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Trazado el gráfico de la función/ (cos^3)*x+1

Gráfico de la función y = (cos^3)*x+1

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          3         
f(x) = cos (x)*x + 1
f(x)=xcos⁡3(x)+1f{\left(x \right)} = x \cos^{3}{\left(x \right)} + 1 
f = x*cos(x)^3 + 1
Gráfico de la función
02468-8-6-4-2-1010-2020
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:
xcos⁡3(x)+1=0x \cos^{3}{\left(x \right)} + 1 = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−17.6727887878944x_{1} = -17.6727887878944
x2=−23.9164050080469x_{2} = -23.9164050080469
x3=39.5677551541366x_{3} = 39.5677551541366
x4=8.3689886401159x_{4} = 8.3689886401159
x5=−57.8579425332352x_{5} = -57.8579425332352
x6=−32.6685657073184x_{6} = -32.6685657073184
x7=−42.7016554342642x_{7} = -42.7016554342642
x8=58.3801852473287x_{8} = 58.3801852473287
x9=27.0431861705432x_{9} = 27.0431861705432
x10=98.7420896301744x_{10} = 98.7420896301744
x11=−99.1779224726838x_{11} = -99.1779224726838
x12=−13.7060804054404x_{12} = -13.7060804054404
x13=29.5156187929179x_{13} = 29.5156187929179
x14=23.2037313596922x_{14} = 23.2037313596922
x15=−95.5980931537388x_{15} = -95.5980931537388
x16=42.1199777411493x_{16} = 42.1199777411493
x17=−86.6217759238088x_{17} = -86.6217759238088
x18=10.5216772572525x_{18} = 10.5216772572525
x19=−92.8996159864853x_{19} = -92.8996159864853
x20=16.8782978039087x_{20} = 16.8782978039087
x21=−70.4412689534459x_{21} = -70.4412689534459
x22=−45.2686879771139x_{22} = -45.2686879771139
x23=64.6544556021531x_{23} = 64.6544556021531
x24=54.7113106241164x_{24} = 54.7113106241164
x25=−11.4551945422505x_{25} = -11.4551945422505
x26=−67.7920185219161x_{26} = -67.7920185219161
x27=−64.1501707704071x_{27} = -64.1501707704071
x28=52.1073172028376x_{28} = 52.1073172028376
x29=20.7925478598529x_{29} = 20.7925478598529
x30=−76.7314629197398x_{30} = -76.7314629197398
x31=48.4167528419446x_{31} = 48.4167528419446
x32=70.9298261944772x_{32} = 70.9298261944772
x33=33.3027766701391x_{33} = 33.3027766701391
x34=83.4830561861961x_{34} = 83.4830561861961
x35=−83.0209191187711x_{35} = -83.0209191187711
x36=4.03311044914193x_{36} = 4.03311044914193
x37=−20.0433473349474x_{37} = -20.0433473349474
x38=−74.0678538003873x_{38} = -74.0678538003873
x39=−89.309763432447x_{39} = -89.309763432447
x40=−5.32225071111272x_{40} = -5.32225071111272
x41=77.206079960794x_{41} = 77.206079960794
x42=2.41227823457333x_{42} = 2.41227823457333
x43=89.7606328734011x_{43} = 89.7606328734011
x44=67.2958453267276x_{44} = 67.2958453267276
x45=35.8200990876682x_{45} = 35.8200990876682
x46=96.038715441529x_{46} = 96.038715441529
x47=−38.9705005218809x_{47} = -38.9705005218809
x48=−51.5642681536145x_{48} = -51.5642681536145
x49=45.8362875156384x_{49} = 45.8362875156384
x50=−61.5171664503898x_{50} = -61.5171664503898
x51=14.5591085347947x_{51} = 14.5591085347947
x52=61.0042150700563x_{52} = 61.0042150700563
x53=−36.434732102676x_{53} = -36.434732102676
x54=−26.3608646341664x_{54} = -26.3608646341664
x55=79.8762745870391x_{55} = 79.8762745870391
x56=86.1654111182467x_{56} = 86.1654111182467
x57=92.4539874170225x_{57} = 92.4539874170225
x58=−48.9715380613435x_{58} = -48.9715380613435
x59=−55.2435525999189x_{59} = -55.2435525999189
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)^3*x + 1.
0cos⁡3(0)+10 \cos^{3}{\left(0 \right)} + 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
−3xsin⁡(x)cos⁡2(x)+cos⁡3(x)=0- 3 x \sin{\left(x \right)} \cos^{2}{\left(x \right)} + \cos^{3}{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−22.0062945981151x_{1} = -22.0062945981151
x2=−94.2513162367499x_{2} = -94.2513162367499
x3=87.9683835225577x_{3} = 87.9683835225577
x4=−65.978497833395x_{4} = -65.978497833395
x5=−37.7079514813517x_{5} = -37.7079514813517
x6=37.7079514813517x_{6} = 37.7079514813517
x7=14.1371670969671x_{7} = 14.1371670969671
x8=−53.4133156711182x_{8} = -53.4133156711182
x9=−75.402644368808x_{9} = -75.402644368808
x10=−43.9898745065796x_{10} = -43.9898745065796
x11=59.6958442217916x_{11} = 59.6958442217916
x12=94.2513162367499x_{12} = 94.2513162367499
x13=50.2721129415958x_{13} = 50.2721129415958
x14=4.71238890099015x_{14} = 4.71238890099015
x15=22.0062945981151x_{15} = 22.0062945981151
x16=72.2612438921163x_{16} = 72.2612438921163
x17=−31.426532886749x_{17} = -31.426532886749
x18=−9.45999947251134x_{18} = -9.45999947251134
x19=65.978497833395x_{19} = 65.978497833395
x20=0.54716075726033x_{20} = 0.54716075726033
x21=7.85398173981436x_{21} = 7.85398173981436
x22=12.5928345144433x_{22} = 12.5928345144433
x23=−29.8451300993981x_{23} = -29.8451300993981
x24=−12.5928345144433x_{24} = -12.5928345144433
x25=−50.2721129415958x_{25} = -50.2721129415958
x26=56.5545617095679x_{26} = 56.5545617095679
x27=−14.137166846121x_{27} = -14.137166846121
x28=−97.3927948145887x_{28} = -97.3927948145887
x29=15.7291521688357x_{29} = 15.7291521688357
x30=−72.2612438921163x_{30} = -72.2612438921163
x31=−15.7291521688357x_{31} = -15.7291521688357
x32=−0.54716075726033x_{32} = -0.54716075726033
x33=−1.57079644420084x_{33} = -1.57079644420084
x34=1.57079657923741x_{34} = 1.57079657923741
x35=29.8451303157785x_{35} = 29.8451303157785
x36=−59.6958442217916x_{36} = -59.6958442217916
x37=20.4203521553001x_{37} = 20.4203521553001
x38=81.6854896627962x_{38} = 81.6854896627962
x39=78.5440602167642x_{39} = 78.5440602167642
x40=−81.6854896627962x_{40} = -81.6854896627962
x41=−17.2787597501589x_{41} = -17.2787597501589
x42=43.9898745065796x_{42} = 43.9898745065796
x43=42.4115007334065x_{43} = 42.4115007334065
x44=−95.8185758684232x_{44} = -95.8185758684232
x45=−28.2861176805762x_{45} = -28.2861176805762
x46=−23.5619450053232x_{46} = -23.5619450053232
x47=34.5671619539785x_{47} = 34.5671619539785
x48=−36.1283154240079x_{48} = -36.1283154240079
x49=−6.33574836234573x_{49} = -6.33574836234573
x50=100.534280521352x_{50} = 100.534280521352
x51=−87.9683835225577x_{51} = -87.9683835225577
x52=28.2861176805762x_{52} = 28.2861176805762
x53=6.33574836234573x_{53} = 6.33574836234573
x54=−7.85398150906577x_{54} = -7.85398150906577
Signos de extremos en los puntos:
(-22.006294598115065, 22.9987231789777)

(-94.25131623674994, -93.2495479424962)

(87.96838352255773, 88.9664889364721)

(-65.97849783339504, 66.9759718384914)

(-37.707951481351664, -36.7035319787864)

(37.707951481351664, 38.7035319787864)

(14.13716709696708, 1)

(-53.41331567111824, 54.4101955024623)

(-75.402644368808, -74.4004340670771)

(-43.98987450657957, -42.9860860278184)

(59.695844221791624, -58.6930523997922)

(94.25131623674994, 95.2495479424962)

(50.27211294159583, 51.2687978331174)

(4.712388900990145, 1)

(22.006294598115065, -20.9987231789777)

(72.26124389211625, -71.2589375073279)

(-31.426532886749044, -30.4212302581647)

(-9.459999472511342, 10.4424087338996)

(65.97849783339504, -64.9759718384914)

(0.5471607572603301, 1.34079747220024)

(7.853981739814361, 1)

(12.592834514443268, 13.5796110567852)

(-29.84513009939808, 1)

(-12.592834514443268, -11.5796110567852)

(-50.27211294159583, -49.2687978331174)

(56.554561709567935, 57.5516148309334)

(-14.13716684612103, 1)

(-97.39279481458874, 98.3910835563113)

(15.7291521688357, -14.718562077866)

(-72.26124389211625, 73.2589375073279)

(-15.7291521688357, 16.718562077866)

(-0.5471607572603301, 0.659202527799765)

(-1.5707964442008373, 1)

(1.570796579237414, 1)

(29.845130315778494, 1)

(-59.695844221791624, 60.6930523997922)

(20.42035215530012, 1)

(81.68548966279624, 82.6834493592093)

(78.54406021676424, -77.5419383132867)

(-81.68548966279624, -80.6834493592093)

(-17.278759750158898, 1)

(43.98987450657957, 44.9860860278184)

(42.4115007334065, 1)

(-95.81857586842315, 1)

(-28.286117680576208, 29.280226531362)

(-23.5619450053232, 1)

(34.567161953978456, -33.5623409826661)

(-36.12831542400792, 1)

(-6.335748362345733, -5.30953332777999)

(100.53428052135241, 101.532622734819)

(-87.96838352255773, -86.9664889364721)

(28.286117680576208, -27.280226531362)

(6.335748362345733, 7.30953332777999)

(-7.8539815090657745, 1)


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=−94.2513162367499x_{1} = -94.2513162367499
x2=−37.7079514813517x_{2} = -37.7079514813517
x3=−75.402644368808x_{3} = -75.402644368808
x4=−43.9898745065796x_{4} = -43.9898745065796
x5=59.6958442217916x_{5} = 59.6958442217916
x6=22.0062945981151x_{6} = 22.0062945981151
x7=72.2612438921163x_{7} = 72.2612438921163
x8=−31.426532886749x_{8} = -31.426532886749
x9=65.978497833395x_{9} = 65.978497833395
x10=−12.5928345144433x_{10} = -12.5928345144433
x11=−50.2721129415958x_{11} = -50.2721129415958
x12=15.7291521688357x_{12} = 15.7291521688357
x13=−0.54716075726033x_{13} = -0.54716075726033
x14=78.5440602167642x_{14} = 78.5440602167642
x15=−81.6854896627962x_{15} = -81.6854896627962
x16=34.5671619539785x_{16} = 34.5671619539785
x17=−6.33574836234573x_{17} = -6.33574836234573
x18=−87.9683835225577x_{18} = -87.9683835225577
x19=28.2861176805762x_{19} = 28.2861176805762
Puntos máximos de la función:
x19=−22.0062945981151x_{19} = -22.0062945981151
x19=87.9683835225577x_{19} = 87.9683835225577
x19=−65.978497833395x_{19} = -65.978497833395
x19=37.7079514813517x_{19} = 37.7079514813517
x19=−53.4133156711182x_{19} = -53.4133156711182
x19=94.2513162367499x_{19} = 94.2513162367499
x19=50.2721129415958x_{19} = 50.2721129415958
x19=−9.45999947251134x_{19} = -9.45999947251134
x19=0.54716075726033x_{19} = 0.54716075726033
x19=12.5928345144433x_{19} = 12.5928345144433
x19=56.5545617095679x_{19} = 56.5545617095679
x19=−97.3927948145887x_{19} = -97.3927948145887
x19=−72.2612438921163x_{19} = -72.2612438921163
x19=−15.7291521688357x_{19} = -15.7291521688357
x19=−59.6958442217916x_{19} = -59.6958442217916
x19=81.6854896627962x_{19} = 81.6854896627962
x19=43.9898745065796x_{19} = 43.9898745065796
x19=−28.2861176805762x_{19} = -28.2861176805762
x19=100.534280521352x_{19} = 100.534280521352
x19=6.33574836234573x_{19} = 6.33574836234573
Decrece en los intervalos
[78.5440602167642,∞)\left[78.5440602167642, \infty\right)
Crece en los intervalos
(−∞,−94.2513162367499]\left(-\infty, -94.2513162367499\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
3(2xsin⁡2(x)−xcos⁡2(x)−2sin⁡(x)cos⁡(x))cos⁡(x)=03 \left(2 x \sin^{2}{\left(x \right)} - x \cos^{2}{\left(x \right)} - 2 \sin{\left(x \right)} \cos{\left(x \right)}\right) \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−54.0287375186797x_{1} = -54.0287375186797
x2=−86.3937979737193x_{2} = -86.3937979737193
x3=29.845130209103x_{3} = 29.845130209103
x4=14.1371669411541x_{4} = 14.1371669411541
x5=46.5155575128945x_{5} = 46.5155575128945
x6=−80.1106126665397x_{6} = -80.1106126665397
x7=−7.85398163397448x_{7} = -7.85398163397448
x8=−49.6566991728001x_{8} = -49.6566991728001
x9=−38.3233153411839x_{9} = -38.3233153411839
x10=−58.1194640914112x_{10} = -58.1194640914112
x11=95.8185759344887x_{11} = 95.8185759344887
x12=−40.2334847398618x_{12} = -40.2334847398618
x13=−19.4822419590393x_{13} = -19.4822419590393
x14=−71.6457960548402x_{14} = -71.6457960548402
x15=80.1106126665397x_{15} = 80.1106126665397
x16=44.6052690020543x_{16} = 44.6052690020543
x17=−14.1371669411541x_{17} = -14.1371669411541
x18=7.85398163397448x_{18} = 7.85398163397448
x19=−29.845130209103x_{19} = -29.845130209103
x20=55.9391340788625x_{20} = 55.9391340788625
x21=25.761216560864x_{21} = 25.761216560864
x22=68.5044160127206x_{22} = 68.5044160127206
x23=18.2522129281291x_{23} = 18.2522129281291
x24=42.4115008234622x_{24} = 42.4115008234622
x25=11.9784169281255x_{25} = 11.9784169281255
x26=−69.7353059990218x_{26} = -69.7353059990218
x27=−55.9391340788625x_{27} = -55.9391340788625
x28=−25.761216560864x_{28} = -25.761216560864
x29=−32.0418460564325x_{29} = -32.0418460564325
x30=0x_{30} = 0
x31=−67.5442420521806x_{31} = -67.5442420521806
x32=40.2334847398618x_{32} = 40.2334847398618
x33=90.494385851764x_{33} = 90.494385851764
x34=−60.3112775828661x_{34} = -60.3112775828661
x35=−93.635855248412x_{35} = -93.635855248412
x36=−77.9286074777235x_{36} = -77.9286074777235
x37=66.5939395847984x_{37} = 66.5939395847984
x38=98.0082570809958x_{38} = 98.0082570809958
x39=−45.553093477052x_{39} = -45.553093477052
x40=−23.5619449019235x_{40} = -23.5619449019235
x41=58.1194640914112x_{41} = 58.1194640914112
x42=−0.952446522565002x_{42} = -0.952446522565002
x43=−3.8454964771924x_{43} = -3.8454964771924
x44=−62.2217201983468x_{44} = -62.2217201983468
x45=−26.7035375555132x_{45} = -26.7035375555132
x46=−36.1283155162826x_{46} = -36.1283155162826
x47=10.0736847946507x_{47} = 10.0736847946507
x48=−51.8362787842316x_{48} = -51.8362787842316
x49=−73.8274273593601x_{49} = -73.8274273593601
x50=64.4026493985908x_{50} = 64.4026493985908
x51=27.6708469335931x_{51} = 27.6708469335931
x52=82.3009445890673x_{52} = 82.3009445890673
x53=−16.3439734242303x_{53} = -16.3439734242303
x54=−95.8185759344887x_{54} = -95.8185759344887
x55=88.5838418575576x_{55} = 88.5838418575576
x56=54.0287375186797x_{56} = 54.0287375186797
x57=16.3439734242303x_{57} = 16.3439734242303
x58=5.72448762140597x_{58} = 5.72448762140597
x59=20.4203522483337x_{59} = 20.4203522483337
x60=−82.3009445890673x_{60} = -82.3009445890673
x61=−1.5707963267949x_{61} = -1.5707963267949
x62=−47.746367621976x_{62} = -47.746367621976
x63=−27.6708469335931x_{63} = -27.6708469335931
x64=−84.2114746051039x_{64} = -84.2114746051039
x65=3.8454964771924x_{65} = 3.8454964771924
x66=49.6566991728001x_{66} = 49.6566991728001
x67=86.3937979737193x_{67} = 86.3937979737193
x68=62.2217201983468x_{68} = 62.2217201983468
x69=−10.0736847946507x_{69} = -10.0736847946507
x70=−5.72448762140597x_{70} = -5.72448762140597
x71=−11.9784169281255x_{71} = -11.9784169281255
x72=−21.3911607990677x_{72} = -21.3911607990677
x73=77.9286074777235x_{73} = 77.9286074777235
x74=−64.4026493985908x_{74} = -64.4026493985908
x75=0.952446522565002x_{75} = 0.952446522565002
x76=−76.0180950008633x_{76} = -76.0180950008633
x77=−91.7253053066676x_{77} = -91.7253053066676
x78=36.1283155162826x_{78} = 36.1283155162826
x79=−18.2522129281291x_{79} = -18.2522129281291
x80=33.9518219840686x_{80} = 33.9518219840686
x81=51.8362787842316x_{81} = 51.8362787842316
x82=−99.9188172637041x_{82} = -99.9188172637041
x83=60.3112775828661x_{83} = 60.3112775828661
x84=24.5307813024125x_{84} = 24.5307813024125
x85=73.8274273593601x_{85} = 73.8274273593601
x86=76.0180950008633x_{86} = 76.0180950008633
x87=84.2114746051039x_{87} = 84.2114746051039
x88=2.64424542148422x_{88} = 2.64424542148422
x89=99.9188172637041x_{89} = 99.9188172637041
x90=−98.0082570809958x_{90} = -98.0082570809958
x91=−89.5353906273091x_{91} = -89.5353906273091
x92=32.0418460564325x_{92} = 32.0418460564325
x93=93.635855248412x_{93} = 93.635855248412
x94=38.3233153411839x_{94} = 38.3233153411839
x95=−33.9518219840686x_{95} = -33.9518219840686
x96=71.6457960548402x_{96} = 71.6457960548402
x97=69.7353059990218x_{97} = 69.7353059990218
x98=47.746367621976x_{98} = 47.746367621976

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
[90.494385851764,∞)\left[90.494385851764, \infty\right)
Convexa en los intervalos
(−∞,−86.3937979737193]\left(-\infty, -86.3937979737193\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(xcos⁡3(x)+1)=⟨−∞,∞⟩\lim_{x \to -\infty}\left(x \cos^{3}{\left(x \right)} + 1\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(xcos⁡3(x)+1)=⟨−∞,∞⟩\lim_{x \to \infty}\left(x \cos^{3}{\left(x \right)} + 1\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x)^3*x + 1, dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞(xcos⁡3(x)+1x)y = x \lim_{x \to -\infty}\left(\frac{x \cos^{3}{\left(x \right)} + 1}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(xcos⁡3(x)+1x)y = x \lim_{x \to \infty}\left(\frac{x \cos^{3}{\left(x \right)} + 1}{x}\right)
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:
xcos⁡3(x)+1=−xcos⁡3(x)+1x \cos^{3}{\left(x \right)} + 1 = - x \cos^{3}{\left(x \right)} + 1
- No
xcos⁡3(x)+1=xcos⁡3(x)−1x \cos^{3}{\left(x \right)} + 1 = x \cos^{3}{\left(x \right)} - 1
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор