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Trazado el gráfico de la función/ (1+cos(x)*x/3)^3

Gráfico de la función y = (1+cos(x)*x/3)^3

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                     3
       /    cos(x)*x\ 
f(x) = |1 + --------| 
       \       3    /
f(x)=(xcos⁡(x)3+1)3f{\left(x \right)} = \left(\frac{x \cos{\left(x \right)}}{3} + 1\right)^{3} 
f = ((x*cos(x))/3 + 1)^3
Gráfico de la función
02468-8-6-4-2-1010-100100
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:
(xcos⁡(x)3+1)3=0\left(\frac{x \cos{\left(x \right)}}{3} + 1\right)^{3} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−13.9199614381386x_{1} = -13.9199614381386
x2=3.80376723179969x_{2} = 3.80376723179969
x3=8.22723664765616x_{3} = 8.22723664765616
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 (1 + (cos(x)*x)/3)^3.
(0cos⁡(0)3+1)3\left(\frac{0 \cos{\left(0 \right)}}{3} + 1\right)^{3}
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
(xcos⁡(x)3+1)2(−xsin⁡(x)+cos⁡(x))=0\left(\frac{x \cos{\left(x \right)}}{3} + 1\right)^{2} \left(- x \sin{\left(x \right)} + \cos{\left(x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−12.6452872238566x_{1} = -12.6452872238566
x2=−72.270467060309x_{2} = -72.270467060309
x3=72.270467060309x_{3} = 72.270467060309
x4=9.52933440536196x_{4} = 9.52933440536196
x5=78.5525459842429x_{5} = 78.5525459842429
x6=37.7256128277765x_{6} = 37.7256128277765
x7=−37.7256128277765x_{7} = -37.7256128277765
x8=22.0364967279386x_{8} = 22.0364967279386
x9=−28.309642854452x_{9} = -28.309642854452
x10=47.145097736761x_{10} = 47.145097736761
x11=−97.3996388790738x_{11} = -97.3996388790738
x12=−59.7070073053355x_{12} = -59.7070073053355
x13=−9.52933440536196x_{13} = -9.52933440536196
x14=6.43729817917195x_{14} = 6.43729817917195
x15=−53.4257904773947x_{15} = -53.4257904773947
x16=28.309642854452x_{16} = 28.309642854452
x17=−15.7712848748159x_{17} = -15.7712848748159
x18=44.0050179208308x_{18} = 44.0050179208308
x19=−65.9885986984904x_{19} = -65.9885986984904
x20=14.3478123118765x_{20} = 14.3478123118765
x21=81.6936492356017x_{21} = 81.6936492356017
x22=−91.1171613944647x_{22} = -91.1171613944647
x23=3.02169257863407x_{23} = 3.02169257863407
x24=−3.42561845948173x_{24} = -3.42561845948173
x25=−50.2853663377737x_{25} = -50.2853663377737
x26=−94.2583883450399x_{26} = -94.2583883450399
x27=94.2583883450399x_{27} = 94.2583883450399
x28=−87.9759605524932x_{28} = -87.9759605524932
x29=87.9759605524932x_{29} = 87.9759605524932
x30=15.7712848748159x_{30} = 15.7712848748159
x31=−75.4114834888481x_{31} = -75.4114834888481
x32=−81.6936492356017x_{32} = -81.6936492356017
x33=3.80368245374897x_{33} = 3.80368245374897
x34=62.8477631944545x_{34} = 62.8477631944545
x35=8.22723075438385x_{35} = 8.22723075438385
x36=59.7070073053355x_{36} = 59.7070073053355
x37=12.6452872238566x_{37} = 12.6452872238566
x38=−31.4477146375462x_{38} = -31.4477146375462
x39=34.5864242152889x_{39} = 34.5864242152889
x40=−22.0364967279386x_{40} = -22.0364967279386
x41=−11.2651354100605x_{41} = -11.2651354100605
x42=50.2853663377737x_{42} = 50.2853663377737
x43=56.5663442798215x_{43} = 56.5663442798215
x44=−44.0050179208308x_{44} = -44.0050179208308
x45=100.540910786842x_{45} = 100.540910786842
x46=65.9885986984904x_{46} = 65.9885986984904
x47=−62.8477631944545x_{47} = -62.8477631944545
x48=0.86033358901938x_{48} = 0.86033358901938
x49=−0.86033358901938x_{49} = -0.86033358901938
Signos de extremos en los puntos:
(-12.645287223856643, -32.828774023797)

(-72.27046706030896, 15790.2979968187)

(72.27046706030896, -12306.950929793)

(9.529334405361963, -10.0650773674381)

(78.55254598424293, -15968.8554821785)

(37.7256128277765, 2499.28204434774)

(-37.7256128277765, -1549.13366743501)

(22.036496727938566, -254.59266280595)

(-28.30964285445201, 1134.84382296338)

(47.14509773676103, -3183.98302055827)

(-97.39963887907376, 37477.1083597917)

(-59.70700730533546, 9128.73269755989)

(-9.529334405361963, 71.9444816879506)

(6.437298179171947, 30.3811018488496)

(-53.42579047739466, 6650.48218214952)

(28.30964285445201, -599.219073128636)

(-15.771284874815882, 243.737603303253)

(44.005017920830845, 3843.74038329261)

(-65.98859869849039, 12156.9638942981)

(14.347812311876497, 2.47415879673643e-20)

(81.69364923560168, 22495.4469989332)

(-91.11716139446474, 30872.1401695361)

(3.021692578634074, 9.77129448949653e-20)

(-3.4256184594817283, 9.20981245342452)

(-50.28536633777365, -3913.29501555171)

(-94.25838834503986, -28143.550878091)

(94.25838834503986, 34067.9801353879)

(-87.97596055249322, -22721.5183368302)

(87.97596055249322, 27882.6981797102)

(15.771284874815882, -76.5793160550373)

(-75.41148348884815, -14058.5096584455)

(-81.69364923560168, -18044.8786821028)

(3.8036824537489724, 7.80936685692007e-21)

(62.84776319445445, 10570.6555721547)

(8.227230754383847, -1.09184369580575e-22)

(59.70700730533546, -6750.7813630323)

(12.645287223856643, 140.768443281924)

(-31.447714637546234, -851.237117299668)

(34.58642421528892, -1165.57858096435)

(-22.036496727938566, 579.665491530036)

(-11.265135410060545, -6.71203369428445e-21)

(50.28536633777365, 5600.37399091217)

(56.56634427982152, 7824.30499294003)

(-44.005017920830845, -2551.44563772113)

(100.54091078684232, 41106.4151254204)

(65.98859869849039, -9252.6336357745)

(-62.84776319445445, -7936.09451105009)

(0.8603335890193797, 1.67258194419693)

(-0.8603335890193797, 0.537304122957334)


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=−12.6452872238566x_{1} = -12.6452872238566
x2=72.270467060309x_{2} = 72.270467060309
x3=9.52933440536196x_{3} = 9.52933440536196
x4=78.5525459842429x_{4} = 78.5525459842429
x5=−37.7256128277765x_{5} = -37.7256128277765
x6=22.0364967279386x_{6} = 22.0364967279386
x7=47.145097736761x_{7} = 47.145097736761
x8=28.309642854452x_{8} = 28.309642854452
x9=−50.2853663377737x_{9} = -50.2853663377737
x10=−94.2583883450399x_{10} = -94.2583883450399
x11=−87.9759605524932x_{11} = -87.9759605524932
x12=15.7712848748159x_{12} = 15.7712848748159
x13=−75.4114834888481x_{13} = -75.4114834888481
x14=−81.6936492356017x_{14} = -81.6936492356017
x15=59.7070073053355x_{15} = 59.7070073053355
x16=−31.4477146375462x_{16} = -31.4477146375462
x17=34.5864242152889x_{17} = 34.5864242152889
x18=−44.0050179208308x_{18} = -44.0050179208308
x19=65.9885986984904x_{19} = 65.9885986984904
x20=−62.8477631944545x_{20} = -62.8477631944545
x21=−0.86033358901938x_{21} = -0.86033358901938
Puntos máximos de la función:
x21=−72.270467060309x_{21} = -72.270467060309
x21=37.7256128277765x_{21} = 37.7256128277765
x21=−28.309642854452x_{21} = -28.309642854452
x21=−97.3996388790738x_{21} = -97.3996388790738
x21=−59.7070073053355x_{21} = -59.7070073053355
x21=−9.52933440536196x_{21} = -9.52933440536196
x21=6.43729817917195x_{21} = 6.43729817917195
x21=−53.4257904773947x_{21} = -53.4257904773947
x21=−15.7712848748159x_{21} = -15.7712848748159
x21=44.0050179208308x_{21} = 44.0050179208308
x21=−65.9885986984904x_{21} = -65.9885986984904
x21=81.6936492356017x_{21} = 81.6936492356017
x21=−91.1171613944647x_{21} = -91.1171613944647
x21=−3.42561845948173x_{21} = -3.42561845948173
x21=94.2583883450399x_{21} = 94.2583883450399
x21=87.9759605524932x_{21} = 87.9759605524932
x21=62.8477631944545x_{21} = 62.8477631944545
x21=12.6452872238566x_{21} = 12.6452872238566
x21=−22.0364967279386x_{21} = -22.0364967279386
x21=50.2853663377737x_{21} = 50.2853663377737
x21=56.5663442798215x_{21} = 56.5663442798215
x21=100.540910786842x_{21} = 100.540910786842
x21=0.86033358901938x_{21} = 0.86033358901938
Decrece en los intervalos
[78.5525459842429,∞)\left[78.5525459842429, \infty\right)
Crece en los intervalos
(−∞,−94.2583883450399]\left(-\infty, -94.2583883450399\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
(xcos⁡(x)9+13)(2(xsin⁡(x)−cos⁡(x))2−(xcos⁡(x)+3)(xcos⁡(x)+2sin⁡(x)))=0\left(\frac{x \cos{\left(x \right)}}{9} + \frac{1}{3}\right) \left(2 \left(x \sin{\left(x \right)} - \cos{\left(x \right)}\right)^{2} - \left(x \cos{\left(x \right)} + 3\right) \left(x \cos{\left(x \right)} + 2 \sin{\left(x \right)}\right)\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=30.8063348042047x_{1} = 30.8063348042047
x2=25.8200738854295x_{2} = 25.8200738854295
x3=36.0449898053462x_{3} = 36.0449898053462
x4=77.9483270062296x_{4} = 77.9483270062296
x5=62.2189123701449x_{5} = 62.2189123701449
x6=21.4633503206057x_{6} = 21.4633503206057
x7=−5.99435491917131x_{7} = -5.99435491917131
x8=33.9972079976884x_{8} = 33.9972079976884
x9=76.0381934797549x_{9} = 76.0381934797549
x10=90.5113614608063x_{10} = 90.5113614608063
x11=0.189338882824807x_{11} = 0.189338882824807
x12=−43.409972828557x_{12} = -43.409972828557
x13=−28.9539273230921x_{13} = -28.9539273230921
x14=−13.9199445744068x_{14} = -13.9199445744068
x15=−77.926302850514x_{15} = -77.926302850514
x16=54.0244988951785x_{16} = 54.0244988951785
x17=27.7265855802507x_{17} = 27.7265855802507
x18=80.0731381481904x_{18} = 80.0731381481904
x19=46.5486443495984x_{19} = 46.5486443495984
x20=−25.7510005611425x_{20} = -25.7510005611425
x21=73.7867584479877x_{21} = 73.7867584479877
x22=82.3195135621576x_{22} = 82.3195135621576
x23=38.3630393167073x_{23} = 38.3630393167073
x24=49.6533033106206x_{24} = 49.6533033106206
x25=−20.2718179788495x_{25} = -20.2718179788495
x26=95.8498799906678x_{26} = 95.8498799906678
x27=−73.8680514892562x_{27} = -73.8680514892562
x28=−4.18779481587838x_{28} = -4.18779481587838
x29=32.0892834125946x_{29} = 32.0892834125946
x30=−41.5009816639229x_{30} = -41.5009816639229
x31=−8.84115409558241x_{31} = -8.84115409558241
x32=−27.6655956331173x_{32} = -27.6655956331173
x33=−99.9341886251782x_{33} = -99.9341886251782
x34=99.9169768585517x_{34} = 99.9169768585517
x35=−98.0238578013247x_{35} = -98.0238578013247
x36=−51.7783069964833x_{36} = -51.7783069964833
x37=−7.43887584016775x_{37} = -7.43887584016775
x38=42.3405874040572x_{38} = 42.3405874040572
x39=11.9715815739034x_{39} = 11.9715815739034
x40=29.7440980707673x_{40} = 29.7440980707673
x41=60.3075327719793x_{41} = 60.3075327719793
x42=−23.6889273278962x_{42} = -23.6889273278962
x43=24.5251253678905x_{43} = 24.5251253678905
x44=86.3590522969634x_{44} = 86.3590522969634
x45=68.5018332987326x_{45} = 68.5018332987326
x46=40.2717582834306x_{46} = 40.2717582834306
x47=−38.3170120646221x_{47} = -38.3170120646221
x48=−33.9472586172655x_{48} = -33.9472586172655
x49=18.2456748774413x_{49} = 18.2456748774413
x50=−15.1074626403289x_{50} = -15.1074626403289
x51=−67.5886427949036x_{51} = -67.5886427949036
x52=−76.0151971497371x_{52} = -76.0151971497371
x53=14.347812373266x_{53} = 14.347812373266
x54=10.0362785858514x_{54} = 10.0362785858514
x55=58.1710590164887x_{55} = 58.1710590164887
x56=64.4492145034092x_{56} = 64.4492145034092
x57=−12.1076393884897x_{57} = -12.1076393884897
x58=−62.2464322162983x_{58} = -62.2464322162983
x59=−93.6522601596371x_{59} = -93.6522601596371
x60=−82.2982876712779x_{60} = -82.2982876712779
x61=−89.5018654932298x_{61} = -89.5018654932298
x62=−86.428515713816x_{62} = -86.428515713816
x63=8.22723073651729x_{63} = 8.22723073651729
x64=−10.2203615035817x_{64} = -10.2203615035817
x65=−40.229466859082x_{65} = -40.229466859082
x66=88.6010976042828x_{66} = 88.6010976042828
x67=−84.2093246737813x_{67} = -84.2093246737813
x68=−45.4870927949251x_{68} = -45.4870927949251
x69=3.80368206966899x_{69} = 3.80368206966899
x70=93.6339022881948x_{70} = 93.6339022881948
x71=−32.0340385445269x_{71} = -32.0340385445269
x72=16.3254368168691x_{72} = 16.3254368168691
x73=−80.1480521412568x_{73} = -80.1480521412568
x74=−95.787251404081x_{74} = -95.787251404081
x75=−49.6876860427652x_{75} = -49.6876860427652
x76=−58.0677773157746x_{76} = -58.0677773157746
x77=47.7414874127795x_{77} = 47.7414874127795
x78=5.72745367001681x_{78} = 5.72745367001681
x79=−29.9454806113033x_{79} = -29.9454806113033
x80=−21.3850660677235x_{80} = -21.3850660677235
x81=−60.3365873514726x_{81} = -60.3365873514726
x82=69.7572084651731x_{82} = 69.7572084651731
x83=−36.211257746307x_{83} = -36.211257746307
x84=−47.7782994419632x_{84} = -47.7782994419632
x85=−55.9666302569796x_{85} = -55.9666302569796
x86=−69.7321195360243x_{86} = -69.7321195360243
x87=71.667249315451x_{87} = 71.667249315451
x88=98.0060579528609x_{88} = 98.0060579528609
x89=84.2297196084685x_{89} = 84.2297196084685
x90=−71.6433130948744x_{90} = -71.6433130948744
x91=51.8941210495273x_{91} = 51.8941210495273
x92=−91.741971701766x_{92} = -91.741971701766
x93=−54.0569755172093x_{93} = -54.0569755172093
x94=55.936059285644x_{94} = 55.936059285644
x95=66.5905863172244x_{95} = 66.5905863172244

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
[90.5113614608063,∞)\left[90.5113614608063, \infty\right)
Convexa en los intervalos
(−∞,−86.428515713816]\left(-\infty, -86.428515713816\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(xcos⁡(x)3+1)3y = \lim_{x \to -\infty} \left(\frac{x \cos{\left(x \right)}}{3} + 1\right)^{3}
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(xcos⁡(x)3+1)3y = \lim_{x \to \infty} \left(\frac{x \cos{\left(x \right)}}{3} + 1\right)^{3}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (1 + (cos(x)*x)/3)^3, dividida por x con x->+oo y x ->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la izquierda:
y=xlim⁡x→−∞((xcos⁡(x)3+1)3x)y = x \lim_{x \to -\infty}\left(\frac{\left(\frac{x \cos{\left(x \right)}}{3} + 1\right)^{3}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((xcos⁡(x)3+1)3x)y = x \lim_{x \to \infty}\left(\frac{\left(\frac{x \cos{\left(x \right)}}{3} + 1\right)^{3}}{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:
(xcos⁡(x)3+1)3=(−xcos⁡(x)3+1)3\left(\frac{x \cos{\left(x \right)}}{3} + 1\right)^{3} = \left(- \frac{x \cos{\left(x \right)}}{3} + 1\right)^{3}
- No
(xcos⁡(x)3+1)3=−(−xcos⁡(x)3+1)3\left(\frac{x \cos{\left(x \right)}}{3} + 1\right)^{3} = - \left(- \frac{x \cos{\left(x \right)}}{3} + 1\right)^{3}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор