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Trazado el gráfico de la función/ (2+sin((x)/(2)))cos(4x)

Gráfico de la función y = (2+sin((x)/(2)))cos(4x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       /       /x\\         
f(x) = |2 + sin|-||*cos(4*x)
       \       \2//
f(x)=(sin⁡(x2)+2)cos⁡(4x)f{\left(x \right)} = \left(\sin{\left(\frac{x}{2} \right)} + 2\right) \cos{\left(4 x \right)} 
f = (sin(x/2) + 2)*cos(4*x)
Gráfico de la función
02468-8-6-4-2-10105-5
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:
(sin⁡(x2)+2)cos⁡(4x)=0\left(\sin{\left(\frac{x}{2} \right)} + 2\right) \cos{\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=66.3661448070844x_{1} = 66.3661448070844
x2=0.392699081698724x_{2} = 0.392699081698724
x3=93.8550805259951x_{3} = 93.8550805259951
x4=34.164820107789x_{4} = 34.164820107789
x5=−75.0055246044563x_{5} = -75.0055246044563
x6=−96.9966731795849x_{6} = -96.9966731795849
x7=38.0918109247762x_{7} = 38.0918109247762
x8=84.4303025652257x_{8} = 84.4303025652257
x9=67.9369411338793x_{9} = 67.9369411338793
x10=−31.0232274541992x_{10} = -31.0232274541992
x11=−13.7444678594553x_{11} = -13.7444678594553
x12=−34.164820107789x_{12} = -34.164820107789
x13=86.0010988920206x_{13} = 86.0010988920206
x14=−78.1471172580461x_{14} = -78.1471172580461
x15=−56.1559686829176x_{15} = -56.1559686829176
x16=64.009950316892x_{16} = 64.009950316892
x17=−64.7953484802895x_{17} = -64.7953484802895
x18=12.1736715326604x_{18} = 12.1736715326604
x19=−1.96349540849362x_{19} = -1.96349540849362
x20=1.96349540849362x_{20} = 1.96349540849362
x21=53.7997741927252x_{21} = 53.7997741927252
x22=22.3838476568273x_{22} = 22.3838476568273
x23=−31.8086256175967x_{23} = -31.8086256175967
x24=100.138265833175x_{24} = 100.138265833175
x25=75.7909227678538x_{25} = 75.7909227678538
x26=−9.8174770424681x_{26} = -9.8174770424681
x27=82.0741080750334x_{27} = 82.0741080750334
x28=−79.717913584841x_{28} = -79.717913584841
x29=−82.0741080750334x_{29} = -82.0741080750334
x30=−86.0010988920206x_{30} = -86.0010988920206
x31=−35.7356164345839x_{31} = -35.7356164345839
x32=52.2289778659303x_{32} = 52.2289778659303
x33=−93.8550805259951x_{33} = -93.8550805259951
x34=−89.9280897090078x_{34} = -89.9280897090078
x35=−61.6537558266997x_{35} = -61.6537558266997
x36=45.9457925587507x_{36} = 45.9457925587507
x37=−11.388273369263x_{37} = -11.388273369263
x38=−64.009950316892x_{38} = -64.009950316892
x39=−65.5807466436869x_{39} = -65.5807466436869
x40=−27.8816348006094x_{40} = -27.8816348006094
x41=−67.9369411338793x_{41} = -67.9369411338793
x42=−90.7134878724053x_{42} = -90.7134878724053
x43=74.2201264410589x_{43} = 74.2201264410589
x44=−39.6626072515711x_{44} = -39.6626072515711
x45=16.1006623496477x_{45} = 16.1006623496477
x46=−16.1006623496477x_{46} = -16.1006623496477
x47=40.4480054149686x_{47} = 40.4480054149686
x48=48.3019870489431x_{48} = 48.3019870489431
x49=60.0829594999048x_{49} = 60.0829594999048
x50=70.2931356240716x_{50} = 70.2931356240716
x51=9.8174770424681x_{51} = 9.8174770424681
x52=−71.8639319508665x_{52} = -71.8639319508665
x53=20.0276531666349x_{53} = 20.0276531666349
x54=−53.7997741927252x_{54} = -53.7997741927252
x55=8.24668071567321x_{55} = 8.24668071567321
x56=−12.1736715326604x_{56} = -12.1736715326604
x57=42.0188017417635x_{57} = 42.0188017417635
x58=−20.0276531666349x_{58} = -20.0276531666349
x59=−97.7820713429823x_{59} = -97.7820713429823
x60=−60.0829594999048x_{60} = -60.0829594999048
x61=56.1559686829176x_{61} = 56.1559686829176
x62=96.2112750161874x_{62} = 96.2112750161874
x63=88.3572933822129x_{63} = 88.3572933822129
x64=−57.7267650097125x_{64} = -57.7267650097125
x65=−23.9546439836222x_{65} = -23.9546439836222
x66=−75.7909227678538x_{66} = -75.7909227678538
x67=−38.0918109247762x_{67} = -38.0918109247762
x68=30.2378292908018x_{68} = 30.2378292908018
x69=−38.8772090881737x_{69} = -38.8772090881737
x70=78.1471172580461x_{70} = 78.1471172580461
x71=−83.6449044018282x_{71} = -83.6449044018282
x72=−100.138265833175x_{72} = -100.138265833175
x73=49.872783375738x_{73} = 49.872783375738
x74=18.45685683984x_{74} = 18.45685683984
x75=−17.6714586764426x_{75} = -17.6714586764426
x76=−53.0143760293278x_{76} = -53.0143760293278
x77=27.8816348006094x_{77} = 27.8816348006094
x78=−21.5984494934298x_{78} = -21.5984494934298
x79=−87.5718952188155x_{79} = -87.5718952188155
x80=60.8683576633022x_{80} = 60.8683576633022
x81=62.4391539900971x_{81} = 62.4391539900971
x82=92.2842841992002x_{82} = 92.2842841992002
x83=71.8639319508665x_{83} = 71.8639319508665
x84=23.9546439836222x_{84} = 23.9546439836222
x85=110.348441957341x_{85} = 110.348441957341
x86=−43.5895980685584x_{86} = -43.5895980685584
x87=−42.0188017417635x_{87} = -42.0188017417635
x88=−45.9457925587507x_{88} = -45.9457925587507
x89=4.31968989868597x_{89} = 4.31968989868597
x90=−5.89048622548086x_{90} = -5.89048622548086
x91=44.3749962319558x_{91} = 44.3749962319558
x92=89.9280897090078x_{92} = 89.9280897090078
x93=−49.872783375738x_{93} = -49.872783375738
x94=26.3108384738145x_{94} = 26.3108384738145
x95=31.8086256175967x_{95} = 31.8086256175967
x96=5.89048622548086x_{96} = 5.89048622548086
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 (2 + sin(x/2))*cos(4*x).
(sin⁡(02)+2)cos⁡(0⋅4)\left(\sin{\left(\frac{0}{2} \right)} + 2\right) \cos{\left(0 \cdot 4 \right)}
Resultado:
f(0)=2f{\left(0 \right)} = 2
Punto:
(0, 2)
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
−4(sin⁡(x2)+2)sin⁡(4x)+cos⁡(4x)cos⁡(x2)2=0- 4 \left(\sin{\left(\frac{x}{2} \right)} + 2\right) \sin{\left(4 x \right)} + \frac{\cos{\left(4 x \right)} \cos{\left(\frac{x}{2} \right)}}{2} = 0
Resolvermos esta ecuación
Soluciones no halladas,
tal vez la función no tenga extremos
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
−(16(sin⁡(x2)+2)cos⁡(4x)+sin⁡(x2)cos⁡(4x)4+4sin⁡(4x)cos⁡(x2))=0- (16 \left(\sin{\left(\frac{x}{2} \right)} + 2\right) \cos{\left(4 x \right)} + \frac{\sin{\left(\frac{x}{2} \right)} \cos{\left(4 x \right)}}{4} + 4 \sin{\left(4 x \right)} \cos{\left(\frac{x}{2} \right)}) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=93.8275565445474x_{1} = 93.8275565445474
x2=66.3621172399731x_{2} = 66.3621172399731
x3=59.2850601140937x_{3} = 59.2850601140937
x4=0.420223063146415x_{4} = 0.420223063146415
x5=−13.7084800918652x_{5} = -13.7084800918652
x6=64.0299708244177x_{6} = 64.0299708244177
x7=49.9063935313716x_{7} = 49.9063935313716
x8=74.2561142086491x_{8} = 74.2561142086491
x9=56.1284447014699x_{9} = 56.1284447014699
x10=23.9906317512124x_{10} = 23.9906317512124
x11=−27.8691335781958x_{11} = -27.8691335781958
x12=97.794572565396x_{12} = 97.794572565396
x13=88.3848173636606x_{13} = 88.3848173636606
x14=−71.8679595179778x_{14} = -71.8679595179778
x15=18.4293328583923x_{15} = 18.4293328583923
x16=−75.7573126122201x_{16} = -75.7573126122201
x17=−65.5682454212733x_{17} = -65.5682454212733
x18=100.171875988808x_{18} = 100.171875988808
x19=34.9627194936001x_{19} = 34.9627194936001
x20=96.1809252856525x_{20} = 96.1809252856525
x21=82.0404979193997x_{21} = 82.0404979193997
x22=40.4520329820798x_{22} = 40.4520329820798
x23=27.1083096166953x_{23} = 27.1083096166953
x24=16.8739875335618x_{24} = 16.8739875335618
x25=−67.9729289014695x_{25} = -67.9729289014695
x26=−49.8452593942903x_{26} = -49.8452593942903
x27=42.0067287622802x_{27} = 42.0067287622802
x28=38.1193349062239x_{28} = 38.1193349062239
x29=22.3963488792409x_{29} = 22.3963488792409
x30=75.8184467493015x_{30} = 75.8184467493015
x31=78.1511448251574x_{31} = 78.1511448251574
x32=−97.7780437758711x_{32} = -97.7780437758711
x33=30.2178087832761x_{33} = 30.2178087832761
x34=19.9916653990448x_{34} = 19.9916653990448
x35=−52.1986281353954x_{35} = -52.1986281353954
x36=86.0314486225555x_{36} = 86.0314486225555
x37=−68.7559494529104x_{37} = -68.7559494529104
x38=−61.6337353191741x_{38} = -61.6337353191741
x39=−16.1131635720613x_{39} = -16.1131635720613
x40=−35.7235434551006x_{40} = -35.7235434551006
x41=−744.152258596669x_{41} = -744.152258596669
x42=60.0954607223184x_{42} = 60.0954607223184
x43=−63.9739625493019x_{43} = -63.9739625493019
x44=9.82997826488173x_{44} = 9.82997826488173
x45=8.21633098513826x_{45} = 8.21633098513826
x46=7.42529478468558x_{46} = 7.42529478468558
x47=−9.81344947535686x_{47} = -9.81344947535686
x48=−27.065886906677x_{48} = -27.065886906677
x49=−85.9890259125373x_{49} = -85.9890259125373
x50=48.332336779478x_{50} = 48.332336779478
x51=−89.8977399784729x_{51} = -89.8977399784729
x52=16.0966347825364x_{52} = 16.0966347825364
x53=70.2571478564814x_{53} = 70.2571478564814
x54=12.2072816882941x_{54} = 12.2072816882941
x55=−79.748263315376x_{55} = -79.748263315376
x56=−31.8361495990443x_{56} = -31.8361495990443
x57=−56.1895788385512x_{57} = -56.1895788385512
x58=−83.6569773813116x_{58} = -83.6569773813116
x59=−20.0476736741606x_{59} = -20.0476736741606
x60=−74.2001059335332x_{60} = -74.2001059335332
x61=−90.7009866499916x_{61} = -90.7009866499916
x62=5.86296224403317x_{62} = 5.86296224403317
x63=1.97556838797694x_{63} = 1.97556838797694
x64=89.9401626884912x_{64} = 89.9401626884912
x65=45.9154428282158x_{65} = 45.9154428282158
x66=71.8514307284529x_{66} = 71.8514307284529
x67=−39.6322575210362x_{67} = -39.6322575210362
x68=−12.1461475512128x_{68} = -12.1461475512128
x69=52.2410508454136x_{69} = 52.2410508454136
x70=−5.92409638111454x_{70} = -5.92409638111454
x71=−34.1688476749002x_{71} = -34.1688476749002
x72=−60.0789319327936x_{72} = -60.0789319327936
x73=−78.1346160356325x_{73} = -78.1346160356325
x74=53.018403596439x_{74} = 53.018403596439
x75=−82.101632056481x_{75} = -82.101632056481
x76=−42.0491514722984x_{76} = -42.0491514722984
x77=−53.8122754151388x_{77} = -53.8122754151388
x78=−57.7467855172381x_{78} = -57.7467855172381
x79=−3.54679295770215x_{79} = -3.54679295770215
x80=44.3413860763222x_{80} = 44.3413860763222
x81=−43.6232082241921x_{81} = -43.6232082241921
x82=−1.93314567795867x_{82} = -1.93314567795867
x83=4.30761691920265x_{83} = 4.30761691920265
x84=92.2722112197169x_{84} = 92.2722112197169
x85=−17.7074464440328x_{85} = -17.7074464440328
x86=−45.957865538234x_{86} = -45.957865538234
x87=26.3308589813402x_{87} = 26.3308589813402
x88=−38.0582007691426x_{88} = -38.0582007691426
x89=67.9169206263536x_{89} = 67.9169206263536
x90=34.1523188853754x_{90} = 34.1523188853754
x91=−21.6024770605411x_{91} = -21.6024770605411
x92=−100.110741851727x_{92} = -100.110741851727
x93=84.4178013428121x_{93} = 84.4178013428121
x94=27.8856623677207x_{94} = 27.8856623677207
x95=−93.8886906816288x_{95} = -93.8886906816288
x96=−23.9346234760965x_{96} = -23.9346234760965

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
[97.794572565396,∞)\left[97.794572565396, \infty\right)
Convexa en los intervalos
(−∞,−744.152258596669]\left(-\infty, -744.152258596669\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((sin⁡(x2)+2)cos⁡(4x))=⟨−3,3⟩\lim_{x \to -\infty}\left(\left(\sin{\left(\frac{x}{2} \right)} + 2\right) \cos{\left(4 x \right)}\right) = \left\langle -3, 3\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
lim⁡x→∞((sin⁡(x2)+2)cos⁡(4x))=⟨−3,3⟩\lim_{x \to \infty}\left(\left(\sin{\left(\frac{x}{2} \right)} + 2\right) \cos{\left(4 x \right)}\right) = \left\langle -3, 3\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (2 + sin(x/2))*cos(4*x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((sin⁡(x2)+2)cos⁡(4x)x)=0\lim_{x \to -\infty}\left(\frac{\left(\sin{\left(\frac{x}{2} \right)} + 2\right) \cos{\left(4 x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((sin⁡(x2)+2)cos⁡(4x)x)=0\lim_{x \to \infty}\left(\frac{\left(\sin{\left(\frac{x}{2} \right)} + 2\right) \cos{\left(4 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⁡(x2)+2)cos⁡(4x)=(2−sin⁡(x2))cos⁡(4x)\left(\sin{\left(\frac{x}{2} \right)} + 2\right) \cos{\left(4 x \right)} = \left(2 - \sin{\left(\frac{x}{2} \right)}\right) \cos{\left(4 x \right)}
- No
(sin⁡(x2)+2)cos⁡(4x)=−(2−sin⁡(x2))cos⁡(4x)\left(\sin{\left(\frac{x}{2} \right)} + 2\right) \cos{\left(4 x \right)} = - \left(2 - \sin{\left(\frac{x}{2} \right)}\right) \cos{\left(4 x \right)}
- No
es decir, función
no es
par ni impar
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