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Trazado el gráfico de la función/ abs(cos(2*x)-sin(x))

Gráfico de la función y = abs(cos(2*x)-sin(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = |cos(2*x) - sin(x)|
f(x)=∣−sin⁡(x)+cos⁡(2x)∣f{\left(x \right)} = \left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| 
f = Abs(-sin(x) + cos(2*x))
Gráfico de la función
02468-8-6-4-2-101004
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:
∣−sin⁡(x)+cos⁡(2x)∣=0\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π6x_{2} = \frac{\pi}{6}
x3=5π6x_{3} = \frac{5 \pi}{6}
Solución numérica
x1=86.393797888715x_{1} = 86.393797888715
x2=4.71238877821279x_{2} = 4.71238877821279
x3=84.2994028713261x_{3} = 84.2994028713261
x4=−39.2699083757319x_{4} = -39.2699083757319
x5=−93.7241808320955x_{5} = -93.7241808320955
x6=92.6769830871924x_{6} = 92.6769830871924
x7=54.9778712411975x_{7} = 54.9778712411975
x8=90.5825881785057x_{8} = 90.5825881785057
x9=−64.4026491963026x_{9} = -64.4026491963026
x10=40.317105721069x_{10} = 40.317105721069
x11=69.6386371545737x_{11} = 69.6386371545737
x12=17.2787597959772x_{12} = 17.2787597959772
x13=48.6946859325274x_{13} = 48.6946859325274
x14=67.5442422659503x_{14} = 67.5442422659503
x15=25.6563400043166x_{15} = 25.6563400043166
x16=80.1106131458253x_{16} = 80.1106131458253
x17=−70.6858344924983x_{17} = -70.6858344924983
x18=−18.3259571459405x_{18} = -18.3259571459405
x19=−5.75958653158129x_{19} = -5.75958653158129
x20=−97.9129710368819x_{20} = -97.9129710368819
x21=−3.66519142918809x_{21} = -3.66519142918809
x22=−26.7035373476123x_{22} = -26.7035373476123
x23=−49.7418836818384x_{23} = -49.7418836818384
x24=0.523598775598299x_{24} = 0.523598775598299
x25=61.2610569380464x_{25} = 61.2610569380464
x26=−16.2315620435473x_{26} = -16.2315620435473
x27=23.5619451122289x_{27} = 23.5619451122289
x28=−64.4026494629427x_{28} = -64.4026494629427
x29=73.8274274783337x_{29} = 73.8274274783337
x30=10.9955740992967x_{30} = 10.9955740992967
x31=34.0339204138894x_{31} = 34.0339204138894
x32=−26.7035379915215x_{32} = -26.7035379915215
x33=10.995574056153x_{33} = 10.995574056153
x34=−14.1371670557608x_{34} = -14.1371670557608
x35=27.7507351067098x_{35} = 27.7507351067098
x36=92.6769826185806x_{36} = 92.6769826185806
x37=−95.8185758681551x_{37} = -95.8185758681551
x38=−32.9867230405965x_{38} = -32.9867230405965
x39=−62.3082542961976x_{39} = -62.3082542961976
x40=54.9778721441305x_{40} = 54.9778721441305
x41=54.9778708860144x_{41} = 54.9778708860144
x42=−14.1371668400256x_{42} = -14.1371668400256
x43=2.61799387799149x_{43} = 2.61799387799149
x44=−41.3643032722656x_{44} = -41.3643032722656
x45=98.9601685995222x_{45} = 98.9601685995222
x46=−76.9690201780717x_{46} = -76.9690201780717
x47=78.0162175641465x_{47} = 78.0162175641465
x48=−76.9690204511548x_{48} = -76.9690204511548
x49=−56.025068989018x_{49} = -56.025068989018
x50=−12.0427718387609x_{50} = -12.0427718387609
x51=88.4881930761125x_{51} = 88.4881930761125
x52=−32.98672341235x_{52} = -32.98672341235
x53=31.9395253114962x_{53} = 31.9395253114962
x54=−60.2138591938044x_{54} = -60.2138591938044
x55=46.6002910282486x_{55} = 46.6002910282486
x56=61.2610562112906x_{56} = 61.2610562112906
x57=63.3554518473942x_{57} = 63.3554518473942
x58=−51.8362786898924x_{58} = -51.8362786898924
x59=98.9601683847854x_{59} = 98.9601683847854
x60=−20.4203520418601x_{60} = -20.4203520418601
x61=−7.85398149924071x_{61} = -7.85398149924071
x62=−58.1194639999037x_{62} = -58.1194639999037
x63=−91.6297857297023x_{63} = -91.6297857297023
x64=−47.6474885794452x_{64} = -47.6474885794452
x65=−83.2522055292846x_{65} = -83.2522055292846
x66=−89.5353907455655x_{66} = -89.5353907455655
x67=36.1283156017834x_{67} = 36.1283156017834
x68=75.9218224617533x_{68} = 75.9218224617533
x69=82.2050077689329x_{69} = 82.2050077689329
x70=−1.57079642893127x_{70} = -1.57079642893127
x71=−76.9690198122422x_{71} = -76.9690198122422
x72=44.5058959258554x_{72} = 44.5058959258554
x73=71.733032256967x_{73} = 71.733032256967
x74=−100.007366139275x_{74} = -100.007366139275
x75=−45.5530935873709x_{75} = -45.5530935873709
x76=29.8451303193672x_{76} = 29.8451303193672
x77=17.2787598104547x_{77} = 17.2787598104547
x78=−85.3466004225227x_{78} = -85.3466004225227
x79=42.4115007297604x_{79} = 42.4115007297604
x80=38.2227106186758x_{80} = 38.2227106186758
x81=−95.8185760435073x_{81} = -95.8185760435073
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 Abs(cos(2*x) - sin(x)).
∣−sin⁡(0)+cos⁡(0⋅2)∣\left|{- \sin{\left(0 \right)} + \cos{\left(0 \cdot 2 \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
(−2sin⁡(2x)−cos⁡(x))sign⁡(−sin⁡(x)+cos⁡(2x))=0\left(- 2 \sin{\left(2 x \right)} - \cos{\left(x \right)}\right) \operatorname{sign}{\left(- \sin{\left(x \right)} + \cos{\left(2 x \right)} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=1.5707963267949x_{1} = 1.5707963267949
x2=56.2959875094742x_{2} = 56.2959875094742
x3=92.6769832808989x_{3} = 92.6769832808989
x4=14.1371669411541x_{4} = 14.1371669411541
x5=−21.7384683199865x_{5} = -21.7384683199865
x6=−70.6858347057703x_{6} = -70.6858347057703
x7=53.6597553661686x_{7} = 53.6597553661686
x8=−36.1283155162826x_{8} = -36.1283155162826
x9=−81.9340892484767x_{9} = -81.9340892484767
x10=7.85398163397448x_{10} = 7.85398163397448
x11=−59.437580163064x_{11} = -59.437580163064
x12=42.4115008234622x_{12} = 42.4115008234622
x13=−78.2871360846027x_{13} = -78.2871360846027
x14=28.5270141374502x_{14} = 28.5270141374502
x15=12.3136903592171x_{15} = 12.3136903592171
x16=−89.5353906273091x_{16} = -89.5353906273091
x17=−0.252680255142079x_{17} = -0.252680255142079
x18=43.729616895115x_{18} = 43.729616895115
x19=22.2438288302706x_{19} = 22.2438288302706
x20=−67.5442420521806x_{20} = -67.5442420521806
x21=−51.8362787842316x_{21} = -51.8362787842316
x22=−83.2522053201295x_{22} = -83.2522053201295
x23=−25.3854214838604x_{23} = -25.3854214838604
x24=−14.1371669411541x_{24} = -14.1371669411541
x25=−65.7207654702436x_{25} = -65.7207654702436
x26=−28.0216536271661x_{26} = -28.0216536271661
x27=64.4026493985908x_{27} = 64.4026493985908
x28=81.4287287381925x_{28} = 81.4287287381925
x29=48.6946861306418x_{29} = 48.6946861306418
x30=67.5442420521806x_{30} = 67.5442420521806
x31=23.5619449019235x_{31} = 23.5619449019235
x32=−117.809724509617x_{32} = -117.809724509617
x33=−9.1720977056273x_{33} = -9.1720977056273
x34=−94.5004598628359x_{34} = -94.5004598628359
x35=50.0128022022946x_{35} = 50.0128022022946
x36=18.5968756663967x_{36} = 18.5968756663967
x37=−80.1106126665397x_{37} = -80.1106126665397
x38=87.7119140453721x_{38} = 87.7119140453721
x39=66.2261259805277x_{39} = 66.2261259805277
x40=62.5791728166538x_{40} = 62.5791728166538
x41=−29.845130209103x_{41} = -29.845130209103
x42=−64.4026493985908x_{42} = -64.4026493985908
x43=15.960643523091x_{43} = 15.960643523091
x44=−86.3937979737193x_{44} = -86.3937979737193
x45=20.4203522483337x_{45} = 20.4203522483337
x46=−88.2172745556563x_{46} = -88.2172745556563
x47=−102.101761241668x_{47} = -102.101761241668
x48=−1.5707963267949x_{48} = -1.5707963267949
x49=86.3937979737193x_{49} = 86.3937979737193
x50=−72.0039507774232x_{50} = -72.0039507774232
x51=36.1283155162826x_{51} = 36.1283155162826
x52=100.278284659731x_{52} = 100.278284659731
x53=−44.2349774053992x_{53} = -44.2349774053992
x54=51.8362787842316x_{54} = 51.8362787842316
x55=80.1106126665397x_{55} = 80.1106126665397
x56=−15.4552830128069x_{56} = -15.4552830128069
x57=34.8101994446298x_{57} = 34.8101994446298
x58=93.9950993525517x_{58} = 93.9950993525517
x59=89.5353906273091x_{59} = 89.5353906273091
x60=58.1194640914112x_{60} = 58.1194640914112
x61=97.6420525164257x_{61} = 97.6420525164257
x62=73.8274273593601x_{62} = 73.8274273593601
x63=−58.1194640914112x_{63} = -58.1194640914112
x64=−20.4203522483337x_{64} = -20.4203522483337
x65=−50.5181627125788x_{65} = -50.5181627125788
x66=−6.53586556232167x_{66} = -6.53586556232167
x67=−95.8185759344887x_{67} = -95.8185759344887
x68=29.845130209103x_{68} = 29.845130209103
x69=26.7035375555132x_{69} = 26.7035375555132
x70=−23.5619449019235x_{70} = -23.5619449019235
x71=−42.4115008234622x_{71} = -42.4115008234622
x72=37.4464315879354x_{72} = 37.4464315879354
x73=−34.3048389343456x_{73} = -34.3048389343456
x74=−31.66860679104x_{74} = -31.66860679104
x75=78.7924965948869x_{75} = 78.7924965948869
x76=−97.1366920061415x_{76} = -97.1366920061415
x77=−61.261056745001x_{77} = -61.261056745001
x78=70.6858347057703x_{78} = 70.6858347057703
x79=−53.1543948558844x_{79} = -53.1543948558844
x80=59.9429406733481x_{80} = 59.9429406733481
x81=−103.419877313321x_{81} = -103.419877313321
x82=72.5093112877073x_{82} = 72.5093112877073
x83=6.03050505203751x_{83} = 6.03050505203751
x84=45.553093477052x_{84} = 45.553093477052
x85=−37.9517920982196x_{85} = -37.9517920982196
x86=−73.8274273593601x_{86} = -73.8274273593601
x87=−75.6509039412971x_{87} = -75.6509039412971
x88=−45.553093477052x_{88} = -45.553093477052
x89=9.67745821591146x_{89} = 9.67745821591146
x90=−7.85398163397448x_{90} = -7.85398163397448
x91=−17.2787595947439x_{91} = -17.2787595947439
x92=95.8185759344887x_{92} = 95.8185759344887
Signos de extremos en los puntos:
(1.5707963267948966, 2)

(56.2959875094742, 1.125)

(92.67698328089891, 0)

(14.137166941154069, 2)

(-21.738468319986474, 1.125)

(-70.68583470577035, 0)

(53.65975536616856, 1.125)

(-36.12831551628262, 2)

(-81.93408924847671, 1.125)

(7.853981633974483, 2)

(-59.43758016306399, 1.125)

(42.411500823462205, 0)

(-78.28713608460275, 1.125)

(28.52701413745022, 1.125)

(12.313690359217095, 1.125)

(-89.53539062730911, 0)

(-0.25268025514207865, 1.125)

(43.72961689511503, 1.125)

(22.24382883027063, 1.125)

(-67.54424205218055, 2)

(-51.83627878423159, 0)

(-83.25220532012952, 0)

(-25.385421483860423, 1.125)

(-14.137166941154069, 0)

(-65.72076547024358, 1.125)

(-28.02165362716606, 1.125)

(64.40264939859077, 2)

(81.42872873819255, 1.125)

(48.6946861306418, 0)

(67.54424205218055, 0)

(23.56194490192345, 0)

(-117.80972450961724, 2)

(-9.172097705627301, 1.125)

(-94.50045986283588, 1.125)

(50.012802202294615, 1.125)

(18.59687566639668, 1.125)

(-80.11061266653972, 2)

(87.71191404537213, 1.125)

(66.22612598052774, 1.125)

(62.57917281665379, 1.125)

(-29.845130209103036, 2)

(-64.40264939859077, 0)

(15.960643523091045, 1.125)

(-86.39379797371932, 2)

(20.420352248333657, 2)

(-88.21727455565629, 1.125)

(-102.10176124166829, 0)

(-1.5707963267948966, 0)

(86.39379797371932, 0)

(-72.00395077742317, 1.125)

(36.12831551628262, 0)

(100.2782846597313, 1.125)

(-44.234977405399185, 1.125)

(51.83627878423159, 2)

(80.11061266653972, 0)

(-15.455283012806888, 1.125)

(34.8101994446298, 1.125)

(93.99509935255172, 1.125)

(89.53539062730911, 2)

(58.119464091411174, 2)

(97.64205251642566, 1.125)

(73.82742735936014, 0)

(-58.119464091411174, 0)

(-20.420352248333657, 0)

(-50.51816271257877, 1.125)

(-6.535865562321665, 1.125)

(-95.81857593448869, 0)

(29.845130209103036, 0)

(26.703537555513243, 2)

(-23.56194490192345, 2)

(-42.411500823462205, 2)

(37.44643158793544, 1.125)

(-34.304838934345646, 1.125)

(-31.668606791040013, 1.125)

(78.79249659488691, 1.125)

(-97.13669200614152, 1.125)

(-61.26105674500097, 2)

(70.68583470577035, 2)

(-53.154394855884405, 1.125)

(59.94294067334815, 1.125)

(-103.4198773133211, 1.125)

(72.50931128770732, 1.125)

(6.030505052037507, 1.125)

(45.553093477052, 2)

(-37.9517920982196, 1.125)

(-73.82742735936014, 2)

(-75.65090394129712, 1.125)

(-45.553093477052, 0)

(9.677458215911459, 1.125)

(-7.853981633974483, 0)

(-17.278759594743864, 2)

(95.81857593448869, 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=92.6769832808989x_{1} = 92.6769832808989
x2=−70.6858347057703x_{2} = -70.6858347057703
x3=42.4115008234622x_{3} = 42.4115008234622
x4=−89.5353906273091x_{4} = -89.5353906273091
x5=−51.8362787842316x_{5} = -51.8362787842316
x6=−83.2522053201295x_{6} = -83.2522053201295
x7=−14.1371669411541x_{7} = -14.1371669411541
x8=48.6946861306418x_{8} = 48.6946861306418
x9=67.5442420521806x_{9} = 67.5442420521806
x10=23.5619449019235x_{10} = 23.5619449019235
x11=−64.4026493985908x_{11} = -64.4026493985908
x12=−102.101761241668x_{12} = -102.101761241668
x13=−1.5707963267949x_{13} = -1.5707963267949
x14=86.3937979737193x_{14} = 86.3937979737193
x15=36.1283155162826x_{15} = 36.1283155162826
x16=80.1106126665397x_{16} = 80.1106126665397
x17=73.8274273593601x_{17} = 73.8274273593601
x18=−58.1194640914112x_{18} = -58.1194640914112
x19=−20.4203522483337x_{19} = -20.4203522483337
x20=−95.8185759344887x_{20} = -95.8185759344887
x21=29.845130209103x_{21} = 29.845130209103
x22=−45.553093477052x_{22} = -45.553093477052
x23=−7.85398163397448x_{23} = -7.85398163397448
Puntos máximos de la función:
x23=1.5707963267949x_{23} = 1.5707963267949
x23=56.2959875094742x_{23} = 56.2959875094742
x23=14.1371669411541x_{23} = 14.1371669411541
x23=−21.7384683199865x_{23} = -21.7384683199865
x23=53.6597553661686x_{23} = 53.6597553661686
x23=−36.1283155162826x_{23} = -36.1283155162826
x23=−81.9340892484767x_{23} = -81.9340892484767
x23=7.85398163397448x_{23} = 7.85398163397448
x23=−59.437580163064x_{23} = -59.437580163064
x23=−78.2871360846027x_{23} = -78.2871360846027
x23=28.5270141374502x_{23} = 28.5270141374502
x23=12.3136903592171x_{23} = 12.3136903592171
x23=−0.252680255142079x_{23} = -0.252680255142079
x23=43.729616895115x_{23} = 43.729616895115
x23=22.2438288302706x_{23} = 22.2438288302706
x23=−67.5442420521806x_{23} = -67.5442420521806
x23=−25.3854214838604x_{23} = -25.3854214838604
x23=−65.7207654702436x_{23} = -65.7207654702436
x23=−28.0216536271661x_{23} = -28.0216536271661
x23=64.4026493985908x_{23} = 64.4026493985908
x23=81.4287287381925x_{23} = 81.4287287381925
x23=−117.809724509617x_{23} = -117.809724509617
x23=−9.1720977056273x_{23} = -9.1720977056273
x23=−94.5004598628359x_{23} = -94.5004598628359
x23=50.0128022022946x_{23} = 50.0128022022946
x23=18.5968756663967x_{23} = 18.5968756663967
x23=−80.1106126665397x_{23} = -80.1106126665397
x23=87.7119140453721x_{23} = 87.7119140453721
x23=66.2261259805277x_{23} = 66.2261259805277
x23=62.5791728166538x_{23} = 62.5791728166538
x23=−29.845130209103x_{23} = -29.845130209103
x23=15.960643523091x_{23} = 15.960643523091
x23=−86.3937979737193x_{23} = -86.3937979737193
x23=20.4203522483337x_{23} = 20.4203522483337
x23=−88.2172745556563x_{23} = -88.2172745556563
x23=−72.0039507774232x_{23} = -72.0039507774232
x23=100.278284659731x_{23} = 100.278284659731
x23=−44.2349774053992x_{23} = -44.2349774053992
x23=51.8362787842316x_{23} = 51.8362787842316
x23=−15.4552830128069x_{23} = -15.4552830128069
x23=34.8101994446298x_{23} = 34.8101994446298
x23=93.9950993525517x_{23} = 93.9950993525517
x23=89.5353906273091x_{23} = 89.5353906273091
x23=58.1194640914112x_{23} = 58.1194640914112
x23=97.6420525164257x_{23} = 97.6420525164257
x23=−50.5181627125788x_{23} = -50.5181627125788
x23=−6.53586556232167x_{23} = -6.53586556232167
x23=26.7035375555132x_{23} = 26.7035375555132
x23=−23.5619449019235x_{23} = -23.5619449019235
x23=−42.4115008234622x_{23} = -42.4115008234622
x23=37.4464315879354x_{23} = 37.4464315879354
x23=−34.3048389343456x_{23} = -34.3048389343456
x23=−31.66860679104x_{23} = -31.66860679104
x23=78.7924965948869x_{23} = 78.7924965948869
x23=−97.1366920061415x_{23} = -97.1366920061415
x23=−61.261056745001x_{23} = -61.261056745001
x23=70.6858347057703x_{23} = 70.6858347057703
x23=−53.1543948558844x_{23} = -53.1543948558844
x23=59.9429406733481x_{23} = 59.9429406733481
x23=−103.419877313321x_{23} = -103.419877313321
x23=72.5093112877073x_{23} = 72.5093112877073
x23=6.03050505203751x_{23} = 6.03050505203751
x23=45.553093477052x_{23} = 45.553093477052
x23=−37.9517920982196x_{23} = -37.9517920982196
x23=−73.8274273593601x_{23} = -73.8274273593601
x23=−75.6509039412971x_{23} = -75.6509039412971
x23=9.67745821591146x_{23} = 9.67745821591146
x23=−17.2787595947439x_{23} = -17.2787595947439
x23=95.8185759344887x_{23} = 95.8185759344887
Decrece en los intervalos
[92.6769832808989,∞)\left[92.6769832808989, \infty\right)
Crece en los intervalos
(−∞,−102.101761241668]\left(-\infty, -102.101761241668\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
−(sin⁡(x)−4cos⁡(2x))sign⁡(sin⁡(x)−cos⁡(2x))+2(2sin⁡(2x)+cos⁡(x))2δ(sin⁡(x)−cos⁡(2x))=0- \left(\sin{\left(x \right)} - 4 \cos{\left(2 x \right)}\right) \operatorname{sign}{\left(\sin{\left(x \right)} - \cos{\left(2 x \right)} \right)} + 2 \left(2 \sin{\left(2 x \right)} + \cos{\left(x \right)}\right)^{2} \delta\left(\sin{\left(x \right)} - \cos{\left(2 x \right)}\right) = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga flexiones
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞∣−sin⁡(x)+cos⁡(2x)∣=∣⟨−2,2⟩∣\lim_{x \to -\infty} \left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = \left|{\left\langle -2, 2\right\rangle}\right|
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=∣⟨−2,2⟩∣y = \left|{\left\langle -2, 2\right\rangle}\right|
lim⁡x→∞∣−sin⁡(x)+cos⁡(2x)∣=∣⟨−2,2⟩∣\lim_{x \to \infty} \left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = \left|{\left\langle -2, 2\right\rangle}\right|
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=∣⟨−2,2⟩∣y = \left|{\left\langle -2, 2\right\rangle}\right|
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función Abs(cos(2*x) - sin(x)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(∣−sin⁡(x)+cos⁡(2x)∣x)=0\lim_{x \to -\infty}\left(\frac{\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(∣−sin⁡(x)+cos⁡(2x)∣x)=0\lim_{x \to \infty}\left(\frac{\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|}{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:
∣−sin⁡(x)+cos⁡(2x)∣=∣sin⁡(x)+cos⁡(2x)∣\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = \left|{\sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|
- No
∣−sin⁡(x)+cos⁡(2x)∣=−∣sin⁡(x)+cos⁡(2x)∣\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = - \left|{\sin{\left(x \right)} + \cos{\left(2 x \right)}}\right|
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор