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Trazado el gráfico de la función/ sin(x)^(32)*cos(8*x)^5

Gráfico de la función y = sin(x)^(32)*cos(8*x)^5

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          32       5     
f(x) = sin  (x)*cos (8*x)
f(x)=sin⁡32(x)cos⁡5(8x)f{\left(x \right)} = \sin^{32}{\left(x \right)} \cos^{5}{\left(8 x \right)} 
f = sin(x)^32*cos(8*x)^5
Gráfico de la función
02468-8-6-4-2-10101-1
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:
sin⁡32(x)cos⁡5(8x)=0\sin^{32}{\left(x \right)} \cos^{5}{\left(8 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−π16x_{2} = - \frac{\pi}{16}
x3=π16x_{3} = \frac{\pi}{16}
Solución numérica
x1=30.0410851833975x_{1} = 30.0410851833975
x2=75.9874560862425x_{2} = 75.9874560862425
x3=8.04995274214039x_{3} = 8.04995274214039
x4=46.1417889744877x_{4} = 46.1417889744877
x5=−25.7221999777861x_{5} = -25.7221999777861
x6=−75.6241504560321x_{6} = -75.6241504560321
x7=−66x_{7} = -66
x8=86.1978114165678x_{8} = 86.1978114165678
x9=84.2340935641358x_{9} = 84.2340935641358
x10=−31.6124611297845x_{10} = -31.6124611297845
x11=−84.2256248075567x_{11} = -84.2256248075567
x12=64.206689110968x_{12} = 64.206689110968
x13=97.9787631115015x_{13} = 97.9787631115015
x14=92.0874236631198x_{14} = 92.0874236631198
x15=−79.913938971999x_{15} = -79.913938971999
x16=−67.7406905449611x_{16} = -67.7406905449611
x17=32.0049165937106x_{17} = 32.0049165937106
x18=−22x_{18} = -22
x19=88x_{19} = 88
x20=35.539378097939x_{20} = 35.539378097939
x21=−89.7319245985772x_{21} = -89.7319245985772
x22=66x_{22} = 66
x23=52.032267211443x_{23} = 52.032267211443
x24=−97.6178269959692x_{24} = -97.6178269959692
x25=12.352172581576x_{25} = 12.352172581576
x26=26.1132692431213x_{26} = 26.1132692431213
x27=22x_{27} = 22
x28=−94.0483079639601x_{28} = -94.0483079639601
x29=−50.0631212262217x_{29} = -50.0631212262217
x30=−57.9230094030261x_{30} = -57.9230094030261
x31=−53.634267240707x_{31} = -53.634267240707
x32=100.326228617535x_{32} = 100.326228617535
x33=−81.7166660167779x_{33} = -81.7166660167779
x34=−47.7135252508252x_{34} = -47.7135252508252
x35=53.9961744173886x_{35} = 53.9961744173886
x36=37.9102905341884x_{36} = 37.9102905341884
x37=−87.9834821668133x_{37} = -87.9834821668133
x38=−55.959906099173x_{38} = -55.959906099173
x39=−77.9510602327158x_{39} = -77.9510602327158
x40=78.3329887296144x_{40} = 78.3329887296144
x41=13.9407962014931x_{41} = 13.9407962014931
x42=−59.7480556116346x_{42} = -59.7480556116346
x43=−11.977533910082x_{43} = -11.977533910082
x44=59.904124264314x_{44} = 59.904124264314
x45=2.16026982189269x_{45} = 2.16026982189269
x46=−96.0147355305124x_{46} = -96.0147355305124
x47=−45.7494489755566x_{47} = -45.7494489755566
x48=−18.2594685868023x_{48} = -18.2594685868023
x49=10.0136810100772x_{49} = 10.0136810100772
x50=40.2516377974999x_{50} = 40.2516377974999
x51=−9.64164615826976x_{51} = -9.64164615826976
x52=−99.9421582198804x_{52} = -99.9421582198804
x53=80.3067739093174x_{53} = 80.3067739093174
x54=−63.8131665502363x_{54} = -63.8131665502363
x55=−28.0701180543067x_{55} = -28.0701180543067
x56=24.1512895630134x_{56} = 24.1512895630134
x57=−62.235021897431x_{57} = -62.235021897431
x58=−3.73089842840719x_{58} = -3.73089842840719
x59=34.345929542388x_{59} = 34.345929542388
x60=70.0963710870434x_{60} = 70.0963710870434
x61=−33.9687294814307x_{61} = -33.9687294814307
x62=−19.8308506170992x_{62} = -19.8308506170992
x63=56.339545752106x_{63} = 56.339545752106
x64=−72.0558536590819x_{64} = -72.0558536590819
x65=−69.7048750161357x_{65} = -69.7048750161357
x66=−52.0322235739787x_{66} = -52.0322235739787
x67=−6.07686303999633x_{67} = -6.07686303999633
x68=−85.8043443268549x_{68} = -85.8043443268549
x69=20.2242801063138x_{69} = 20.2242801063138
x70=15.9166169375401x_{70} = 15.9166169375401
x71=44x_{71} = 44
x72=−35.9320804938058x_{72} = -35.9320804938058
x73=−8.04989652647757x_{73} = -8.04989652647757
x74=−91.6962497054981x_{74} = -91.6962497054981
x75=3.42250408888482x_{75} = 3.42250408888482
x76=48.1056803453238x_{76} = 48.1056803453238
x77=−13.94110449213x_{77} = -13.94110449213
x78=−44x_{78} = -44
x79=−74.0234742375586x_{79} = -74.0234742375586
x80=50.25x_{80} = 50.25
x81=58.3155002488978x_{81} = 58.3155002488978
x82=28.25x_{82} = 28.25
x83=0x_{83} = 0
x84=96.0147373740838x_{84} = 96.0147373740838
x85=18.2603804233025x_{85} = 18.2603804233025
x86=72.25x_{86} = 72.25
x87=−41.8220007305395x_{87} = -41.8220007305395
x88=−37.7483633047803x_{88} = -37.7483633047803
x89=42.2155028832606x_{89} = 42.2155028832606
x90=62.2428750440071x_{90} = 62.2428750440071
x91=81.8980755427455x_{91} = 81.8980755427455
x92=6.25x_{92} = 6.25
x93=−40.2455515993211x_{93} = -40.2455515993211
x94=−15.75x_{94} = -15.75
x95=−23.7582046392524x_{95} = -23.7582046392524
x96=90.8708362699616x_{96} = 90.8708362699616
x97=94.25x_{97} = 94.25
x98=−1.76695964871366x_{98} = -1.76695964871366
x99=74.0234881584279x_{99} = 74.0234881584279
x100=−30.0410117428983x_{100} = -30.0410117428983
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 sin(x)^32*cos(8*x)^5.
sin⁡32(0)cos⁡5(0⋅8)\sin^{32}{\left(0 \right)} \cos^{5}{\left(0 \cdot 8 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(sin⁡32(x)cos⁡5(8x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin^{32}{\left(x \right)} \cos^{5}{\left(8 x \right)}\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→∞(sin⁡32(x)cos⁡5(8x))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin^{32}{\left(x \right)} \cos^{5}{\left(8 x \right)}\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 sin(x)^32*cos(8*x)^5, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡32(x)cos⁡5(8x)x)=0\lim_{x \to -\infty}\left(\frac{\sin^{32}{\left(x \right)} \cos^{5}{\left(8 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡32(x)cos⁡5(8x)x)=0\lim_{x \to \infty}\left(\frac{\sin^{32}{\left(x \right)} \cos^{5}{\left(8 x \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⁡32(x)cos⁡5(8x)=sin⁡32(x)cos⁡5(8x)\sin^{32}{\left(x \right)} \cos^{5}{\left(8 x \right)} = \sin^{32}{\left(x \right)} \cos^{5}{\left(8 x \right)}
- Sí
sin⁡32(x)cos⁡5(8x)=−sin⁡32(x)cos⁡5(8x)\sin^{32}{\left(x \right)} \cos^{5}{\left(8 x \right)} = - \sin^{32}{\left(x \right)} \cos^{5}{\left(8 x \right)}
- No
es decir, función
es
par
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