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Trazado el gráfico de la función/ (sinx)(sin3x)

Gráfico de la función y = (sinx)(sin3x)

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
02468-8-6-4-2-10102-2
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=−12.5663703728218x_{1} = -12.5663703728218
x2=−50.2654823521144x_{2} = -50.2654823521144
x3=−68.0678408277789x_{3} = -68.0678408277789
x4=−79.5870138909414x_{4} = -79.5870138909414
x5=−21.9911485865076x_{5} = -21.9911485865076
x6=63.8790506229925x_{6} = 63.8790506229925
x7=34.5575190978177x_{7} = 34.5575190978177
x8=−85.870199198121x_{8} = -85.870199198121
x9=−4.18879020478639x_{9} = -4.18879020478639
x10=−75.3982232420527x_{10} = -75.3982232420527
x11=74.3510261349584x_{11} = 74.3510261349584
x12=−35.6047167406843x_{12} = -35.6047167406843
x13=720.471915223259x_{13} = 720.471915223259
x14=−28.2743337784666x_{14} = -28.2743337784666
x15=2.0943951023932x_{15} = 2.0943951023932
x16=33.5103216382911x_{16} = 33.5103216382911
x17=75.3982236924273x_{17} = 75.3982236924273
x18=30.3687289847013x_{18} = 30.3687289847013
x19=−63.8790506229925x_{19} = -63.8790506229925
x20=39.7935069454707x_{20} = 39.7935069454707
x21=−70.162235930172x_{21} = -70.162235930172
x22=70.162235930172x_{22} = 70.162235930172
x23=−94.2477795007104x_{23} = -94.2477795007104
x24=15.7079633672581x_{24} = 15.7079633672581
x25=56.5486676696959x_{25} = 56.5486676696959
x26=76.4454212373516x_{26} = 76.4454212373516
x27=−39.7935069454707x_{27} = -39.7935069454707
x28=−21.9911487680291x_{28} = -21.9911487680291
x29=85.870199198121x_{29} = 85.870199198121
x30=98.4365698124802x_{30} = 98.4365698124802
x31=−24.0855436775217x_{31} = -24.0855436775217
x32=−15.707963296108x_{32} = -15.707963296108
x33=−97.3893723140246x_{33} = -97.3893723140246
x34=−90.0589894029074x_{34} = -90.0589894029074
x35=−41.8879020478639x_{35} = -41.8879020478639
x36=60.7374579694027x_{36} = 60.7374579694027
x37=90.0589894029074x_{37} = 90.0589894029074
x38=−53.4070751780852x_{38} = -53.4070751780852
x39=−2.0943951023932x_{39} = -2.0943951023932
x40=41.8879020478639x_{40} = 41.8879020478639
x41=26.1799387799149x_{41} = 26.1799387799149
x42=50.2654824463941x_{42} = 50.2654824463941
x43=87.9645943349125x_{43} = 87.9645943349125
x44=72.2566310277269x_{44} = 72.2566310277269
x45=84.823001664144x_{45} = 84.823001664144
x46=−61.7846555205993x_{46} = -61.7846555205993
x47=37.6991119391801x_{47} = 37.6991119391801
x48=68.0678408277789x_{48} = 68.0678408277789
x49=−13.6135681655558x_{49} = -13.6135681655558
x50=−75.3982237427215x_{50} = -75.3982237427215
x51=−26.1799387799149x_{51} = -26.1799387799149
x52=8.37758040957278x_{52} = 8.37758040957278
x53=−6.28318520531977x_{53} = -6.28318520531977
x54=59.6902605103337x_{54} = 59.6902605103337
x55=52.3598775598299x_{55} = 52.3598775598299
x56=81.6814090805726x_{56} = 81.6814090805726
x57=43.9822971692115x_{57} = 43.9822971692115
x58=−59.6902604567055x_{58} = -59.6902604567055
x59=−46.0766922526503x_{59} = -46.0766922526503
x60=−81.6814090367173x_{60} = -81.6814090367173
x61=0x_{61} = 0
x62=96.342174710087x_{62} = 96.342174710087
x63=−37.6991118765052x_{63} = -37.6991118765052
x64=−75.3982234563503x_{64} = -75.3982234563503
x65=−31.4159264982662x_{65} = -31.4159264982662
x66=−9.42477804291614x_{66} = -9.42477804291614
x67=−83.7758040957278x_{67} = -83.7758040957278
x68=6.28318528449741x_{68} = 6.28318528449741
x69=12.5663705268849x_{69} = 12.5663705268849
x70=−31.4159266113445x_{70} = -31.4159266113445
x71=19.8967534727354x_{71} = 19.8967534727354
x72=48.1710873550435x_{72} = 48.1710873550435
x73=100.530964815737x_{73} = 100.530964815737
x74=−65.973445765529x_{74} = -65.973445765529
x75=94.2477796093533x_{75} = 94.2477796093533
x76=4.18879020478639x_{76} = 4.18879020478639
x77=78.539816242374x_{77} = 78.539816242374
x78=−43.9822971748057x_{78} = -43.9822971748057
x79=−19.8967534727354x_{79} = -19.8967534727354
x80=54.4542726622231x_{80} = 54.4542726622231
x81=21.9911485851418x_{81} = 21.9911485851418
x82=92.1533845053006x_{82} = 92.1533845053006
x83=−48.1710873550435x_{83} = -48.1710873550435
x84=46.0766922526503x_{84} = 46.0766922526503
x85=24.0855436775217x_{85} = 24.0855436775217
x86=65.9734457524179x_{86} = 65.9734457524179
x87=−97.3893723046965x_{87} = -97.3893723046965
x88=28.2743338653253x_{88} = 28.2743338653253
x89=−72.2566309262098x_{89} = -72.2566309262098
x90=17.8023583703422x_{90} = 17.8023583703422
x91=−17.8023583703422x_{91} = -17.8023583703422
x92=−57.5958653158129x_{92} = -57.5958653158129
x93=−92.1533845053006x_{93} = -92.1533845053006
x94=−87.9645943596624x_{94} = -87.9645943596624
x95=−52.3598775598299x_{95} = -52.3598775598299
x96=6.2831852928604x_{96} = 6.2831852928604
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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