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Trazado el gráfico de la función/ sin(0.5*x^2)

Gráfico de la función y = sin(0.5*x^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          / 2\
          |x |
f(x) = sin|--|
          \2 /
f(x)=sin⁡(x22)f{\left(x \right)} = \sin{\left(\frac{x^{2}}{2} \right)} 
f = sin(x^2/2)
Gráfico de la función
05-30-25-20-15-10-530101520252-2
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⁡(x22)=0\sin{\left(\frac{x^{2}}{2} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=0x_{1} = 0
x2=−2πx_{2} = - \sqrt{2} \sqrt{\pi}
x3=2πx_{3} = \sqrt{2} \sqrt{\pi}
Solución numérica
x1=2.506628274631x_{1} = 2.506628274631
x2=88.1250080265186x_{2} = 88.1250080265186
x3=−47.625937217989x_{3} = -47.625937217989
x4=−11.7571287633483x_{4} = -11.7571287633483
x5=−91.9628589348098x_{5} = -91.9628589348098
x6=−48.669917411135x_{6} = -48.669917411135
x7=−33.9090387833961x_{7} = -33.9090387833961
x8=−92.3378712221735x_{8} = -92.3378712221735
x9=−99.9827376286663x_{9} = -99.9827376286663
x10=36.1511070908396x_{10} = 36.1511070908396
x11=14.1796308072441x_{11} = 14.1796308072441
x12=−45.8103033454045x_{12} = -45.8103033454045
x13=−69.9165085583206x_{13} = -69.9165085583206
x14=−81.9939526958065x_{14} = -81.9939526958065
x15=−55.8815094433593x_{15} = -55.8815094433593
x16=−35.8894551083158x_{16} = -35.8894551083158
x17=12.2799204953579x_{17} = 12.2799204953579
x18=−89.7498945058111x_{18} = -89.7498945058111
x19=−39.949080940913x_{19} = -39.949080940913
x20=−118.104493487499x_{20} = -118.104493487499
x21=−7.92665459521202x_{21} = -7.92665459521202
x22=−71.6474870177993x_{22} = -71.6474870177993
x23=4.34160752734961x_{23} = 4.34160752734961
x24=67.3065866500187x_{24} = 67.3065866500187
x25=42.2424505354389x_{25} = 42.2424505354389
x26=−31.9041693162993x_{26} = -31.9041693162993
x27=22.137941502317x_{27} = 22.137941502317
x28=31.9041693162993x_{28} = 31.9041693162993
x29=72.3022141816792x_{29} = 72.3022141816792
x30=53.6441876691991x_{30} = 53.6441876691991
x31=−19.7372107716651x_{31} = -19.7372107716651
x32=−53.7611871269842x_{32} = -53.7611871269842
x33=−29.3393317422808x_{33} = -29.3393317422808
x34=45.7416736389346x_{34} = 45.7416736389346
x35=−67.1664131136608x_{35} = -67.1664131136608
x36=33.7232344326505x_{36} = 33.7232344326505
x37=−65.7482432566799x_{37} = -65.7482432566799
x38=−59.2646655834097x_{38} = -59.2646655834097
x39=−57.7069165075139x_{39} = -57.7069165075139
x40=21.4166413665661x_{40} = 21.4166413665661
x41=−15.853309190424x_{41} = -15.853309190424
x42=−87.875108776739x_{42} = -87.875108776739
x43=−51.4927587106166x_{43} = -51.4927587106166
x44=−28.4698244572419x_{44} = -28.4698244572419
x45=18.2485292908913x_{45} = 18.2485292908913
x46=64.2499240996983x_{46} = 64.2499240996983
x47=−17.7245385090552x_{47} = -17.7245385090552
x48=80.2512612976212x_{48} = 80.2512612976212
x49=40.1843081804969x_{49} = 40.1843081804969
x50=−78.3095706782209x_{50} = -78.3095706782209
x51=62.1118006389146x_{51} = 62.1118006389146
x52=−3.54490770181103x_{52} = -3.54490770181103
x53=10.026513098524x_{53} = 10.026513098524
x54=−85.9228883123468x_{54} = -85.9228883123468
x55=90.1689624399127x_{55} = 90.1689624399127
x56=47.2284954667069x_{56} = 47.2284954667069
x57=−62.4648547245451x_{57} = -62.4648547245451
x58=6.13996024767893x_{58} = 6.13996024767893
x59=16.244807875181x_{59} = 16.244807875181
x60=83.9252687177605x_{60} = 83.9252687177605
x61=−13.7293684929565x_{61} = -13.7293684929565
x62=−92.405891762685x_{62} = -92.405891762685
x63=28.2482660354898x_{63} = 28.2482660354898
x64=58.248777376671x_{64} = 58.248777376671
x65=−64.8824641625942x_{65} = -64.8824641625942
x66=94.0570301067861x_{66} = 94.0570301067861
x67=−94.7558536659326x_{67} = -94.7558536659326
x68=−96.3993312700799x_{68} = -96.3993312700799
x69=66.176885383827x_{69} = 66.176885383827
x70=−67.9569069394495x_{70} = -67.9569069394495
x71=0x_{71} = 0
x72=20.9719567876384x_{72} = 20.9719567876384
x73=38.1798244006657x_{73} = 38.1798244006657
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).
sin⁡(022)\sin{\left(\frac{0^{2}}{2} \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
xcos⁡(x22)=0x \cos{\left(\frac{x^{2}}{2} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=0x_{1} = 0
x2=−πx_{2} = - \sqrt{\pi}
x3=πx_{3} = \sqrt{\pi}
Signos de extremos en los puntos:
(0, 0)

    ____    
(-\/ pi, 1)

   ____    
(\/ pi, 1)


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=0x_{1} = 0
Puntos máximos de la función:
x1=−πx_{1} = - \sqrt{\pi}
x1=πx_{1} = \sqrt{\pi}
Decrece en los intervalos
(−∞,−π]∪[0,∞)\left(-\infty, - \sqrt{\pi}\right] \cup \left[0, \infty\right)
Crece en los intervalos
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\sqrt{\pi}, \infty\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
−x2sin⁡(x22)+cos⁡(x22)=0- x^{2} \sin{\left(\frac{x^{2}}{2} \right)} + \cos{\left(\frac{x^{2}}{2} \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−55.8815151738985x_{1} = -55.8815151738985
x2=83.9252704094543x_{2} = 83.9252704094543
x3=−53.7611935626473x_{3} = -53.7611935626473
x4=−59.9498336293975x_{4} = -59.9498336293975
x5=−95.8764847185019x_{5} = -95.8764847185019
x6=62.1118048121949x_{6} = 62.1118048121949
x7=22.1380336709551x_{7} = 22.1380336709551
x8=−91.449001541881x_{8} = -91.449001541881
x9=40.1843235914693x_{9} = 40.1843235914693
x10=12.2804604524256x_{10} = 12.2804604524256
x11=18.2486938436992x_{11} = 18.2486938436992
x12=−74.2317578759226x_{12} = -74.2317578759226
x13=−11.7577439852518x_{13} = -11.7577439852518
x14=28.3593054588084x_{14} = 28.3593054588084
x15=−17.724718091022x_{15} = -17.724718091022
x16=53.5855986427455x_{16} = 53.5855986427455
x17=43.9911594310076x_{17} = 43.9911594310076
x18=88.1250094876944x_{18} = 88.1250094876944
x19=14.1799815387615x_{19} = 14.1799815387615
x20=−22.6985699325417x_{20} = -22.6985699325417
x21=−16.4372859842373x_{21} = -16.4372859842373
x22=−10.0275049085882x_{22} = -10.0275049085882
x23=−15.8535601598001x_{23} = -15.8535601598001
x24=72.9510680592798x_{24} = 72.9510680592798
x25=−63.9067362597573x_{25} = -63.9067362597573
x26=−1.14304084537203x_{26} = -1.14304084537203
x27=90.8285337917345x_{27} = 90.8285337917345
x28=−30.9038227543119x_{28} = -30.9038227543119
x29=−9.38015585525559x_{29} = -9.38015585525559
x30=1.14304084537203x_{30} = 1.14304084537203
x31=−23.5143344401239x_{31} = -23.5143344401239
x32=45.6729513032366x_{32} = 45.6729513032366
x33=6.14427185889196x_{33} = 6.14427185889196
x34=−57.4340743163298x_{34} = -57.4340743163298
x35=66.1768888343185x_{35} = 66.1768888343185
x36=−16.815183976682x_{36} = -16.815183976682
x37=−19.2539172148114x_{37} = -19.2539172148114
x38=−4.35373074624989x_{38} = -4.35373074624989
x39=29.7645995407721x_{39} = 29.7645995407721
x40=−87.8751102504162x_{40} = -87.8751102504162
x41=−7.52223435955955x_{41} = -7.52223435955955
x42=10.0275049085882x_{42} = 10.0275049085882
x43=−79.4249228230661x_{43} = -79.4249228230661
x44=−2.56605144971105x_{44} = -2.56605144971105
x45=55.8815151738985x_{45} = 55.8815151738985
x46=−33.9090644313446x_{46} = -33.9090644313446
x47=−39.9490966257185x_{47} = -39.9490966257185
x48=85.4462497765434x_{48} = 85.4462497765434
x49=132.09241353182x_{49} = 132.09241353182
x50=−7.92866100102534x_{50} = -7.92866100102534
x51=64.2499278700518x_{51} = 64.2499278700518
x52=82.2617217871396x_{52} = 82.2617217871396
x53=−6.14427185889196x_{53} = -6.14427185889196
x54=9.03913081064472x_{54} = 9.03913081064472
x55=−3.56696520646492x_{55} = -3.56696520646492
x56=−65.7482467750982x_{56} = -65.7482467750982
x57=−71.7789129206483x_{57} = -71.7789129206483
x58=−45.8103137472457x_{58} = -45.8103137472457
x59=−100.98321617812x_{59} = -100.98321617812
x60=5.02115792308416x_{60} = 5.02115792308416
x61=−19.7373408289116x_{61} = -19.7373408289116
x62=62.2633588110138x_{62} = 62.2633588110138
x63=38.1798423685789x_{63} = 38.1798423685789
x64=80.2512632324581x_{64} = 80.2512632324581
x65=58.2487824365421x_{65} = 58.2487824365421
x66=−85.922889888772x_{66} = -85.922889888772
x67=−47.559936866846x_{67} = -47.559936866846
x68=−67.9569101258409x_{68} = -67.9569101258409
x69=42.093460126037x_{69} = 42.093460126037
x70=−28.2483103986713x_{70} = -28.2483103986713
x71=90.1689638039579x_{71} = 90.1689638039579
x72=13.4990129024273x_{72} = 13.4990129024273
x73=−91.9286923371488x_{73} = -91.9286923371488
x74=36.1511282566292x_{74} = 36.1511282566292
x75=4.35373074624989x_{75} = 4.35373074624989
x76=−29.7645995407721x_{76} = -29.7645995407721
x77=−13.7297548721099x_{77} = -13.7297548721099
x78=95.3837121558146x_{78} = 95.3837121558146
x79=−35.8894767404149x_{79} = -35.8894767404149
x80=42.3909437359466x_{80} = 42.3909437359466
x81=46.894730695135x_{81} = 46.894730695135
x82=28.2483103986713x_{82} = 28.2483103986713
x83=−31.9042001096174x_{83} = -31.9042001096174
x84=16.2450411342797x_{84} = 16.2450411342797
x85=16.815183976682x_{85} = 16.815183976682
x86=−97.0489308947314x_{86} = -97.0489308947314
x87=−70.587295953259x_{87} = -70.587295953259
x88=31.805578308395x_{88} = 31.805578308395
x89=3.56696520646492x_{89} = 3.56696520646492
x90=79.8588342038191x_{90} = 79.8588342038191
x91=33.8162900812461x_{91} = 33.8162900812461
x92=82.2999031373535x_{92} = 82.2999031373535
x93=−89.7498958890531x_{93} = -89.7498958890531

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
[132.09241353182,∞)\left[132.09241353182, \infty\right)
Convexa en los intervalos
(−∞,−89.7498958890531]\left(-\infty, -89.7498958890531\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞sin⁡(x22)=⟨−1,1⟩\lim_{x \to -\infty} \sin{\left(\frac{x^{2}}{2} \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞sin⁡(x22)=⟨−1,1⟩\lim_{x \to \infty} \sin{\left(\frac{x^{2}}{2} \right)} = \left\langle -1, 1\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x^2/2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(sin⁡(x22)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{x^{2}}{2} \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(sin⁡(x22)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{x^{2}}{2} \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⁡(x22)=sin⁡(x22)\sin{\left(\frac{x^{2}}{2} \right)} = \sin{\left(\frac{x^{2}}{2} \right)}
- Sí
sin⁡(x22)=−sin⁡(x22)\sin{\left(\frac{x^{2}}{2} \right)} = - \sin{\left(\frac{x^{2}}{2} \right)}
- No
es decir, función
es
par
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