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Trazado el gráfico de la función/ ln(2𝑥+1)−sin𝑥−2

Gráfico de la función y = ln(2𝑥+1)−sin𝑥−2

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = log(2*x + 1) - sin(x) - 2
f(x)=(log⁡(2x+1)−sin⁡(x))−2f{\left(x \right)} = \left(\log{\left(2 x + 1 \right)} - \sin{\left(x \right)}\right) - 2 
f = log(2*x + 1) - sin(x) - 2
Gráfico de la función
-0.55.00.00.51.01.52.02.53.03.54.04.5-1010
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:
(log⁡(2x+1)−sin⁡(x))−2=0\left(\log{\left(2 x + 1 \right)} - \sin{\left(x \right)}\right) - 2 = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=8.35997888477279x_{1} = 8.35997888477279
x2=3.1529191108479x_{2} = 3.1529191108479
x3=7.08624407834269x_{3} = 7.08624407834269
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 log(2*x + 1) - sin(x) - 2.
−2+(log⁡(0⋅2+1)−sin⁡(0))-2 + \left(\log{\left(0 \cdot 2 + 1 \right)} - \sin{\left(0 \right)}\right)
Resultado:
f(0)=−2f{\left(0 \right)} = -2
Punto:
(0, -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
−cos⁡(x)+22x+1=0- \cos{\left(x \right)} + \frac{2}{2 x + 1} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−14.210170323167x_{1} = -14.210170323167
x2=−86.3821538479823x_{2} = -86.3821538479823
x3=−10.8992648262709x_{3} = -10.8992648262709
x4=−64.4182950098909x_{4} = -64.4182950098909
x5=45.5313674513061x_{5} = 45.5313674513061
x6=−26.7416541384627x_{6} = -26.7416541384627
x7=20.3724237964937x_{7} = 20.3724237964937
x8=11.0820226283739x_{8} = 11.0820226283739
x9=−54.9595081393237x_{9} = -54.9595081393237
x10=70.6717837314473x_{10} = 70.6717837314473
x11=−98.9500109742144x_{11} = -98.9500109742144
x12=−548.209743837005x_{12} = -548.209743837005
x13=7.73220677712066x_{13} = 7.73220677712066
x14=−61.2445936309359x_{14} = -61.2445936309359
x15=−4.45689497108666x_{15} = -4.45689497108666
x16=−23.5184878848326x_{16} = -23.5184878848326
x17=58.1023991153639x_{17} = 58.1023991153639
x18=−45.5752804060962x_{18} = -45.5752804060962
x19=−48.6739265226404x_{19} = -48.6739265226404
x20=−67.5293226548269x_{20} = -67.5293226548269
x21=83.2402633494593x_{21} = 83.2402633494593
x22=−29.8110067105449x_{22} = -29.8110067105449
x23=26.6667194934628x_{23} = 26.6667194934628
x24=42.4347940628202x_{24} = 42.4347940628202
x25=−51.8557520304499x_{25} = -51.8557520304499
x26=−79157.1393088084x_{26} = -79157.1393088084
x27=92.6877145151468x_{27} = 92.6877145151468
x28=86.4053050050669x_{28} = 86.4053050050669
x29=54.9958917649533x_{29} = 54.9958917649533
x30=−36.1002221086352x_{30} = -36.1002221086352
x31=61.2772446420462x_{31} = 61.2772446420462
x32=95.8081924152394x_{32} = 95.8081924152394
x33=64.3872374373379x_{33} = 64.3872374373379
x34=80.1230163905294x_{34} = 80.1230163905294
x35=4.89869484147589x_{35} = 4.89869484147589
x36=−17.2189113668301x_{36} = -17.2189113668301
x37=89.5242822847509x_{37} = 89.5242822847509
x38=−89.5466209357674x_{38} = -89.5466209357674
x39=−33.0174804022092x_{39} = -33.0174804022092
x40=−39.2956870860989x_{40} = -39.2956870860989
x41=−83.264288120036x_{41} = -83.264288120036
x42=29.8780546574922x_{42} = 29.8780546574922
x43=−73.8137869371062x_{43} = -73.8137869371062
x44=23.6034446589144x_{44} = 23.6034446589144
x45=−42.3876251558751x_{45} = -42.3876251558751
x46=−20.4704471892992x_{46} = -20.4704471892992
x47=39.2447449554615x_{47} = 39.2447449554615
x48=73.8408793143147x_{48} = 73.8408793143147
x49=−92.666133095611x_{49} = -92.666133095611
x50=51.8171634340025x_{50} = 51.8171634340025
x51=76.9561091167815x_{51} = 76.9561091167815
x52=−58.1368149838507x_{52} = -58.1368149838507
x53=36.1555998626613x_{53} = 36.1555998626613
x54=98.9702220173879x_{54} = 98.9702220173879
x55=14.0684715437106x_{55} = 14.0684715437106
x56=−7.98793008497043x_{56} = -7.98793008497043
x57=−80.0980492140798x_{57} = -80.0980492140798
x58=48.7150065347936x_{58} = 48.7150065347936
x59=−2.19976977669838x_{59} = -2.19976977669838
x60=17.3348589851056x_{60} = 17.3348589851056
x61=32.9568291465193x_{61} = 32.9568291465193
x62=−76.9820953411054x_{62} = -76.9820953411054
x63=−70.7000801855538x_{63} = -70.7000801855538
x64=−95.8290661069166x_{64} = -95.8290661069166
x65=67.5589357286864x_{65} = 67.5589357286864
Signos de extremos en los puntos:
(-14.210170323167045, 2.30862153371331 + pi*I)

(-86.38215384798234, 2.14619102506111 + pi*I)

(-10.899264826270889, 0.0395164669054306 + pi*I)

(-64.41829500989087, 3.85063041776965 + pi*I)

(45.531367451306124, 1.5227062465895)

(-26.741654138462703, 2.95976883246198 + pi*I)

(20.372423796493738, 0.73272438113125)

(11.08202262837387, 2.13886697182315)

(-54.959508139323695, 1.69077323679749 + pi*I)

(70.67178373144728, 1.95834233784088)

(-98.95001097421438, 2.28274768523115 + pi*I)

(-548.2097438370049, 5.9988909959091 + pi*I)

(7.732206777120656, -0.191393302728559)

(-61.244593630935896, 1.79996077909255 + pi*I)

(-4.456894971086657, -0.898931765989503 + pi*I)

(-23.518487884832552, 0.830389002250915 + pi*I)

(58.102399115363895, 1.76406842001032)

(-45.575280406096155, 3.50123505069471 + pi*I)

(-48.67392652264038, 1.56818058466373 + pi*I)

(-67.52932265482691, 1.89838864791829 + pi*I)

(83.24026334945925, 2.120938390477)

(-29.811006710544856, 1.07169243177708 + pi*I)

(26.66671949346278, 0.995817564966056)

(42.434794062820195, 3.45255845295168)

(-51.855752030449906, 3.63173452916589 + pi*I)

(-79157.1393088084, 10.9723311248842 + pi*I)

(92.68771451514681, 4.22770549591399)

(86.40530500506695, 4.15790005305228)

(54.995891764953306, 3.70929381461823)

(-36.10022210863515, 1.26589365122825 + pi*I)

(61.277244642046206, 3.81655124639795)

(95.80819241523943, 2.26075447580661)

(64.38723743733792, 1.86591689720012)

(80.12301639052939, 4.08285442726693)

(4.898694841475885, 1.36199961288421)

(-17.21891136683013, 0.511478047156555 + pi*I)

(89.52428228475091, 2.19328831466655)

(-89.54662093576744, 4.18224418462834 + pi*I)

(-33.01748040220919, 3.17445200800161 + pi*I)

(-39.2956870860989, 3.35112400557588 + pi*I)

(-83.264288120036, 4.1090708498765 + pi*I)

(29.87805465749225, 3.10632568132984)

(-73.81378693710617, 1.98798889006609 + pi*I)

(23.603444658914377, 2.87464095141216)

(-42.38762515587508, 1.42842263173588 + pi*I)

(-20.47044718929924, 2.68614623161461 + pi*I)

(39.244744955461464, 1.37594138751729)

(73.8408793143147, 4.00171769745713)

(-92.666133095611, 2.21679878632267 + pi*I)

(51.81716343400253, 1.65065436327235)

(76.95610911678155, 2.04294196697677)

(-58.13681498385068, 3.74715817016035 + pi*I)

(36.155599862661255, 3.29434119551121)

(98.97022201738785, 4.29295496843048)

(14.068471543710626, 0.374365491985265)

(-7.9879300849704284, 1.6974818943874 + pi*I)

(-80.09804921407977, 2.07021568487996 + pi*I)

(48.71500653479358, 3.58913931611334)

(-2.19976977669838, 0.0322718660420582 + pi*I)

(17.33485898510556, 2.57272893454596)

(32.956829146519254, 1.20384988999262)

(-76.9820953411054, 4.03011836528495 + pi*I)

(-70.70008018555384, 3.94439516870419 + pi*I)

(-95.82906610691663, 4.2504269192394 + pi*I)

(67.55893572868638, 3.91341326213692)


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=45.5313674513061x_{1} = 45.5313674513061
x2=20.3724237964937x_{2} = 20.3724237964937
x3=70.6717837314473x_{3} = 70.6717837314473
x4=7.73220677712066x_{4} = 7.73220677712066
x5=58.1023991153639x_{5} = 58.1023991153639
x6=83.2402633494593x_{6} = 83.2402633494593
x7=26.6667194934628x_{7} = 26.6667194934628
x8=95.8081924152394x_{8} = 95.8081924152394
x9=64.3872374373379x_{9} = 64.3872374373379
x10=89.5242822847509x_{10} = 89.5242822847509
x11=39.2447449554615x_{11} = 39.2447449554615
x12=51.8171634340025x_{12} = 51.8171634340025
x13=76.9561091167815x_{13} = 76.9561091167815
x14=14.0684715437106x_{14} = 14.0684715437106
x15=32.9568291465193x_{15} = 32.9568291465193
Puntos máximos de la función:
x15=11.0820226283739x_{15} = 11.0820226283739
x15=42.4347940628202x_{15} = 42.4347940628202
x15=92.6877145151468x_{15} = 92.6877145151468
x15=86.4053050050669x_{15} = 86.4053050050669
x15=54.9958917649533x_{15} = 54.9958917649533
x15=61.2772446420462x_{15} = 61.2772446420462
x15=80.1230163905294x_{15} = 80.1230163905294
x15=4.89869484147589x_{15} = 4.89869484147589
x15=29.8780546574922x_{15} = 29.8780546574922
x15=23.6034446589144x_{15} = 23.6034446589144
x15=73.8408793143147x_{15} = 73.8408793143147
x15=36.1555998626613x_{15} = 36.1555998626613
x15=98.9702220173879x_{15} = 98.9702220173879
x15=48.7150065347936x_{15} = 48.7150065347936
x15=17.3348589851056x_{15} = 17.3348589851056
x15=67.5589357286864x_{15} = 67.5589357286864
Decrece en los intervalos
[95.8081924152394,∞)\left[95.8081924152394, \infty\right)
Crece en los intervalos
(−∞,7.73220677712066]\left(-\infty, 7.73220677712066\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
sin⁡(x)−4(2x+1)2=0\sin{\left(x \right)} - \frac{4}{\left(2 x + 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=25.1342630319257x_{1} = 25.1342630319257
x2=−78.539980536926x_{2} = -78.539980536926
x3=−65.9736789994812x_{3} = -65.9736789994812
x4=−97.3894787851108x_{4} = -97.3894787851108
x5=56.5489750234055x_{5} = 56.5489750234055
x6=−3.2720946952727x_{6} = -3.2720946952727
x7=−59.690545845379x_{7} = -59.690545845379
x8=40.840119360999x_{8} = 40.840119360999
x9=−43.9817682355704x_{9} = -43.9817682355704
x10=−34.5583812783767x_{10} = -34.5583812783767
x11=94.2478910014625x_{11} = 94.2478910014625
x12=−9.43729780388242x_{12} = -9.43729780388242
x13=72.256442121631x_{13} = 72.256442121631
x14=91.1060677883354x_{14} = 91.1060677883354
x15=−56.5483494369744x_{15} = -56.5483494369744
x16=87.964722079597x_{16} = 87.964722079597
x17=−91.1063087641551x_{17} = -91.1063087641551
x18=−411.548643538789x_{18} = -411.548643538789
x19=6.30478289340424x_{19} = 6.30478289340424
x20=84.82286428412x_{20} = 84.82286428412
x21=18.8522260899739x_{21} = 18.8522260899739
x22=−47.1243498214366x_{22} = -47.1243498214366
x23=−471.238893525734x_{23} = -471.238893525734
x24=3.06272751415172x_{24} = 3.06272751415172
x25=65.9732194139093x_{25} = 65.9732194139093
x26=37.6997971386419x_{26} = 37.6997971386419
x27=−37.6983891535344x_{27} = -37.6983891535344
x28=47.1234488853416x_{28} = 47.1234488853416
x29=−62.831595686629x_{29} = -62.831595686629
x30=−6.25296614180442x_{30} = -6.25296614180442
x31=100.531062884208x_{31} = 100.531062884208
x32=50.2658704794244x_{32} = 50.2658704794244
x33=−72.2568252434519x_{33} = -72.2568252434519
x34=0.726774476611794x_{34} = 0.726774476611794
x35=34.5567055016074x_{35} = 34.5567055016074
x36=−100.530864976575x_{36} = -100.530864976575
x37=−84.8231422861406x_{37} = -84.8231422861406
x38=−28.2756300831x_{38} = -28.2756300831
x39=−53.4074323564208x_{39} = -53.4074323564208
x40=−87.964463582113x_{40} = -87.964463582113
x41=−81.6812572573959x_{41} = -81.6812572573959
x42=−119.38045007783x_{42} = -119.38045007783
x43=43.9828025275635x_{43} = 43.9828025275635
x44=75.3983972801876x_{44} = 75.3983972801876
x45=−25.1310929417285x_{45} = -25.1310929417285
x46=78.5396562697555x_{46} = 78.5396562697555
x47=−12.5594944663359x_{47} = -12.5594944663359
x48=−1709.02640321027x_{48} = -1709.02640321027
x49=59.6899843912195x_{49} = 59.6899843912195
x50=−40.8413189654733x_{50} = -40.8413189654733
x51=69.1152447227139x_{51} = 69.1152447227139
x52=15.7041548177396x_{52} = 15.7041548177396
x53=28.2731259961328x_{53} = 28.2731259961328
x54=−94.2476658242499x_{54} = -94.2476658242499
x55=−21.9933132546193x_{55} = -21.9933132546193
x56=−75.3980454240487x_{56} = -75.3980454240487
x57=9.41460478197694x_{57} = 9.41460478197694
x58=−69.1148259743006x_{58} = -69.1148259743006
x59=97.3892679023002x_{59} = 97.3892679023002
x60=12.572222604816x_{60} = 12.572222604816
x61=−50.2650786720273x_{61} = -50.2650786720273
x62=53.4067309877761x_{62} = 53.4067309877761
x63=31.416908189899x_{63} = 31.416908189899
x64=21.9891713625003x_{64} = 21.9891713625003
x65=62.832102388969x_{65} = 62.832102388969
x66=81.6815570579459x_{66} = 81.6815570579459
x67=−15.7122845485786x_{67} = -15.7122845485786
x68=−31.414880214895x_{68} = -31.414880214895
x69=−18.8465850069671x_{69} = -18.8465850069671

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
[100.531062884208,∞)\left[100.531062884208, \infty\right)
Convexa en los intervalos
(−∞,−1709.02640321027]\left(-\infty, -1709.02640321027\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((log⁡(2x+1)−sin⁡(x))−2)=∞\lim_{x \to -\infty}\left(\left(\log{\left(2 x + 1 \right)} - \sin{\left(x \right)}\right) - 2\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞((log⁡(2x+1)−sin⁡(x))−2)=∞\lim_{x \to \infty}\left(\left(\log{\left(2 x + 1 \right)} - \sin{\left(x \right)}\right) - 2\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 log(2*x + 1) - sin(x) - 2, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((log⁡(2x+1)−sin⁡(x))−2x)=0\lim_{x \to -\infty}\left(\frac{\left(\log{\left(2 x + 1 \right)} - \sin{\left(x \right)}\right) - 2}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((log⁡(2x+1)−sin⁡(x))−2x)=0\lim_{x \to \infty}\left(\frac{\left(\log{\left(2 x + 1 \right)} - \sin{\left(x \right)}\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:
(log⁡(2x+1)−sin⁡(x))−2=log⁡(1−2x)+sin⁡(x)−2\left(\log{\left(2 x + 1 \right)} - \sin{\left(x \right)}\right) - 2 = \log{\left(1 - 2 x \right)} + \sin{\left(x \right)} - 2
- No
(log⁡(2x+1)−sin⁡(x))−2=−log⁡(1−2x)−sin⁡(x)+2\left(\log{\left(2 x + 1 \right)} - \sin{\left(x \right)}\right) - 2 = - \log{\left(1 - 2 x \right)} - \sin{\left(x \right)} + 2
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор