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cos(x^2)
Trazado el gráfico de la función/ cos(x^2)

Gráfico de la función y = cos(x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

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

Solución analítica
x1=−2π2x_{1} = - \frac{\sqrt{2} \sqrt{\pi}}{2}
x2=2π2x_{2} = \frac{\sqrt{2} \sqrt{\pi}}{2}
x3=−6π2x_{3} = - \frac{\sqrt{6} \sqrt{\pi}}{2}
x4=6π2x_{4} = \frac{\sqrt{6} \sqrt{\pi}}{2}
Solución numérica
x1=−97.7504683154682x_{1} = -97.7504683154682
x2=40.1647585280723x_{2} = 40.1647585280723
x3=36.042316272118x_{3} = 36.042316272118
x4=−76.1639733394572x_{4} = -76.1639733394572
x5=16.0012437412711x_{5} = 16.0012437412711
x6=34.2088185043301x_{6} = 34.2088185043301
x7=−3.31595752197827x_{7} = -3.31595752197827
x8=58.181320750246x_{8} = 58.181320750246
x9=96.1791015163645x_{9} = 96.1791015163645
x10=−53.6295447898027x_{10} = -53.6295447898027
x11=23.3466555505886x_{11} = 23.3466555505886
x12=−13.7864555104705x_{12} = -13.7864555104705
x13=63.9435910678085x_{13} = 63.9435910678085
x14=−93.7978156802077x_{14} = -93.7978156802077
x15=60.2503967723284x_{15} = 60.2503967723284
x16=−43.9375543883695x_{16} = -43.9375543883695
x17=82.2330723422884x_{17} = 82.2330723422884
x18=−92.1420324625326x_{18} = -92.1420324625326
x19=92.0738169932802x_{19} = 92.0738169932802
x20=67.736962365963x_{20} = 67.736962365963
x21=54.1251803643413x_{21} = 54.1251803643413
x22=36.7757479512029x_{22} = 36.7757479512029
x23=−63.2520216621303x_{23} = -63.2520216621303
x24=−22.2441192889355x_{24} = -22.2441192889355
x25=−37.7869535826396x_{25} = -37.7869535826396
x26=−46.338737344997x_{26} = -46.338737344997
x27=−5.16754657023168x_{27} = -5.16754657023168
x28=−69.7703218907394x_{28} = -69.7703218907394
x29=89.4957567168811x_{29} = 89.4957567168811
x30=23.5476347720657x_{30} = 23.5476347720657
x31=2.1708037636748x_{31} = 2.1708037636748
x32=103.115082970798x_{32} = 103.115082970798
x33=56.1199308198413x_{33} = 56.1199308198413
x34=−60.5883736018691x_{34} = -60.5883736018691
x35=−75.8953893880049x_{35} = -75.8953893880049
x36=−17.8569777493103x_{36} = -17.8569777493103
x37=41.9626343588803x_{37} = 41.9626343588803
x38=−65.8556653468835x_{38} = -65.8556653468835
x39=57.4750787320006x_{39} = 57.4750787320006
x40=70.1296178457777x_{40} = 70.1296178457777
x41=−42.0000508927152x_{41} = -42.0000508927152
x42=−58.8524077721783x_{42} = -58.8524077721783
x43=−71.6803654074538x_{43} = -71.6803654074538
x44=−81.792551958106x_{44} = -81.792551958106
x45=−29.8962056111858x_{45} = -29.8962056111858
x46=−57.7477324384739x_{46} = -57.7477324384739
x47=114.587462209783x_{47} = 114.587462209783
x48=6.2665706865775x_{48} = 6.2665706865775
x49=−68.2222069145618x_{49} = -68.2222069145618
x50=18.1189522958733x_{50} = 18.1189522958733
x51=10.2588183479024x_{51} = 10.2588183479024
x52=−79.6323098451313x_{52} = -79.6323098451313
x53=−89.407955502065x_{53} = -89.407955502065
x54=13.5566651590649x_{54} = 13.5566651590649
x55=−50.6779241318116x_{55} = -50.6779241318116
x56=26.0797777885892x_{56} = 26.0797777885892
x57=31.3829459563694x_{57} = 31.3829459563694
x58=−8.7731989612085x_{58} = -8.7731989612085
x59=17.4115964538412x_{59} = 17.4115964538412
x60=−33.7930307841704x_{60} = -33.7930307841704
x61=68.1530977897007x_{61} = 68.1530977897007
x62=−85.7673550856064x_{62} = -85.7673550856064
x63=66.1412713071331x_{63} = 66.1412713071331
x64=20.2479095536667x_{64} = 20.2479095536667
x65=80.1827248574873x_{65} = 80.1827248574873
x66=62.2255007657586x_{66} = 62.2255007657586
x67=−16.0012437412711x_{67} = -16.0012437412711
x68=−7.82694427889971x_{68} = -7.82694427889971
x69=27.9969170993996x_{69} = 27.9969170993996
x70=32.1249620498491x_{70} = 32.1249620498491
x71=−3.7599424119465x_{71} = -3.7599424119465
x72=4.15677273792348x_{72} = 4.15677273792348
x73=−32.0270198644374x_{73} = -32.0270198644374
x74=22.3146238057912x_{74} = 22.3146238057912
x75=−74.2423350301113x_{75} = -74.2423350301113
x76=−67.8759576469418x_{76} = -67.8759576469418
x77=46.0326432528734x_{77} = 46.0326432528734
x78=−21.7441875995693x_{78} = -21.7441875995693
x79=−2.1708037636748x_{79} = -2.1708037636748
x80=−47.8726627710821x_{80} = -47.8726627710821
x81=84.0842093790732x_{81} = 84.0842093790732
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^2).
cos⁡(02)\cos{\left(0^{2} \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
−2xsin⁡(x2)=0- 2 x \sin{\left(x^{2} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=−πx_{2} = - \sqrt{\pi}
x3=πx_{3} = \sqrt{\pi}
Signos de extremos en los puntos:
(0, 1)

    ____     
(-\/ pi, -1)

   ____     
(\/ pi, -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=−πx_{1} = - \sqrt{\pi}
x2=πx_{2} = \sqrt{\pi}
La función no tiene puntos máximos
Decrece en los intervalos
[π,∞)\left[\sqrt{\pi}, \infty\right)
Crece en los intervalos
(−∞,−π]\left(-\infty, - \sqrt{\pi}\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
−2(2x2cos⁡(x2)+sin⁡(x2))=0- 2 \left(2 x^{2} \cos{\left(x^{2} \right)} + \sin{\left(x^{2} \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−90.0556577728139x_{1} = -90.0556577728139
x2=80.182725342438x_{2} = 80.182725342438
x3=40.1647623864471x_{3} = 40.1647623864471
x4=55.698500038955x_{4} = 55.698500038955
x5=−85.7673554818607x_{5} = -85.7673554818607
x6=56.1199322342945x_{6} = 56.1199322342945
x7=−9.78896285608669x_{7} = -9.78896285608669
x8=2.19450274956445x_{8} = 2.19450274956445
x9=−53.6002486537402x_{9} = -53.6002486537402
x10=0x_{10} = 0
x11=46.0326458158356x_{11} = 46.0326458158356
x12=−42.3724023394102x_{12} = -42.3724023394102
x13=−1.35521112862614x_{13} = -1.35521112862614
x14=−3.76462907532733x_{14} = -3.76462907532733
x15=−8.95080183389482x_{15} = -8.95080183389482
x16=18.2915584905206x_{16} = 18.2915584905206
x17=6.01183407098084x_{17} = 6.01183407098084
x18=33.0409428606701x_{18} = 33.0409428606701
x19=62.2255018033701x_{19} = 62.2255018033701
x20=−29.8962149672115x_{20} = -29.8962149672115
x21=41.2071732071487x_{21} = 41.2071732071487
x22=−91.8517671603543x_{22} = -91.8517671603543
x23=27.996928491633x_{23} = 27.996928491633
x24=58.5580782403786x_{24} = 58.5580782403786
x25=−7.82746557122563x_{25} = -7.82746557122563
x26=70.1520134668099x_{26} = 70.1520134668099
x27=14.1242217429234x_{27} = 14.1242217429234
x28=−33.7930372624299x_{28} = -33.7930372624299
x29=26.438704217983x_{29} = 26.438704217983
x30=37.0735370544564x_{30} = 37.0735370544564
x31=58.2083140493455x_{31} = 58.2083140493455
x32=94.2155512590465x_{32} = 94.2155512590465
x33=−83.5970868479093x_{33} = -83.5970868479093
x34=−75.8953899598703x_{34} = -75.8953899598703
x35=77.3509100937384x_{35} = 77.3509100937384
x36=−52.1143649402824x_{36} = -52.1143649402824
x37=20.2479396696885x_{37} = 20.2479396696885
x38=91.8517671603543x_{38} = 91.8517671603543
x39=−13.3230177428884x_{39} = -13.3230177428884
x40=−5.74472561217197x_{40} = -5.74472561217197
x41=34.2088247492311x_{41} = 34.2088247492311
x42=22.3146463051457x_{42} = 22.3146463051457
x43=−68.7040251218618x_{43} = -68.7040251218618
x44=8.40790743485922x_{44} = 8.40790743485922
x45=−14.3449206558669x_{45} = -14.3449206558669
x46=−65.8556662221908x_{46} = -65.8556662221908
x47=−44.0803279657641x_{47} = -44.0803279657641
x48=−97.7504685831282x_{48} = -97.7504685831282
x49=10.2590498848041x_{49} = 10.2590498848041
x50=−42.0000542670678x_{50} = -42.0000542670678
x51=54.1251819410153x_{51} = 54.1251819410153
x52=−18.0320929835385x_{52} = -18.0320929835385
x53=82.5952088232899x_{53} = 82.5952088232899
x54=96.048356995137x_{54} = 96.048356995137
x55=−21.7442119165177x_{55} = -21.7442119165177
x56=−43.3979845304653x_{56} = -43.3979845304653
x57=−11.1398805605465x_{57} = -11.1398805605465
x58=6.26758611849278x_{58} = 6.26758611849278
x59=−38.3235554977812x_{59} = -38.3235554977812
x60=35.4269200396297x_{60} = 35.4269200396297
x61=−23.4138597867238x_{61} = -23.4138597867238
x62=26.6163455262094x_{62} = 26.6163455262094
x63=1.35521112862614x_{63} = 1.35521112862614
x64=−17.8570216542223x_{64} = -17.8570216542223
x65=84.2708150182891x_{65} = 84.2708150182891
x66=−4.16024524967154x_{66} = -4.16024524967154
x67=90.2299142368658x_{67} = 90.2299142368658
x68=−35.6038344867429x_{68} = -35.6038344867429
x69=5.16935647582827x_{69} = 5.16935647582827
x70=−47.8726650497299x_{70} = -47.8726650497299
x71=4.16024524967154x_{71} = 4.16024524967154
x72=18.7997496853775x_{72} = 18.7997496853775
x73=12.7198707532056x_{73} = 12.7198707532056
x74=−81.0594911844327x_{74} = -81.0594911844327
x75=−26.7928090700661x_{75} = -26.7928090700661
x76=−6.97889329812938x_{76} = -6.97889329812938
x77=60.2503979153653x_{77} = 60.2503979153653
x78=−69.7703226268241x_{78} = -69.7703226268241
x79=32.1249695905524x_{79} = 32.1249695905524
x80=36.0423216116322x_{80} = 36.0423216116322
x81=18.1189943237946x_{81} = 18.1189943237946
x82=−2.19450274956445x_{82} = -2.19450274956445
x83=−93.7978159831513x_{83} = -93.7978159831513
x84=−70.1072165206277x_{84} = -70.1072165206277
x85=−16.0013047615368x_{85} = -16.0013047615368

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
[96.048356995137,∞)\left[96.048356995137, \infty\right)
Convexa en los intervalos
(−∞,−97.7504685831282]\left(-\infty, -97.7504685831282\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞cos⁡(x2)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\left(x^{2} \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞cos⁡(x2)=⟨−1,1⟩\lim_{x \to \infty} \cos{\left(x^{2} \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x^2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(cos⁡(x2)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(x^{2} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(cos⁡(x2)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(x^{2} \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:
cos⁡(x2)=cos⁡(x2)\cos{\left(x^{2} \right)} = \cos{\left(x^{2} \right)}
- Sí
cos⁡(x2)=−cos⁡(x2)\cos{\left(x^{2} \right)} = - \cos{\left(x^{2} \right)}
- No
es decir, función
es
par
Gráfico
Gráfico de la función y = cos(x^2)
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