -43.9822971748057
x20=−75.3982234563503x_{20} = -75.3982234563503
x21=17.8023583703422x_{21} = 17.8023583703422
x22=−31.4159266113445x_{22} = -31.4159266113445
x23=−53.4070751780852x_{23} = -53.4070751780852
x24=−79.5870138909414x_{24} = -79.5870138909414
x25=−48.1710873550435x_{25} = -48.1710873550435
x26=85.870199198121x_{26} = 85.870199198121
x27=−19.8967534727354x_{27} = -19.8967534727354
x28=24.0855436775217x_{28} = 24.0855436775217
x29=−13.6135681655558x_{29} = -13.6135681655558
x30=34.5575190978177x_{30} = 34.5575190978177
x31=84.823001664144x_{31} = 84.823001664144
x32=4.18879020478639x_{32} = 4.18879020478639
x33=−92.1533845053006x_{33} = -92.1533845053006
x34=−17.8023583703422x_{34} = -17.8023583703422
x35=75.3982236924273x_{35} = 75.3982236924273
x36=−2.0943951023932x_{36} = -2.0943951023932
x37=98.4365698124802x_{37} = 98.4365698124802
x38=−12.5663703728218x_{38} = -12.5663703728218
x39=−35.6047167406843x_{39} = -35.6047167406843
x40=50.2654824463941x_{40} = 50.2654824463941
x41=−72.2566309262098x_{41} = -72.2566309262098
x42=−24.0855436775217x_{42} = -24.0855436775217
x43=−28.2743337784666x_{43} = -28.2743337784666
x44=−75.3982232420527x_{44} = -75.3982232420527
x45=72.2566310277269x_{45} = 72.2566310277269
x46=54.4542726622231x_{46} = 54.4542726622231
x47=26.1799387799149x_{47} = 26.1799387799149
x48=−50.2654823521144x_{48} = -50.2654823521144
x49=19.8967534727354x_{49} = 19.8967534727354
x50=−26.1799387799149x_{50} = -26.1799387799149
x51=96.342174710087x_{51} = 96.342174710087
x52=−97.3893723140246x_{52} = -97.3893723140246
x53=−90.0589894029074x_{53} = -90.0589894029074
x54=60.7374579694027x_{54} = 60.7374579694027
x55=21.9911485851418x_{55} = 21.9911485851418
x56=15.7079633672581x_{56} = 15.7079633672581
x57=−57.5958653158129x_{57} = -57.5958653158129
x58=−97.3893723046965x_{58} = -97.3893723046965
x59=94.2477796093533x_{59} = 94.2477796093533
x60=−52.3598775598299x_{60} = -52.3598775598299
x61=6.28318528449741x_{61} = 6.28318528449741
x62=−87.9645943596624x_{62} = -87.9645943596624
x63=−94.2477795007104x_{63} = -94.2477795007104
x64=−70.162235930172x_{64} = -70.162235930172
x65=−61.7846555205993x_{65} = -61.7846555205993
x66=46.0766922526503x_{66} = 46.0766922526503
x67=−6.28318520531977x_{67} = -6.28318520531977
x68=0x_{68} = 0
x69=−21.9911487680291x_{69} = -21.9911487680291
x70=−85.870199198121x_{70} = -85.870199198121
x71=56.5486676696959x_{71} = 56.5486676696959
x72=52.3598775598299x_{72} = 52.3598775598299
x73=−65.973445765529x_{73} = -65.973445765529
x74=−9.42477804291614x_{74} = -9.42477804291614
x75=87.9645943349125x_{75} = 87.9645943349125
x76=−68.0678408277789x_{76} = -68.0678408277789
x77=−75.3982237427215x_{77} = -75.3982237427215
x78=65.9734457524179x_{78} = 65.9734457524179
x79=−37.6991118765052x_{79} = -37.6991118765052
x80=2.0943951023932x_{80} = 2.0943951023932
x81=720.471915223259x_{81} = 720.471915223259
x82=37.6991119391801x_{82} = 37.6991119391801
x83=43.9822971692115x_{83} = 43.9822971692115
x84=12.5663705268849x_{84} = 12.5663705268849
x85=90.0589894029074x_{85} = 90.0589894029074
x86=−15.707963296108x_{86} = -15.707963296108
x87=76.4454212373516x_{87} = 76.4454212373516
x88=−83.7758040957278x_{88} = -83.7758040957278
x89=68.0678408277789x_{89} = 68.0678408277789
x90=−21.9911485865076x_{90} = -21.9911485865076
x91=100.530964815737x_{91} = 100.530964815737
x92=92.1533845053006x_{92} = 92.1533845053006
x93=−39.7935069454707x_{93} = -39.7935069454707
x94=−59.6902604567055x_{94} = -59.6902604567055
x95=74.3510261349584x_{95} = 74.3510261349584
x96=70.162235930172x_{96} = 70.162235930172
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(3*x).
sin⁡(0)sin⁡(0⋅3)\sin{\left(0 \right)} \sin{\left(0 \cdot 3 \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
3sin⁡(x)cos⁡(3x)+sin⁡(3x)cos⁡(x)=03 \sin{\left(x \right)} \cos{\left(3 x \right)} + \sin{\left(3 x \right)} \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
x4=πx_{4} = \pi
x5=i(log⁡(4)−log⁡(1−15i))2x_{5} = \frac{i \left(\log{\left(4 \right)} - \log{\left(1 - \sqrt{15} i \right)}\right)}{2}
x6=i(log⁡(4)−log⁡(1+15i))2x_{6} = \frac{i \left(\log{\left(4 \right)} - \log{\left(1 + \sqrt{15} i \right)}\right)}{2}
x7=−ilog⁡(−1−15i2)x_{7} = - i \log{\left(- \frac{\sqrt{1 - \sqrt{15} i}}{2} \right)}
x8=−ilog⁡(−1+15i2)x_{8} = - i \log{\left(- \frac{\sqrt{1 + \sqrt{15} i}}{2} \right)}
Signos de extremos en los puntos:
(0, 0)

 -pi      
(----, -1)
  2       

 pi     
(--, -1)
 2      

(pi, 0)

   /     /        ____\         \     /  /     /        ____\         \\    /    /     /        ____\         \\ 
 I*\- log\1 - I*\/ 15 / + log(4)/     |I*\- log\1 - I*\/ 15 / + log(4)/|    |3*I*\- log\1 - I*\/ 15 / + log(4)/| 
(--------------------------------, sin|--------------------------------|*sin|----------------------------------|)
                2                     \               2                /    \                2                 / 

   /     /        ____\         \     /  /     /        ____\         \\    /    /     /        ____\         \\ 
 I*\- log\1 + I*\/ 15 / + log(4)/     |I*\- log\1 + I*\/ 15 / + log(4)/|    |3*I*\- log\1 + I*\/ 15 / + log(4)/| 
(--------------------------------, sin|--------------------------------|*sin|----------------------------------|)
                2                     \               2                /    \                2                 / 

       /    ______________ \     /     /    ______________ \\    /       /    ______________ \\ 
       |   /         ____  |     |     |   /         ____  ||    |       |   /         ____  || 
       |-\/  1 - I*\/ 15   |     |     |-\/  1 - I*\/ 15   ||    |       |-\/  1 - I*\/ 15   || 
(-I*log|-------------------|, sin|I*log|-------------------||*sin|3*I*log|-------------------||)
       \         2         /     \     \         2         //    \       \         2         // 

       /    ______________ \     /     /    ______________ \\    /       /    ______________ \\ 
       |   /         ____  |     |     |   /         ____  ||    |       |   /         ____  || 
       |-\/  1 + I*\/ 15   |     |     |-\/  1 + I*\/ 15   ||    |       |-\/  1 + I*\/ 15   || 
(-I*log|-------------------|, sin|I*log|-------------------||*sin|3*I*log|-------------------||)
       \         2         /     \     \         2         //    \       \         2         //


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=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
x4=πx_{4} = \pi
Puntos máximos de la función:
x4=−atan⁡(15)2x_{4} = - \frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2}
x4=atan⁡(15)2x_{4} = \frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2}
x4=π−atan⁡(sin⁡(atan⁡(15)2)cos⁡(atan⁡(15)2))x_{4} = \pi - \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}} \right)}
x4=−π+atan⁡(sin⁡(atan⁡(15)2)cos⁡(atan⁡(15)2))x_{4} = - \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\sqrt{15} \right)}}{2} \right)}} \right)}
Decrece en los intervalos
[π,∞)\left[\pi, \infty\right)
Crece en los intervalos
(−∞,−π2]\left(-\infty, - \frac{\pi}{2}\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(x)sin⁡(3x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} \sin{\left(3 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⁡(x)sin⁡(3x))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} \sin{\left(3 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 sin(x)*sin(3*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x)sin⁡(3x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(x \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⁡(3x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(x \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⁡(3x)=sin⁡(x)sin⁡(3x)\sin{\left(x \right)} \sin{\left(3 x \right)} = \sin{\left(x \right)} \sin{\left(3 x \right)}
- Sí
sin⁡(x)sin⁡(3x)=−sin⁡(x)sin⁡(3x)\sin{\left(x \right)} \sin{\left(3 x \right)} = - \sin{\left(x \right)} \sin{\left(3 x \right)}
- No
es decir, función
es
par
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