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Trazado el gráfico de la función/ cos(x)-(1+2x^2)^(1/4)

Gráfico de la función y = cos(x)-(1+2x^2)^(1/4)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
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                4 /        2 
f(x) = cos(x) - \/  1 + 2*x
f(x)=−2x2+14+cos⁡(x)f{\left(x \right)} = - \sqrt[4]{2 x^{2} + 1} + \cos{\left(x \right)} 
f = -(2*x^2 + 1)^(1/4) + cos(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:
−2x2+14+cos⁡(x)=0- \sqrt[4]{2 x^{2} + 1} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=0x_{1} = 0
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 cos(x) - (1 + 2*x^2)^(1/4).
−2⋅02+14+cos⁡(0)- \sqrt[4]{2 \cdot 0^{2} + 1} + \cos{\left(0 \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
−x(2x2+1)34−sin⁡(x)=0- \frac{x}{\left(2 x^{2} + 1\right)^{\frac{3}{4}}} - \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=72.326599403104x_{1} = 72.326599403104
x2=43.8924441386092x_{2} = 43.8924441386092
x3=−22.1178220437292x_{3} = -22.1178220437292
x4=−43.8924441386092x_{4} = -43.8924441386092
x5=0x_{5} = 0
x6=−100.471611637434x_{6} = -100.471611637434
x7=−47.2105218186528x_{7} = -47.2105218186528
x8=−91.1684982214444x_{8} = -91.1684982214444
x9=−72.326599403104x_{9} = -72.326599403104
x10=87.9011342047635x_{10} = 87.9011342047635
x11=18.7118099053195x_{11} = 18.7118099053195
x12=66.0466696283717x_{12} = 66.0466696283717
x13=37.6020185275753x_{13} = 37.6020185275753
x14=−18.7118099053195x_{14} = -18.7118099053195
x15=−17131.1092829573x_{15} = -17131.1092829573
x16=−6.04137871999288x_{16} = -6.04137871999288
x17=−78.6069278159226x_{17} = -78.6069278159226
x18=3.45691388964406x_{18} = 3.45691388964406
x19=−50.1814586542191x_{19} = -50.1814586542191
x20=−69.0434234652544x_{20} = -69.0434234652544
x21=97.4496397606833x_{21} = 97.4496397606833
x22=69.0434234652544x_{22} = 69.0434234652544
x23=53.4884555510157x_{23} = 53.4884555510157
x24=−9.61692286699886x_{24} = -9.61692286699886
x25=31.3095015833362x_{25} = 31.3095015833362
x26=12.3970984770679x_{26} = 12.3970984770679
x27=15.8576157177088x_{27} = 15.8576157177088
x28=−62.756731521076x_{28} = -62.756731521076
x29=91.1684982214444x_{29} = 91.1684982214444
x30=−31.3095015833362x_{30} = -31.3095015833362
x31=81.6155476438615x_{31} = 81.6155476438615
x32=56.4694680335143x_{32} = 56.4694680335143
x33=28.3861172747639x_{33} = 28.3861172747639
x34=62.756731521076x_{34} = 62.756731521076
x35=−12.3970984770679x_{35} = -12.3970984770679
x36=−53.4884555510157x_{36} = -53.4884555510157
x37=25.0136427998438x_{37} = 25.0136427998438
x38=1492.27190336053x_{38} = 1492.27190336053
x39=−94.1864758735148x_{39} = -94.1864758735148
x40=−34.6586600758426x_{40} = -34.6586600758426
x41=−25.0136427998438x_{41} = -25.0136427998438
x42=78.6069278159226x_{42} = 78.6069278159226
x43=75.3296660167847x_{43} = 75.3296660167847
x44=1957.22566383068x_{44} = 1957.22566383068
x45=113.041382621405x_{45} = 113.041382621405
x46=50.1814586542191x_{46} = 50.1814586542191
x47=59.7672406707413x_{47} = 59.7672406707413
x48=−40.9337545058814x_{48} = -40.9337545058814
x49=−15.8576157177088x_{49} = -15.8576157177088
x50=−56.4694680335143x_{50} = -56.4694680335143
x51=−87.9011342047635x_{51} = -87.9011342047635
x52=−106.756572312523x_{52} = -106.756572312523
x53=−59.7672406707413x_{53} = -59.7672406707413
x54=9.61692286699886x_{54} = 9.61692286699886
x55=−28.3861172747639x_{55} = -28.3861172747639
x56=−66.0466696283717x_{56} = -66.0466696283717
x57=94.1864758735148x_{57} = 94.1864758735148
x58=22.1178220437292x_{58} = 22.1178220437292
x59=34.6586600758426x_{59} = 34.6586600758426
x60=−97.4496397606833x_{60} = -97.4496397606833
x61=−634.578109889878x_{61} = -634.578109889878
x62=−37.6020185275753x_{62} = -37.6020185275753
x63=6.04137871999288x_{63} = 6.04137871999288
x64=−3.45691388964406x_{64} = -3.45691388964406
x65=100.471611637434x_{65} = 100.471611637434
x66=40.9337545058814x_{66} = 40.9337545058814
x67=−84.8875796672537x_{67} = -84.8875796672537
x68=84.8875796672537x_{68} = 84.8875796672537
x69=−75.3296660167847x_{69} = -75.3296660167847
x70=47.2105218186528x_{70} = 47.2105218186528
x71=−81.6155476438615x_{71} = -81.6155476438615
x72=−125.610628953851x_{72} = -125.610628953851
Signos de extremos en los puntos:
(72.32659940310404, -11.1114123157025)

(43.89244413860915, -6.88320561309394)

(-22.117822043729237, -6.58620830138484)

(-43.89244413860915, -6.88320561309394)

(0, 0)

(-100.47161163743417, -10.9219888524059)

(-47.21052181865279, -9.16773978547441)

(-91.16849822144442, -12.3530407697535)

(-72.32659940310404, -11.1114123157025)

(87.90113420476347, -10.1516765096205)

(18.711809905319484, -4.15547852759428)

(66.04666962837173, -10.6621767088309)

(37.602018527575254, -6.29763013718237)

(-18.711809905319484, -4.15547852759428)

(-17131.109282957324, -156.650389960155)

(-6.04137871999288, -1.9620305827026)

(-78.6069278159226, -11.5415376131808)

(3.456913889644064, -3.18453673041613)

(-50.18145865421914, -7.42815518860757)

(-69.04342346525443, -8.88422639467584)

(97.44963976068325, -12.7377854031322)

(69.04342346525443, -8.88422639467584)

(53.48845555101569, -9.6944320358275)

(-9.616922866998857, -4.67444089933777)

(31.309501583336207, -5.66070193457465)

(12.397098477067875, -3.20483416558351)

(15.857615717708844, -5.72679083611434)

(-62.75673152107604, -8.42391653795878)

(91.16849822144442, -12.3530407697535)

(-31.309501583336207, -5.66070193457465)

(81.61554764386152, -9.7458241672997)

(56.469468033514325, -7.93992089817508)

(28.38611727476391, -7.33067322958811)

(62.75673152107604, -8.42391653795878)

(-12.397098477067875, -3.20483416558351)

(-53.48845555101569, -9.6944320358275)

(25.01364279984378, -4.95592948219582)

(1492.2719033605251, -46.9388780530851)

(-94.1864758735148, -10.5432625133873)

(-34.658660075842604, -7.99667132692243)

(-25.01364279984378, -4.95592948219582)

(78.6069278159226, -11.5415376131808)

(75.3296660167847, -9.32402192041903)

(1957.2256638306803, -53.6110801537957)

(113.04138262140516, -11.6454482209407)

(50.18145865421914, -7.42815518860757)

(59.767240670741266, -10.1910342217948)

(-40.93375450588142, -8.60472827094001)

(-15.857615717708844, -5.72679083611434)

(-56.469468033514325, -7.93992089817508)

(-87.90113420476347, -10.1516765096205)

(-106.75657231252316, -11.2890451625389)

(-59.767240670741266, -10.1910342217948)

(9.616922866998857, -4.67444089933777)

(-28.38611727476391, -7.33067322958811)

(-66.04666962837173, -10.6621767088309)

(94.1864758735148, -10.5432625133873)

(22.117822043729237, -6.58620830138484)

(34.658660075842604, -7.99667132692243)

(-97.44963976068325, -12.7377854031322)

(-634.5781098898784, -28.9574067542419)

(-37.602018527575254, -6.29763013718237)

(6.04137871999288, -1.9620305827026)

(-3.456913889644064, -3.18453673041613)

(100.47161163743417, -10.9219888524059)

(40.93375450588142, -8.60472827094001)

(-84.88757966725373, -11.954800674954)

(84.88757966725373, -11.954800674954)

(-75.3296660167847, -9.32402192041903)

(47.21052181865279, -9.16773978547441)

(-81.61554764386152, -9.7458241672997)

(-125.61062895385105, -12.3296890864909)


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=72.326599403104x_{1} = 72.326599403104
x2=−22.1178220437292x_{2} = -22.1178220437292
x3=−47.2105218186528x_{3} = -47.2105218186528
x4=−91.1684982214444x_{4} = -91.1684982214444
x5=−72.326599403104x_{5} = -72.326599403104
x6=66.0466696283717x_{6} = 66.0466696283717
x7=−17131.1092829573x_{7} = -17131.1092829573
x8=−78.6069278159226x_{8} = -78.6069278159226
x9=3.45691388964406x_{9} = 3.45691388964406
x10=97.4496397606833x_{10} = 97.4496397606833
x11=53.4884555510157x_{11} = 53.4884555510157
x12=−9.61692286699886x_{12} = -9.61692286699886
x13=15.8576157177088x_{13} = 15.8576157177088
x14=91.1684982214444x_{14} = 91.1684982214444
x15=28.3861172747639x_{15} = 28.3861172747639
x16=−53.4884555510157x_{16} = -53.4884555510157
x17=1492.27190336053x_{17} = 1492.27190336053
x18=−34.6586600758426x_{18} = -34.6586600758426
x19=78.6069278159226x_{19} = 78.6069278159226
x20=1957.22566383068x_{20} = 1957.22566383068
x21=59.7672406707413x_{21} = 59.7672406707413
x22=−40.9337545058814x_{22} = -40.9337545058814
x23=−15.8576157177088x_{23} = -15.8576157177088
x24=−59.7672406707413x_{24} = -59.7672406707413
x25=9.61692286699886x_{25} = 9.61692286699886
x26=−28.3861172747639x_{26} = -28.3861172747639
x27=−66.0466696283717x_{27} = -66.0466696283717
x28=22.1178220437292x_{28} = 22.1178220437292
x29=34.6586600758426x_{29} = 34.6586600758426
x30=−97.4496397606833x_{30} = -97.4496397606833
x31=−3.45691388964406x_{31} = -3.45691388964406
x32=40.9337545058814x_{32} = 40.9337545058814
x33=−84.8875796672537x_{33} = -84.8875796672537
x34=84.8875796672537x_{34} = 84.8875796672537
x35=47.2105218186528x_{35} = 47.2105218186528
Puntos máximos de la función:
x35=43.8924441386092x_{35} = 43.8924441386092
x35=−43.8924441386092x_{35} = -43.8924441386092
x35=0x_{35} = 0
x35=−100.471611637434x_{35} = -100.471611637434
x35=87.9011342047635x_{35} = 87.9011342047635
x35=18.7118099053195x_{35} = 18.7118099053195
x35=37.6020185275753x_{35} = 37.6020185275753
x35=−18.7118099053195x_{35} = -18.7118099053195
x35=−6.04137871999288x_{35} = -6.04137871999288
x35=−50.1814586542191x_{35} = -50.1814586542191
x35=−69.0434234652544x_{35} = -69.0434234652544
x35=69.0434234652544x_{35} = 69.0434234652544
x35=31.3095015833362x_{35} = 31.3095015833362
x35=12.3970984770679x_{35} = 12.3970984770679
x35=−62.756731521076x_{35} = -62.756731521076
x35=−31.3095015833362x_{35} = -31.3095015833362
x35=81.6155476438615x_{35} = 81.6155476438615
x35=56.4694680335143x_{35} = 56.4694680335143
x35=62.756731521076x_{35} = 62.756731521076
x35=−12.3970984770679x_{35} = -12.3970984770679
x35=25.0136427998438x_{35} = 25.0136427998438
x35=−94.1864758735148x_{35} = -94.1864758735148
x35=−25.0136427998438x_{35} = -25.0136427998438
x35=75.3296660167847x_{35} = 75.3296660167847
x35=113.041382621405x_{35} = 113.041382621405
x35=50.1814586542191x_{35} = 50.1814586542191
x35=−56.4694680335143x_{35} = -56.4694680335143
x35=−87.9011342047635x_{35} = -87.9011342047635
x35=−106.756572312523x_{35} = -106.756572312523
x35=94.1864758735148x_{35} = 94.1864758735148
x35=−634.578109889878x_{35} = -634.578109889878
x35=−37.6020185275753x_{35} = -37.6020185275753
x35=6.04137871999288x_{35} = 6.04137871999288
x35=100.471611637434x_{35} = 100.471611637434
x35=−75.3296660167847x_{35} = -75.3296660167847
x35=−81.6155476438615x_{35} = -81.6155476438615
x35=−125.610628953851x_{35} = -125.610628953851
Decrece en los intervalos
[1957.22566383068,∞)\left[1957.22566383068, \infty\right)
Crece en los intervalos
(−∞,−17131.1092829573]\left(-\infty, -17131.1092829573\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
3x2(2x2+1)74−cos⁡(x)−1(2x2+1)34=0\frac{3 x^{2}}{\left(2 x^{2} + 1\right)^{\frac{7}{4}}} - \cos{\left(x \right)} - \frac{1}{\left(2 x^{2} + 1\right)^{\frac{3}{4}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−29.8469496316858x_{1} = -29.8469496316858
x2=86.3941681101022x_{2} = 86.3941681101022
x3=−4.73888710828364x_{3} = -4.73888710828364
x4=67.5447773942523x_{4} = 67.5447773942523
x5=−26.7013884650488x_{5} = -26.7013884650488
x6=−64.4020744203495x_{6} = -64.4020744203495
x7=39.2687014688524x_{7} = 39.2687014688524
x8=45.5521273324859x_{8} = 45.5521273324859
x9=7.84084793272244x_{9} = 7.84084793272244
x10=32.985156215017x_{10} = 32.985156215017
x11=89.5350397895001x_{11} = 89.5350397895001
x12=−70.6853346245265x_{12} = -70.6853346245265
x13=−89.5350397895001x_{13} = -89.5350397895001
x14=−67.5447773942523x_{14} = -67.5447773942523
x15=73.8278958677635x_{15} = 73.8278958677635
x16=70.6853346245265x_{16} = 70.6853346245265
x17=−39.2687014688524x_{17} = -39.2687014688524
x18=42.4125760572205x_{18} = 42.4125760572205
x19=54.9786002869082x_{19} = 54.9786002869082
x20=4.73888710828364x_{20} = 4.73888710828364
x21=−36.1296825446776x_{21} = -36.1296825446776
x22=−80.1110271738407x_{22} = -80.1110271738407
x23=11.003594142486x_{23} = 11.003594142486
x24=−23.5645351587588x_{24} = -23.5645351587588
x25=80.1110271738407x_{25} = 80.1110271738407
x26=−11.003594142486x_{26} = -11.003594142486
x27=29.8469496316858x_{27} = 29.8469496316858
x28=−54.9786002869082x_{28} = -54.9786002869082
x29=48.6955603493265x_{29} = 48.6955603493265
x30=3617.54394197504x_{30} = 3617.54394197504
x31=26.7013884650488x_{31} = 26.7013884650488
x32=−83.2518140384908x_{32} = -83.2518140384908
x33=58.118793463585x_{33} = 58.118793463585
x34=−281.172605552473x_{34} = -281.172605552473
x35=−51.8354827096562x_{35} = -51.8354827096562
x36=−1.50725212436071x_{36} = -1.50725212436071
x37=−86.3941681101022x_{37} = -86.3941681101022
x38=36.1296825446776x_{38} = 36.1296825446776
x39=98.9604705288327x_{39} = 98.9604705288327
x40=−61.2616764674615x_{40} = -61.2616764674615
x41=−73.8278958677635x_{41} = -73.8278958677635
x42=−48.6955603493265x_{42} = -48.6955603493265
x43=−92.6773164329816x_{43} = -92.6773164329816
x44=−58.118793463585x_{44} = -58.118793463585
x45=23.5645351587588x_{45} = 23.5645351587588
x46=−17.2828715400328x_{46} = -17.2828715400328
x47=−76.968579873949x_{47} = -76.968579873949
x48=−136.659466508855x_{48} = -136.659466508855
x49=51.8354827096562x_{49} = 51.8354827096562
x50=17.2828715400328x_{50} = 17.2828715400328
x51=−7.84084793272244x_{51} = -7.84084793272244
x52=−98.9604705288327x_{52} = -98.9604705288327
x53=76.968579873949x_{53} = 76.968579873949
x54=−95.8182590241504x_{54} = -95.8182590241504
x55=64.4020744203495x_{55} = 64.4020744203495
x56=20.4171441207174x_{56} = 20.4171441207174
x57=−32.985156215017x_{57} = -32.985156215017
x58=−14.1316228416072x_{58} = -14.1316228416072
x59=−42.4125760572205x_{59} = -42.4125760572205
x60=83.2518140384908x_{60} = 83.2518140384908
x61=95.8182590241504x_{61} = 95.8182590241504
x62=−20.4171441207174x_{62} = -20.4171441207174
x63=61.2616764674615x_{63} = 61.2616764674615
x64=92.6773164329816x_{64} = 92.6773164329816
x65=14.1316228416072x_{65} = 14.1316228416072
x66=−45.5521273324859x_{66} = -45.5521273324859
x67=1.50725212436071x_{67} = 1.50725212436071

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
[95.8182590241504,∞)\left[95.8182590241504, \infty\right)
Convexa en los intervalos
(−∞,−281.172605552473]\left(-\infty, -281.172605552473\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−2x2+14+cos⁡(x))=−∞\lim_{x \to -\infty}\left(- \sqrt[4]{2 x^{2} + 1} + \cos{\left(x \right)}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(−2x2+14+cos⁡(x))=−∞\lim_{x \to \infty}\left(- \sqrt[4]{2 x^{2} + 1} + \cos{\left(x \right)}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la derecha
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función cos(x) - (1 + 2*x^2)^(1/4), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−2x2+14+cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{- \sqrt[4]{2 x^{2} + 1} + \cos{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−2x2+14+cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{- \sqrt[4]{2 x^{2} + 1} + \cos{\left(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:
−2x2+14+cos⁡(x)=−2x2+14+cos⁡(x)- \sqrt[4]{2 x^{2} + 1} + \cos{\left(x \right)} = - \sqrt[4]{2 x^{2} + 1} + \cos{\left(x \right)}
- Sí
−2x2+14+cos⁡(x)=2x2+14−cos⁡(x)- \sqrt[4]{2 x^{2} + 1} + \cos{\left(x \right)} = \sqrt[4]{2 x^{2} + 1} - \cos{\left(x \right)}
- No
es decir, función
es
par
    © Sr Examen – Калькулятор