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Trazado el gráfico de la función/ cosx/(x^2+1)

Gráfico de la función y = cosx/(x^2+1)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       cos(x)
f(x) = ------
        2    
       x  + 1
f(x)=cos⁡(x)x2+1f{\left(x \right)} = \frac{\cos{\left(x \right)}}{x^{2} + 1} 
f = cos(x)/(x^2 + 1)
Gráfico de la función
02468-8-6-4-2-10102-1
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:
cos⁡(x)x2+1=0\frac{\cos{\left(x \right)}}{x^{2} + 1} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Solución numérica
x1=4.71238898038469x_{1} = 4.71238898038469
x2=17.2787595947439x_{2} = 17.2787595947439
x3=−89.5353906273091x_{3} = -89.5353906273091
x4=64.4026493985908x_{4} = 64.4026493985908
x5=70.6858347057703x_{5} = 70.6858347057703
x6=36.1283155162826x_{6} = 36.1283155162826
x7=−98.9601685880785x_{7} = -98.9601685880785
x8=48.6946861306418x_{8} = 48.6946861306418
x9=−58.1194640914112x_{9} = -58.1194640914112
x10=7.85398163397448x_{10} = 7.85398163397448
x11=39.2699081698724x_{11} = 39.2699081698724
x12=−95.8185759344887x_{12} = -95.8185759344887
x13=−1.5707963267949x_{13} = -1.5707963267949
x14=−92.6769832808989x_{14} = -92.6769832808989
x15=−23.5619449019235x_{15} = -23.5619449019235
x16=23.5619449019235x_{16} = 23.5619449019235
x17=61.261056745001x_{17} = 61.261056745001
x18=−177.499984927823x_{18} = -177.499984927823
x19=29.845130209103x_{19} = 29.845130209103
x20=−32.9867228626928x_{20} = -32.9867228626928
x21=−51.8362787842316x_{21} = -51.8362787842316
x22=−80.1106126665397x_{22} = -80.1106126665397
x23=−83.2522053201295x_{23} = -83.2522053201295
x24=67.5442420521806x_{24} = 67.5442420521806
x25=98.9601685880785x_{25} = 98.9601685880785
x26=92.6769832808989x_{26} = 92.6769832808989
x27=−39.2699081698724x_{27} = -39.2699081698724
x28=86.3937979737193x_{28} = 86.3937979737193
x29=45.553093477052x_{29} = 45.553093477052
x30=−306.305283725005x_{30} = -306.305283725005
x31=−67.5442420521806x_{31} = -67.5442420521806
x32=51.8362787842316x_{32} = 51.8362787842316
x33=76.9690200129499x_{33} = 76.9690200129499
x34=−26.7035375555132x_{34} = -26.7035375555132
x35=−4.71238898038469x_{35} = -4.71238898038469
x36=95.8185759344887x_{36} = 95.8185759344887
x37=108.384946548848x_{37} = 108.384946548848
x38=−86.3937979737193x_{38} = -86.3937979737193
x39=−10.9955742875643x_{39} = -10.9955742875643
x40=83.2522053201295x_{40} = 83.2522053201295
x41=−7.85398163397448x_{41} = -7.85398163397448
x42=−36.1283155162826x_{42} = -36.1283155162826
x43=−17.2787595947439x_{43} = -17.2787595947439
x44=−14.1371669411541x_{44} = -14.1371669411541
x45=54.9778714378214x_{45} = 54.9778714378214
x46=20.4203522483337x_{46} = 20.4203522483337
x47=−70.6858347057703x_{47} = -70.6858347057703
x48=−48.6946861306418x_{48} = -48.6946861306418
x49=−54.9778714378214x_{49} = -54.9778714378214
x50=−45.553093477052x_{50} = -45.553093477052
x51=14.1371669411541x_{51} = 14.1371669411541
x52=−73.8274273593601x_{52} = -73.8274273593601
x53=26.7035375555132x_{53} = 26.7035375555132
x54=89.5353906273091x_{54} = 89.5353906273091
x55=10.9955742875643x_{55} = 10.9955742875643
x56=80.1106126665397x_{56} = 80.1106126665397
x57=73.8274273593601x_{57} = 73.8274273593601
x58=58.1194640914112x_{58} = 58.1194640914112
x59=−61.261056745001x_{59} = -61.261056745001
x60=1.5707963267949x_{60} = 1.5707963267949
x61=−20.4203522483337x_{61} = -20.4203522483337
x62=−42.4115008234622x_{62} = -42.4115008234622
x63=32.9867228626928x_{63} = 32.9867228626928
x64=42.4115008234622x_{64} = 42.4115008234622
x65=−76.9690200129499x_{65} = -76.9690200129499
x66=−64.4026493985908x_{66} = -64.4026493985908
x67=−29.845130209103x_{67} = -29.845130209103
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^2 + 1).
cos⁡(0)02+1\frac{\cos{\left(0 \right)}}{0^{2} + 1}
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
−2xcos⁡(x)(x2+1)2−sin⁡(x)x2+1=0- \frac{2 x \cos{\left(x \right)}}{\left(x^{2} + 1\right)^{2}} - \frac{\sin{\left(x \right)}}{x^{2} + 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−2.54373214752609x_{1} = -2.54373214752609
x2=−94.2265597456126x_{2} = -94.2265597456126
x3=50.2256989863186x_{3} = 50.2256989863186
x4=37.6460727029451x_{4} = 37.6460727029451
x5=−21.9002665401996x_{5} = -21.9002665401996
x6=−15.5808165061202x_{6} = -15.5808165061202
x7=0x_{7} = 0
x8=9.21343494397267x_{8} = 9.21343494397267
x9=−37.6460727029451x_{9} = -37.6460727029451
x10=53.3696312768345x_{10} = 53.3696312768345
x11=−56.513303694752x_{11} = -56.513303694752
x12=−84.7994242303256x_{12} = -84.7994242303256
x13=94.2265597456126x_{13} = 94.2265597456126
x14=−5.96808139239822x_{14} = -5.96808139239822
x15=12.4075674897868x_{15} = 12.4075674897868
x16=25.0532062442974x_{16} = 25.0532062442974
x17=−53.3696312768345x_{17} = -53.3696312768345
x18=78.5143529265667x_{18} = 78.5143529265667
x19=72.2289536816917x_{19} = 72.2289536816917
x20=−25.0532062442974x_{20} = -25.0532062442974
x21=−106.7954266585x_{21} = -106.7954266585
x22=−34.4996609189666x_{22} = -34.4996609189666
x23=15.5808165061202x_{23} = 15.5808165061202
x24=−87.9418588604656x_{24} = -87.9418588604656
x25=87.9418588604656x_{25} = 87.9418588604656
x26=97.3688368618863x_{26} = 97.3688368618863
x27=−75.3716994196716x_{27} = -75.3716994196716
x28=−100.511071203627x_{28} = -100.511071203627
x29=−62.8000247758447x_{29} = -62.8000247758447
x30=−43.9368321750172x_{30} = -43.9368321750172
x31=31.3522862210969x_{31} = 31.3522862210969
x32=−65.9431328237524x_{32} = -65.9431328237524
x33=163.350575451696x_{33} = 163.350575451696
x34=−47.0814548776037x_{34} = -47.0814548776037
x35=21.9002665401996x_{35} = 21.9002665401996
x36=−103.653266658919x_{36} = -103.653266658919
x37=100.511071203627x_{37} = 100.511071203627
x38=−91.0842354305333x_{38} = -91.0842354305333
x39=−97.3688368618863x_{39} = -97.3688368618863
x40=84.7994242303256x_{40} = 84.7994242303256
x41=2.54373214752609x_{41} = 2.54373214752609
x42=−81.6569248421486x_{42} = -81.6569248421486
x43=47.0814548776037x_{43} = 47.0814548776037
x44=43.9368321750172x_{44} = 43.9368321750172
x45=69.0861031389786x_{45} = 69.0861031389786
x46=34.4996609189666x_{46} = 34.4996609189666
x47=18.7435542483014x_{47} = 18.7435542483014
x48=−122.505790268738x_{48} = -122.505790268738
x49=75.3716994196716x_{49} = 75.3716994196716
x50=−69.0861031389786x_{50} = -69.0861031389786
x51=65.9431328237524x_{51} = 65.9431328237524
x52=62.8000247758447x_{52} = 62.8000247758447
x53=56.513303694752x_{53} = 56.513303694752
x54=40.7917435749351x_{54} = 40.7917435749351
x55=−31.3522862210969x_{55} = -31.3522862210969
x56=−78.5143529265667x_{56} = -78.5143529265667
x57=−40.7917435749351x_{57} = -40.7917435749351
x58=−9.21343494397267x_{58} = -9.21343494397267
x59=−72.2289536816917x_{59} = -72.2289536816917
x60=−113.079652107775x_{60} = -113.079652107775
x61=81.6569248421486x_{61} = 81.6569248421486
x62=5.96808139239822x_{62} = 5.96808139239822
x63=91.0842354305333x_{63} = 91.0842354305333
x64=−12.4075674897868x_{64} = -12.4075674897868
x65=−59.656757255627x_{65} = -59.656757255627
x66=−50.2256989863186x_{66} = -50.2256989863186
x67=59.656757255627x_{67} = 59.656757255627
x68=−18.7435542483014x_{68} = -18.7435542483014
x69=−210.477206074369x_{69} = -210.477206074369
x70=−28.203628119338x_{70} = -28.203628119338
x71=28.203628119338x_{71} = 28.203628119338
Signos de extremos en los puntos:
(-2.5437321475260917, -0.110639672191836)

(-94.22655974561256, 0.000112591766511704)

(50.22569898631863, 0.000395942499274958)

(37.64607270294512, 0.000704114293701762)

(-21.90026654019963, -0.00207205193264381)

(-15.580816506120234, -0.00406924940329345)

(0, 1)

(9.213434943972674, -0.011384094242491)

(-37.64607270294512, 0.000704114293701762)

(53.36963127683454, -0.000350715231486932)

(-56.51330369475196, 0.000312817485971633)

(-84.79942423032556, -0.00013900585084881)

(94.22655974561256, 0.000112591766511704)

(-5.968081392398221, 0.0259643971802455)

(12.40756748978677, 0.00637258289495849)

(25.053206244297428, 0.00158564848443144)

(-53.36963127683454, -0.000350715231486932)

(78.51435292656672, -0.000162140173318783)

(72.22895368169175, -0.000191570111140939)

(-25.053206244297428, 0.00158564848443144)

(-106.79542665849998, 8.76557646972633e-5)

(-34.49966091896661, -0.000838066136358659)

(15.580816506120234, -0.00406924940329345)

(-87.94185886046559, 0.0001292529049009)

(87.94185886046559, 0.0001292529049009)

(97.3688368618863, -0.000105444189250915)

(-75.37169941967161, 0.00017593577340359)

(-100.51107120362654, 9.89562584809543e-5)

(-62.80002477584475, 0.000253367116057383)

(-43.936832175017194, 0.000517212000046997)

(31.352286221096882, 0.00101423808278872)

(-65.94313282375245, -0.000229806033389755)

(163.35057545169616, 3.74722559859326e-5)

(-47.08145487760369, -0.000450519125938963)

(21.90026654019963, -0.00207205193264381)

(-103.65326665891925, -9.30492289536518e-5)

(100.51107120362654, 9.89562584809543e-5)

(-91.08423543053327, -0.000120491545810595)

(-97.3688368618863, -0.000105444189250915)

(84.79942423032556, -0.00013900585084881)

(2.5437321475260917, -0.110639672191836)

(-81.6569248421486, 0.000149905871666022)

(47.08145487760369, -0.000450519125938963)

(43.936832175017194, 0.000517212000046997)

(69.0861031389786, 0.000209385109224912)

(34.49966091896661, -0.000838066136358659)

(18.7435542483014, 0.00282239086745388)

(-122.50579026873812, -6.66192853304118e-5)

(75.37169941967161, 0.00017593577340359)

(-69.0861031389786, 0.000209385109224912)

(65.94313282375245, -0.000229806033389755)

(62.80002477584475, 0.000253367116057383)

(56.51330369475196, 0.000312817485971633)

(40.79174357493512, -0.000599892999132703)

(-31.352286221096882, 0.00101423808278872)

(-78.51435292656672, -0.000162140173318783)

(-40.79174357493512, -0.000599892999132703)

(-9.213434943972674, -0.011384094242491)

(-72.22895368169175, -0.000191570111140939)

(-113.07965210777498, 7.81860375912636e-5)

(81.6569248421486, 0.000149905871666022)

(5.968081392398221, 0.0259643971802455)

(91.08423543053327, -0.000120491545810595)

(-12.40756748978677, 0.00637258289495849)

(-59.65675725562702, -0.000280746865913829)

(-50.22569898631863, 0.000395942499274958)

(59.65675725562702, -0.000280746865913829)

(-18.7435542483014, 0.00282239086745388)

(-210.47720607436906, -2.25715015693393e-5)

(-28.203628119338006, -0.00125244284383629)

(28.203628119338006, -0.00125244284383629)


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=−2.54373214752609x_{1} = -2.54373214752609
x2=−21.9002665401996x_{2} = -21.9002665401996
x3=−15.5808165061202x_{3} = -15.5808165061202
x4=9.21343494397267x_{4} = 9.21343494397267
x5=53.3696312768345x_{5} = 53.3696312768345
x6=−84.7994242303256x_{6} = -84.7994242303256
x7=−53.3696312768345x_{7} = -53.3696312768345
x8=78.5143529265667x_{8} = 78.5143529265667
x9=72.2289536816917x_{9} = 72.2289536816917
x10=−34.4996609189666x_{10} = -34.4996609189666
x11=15.5808165061202x_{11} = 15.5808165061202
x12=97.3688368618863x_{12} = 97.3688368618863
x13=−65.9431328237524x_{13} = -65.9431328237524
x14=−47.0814548776037x_{14} = -47.0814548776037
x15=21.9002665401996x_{15} = 21.9002665401996
x16=−103.653266658919x_{16} = -103.653266658919
x17=−91.0842354305333x_{17} = -91.0842354305333
x18=−97.3688368618863x_{18} = -97.3688368618863
x19=84.7994242303256x_{19} = 84.7994242303256
x20=2.54373214752609x_{20} = 2.54373214752609
x21=47.0814548776037x_{21} = 47.0814548776037
x22=34.4996609189666x_{22} = 34.4996609189666
x23=−122.505790268738x_{23} = -122.505790268738
x24=65.9431328237524x_{24} = 65.9431328237524
x25=40.7917435749351x_{25} = 40.7917435749351
x26=−78.5143529265667x_{26} = -78.5143529265667
x27=−40.7917435749351x_{27} = -40.7917435749351
x28=−9.21343494397267x_{28} = -9.21343494397267
x29=−72.2289536816917x_{29} = -72.2289536816917
x30=91.0842354305333x_{30} = 91.0842354305333
x31=−59.656757255627x_{31} = -59.656757255627
x32=59.656757255627x_{32} = 59.656757255627
x33=−210.477206074369x_{33} = -210.477206074369
x34=−28.203628119338x_{34} = -28.203628119338
x35=28.203628119338x_{35} = 28.203628119338
Puntos máximos de la función:
x35=−94.2265597456126x_{35} = -94.2265597456126
x35=50.2256989863186x_{35} = 50.2256989863186
x35=37.6460727029451x_{35} = 37.6460727029451
x35=0x_{35} = 0
x35=−37.6460727029451x_{35} = -37.6460727029451
x35=−56.513303694752x_{35} = -56.513303694752
x35=94.2265597456126x_{35} = 94.2265597456126
x35=−5.96808139239822x_{35} = -5.96808139239822
x35=12.4075674897868x_{35} = 12.4075674897868
x35=25.0532062442974x_{35} = 25.0532062442974
x35=−25.0532062442974x_{35} = -25.0532062442974
x35=−106.7954266585x_{35} = -106.7954266585
x35=−87.9418588604656x_{35} = -87.9418588604656
x35=87.9418588604656x_{35} = 87.9418588604656
x35=−75.3716994196716x_{35} = -75.3716994196716
x35=−100.511071203627x_{35} = -100.511071203627
x35=−62.8000247758447x_{35} = -62.8000247758447
x35=−43.9368321750172x_{35} = -43.9368321750172
x35=31.3522862210969x_{35} = 31.3522862210969
x35=163.350575451696x_{35} = 163.350575451696
x35=100.511071203627x_{35} = 100.511071203627
x35=−81.6569248421486x_{35} = -81.6569248421486
x35=43.9368321750172x_{35} = 43.9368321750172
x35=69.0861031389786x_{35} = 69.0861031389786
x35=18.7435542483014x_{35} = 18.7435542483014
x35=75.3716994196716x_{35} = 75.3716994196716
x35=−69.0861031389786x_{35} = -69.0861031389786
x35=62.8000247758447x_{35} = 62.8000247758447
x35=56.513303694752x_{35} = 56.513303694752
x35=−31.3522862210969x_{35} = -31.3522862210969
x35=−113.079652107775x_{35} = -113.079652107775
x35=81.6569248421486x_{35} = 81.6569248421486
x35=5.96808139239822x_{35} = 5.96808139239822
x35=−12.4075674897868x_{35} = -12.4075674897868
x35=−50.2256989863186x_{35} = -50.2256989863186
x35=−18.7435542483014x_{35} = -18.7435542483014
Decrece en los intervalos
[97.3688368618863,∞)\left[97.3688368618863, \infty\right)
Crece en los intervalos
(−∞,−210.477206074369]\left(-\infty, -210.477206074369\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
4xsin⁡(x)x2+1−cos⁡(x)+2(4x2x2+1−1)cos⁡(x)x2+1x2+1=0\frac{\frac{4 x \sin{\left(x \right)}}{x^{2} + 1} - \cos{\left(x \right)} + \frac{2 \left(\frac{4 x^{2}}{x^{2} + 1} - 1\right) \cos{\left(x \right)}}{x^{2} + 1}}{x^{2} + 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−51.7590072918019x_{1} = -51.7590072918019
x2=32.8650518133113x_{2} = 32.8650518133113
x3=−17.0443794915286x_{3} = -17.0443794915286
x4=−7.31236957667153x_{4} = -7.31236957667153
x5=149.198841587983x_{5} = 149.198841587983
x6=−54.9050265131646x_{6} = -54.9050265131646
x7=−76.9170188649185x_{7} = -76.9170188649185
x8=−45.4651283904817x_{8} = -45.4651283904817
x9=−73.7732104776975x_{9} = -73.7732104776975
x10=45.4651283904817x_{10} = 45.4651283904817
x11=−83.2041331118179x_{11} = -83.2041331118179
x12=−86.3474755966677x_{12} = -86.3474755966677
x13=−168.051405007597x_{13} = -168.051405007597
x14=−39.1678061810769x_{14} = -39.1678061810769
x15=13.8489274699853x_{15} = 13.8489274699853
x16=20.2227299740537x_{16} = 20.2227299740537
x17=36.0172870463513x_{17} = 36.0172870463513
x18=−89.4906951115164x_{18} = -89.4906951115164
x19=26.5529700363992x_{19} = 26.5529700363992
x20=86.3474755966677x_{20} = 86.3474755966677
x21=29.7105507660002x_{21} = 29.7105507660002
x22=3.69928083952331x_{22} = 3.69928083952331
x23=−67.4849739703609x_{23} = -67.4849739703609
x24=10.6203828275874x_{24} = 10.6203828275874
x25=−32.8650518133113x_{25} = -32.8650518133113
x26=−26.5529700363992x_{26} = -26.5529700363992
x27=0.5599347473979x_{27} = 0.5599347473979
x28=−13.8489274699853x_{28} = -13.8489274699853
x29=58.0505655345866x_{29} = 58.0505655345866
x30=61.1956985466846x_{30} = 61.1956985466846
x31=114.633238850285x_{31} = 114.633238850285
x32=−29.7105507660002x_{32} = -29.7105507660002
x33=17.0443794915286x_{33} = 17.0443794915286
x34=−61.1956985466846x_{34} = -61.1956985466846
x35=95.7768136964885x_{35} = 95.7768136964885
x36=−64.3404851927513x_{36} = -64.3404851927513
x37=23.3910501690508x_{37} = 23.3910501690508
x38=−20.2227299740537x_{38} = -20.2227299740537
x39=64.3404851927513x_{39} = 64.3404851927513
x40=98.9197331449288x_{40} = 98.9197331449288
x41=−10.6203828275874x_{41} = -10.6203828275874
x42=70.629204720493x_{42} = 70.629204720493
x43=39.1678061810769x_{43} = 39.1678061810769
x44=76.9170188649185x_{44} = 76.9170188649185
x45=7.31236957667153x_{45} = 7.31236957667153
x46=−70.629204720493x_{46} = -70.629204720493
x47=−42.316994092934x_{47} = -42.316994092934
x48=48.612414402203x_{48} = 48.612414402203
x49=−92.6338041843149x_{49} = -92.6338041843149
x50=−3.69928083952331x_{50} = -3.69928083952331
x51=−23.3910501690508x_{51} = -23.3910501690508
x52=54.9050265131646x_{52} = 54.9050265131646
x53=463.376284124582x_{53} = 463.376284124582
x54=−98.9197331449288x_{54} = -98.9197331449288
x55=67.4849739703609x_{55} = 67.4849739703609
x56=−95.7768136964885x_{56} = -95.7768136964885
x57=89.4906951115164x_{57} = 89.4906951115164
x58=51.7590072918019x_{58} = 51.7590072918019
x59=−48.612414402203x_{59} = -48.612414402203
x60=83.2041331118179x_{60} = 83.2041331118179
x61=42.316994092934x_{61} = 42.316994092934
x62=−80.0606531586329x_{62} = -80.0606531586329
x63=80.0606531586329x_{63} = 80.0606531586329
x64=73.7732104776975x_{64} = 73.7732104776975
x65=−36.0172870463513x_{65} = -36.0172870463513
x66=92.6338041843149x_{66} = 92.6338041843149
x67=−58.0505655345866x_{67} = -58.0505655345866

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
[114.633238850285,∞)\left[114.633238850285, \infty\right)
Convexa en los intervalos
(−∞,−168.051405007597]\left(-\infty, -168.051405007597\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(cos⁡(x)x2+1)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x^{2} + 1}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(cos⁡(x)x2+1)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x^{2} + 1}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=0y = 0
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x)/(x^2 + 1), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x)x(x2+1))=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x \right)}}{x \left(x^{2} + 1\right)}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x)x(x2+1))=0\lim_{x \to \infty}\left(\frac{\cos{\left(x \right)}}{x \left(x^{2} + 1\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:
cos⁡(x)x2+1=cos⁡(x)x2+1\frac{\cos{\left(x \right)}}{x^{2} + 1} = \frac{\cos{\left(x \right)}}{x^{2} + 1}
- Sí
cos⁡(x)x2+1=−cos⁡(x)x2+1\frac{\cos{\left(x \right)}}{x^{2} + 1} = - \frac{\cos{\left(x \right)}}{x^{2} + 1}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор