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Trazado el gráfico de la función/ (1-log(cos(3*x)))*sin(3*x)+(1+3*x)*cos(3*x)

Gráfico de la función y = (1-log(cos(3*x)))*sin(3*x)+(1+3*x)*cos(3*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = (1 - log(cos(3*x)))*sin(3*x) + (1 + 3*x)*cos(3*x)
f(x)=(1−log⁡(cos⁡(3x)))sin⁡(3x)+(3x+1)cos⁡(3x)f{\left(x \right)} = \left(1 - \log{\left(\cos{\left(3 x \right)} \right)}\right) \sin{\left(3 x \right)} + \left(3 x + 1\right) \cos{\left(3 x \right)} 
f = (1 - log(cos(3*x)))*sin(3*x) + (3*x + 1)*cos(3*x)
Gráfico de la función
02468-8-6-4-2-1010-5050
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:
(1−log⁡(cos⁡(3x)))sin⁡(3x)+(3x+1)cos⁡(3x)=0\left(1 - \log{\left(\cos{\left(3 x \right)} \right)}\right) \sin{\left(3 x \right)} + \left(3 x + 1\right) \cos{\left(3 x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−16.2565715171551x_{1} = -16.2565715171551
x2=20.4406481733535x_{2} = 20.4406481733535
x3=−41.376060812236x_{3} = -41.376060812236
x4=−91.6358733990181x_{4} = -91.6358733990181
x5=89.5415587517975x_{5} = 89.5415587517975
x6=9.98332679979598x_{6} = 9.98332679979598
x7=−56.0342310918299x_{7} = -56.0342310918299
x8=−0.152083806860886x_{8} = -0.152083806860886
x9=−43.4699896829381x_{9} = -43.4699896829381
x10=66.5049297916402x_{10} = 66.5049297916402
x11=49.751879483377x_{11} = 49.751879483377
x12=56.034142033576x_{12} = 56.034142033576
x13=−97.9187300149069x_{13} = -97.9187300149069
x14=−58.1283526037383x_{14} = -58.1283526037383
x15=−95.8244401325115x_{15} = -95.8244401325115
x16=53.9400332800438x_{16} = 53.9400332800438
x17=24.6266870278461x_{17} = 24.6266870278461
x18=−100.013023900746x_{18} = -100.013023900746
x19=3.73455949366956x_{19} = 3.73455949366956
x20=95.8244061908117x_{20} = 95.8244061908117
x21=62.3165719367185x_{21} = 62.3165719367185
x22=−12.0744317499687x_{22} = -12.0744317499687
x23=−5.81557424061277x_{23} = -5.81557424061277
x24=60.2224130207926x_{24} = 60.2224130207926
x25=−89.5415971147709x_{25} = -89.5415971147709
x26=97.9186973736816x_{26} = 97.9186973736816
x27=−53.9401286136827x_{27} = -53.9401286136827
x28=18.3480250187175x_{28} = 18.3480250187175
x29=64.4107445930505x_{29} = 64.4107445930505
x30=−93.7301545034054x_{30} = -93.7301545034054
x31=−1.71304882143984x_{31} = -1.71304882143984
x32=5.8110960289627x_{32} = 5.8110960289627
x33=−45.5639571355696x_{33} = -45.5639571355696
x34=14.164070751232x_{34} = 14.164070751232
x35=−60.2224912996033x_{35} = -60.2224912996033
x36=−7.89814141021579x_{36} = -7.89814141021579
x37=−14.1650694353358x_{37} = -14.1650694353358
x38=−51.8460472997267x_{38} = -51.8460472997267
x39=−87.4473259728163x_{39} = -87.4473259728163
x40=58.1282692039559x_{40} = 58.1282692039559
x41=22.5335574852964x_{41} = 22.5335574852964
x42=−9.98514150112233x_{42} = -9.98514150112233
x43=12.0731184733638x_{43} = 12.0731184733638
x44=−47.6579584393005x_{44} = -47.6579584393005
x45=93.7301191794675x_{45} = 93.7301191794675
x46=7.89544741855065x_{46} = 7.89544741855065
x47=−3.74369873535378x_{47} = -3.74369873535378
x48=100.012992484645x_{48} = 100.012992484645
x49=−49.7519896170135x_{49} = -49.7519896170135
x50=51.8459449789627x_{50} = 51.8459449789627
x51=−62.3166455651292x_{51} = -62.3166455651292
x52=16.2557839847342x_{52} = 16.2557839847342
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 - log(cos(3*x)))*sin(3*x) + (1 + 3*x)*cos(3*x).
(1−log⁡(cos⁡(0⋅3)))sin⁡(0⋅3)+(0⋅3+1)cos⁡(0⋅3)\left(1 - \log{\left(\cos{\left(0 \cdot 3 \right)} \right)}\right) \sin{\left(0 \cdot 3 \right)} + \left(0 \cdot 3 + 1\right) \cos{\left(0 \cdot 3 \right)}
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
3(1−log⁡(cos⁡(3x)))cos⁡(3x)−3(3x+1)sin⁡(3x)+3sin⁡2(3x)cos⁡(3x)+3cos⁡(3x)=03 \left(1 - \log{\left(\cos{\left(3 x \right)} \right)}\right) \cos{\left(3 x \right)} - 3 \left(3 x + 1\right) \sin{\left(3 x \right)} + \frac{3 \sin^{2}{\left(3 x \right)}}{\cos{\left(3 x \right)}} + 3 \cos{\left(3 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−48.1757326113108x_{1} = -48.1757326113108
x2=79.5897944210754x_{2} = 79.5897944210754
x3=−77.4954988157363x_{3} = -77.4954988157363
x4=4.23784503455049x_{4} = 4.23784503455049
x5=−31.4230756726269x_{5} = -31.4230756726269
x6=43.987311589411x_{6} = 43.987311589411
x7=−94.2501458175575x_{7} = -94.2501458175575
x8=−33.5170194960659x_{8} = -33.5170194960659
x9=−43.9873881738297x_{9} = -43.9873881738297
x10=90.0614478109698x_{10} = 90.0614478109698
x11=41.8931652194601x_{11} = 41.8931652194601
x12=−81.6841407233331x_{12} = -81.6841407233331
x13=6.3167432708165x_{13} = 6.3167432708165
x14=−41.8932496537821x_{14} = -41.8932496537821
x15=10.492535045022x_{15} = 10.492535045022
x16=−6.32049612044864x_{16} = -6.32049612044864
x17=−92.1558046874203x_{17} = -92.1558046874203
x18=−2.2193805726078x_{18} = -2.2193805726078
x19=−79.5898178100837x_{19} = -79.5898178100837
x20=37.7049546579058x_{20} = 37.7049546579058
x21=−75.4011840683081x_{21} = -75.4011840683081
x22=−35.6110169077977x_{22} = -35.6110169077977
x23=−4.24629097079394x_{23} = -4.24629097079394
x24=−85.8727971550974x_{24} = -85.8727971550974
x25=−37.7050588978111x_{25} = -37.7050588978111
x26=−39.7991383687072x_{26} = -39.7991383687072
x27=−98.4388349912355x_{27} = -98.4388349912355
x28=46.081480407781x_{28} = 46.081480407781
x29=−50.2699328751698x_{29} = -50.2699328751698
x30=92.1557872423129x_{30} = 92.1557872423129
x31=81.6841185183838x_{31} = 81.6841185183838
x32=54.4583286795797x_{32} = 54.4583286795797
x33=−83.7784672603994x_{33} = -83.7784672603994
x34=48.1756687668336x_{34} = 48.1756687668336
x35=87.9671110205846x_{35} = 87.9671110205846
x36=83.7784461518194x_{36} = 83.7784461518194
x37=−52.3641488253065x_{37} = -52.3641488253065
x38=96.3444733413138x_{38} = 96.3444733413138
x39=50.2698742402247x_{39} = 50.2698742402247
x40=98.4388197020875x_{40} = 98.4388197020875
x41=94.2501291391734x_{41} = 94.2501291391734
x42=−87.9671301666889x_{42} = -87.9671301666889
x43=2.18541589252129x_{43} = 2.18541589252129
x44=8.40307884195836x_{44} = 8.40307884195836
x45=−46.0815501882118x_{45} = -46.0815501882118
x46=39.7990448126711x_{46} = 39.7990448126711
x47=52.3640947873412x_{47} = 52.3640947873412
x48=85.8727770636471x_{48} = 85.8727770636471
x49=−90.0614660769125x_{49} = -90.0614660769125
Signos de extremos en los puntos:
(-48.17573261131079, -143.5271978339)

(79.58979442107541, 239.769383263224)

(-77.49549881573631, -231.486496447206)

(4.237845034550495, 13.7135307344464)

(-31.423075672626926, -93.2692270175971)

(43.98731158941098, 132.961934768185)

(-94.25014581755754, -281.750437452672)

(-33.517019496065934, -99.551058487993)

(-43.98738817382971, -130.962164521437)

(90.06144781096981, 271.184343432908)

(41.893165219460066, 126.679495658319)

(-81.68414072333312, -244.052422169997)

(6.316743270816496, 19.9502291622653)

(-41.893249653782135, -124.67974896128)

(10.492535045021969, 32.4776050791524)

(-6.320496120448643, -17.9614872551277)

(-92.15580468742026, -275.467414062259)

(-2.2193805726078013, -5.65763787178668)

(-79.58981781008369, -237.769453430248)

(37.704954657905766, 114.114863973614)

(-75.40118406830808, -225.203552204921)

(-35.611016907797705, -105.833050723242)

(-4.2462909707939405, -11.7388631976221)

(-85.87279715509737, -256.61839146529)

(-37.70505889781109, -112.11517669332)

(-39.7991383687072, -118.397415106036)

(-98.43883499123548, -294.316504973705)

(46.08148040778095, 139.244441223305)

(-50.26993287516975, -149.809798625483)

(92.15578724231285, 277.467361726937)

(81.68411851838378, 246.052355555149)

(54.458328679579665, 164.374986038722)

(-83.77846726039937, -250.335401781196)

(48.175668766833624, 145.52700630047)

(87.9671110205846, 264.901333061752)

(83.77844615181941, 252.335338455456)

(-52.36414882530646, -156.092446475898)

(96.3444733413138, 290.03342002394)

(50.26987424022472, 151.809622720649)

(98.4388197020875, 296.316459106262)

(94.25012913917338, 283.750387417519)

(-87.96713016668886, -262.901390500065)

(2.185415892521294, 7.55614851978157)

(8.403078841958362, 26.2092363616129)

(-46.08155018821179, -137.244650564594)

(39.79904481267106, 120.397134437934)

(52.36409478734123, 158.092284362003)

(85.87277706364709, 258.61833119094)

(-90.06146607691245, -269.184398230736)


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=−48.1757326113108x_{1} = -48.1757326113108
x2=−77.4954988157363x_{2} = -77.4954988157363
x3=−31.4230756726269x_{3} = -31.4230756726269
x4=−94.2501458175575x_{4} = -94.2501458175575
x5=−33.5170194960659x_{5} = -33.5170194960659
x6=−43.9873881738297x_{6} = -43.9873881738297
x7=−81.6841407233331x_{7} = -81.6841407233331
x8=−41.8932496537821x_{8} = -41.8932496537821
x9=−6.32049612044864x_{9} = -6.32049612044864
x10=−92.1558046874203x_{10} = -92.1558046874203
x11=−2.2193805726078x_{11} = -2.2193805726078
x12=−79.5898178100837x_{12} = -79.5898178100837
x13=−75.4011840683081x_{13} = -75.4011840683081
x14=−35.6110169077977x_{14} = -35.6110169077977
x15=−4.24629097079394x_{15} = -4.24629097079394
x16=−85.8727971550974x_{16} = -85.8727971550974
x17=−37.7050588978111x_{17} = -37.7050588978111
x18=−39.7991383687072x_{18} = -39.7991383687072
x19=−98.4388349912355x_{19} = -98.4388349912355
x20=−50.2699328751698x_{20} = -50.2699328751698
x21=−83.7784672603994x_{21} = -83.7784672603994
x22=−52.3641488253065x_{22} = -52.3641488253065
x23=−87.9671301666889x_{23} = -87.9671301666889
x24=−46.0815501882118x_{24} = -46.0815501882118
x25=−90.0614660769125x_{25} = -90.0614660769125
Puntos máximos de la función:
x25=79.5897944210754x_{25} = 79.5897944210754
x25=4.23784503455049x_{25} = 4.23784503455049
x25=43.987311589411x_{25} = 43.987311589411
x25=90.0614478109698x_{25} = 90.0614478109698
x25=41.8931652194601x_{25} = 41.8931652194601
x25=6.3167432708165x_{25} = 6.3167432708165
x25=10.492535045022x_{25} = 10.492535045022
x25=37.7049546579058x_{25} = 37.7049546579058
x25=46.081480407781x_{25} = 46.081480407781
x25=92.1557872423129x_{25} = 92.1557872423129
x25=81.6841185183838x_{25} = 81.6841185183838
x25=54.4583286795797x_{25} = 54.4583286795797
x25=48.1756687668336x_{25} = 48.1756687668336
x25=87.9671110205846x_{25} = 87.9671110205846
x25=83.7784461518194x_{25} = 83.7784461518194
x25=96.3444733413138x_{25} = 96.3444733413138
x25=50.2698742402247x_{25} = 50.2698742402247
x25=98.4388197020875x_{25} = 98.4388197020875
x25=94.2501291391734x_{25} = 94.2501291391734
x25=2.18541589252129x_{25} = 2.18541589252129
x25=8.40307884195836x_{25} = 8.40307884195836
x25=39.7990448126711x_{25} = 39.7990448126711
x25=52.3640947873412x_{25} = 52.3640947873412
x25=85.8727770636471x_{25} = 85.8727770636471
Decrece en los intervalos
[−2.2193805726078,2.18541589252129]\left[-2.2193805726078, 2.18541589252129\right]
Crece en los intervalos
(−∞,−98.4388349912355]\left(-\infty, -98.4388349912355\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
9(−(3x+1)cos⁡(3x)+(log⁡(cos⁡(3x))−1)sin⁡(3x)+sin⁡3(3x)cos⁡2(3x)+sin⁡(3x))=09 \left(- \left(3 x + 1\right) \cos{\left(3 x \right)} + \left(\log{\left(\cos{\left(3 x \right)} \right)} - 1\right) \sin{\left(3 x \right)} + \frac{\sin^{3}{\left(3 x \right)}}{\cos^{2}{\left(3 x \right)}} + \sin{\left(3 x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−75.8684932214368x_{1} = -75.8684932214368
x2=86.3427675652434x_{2} = 86.3427675652434
x3=−38.1563012053924x_{3} = -38.1563012053924
x4=−27.6772385476738x_{4} = -27.6772385476738
x5=77.9634974804464x_{5} = 77.9634974804464
x6=−25.5810009740293x_{6} = -25.5810009740293
x7=88.4375521203248x_{7} = 88.4375521203248
x8=−63.2989329954189x_{8} = -63.2989329954189
x9=−84.2478399206619x_{9} = -84.2478399206619
x10=10.8991326824713x_{10} = 10.8991326824713
x11=84.2479705711905x_{11} = 84.2479705711905
x12=−77.9633529808512x_{12} = -77.9633529808512
x13=71.6788826074216x_{13} = 71.6788826074216
x14=−23.4845572720462x_{14} = -23.4845572720462
x15=27.6777875292742x_{15} = 27.6777875292742
x16=−80.0581967329959x_{16} = -80.0581967329959
x17=−73.7736164214155x_{17} = -73.7736164214155
x18=73.7737716660713x_{18} = 73.7737716660713
x19=0.315961226644268x_{19} = 0.315961226644268
x20=−69.5838070617879x_{20} = -69.5838070617879
x21=75.8686429240397x_{21} = 75.8686429240397
x22=−31.8692240131575x_{22} = -31.8692240131575
x23=−29.773304055445x_{23} = -29.773304055445
x24=44.4429240536229x_{24} = 44.4429240536229
x25=−33.9650194344981x_{25} = -33.9650194344981
x26=−36.0607072325407x_{26} = -36.0607072325407
x27=−82.1530254236923x_{27} = -82.1530254236923
x28=82.1531604217036x_{28} = 82.1531604217036
x29=29.7738041263301x_{29} = 29.7738041263301
x30=40.2521524079289x_{30} = 40.2521524079289
x31=38.1566650349979x_{31} = 38.1566650349979

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
[88.4375521203248,∞)\left[88.4375521203248, \infty\right)
Convexa en los intervalos
(−∞,−84.2478399206619]\left(-\infty, -84.2478399206619\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((1−log⁡(cos⁡(3x)))sin⁡(3x)+(3x+1)cos⁡(3x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(1 - \log{\left(\cos{\left(3 x \right)} \right)}\right) \sin{\left(3 x \right)} + \left(3 x + 1\right) \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞((1−log⁡(cos⁡(3x)))sin⁡(3x)+(3x+1)cos⁡(3x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(1 - \log{\left(\cos{\left(3 x \right)} \right)}\right) \sin{\left(3 x \right)} + \left(3 x + 1\right) \cos{\left(3 x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (1 - log(cos(3*x)))*sin(3*x) + (1 + 3*x)*cos(3*x), 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→−∞((1−log⁡(cos⁡(3x)))sin⁡(3x)+(3x+1)cos⁡(3x)x)y = x \lim_{x \to -\infty}\left(\frac{\left(1 - \log{\left(\cos{\left(3 x \right)} \right)}\right) \sin{\left(3 x \right)} + \left(3 x + 1\right) \cos{\left(3 x \right)}}{x}\right)
True

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