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Trazado el gráfico de la función/ 1/2*(e^(sin(2*x))+cos(2*x)-log(sqrt(3)))

Gráfico de la función y = 1/2*(e^(sin(2*x))+cos(2*x)-log(sqrt(3)))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        sin(2*x)                 /  ___\
       E         + cos(2*x) - log\\/ 3 /
f(x) = ---------------------------------
                       2
f(x)=(esin⁡(2x)+cos⁡(2x))−log⁡(3)2f{\left(x \right)} = \frac{\left(e^{\sin{\left(2 x \right)}} + \cos{\left(2 x \right)}\right) - \log{\left(\sqrt{3} \right)}}{2} 
f = (E^sin(2*x) + cos(2*x) - log(sqrt(3)))/2
Gráfico de la función
02468-8-6-4-2-10102-2
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:
(esin⁡(2x)+cos⁡(2x))−log⁡(3)2=0\frac{\left(e^{\sin{\left(2 x \right)}} + \cos{\left(2 x \right)}\right) - \log{\left(\sqrt{3} \right)}}{2} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=71.559521804308x_{1} = 71.559521804308
x2=64.2042970386951x_{2} = 64.2042970386951
x3=−19.546665149796x_{3} = -19.546665149796
x4=−82.3785182215919x_{4} = -82.3785182215919
x5=49.5683732291795x_{5} = 49.5683732291795
x6=−23.7602972618191x_{6} = -23.7602972618191
x7=2.44448342533256x_{7} = 2.44448342533256
x8=33.8604099612305x_{8} = 33.8604099612305
x9=26.5051851956176x_{9} = 26.5051851956176
x10=42.2131484635666x_{10} = 42.2131484635666
x11=−94.944888835951x_{11} = -94.944888835951
x12=−25.8298504569756x_{12} = -25.8298504569756
x13=−98.0864814895408x_{13} = -98.0864814895408
x14=92.4786309210033x_{14} = 92.4786309210033
x15=70.4874823458747x_{15} = 70.4874823458747
x16=13.9388145812584x_{16} = 13.9388145812584
x17=−85.5201108751817x_{17} = -85.5201108751817
x18=−89.7337429872047x_{18} = -89.7337429872047
x19=−32.1130357641552x_{19} = -32.1130357641552
x20=90.4090777258468x_{20} = 90.4090777258468
x21=79.9122603066441x_{21} = 79.9122603066441
x22=65.2763364971284x_{22} = 65.2763364971284
x23=21.2940393468713x_{23} = 21.2940393468713
x24=43.2851879219999x_{24} = 43.2851879219999
x25=27.5772246540509x_{25} = 27.5772246540509
x26=37.0020026148203x_{26} = 37.0020026148203
x27=93.5506703794366x_{27} = 93.5506703794366
x28=−47.8209990321041x_{28} = -47.8209990321041
x29=−76.0953329144123x_{29} = -76.0953329144123
x30=101.903408881773x_{30} = 101.903408881773
x31=−42.6098531833578x_{31} = -42.6098531833578
x32=−1.76914868669053x_{32} = -1.76914868669053
x33=−28.9714431105654x_{33} = -28.9714431105654
x34=−30.0434825689987x_{34} = -30.0434825689987
x35=−6104.81163515323x_{35} = -6104.81163515323
x36=−74.0257797192558x_{36} = -74.0257797192558
x37=−91.8032961823612x_{37} = -91.8032961823612
x38=−44.6794063785143x_{38} = -44.6794063785143
x39=68.4179291507182x_{39} = 68.4179291507182
x40=4.51403662048906x_{40} = 4.51403662048906
x41=46.4267805755897x_{41} = 46.4267805755897
x42=−41.5378137249245x_{42} = -41.5378137249245
x43=87.267485072257x_{43} = 87.267485072257
x44=−99.1585209479741x_{44} = -99.1585209479741
x45=−39.468260529768x_{45} = -39.468260529768
x46=−54.1041843392837x_{46} = -54.1041843392837
x47=20.221999888438x_{47} = 20.221999888438
x48=24.4356320004611x_{48} = 24.4356320004611
x49=84.1258924186672x_{49} = 84.1258924186672
x50=55.851558536359x_{50} = 55.851558536359
x51=−38.3962210713348x_{51} = -38.3962210713348
x52=−69.8121476072327x_{52} = -69.8121476072327
x53=−88.6617035287715x_{53} = -88.6617035287715
x54=5.58607607892235x_{54} = 5.58607607892235
x55=48.4963337707462x_{55} = 48.4963337707462
x56=15.0108540396917x_{56} = 15.0108540396917
x57=−60.3873696464633x_{57} = -60.3873696464633
x58=−0.697109228257237x_{58} = -0.697109228257237
x59=99.8338556866161x_{59} = 99.8338556866161
x60=−96.0169282943843x_{60} = -96.0169282943843
x61=−17.4771119546395x_{61} = -17.4771119546395
x62=−66.6705549536429x_{62} = -66.6705549536429
x63=11.8692613861019x_{63} = 11.8692613861019
x64=−16.4050724962062x_{64} = -16.4050724962062
x65=86.1954456138237x_{65} = 86.1954456138237
x66=−72.9537402608225x_{66} = -72.9537402608225
x67=−50.9625916856939x_{67} = -50.9625916856939
x68=−52.0346311441272x_{68} = -52.0346311441272
x69=35.929963156387x_{69} = 35.929963156387
x70=−67.7425944120762x_{70} = -67.7425944120762
x71=18.1524466932815x_{71} = 18.1524466932815
x72=−63.5289623000531x_{72} = -63.5289623000531
x73=−10.1218871890266x_{73} = -10.1218871890266
x74=−3.83870188184703x_{74} = -3.83870188184703
x75=−22.6882578033858x_{75} = -22.6882578033858
x76=−45.7514458369476x_{76} = -45.7514458369476
x77=−83.4505576800252x_{77} = -83.4505576800252
x78=40.1435952684101x_{78} = 40.1435952684101
x79=57.9211117315155x_{79} = 57.9211117315155
x80=−8.05233399387011x_{80} = -8.05233399387011
x81=−6.98029453543682x_{81} = -6.98029453543682
x82=−469.86645407157x_{82} = -469.86645407157
x83=58.9931511899488x_{83} = 58.9931511899488
x84=77.8427071114876x_{84} = 77.8427071114876
x85=62.1347438435386x_{85} = 62.1347438435386
x86=54.7795190779257x_{86} = 54.7795190779257
x87=−61.4594091048966x_{87} = -61.4594091048966
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 (E^sin(2*x) + cos(2*x) - log(sqrt(3)))/2.
−log⁡(3)+(esin⁡(0⋅2)+cos⁡(0⋅2))2\frac{- \log{\left(\sqrt{3} \right)} + \left(e^{\sin{\left(0 \cdot 2 \right)}} + \cos{\left(0 \cdot 2 \right)}\right)}{2}
Resultado:
f(0)=1−log⁡(3)2f{\left(0 \right)} = 1 - \frac{\log{\left(\sqrt{3} \right)}}{2}
Punto:
(0, 1 - log(sqrt(3))/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
esin⁡(2x)cos⁡(2x)−sin⁡(2x)=0e^{\sin{\left(2 x \right)}} \cos{\left(2 x \right)} - \sin{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=25.7302797812042x_{1} = 25.7302797812042
x2=107.411688774539x_{2} = 107.411688774539
x3=−51.5666560890822x_{3} = -51.5666560890822
x4=−64.1330267034414x_{4} = -64.1330267034414
x5=−49.6679439049508x_{5} = -49.6679439049508
x6=96.0881986296381x_{6} = 96.0881986296381
x7=−57.8498413962618x_{7} = -57.8498413962618
x8=67.81386474733x_{8} = 67.81386474733
x9=60.2877989706919x_{9} = 60.2877989706919
x10=−87.3670557480283x_{10} = -87.3670557480283
x11=−11.9688320618733x_{11} = -11.9688320618733
x12=47.7214283563328x_{12} = 47.7214283563328
x13=74.0970500545096x_{13} = 74.0970500545096
x14=32.0134650883838x_{14} = 32.0134650883838
x15=64.6722720937402x_{15} = 64.6722720937402
x16=−2008.77887927552x_{16} = -2008.77887927552
x17=−84.2254630944386x_{17} = -84.2254630944386
x18=54.0046136635124x_{18} = 54.0046136635124
x19=0.597538552485871x_{19} = 0.597538552485871
x20=91.7037255065899x_{20} = 91.7037255065899
x21=−13.8675442460047x_{21} = -13.8675442460047
x22=3.73913120607566x_{22} = 3.73913120607566
x23=−48.4250634354924x_{23} = -48.4250634354924
x24=23.8315675970729x_{24} = 23.8315675970729
x25=−92.4073605857495x_{25} = -92.4073605857495
x26=−73.5578046642107x_{26} = -73.5578046642107
x27=1.8404190219443x_{27} = 1.8404190219443
x28=−39.000285474723x_{28} = -39.000285474723
x29=−79.8409899713903x_{29} = -79.8409899713903
x30=−7.58435893882508x_{30} = -7.58435893882508
x31=−33.9599806370019x_{31} = -33.9599806370019
x32=−20.1507295531843x_{32} = -20.1507295531843
x33=75.9957622386409x_{33} = 75.9957622386409
x34=−70.4162120106209x_{34} = -70.4162120106209
x35=−77.942277787259x_{35} = -77.942277787259
x36=89.8050133224585x_{36} = 89.8050133224585
x37=−5.68564675469371x_{37} = -5.68564675469371
x38=−18.2520173690529x_{38} = -18.2520173690529
x39=52.105901479381x_{39} = 52.105901479381
x40=−42.1418781283128x_{40} = -42.1418781283128
x41=−1.30117363164549x_{41} = -1.30117363164549
x42=30.1147529042524x_{42} = 30.1147529042524
x43=86.6634206688687x_{43} = 86.6634206688687
x44=−23.292322206774x_{44} = -23.292322206774
x45=82.2789475458205x_{45} = 82.2789475458205
x46=104.270096120949x_{46} = 104.270096120949
x47=−90.5086484016181x_{47} = -90.5086484016181
x48=42.6811235186116x_{48} = 42.6811235186116
x49=20.6899749434831x_{49} = 20.6899749434831
x50=−99.9334263623875x_{50} = -99.9334263623875
x51=−35.8586928211332x_{51} = -35.8586928211332
x52=−95.5489532393393x_{52} = -95.5489532393393
x53=−67.2746193570312x_{53} = -67.2746193570312
x54=−93.6502410552079x_{54} = -93.6502410552079
x55=14.4067896363035x_{55} = 14.4067896363035
x56=−111.256916507288x_{56} = -111.256916507288
x57=58.3890867865606x_{57} = 58.3890867865606
x58=97.9869108137695x_{58} = 97.9869108137695
x59=−40.2431659441814x_{59} = -40.2431659441814
x60=−29.5755075139536x_{60} = -29.5755075139536
x61=16.3055018204348x_{61} = 16.3055018204348
x62=−54.708248742672x_{62} = -54.708248742672
x63=−55.9511292121304x_{63} = -55.9511292121304
x64=8.12360432912389x_{64} = 8.12360432912389
x65=−27.6767953298223x_{65} = -27.6767953298223
x66=−71.6590924800794x_{66} = -71.6590924800794
x67=−62.23431451931x_{67} = -62.23431451931
x68=−86.1241752785699x_{68} = -86.1241752785699
x69=36.397938211432x_{69} = 36.397938211432
x70=38.2966503955634x_{70} = 38.2966503955634
x71=45.8227161722014x_{71} = 45.8227161722014
x72=80.3802353616891x_{72} = 80.3802353616891
x73=10.0223165132553x_{73} = 10.0223165132553
x74=−45.2834707819026x_{74} = -45.2834707819026
x75=69.7125769314613x_{75} = 69.7125769314613
Signos de extremos en los puntos:
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                                        log\\/ 3 / 
(25.730279781204217, 1.45103461121082 - ----------)
                                            2      

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                                        log\\/ 3 / 
(107.41168877453885, 1.45103461121082 - ----------)
                                            2      

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                                          log\\/ 3 / 
(-51.56665608908219, -0.129846037665657 - ----------)
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                                          log\\/ 3 / 
(-64.13302670344136, -0.129846037665657 - ----------)
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                                        log\\/ 3 / 
(-49.66794390495082, 1.45103461121082 - ----------)
                                            2      

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                                        log\\/ 3 / 
(96.0881986296381, -0.129846037665657 - ----------)
                                            2      

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                                          log\\/ 3 / 
(-57.84984139626177, -0.129846037665657 - ----------)
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                                         log\\/ 3 / 
(67.81386474732996, -0.129846037665657 - ----------)
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                                        log\\/ 3 / 
(60.287798970691945, 1.45103461121082 - ----------)
                                            2      

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                                        log\\/ 3 / 
(-87.36705574802833, 1.45103461121082 - ----------)
                                            2      

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                                         log\\/ 3 / 
(-11.968832061873302, 1.45103461121082 - ----------)
                                             2      

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                                       log\\/ 3 / 
(47.72142835633277, 1.45103461121082 - ----------)
                                           2      

                                            /  ___\ 
                                         log\\/ 3 / 
(74.09705005450955, -0.129846037665657 - ----------)
                                             2      

                                         /  ___\ 
                                      log\\/ 3 / 
(32.0134650883838, 1.45103461121082 - ----------)
                                          2      

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                                         log\\/ 3 / 
(64.67227209374016, -0.129846037665657 - ----------)
                                             2      

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                                           log\\/ 3 / 
(-2008.7788792755234, -0.129846037665657 - ----------)
                                               2      

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                                        log\\/ 3 / 
(-84.22546309443855, 1.45103461121082 - ----------)
                                            2      

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                                       log\\/ 3 / 
(54.00461366351236, 1.45103461121082 - ----------)
                                           2      

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                                        log\\/ 3 / 
(0.5975385524858713, 1.45103461121082 - ----------)
                                            2      

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                                       log\\/ 3 / 
(91.70372550658988, 1.45103461121082 - ----------)
                                           2      

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                                           log\\/ 3 / 
(-13.867544246004664, -0.129846037665657 - ----------)
                                               2      

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                                        log\\/ 3 / 
(3.7391312060756645, 1.45103461121082 - ----------)
                                            2      

                                             /  ___\ 
                                          log\\/ 3 / 
(-48.42506343549239, -0.129846037665657 - ----------)
                                              2      

                                             /  ___\ 
                                          log\\/ 3 / 
(23.831567597072855, -0.129846037665657 - ----------)
                                              2      

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                                         log\\/ 3 / 
(-92.4073605857495, -0.129846037665657 - ----------)
                                             2      

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                                          log\\/ 3 / 
(-73.55780466421074, -0.129846037665657 - ----------)
                                              2      

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                                         log\\/ 3 / 
(1.840419021944301, -0.129846037665657 - ----------)
                                             2      

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                                           log\\/ 3 / 
(-39.000285474723015, -0.129846037665657 - ----------)
                                               2      

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                                          log\\/ 3 / 
(-79.84098997139033, -0.129846037665657 - ----------)
                                              2      

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                                          log\\/ 3 / 
(-7.584358938825079, -0.129846037665657 - ----------)
                                              2      

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                                         log\\/ 3 / 
(-33.959980637001856, 1.45103461121082 - ----------)
                                             2      

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                                           log\\/ 3 / 
(-20.150729553184252, -0.129846037665657 - ----------)
                                               2      

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                                      log\\/ 3 / 
(75.9957622386409, 1.45103461121082 - ----------)
                                          2      

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                                          log\\/ 3 / 
(-70.41621201062094, -0.129846037665657 - ----------)
                                              2      

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(-77.94227778725896, 1.45103461121082 - ----------)
                                            2      

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                                        log\\/ 3 / 
(89.8050133224585, -0.129846037665657 - ----------)
                                            2      

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                                        log\\/ 3 / 
(-5.685646754693715, 1.45103461121082 - ----------)
                                            2      

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                                         log\\/ 3 / 
(-18.252017369052886, 1.45103461121082 - ----------)
                                             2      

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                                         log\\/ 3 / 
(52.10590147938099, -0.129846037665657 - ----------)
                                             2      

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                                           log\\/ 3 / 
(-42.141878128312804, -0.129846037665657 - ----------)
                                               2      

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                                           log\\/ 3 / 
(-1.3011736316454923, -0.129846037665657 - ----------)
                                               2      

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                                         log\\/ 3 / 
(30.11475290425244, -0.129846037665657 - ----------)
                                             2      

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                                         log\\/ 3 / 
(86.66342066886872, -0.129846037665657 - ----------)
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                                           log\\/ 3 / 
(-23.292322206774045, -0.129846037665657 - ----------)
                                               2      

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                                      log\\/ 3 / 
(82.2789475458205, 1.45103461121082 - ----------)
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(104.27009612094905, 1.45103461121082 - ----------)
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                                        log\\/ 3 / 
(-90.50864840161813, 1.45103461121082 - ----------)
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                                          log\\/ 3 / 
(42.681123518611614, -0.129846037665657 - ----------)
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                                         log\\/ 3 / 
(20.68997494348306, -0.129846037665657 - ----------)
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                                        log\\/ 3 / 
(-99.93342636238751, 1.45103461121082 - ----------)
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(-35.85869282113322, -0.129846037665657 - ----------)
                                              2      

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                                          log\\/ 3 / 
(-95.54895323933928, -0.129846037665657 - ----------)
                                              2      

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                                          log\\/ 3 / 
(-67.27461935703116, -0.129846037665657 - ----------)
                                              2      

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                                        log\\/ 3 / 
(-93.65024105520793, 1.45103461121082 - ----------)
                                            2      

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                                          log\\/ 3 / 
(14.406789636303474, -0.129846037665657 - ----------)
                                              2      

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                                           log\\/ 3 / 
(-111.25691650728825, -0.129846037665657 - ----------)
                                               2      

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                                         log\\/ 3 / 
(58.38908678656058, -0.129846037665657 - ----------)
                                             2      

                                          /  ___\ 
                                       log\\/ 3 / 
(97.98691081376946, 1.45103461121082 - ----------)
                                           2      

                                           /  ___\ 
                                        log\\/ 3 / 
(-40.24316594418144, 1.45103461121082 - ----------)
                                            2      

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                                          log\\/ 3 / 
(-29.57550751395363, -0.129846037665657 - ----------)
                                              2      

                                           /  ___\ 
                                        log\\/ 3 / 
(16.305501820434838, 1.45103461121082 - ----------)
                                            2      

                                             /  ___\ 
                                          log\\/ 3 / 
(-54.70824874267198, -0.129846037665657 - ----------)
                                              2      

                                            /  ___\ 
                                         log\\/ 3 / 
(-55.951129212130404, 1.45103461121082 - ----------)
                                             2      

                                            /  ___\ 
                                         log\\/ 3 / 
(8.123604329123888, -0.129846037665657 - ----------)
                                             2      

                                           /  ___\ 
                                        log\\/ 3 / 
(-27.67679532982227, 1.45103461121082 - ----------)
                                            2      

                                           /  ___\ 
                                        log\\/ 3 / 
(-71.65909248007938, 1.45103461121082 - ----------)
                                            2      

                                           /  ___\ 
                                        log\\/ 3 / 
(-62.23431451930999, 1.45103461121082 - ----------)
                                            2      

                                             /  ___\ 
                                          log\\/ 3 / 
(-86.12417527856991, -0.129846037665657 - ----------)
                                              2      

                                            /  ___\ 
                                         log\\/ 3 / 
(36.39793821143203, -0.129846037665657 - ----------)
                                             2      

                                          /  ___\ 
                                       log\\/ 3 / 
(38.29665039556339, 1.45103461121082 - ----------)
                                           2      

                                           /  ___\ 
                                        log\\/ 3 / 
(45.8227161722014, -0.129846037665657 - ----------)
                                            2      

                                            /  ___\ 
                                         log\\/ 3 / 
(80.38023536168913, -0.129846037665657 - ----------)
                                             2      

                                           /  ___\ 
                                        log\\/ 3 / 
(10.022316513255252, 1.45103461121082 - ----------)
                                            2      

                                            /  ___\ 
                                         log\\/ 3 / 
(-45.2834707819026, -0.129846037665657 - ----------)
                                             2      

                                          /  ___\ 
                                       log\\/ 3 / 
(69.71257693146133, 1.45103461121082 - ----------)
                                           2


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=−51.5666560890822x_{1} = -51.5666560890822
x2=−64.1330267034414x_{2} = -64.1330267034414
x3=96.0881986296381x_{3} = 96.0881986296381
x4=−57.8498413962618x_{4} = -57.8498413962618
x5=67.81386474733x_{5} = 67.81386474733
x6=74.0970500545096x_{6} = 74.0970500545096
x7=64.6722720937402x_{7} = 64.6722720937402
x8=−2008.77887927552x_{8} = -2008.77887927552
x9=−13.8675442460047x_{9} = -13.8675442460047
x10=−48.4250634354924x_{10} = -48.4250634354924
x11=23.8315675970729x_{11} = 23.8315675970729
x12=−92.4073605857495x_{12} = -92.4073605857495
x13=−73.5578046642107x_{13} = -73.5578046642107
x14=1.8404190219443x_{14} = 1.8404190219443
x15=−39.000285474723x_{15} = -39.000285474723
x16=−79.8409899713903x_{16} = -79.8409899713903
x17=−7.58435893882508x_{17} = -7.58435893882508
x18=−20.1507295531843x_{18} = -20.1507295531843
x19=−70.4162120106209x_{19} = -70.4162120106209
x20=89.8050133224585x_{20} = 89.8050133224585
x21=52.105901479381x_{21} = 52.105901479381
x22=−42.1418781283128x_{22} = -42.1418781283128
x23=−1.30117363164549x_{23} = -1.30117363164549
x24=30.1147529042524x_{24} = 30.1147529042524
x25=86.6634206688687x_{25} = 86.6634206688687
x26=−23.292322206774x_{26} = -23.292322206774
x27=42.6811235186116x_{27} = 42.6811235186116
x28=20.6899749434831x_{28} = 20.6899749434831
x29=−35.8586928211332x_{29} = -35.8586928211332
x30=−95.5489532393393x_{30} = -95.5489532393393
x31=−67.2746193570312x_{31} = -67.2746193570312
x32=14.4067896363035x_{32} = 14.4067896363035
x33=−111.256916507288x_{33} = -111.256916507288
x34=58.3890867865606x_{34} = 58.3890867865606
x35=−29.5755075139536x_{35} = -29.5755075139536
x36=−54.708248742672x_{36} = -54.708248742672
x37=8.12360432912389x_{37} = 8.12360432912389
x38=−86.1241752785699x_{38} = -86.1241752785699
x39=36.397938211432x_{39} = 36.397938211432
x40=45.8227161722014x_{40} = 45.8227161722014
x41=80.3802353616891x_{41} = 80.3802353616891
x42=−45.2834707819026x_{42} = -45.2834707819026
Puntos máximos de la función:
x42=25.7302797812042x_{42} = 25.7302797812042
x42=107.411688774539x_{42} = 107.411688774539
x42=−49.6679439049508x_{42} = -49.6679439049508
x42=60.2877989706919x_{42} = 60.2877989706919
x42=−87.3670557480283x_{42} = -87.3670557480283
x42=−11.9688320618733x_{42} = -11.9688320618733
x42=47.7214283563328x_{42} = 47.7214283563328
x42=32.0134650883838x_{42} = 32.0134650883838
x42=−84.2254630944386x_{42} = -84.2254630944386
x42=54.0046136635124x_{42} = 54.0046136635124
x42=0.597538552485871x_{42} = 0.597538552485871
x42=91.7037255065899x_{42} = 91.7037255065899
x42=3.73913120607566x_{42} = 3.73913120607566
x42=−33.9599806370019x_{42} = -33.9599806370019
x42=75.9957622386409x_{42} = 75.9957622386409
x42=−77.942277787259x_{42} = -77.942277787259
x42=−5.68564675469371x_{42} = -5.68564675469371
x42=−18.2520173690529x_{42} = -18.2520173690529
x42=82.2789475458205x_{42} = 82.2789475458205
x42=104.270096120949x_{42} = 104.270096120949
x42=−90.5086484016181x_{42} = -90.5086484016181
x42=−99.9334263623875x_{42} = -99.9334263623875
x42=−93.6502410552079x_{42} = -93.6502410552079
x42=97.9869108137695x_{42} = 97.9869108137695
x42=−40.2431659441814x_{42} = -40.2431659441814
x42=16.3055018204348x_{42} = 16.3055018204348
x42=−55.9511292121304x_{42} = -55.9511292121304
x42=−27.6767953298223x_{42} = -27.6767953298223
x42=−71.6590924800794x_{42} = -71.6590924800794
x42=−62.23431451931x_{42} = -62.23431451931
x42=38.2966503955634x_{42} = 38.2966503955634
x42=10.0223165132553x_{42} = 10.0223165132553
x42=69.7125769314613x_{42} = 69.7125769314613
Decrece en los intervalos
[96.0881986296381,∞)\left[96.0881986296381, \infty\right)
Crece en los intervalos
(−∞,−2008.77887927552]\left(-\infty, -2008.77887927552\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
2(−esin⁡(2x)sin⁡(2x)+esin⁡(2x)cos⁡2(2x)−cos⁡(2x))=02 \left(- e^{\sin{\left(2 x \right)}} \sin{\left(2 x \right)} + e^{\sin{\left(2 x \right)}} \cos^{2}{\left(2 x \right)} - \cos{\left(2 x \right)}\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=96.8360106018888x_{1} = 96.8360106018888
x2=81.6814092797807x_{2} = 81.6814092797807
x3=28.2743338676393x_{3} = 28.2743338676393
x4=49.7121207980419x_{4} = 49.7121207980419
x5=−79.0931779991396x_{5} = -79.0931779991396
x6=−41.3940661560621x_{6} = -41.3940661560621
x7=−28.2743336158059x_{7} = -28.2743336158059
x8=15.7079635439569x_{8} = 15.7079635439569
x9=19.9963345057749x_{9} = 19.9963345057749
x10=4.28837123782592x_{10} = 4.28837123782592
x11=−83.6762230626883x_{11} = -83.6762230626883
x12=−23.9859626444822x_{12} = -23.9859626444822
x13=−47.6772514632417x_{13} = -47.6772514632417
x14=−67.9682597947393x_{14} = -67.9682597947393
x15=−25.6861028881131x_{15} = -25.6861028881131
x16=87.9645943415359x_{16} = 87.9645943415359
x17=63.978631656032x_{17} = 63.978631656032
x18=74.8448620267603x_{18} = 74.8448620267603
x19=70.2618169632116x_{19} = 70.2618169632116
x20=−69.6684000383702x_{20} = -69.6684000383702
x21=55.9953061052215x_{21} = 55.9953061052215
x22=−87.9645943455885x_{22} = -87.9645943455885
x23=2.58823099419503x_{23} = 2.58823099419503
x24=−94.247779358601x_{24} = -94.247779358601
x25=−100.530965007034x_{25} = -100.530965007034
x26=48.270668388083x_{26} = 48.270668388083
x27=−352.411738861452x_{27} = -352.411738861452
x28=−40.8407044595488x_{28} = -40.8407044595488
x29=59.6902607017326x_{29} = 59.6902607017326
x30=12.0130089549644x_{30} = 12.0130089549644
x31=−53.9604367704212x_{31} = -53.9604367704212
x32=−91.6595486134988x_{32} = -91.6595486134988
x33=50.2654824473634x_{33} = 50.2654824473634
x34=−59.6902604436943x_{34} = -59.6902604436943
x35=24.5793795693236x_{35} = 24.5793795693236
x36=40.2873428372725x_{36} = 40.2873428372725
x37=−45.9771112196108x_{37} = -45.9771112196108
x38=85.9697802311605x_{38} = 85.9697802311605
x39=30.8625648765032x_{39} = 30.8625648765032
x40=−6.28318503569359x_{40} = -6.28318503569359
x41=94.247779609375x_{41} = 94.247779609375
x42=26.2795198129545x_{42} = 26.2795198129545
x43=−97.389372310019x_{43} = -97.389372310019
x44=62.2784914124011x_{44} = 62.2784914124011
x45=−19.4029175809335x_{45} = -19.4029175809335
x46=−78.5398167377463x_{46} = -78.5398167377463
x47=99.9776032554786x_{47} = 99.9776032554786
x48=−37.6991118670078x_{48} = -37.6991118670078
x49=69.1150383073985x_{49} = 69.1150383073985
x50=−13.1197322737539x_{50} = -13.1197322737539
x51=−65.9734457586377x_{51} = -65.9734457586377
x52=−31.9692881952927x_{52} = -31.9692881952927
x53=52.8537134516317x_{53} = 52.8537134516317
x54=84.2696399875297x_{54} = 84.2696399875297
x55=72.256631027912x_{55} = 72.256631027912
x56=−35.1108808488825x_{56} = -35.1108808488825
x57=−17.7027773373026x_{57} = -17.7027773373026
x58=90.5528252947092x_{58} = 90.5528252947092
x59=−72.2566307772889x_{59} = -72.2566307772889
x60=−21.9911485858974x_{60} = -21.9911485858974
x61=0x_{61} = 0
x62=92.2529655383401x_{62} = 92.2529655383401
x63=−81.6814090194997x_{63} = -81.6814090194997
x64=−57.102029424011x_{64} = -57.102029424011
x65=25.1327414331326x_{65} = 25.1327414331326
x66=37.6991121231114x_{66} = 37.6991121231114
x67=−61.6850744875597x_{67} = -61.6850744875597
x68=6.28318528868238x_{68} = 6.28318528868238
x69=−43.9822971720912x_{69} = -43.9822971720912
x70=−9.97813962016414x_{70} = -9.97813962016414
x71=−39.6939259124312x_{71} = -39.6939259124312
x72=8.87141630137462x_{72} = 8.87141630137462
x73=34.004157530093x_{73} = 34.004157530093
x74=−1.99481406935366x_{74} = -1.99481406935366
x75=−50.2654821963477x_{75} = -50.2654821963477
x76=43.9822971711027x_{76} = 43.9822971711027
x77=−3.69495431298456x_{77} = -3.69495431298456
x78=77.9864546803501x_{78} = 77.9864546803501
x79=−63.3852147311906x_{79} = -63.3852147311906
x80=−89.9594083698679x_{80} = -89.9594083698679
x81=−75.9515853455498x_{81} = -75.9515853455498
x82=−85.3763633063192x_{82} = -85.3763633063192
x83=21.9911485856521x_{83} = 21.9911485856521
x84=−15.7079632895347x_{84} = -15.7079632895347
x85=−97.9427339206784x_{85} = -97.9427339206784
x86=46.5705281444521x_{86} = 46.5705281444521
x87=68.5616767195807x_{87} = 68.5616767195807
x88=41.9874830809034x_{88} = 41.9874830809034
x89=65.9734457563877x_{89} = 65.9734457563877
x90=18.296194262144x_{90} = 18.296194262144
x91=93.694417948299x_{91} = 93.694417948299

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
[92.2529655383401,∞)\left[92.2529655383401, \infty\right)
Convexa en los intervalos
(−∞,−89.9594083698679]\left(-\infty, -89.9594083698679\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((esin⁡(2x)+cos⁡(2x))−log⁡(3)2)=⟨−12+12e,12+e2⟩−log⁡(3)2\lim_{x \to -\infty}\left(\frac{\left(e^{\sin{\left(2 x \right)}} + \cos{\left(2 x \right)}\right) - \log{\left(\sqrt{3} \right)}}{2}\right) = \left\langle - \frac{1}{2} + \frac{1}{2 e}, \frac{1}{2} + \frac{e}{2}\right\rangle - \frac{\log{\left(\sqrt{3} \right)}}{2}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−12+12e,12+e2⟩−log⁡(3)2y = \left\langle - \frac{1}{2} + \frac{1}{2 e}, \frac{1}{2} + \frac{e}{2}\right\rangle - \frac{\log{\left(\sqrt{3} \right)}}{2}
lim⁡x→∞((esin⁡(2x)+cos⁡(2x))−log⁡(3)2)=⟨−12+12e,12+e2⟩−log⁡(3)2\lim_{x \to \infty}\left(\frac{\left(e^{\sin{\left(2 x \right)}} + \cos{\left(2 x \right)}\right) - \log{\left(\sqrt{3} \right)}}{2}\right) = \left\langle - \frac{1}{2} + \frac{1}{2 e}, \frac{1}{2} + \frac{e}{2}\right\rangle - \frac{\log{\left(\sqrt{3} \right)}}{2}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−12+12e,12+e2⟩−log⁡(3)2y = \left\langle - \frac{1}{2} + \frac{1}{2 e}, \frac{1}{2} + \frac{e}{2}\right\rangle - \frac{\log{\left(\sqrt{3} \right)}}{2}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (E^sin(2*x) + cos(2*x) - log(sqrt(3)))/2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((esin⁡(2x)+cos⁡(2x))−log⁡(3)2x)=0\lim_{x \to -\infty}\left(\frac{\left(e^{\sin{\left(2 x \right)}} + \cos{\left(2 x \right)}\right) - \log{\left(\sqrt{3} \right)}}{2 x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((esin⁡(2x)+cos⁡(2x))−log⁡(3)2x)=0\lim_{x \to \infty}\left(\frac{\left(e^{\sin{\left(2 x \right)}} + \cos{\left(2 x \right)}\right) - \log{\left(\sqrt{3} \right)}}{2 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:
(esin⁡(2x)+cos⁡(2x))−log⁡(3)2=cos⁡(2x)2−log⁡(3)2+e−sin⁡(2x)2\frac{\left(e^{\sin{\left(2 x \right)}} + \cos{\left(2 x \right)}\right) - \log{\left(\sqrt{3} \right)}}{2} = \frac{\cos{\left(2 x \right)}}{2} - \frac{\log{\left(\sqrt{3} \right)}}{2} + \frac{e^{- \sin{\left(2 x \right)}}}{2}
- No
(esin⁡(2x)+cos⁡(2x))−log⁡(3)2=−cos⁡(2x)2+log⁡(3)2−e−sin⁡(2x)2\frac{\left(e^{\sin{\left(2 x \right)}} + \cos{\left(2 x \right)}\right) - \log{\left(\sqrt{3} \right)}}{2} = - \frac{\cos{\left(2 x \right)}}{2} + \frac{\log{\left(\sqrt{3} \right)}}{2} - \frac{e^{- \sin{\left(2 x \right)}}}{2}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор