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Trazado el gráfico de la función/ (sin(x/2))^2((-(cos(3x))^2))^1/2+2

Gráfico de la función y = (sin(x/2))^2((-(cos(3x))^2))^1/2+2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                  ____________    
          2/x\   /     2          
f(x) = sin |-|*\/  -cos (3*x)  + 2
           \2/
f(x)=−cos⁡2(3x)sin⁡2(x2)+2f{\left(x \right)} = \sqrt{- \cos^{2}{\left(3 x \right)}} \sin^{2}{\left(\frac{x}{2} \right)} + 2 
f = sqrt(-cos(3*x)^2)*sin(x/2)^2 + 2
Gráfico de la función
02468-8-6-4-2-10100.02-0.02
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⁡2(3x)sin⁡2(x2)+2=0\sqrt{- \cos^{2}{\left(3 x \right)}} \sin^{2}{\left(\frac{x}{2} \right)} + 2 = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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/2)^2*sqrt(-cos(3*x)^2) + 2.
−cos⁡2(0⋅3)sin⁡2(02)+2\sqrt{- \cos^{2}{\left(0 \cdot 3 \right)}} \sin^{2}{\left(\frac{0}{2} \right)} + 2
Resultado:
f(0)=2f{\left(0 \right)} = 2
Punto:
(0, 2)
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
i(9sin⁡2(x2)sin⁡2(3x)sign⁡(cos⁡(3x))cos⁡(3x)−9sin⁡2(x2)sin⁡2(3x)∣cos⁡(3x)∣cos⁡2(3x)−19sin⁡2(x2)∣cos⁡(3x)∣2−3sin⁡(x2)sin⁡(3x)cos⁡(x2)sign⁡(cos⁡(3x))−3sin⁡(x2)sin⁡(3x)cos⁡(x2)∣cos⁡(3x)∣cos⁡(3x)+cos⁡2(x2)∣cos⁡(3x)∣2)=0i \left(\frac{9 \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(3 x \right)} \operatorname{sign}{\left(\cos{\left(3 x \right)} \right)}}{\cos{\left(3 x \right)}} - \frac{9 \sin^{2}{\left(\frac{x}{2} \right)} \sin^{2}{\left(3 x \right)} \left|{\cos{\left(3 x \right)}}\right|}{\cos^{2}{\left(3 x \right)}} - \frac{19 \sin^{2}{\left(\frac{x}{2} \right)} \left|{\cos{\left(3 x \right)}}\right|}{2} - 3 \sin{\left(\frac{x}{2} \right)} \sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)} \operatorname{sign}{\left(\cos{\left(3 x \right)} \right)} - \frac{3 \sin{\left(\frac{x}{2} \right)} \sin{\left(3 x \right)} \cos{\left(\frac{x}{2} \right)} \left|{\cos{\left(3 x \right)}}\right|}{\cos{\left(3 x \right)}} + \frac{\cos^{2}{\left(\frac{x}{2} \right)} \left|{\cos{\left(3 x \right)}}\right|}{2}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=17.9734625556164x_{1} = 17.9734625556164
x2=−12.7650465573008x_{2} = -12.7650465573008
x3=37.8977877860192x_{3} = 37.8977877860192
x4=0.198675942941671x_{4} = 0.198675942941671
x5=−24.256647862796x_{5} = -24.256647862796
x6=−55.672574398694x_{6} = -55.672574398694
x7=−87.7659183575725x_{7} = -87.7659183575725
x8=−60.1633453220313x_{8} = -60.1633453220313
x9=−89.7040076030613x_{9} = -89.7040076030613
x10=−69.9911317448978x_{10} = -69.9911317448978
x11=−95.9871929102409x_{11} = -95.9871929102409
x12=−99.654871548951x_{12} = -99.654871548951
x13=37.5004359001358x_{13} = 37.5004359001358
x14=−79.9419956907876x_{14} = -79.9419956907876
x15=−26.0088345946407x_{15} = -26.0088345946407
x16=−81.8800849362763x_{16} = -81.8800849362763
x17=−43.7836212073154x_{17} = -43.7836212073154
x18=−1.73941330254706x_{18} = -1.73941330254706
x19=53.8801600148517x_{19} = 53.8801600148517
x20=9.89786286459462x_{20} = 9.89786286459462
x21=−93.3716862417715x_{21} = -93.3716862417715
x22=−17.9734625556164x_{22} = -17.9734625556164
x23=100.332288971932x_{23} = 100.332288971932
x24=26.0088345946407x_{24} = 26.0088345946407
x25=−27.8012489784829x_{25} = -27.8012489784829
x26=32.2920199018203x_{26} = 32.2920199018203
x27=−29.6765132333509x_{27} = -29.6765132333509
x28=86.2251809979671x_{28} = 86.2251809979671
x29=50.066806514495x_{29} = 50.066806514495
x30=−88.8406876664365x_{30} = -88.8406876664365
x31=−271.053061574645x_{31} = -271.053061574645
x32=−11.6902772484368x_{32} = -11.6902772484368
x33=68.2389450130531x_{33} = 68.2389450130531
x34=−68.2389450130531x_{34} = -68.2389450130531
x35=−188.694235158329x_{35} = -188.694235158329
x36=−50.066806514495x_{36} = -50.066806514495
x37=−75.5968996290967x_{37} = -75.5968996290967
x38=52.0048957599838x_{38} = 52.0048957599838
x39=94.0491036647521x_{39} = 94.0491036647521
x40=42.24288384771x_{40} = 42.24288384771
x41=31.2172505929563x_{41} = 31.2172505929563
x42=6.48186125012126x_{42} = 6.48186125012126
x43=−31.6146024788396x_{43} = -31.6146024788396
x44=56.3499918216746x_{44} = 56.3499918216746
x45=−38.5752052089998x_{45} = -38.5752052089998
x46=75.5968996290967x_{46} = 75.5968996290967
x47=60.1633453220313x_{47} = 60.1633453220313
x48=40.3676195928421x_{48} = 40.3676195928421
x49=−34.0844342856625x_{49} = -34.0844342856625
x50=−125.862382086533x_{50} = -125.862382086533
x51=68.9163624360338x_{51} = 68.9163624360338
x52=−37.8977877860192x_{52} = -37.8977877860192
x53=19.0482318644804x_{53} = 19.0482318644804
x54=−8.02259860972665x_{54} = -8.02259860972665
x55=26.8721545312654x_{55} = 26.8721545312654
x56=12.3676946714175x_{56} = 12.3676946714175
x57=−31.2172505929563x_{57} = -31.2172505929563
x58=−94.4464555506355x_{58} = -94.4464555506355
x59=76.2743170520774x_{59} = 76.2743170520774
x60=−6.08450936423792x_{60} = -6.08450936423792
x61=−73.658810383608x_{61} = -73.658810383608
x62=61.9557597058735x_{62} = 61.9557597058735
x63=50.4641584003784x_{63} = 50.4641584003784
x64=88.8406876664365x_{64} = 88.8406876664365
x65=81.8800849362763x_{65} = 81.8800849362763
x66=−61.9557597058735x_{66} = -61.9557597058735
x67=69.9911317448978x_{67} = 69.9911317448978
x68=6.08450936423792x_{68} = 6.08450936423792
x69=24.256647862796x_{69} = 24.256647862796
x70=97.8624571651088x_{70} = 97.8624571651088
x71=8.02259860972665x_{71} = 8.02259860972665
x72=−78.0667314359196x_{72} = -78.0667314359196
x73=79.9419956907876x_{73} = 79.9419956907876
x74=−52.0048957599838x_{74} = -52.0048957599838
x75=63.0305290147375x_{75} = 63.0305290147375
x76=58.2880810671633x_{76} = 58.2880810671633
x77=44.1809730931988x_{77} = 44.1809730931988
x78=−43.1062037843348x_{78} = -43.1062037843348
x79=−94.0491036647521x_{79} = -94.0491036647521
x80=−71.78354612874x_{80} = -71.78354612874
x81=35.9596985405305x_{81} = 35.9596985405305
x82=16.1810481717742x_{82} = 16.1810481717742
x83=84.3499167430992x_{83} = 84.3499167430992
x84=−6.48186125012126x_{84} = -6.48186125012126
x85=14.3057839169062x_{85} = 14.3057839169062
x86=−19.7256492874611x_{86} = -19.7256492874611
x87=95.9871929102409x_{87} = 95.9871929102409
x88=94.4464555506355x_{88} = 94.4464555506355
x89=−35.9596985405305x_{89} = -35.9596985405305
x90=−45.7217104528042x_{90} = -45.7217104528042
x91=88.1632702434559x_{91} = 88.1632702434559
x92=−63.7079464377182x_{92} = -63.7079464377182

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:
No tiene corvaduras en todo el eje numérico
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−cos⁡2(3x)sin⁡2(x2)+2)=⟨0,1⟩i∣⟨−1,1⟩∣+2\lim_{x \to -\infty}\left(\sqrt{- \cos^{2}{\left(3 x \right)}} \sin^{2}{\left(\frac{x}{2} \right)} + 2\right) = \left\langle 0, 1\right\rangle i \left|{\left\langle -1, 1\right\rangle}\right| + 2
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨0,1⟩i∣⟨−1,1⟩∣+2y = \left\langle 0, 1\right\rangle i \left|{\left\langle -1, 1\right\rangle}\right| + 2
lim⁡x→∞(−cos⁡2(3x)sin⁡2(x2)+2)=⟨0,1⟩i∣⟨−1,1⟩∣+2\lim_{x \to \infty}\left(\sqrt{- \cos^{2}{\left(3 x \right)}} \sin^{2}{\left(\frac{x}{2} \right)} + 2\right) = \left\langle 0, 1\right\rangle i \left|{\left\langle -1, 1\right\rangle}\right| + 2
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨0,1⟩i∣⟨−1,1⟩∣+2y = \left\langle 0, 1\right\rangle i \left|{\left\langle -1, 1\right\rangle}\right| + 2
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x/2)^2*sqrt(-cos(3*x)^2) + 2, dividida por x con x->+oo y x ->-oo
No se ha logrado calcular el límite a la izquierda
lim⁡x→−∞(−cos⁡2(3x)sin⁡2(x2)+2x)\lim_{x \to -\infty}\left(\frac{\sqrt{- \cos^{2}{\left(3 x \right)}} \sin^{2}{\left(\frac{x}{2} \right)} + 2}{x}\right)
No se ha logrado calcular el límite a la derecha
lim⁡x→∞(−cos⁡2(3x)sin⁡2(x2)+2x)\lim_{x \to \infty}\left(\frac{\sqrt{- \cos^{2}{\left(3 x \right)}} \sin^{2}{\left(\frac{x}{2} \right)} + 2}{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⁡2(3x)sin⁡2(x2)+2=i∣cos⁡(3x)∣sin⁡2(x2)+2\sqrt{- \cos^{2}{\left(3 x \right)}} \sin^{2}{\left(\frac{x}{2} \right)} + 2 = i \left|{\cos{\left(3 x \right)}}\right| \sin^{2}{\left(\frac{x}{2} \right)} + 2
- No
−cos⁡2(3x)sin⁡2(x2)+2=−i∣cos⁡(3x)∣sin⁡2(x2)−2\sqrt{- \cos^{2}{\left(3 x \right)}} \sin^{2}{\left(\frac{x}{2} \right)} + 2 = - i \left|{\cos{\left(3 x \right)}}\right| \sin^{2}{\left(\frac{x}{2} \right)} - 2
- No
es decir, función
no es
par ni impar
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