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Trazado el gráfico de la función/ x*sin(x+5)*cos(x-6)*sin(x+7)*cos(x-8)*sin(x/3)

Gráfico de la función y = x*sin(x+5)*cos(x-6)*sin(x+7)*cos(x-8)*sin(x/3)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                                        /x\
f(x) = x*sin(x + 5)*cos(x - 6)*sin(x + 7)*cos(x - 8)*sin|-|
                                                        \3/
f(x)=xsin⁡(x+5)cos⁡(x−6)sin⁡(x+7)cos⁡(x−8)sin⁡(x3)f{\left(x \right)} = x \sin{\left(x + 5 \right)} \cos{\left(x - 6 \right)} \sin{\left(x + 7 \right)} \cos{\left(x - 8 \right)} \sin{\left(\frac{x}{3} \right)} 
f = ((((x*sin(x + 5))*cos(x - 6))*sin(x + 7))*cos(x - 8))*sin(x/3)
Gráfico de la función
2.8802.9002.8822.8842.8862.8882.8902.8922.8942.8962.8980.4100.412
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:
xsin⁡(x+5)cos⁡(x−6)sin⁡(x+7)cos⁡(x−8)sin⁡(x3)=0x \sin{\left(x + 5 \right)} \cos{\left(x - 6 \right)} \sin{\left(x + 7 \right)} \cos{\left(x - 8 \right)} \sin{\left(\frac{x}{3} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−7x_{1} = -7
x2=−5x_{2} = -5
x3=0x_{3} = 0
x4=6−π2x_{4} = 6 - \frac{\pi}{2}
x5=8−π2x_{5} = 8 - \frac{\pi}{2}
x6=π2+6x_{6} = \frac{\pi}{2} + 6
x7=π2+8x_{7} = \frac{\pi}{2} + 8
Solución numérica
x1=13.8539816339745x_{1} = 13.8539816339745
x2=88.1106126665397x_{2} = 88.1106126665397
x3=−39.5575191894877x_{3} = -39.5575191894877
x4=6.4292036732051x_{4} = 6.4292036732051
x5=−56.5486677646163x_{5} = -56.5486677646163
x6=−36.4115008234622x_{6} = -36.4115008234622
x7=98.6769832808989x_{7} = 98.6769832808989
x8=−32.1327412287183x_{8} = -32.1327412287183
x9=−54.1238898038469x_{9} = -54.1238898038469
x10=−10.1415926535898x_{10} = -10.1415926535898
x11=−72.1106126665397x_{11} = -72.1106126665397
x12=−45.8362787842316x_{12} = -45.8362787842316
x13=92.3893722612836x_{13} = 92.3893722612836
x14=−85.5398163397448x_{14} = -85.5398163397448
x15=−87.8185759344887x_{15} = -87.8185759344887
x16=79.8230016469244x_{16} = 79.8230016469244
x17=−50.1194640914112x_{17} = -50.1194640914112
x18=20.1371669411541x_{18} = 20.1371669411541
x19=51.5486677646163x_{19} = 51.5486677646163
x20=−28.9911485751286x_{20} = -28.9911485751286
x21=−8.13716694115407x_{21} = -8.13716694115407
x22=84.106186954104x_{22} = 84.106186954104
x23=37.845130209103x_{23} = 37.845130209103
x24=−6.13716694115407x_{24} = -6.13716694115407
x25=33.8407044966673x_{25} = 33.8407044966673
x26=−41.5575191894877x_{26} = -41.5575191894877
x27=50.4115008234622x_{27} = 50.4115008234622
x28=−63.5486677646163x_{28} = -63.5486677646163
x29=−14.4247779607694x_{29} = -14.4247779607694
x30=95.5309649148734x_{30} = 95.5309649148734
x31=−28.1283155162826x_{31} = -28.1283155162826
x32=−91.8230016469244x_{32} = -91.8230016469244
x33=−17.5663706143592x_{33} = -17.5663706143592
x34=81.8274273593601x_{34} = 81.8274273593601
x35=70.398223686155x_{35} = 70.398223686155
x36=18.1327412287183x_{36} = 18.1327412287183
x37=−15.5619449019235x_{37} = -15.5619449019235
x38=40.1238898038469x_{38} = 40.1238898038469
x39=0x_{39} = 0
x40=35.845130209103x_{40} = 35.845130209103
x41=57.8362787842316x_{41} = 57.8362787842316
x42=−43.8362787842316x_{42} = -43.8362787842316
x43=−76.1150383789755x_{43} = -76.1150383789755
x44=−3.85840734641021x_{44} = -3.85840734641021
x45=−80.3937979737193x_{45} = -80.3937979737193
x46=−61.5486677646163x_{46} = -61.5486677646163
x47=22.1371669411541x_{47} = 22.1371669411541
x48=57.8318530717959x_{48} = 57.8318530717959
x49=−59.5442420521806x_{49} = -59.5442420521806
x50=59.8362787842316x_{50} = 59.8362787842316
x51=−94.1017612416683x_{51} = -94.1017612416683
x52=−25.8495559215388x_{52} = -25.8495559215388
x53=28.2743338823081x_{53} = 28.2743338823081
x54=65.9734457253857x_{54} = 65.9734457253857
x55=−21.845130209103x_{55} = -21.845130209103
x56=−19.5663706143592x_{56} = -19.5663706143592
x57=99.814150222053x_{57} = 99.814150222053
x58=−74.1106126665397x_{58} = -74.1106126665397
x59=11.8495559215388x_{59} = 11.8495559215388
x60=−47.8407044966673x_{60} = -47.8407044966673
x61=−96.1017612416683x_{61} = -96.1017612416683
x62=2.42477796076938x_{62} = 2.42477796076938
x63=4.42477796076938x_{63} = 4.42477796076938
x64=90.3893722612836x_{64} = 90.3893722612836
x65=54.6946861306418x_{65} = 54.6946861306418
x66=32.7035375555132x_{66} = 32.7035375555132
x67=−83.5398163397448x_{67} = -83.5398163397448
x68=26.4159265358979x_{68} = 26.4159265358979
x69=−77.2522053201295x_{69} = -77.2522053201295
x70=29.5619449019235x_{70} = 29.5619449019235
x71=72.4026493985908x_{71} = 72.4026493985908
x72=−1.85398163397448x_{72} = -1.85398163397448
x73=−98.106186954104x_{73} = -98.106186954104
x74=94.2477796076938x_{74} = 94.2477796076938
x75=62.1150383789754x_{75} = 62.1150383789754
x76=−30.1283155162826x_{76} = -30.1283155162826
x77=0.146018366025517x_{77} = 0.146018366025517
x78=44.1283155162826x_{78} = 44.1283155162826
x79=−23.845130209103x_{79} = -23.845130209103
x80=−11.2787595947439x_{80} = -11.2787595947439
x81=−89.8185759344887x_{81} = -89.8185759344887
x82=24.4159265358979x_{82} = 24.4159265358979
x83=79.8274273593601x_{83} = 79.8274273593601
x84=48.4070751110265x_{84} = 48.4070751110265
x85=−236.336263712055x_{85} = -236.336263712055
x86=68.398223686155x_{86} = 68.398223686155
x87=46.4070751110265x_{87} = 46.4070751110265
x88=−67.8274273593601x_{88} = -67.8274273593601
x89=15.8539816339745x_{89} = 15.8539816339745
x90=7.5707963267949x_{90} = 7.5707963267949
x91=−81.5353906273091x_{91} = -81.5353906273091
x92=77.8230016469244x_{92} = 77.8230016469244
x93=−52.1194640914112x_{93} = -52.1194640914112
x94=−37.6991118430775x_{94} = -37.6991118430775
x95=−69.8318530717959x_{95} = -69.8318530717959
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 ((((x*sin(x + 5))*cos(x - 6))*sin(x + 7))*cos(x - 8))*sin(x/3).
0sin⁡(5)cos⁡(−6)sin⁡(7)cos⁡(−8)sin⁡(03)0 \sin{\left(5 \right)} \cos{\left(-6 \right)} \sin{\left(7 \right)} \cos{\left(-8 \right)} \sin{\left(\frac{0}{3} \right)}
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→−∞(xsin⁡(x+5)cos⁡(x−6)sin⁡(x+7)cos⁡(x−8)sin⁡(x3))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \sin{\left(x + 5 \right)} \cos{\left(x - 6 \right)} \sin{\left(x + 7 \right)} \cos{\left(x - 8 \right)} \sin{\left(\frac{x}{3} \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(xsin⁡(x+5)cos⁡(x−6)sin⁡(x+7)cos⁡(x−8)sin⁡(x3))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \sin{\left(x + 5 \right)} \cos{\left(x - 6 \right)} \sin{\left(x + 7 \right)} \cos{\left(x - 8 \right)} \sin{\left(\frac{x}{3} \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función ((((x*sin(x + 5))*cos(x - 6))*sin(x + 7))*cos(x - 8))*sin(x/3), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x3)sin⁡(x+5)sin⁡(x+7)cos⁡(x−8)cos⁡(x−6))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin{\left(\frac{x}{3} \right)} \sin{\left(x + 5 \right)} \sin{\left(x + 7 \right)} \cos{\left(x - 8 \right)} \cos{\left(x - 6 \right)}\right) = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
lim⁡x→∞(sin⁡(x3)sin⁡(x+5)sin⁡(x+7)cos⁡(x−8)cos⁡(x−6))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin{\left(\frac{x}{3} \right)} \sin{\left(x + 5 \right)} \sin{\left(x + 7 \right)} \cos{\left(x - 8 \right)} \cos{\left(x - 6 \right)}\right) = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−1,1⟩xy = \left\langle -1, 1\right\rangle x
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:
xsin⁡(x+5)cos⁡(x−6)sin⁡(x+7)cos⁡(x−8)sin⁡(x3)=xsin⁡(x3)sin⁡(x−7)sin⁡(x−5)cos⁡(x+6)cos⁡(x+8)x \sin{\left(x + 5 \right)} \cos{\left(x - 6 \right)} \sin{\left(x + 7 \right)} \cos{\left(x - 8 \right)} \sin{\left(\frac{x}{3} \right)} = x \sin{\left(\frac{x}{3} \right)} \sin{\left(x - 7 \right)} \sin{\left(x - 5 \right)} \cos{\left(x + 6 \right)} \cos{\left(x + 8 \right)}
- No
xsin⁡(x+5)cos⁡(x−6)sin⁡(x+7)cos⁡(x−8)sin⁡(x3)=−xsin⁡(x3)sin⁡(x−7)sin⁡(x−5)cos⁡(x+6)cos⁡(x+8)x \sin{\left(x + 5 \right)} \cos{\left(x - 6 \right)} \sin{\left(x + 7 \right)} \cos{\left(x - 8 \right)} \sin{\left(\frac{x}{3} \right)} = - x \sin{\left(\frac{x}{3} \right)} \sin{\left(x - 7 \right)} \sin{\left(x - 5 \right)} \cos{\left(x + 6 \right)} \cos{\left(x + 8 \right)}
- No
es decir, función
no es
par ni impar
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