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Trazado el gráfico de la función/ (cos(x))/x+1

Gráfico de la función y = (cos(x))/x+1

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x)    
f(x) = ------ + 1
         x
f(x)=1+cos⁡(x)xf{\left(x \right)} = 1 + \frac{\cos{\left(x \right)}}{x} 
f = 1 + cos(x)/x
Gráfico de la función
0-50-40-30-20-105010203040-10001000
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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:
1+cos⁡(x)x=01 + \frac{\cos{\left(x \right)}}{x} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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)/x + 1.
cos⁡(0)0+1\frac{\cos{\left(0 \right)}}{0} + 1
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
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
−sin⁡(x)x−cos⁡(x)x2=0- \frac{\sin{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=47.1026627703624x_{1} = 47.1026627703624
x2=84.811211299318x_{2} = 84.811211299318
x3=−50.2455828375744x_{3} = -50.2455828375744
x4=−100.521017074687x_{4} = -100.521017074687
x5=−65.9582857893902x_{5} = -65.9582857893902
x6=65.9582857893902x_{6} = 65.9582857893902
x7=−78.5270825679419x_{7} = -78.5270825679419
x8=−21.945612879981x_{8} = -21.945612879981
x9=28.2389365752603x_{9} = 28.2389365752603
x10=53.3883466217256x_{10} = 53.3883466217256
x11=−87.9532251106725x_{11} = -87.9532251106725
x12=−25.0929104121121x_{12} = -25.0929104121121
x13=−34.5285657554621x_{13} = -34.5285657554621
x14=−69.100567727981x_{14} = -69.100567727981
x15=94.2371684817036x_{15} = 94.2371684817036
x16=−135.08108127842x_{16} = -135.08108127842
x17=−62.8159348889734x_{17} = -62.8159348889734
x18=50.2455828375744x_{18} = 50.2455828375744
x19=−91.0952098694071x_{19} = -91.0952098694071
x20=91.0952098694071x_{20} = 91.0952098694071
x21=56.5309801938186x_{21} = 56.5309801938186
x22=−53.3883466217256x_{22} = -53.3883466217256
x23=−169.640108529775x_{23} = -169.640108529775
x24=−109.946647805931x_{24} = -109.946647805931
x25=−6.12125046689807x_{25} = -6.12125046689807
x26=2.79838604578389x_{26} = 2.79838604578389
x27=−47.1026627703624x_{27} = -47.1026627703624
x28=62.8159348889734x_{28} = 62.8159348889734
x29=9.31786646179107x_{29} = 9.31786646179107
x30=−2.79838604578389x_{30} = -2.79838604578389
x31=−81.6691650818489x_{31} = -81.6691650818489
x32=12.4864543952238x_{32} = 12.4864543952238
x33=−31.3840740178899x_{33} = -31.3840740178899
x34=−94.2371684817036x_{34} = -94.2371684817036
x35=197.91528455229x_{35} = 197.91528455229
x36=59.6735041304405x_{36} = 59.6735041304405
x37=97.3791034786112x_{37} = 97.3791034786112
x38=75.3849592185347x_{38} = 75.3849592185347
x39=−40.8162093266346x_{39} = -40.8162093266346
x40=−15.644128370333x_{40} = -15.644128370333
x41=−37.672573565113x_{41} = -37.672573565113
x42=−12.4864543952238x_{42} = -12.4864543952238
x43=15.644128370333x_{43} = 15.644128370333
x44=69.100567727981x_{44} = 69.100567727981
x45=−84.811211299318x_{45} = -84.811211299318
x46=31.3840740178899x_{46} = 31.3840740178899
x47=37.672573565113x_{47} = 37.672573565113
x48=−97.3791034786112x_{48} = -97.3791034786112
x49=6.12125046689807x_{49} = 6.12125046689807
x50=72.2427897046973x_{50} = 72.2427897046973
x51=−75.3849592185347x_{51} = -75.3849592185347
x52=34.5285657554621x_{52} = 34.5285657554621
x53=43.9595528888955x_{53} = 43.9595528888955
x54=78.5270825679419x_{54} = 78.5270825679419
x55=40.8162093266346x_{55} = 40.8162093266346
x56=100.521017074687x_{56} = 100.521017074687
x57=21.945612879981x_{57} = 21.945612879981
x58=−9.31786646179107x_{58} = -9.31786646179107
x59=25.0929104121121x_{59} = 25.0929104121121
x60=−72.2427897046973x_{60} = -72.2427897046973
x61=−18.7964043662102x_{61} = -18.7964043662102
x62=87.9532251106725x_{62} = 87.9532251106725
x63=−59.6735041304405x_{63} = -59.6735041304405
x64=−28.2389365752603x_{64} = -28.2389365752603
x65=18.7964043662102x_{65} = 18.7964043662102
x66=81.6691650818489x_{66} = 81.6691650818489
x67=−56.5309801938186x_{67} = -56.5309801938186
x68=−43.9595528888955x_{68} = -43.9595528888955
Signos de extremos en los puntos:
(47.10266277036235, 0.978774560583586)

(84.81121129931802, 0.988209925558923)

(-50.24558283757444, 0.980101693469645)

(-100.52101707468658, 0.990052323884637)

(-65.95828578939016, 1.01515935531684)

(65.95828578939016, 0.984840644683159)

(-78.52708256794193, 1.01273342767775)

(-21.945612879981045, 1.04551996040513)

(28.238936575260272, 0.964610084445831)

(53.38834662172563, 0.981272605535913)

(-87.95322511067255, 0.988631055084119)

(-25.092910412112097, 0.960179714449949)

(-34.52856575546206, 1.02894938891145)

(-69.10056772798097, 0.985529854025324)

(94.23716848170359, 1.01061092686295)

(-135.0810812784199, 1.00740275832667)

(-62.81593488897342, 0.984082489416574)

(50.24558283757444, 1.01989830653036)

(-91.09520986940714, 1.01097686424834)

(91.09520986940714, 0.989023135751658)

(56.53098019381864, 1.01768664855217)

(-53.38834662172563, 1.01872739446409)

(-169.6401085297751, 0.994105270064991)

(-109.94664780593057, 1.00909494432157)

(-6.1212504668980685, 0.838771965674936)

(2.798386045783887, 0.663491583081605)

(-47.10266277036235, 1.02122543941641)

(62.81593488897342, 1.01591751058343)

(9.317866461791066, 0.893292052284763)

(-2.798386045783887, 1.3365084169184)

(-81.66916508184887, 0.987756394432953)

(12.486454395223781, 1.07983118078)

(-31.38407401788986, 0.968152867888731)

(-94.23716848170359, 0.98938907313705)

(197.91528455229027, 0.994947397631339)

(59.67350413044053, 0.983244496342811)

(97.3791034786112, 0.989731397796919)

(75.38495921853475, 1.01326407865182)

(-40.81620932663458, 1.0244927205347)

(-15.644128370333028, 1.06379155303959)

(-37.67257356511297, 0.973464836989695)

(-12.486454395223781, 0.920168819219997)

(15.644128370333028, 0.936208446960406)

(69.10056772798097, 1.01447014597468)

(-84.81121129931802, 1.01179007444108)

(31.38407401788986, 1.03184713211127)

(37.67257356511297, 1.0265351630103)

(-97.3791034786112, 1.01026860220308)

(6.1212504668980685, 1.16122803432506)

(72.24278970469729, 0.986159114086845)

(-75.38495921853475, 0.986735921348175)

(34.52856575546206, 0.97105061108855)

(43.959552888895495, 1.02274230047253)

(78.52708256794193, 0.987266572322253)

(40.81620932663458, 0.975507279465304)

(100.52101707468658, 1.00994767611536)

(21.945612879981045, 0.954480039594871)

(-9.317866461791066, 1.10670794771524)

(25.092910412112097, 1.03982028555005)

(-72.24278970469729, 1.01384088591315)

(-18.796404366210158, 0.946873467438612)

(87.95322511067255, 1.01136894491588)

(-59.67350413044053, 1.01675550365719)

(-28.238936575260272, 1.03538991555417)

(18.796404366210158, 1.05312653256139)

(81.66916508184887, 1.01224360556705)

(-56.53098019381864, 0.98231335144783)

(-43.959552888895495, 0.977257699527469)


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=47.1026627703624x_{1} = 47.1026627703624
x2=84.811211299318x_{2} = 84.811211299318
x3=−50.2455828375744x_{3} = -50.2455828375744
x4=−100.521017074687x_{4} = -100.521017074687
x5=65.9582857893902x_{5} = 65.9582857893902
x6=28.2389365752603x_{6} = 28.2389365752603
x7=53.3883466217256x_{7} = 53.3883466217256
x8=−87.9532251106725x_{8} = -87.9532251106725
x9=−25.0929104121121x_{9} = -25.0929104121121
x10=−69.100567727981x_{10} = -69.100567727981
x11=−62.8159348889734x_{11} = -62.8159348889734
x12=91.0952098694071x_{12} = 91.0952098694071
x13=−169.640108529775x_{13} = -169.640108529775
x14=−6.12125046689807x_{14} = -6.12125046689807
x15=2.79838604578389x_{15} = 2.79838604578389
x16=9.31786646179107x_{16} = 9.31786646179107
x17=−81.6691650818489x_{17} = -81.6691650818489
x18=−31.3840740178899x_{18} = -31.3840740178899
x19=−94.2371684817036x_{19} = -94.2371684817036
x20=197.91528455229x_{20} = 197.91528455229
x21=59.6735041304405x_{21} = 59.6735041304405
x22=97.3791034786112x_{22} = 97.3791034786112
x23=−37.672573565113x_{23} = -37.672573565113
x24=−12.4864543952238x_{24} = -12.4864543952238
x25=15.644128370333x_{25} = 15.644128370333
x26=72.2427897046973x_{26} = 72.2427897046973
x27=−75.3849592185347x_{27} = -75.3849592185347
x28=34.5285657554621x_{28} = 34.5285657554621
x29=78.5270825679419x_{29} = 78.5270825679419
x30=40.8162093266346x_{30} = 40.8162093266346
x31=21.945612879981x_{31} = 21.945612879981
x32=−18.7964043662102x_{32} = -18.7964043662102
x33=−56.5309801938186x_{33} = -56.5309801938186
x34=−43.9595528888955x_{34} = -43.9595528888955
Puntos máximos de la función:
x34=−65.9582857893902x_{34} = -65.9582857893902
x34=−78.5270825679419x_{34} = -78.5270825679419
x34=−21.945612879981x_{34} = -21.945612879981
x34=−34.5285657554621x_{34} = -34.5285657554621
x34=94.2371684817036x_{34} = 94.2371684817036
x34=−135.08108127842x_{34} = -135.08108127842
x34=50.2455828375744x_{34} = 50.2455828375744
x34=−91.0952098694071x_{34} = -91.0952098694071
x34=56.5309801938186x_{34} = 56.5309801938186
x34=−53.3883466217256x_{34} = -53.3883466217256
x34=−109.946647805931x_{34} = -109.946647805931
x34=−47.1026627703624x_{34} = -47.1026627703624
x34=62.8159348889734x_{34} = 62.8159348889734
x34=−2.79838604578389x_{34} = -2.79838604578389
x34=12.4864543952238x_{34} = 12.4864543952238
x34=75.3849592185347x_{34} = 75.3849592185347
x34=−40.8162093266346x_{34} = -40.8162093266346
x34=−15.644128370333x_{34} = -15.644128370333
x34=69.100567727981x_{34} = 69.100567727981
x34=−84.811211299318x_{34} = -84.811211299318
x34=31.3840740178899x_{34} = 31.3840740178899
x34=37.672573565113x_{34} = 37.672573565113
x34=−97.3791034786112x_{34} = -97.3791034786112
x34=6.12125046689807x_{34} = 6.12125046689807
x34=43.9595528888955x_{34} = 43.9595528888955
x34=100.521017074687x_{34} = 100.521017074687
x34=−9.31786646179107x_{34} = -9.31786646179107
x34=25.0929104121121x_{34} = 25.0929104121121
x34=−72.2427897046973x_{34} = -72.2427897046973
x34=87.9532251106725x_{34} = 87.9532251106725
x34=−59.6735041304405x_{34} = -59.6735041304405
x34=−28.2389365752603x_{34} = -28.2389365752603
x34=18.7964043662102x_{34} = 18.7964043662102
x34=81.6691650818489x_{34} = 81.6691650818489
Decrece en los intervalos
[197.91528455229,∞)\left[197.91528455229, \infty\right)
Crece en los intervalos
(−∞,−169.640108529775]\left(-\infty, -169.640108529775\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
−cos⁡(x)+2sin⁡(x)x+2cos⁡(x)x2x=0\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \right)}}{x^{2}}}{x} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−4.2222763997912x_{1} = -4.2222763997912
x2=−10.8095072981602x_{2} = -10.8095072981602
x3=42.3642737086586x_{3} = 42.3642737086586
x4=−64.3715747870554x_{4} = -64.3715747870554
x5=−13.9937625671267x_{5} = -13.9937625671267
x6=76.9430238267933x_{6} = 76.9430238267933
x7=−7.5873993379941x_{7} = -7.5873993379941
x8=48.6535676048409x_{8} = 48.6535676048409
x9=−80.0856368040887x_{9} = -80.0856368040887
x10=23.4766510546492x_{10} = 23.4766510546492
x11=89.5130456566371x_{11} = 89.5130456566371
x12=4.2222763997912x_{12} = 4.2222763997912
x13=73.8003238908837x_{13} = 73.8003238908837
x14=−23.4766510546492x_{14} = -23.4766510546492
x15=−86.370639887736x_{15} = -86.370639887736
x16=58.085025007445x_{16} = 58.085025007445
x17=−20.3217772482235x_{17} = -20.3217772482235
x18=−92.655396245836x_{18} = -92.655396245836
x19=54.9414610202918x_{19} = 54.9414610202918
x20=29.7779159141436x_{20} = 29.7779159141436
x21=230.898398112111x_{21} = 230.898398112111
x22=17.1619600917303x_{22} = 17.1619600917303
x23=32.9259431758392x_{23} = 32.9259431758392
x24=−83.2281726832512x_{24} = -83.2281726832512
x25=−73.8003238908837x_{25} = -73.8003238908837
x26=26.6283591640252x_{26} = 26.6283591640252
x27=−271.740404503579x_{27} = -271.740404503579
x28=−61.2283863503723x_{28} = -61.2283863503723
x29=70.6575253785884x_{29} = 70.6575253785884
x30=−95.7976970894915x_{30} = -95.7976970894915
x31=61.2283863503723x_{31} = 61.2283863503723
x32=83.2281726832512x_{32} = 83.2281726832512
x33=7.5873993379941x_{33} = 7.5873993379941
x34=39.218890250481x_{34} = 39.218890250481
x35=−26.6283591640252x_{35} = -26.6283591640252
x36=13.9937625671267x_{36} = 13.9937625671267
x37=−29.7779159141436x_{37} = -29.7779159141436
x38=−70.6575253785884x_{38} = -70.6575253785884
x39=−42.3642737086586x_{39} = -42.3642737086586
x40=−51.7976574095537x_{40} = -51.7976574095537
x41=−48.6535676048409x_{41} = -48.6535676048409
x42=−54.9414610202918x_{42} = -54.9414610202918
x43=51.7976574095537x_{43} = 51.7976574095537
x44=−98.9399529307048x_{44} = -98.9399529307048
x45=−67.5146145048817x_{45} = -67.5146145048817
x46=36.0728437679879x_{46} = 36.0728437679879
x47=−39.218890250481x_{47} = -39.218890250481
x48=20.3217772482235x_{48} = 20.3217772482235
x49=−45.5091321154553x_{49} = -45.5091321154553
x50=−32.9259431758392x_{50} = -32.9259431758392
x51=−58.085025007445x_{51} = -58.085025007445
x52=98.9399529307048x_{52} = 98.9399529307048
x53=45.5091321154553x_{53} = 45.5091321154553
x54=67.5146145048817x_{54} = 67.5146145048817
x55=92.655396245836x_{55} = 92.655396245836
x56=−36.0728437679879x_{56} = -36.0728437679879
x57=80.0856368040887x_{57} = 80.0856368040887
x58=86.370639887736x_{58} = 86.370639887736
x59=95.7976970894915x_{59} = 95.7976970894915
x60=−89.5130456566371x_{60} = -89.5130456566371
x61=−17.1619600917303x_{61} = -17.1619600917303
x62=64.3715747870554x_{62} = 64.3715747870554
x63=10.8095072981602x_{63} = 10.8095072981602
x64=−76.9430238267933x_{64} = -76.9430238267933
Además hay que calcular los límites de y'' para los argumentos tendientes a los puntos de indeterminación de la función:
Puntos donde hay indeterminación:
x1=0x_{1} = 0

lim⁡x→0−(−cos⁡(x)+2sin⁡(x)x+2cos⁡(x)x2x)=−∞\lim_{x \to 0^-}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \right)}}{x^{2}}}{x}\right) = -\infty
lim⁡x→0+(−cos⁡(x)+2sin⁡(x)x+2cos⁡(x)x2x)=∞\lim_{x \to 0^+}\left(\frac{- \cos{\left(x \right)} + \frac{2 \sin{\left(x \right)}}{x} + \frac{2 \cos{\left(x \right)}}{x^{2}}}{x}\right) = \infty
- los límites no son iguales, signo
x1=0x_{1} = 0
- es el punto de flexión

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
[95.7976970894915,∞)\left[95.7976970894915, \infty\right)
Convexa en los intervalos
(−∞,−271.740404503579]\left(-\infty, -271.740404503579\right]
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(1+cos⁡(x)x)=1\lim_{x \to -\infty}\left(1 + \frac{\cos{\left(x \right)}}{x}\right) = 1
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=1y = 1
lim⁡x→∞(1+cos⁡(x)x)=1\lim_{x \to \infty}\left(1 + \frac{\cos{\left(x \right)}}{x}\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 cos(x)/x + 1, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(1+cos⁡(x)xx)=0\lim_{x \to -\infty}\left(\frac{1 + \frac{\cos{\left(x \right)}}{x}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(1+cos⁡(x)xx)=0\lim_{x \to \infty}\left(\frac{1 + \frac{\cos{\left(x \right)}}{x}}{x}\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:
1+cos⁡(x)x=1−cos⁡(x)x1 + \frac{\cos{\left(x \right)}}{x} = 1 - \frac{\cos{\left(x \right)}}{x}
- No
1+cos⁡(x)x=−1+cos⁡(x)x1 + \frac{\cos{\left(x \right)}}{x} = -1 + \frac{\cos{\left(x \right)}}{x}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор