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Trazado el gráfico de la función/ y=((4x−3)sin(x))+4cos(x)−4

Gráfico de la función y = y=((4x−3)sin(x))+4cos(x)−4

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = (4*x - 3)*sin(x) + 4*cos(x) - 4
f(x)=((4x−3)sin⁡(x)+4cos⁡(x))−4f{\left(x \right)} = \left(\left(4 x - 3\right) \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) - 4 
f = (4*x - 3)*sin(x) + 4*cos(x) - 4
Gráfico de la función
0.00.10.20.30.40.50.60.70.80.91.01.11.21.31.41.51-2
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:
((4x−3)sin⁡(x)+4cos⁡(x))−4=0\left(\left(4 x - 3\right) \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) - 4 = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=12.5663706143592x_{1} = 12.5663706143592
x2=37.6991118430775x_{2} = 37.6991118430775
x3=−21.9029170081886x_{3} = -21.9029170081886
x4=−59.6571547812987x_{4} = -59.6571547812987
x5=28.2015103465998x_{5} = 28.2015103465998
x6=97.3686730679853x_{6} = 97.3686730679853
x7=−53.3701244858071x_{7} = -53.3701244858071
x8=−31.4159265358979x_{8} = -31.4159265358979
x9=0x_{9} = 0
x10=15.5732441612579x_{10} = 15.5732441612579
x11=−50.2654824574367x_{11} = -50.2654824574367
x12=−94.2477796076938x_{12} = -94.2477796076938
x13=6.28318530717959x_{13} = 6.28318530717959
x14=69.1150383789755x_{14} = 69.1150383789755
x15=−2.55375438589838x_{15} = -2.55375438589838
x16=−69.1150383789755x_{16} = -69.1150383789755
x17=72.2286524758575x_{17} = 72.2286524758575
x18=91.0840478121359x_{18} = 91.0840478121359
x19=47.0807286110731x_{19} = 47.0807286110731
x20=62.8318530717959x_{20} = 62.8318530717959
x21=128.789678956888x_{21} = 128.789678956888
x22=−119.380520836412x_{22} = -119.380520836412
x23=50.2654824574367x_{23} = 50.2654824574367
x24=81.6814089933346x_{24} = 81.6814089933346
x25=100.530964914873x_{25} = 100.530964914873
x26=−65.9434600245392x_{26} = -65.9434600245392
x27=−15.5856845007186x_{27} = -15.5856845007186
x28=−87.9645943005142x_{28} = -87.9645943005142
x29=191.626673995173x_{29} = 191.626673995173
x30=−62.8318530717959x_{30} = -62.8318530717959
x31=−18.8495559215388x_{31} = -18.8495559215388
x32=34.4982742349829x_{32} = 34.4982742349829
x33=59.6563114599685x_{33} = 59.6563114599685
x34=−56.5486677646163x_{34} = -56.5486677646163
x35=78.5140989483634x_{35} = 78.5140989483634
x36=−37.6991118430775x_{36} = -37.6991118430775
x37=−25.1327412287183x_{37} = -25.1327412287183
x38=−100.530964914873x_{38} = -100.530964914873
x39=−9.22494324849917x_{39} = -9.22494324849917
x40=9.18887973892067x_{40} = 9.18887973892067
x41=−40.7925704063333x_{41} = -40.7925704063333
x42=−75.398223686155x_{42} = -75.398223686155
x43=18.8495559215388x_{43} = 18.8495559215388
x44=−84.7996244675442x_{44} = -84.7996244675442
x45=−78.5145857290857x_{45} = -78.5145857290857
x46=−47.0820829544293x_{46} = -47.0820829544293
x47=−97.3689895528378x_{47} = -97.3689895528378
x48=−6.28318530717959x_{48} = -6.28318530717959
x49=25.1327412287183x_{49} = 25.1327412287183
x50=−28.2052893153649x_{50} = -28.2052893153649
x51=21.8966413129579x_{51} = 21.8966413129579
x52=40.7907657830147x_{52} = 40.7907657830147
x53=56.5486677646163x_{53} = 56.5486677646163
x54=−43.9822971502571x_{54} = -43.9822971502571
x55=84.7992071852828x_{55} = 84.7992071852828
x56=−72.2292276891132x_{56} = -72.2292276891132
x57=31.4159265358979x_{57} = 31.4159265358979
x58=94.2477796076938x_{58} = 94.2477796076938
x59=−12.5663706143592x_{59} = -12.5663706143592
x60=75.398223686155x_{60} = 75.398223686155
x61=−91.0844094856414x_{61} = -91.0844094856414
x62=−34.5007980976135x_{62} = -34.5007980976135
x63=65.9427698825493x_{63} = 65.9427698825493
x64=−81.6814089933346x_{64} = -81.6814089933346
x65=43.9822971502571x_{65} = 43.9822971502571
x66=53.3690706528281x_{66} = 53.3690706528281
x67=87.9645943005142x_{67} = 87.9645943005142
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 (4*x - 3)*sin(x) + 4*cos(x) - 4.
−4+((−3+0⋅4)sin⁡(0)+4cos⁡(0))-4 + \left(\left(-3 + 0 \cdot 4\right) \sin{\left(0 \right)} + 4 \cos{\left(0 \right)}\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
(4x−3)cos⁡(x)=0\left(4 x - 3\right) \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=34x_{1} = \frac{3}{4}
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Signos de extremos en los puntos:
(3/4, -4 + 4*cos(3/4))

 -pi             
(----, -1 + 2*pi)
  2              

 pi            
(--, -7 + 2*pi)
 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=34x_{1} = \frac{3}{4}
Puntos máximos de la función:
x1=−π2x_{1} = - \frac{\pi}{2}
x1=π2x_{1} = \frac{\pi}{2}
Decrece en los intervalos
(−∞,−π2]∪[34,∞)\left(-\infty, - \frac{\pi}{2}\right] \cup \left[\frac{3}{4}, \infty\right)
Crece en los intervalos
(−∞,34]∪[π2,∞)\left(-\infty, \frac{3}{4}\right] \cup \left[\frac{\pi}{2}, \infty\right)
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
−(4x−3)sin⁡(x)+4cos⁡(x)=0- \left(4 x - 3\right) \sin{\left(x \right)} + 4 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=47.145440309251x_{1} = 47.145440309251
x2=−122.530224913135x_{2} = -122.530224913135
x3=75.4116166483449x_{3} = 75.4116166483449
x4=−56.5661130953866x_{4} = -56.5661130953866
x5=−69.1293477819365x_{5} = -69.1293477819365
x6=−25.1713004186901x_{6} = -25.1713004186901
x7=53.4260567880298x_{7} = 53.4260567880298
x8=−78.5524256262012x_{8} = -78.5524256262012
x9=−87.9758644940058x_{9} = -87.9758644940058
x10=78.5526686615979x_{10} = 78.5526686615979
x11=72.2706121047755x_{11} = 72.2706121047755
x12=−100.540837155451x_{12} = -100.540837155451
x13=15.7744235378729x_{13} = 15.7744235378729
x14=31.4484899134444x_{14} = 31.4484899134444
x15=−75.4113529505576x_{15} = -75.4113529505576
x16=−9.52182580963093x_{16} = -9.52182580963093
x17=37.7261497124547x_{17} = 37.7261497124547
x18=91.1172524580044x_{18} = 91.1172524580044
x19=−81.6935379119613x_{19} = -81.6935379119613
x20=3.49137388507773x_{20} = 3.49137388507773
x21=1.17171304809526x_{21} = 1.17171304809526
x22=−59.7067996465833x_{22} = -59.7067996465833
x23=94.2584734180402x_{23} = 94.2584734180402
x24=−94.258304614729x_{24} = -94.258304614729
x25=−47.1447658788681x_{25} = -47.1447658788681
x26=−22.0350089342327x_{26} = -22.0350089342327
x27=12.6502058132628x_{27} = 12.6502058132628
x28=−50.2850743178012x_{28} = -50.2850743178012
x29=−65.9884284751167x_{29} = -65.9884284751167
x30=40.8656272764165x_{30} = 40.8656272764165
x31=84.834893828893x_{31} = 84.834893828893
x32=50.2856671898775x_{32} = 50.2856671898775
x33=69.1296615680471x_{33} = 69.1296615680471
x34=−0.627821506324058x_{34} = -0.627821506324058
x35=−62.847575645946x_{35} = -62.847575645946
x36=−84.8346854486537x_{36} = -84.8346854486537
x37=59.7072202428974x_{37} = 59.7072202428974
x38=25.1736622727957x_{38} = 25.1736622727957
x39=−18.9004016045119x_{39} = -18.9004016045119
x40=6.45665792711158x_{40} = 6.45665792711158
x41=−72.2703249936797x_{41} = -72.2703249936797
x42=−37.7250968307455x_{42} = -37.7250968307455
x43=−53.4255315326367x_{43} = -53.4255315326367
x44=28.3106016504755x_{44} = 28.3106016504755
x45=87.9760582622255x_{45} = 87.9760582622255
x46=97.3997185337108x_{46} = 97.3997185337108
x47=−28.3087333695327x_{47} = -28.3087333695327
x48=34.5870639815152x_{48} = 34.5870639815152
x49=62.8479552698028x_{49} = 62.8479552698028
x50=−97.3995604413997x_{50} = -97.3995604413997
x51=−34.5858115400283x_{51} = -34.5858115400283
x52=65.9887728330029x_{52} = 65.9887728330029
x53=−3.37919590114709x_{53} = -3.37919590114709
x54=22.0380886949359x_{54} = 22.0380886949359
x55=−91.1170718172709x_{55} = -91.1170718172709
x56=9.53808109513302x_{56} = 9.53808109513302
x57=−31.4469753727638x_{57} = -31.4469753727638
x58=100.540985524687x_{58} = 100.540985524687
x59=56.5665816711728x_{59} = 56.5665816711728
x60=−6.42172844075492x_{60} = -6.42172844075492
x61=−15.7684279229008x_{61} = -15.7684279229008
x62=−40.8647298259508x_{62} = -40.8647298259508
x63=81.6937626212905x_{63} = 81.6937626212905
x64=−44.0046374860768x_{64} = -44.0046374860768
x65=44.0054115275539x_{65} = 44.0054115275539
x66=−12.6409097831365x_{66} = -12.6409097831365
x67=18.9045828249819x_{67} = 18.9045828249819

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
[97.3997185337108,∞)\left[97.3997185337108, \infty\right)
Convexa en los intervalos
(−∞,−100.540837155451]\left(-\infty, -100.540837155451\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((4x−3)sin⁡(x)+4cos⁡(x))−4)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(\left(4 x - 3\right) \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) - 4\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→∞(((4x−3)sin⁡(x)+4cos⁡(x))−4)=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(\left(4 x - 3\right) \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) - 4\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 (4*x - 3)*sin(x) + 4*cos(x) - 4, dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(((4x−3)sin⁡(x)+4cos⁡(x))−4x)y = x \lim_{x \to -\infty}\left(\frac{\left(\left(4 x - 3\right) \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) - 4}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(((4x−3)sin⁡(x)+4cos⁡(x))−4x)y = x \lim_{x \to \infty}\left(\frac{\left(\left(4 x - 3\right) \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) - 4}{x}\right)
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:
((4x−3)sin⁡(x)+4cos⁡(x))−4=−(−4x−3)sin⁡(x)+4cos⁡(x)−4\left(\left(4 x - 3\right) \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) - 4 = - \left(- 4 x - 3\right) \sin{\left(x \right)} + 4 \cos{\left(x \right)} - 4
- No
((4x−3)sin⁡(x)+4cos⁡(x))−4=(−4x−3)sin⁡(x)−4cos⁡(x)+4\left(\left(4 x - 3\right) \sin{\left(x \right)} + 4 \cos{\left(x \right)}\right) - 4 = \left(- 4 x - 3\right) \sin{\left(x \right)} - 4 \cos{\left(x \right)} + 4
- No
es decir, función
no es
par ni impar
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