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Trazado el gráfico de la función/ ctg(4x)*cos(4x)

Gráfico de la función y = ctg(4x)*cos(4x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = cot(4*x)*cos(4*x)
f(x)=cos⁡(4x)cot⁡(4x)f{\left(x \right)} = \cos{\left(4 x \right)} \cot{\left(4 x \right)} 
f = cos(4*x)*cot(4*x)
Gráfico de la función
02468-8-6-4-2-1010-500500
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⁡(4x)cot⁡(4x)=0\cos{\left(4 x \right)} \cot{\left(4 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−π8x_{1} = - \frac{\pi}{8}
x2=π8x_{2} = \frac{\pi}{8}
Solución numérica
x1=−27.8816347272427x_{1} = -27.8816347272427
x2=89.928089784421x_{2} = 89.928089784421
x3=48.3019870121468x_{3} = 48.3019870121468
x4=−64.009950305446x_{4} = -64.009950305446
x5=−35.7356164450251x_{5} = -35.7356164450251
x6=49.8727835290591x_{6} = 49.8727835290591
x7=−56.1559684590926x_{7} = -56.1559684590926
x8=45.9457926155531x_{8} = 45.9457926155531
x9=92.2842841747879x_{9} = 92.2842841747879
x10=26.3108384307197x_{10} = 26.3108384307197
x11=67.9369411998163x_{11} = 67.9369411998163
x12=40.4480052052875x_{12} = 40.4480052052875
x13=−67.9369412987172x_{13} = -67.9369412987172
x14=18.4568566055691x_{14} = 18.4568566055691
x15=23.9546440314963x_{15} = 23.9546440314963
x16=52.2289778823416x_{16} = 52.2289778823416
x17=20.0276531264367x_{17} = 20.0276531264367
x18=−1.96349546669253x_{18} = -1.96349546669253
x19=8.24668071815482x_{19} = 8.24668071815482
x20=−16.1006622392305x_{20} = -16.1006622392305
x21=64.0099503042551x_{21} = 64.0099503042551
x22=96.2112750483112x_{22} = 96.2112750483112
x23=−75.7909227335983x_{23} = -75.7909227335983
x24=30.2378293000822x_{24} = 30.2378293000822
x25=−61.653755896837x_{25} = -61.653755896837
x26=27.8816349380003x_{26} = 27.8816349380003
x27=93.8550807184008x_{27} = 93.8550807184008
x28=34.1648202052439x_{28} = 34.1648202052439
x29=−9.81747697702242x_{29} = -9.81747697702242
x30=−13.7444678635245x_{30} = -13.7444678635245
x31=−39.6626073145636x_{31} = -39.6626073145636
x32=44.3749961357343x_{32} = 44.3749961357343
x33=5.89048634868838x_{33} = 5.89048634868838
x34=−31.808625563961x_{34} = -31.808625563961
x35=62.4391538011772x_{35} = 62.4391538011772
x36=56.1559688055632x_{36} = 56.1559688055632
x37=78.1471174281992x_{37} = 78.1471174281992
x38=−82.0741079971979x_{38} = -82.0741079971979
x39=−79.7179136077217x_{39} = -79.7179136077217
x40=−5.89048609822403x_{40} = -5.89048609822403
x41=−57.7267650264066x_{41} = -57.7267650264066
x42=88.3572933026447x_{42} = 88.3572933026447
x43=42.0188017163919x_{43} = 42.0188017163919
x44=−42.0188017207944x_{44} = -42.0188017207944
x45=−17.6714587325842x_{45} = -17.6714587325842
x46=12.1736716122318x_{46} = 12.1736716122318
x47=−86.0010988907069x_{47} = -86.0010988907069
x48=−100.138265656808x_{48} = -100.138265656808
x49=−43.5895982070279x_{49} = -43.5895982070279
x50=−97.7820713170403x_{50} = -97.7820713170403
x51=−53.7997741493363x_{51} = -53.7997741493363
x52=−34.1648198530086x_{52} = -34.1648198530086
x53=74.2201264650387x_{53} = 74.2201264650387
x54=84.430302394159x_{54} = 84.430302394159
x55=22.3838475513636x_{55} = 22.3838475513636
x56=−20.0276531365334x_{56} = -20.0276531365334
x57=−23.9546440649227x_{57} = -23.9546440649227
x58=71.8639321223223x_{58} = 71.8639321223223
x59=−38.0918108260377x_{59} = -38.0918108260377
x60=−60.0829594119627x_{60} = -60.0829594119627
x61=−21.5984496198436x_{61} = -21.5984496198436
x62=4.31968984915916x_{62} = 4.31968984915916
x63=−93.8550803321541x_{63} = -93.8550803321541
x64=86.0010988906926x_{64} = 86.0010988906926
x65=−83.6449044794693x_{65} = -83.6449044794693
x66=−3.5342919993209x_{66} = -3.5342919993209
x67=−65.5807467955783x_{67} = -65.5807467955783
x68=−78.1471170599108x_{68} = -78.1471170599108
x69=70.2931355934875x_{69} = 70.2931355934875
x70=1.96349544751322x_{70} = 1.96349544751322
x71=−45.9457926721836x_{71} = -45.9457926721836
x72=66.3661447194599x_{72} = 66.3661447194599
x73=−87.57189538582x_{73} = -87.57189538582
x74=16.8860606369495x_{74} = 16.8860606369495
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 cot(4*x)*cos(4*x).
cos⁡(0⋅4)cot⁡(0⋅4)\cos{\left(0 \cdot 4 \right)} \cot{\left(0 \cdot 4 \right)}
Resultado:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
signof no cruza Y
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(cos⁡(4x)cot⁡(4x))y = \lim_{x \to -\infty}\left(\cos{\left(4 x \right)} \cot{\left(4 x \right)}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(cos⁡(4x)cot⁡(4x))y = \lim_{x \to \infty}\left(\cos{\left(4 x \right)} \cot{\left(4 x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cot(4*x)*cos(4*x), 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→−∞(cos⁡(4x)cot⁡(4x)x)y = x \lim_{x \to -\infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(4 x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(cos⁡(4x)cot⁡(4x)x)y = x \lim_{x \to \infty}\left(\frac{\cos{\left(4 x \right)} \cot{\left(4 x \right)}}{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:
cos⁡(4x)cot⁡(4x)=−cos⁡(4x)cot⁡(4x)\cos{\left(4 x \right)} \cot{\left(4 x \right)} = - \cos{\left(4 x \right)} \cot{\left(4 x \right)}
- No
cos⁡(4x)cot⁡(4x)=cos⁡(4x)cot⁡(4x)\cos{\left(4 x \right)} \cot{\left(4 x \right)} = \cos{\left(4 x \right)} \cot{\left(4 x \right)}
- No
es decir, función
no es
par ni impar
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