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Trazado el gráfico de la función/ sin^2(x/3)-cos(x/5)+2

Gráfico de la función y = sin^2(x/3)-cos(x/5)+2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          2/x\      /x\    
f(x) = sin |-| - cos|-| + 2
           \3/      \5/
f(x)=(sin⁡2(x3)−cos⁡(x5))+2f{\left(x \right)} = \left(\sin^{2}{\left(\frac{x}{3} \right)} - \cos{\left(\frac{x}{5} \right)}\right) + 2 
f = sin(x/3)^2 - cos(x/5) + 2
Gráfico de la función
02468-8-6-4-2-101004
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⁡2(x3)−cos⁡(x5))+2=0\left(\sin^{2}{\left(\frac{x}{3} \right)} - \cos{\left(\frac{x}{5} \right)}\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/3)^2 - cos(x/5) + 2.
(−cos⁡(05)+sin⁡2(03))+2\left(- \cos{\left(\frac{0}{5} \right)} + \sin^{2}{\left(\frac{0}{3} \right)}\right) + 2
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
2sin⁡(x3)cos⁡(x3)3+sin⁡(x5)5=0\frac{2 \sin{\left(\frac{x}{3} \right)} \cos{\left(\frac{x}{3} \right)}}{3} + \frac{\sin{\left(\frac{x}{5} \right)}}{5} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−79.8725525823422x_{1} = -79.8725525823422
x2=−22.6166405439692x_{2} = -22.6166405439692
x3=−14.3752270253516x_{3} = -14.3752270253516
x4=−43.08439670267x_{4} = -43.08439670267
x5=79.8725525823422x_{5} = 79.8725525823422
x6=60.9228693229623x_{6} = 60.9228693229623
x7=−5.5648558902089x_{7} = -5.5648558902089
x8=65.5070766722175x_{8} = 65.5070766722175
x9=85.7797994162419x_{9} = 85.7797994162419
x10=−748.417380971341x_{10} = -748.417380971341
x11=99.8126354979027x_{11} = 99.8126354979027
x12=−57.3893306393696x_{12} = -57.3893306393696
x13=33.3249102847315x_{13} = 33.3249102847315
x14=−99.8126354979027x_{14} = -99.8126354979027
x15=−36.8584489683242x_{15} = -36.8584489683242
x16=−19.4855370618986x_{16} = -19.4855370618986
x17=94.2477796076938x_{17} = 94.2477796076938
x18=−60.9228693229623x_{18} = -60.9228693229623
x19=0x_{19} = 0
x20=−71.6311390637246x_{20} = -71.6311390637246
x21=57.3893306393696x_{21} = 57.3893306393696
x22=−85.7797994162419x_{22} = -85.7797994162419
x23=−65.5070766722175x_{23} = -65.5070766722175
x24=−51.1633829050238x_{24} = -51.1633829050238
x25=−74.7622425457952x_{25} = -74.7622425457952
x26=28.7407029354763x_{26} = 28.7407029354763
x27=71.6311390637246x_{27} = 71.6311390637246
x28=−28.7407029354763x_{28} = -28.7407029354763
x29=−88.6829237174849x_{29} = -88.6829237174849
x30=−47.1238898038469x_{30} = -47.1238898038469
x31=74.7622425457952x_{31} = 74.7622425457952
x32=14.3752270253516x_{32} = 14.3752270253516
x33=19.4855370618986x_{33} = 19.4855370618986
x34=36.8584489683242x_{34} = 36.8584489683242
x35=−8.46798019145186x_{35} = -8.46798019145186
x36=−33.3249102847315x_{36} = -33.3249102847315
x37=471.238898038469x_{37} = 471.238898038469
x38=8.46798019145186x_{38} = 8.46798019145186
x39=22.6166405439692x_{39} = 22.6166405439692
x40=51.1633829050238x_{40} = 51.1633829050238
x41=5.5648558902089x_{41} = 5.5648558902089
x42=47.1238898038469x_{42} = 47.1238898038469
x43=−94.2477796076938x_{43} = -94.2477796076938
x44=43.08439670267x_{44} = 43.08439670267
x45=88.6829237174849x_{45} = 88.6829237174849
Signos de extremos en los puntos:
(-79.87255258234218, 3.95840235727146)

(-22.616640543969176, 3.09189055929575)

(-14.375227025351618, 3.95840235727146)

(-43.08439670267002, 3.64154709336703)

(79.87255258234218, 3.95840235727146)

(60.92286932296227, 2.05934919395075)

(-5.564855890208903, 2.47940725500054)

(65.50707667221747, 1.16372679891789)

(85.77979941624194, 2.22080689513273)

(-748.4173809713415, 2.47940725500051)

(99.8126354979027, 2.47940725500054)

(-57.3893306393696, 1.61267904410199)

(33.32491028473153, 2.05934919395075)

(-99.8126354979027, 2.47940725500054)

(-36.858448968324204, 1.61267904410199)

(-19.4855370618986, 2.77219080296186)

(94.2477796076938, 1)

(-60.92286932296227, 2.05934919395075)

(0, 1)

(-71.63113906372462, 3.09189055929575)

(57.3893306393696, 1.61267904410199)

(-85.77979941624194, 2.22080689513273)

(-65.50707667221747, 1.16372679891789)

(-51.16338290502378, 3.64154709336703)

(-74.76224254579519, 2.77219080296186)

(28.740702935476335, 1.16372679891789)

(71.63113906372462, 3.09189055929575)

(-28.740702935476335, 1.16372679891789)

(-88.6829237174849, 2.47940725500054)

(-47.1238898038469, 3)

(74.76224254579519, 2.77219080296186)

(14.375227025351618, 3.95840235727146)

(19.4855370618986, 2.77219080296186)

(36.858448968324204, 1.61267904410199)

(-8.467980191451856, 2.22080689513273)

(-33.32491028473153, 2.05934919395075)

(471.23889803846896, 1)

(8.467980191451856, 2.22080689513273)

(22.616640543969176, 3.09189055929575)

(51.16338290502378, 3.64154709336703)

(5.564855890208903, 2.47940725500054)

(47.1238898038469, 3)

(-94.2477796076938, 1)

(43.08439670267002, 3.64154709336703)

(88.6829237174849, 2.47940725500054)


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=65.5070766722175x_{1} = 65.5070766722175
x2=85.7797994162419x_{2} = 85.7797994162419
x3=−57.3893306393696x_{3} = -57.3893306393696
x4=−36.8584489683242x_{4} = -36.8584489683242
x5=−19.4855370618986x_{5} = -19.4855370618986
x6=94.2477796076938x_{6} = 94.2477796076938
x7=0x_{7} = 0
x8=57.3893306393696x_{8} = 57.3893306393696
x9=−85.7797994162419x_{9} = -85.7797994162419
x10=−65.5070766722175x_{10} = -65.5070766722175
x11=−74.7622425457952x_{11} = -74.7622425457952
x12=28.7407029354763x_{12} = 28.7407029354763
x13=−28.7407029354763x_{13} = -28.7407029354763
x14=−47.1238898038469x_{14} = -47.1238898038469
x15=74.7622425457952x_{15} = 74.7622425457952
x16=19.4855370618986x_{16} = 19.4855370618986
x17=36.8584489683242x_{17} = 36.8584489683242
x18=−8.46798019145186x_{18} = -8.46798019145186
x19=471.238898038469x_{19} = 471.238898038469
x20=8.46798019145186x_{20} = 8.46798019145186
x21=47.1238898038469x_{21} = 47.1238898038469
x22=−94.2477796076938x_{22} = -94.2477796076938
Puntos máximos de la función:
x22=−79.8725525823422x_{22} = -79.8725525823422
x22=−22.6166405439692x_{22} = -22.6166405439692
x22=−14.3752270253516x_{22} = -14.3752270253516
x22=−43.08439670267x_{22} = -43.08439670267
x22=79.8725525823422x_{22} = 79.8725525823422
x22=60.9228693229623x_{22} = 60.9228693229623
x22=−5.5648558902089x_{22} = -5.5648558902089
x22=−748.417380971341x_{22} = -748.417380971341
x22=99.8126354979027x_{22} = 99.8126354979027
x22=33.3249102847315x_{22} = 33.3249102847315
x22=−99.8126354979027x_{22} = -99.8126354979027
x22=−60.9228693229623x_{22} = -60.9228693229623
x22=−71.6311390637246x_{22} = -71.6311390637246
x22=−51.1633829050238x_{22} = -51.1633829050238
x22=71.6311390637246x_{22} = 71.6311390637246
x22=−88.6829237174849x_{22} = -88.6829237174849
x22=14.3752270253516x_{22} = 14.3752270253516
x22=−33.3249102847315x_{22} = -33.3249102847315
x22=22.6166405439692x_{22} = 22.6166405439692
x22=51.1633829050238x_{22} = 51.1633829050238
x22=5.5648558902089x_{22} = 5.5648558902089
x22=43.08439670267x_{22} = 43.08439670267
x22=88.6829237174849x_{22} = 88.6829237174849
Decrece en los intervalos
[471.238898038469,∞)\left[471.238898038469, \infty\right)
Crece en los intervalos
(−∞,−94.2477796076938]\left(-\infty, -94.2477796076938\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
−50sin⁡2(x3)+9cos⁡(x5)+50cos⁡2(x3)225=0\frac{- 50 \sin^{2}{\left(\frac{x}{3} \right)} + 9 \cos{\left(\frac{x}{5} \right)} + 50 \cos^{2}{\left(\frac{x}{3} \right)}}{225} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−25.8011072454532x_{1} = -25.8011072454532
x2=−96.8394740843314x_{2} = -96.8394740843314
x3=16.7588035498417x_{3} = 16.7588035498417
x4=−30.9005513505339x_{4} = -30.9005513505339
x5=−59.1038071913449x_{5} = -59.1038071913449
x6=−77.4889760578521x_{6} = -77.4889760578521
x7=−101.271740977886x_{7} = -101.271740977886
x8=−2.59169447663759x_{8} = -2.59169447663759
x9=73.1709255527581x_{9} = 73.1709255527581
x10=−40.0152044282997x_{10} = -40.0152044282997
x11=54.2325751793941x_{11} = 54.2325751793941
x12=−73.1709255527581x_{12} = -73.1709255527581
x13=40.0152044282997x_{13} = 40.0152044282997
x14=−82.6510265676321x_{14} = -82.6510265676321
x15=3343.68744393989x_{15} = 3343.68744393989
x16=200.092312255449x_{16} = 200.092312255449
x17=25.8011072454532x_{17} = 25.8011072454532
x18=87.2238182375017x_{18} = 87.2238182375017
x19=11.5967530400617x_{19} = 11.5967530400617
x20=35.1439724163489x_{20} = 35.1439724163489
x21=96.8394740843314x_{21} = 96.8394740843314
x22=2.59169447663759x_{22} = 2.59169447663759
x23=91.6560851310562x_{23} = 91.6560851310562
x24=45.0151576706041x_{24} = 45.0151576706041
x25=−35.1439724163489x_{25} = -35.1439724163489
x26=−63.3472282571599x_{26} = -63.3472282571599
x27=82.6510265676321x_{27} = 82.6510265676321
x28=7.02396137019208x_{28} = 7.02396137019208
x29=−7.02396137019208x_{29} = -7.02396137019208
x30=30.9005513505339x_{30} = 30.9005513505339
x31=−54.2325751793941x_{31} = -54.2325751793941
x32=−11.5967530400617x_{32} = -11.5967530400617
x33=−49.2326219370897x_{33} = -49.2326219370897
x34=459.642144998407x_{34} = 459.642144998407
x35=−21.0768540549357x_{35} = -21.0768540549357
x36=−91.6560851310562x_{36} = -91.6560851310562
x37=49.2326219370897x_{37} = 49.2326219370897
x38=−87.2238182375017x_{38} = -87.2238182375017
x39=−45.0151576706041x_{39} = -45.0151576706041
x40=59.1038071913449x_{40} = 59.1038071913449
x41=−16.7588035498417x_{41} = -16.7588035498417
x42=68.4466723622406x_{42} = 68.4466723622406
x43=21.0768540549357x_{43} = 21.0768540549357
x44=−68.4466723622406x_{44} = -68.4466723622406
x45=63.3472282571599x_{45} = 63.3472282571599
x46=77.4889760578521x_{46} = 77.4889760578521
x47=101.271740977886x_{47} = 101.271740977886

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
[3343.68744393989,∞)\left[3343.68744393989, \infty\right)
Convexa en los intervalos
(−∞,−96.8394740843314]\left(-\infty, -96.8394740843314\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((sin⁡2(x3)−cos⁡(x5))+2)=⟨1,4⟩\lim_{x \to -\infty}\left(\left(\sin^{2}{\left(\frac{x}{3} \right)} - \cos{\left(\frac{x}{5} \right)}\right) + 2\right) = \left\langle 1, 4\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨1,4⟩y = \left\langle 1, 4\right\rangle
lim⁡x→∞((sin⁡2(x3)−cos⁡(x5))+2)=⟨1,4⟩\lim_{x \to \infty}\left(\left(\sin^{2}{\left(\frac{x}{3} \right)} - \cos{\left(\frac{x}{5} \right)}\right) + 2\right) = \left\langle 1, 4\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨1,4⟩y = \left\langle 1, 4\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x/3)^2 - cos(x/5) + 2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((sin⁡2(x3)−cos⁡(x5))+2x)=0\lim_{x \to -\infty}\left(\frac{\left(\sin^{2}{\left(\frac{x}{3} \right)} - \cos{\left(\frac{x}{5} \right)}\right) + 2}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((sin⁡2(x3)−cos⁡(x5))+2x)=0\lim_{x \to \infty}\left(\frac{\left(\sin^{2}{\left(\frac{x}{3} \right)} - \cos{\left(\frac{x}{5} \right)}\right) + 2}{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⁡2(x3)−cos⁡(x5))+2=sin⁡2(x3)−cos⁡(x5)+2\left(\sin^{2}{\left(\frac{x}{3} \right)} - \cos{\left(\frac{x}{5} \right)}\right) + 2 = \sin^{2}{\left(\frac{x}{3} \right)} - \cos{\left(\frac{x}{5} \right)} + 2
- No
(sin⁡2(x3)−cos⁡(x5))+2=−sin⁡2(x3)+cos⁡(x5)−2\left(\sin^{2}{\left(\frac{x}{3} \right)} - \cos{\left(\frac{x}{5} \right)}\right) + 2 = - \sin^{2}{\left(\frac{x}{3} \right)} + \cos{\left(\frac{x}{5} \right)} - 2
- No
es decir, función
no es
par ni impar
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