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Trazado el gráfico de la función/ xsin(x)+cos(x)

Gráfico de la función y = xsin(x)+cos(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = x*sin(x) + cos(x)
f(x)=xsin⁡(x)+cos⁡(x)f{\left(x \right)} = x \sin{\left(x \right)} + \cos{\left(x \right)} 
f = x*sin(x) + cos(x)
Gráfico de la función
02468-8-6-4-2-1010-2020
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)+cos⁡(x)=0x \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=47.1026627703624x_{1} = 47.1026627703624
x2=−84.811211299318x_{2} = -84.811211299318
x3=25.0929104121121x_{3} = 25.0929104121121
x4=81.6691650818489x_{4} = 81.6691650818489
x5=59.6735041304405x_{5} = 59.6735041304405
x6=−28.2389365752603x_{6} = -28.2389365752603
x7=−43.9595528888955x_{7} = -43.9595528888955
x8=−37.672573565113x_{8} = -37.672573565113
x9=−18.7964043662102x_{9} = -18.7964043662102
x10=91.0952098694071x_{10} = 91.0952098694071
x11=87.9532251106725x_{11} = 87.9532251106725
x12=56.5309801938186x_{12} = 56.5309801938186
x13=100.521017074687x_{13} = 100.521017074687
x14=69.100567727981x_{14} = 69.100567727981
x15=−91.0952098694071x_{15} = -91.0952098694071
x16=12.4864543952238x_{16} = 12.4864543952238
x17=−47.1026627703624x_{17} = -47.1026627703624
x18=50.2455828375744x_{18} = 50.2455828375744
x19=−2.79838604578389x_{19} = -2.79838604578389
x20=31.3840740178899x_{20} = 31.3840740178899
x21=62.8159348889734x_{21} = 62.8159348889734
x22=6.12125046689807x_{22} = 6.12125046689807
x23=34.5285657554621x_{23} = 34.5285657554621
x24=78.5270825679419x_{24} = 78.5270825679419
x25=−97.3791034786112x_{25} = -97.3791034786112
x26=94.2371684817036x_{26} = 94.2371684817036
x27=40.8162093266346x_{27} = 40.8162093266346
x28=53.3883466217256x_{28} = 53.3883466217256
x29=−15.644128370333x_{29} = -15.644128370333
x30=−34.5285657554621x_{30} = -34.5285657554621
x31=−25.0929104121121x_{31} = -25.0929104121121
x32=−75.3849592185347x_{32} = -75.3849592185347
x33=84.811211299318x_{33} = 84.811211299318
x34=−62.8159348889734x_{34} = -62.8159348889734
x35=−65.9582857893902x_{35} = -65.9582857893902
x36=−94.2371684817036x_{36} = -94.2371684817036
x37=−9.31786646179107x_{37} = -9.31786646179107
x38=18.7964043662102x_{38} = 18.7964043662102
x39=−81.6691650818489x_{39} = -81.6691650818489
x40=−53.3883466217256x_{40} = -53.3883466217256
x41=2.79838604578389x_{41} = 2.79838604578389
x42=−72.2427897046973x_{42} = -72.2427897046973
x43=−78.5270825679419x_{43} = -78.5270825679419
x44=97.3791034786112x_{44} = 97.3791034786112
x45=−56.5309801938186x_{45} = -56.5309801938186
x46=75.3849592185347x_{46} = 75.3849592185347
x47=−31.3840740178899x_{47} = -31.3840740178899
x48=−21.945612879981x_{48} = -21.945612879981
x49=−100.521017074687x_{49} = -100.521017074687
x50=−69.100567727981x_{50} = -69.100567727981
x51=−87.9532251106725x_{51} = -87.9532251106725
x52=−40.8162093266346x_{52} = -40.8162093266346
x53=65.9582857893902x_{53} = 65.9582857893902
x54=−12.4864543952238x_{54} = -12.4864543952238
x55=9.31786646179107x_{55} = 9.31786646179107
x56=15.644128370333x_{56} = 15.644128370333
x57=43.9595528888955x_{57} = 43.9595528888955
x58=−59.6735041304405x_{58} = -59.6735041304405
x59=28.2389365752603x_{59} = 28.2389365752603
x60=21.945612879981x_{60} = 21.945612879981
x61=−113.088493127061x_{61} = -113.088493127061
x62=−50.2455828375744x_{62} = -50.2455828375744
x63=−6.12125046689807x_{63} = -6.12125046689807
x64=72.2427897046973x_{64} = 72.2427897046973
x65=37.672573565113x_{65} = 37.672573565113
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) + cos(x).
0sin⁡(0)+cos⁡(0)0 \sin{\left(0 \right)} + \cos{\left(0 \right)}
Resultado:
f(0)=1f{\left(0 \right)} = 1
Punto:
(0, 1)
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
xcos⁡(x)=0x \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}
Signos de extremos en los puntos:
(0, 1)

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

 pi  pi 
(--, --)
 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
Puntos máximos de la función:
x1=−π2x_{1} = - \frac{\pi}{2}
x1=π2x_{1} = \frac{\pi}{2}
Decrece en los intervalos
(−∞,−π2]∪[0,∞)\left(-\infty, - \frac{\pi}{2}\right] \cup \left[0, \infty\right)
Crece en los intervalos
(−∞,0]∪[π2,∞)\left(-\infty, 0\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
−xsin⁡(x)+cos⁡(x)=0- x \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=87.9759605524932x_{1} = 87.9759605524932
x2=34.5864242152889x_{2} = 34.5864242152889
x3=62.8477631944545x_{3} = 62.8477631944545
x4=−69.1295029738953x_{4} = -69.1295029738953
x5=50.2853663377737x_{5} = 50.2853663377737
x6=18.90240995686x_{6} = 18.90240995686
x7=56.5663442798215x_{7} = 56.5663442798215
x8=6.43729817917195x_{8} = 6.43729817917195
x9=−62.8477631944545x_{9} = -62.8477631944545
x10=37.7256128277765x_{10} = 37.7256128277765
x11=−9.52933440536196x_{11} = -9.52933440536196
x12=15.7712848748159x_{12} = 15.7712848748159
x13=59.7070073053355x_{13} = 59.7070073053355
x14=12.6452872238566x_{14} = 12.6452872238566
x15=100.540910786842x_{15} = 100.540910786842
x16=78.5525459842429x_{16} = 78.5525459842429
x17=31.4477146375462x_{17} = 31.4477146375462
x18=−72.270467060309x_{18} = -72.270467060309
x19=−15.7712848748159x_{19} = -15.7712848748159
x20=44.0050179208308x_{20} = 44.0050179208308
x21=−94.2583883450399x_{21} = -94.2583883450399
x22=−0.86033358901938x_{22} = -0.86033358901938
x23=−34.5864242152889x_{23} = -34.5864242152889
x24=−31.4477146375462x_{24} = -31.4477146375462
x25=84.8347887180423x_{25} = 84.8347887180423
x26=−75.4114834888481x_{26} = -75.4114834888481
x27=−28.309642854452x_{27} = -28.309642854452
x28=−59.7070073053355x_{28} = -59.7070073053355
x29=−81.6936492356017x_{29} = -81.6936492356017
x30=28.309642854452x_{30} = 28.309642854452
x31=−22.0364967279386x_{31} = -22.0364967279386
x32=−25.1724463266467x_{32} = -25.1724463266467
x33=81.6936492356017x_{33} = 81.6936492356017
x34=65.9885986984904x_{34} = 65.9885986984904
x35=−65.9885986984904x_{35} = -65.9885986984904
x36=−84.8347887180423x_{36} = -84.8347887180423
x37=−78.5525459842429x_{37} = -78.5525459842429
x38=3.42561845948173x_{38} = 3.42561845948173
x39=−50.2853663377737x_{39} = -50.2853663377737
x40=9.52933440536196x_{40} = 9.52933440536196
x41=91.1171613944647x_{41} = 91.1171613944647
x42=−100.540910786842x_{42} = -100.540910786842
x43=−47.145097736761x_{43} = -47.145097736761
x44=−147.661626855354x_{44} = -147.661626855354
x45=75.4114834888481x_{45} = 75.4114834888481
x46=94.2583883450399x_{46} = 94.2583883450399
x47=−87.9759605524932x_{47} = -87.9759605524932
x48=47.145097736761x_{48} = 47.145097736761
x49=−116.247530303932x_{49} = -116.247530303932
x50=−91.1171613944647x_{50} = -91.1171613944647
x51=−12.6452872238566x_{51} = -12.6452872238566
x52=25.1724463266467x_{52} = 25.1724463266467
x53=72.270467060309x_{53} = 72.270467060309
x54=40.8651703304881x_{54} = 40.8651703304881
x55=−40.8651703304881x_{55} = -40.8651703304881
x56=−44.0050179208308x_{56} = -44.0050179208308
x57=0.86033358901938x_{57} = 0.86033358901938
x58=69.1295029738953x_{58} = 69.1295029738953
x59=−3.42561845948173x_{59} = -3.42561845948173
x60=−97.3996388790738x_{60} = -97.3996388790738
x61=−56.5663442798215x_{61} = -56.5663442798215
x62=97.3996388790738x_{62} = 97.3996388790738
x63=−6.43729817917195x_{63} = -6.43729817917195
x64=−53.4257904773947x_{64} = -53.4257904773947
x65=53.4257904773947x_{65} = 53.4257904773947
x66=22.0364967279386x_{66} = 22.0364967279386
x67=−37.7256128277765x_{67} = -37.7256128277765
x68=−18.90240995686x_{68} = -18.90240995686

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
[97.3996388790738,∞)\left[97.3996388790738, \infty\right)
Convexa en los intervalos
(−∞,−100.540910786842]\left(-\infty, -100.540910786842\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xsin⁡(x)+cos⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \sin{\left(x \right)} + \cos{\left(x \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)+cos⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \sin{\left(x \right)} + \cos{\left(x \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) + cos(x), 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→−∞(xsin⁡(x)+cos⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{x \sin{\left(x \right)} + \cos{\left(x \right)}}{x}\right)
True

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