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(x-2*cos(x))/(x+5*cos(x))
Trazado el gráfico de la función/ (x-2*cos(x))/(x+5*cos(x))

Gráfico de la función y = (x-2*cos(x))/(x+5*cos(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       x - 2*cos(x)
f(x) = ------------
       x + 5*cos(x)
f(x)=x−2cos⁡(x)x+5cos⁡(x)f{\left(x \right)} = \frac{x - 2 \cos{\left(x \right)}}{x + 5 \cos{\left(x \right)}} 
f = (x - 2*cos(x))/(x + 5*cos(x))
Gráfico de la función
02468-8-6-4-2-1010-5050
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:
x−2cos⁡(x)x+5cos⁡(x)=0\frac{x - 2 \cos{\left(x \right)}}{x + 5 \cos{\left(x \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=1.02986652932226x_{1} = 1.02986652932226
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 - 2*cos(x))/(x + 5*cos(x)).
(−1)2cos⁡(0)5cos⁡(0)\frac{\left(-1\right) 2 \cos{\left(0 \right)}}{5 \cos{\left(0 \right)}}
Resultado:
f(0)=−25f{\left(0 \right)} = - \frac{2}{5}
Punto:
(0, -2/5)
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
(x−2cos⁡(x))(5sin⁡(x)−1)(x+5cos⁡(x))2+2sin⁡(x)+1x+5cos⁡(x)=0\frac{\left(x - 2 \cos{\left(x \right)}\right) \left(5 \sin{\left(x \right)} - 1\right)}{\left(x + 5 \cos{\left(x \right)}\right)^{2}} + \frac{2 \sin{\left(x \right)} + 1}{x + 5 \cos{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−69.100567727981x_{1} = -69.100567727981
x2=−12.4864543952238x_{2} = -12.4864543952238
x3=235.615204836452x_{3} = 235.615204836452
x4=−2.79838604578389x_{4} = -2.79838604578389
x5=91.0952098694071x_{5} = 91.0952098694071
x6=78.5270825679419x_{6} = 78.5270825679419
x7=−72.2427897046973x_{7} = -72.2427897046973
x8=15.644128370333x_{8} = 15.644128370333
x9=−43.9595528888955x_{9} = -43.9595528888955
x10=81.6691650818489x_{10} = 81.6691650818489
x11=50.2455828375744x_{11} = 50.2455828375744
x12=−62.8159348889734x_{12} = -62.8159348889734
x13=6.12125046689807x_{13} = 6.12125046689807
x14=−163.356696489782x_{14} = -163.356696489782
x15=−84.811211299318x_{15} = -84.811211299318
x16=62.8159348889734x_{16} = 62.8159348889734
x17=56.5309801938186x_{17} = 56.5309801938186
x18=−28.2389365752603x_{18} = -28.2389365752603
x19=−56.5309801938186x_{19} = -56.5309801938186
x20=−25.0929104121121x_{20} = -25.0929104121121
x21=−59.6735041304405x_{21} = -59.6735041304405
x22=69.100567727981x_{22} = 69.100567727981
x23=−47.1026627703624x_{23} = -47.1026627703624
x24=100.521017074687x_{24} = 100.521017074687
x25=72.2427897046973x_{25} = 72.2427897046973
x26=−40.8162093266346x_{26} = -40.8162093266346
x27=47.1026627703624x_{27} = 47.1026627703624
x28=−78.5270825679419x_{28} = -78.5270825679419
x29=97.3791034786112x_{29} = 97.3791034786112
x30=−31.3840740178899x_{30} = -31.3840740178899
x31=−37.672573565113x_{31} = -37.672573565113
x32=18.7964043662102x_{32} = 18.7964043662102
x33=−75.3849592185347x_{33} = -75.3849592185347
x34=40.8162093266346x_{34} = 40.8162093266346
x35=−34.5285657554621x_{35} = -34.5285657554621
x36=−53.3883466217256x_{36} = -53.3883466217256
x37=34.5285657554621x_{37} = 34.5285657554621
x38=37.672573565113x_{38} = 37.672573565113
x39=−100.521017074687x_{39} = -100.521017074687
x40=−6.12125046689807x_{40} = -6.12125046689807
x41=−65.9582857893902x_{41} = -65.9582857893902
x42=59.6735041304405x_{42} = 59.6735041304405
x43=−91.0952098694071x_{43} = -91.0952098694071
x44=75.3849592185347x_{44} = 75.3849592185347
x45=65.9582857893902x_{45} = 65.9582857893902
x46=84.811211299318x_{46} = 84.811211299318
x47=−87.9532251106725x_{47} = -87.9532251106725
x48=150.789815721919x_{48} = 150.789815721919
x49=25.0929104121121x_{49} = 25.0929104121121
x50=94.2371684817036x_{50} = 94.2371684817036
x51=−94.2371684817036x_{51} = -94.2371684817036
x52=2.79838604578389x_{52} = 2.79838604578389
x53=28.2389365752603x_{53} = 28.2389365752603
x54=−21.945612879981x_{54} = -21.945612879981
x55=53.3883466217256x_{55} = 53.3883466217256
x56=−50.2455828375744x_{56} = -50.2455828375744
x57=87.9532251106725x_{57} = 87.9532251106725
x58=43.9595528888955x_{58} = 43.9595528888955
x59=−81.6691650818489x_{59} = -81.6691650818489
x60=−9.31786646179107x_{60} = -9.31786646179107
x61=12.4864543952238x_{61} = 12.4864543952238
x62=−97.3791034786112x_{62} = -97.3791034786112
x63=31.3840740178899x_{63} = 31.3840740178899
x64=−15.644128370333x_{64} = -15.644128370333
x65=21.945612879981x_{65} = 21.945612879981
Signos de extremos en los puntos:
(-69.10056772798097, 1.10919107585662)

(-12.486454395223781, 1.93005534894535)

(235.61520483645214, 1.03035331340083)

(-2.798386045783887, 0.121893023800573)

(91.09520986940714, 1.08130015347894)

(78.52708256794193, 1.09519477249879)

(-72.24278970469729, 0.909384773656961)

(15.644128370333028, 1.65567280332762)

(-43.959552888895495, 1.17962108696221)

(81.66916508184887, 0.919238802280663)

(50.24558283757444, 0.873315854123452)

(-62.81593488897342, 1.12105722215852)

(6.1212504668980685, 0.375133636909343)

(-163.35669648978208, 1.04420314934587)

(-84.81121129931802, 0.922063844299278)

(62.81593488897342, 0.896791535079383)

(56.53098019381864, 0.886252518757009)

(-28.238936575260272, 0.789515699263529)

(-56.53098019381864, 1.1358173045276)

(-25.092910412112097, 1.34803657854872)

(-59.67350413044053, 0.891778041506324)

(69.10056772798097, 0.905543010322104)

(-47.10266277036235, 0.865677223824851)

(100.52101707468658, 0.93366563126969)

(72.24278970469729, 1.10408966745177)

(-40.81620932663458, 0.847256478219416)

(47.10266277036235, 1.1662183651073)

(-78.52708256794193, 0.916201233894233)

(97.3791034786112, 1.07577050109387)

(-31.38407401788986, 1.26515149884497)

(-37.67257356511297, 1.21415999264815)

(18.796404366210158, 0.706166142153014)

(-75.38495921853475, 1.09944369555941)

(40.81620932663458, 1.19537542176621)

(-34.52856575546206, 0.822977712809246)

(-53.38834662172563, 0.88013228834934)

(34.52856575546206, 1.2369424126468)

(37.67257356511297, 0.836011205850915)

(-100.52101707468658, 1.07327848603399)

(-6.1212504668980685, 6.82171277947367)

(-65.95828578939016, 0.901361028841512)

(59.67350413044053, 1.12801314958011)

(-91.09520986940714, 0.927159738569296)

(75.38495921853475, 0.912926216970315)

(65.95828578939016, 1.1148183478799)

(84.81121129931802, 1.0877004980927)

(-87.95322511067255, 1.0843791223833)

(150.78981572191927, 0.955068622538491)

(25.092910412112097, 0.767540933235872)

(94.23716848170359, 0.929465684286188)

(-94.23716848170359, 1.07843798674372)

(2.798386045783887, -2.45115557209475)

(28.238936575260272, 1.30098934668939)

(-21.945612879981045, 0.740436808225191)

(53.38834662172563, 1.14463493903124)

(-50.24558283757444, 1.15467721932994)

(87.95322511067255, 0.924697912039217)

(43.959552888895495, 0.857058041539486)

(-81.66916508184887, 1.09129408266298)

(-9.317866461791066, 0.512920588089769)

(12.486454395223781, 0.600603289524513)

(-97.3791034786112, 0.931630101053182)

(31.38407401788986, 0.807692316269933)

(-15.644128370333028, 0.661444146992734)

(21.945612879981045, 1.41253190207815)


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=−2.79838604578389x_{1} = -2.79838604578389
x2=−72.2427897046973x_{2} = -72.2427897046973
x3=81.6691650818489x_{3} = 81.6691650818489
x4=50.2455828375744x_{4} = 50.2455828375744
x5=6.12125046689807x_{5} = 6.12125046689807
x6=−84.811211299318x_{6} = -84.811211299318
x7=62.8159348889734x_{7} = 62.8159348889734
x8=56.5309801938186x_{8} = 56.5309801938186
x9=−28.2389365752603x_{9} = -28.2389365752603
x10=−59.6735041304405x_{10} = -59.6735041304405
x11=69.100567727981x_{11} = 69.100567727981
x12=−47.1026627703624x_{12} = -47.1026627703624
x13=100.521017074687x_{13} = 100.521017074687
x14=−40.8162093266346x_{14} = -40.8162093266346
x15=−78.5270825679419x_{15} = -78.5270825679419
x16=18.7964043662102x_{16} = 18.7964043662102
x17=−34.5285657554621x_{17} = -34.5285657554621
x18=−53.3883466217256x_{18} = -53.3883466217256
x19=37.672573565113x_{19} = 37.672573565113
x20=−65.9582857893902x_{20} = -65.9582857893902
x21=−91.0952098694071x_{21} = -91.0952098694071
x22=75.3849592185347x_{22} = 75.3849592185347
x23=150.789815721919x_{23} = 150.789815721919
x24=25.0929104121121x_{24} = 25.0929104121121
x25=94.2371684817036x_{25} = 94.2371684817036
x26=−21.945612879981x_{26} = -21.945612879981
x27=87.9532251106725x_{27} = 87.9532251106725
x28=43.9595528888955x_{28} = 43.9595528888955
x29=−9.31786646179107x_{29} = -9.31786646179107
x30=12.4864543952238x_{30} = 12.4864543952238
x31=−97.3791034786112x_{31} = -97.3791034786112
x32=31.3840740178899x_{32} = 31.3840740178899
x33=−15.644128370333x_{33} = -15.644128370333
Puntos máximos de la función:
x33=−69.100567727981x_{33} = -69.100567727981
x33=−12.4864543952238x_{33} = -12.4864543952238
x33=235.615204836452x_{33} = 235.615204836452
x33=91.0952098694071x_{33} = 91.0952098694071
x33=78.5270825679419x_{33} = 78.5270825679419
x33=15.644128370333x_{33} = 15.644128370333
x33=−43.9595528888955x_{33} = -43.9595528888955
x33=−62.8159348889734x_{33} = -62.8159348889734
x33=−163.356696489782x_{33} = -163.356696489782
x33=−56.5309801938186x_{33} = -56.5309801938186
x33=−25.0929104121121x_{33} = -25.0929104121121
x33=72.2427897046973x_{33} = 72.2427897046973
x33=47.1026627703624x_{33} = 47.1026627703624
x33=97.3791034786112x_{33} = 97.3791034786112
x33=−31.3840740178899x_{33} = -31.3840740178899
x33=−37.672573565113x_{33} = -37.672573565113
x33=−75.3849592185347x_{33} = -75.3849592185347
x33=40.8162093266346x_{33} = 40.8162093266346
x33=34.5285657554621x_{33} = 34.5285657554621
x33=−100.521017074687x_{33} = -100.521017074687
x33=−6.12125046689807x_{33} = -6.12125046689807
x33=59.6735041304405x_{33} = 59.6735041304405
x33=65.9582857893902x_{33} = 65.9582857893902
x33=84.811211299318x_{33} = 84.811211299318
x33=−87.9532251106725x_{33} = -87.9532251106725
x33=−94.2371684817036x_{33} = -94.2371684817036
x33=2.79838604578389x_{33} = 2.79838604578389
x33=28.2389365752603x_{33} = 28.2389365752603
x33=53.3883466217256x_{33} = 53.3883466217256
x33=−50.2455828375744x_{33} = -50.2455828375744
x33=−81.6691650818489x_{33} = -81.6691650818489
x33=21.945612879981x_{33} = 21.945612879981
Decrece en los intervalos
[150.789815721919,∞)\left[150.789815721919, \infty\right)
Crece en los intervalos
(−∞,−97.3791034786112]\left(-\infty, -97.3791034786112\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(x−2cos⁡(x)x+5cos⁡(x))=1\lim_{x \to -\infty}\left(\frac{x - 2 \cos{\left(x \right)}}{x + 5 \cos{\left(x \right)}}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=1y = 1
lim⁡x→∞(x−2cos⁡(x)x+5cos⁡(x))=1\lim_{x \to \infty}\left(\frac{x - 2 \cos{\left(x \right)}}{x + 5 \cos{\left(x \right)}}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=1y = 1
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (x - 2*cos(x))/(x + 5*cos(x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(x−2cos⁡(x)x(x+5cos⁡(x)))=0\lim_{x \to -\infty}\left(\frac{x - 2 \cos{\left(x \right)}}{x \left(x + 5 \cos{\left(x \right)}\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(x−2cos⁡(x)x(x+5cos⁡(x)))=0\lim_{x \to \infty}\left(\frac{x - 2 \cos{\left(x \right)}}{x \left(x + 5 \cos{\left(x \right)}\right)}\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:
x−2cos⁡(x)x+5cos⁡(x)=−x−2cos⁡(x)−x+5cos⁡(x)\frac{x - 2 \cos{\left(x \right)}}{x + 5 \cos{\left(x \right)}} = \frac{- x - 2 \cos{\left(x \right)}}{- x + 5 \cos{\left(x \right)}}
- No
x−2cos⁡(x)x+5cos⁡(x)=−−x−2cos⁡(x)−x+5cos⁡(x)\frac{x - 2 \cos{\left(x \right)}}{x + 5 \cos{\left(x \right)}} = - \frac{- x - 2 \cos{\left(x \right)}}{- x + 5 \cos{\left(x \right)}}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = (x-2*cos(x))/(x+5*cos(x))
    © Sr Examen – Калькулятор