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Trazado el gráfico de la función/ tan(cos(x/3))^2

Gráfico de la función y = tan(cos(x/3))^2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          2/   /x\\
f(x) = tan |cos|-||
           \   \3//
f(x)=tan⁡2(cos⁡(x3))f{\left(x \right)} = \tan^{2}{\left(\cos{\left(\frac{x}{3} \right)} \right)} 
f = tan(cos(x/3))^2
Gráfico de la función
02468-8-6-4-2-10100.02.5
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:
tan⁡2(cos⁡(x3))=0\tan^{2}{\left(\cos{\left(\frac{x}{3} \right)} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=3π2x_{1} = \frac{3 \pi}{2}
x2=9π2x_{2} = \frac{9 \pi}{2}
Solución numérica
x1=14.1371671123354x_{1} = 14.1371671123354
x2=42.4115024712385x_{2} = 42.4115024712385
x3=61.2610550384439x_{3} = 61.2610550384439
x4=23.5619431455321x_{4} = 23.5619431455321
x5=−80.1106128223901x_{5} = -80.1106128223901
x6=−32.9867213388375x_{6} = -32.9867213388375
x7=−51.8362777290941x_{7} = -51.8362777290941
x8=80.1106114904059x_{8} = 80.1106114904059
x9=14.1371680134254x_{9} = 14.1371680134254
x10=−14.1371673854295x_{10} = -14.1371673854295
x11=−42.4115015367651x_{11} = -42.4115015367651
x12=51.8362789046501x_{12} = 51.8362789046501
x13=−32.9867244588282x_{13} = -32.9867244588282
x14=−14.1371656898992x_{14} = -14.1371656898992
x15=−89.5353897616727x_{15} = -89.5353897616727
x16=89.5353899446061x_{16} = 89.5353899446061
x17=23.5619442039661x_{17} = 23.5619442039661
x18=42.4115007266732x_{18} = 42.4115007266732
x19=80.1106122306522x_{19} = 80.1106122306522
x20=4.71238873176281x_{20} = 4.71238873176281
x21=80.1106137561685x_{21} = 80.1106137561685
x22=51.8362771427437x_{22} = 51.8362771427437
x23=32.9867245996946x_{23} = 32.9867245996946
x24=23.5619463467761x_{24} = 23.5619463467761
x25=−51.8362798509846x_{25} = -51.8362798509846
x26=42.4115015962664x_{26} = 42.4115015962664
x27=−89.5353907518105x_{27} = -89.5353907518105
x28=−70.6858352648691x_{28} = -70.6858352648691
x29=−51.8362775406568x_{29} = -51.8362775406568
x30=−70.6858331149315x_{30} = -70.6858331149315
x31=−89.5353887271248x_{31} = -89.5353887271248
x32=−61.2610560428319x_{32} = -61.2610560428319
x33=−42.4115025773252x_{33} = -42.4115025773252
x34=89.5353909041566x_{34} = 89.5353909041566
x35=−4.71238952505097x_{35} = -4.71238952505097
x36=98.9601691654567x_{36} = 98.9601691654567
x37=70.6858333013811x_{37} = 70.6858333013811
x38=−80.1106125774887x_{38} = -80.1106125774887
x39=−4.71239069587438x_{39} = -4.71239069587438
x40=−23.5619432312135x_{40} = -23.5619432312135
x41=51.8362779233718x_{41} = 51.8362779233718
x42=−61.2610570068176x_{42} = -61.2610570068176
x43=−80.1106114326689x_{43} = -80.1106114326689
x44=−80.1106142553118x_{44} = -80.1106142553118
x45=−4.71238737523619x_{45} = -4.71238737523619
x46=80.1106116338725x_{46} = 80.1106116338725
x47=−98.9601690018223x_{47} = -98.9601690018223
x48=80.1106137559953x_{48} = 80.1106137559953
x49=−42.4114993952305x_{49} = -42.4114993952305
x50=−61.2610581825348x_{50} = -61.2610581825348
x51=89.5353920886397x_{51} = 89.5353920886397
x52=70.685835438829x_{52} = 70.685835438829
x53=89.535388869777x_{53} = 89.535388869777
x54=51.8362800528556x_{54} = 51.8362800528556
x55=51.8362779774554x_{55} = 51.8362779774554
x56=−70.6858343164278x_{56} = -70.6858343164278
x57=−42.4115005734436x_{57} = -42.4115005734436
x58=23.5619461277606x_{58} = 23.5619461277606
x59=−14.1371678182824x_{59} = -14.1371678182824
x60=−32.9867211344839x_{60} = -32.9867211344839
x61=−23.5619436612191x_{61} = -23.5619436612191
x62=70.6858364811593x_{62} = 70.6858364811593
x63=32.9867224762249x_{63} = 32.9867224762249
x64=−14.1371668360491x_{64} = -14.1371668360491
x65=−89.5353918896083x_{65} = -89.5353918896083
x66=−51.8362797907412x_{66} = -51.8362797907412
x67=−32.9867232627682x_{67} = -32.9867232627682
x68=4.71238969795282x_{68} = 4.71238969795282
x69=−23.5619461469325x_{69} = -23.5619461469325
x70=−98.960168054169x_{70} = -98.960168054169
x71=61.2610571617859x_{71} = 61.2610571617859
x72=80.1106131336429x_{72} = 80.1106131336429
x73=−98.9601668693224x_{73} = -98.9601668693224
x74=−89.5353889364298x_{74} = -89.5353889364298
x75=4.71238755935612x_{75} = 4.71238755935612
x76=−32.9867223143758x_{76} = -32.9867223143758
x77=42.4115017218334x_{77} = 42.4115017218334
x78=−61.2610549663306x_{78} = -61.2610549663306
x79=61.2610562146846x_{79} = 61.2610562146846
x80=32.9867234254989x_{80} = 32.9867234254989
x81=98.9601670204536x_{81} = 98.9601670204536
x82=23.5619451653612x_{82} = 23.5619451653612
x83=−70.6858364294829x_{83} = -70.6858364294829
x84=−23.5619440199617x_{84} = -23.5619440199617
x85=42.41149959576x_{85} = 42.41149959576
x86=98.96016821516x_{86} = 98.96016821516
x87=14.1371658914709x_{87} = 14.1371658914709
x88=70.6858344703332x_{88} = 70.6858344703332
x89=−23.5619450126749x_{89} = -23.5619450126749
x90=−4.71238857734901x_{90} = -4.71238857734901
x91=−51.8362786891281x_{91} = -51.8362786891281
x92=−80.1106135600202x_{92} = -80.1106135600202
x93=14.1371681119397x_{93} = 14.1371681119397
x94=61.2610583641203x_{94} = 61.2610583641203
x95=−14.1371685579262x_{95} = -14.1371685579262
x96=4.71239075807493x_{96} = 4.71239075807493
x97=14.1371658451151x_{97} = 14.1371658451151
x98=32.9867212806472x_{98} = 32.9867212806472
x99=14.1371672119326x_{99} = 14.1371672119326
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 tan(cos(x/3))^2.
tan⁡2(cos⁡(03))\tan^{2}{\left(\cos{\left(\frac{0}{3} \right)} \right)}
Resultado:
f(0)=tan⁡2(1)f{\left(0 \right)} = \tan^{2}{\left(1 \right)}
Punto:
(0, tan(1)^2)
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞tan⁡2(cos⁡(x3))=⟨0,tan⁡2(1)⟩\lim_{x \to -\infty} \tan^{2}{\left(\cos{\left(\frac{x}{3} \right)} \right)} = \left\langle 0, \tan^{2}{\left(1 \right)}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨0,tan⁡2(1)⟩y = \left\langle 0, \tan^{2}{\left(1 \right)}\right\rangle
lim⁡x→∞tan⁡2(cos⁡(x3))=⟨0,tan⁡2(1)⟩\lim_{x \to \infty} \tan^{2}{\left(\cos{\left(\frac{x}{3} \right)} \right)} = \left\langle 0, \tan^{2}{\left(1 \right)}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨0,tan⁡2(1)⟩y = \left\langle 0, \tan^{2}{\left(1 \right)}\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función tan(cos(x/3))^2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(tan⁡2(cos⁡(x3))x)=0\lim_{x \to -\infty}\left(\frac{\tan^{2}{\left(\cos{\left(\frac{x}{3} \right)} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(tan⁡2(cos⁡(x3))x)=0\lim_{x \to \infty}\left(\frac{\tan^{2}{\left(\cos{\left(\frac{x}{3} \right)} \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:
tan⁡2(cos⁡(x3))=tan⁡2(cos⁡(x3))\tan^{2}{\left(\cos{\left(\frac{x}{3} \right)} \right)} = \tan^{2}{\left(\cos{\left(\frac{x}{3} \right)} \right)}
- No
tan⁡2(cos⁡(x3))=−tan⁡2(cos⁡(x3))\tan^{2}{\left(\cos{\left(\frac{x}{3} \right)} \right)} = - \tan^{2}{\left(\cos{\left(\frac{x}{3} \right)} \right)}
- No
es decir, función
no es
par ni impar
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