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abs(cos(2*x)-sin(x))

  • How to use it?

  • Graphing y =:
  • 2x^2-8x
  • x^2+x+1
  • |x^2-9|/(x^2-x-6)
  • x+x^2

  • Identical expressions

  • abs(cos(two *x)-sin(x))
  • abs( co sinus of e of (2 multiply by x) minus sinus of (x))
  • abs( co sinus of e of (two multiply by x) minus sinus of (x))
  • abs(cos(2x)-sin(x))
  • abscos2x-sinx

  • Similar expressions

  • abs(cos(2*x)+sin(x))
  • abs(cos(2*x)-sinx)
Plotting a function graph/ abs(cos(2*x)-sin(x))

Graphing y = abs(cos(2*x)-sin(x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [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))
The graph of the function
01020304050607080-1004
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
∣−sin⁡(x)+cos⁡(2x)∣=0\left|{- \sin{\left(x \right)} + \cos{\left(2 x \right)}}\right| = 0
Solve this equation
The points of intersection with the axis X:

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

(-58.1194640914112, 0)

(-78.2871360846027, 1.125)

(-6.53586556232167, 1.125)

(-17.2787595947439, 2)

(45.553093477052, 2)

(56.2959875094742, 1.125)

(36.1283155162826, 0)

(-102.101761241668, 0)

(73.8274273593601, 0)

(-51.8362787842316, 0)

(-25.3854214838604, 1.125)

(-53.1543948558844, 1.125)

(-67.5442420521806, 2)

(-9.1720977056273, 1.125)

(1.5707963267949, 2)

(87.7119140453721, 1.125)

(-72.0039507774232, 1.125)

(-117.809724509617, 2)

(-83.2522053201295, 0)

(66.2261259805277, 1.125)

(-64.4026493985908, 0)

(-75.6509039412971, 1.125)

(97.6420525164257, 1.125)

(-45.553093477052, 0)

(6.03050505203751, 1.125)

(48.6946861306418, 0)

(-65.7207654702436, 1.125)

(34.8101994446298, 1.125)

(64.4026493985908, 2)

(-81.9340892484767, 1.125)

(58.1194640914112, 2)

(43.729616895115, 1.125)

(80.1106126665397, 0)

(-28.0216536271661, 1.125)

(51.8362787842316, 2)

(-15.4552830128069, 1.125)

(86.3937979737193, 0)

(-59.437580163064, 1.125)

(28.5270141374502, 1.125)

(-80.1106126665397, 2)

(-50.5181627125788, 1.125)

(-88.2172745556563, 1.125)

(22.2438288302706, 1.125)

(37.4464315879354, 1.125)

(14.1371669411541, 2)

(-21.7384683199865, 1.125)

(95.8185759344887, 2)

(100.278284659731, 1.125)

(92.6769832808989, 0)

(-44.2349774053992, 1.125)

(53.6597553661686, 1.125)

(-86.3937979737193, 2)

(7.85398163397448, 2)

(50.0128022022946, 1.125)

(-20.4203522483337, 0)

(59.9429406733481, 1.125)

(-0.252680255142079, 1.125)

(-89.5353906273091, 0)

(-95.8185759344887, 0)

(-36.1283155162826, 2)

(89.5353906273091, 2)

(-42.4115008234622, 2)

(-14.1371669411541, 0)

(67.5442420521806, 0)

(-31.66860679104, 1.125)

(-34.3048389343456, 1.125)

(-103.419877313321, 1.125)

(-23.5619449019235, 2)

(9.67745821591146, 1.125)

(93.9950993525517, 1.125)

(-73.8274273593601, 2)

(81.4287287381925, 1.125)

(12.3136903592171, 1.125)

(29.845130209103, 0)

(18.5968756663967, 1.125)

(62.5791728166538, 1.125)

(78.7924965948869, 1.125)

(-94.5004598628359, 1.125)

(-1.5707963267949, 0)

(42.4115008234622, 0)

(-97.1366920061415, 1.125)

(20.4203522483337, 2)

(-29.845130209103, 2)

(-37.9517920982196, 1.125)

(-61.261056745001, 2)

(26.7035375555132, 2)

(-70.6858347057703, 0)

(-7.85398163397448, 0)

(70.6858347057703, 2)

(15.960643523091, 1.125)

(72.5093112877073, 1.125)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=23.5619449019235x_{1} = 23.5619449019235
x2=−58.1194640914112x_{2} = -58.1194640914112
x3=36.1283155162826x_{3} = 36.1283155162826
x4=−102.101761241668x_{4} = -102.101761241668
x5=73.8274273593601x_{5} = 73.8274273593601
x6=−51.8362787842316x_{6} = -51.8362787842316
x7=−83.2522053201295x_{7} = -83.2522053201295
x8=−64.4026493985908x_{8} = -64.4026493985908
x9=−45.553093477052x_{9} = -45.553093477052
x10=48.6946861306418x_{10} = 48.6946861306418
x11=80.1106126665397x_{11} = 80.1106126665397
x12=86.3937979737193x_{12} = 86.3937979737193
x13=92.6769832808989x_{13} = 92.6769832808989
x14=−20.4203522483337x_{14} = -20.4203522483337
x15=−89.5353906273091x_{15} = -89.5353906273091
x16=−95.8185759344887x_{16} = -95.8185759344887
x17=−14.1371669411541x_{17} = -14.1371669411541
x18=67.5442420521806x_{18} = 67.5442420521806
x19=29.845130209103x_{19} = 29.845130209103
x20=−1.5707963267949x_{20} = -1.5707963267949
x21=42.4115008234622x_{21} = 42.4115008234622
x22=−70.6858347057703x_{22} = -70.6858347057703
x23=−7.85398163397448x_{23} = -7.85398163397448
Maxima of the function at points:
x23=−78.2871360846027x_{23} = -78.2871360846027
x23=−6.53586556232167x_{23} = -6.53586556232167
x23=−17.2787595947439x_{23} = -17.2787595947439
x23=45.553093477052x_{23} = 45.553093477052
x23=56.2959875094742x_{23} = 56.2959875094742
x23=−25.3854214838604x_{23} = -25.3854214838604
x23=−53.1543948558844x_{23} = -53.1543948558844
x23=−67.5442420521806x_{23} = -67.5442420521806
x23=−9.1720977056273x_{23} = -9.1720977056273
x23=1.5707963267949x_{23} = 1.5707963267949
x23=87.7119140453721x_{23} = 87.7119140453721
x23=−72.0039507774232x_{23} = -72.0039507774232
x23=−117.809724509617x_{23} = -117.809724509617
x23=66.2261259805277x_{23} = 66.2261259805277
x23=−75.6509039412971x_{23} = -75.6509039412971
x23=97.6420525164257x_{23} = 97.6420525164257
x23=6.03050505203751x_{23} = 6.03050505203751
x23=−65.7207654702436x_{23} = -65.7207654702436
x23=34.8101994446298x_{23} = 34.8101994446298
x23=64.4026493985908x_{23} = 64.4026493985908
x23=−81.9340892484767x_{23} = -81.9340892484767
x23=58.1194640914112x_{23} = 58.1194640914112
x23=43.729616895115x_{23} = 43.729616895115
x23=−28.0216536271661x_{23} = -28.0216536271661
x23=51.8362787842316x_{23} = 51.8362787842316
x23=−15.4552830128069x_{23} = -15.4552830128069
x23=−59.437580163064x_{23} = -59.437580163064
x23=28.5270141374502x_{23} = 28.5270141374502
x23=−80.1106126665397x_{23} = -80.1106126665397
x23=−50.5181627125788x_{23} = -50.5181627125788
x23=−88.2172745556563x_{23} = -88.2172745556563
x23=22.2438288302706x_{23} = 22.2438288302706
x23=37.4464315879354x_{23} = 37.4464315879354
x23=14.1371669411541x_{23} = 14.1371669411541
x23=−21.7384683199865x_{23} = -21.7384683199865
x23=95.8185759344887x_{23} = 95.8185759344887
x23=100.278284659731x_{23} = 100.278284659731
x23=−44.2349774053992x_{23} = -44.2349774053992
x23=53.6597553661686x_{23} = 53.6597553661686
x23=−86.3937979737193x_{23} = -86.3937979737193
x23=7.85398163397448x_{23} = 7.85398163397448
x23=50.0128022022946x_{23} = 50.0128022022946
x23=59.9429406733481x_{23} = 59.9429406733481
x23=−0.252680255142079x_{23} = -0.252680255142079
x23=−36.1283155162826x_{23} = -36.1283155162826
x23=89.5353906273091x_{23} = 89.5353906273091
x23=−42.4115008234622x_{23} = -42.4115008234622
x23=−31.66860679104x_{23} = -31.66860679104
x23=−34.3048389343456x_{23} = -34.3048389343456
x23=−103.419877313321x_{23} = -103.419877313321
x23=−23.5619449019235x_{23} = -23.5619449019235
x23=9.67745821591146x_{23} = 9.67745821591146
x23=93.9950993525517x_{23} = 93.9950993525517
x23=−73.8274273593601x_{23} = -73.8274273593601
x23=81.4287287381925x_{23} = 81.4287287381925
x23=12.3136903592171x_{23} = 12.3136903592171
x23=18.5968756663967x_{23} = 18.5968756663967
x23=62.5791728166538x_{23} = 62.5791728166538
x23=78.7924965948869x_{23} = 78.7924965948869
x23=−94.5004598628359x_{23} = -94.5004598628359
x23=−97.1366920061415x_{23} = -97.1366920061415
x23=20.4203522483337x_{23} = 20.4203522483337
x23=−29.845130209103x_{23} = -29.845130209103
x23=−37.9517920982196x_{23} = -37.9517920982196
x23=−61.261056745001x_{23} = -61.261056745001
x23=26.7035375555132x_{23} = 26.7035375555132
x23=70.6858347057703x_{23} = 70.6858347057703
x23=15.960643523091x_{23} = 15.960643523091
x23=72.5093112877073x_{23} = 72.5093112877073
Decreasing at intervals
[92.6769832808989,∞)\left[92.6769832808989, \infty\right)
Increasing at intervals
(−∞,−102.101761241668]\left(-\infty, -102.101761241668\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2(2sin⁡(2x)+cos⁡(x))2δ(sin⁡(x)−cos⁡(2x))−(sin⁡(x)−4cos⁡(2x))sign⁡(sin⁡(x)−cos⁡(2x))=02 \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) - \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)} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and 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|
Let's take the limit
so,
equation of the horizontal asymptote on the left:
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|
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=∣⟨−2,2⟩∣y = \left|{\left\langle -2, 2\right\rangle}\right|
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(cos(2*x) - sin(x)), divided by x at x->+oo and 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
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
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
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
∣−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
so, the function
not is
neither even, nor odd
The graph
Graphing y = abs(cos(2*x)-sin(x))
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