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

Gráfico de la función y = cos(x)+sin(x)+x*sin(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = cos(x) + sin(x) + x*sin(x)
f(x)=xsin⁡(x)+(sin⁡(x)+cos⁡(x))f{\left(x \right)} = x \sin{\left(x \right)} + \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) 
f = x*sin(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)+(sin⁡(x)+cos⁡(x))=0x \sin{\left(x \right)} + \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−62.815677356778x_{1} = -62.815677356778
x2=47.1031041186137x_{2} = 47.1031041186137
x3=−91.0950880256329x_{3} = -91.0950880256329
x4=−37.6718497263809x_{4} = -37.6718497263809
x5=−21.9434371567881x_{5} = -21.9434371567881
x6=−84.8110706151124x_{6} = -84.8110706151124
x7=−43.9590233567938x_{7} = -43.9590233567938
x8=−72.24259540785x_{8} = -72.24259540785
x9=−50.2451786914948x_{9} = -50.2451786914948
x10=−65.9580523911179x_{10} = -65.9580523911179
x11=81.6693131963402x_{11} = 81.6693131963402
x12=−81.6690132946536x_{12} = -81.6690132946536
x13=78.5272426949571x_{13} = 78.5272426949571
x14=−12.4794779911025x_{14} = -12.4794779911025
x15=43.9600588531378x_{15} = 43.9600588531378
x16=59.6737803264459x_{16} = 59.6737803264459
x17=−69.1003552230555x_{17} = -69.1003552230555
x18=21.9475985837942x_{18} = 21.9475985837942
x19=−2.57625015820118x_{19} = -2.57625015820118
x20=37.673259943911x_{20} = 37.673259943911
x21=12.492390025579x_{21} = 12.492390025579
x22=34.5293808983144x_{22} = 34.5293808983144
x23=94.2372799036618x_{23} = 94.2372799036618
x24=−18.7934144113698x_{24} = -18.7934144113698
x25=−100.520917114109x_{25} = -100.520917114109
x26=97.379207861883x_{26} = 97.379207861883
x27=−94.2370546693974x_{27} = -94.2370546693974
x28=−25.0912562079058x_{28} = -25.0912562079058
x29=2.88996969767843x_{29} = 2.88996969767843
x30=−15.6397620877646x_{30} = -15.6397620877646
x31=−9.30494468339504x_{31} = -9.30494468339504
x32=56.5312876685112x_{32} = 56.5312876685112
x33=69.1007741687956x_{33} = 69.1007741687956
x34=53.388691007263x_{34} = 53.388691007263
x35=50.2459712046114x_{35} = 50.2459712046114
x36=84.8113487041494x_{36} = 84.8113487041494
x37=−31.3830252979972x_{37} = -31.3830252979972
x38=6.14411351301787x_{38} = 6.14411351301787
x39=−6.08916120309943x_{39} = -6.08916120309943
x40=62.8161843480611x_{40} = 62.8161843480611
x41=31.38505790634x_{41} = 31.38505790634
x42=−53.3879890840753x_{42} = -53.3879890840753
x43=18.7990914357831x_{43} = 18.7990914357831
x44=−34.527701946778x_{44} = -34.527701946778
x45=100.521115065812x_{45} = 100.521115065812
x46=28.2401476526276x_{46} = 28.2401476526276
x47=−78.5269183093816x_{47} = -78.5269183093816
x48=15.6479679638982x_{48} = 15.6479679638982
x49=9.32825706323943x_{49} = 9.32825706323943
x50=75.3851328811964x_{50} = 75.3851328811964
x51=−87.9530943542027x_{51} = -87.9530943542027
x52=72.242978694986x_{52} = 72.242978694986
x53=−59.6732185170696x_{53} = -59.6732185170696
x54=40.8167952172419x_{54} = 40.8167952172419
x55=−47.1022022669651x_{55} = -47.1022022669651
x56=65.9585122146304x_{56} = 65.9585122146304
x57=−75.3847808857452x_{57} = -75.3847808857452
x58=−97.378996929011x_{58} = -97.378996929011
x59=87.9533529268738x_{59} = 87.9533529268738
x60=−56.5306616416093x_{60} = -56.5306616416093
x61=−40.8155939881502x_{61} = -40.8155939881502
x62=−28.2376364595748x_{62} = -28.2376364595748
x63=91.0953290668266x_{63} = 91.0953290668266
x64=25.0944376288815x_{64} = 25.0944376288815
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 cos(x) + sin(x) + x*sin(x).
0sin⁡(0)+(sin⁡(0)+cos⁡(0))0 \sin{\left(0 \right)} + \left(\sin{\left(0 \right)} + \cos{\left(0 \right)}\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)+cos⁡(x)=0x \cos{\left(x \right)} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−1x_{1} = -1
x2=π2x_{2} = \frac{\pi}{2}
x3=3π2x_{3} = \frac{3 \pi}{2}
Signos de extremos en los puntos:
(-1, cos(1))

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

 3*pi       3*pi 
(----, -1 - ----)
  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=−1x_{1} = -1
x2=3π2x_{2} = \frac{3 \pi}{2}
Puntos máximos de la función:
x2=π2x_{2} = \frac{\pi}{2}
Decrece en los intervalos
[−1,π2]∪[3π2,∞)\left[-1, \frac{\pi}{2}\right] \cup \left[\frac{3 \pi}{2}, \infty\right)
Crece en los intervalos
(−∞,−1]∪[π2,3π2]\left(-\infty, -1\right] \cup \left[\frac{\pi}{2}, \frac{3 \pi}{2}\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)−sin⁡(x)+cos⁡(x)=0- x \sin{\left(x \right)} - \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=72.270278284086x_{1} = 72.270278284086
x2=−62.8480203293155x_{2} = -62.8480203293155
x3=−65.9888317703628x_{3} = -65.9888317703628
x4=−59.7072924300783x_{4} = -59.7072924300783
x5=47.1446575542095x_{5} = 47.1446575542095
x6=−1.28924004659366x_{6} = -1.28924004659366
x7=−87.9760912064584x_{7} = -87.9760912064584
x8=40.8645864924056x_{8} = 40.8645864924056
x9=−78.5527100810745x_{9} = -78.5527100810745
x10=94.2582769975219x_{10} = 94.2582769975219
x11=−47.1455569712193x_{11} = -47.1455569712193
x12=−100.541010687502x_{12} = -100.541010687502
x13=28.3084405150129x_{13} = 28.3084405150129
x14=22.034534408764x_{14} = 22.034534408764
x15=−91.1172831492274x_{15} = -91.1172831492274
x16=−12.6519831342203x_{16} = -12.6519831342203
x17=−28.3109329053961x_{17} = -28.3109329053961
x18=−22.0386444078516x_{18} = -22.0386444078516
x19=−150.803122701426x_{19} = -150.803122701426
x20=75.411310006954x_{20} = 75.411310006954
x21=84.8346514263999x_{21} = 84.8346514263999
x22=−15.7755396389074x_{22} = -15.7755396389074
x23=−18.9053472009705x_{23} = -18.9053472009705
x24=97.399534561137x_{24} = 97.399534561137
x25=50.2849788736454x_{25} = 50.2849788736454
x26=−50.2857695061759x_{26} = -50.2857695061759
x27=6.41719900457425x_{27} = 6.41719900457425
x28=−97.3997453605998x_{28} = -97.3997453605998
x29=−3.51943605007227x_{29} = -3.51943605007227
x30=9.51955422261535x_{30} = 9.51955422261535
x31=12.6395558422743x_{31} = 12.6395558422743
x32=−31.4487567990639x_{32} = -31.4487567990639
x33=44.0045134880243x_{33} = 44.0045134880243
x34=69.129296788477x_{34} = 69.129296788477
x35=−25.1740842709126x_{35} = -25.1740842709126
x36=53.4254468020254x_{36} = 53.4254468020254
x37=−69.1297152082093x_{37} = -69.1297152082093
x38=−81.6938008846843x_{38} = -81.6938008846843
x39=−72.2706611309075x_{39} = -72.2706611309075
x40=31.4467365238542x_{40} = 31.4467365238542
x41=78.5523860109268x_{41} = 78.5523860109268
x42=100.540812853297x_{42} = 100.540812853297
x43=−94.2585020796787x_{43} = -94.2585020796787
x44=−6.46419193189059x_{44} = -6.46419193189059
x45=81.693501252623x_{45} = 81.693501252623
x46=18.8997655344074x_{46} = 18.8997655344074
x47=−56.5666622243178x_{47} = -56.5666622243178
x48=62.8475141081981x_{48} = 62.8475141081981
x49=65.9883725804509x_{49} = 65.9883725804509
x50=−40.8657834053267x_{50} = -40.8657834053267
x51=56.5660373713478x_{51} = 56.5660373713478
x52=15.7675317799203x_{52} = 15.7675317799203
x53=59.7067315662923x_{53} = 59.7067315662923
x54=−84.834929283594x_{54} = -84.834929283594
x55=25.1709329794757x_{55} = 25.1709329794757
x56=−75.4116616310224x_{56} = -75.4116616310224
x57=87.9758328342401x_{57} = 87.9758328342401
x58=−53.4261472496575x_{58} = -53.4261472496575
x59=91.1170422822337x_{59} = 91.1170422822337
x60=34.5856130401369x_{60} = 34.5856130401369
x61=−37.7263335361973x_{61} = -37.7263335361973
x62=−34.58728356964x_{62} = -34.58728356964
x63=−9.54132523092565x_{63} = -9.54132523092565
x64=37.7249292642936x_{64} = 37.7249292642936
x65=0.567782020656099x_{65} = 0.567782020656099
x66=3.36671587754539x_{66} = 3.36671587754539
x67=−44.0055457757373x_{67} = -44.0055457757373

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
[97.399534561137,∞)\left[97.399534561137, \infty\right)
Convexa en los intervalos
(−∞,−150.803122701426]\left(-\infty, -150.803122701426\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xsin⁡(x)+(sin⁡(x)+cos⁡(x)))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \sin{\left(x \right)} + \left(\sin{\left(x \right)} + \cos{\left(x \right)}\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)+(sin⁡(x)+cos⁡(x)))=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \sin{\left(x \right)} + \left(\sin{\left(x \right)} + \cos{\left(x \right)}\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 cos(x) + sin(x) + x*sin(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)+(sin⁡(x)+cos⁡(x))x)y = x \lim_{x \to -\infty}\left(\frac{x \sin{\left(x \right)} + \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{x}\right)
True

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