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Trazado el gráfico de la función/ 3*sin(x)/25+6*cos(x)*sin(x)/((235*sqrt(1-100*sin(x)^2/2209)))

Gráfico de la función y = 3*sin(x)/25+6*cos(x)*sin(x)/((235*sqrt(1-100*sin(x)^2/2209)))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       3*sin(x)        6*cos(x)*sin(x)      
f(x) = -------- + --------------------------
          25               _________________
                          /            2    
                         /      100*sin (x) 
                  235*  /   1 - ----------- 
                      \/            2209
f(x)=sin⁡(x)6cos⁡(x)235−100sin⁡2(x)2209+1+3sin⁡(x)25f{\left(x \right)} = \frac{\sin{\left(x \right)} 6 \cos{\left(x \right)}}{235 \sqrt{- \frac{100 \sin^{2}{\left(x \right)}}{2209} + 1}} + \frac{3 \sin{\left(x \right)}}{25} 
f = (sin(x)*(6*cos(x)))/((235*sqrt(-100*sin(x)^2/2209 + 1))) + (3*sin(x))/25
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)6cos⁡(x)235−100sin⁡2(x)2209+1+3sin⁡(x)25=0\frac{\sin{\left(x \right)} 6 \cos{\left(x \right)}}{235 \sqrt{- \frac{100 \sin^{2}{\left(x \right)}}{2209} + 1}} + \frac{3 \sin{\left(x \right)}}{25} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
Solución numérica
x1=72.2566310325652x_{1} = 72.2566310325652
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=3.14159265358979x_{3} = 3.14159265358979
x4=−43.9822971502571x_{4} = -43.9822971502571
x5=81.6814089933346x_{5} = 81.6814089933346
x6=−100.530964914873x_{6} = -100.530964914873
x7=28.2743338823081x_{7} = 28.2743338823081
x8=65.9734457253857x_{8} = 65.9734457253857
x9=−31.4159265358979x_{9} = -31.4159265358979
x10=−9.42477796076938x_{10} = -9.42477796076938
x11=40.8407044966673x_{11} = 40.8407044966673
x12=56.5486677646163x_{12} = 56.5486677646163
x13=−56.5486677646163x_{13} = -56.5486677646163
x14=12.5663706143592x_{14} = 12.5663706143592
x15=43.9822971502571x_{15} = 43.9822971502571
x16=100.530964914873x_{16} = 100.530964914873
x17=−3.14159265358979x_{17} = -3.14159265358979
x18=−15.707963267949x_{18} = -15.707963267949
x19=59.6902604182061x_{19} = 59.6902604182061
x20=6.28318530717959x_{20} = 6.28318530717959
x21=9.42477796076938x_{21} = 9.42477796076938
x22=−53.4070751110265x_{22} = -53.4070751110265
x23=−47.1238898038469x_{23} = -47.1238898038469
x24=−87.9645943005142x_{24} = -87.9645943005142
x25=69.1150383789755x_{25} = 69.1150383789755
x26=21.9911485751286x_{26} = 21.9911485751286
x27=87.9645943005142x_{27} = 87.9645943005142
x28=18.8495559215388x_{28} = 18.8495559215388
x29=−84.8230016469244x_{29} = -84.8230016469244
x30=−72.2566310325652x_{30} = -72.2566310325652
x31=25.1327412287183x_{31} = 25.1327412287183
x32=37.6991118430775x_{32} = 37.6991118430775
x33=−25.1327412287183x_{33} = -25.1327412287183
x34=0x_{34} = 0
x35=50.2654824574367x_{35} = 50.2654824574367
x36=−6.28318530717959x_{36} = -6.28318530717959
x37=−65.9734457253857x_{37} = -65.9734457253857
x38=−21.9911485751286x_{38} = -21.9911485751286
x39=−62.8318530717959x_{39} = -62.8318530717959
x40=75.398223686155x_{40} = 75.398223686155
x41=84.8230016469244x_{41} = 84.8230016469244
x42=53.4070751110265x_{42} = 53.4070751110265
x43=34.5575191894877x_{43} = 34.5575191894877
x44=−28.2743338823081x_{44} = -28.2743338823081
x45=15.707963267949x_{45} = 15.707963267949
x46=−91.106186954104x_{46} = -91.106186954104
x47=47.1238898038469x_{47} = 47.1238898038469
x48=97.3893722612836x_{48} = 97.3893722612836
x49=−69.1150383789755x_{49} = -69.1150383789755
x50=94.2477796076938x_{50} = 94.2477796076938
x51=−18.8495559215388x_{51} = -18.8495559215388
x52=−50.2654824574367x_{52} = -50.2654824574367
x53=−37.6991118430775x_{53} = -37.6991118430775
x54=534.070751110265x_{54} = 534.070751110265
x55=−81.6814089933346x_{55} = -81.6814089933346
x56=62.8318530717959x_{56} = 62.8318530717959
x57=78.5398163397448x_{57} = 78.5398163397448
x58=31.4159265358979x_{58} = 31.4159265358979
x59=−78.5398163397448x_{59} = -78.5398163397448
x60=−40.8407044966673x_{60} = -40.8407044966673
x61=−97.3893722612836x_{61} = -97.3893722612836
x62=−75.398223686155x_{62} = -75.398223686155
x63=91.106186954104x_{63} = 91.106186954104
x64=−12.5663706143592x_{64} = -12.5663706143592
x65=−94.2477796076938x_{65} = -94.2477796076938
x66=−34.5575191894877x_{66} = -34.5575191894877
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 (3*sin(x))/25 + ((6*cos(x))*sin(x))/((235*sqrt(1 - 100*sin(x)^2/2209))).
3sin⁡(0)25+sin⁡(0)6cos⁡(0)235−100sin⁡2(0)2209+1\frac{3 \sin{\left(0 \right)}}{25} + \frac{\sin{\left(0 \right)} 6 \cos{\left(0 \right)}}{235 \sqrt{- \frac{100 \sin^{2}{\left(0 \right)}}{2209} + 1}}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡(x)6cos⁡(x)235−100sin⁡2(x)2209+1+3sin⁡(x)25)=⟨−325−221093515,221093515+325⟩\lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)} 6 \cos{\left(x \right)}}{235 \sqrt{- \frac{100 \sin^{2}{\left(x \right)}}{2209} + 1}} + \frac{3 \sin{\left(x \right)}}{25}\right) = \left\langle - \frac{3}{25} - \frac{2 \sqrt{2109}}{3515}, \frac{2 \sqrt{2109}}{3515} + \frac{3}{25}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−325−221093515,221093515+325⟩y = \left\langle - \frac{3}{25} - \frac{2 \sqrt{2109}}{3515}, \frac{2 \sqrt{2109}}{3515} + \frac{3}{25}\right\rangle
lim⁡x→∞(sin⁡(x)6cos⁡(x)235−100sin⁡2(x)2209+1+3sin⁡(x)25)=⟨−325−221093515,221093515+325⟩\lim_{x \to \infty}\left(\frac{\sin{\left(x \right)} 6 \cos{\left(x \right)}}{235 \sqrt{- \frac{100 \sin^{2}{\left(x \right)}}{2209} + 1}} + \frac{3 \sin{\left(x \right)}}{25}\right) = \left\langle - \frac{3}{25} - \frac{2 \sqrt{2109}}{3515}, \frac{2 \sqrt{2109}}{3515} + \frac{3}{25}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−325−221093515,221093515+325⟩y = \left\langle - \frac{3}{25} - \frac{2 \sqrt{2109}}{3515}, \frac{2 \sqrt{2109}}{3515} + \frac{3}{25}\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (3*sin(x))/25 + ((6*cos(x))*sin(x))/((235*sqrt(1 - 100*sin(x)^2/2209))), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x)6cos⁡(x)235−100sin⁡2(x)2209+1+3sin⁡(x)25x)=0\lim_{x \to -\infty}\left(\frac{\frac{\sin{\left(x \right)} 6 \cos{\left(x \right)}}{235 \sqrt{- \frac{100 \sin^{2}{\left(x \right)}}{2209} + 1}} + \frac{3 \sin{\left(x \right)}}{25}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(x)6cos⁡(x)235−100sin⁡2(x)2209+1+3sin⁡(x)25x)=0\lim_{x \to \infty}\left(\frac{\frac{\sin{\left(x \right)} 6 \cos{\left(x \right)}}{235 \sqrt{- \frac{100 \sin^{2}{\left(x \right)}}{2209} + 1}} + \frac{3 \sin{\left(x \right)}}{25}}{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)6cos⁡(x)235−100sin⁡2(x)2209+1+3sin⁡(x)25=−61235−100sin⁡2(x)2209+1sin⁡(x)cos⁡(x)−3sin⁡(x)25\frac{\sin{\left(x \right)} 6 \cos{\left(x \right)}}{235 \sqrt{- \frac{100 \sin^{2}{\left(x \right)}}{2209} + 1}} + \frac{3 \sin{\left(x \right)}}{25} = - 6 \frac{1}{235 \sqrt{- \frac{100 \sin^{2}{\left(x \right)}}{2209} + 1}} \sin{\left(x \right)} \cos{\left(x \right)} - \frac{3 \sin{\left(x \right)}}{25}
- No
sin⁡(x)6cos⁡(x)235−100sin⁡2(x)2209+1+3sin⁡(x)25=61235−100sin⁡2(x)2209+1sin⁡(x)cos⁡(x)+3sin⁡(x)25\frac{\sin{\left(x \right)} 6 \cos{\left(x \right)}}{235 \sqrt{- \frac{100 \sin^{2}{\left(x \right)}}{2209} + 1}} + \frac{3 \sin{\left(x \right)}}{25} = 6 \frac{1}{235 \sqrt{- \frac{100 \sin^{2}{\left(x \right)}}{2209} + 1}} \sin{\left(x \right)} \cos{\left(x \right)} + \frac{3 \sin{\left(x \right)}}{25}
- No
es decir, función
no es
par ni impar
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