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Trazado el gráfico de la función/ asin(x-exp(-x)+sin(x))

Gráfico de la función y = asin(x-exp(-x)+sin(x))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
           /     -x         \
f(x) = asin\x - e   + sin(x)/
f(x)=asin⁡((x−e−x)+sin⁡(x))f{\left(x \right)} = \operatorname{asin}{\left(\left(x - e^{- x}\right) + \sin{\left(x \right)} \right)} 
f = asin(x - exp(-x) + sin(x))
Gráfico de la función
02468-8-6-4-2-10102.5-2.5
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:
asin⁡((x−e−x)+sin⁡(x))=0\operatorname{asin}{\left(\left(x - e^{- x}\right) + \sin{\left(x \right)} \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=0.354463104375025x_{1} = 0.354463104375025
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 asin(x - exp(-x) + sin(x)).
asin⁡(−e−0+sin⁡(0))\operatorname{asin}{\left(- e^{- 0} + \sin{\left(0 \right)} \right)}
Resultado:
f(0)=−π2f{\left(0 \right)} = - \frac{\pi}{2}
Punto:
(0, -pi/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)+1+e−x1−((x−e−x)+sin⁡(x))2=0\frac{\cos{\left(x \right)} + 1 + e^{- x}}{\sqrt{1 - \left(\left(x - e^{- x}\right) + \sin{\left(x \right)}\right)^{2}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=34.5575190257521x_{1} = 34.5575190257521
x2=40.8407042055646x_{2} = 40.8407042055646
x3=53.407075678486x_{3} = 53.407075678486
x4=7693647484.09347x_{4} = 7693647484.09347
x5=1994.91133628953x_{5} = 1994.91133628953
x6=91.1061865656557x_{6} = 91.1061865656557
x7=338104271473.553x_{7} = 338104271473.553
x8=72.2566314913255x_{8} = 72.2566314913255
x9=59.690259898483x_{9} = 59.690259898483
x10=45950706697.6697x_{10} = 45950706697.6697
x11=97.3893717929918x_{11} = 97.3893717929918
x12=78.5398161804631x_{12} = 78.5398161804631
x13=65.9734459617666x_{13} = 65.9734459617666
x14=1.30477648400364⋅1019x_{14} = 1.30477648400364 \cdot 10^{19}
x15=59.6902606103509x_{15} = 59.6902606103509
x16=2.83597330859131⋅1017x_{16} = 2.83597330859131 \cdot 10^{17}
x17=493817.52443939x_{17} = 493817.52443939
x18=40.8407056972374x_{18} = 40.8407056972374
x19=3916036.08340469x_{19} = 3916036.08340469
x20=28.2743277002912x_{20} = 28.2743277002912
x21=4.64378437724278⋅1021x_{21} = 4.64378437724278 \cdot 10^{21}
x22=103.672557698635x_{22} = 103.672557698635
x23=91.1061873703038x_{23} = 91.1061873703038
x24=93308.4434039387x_{24} = 93308.4434039387
x25=59.6902597037041x_{25} = 59.6902597037041
x26=40.8407049716024x_{26} = 40.8407049716024
x27=1.87869545653069⋅1018x_{27} = 1.87869545653069 \cdot 10^{18}
x28=25379704.9688094x_{28} = 25379704.9688094
x29=6.75268505481711⋅1020x_{29} = 6.75268505481711 \cdot 10^{20}
x30=84.8230021293147x_{30} = 84.8230021293147
x31=17168.8038510483x_{31} = 17168.8038510483
x32=562.34509408612x_{32} = 562.34509408612
x33=65.9734452146353x_{33} = 65.9734452146353
x34=1.03161669940448⋅1026x_{34} = 1.03161669940448 \cdot 10^{26}
x35=827032452521617x_{35} = 827032452521617
x36=28.2743327433535x_{36} = 28.2743327433535
x37=84.8229984382144x_{37} = 84.8229984382144
x38=34.5575196816551x_{38} = 34.5575196816551
x39=34.5575191109677x_{39} = 34.5575191109677
x40=97.3893725813073x_{40} = 97.3893725813073
x41=53.4070754239199x_{41} = 53.4070754239199
x42=2.45623023667734⋅1023x_{42} = 2.45623023667734 \cdot 10^{23}
x43=15371049140572.2x_{43} = 15371049140572.2
x44=78.5398164201239x_{44} = 78.5398164201239
x45=47.1238894084874x_{45} = 47.1238894084874
x46=298.451301911224x_{46} = 298.451301911224
x47=1.19032696085133⋅1025x_{47} = 1.19032696085133 \cdot 10^{25}
x48=72.2566310277176x_{48} = 72.2566310277176
x49=47.1238902134786x_{49} = 47.1238902134786
x50=223.053081796832x_{50} = 223.053081796832
x51=97.3893730048998x_{51} = 97.3893730048998
x52=1096421984.19966x_{52} = 1096421984.19966
x53=172.787595385667x_{53} = 172.787595385667
x54=72.2566306404027x_{54} = 72.2566306404027
x55=153.938042340335x_{55} = 153.938042340335
x56=3.51927461702689⋅1022x_{56} = 3.51927461702689 \cdot 10^{22}
x57=28.2743346459299x_{57} = 28.2743346459299
x58=78.5398168458379x_{58} = 78.5398168458379
x59=110377634145415x_{59} = 110377634145415
x60=3.74322149756244⋅1016x_{60} = 3.74322149756244 \cdot 10^{16}
x61=65.973445752981x_{61} = 65.973445752981
x62=9.91457705060988⋅1019x_{62} = 9.91457705060988 \cdot 10^{19}
x63=144414612.914214x_{63} = 144414612.914214
x64=84.8230013633188x_{64} = 84.8230013633188
x65=5.3130322669104⋅1015x_{65} = 5.3130322669104 \cdot 10^{15}
x66=1.68198374902905⋅1024x_{66} = 1.68198374902905 \cdot 10^{24}
x67=2475462926024.34x_{67} = 2475462926024.34
x68=53.4070746361996x_{68} = 53.4070746361996
Signos de extremos en los puntos:
(34.5575190257521, 1.5707963267949 - 4.23556293215294*I)

(40.84070420556457, 1.5707963267949 - 4.40267650675435*I)

(53.40707567848595, 1.5707963267949 - 4.6710027508573*I)

(7693647484.093472, 1.5707963267949 - 23.4568080037987*I)

(1994.9113362895323, 1.5707963267949 - 8.29150200248277*I)

(91.1061865656557, 1.5707963267949 - 5.20514277577705*I)

(338104271473.5526, 1.5707963267949 - 27.2397673608983*I)

(72.2566314913255, 1.5707963267949 - 4.9733233955418*I)

(59.69025989848298, 1.5707963267949 - 4.78224587116483*I)

(45950706697.66973, 1.5707963267949 - 25.2439822458997*I)

(97.38937179299177, 1.5707963267949 - 5.27183791158392*I)

(78.53981618046308, 1.5707963267949 - 5.05671236034003*I)

(65.97344596176659, 1.5707963267949 - 4.88234206087068*I)

(1.3047764840036354e+19, 1.5707963267949 - 44.7082956969264*I)

(59.69026061035092, 1.5707963267949 - 4.78224587116483*I)

(2.835973308591315e+17, 1.5707963267949 - 40.8794789584451*I)

(493817.52443938976, 1.5707963267949 - 13.8030685247649*I)

(40.84070569723741, 1.5707963267949 - 4.40267650675435*I)

(3916036.0834046914, 1.5707963267949 - 15.8737376775025*I)

(28.27432770029123, 1.5707963267949 - 4.0347887772505*I)

(4.64378437724278e+21, 1.5707963267949 - 50.582963765797*I)

(103.6725576986345, 1.5707963267949 - 5.33436136692106*I)

(91.10618737030384, 1.5707963267949 - 5.20514277577705*I)

(93308.44340393873, 1.5707963267949 - 12.1368130606383*I)

(59.69025970370411, 1.5707963267949 - 4.78224587116483*I)

(40.840704971602435, 1.5707963267949 - 4.40267650675435*I)

(1.8786954565306913e+18, 1.5707963267949 - 42.7702564843297*I)

(25379704.968809355, 1.5707963267949 - 17.7426075761885*I)

(6.752685054817113e+20, 1.5707963267949 - 48.6547892521283*I)

(84.82300212931469, 1.5707963267949 - 5.13367918396892*I)

(17168.80385104833, 1.5707963267949 - 10.4439964661719*I)

(562.34509408612, 1.5707963267949 - 7.02526208169032*I)

(65.97344521463529, 1.5707963267949 - 4.88234206087068*I)

(1.0316166994044835e+26, 1.5707963267949 - 60.591486781167*I)

(827032452521616.5, 1.5707963267949 - 35.0420122320018*I)

(28.274332743353455, 1.5707963267949 - 4.0347887772505*I)

(84.8229984382144, 1.5707963267949 - 5.13367918396892*I)

(34.55751968165507, 1.5707963267949 - 4.23556293215294*I)

(34.557519110967654, 1.5707963267949 - 4.23556293215294*I)

(97.38937258130734, 1.5707963267949 - 5.27183791158392*I)

(53.40707542391989, 1.5707963267949 - 4.6710027508573*I)

(2.4562302366773418e+23, 1.5707963267949 - 54.5512320698896*I)

(15371049140572.23, 1.5707963267949 - 31.0566541106927*I)

(78.53981642012393, 1.5707963267949 - 5.05671236034003*I)

(47.12388940848737, 1.5707963267949 - 4.54581466940278*I)

(298.45130191122405, 1.5707963267949 - 6.39175115131708*I)

(1.1903269608513273e+25, 1.5707963267949 - 58.4320025318137*I)

(72.25663102771763, 1.5707963267949 - 4.9733233955418*I)

(47.12389021347864, 1.5707963267949 - 4.54581466940278*I)

(223.05308179683172, 1.5707963267949 - 6.10055191855742*I)

(97.38937300489984, 1.5707963267949 - 5.27183791158392*I)

(1096421984.1996603, 1.5707963267949 - 21.5084651540102*I)

(172.78759538566695, 1.5707963267949 - 5.8452018778851*I)

(72.2566306404027, 1.5707963267949 - 4.9733233955418*I)

(153.93804234033496, 1.5707963267949 - 5.72968681445884*I)

(3.519274617026894e+22, 1.5707963267949 - 52.6082741200944*I)

(28.274334645929876, 1.5707963267949 - 4.0347887772505*I)

(78.53981684583788, 1.5707963267949 - 5.05671236034003*I)

(110377634145414.56, 1.5707963267949 - 33.028075820545*I)

(3.743221497562444e+16, 1.5707963267949 - 38.8544552721155*I)

(65.97344575298098, 1.5707963267949 - 4.88234206087068*I)

(9.914577050609882e+19, 1.5707963267949 - 46.7362700509807*I)

(144414612.91421437, 1.5707963267949 - 19.4813461573297*I)

(84.82300136331877, 1.5707963267949 - 5.13367918396892*I)

(5313032266910401, 1.5707963267949 - 36.9020862961208*I)

(1.6819837490290492e+24, 1.5707963267949 - 56.4751633122261*I)

(2475462926024.3364, 1.5707963267949 - 29.2305957158371*I)

(53.40707463619957, 1.5707963267949 - 4.6710027508573*I)


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:
La función no tiene puntos mínimos
La función no tiene puntos máximos
No cambia el valor en todo el eje numérico
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)−e−x+(x+sin⁡(x)−e−x)(cos⁡(x)+1+e−x)21−(x+sin⁡(x)−e−x)21−(x+sin⁡(x)−e−x)2=0\frac{- \sin{\left(x \right)} - e^{- x} + \frac{\left(x + \sin{\left(x \right)} - e^{- x}\right) \left(\cos{\left(x \right)} + 1 + e^{- x}\right)^{2}}{1 - \left(x + \sin{\left(x \right)} - e^{- x}\right)^{2}}}{\sqrt{1 - \left(x + \sin{\left(x \right)} - e^{- x}\right)^{2}}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=81.6324112946227x_{1} = 81.6324112946227
x2=−65.6174811491438x_{2} = -65.6174811491438
x3=−73.9985798423162x_{3} = -73.9985798423162
x4=91.106186954104x_{4} = 91.106186954104
x5=−43.5174717589413x_{5} = -43.5174717589413
x6=62.768131840314x_{6} = 62.768131840314
x7=−100x_{7} = -100
x8=−78x_{8} = -78
x9=−72x_{9} = -72
x10=−74.8332499179354x_{10} = -74.8332499179354
x11=100.491161731562x_{11} = 100.491161731562
x12=−61.3320627505328x_{12} = -61.3320627505328
x13=−84x_{13} = -84
x14=78.5398163397448x_{14} = 78.5398163397448
x15=−68.9420813553217x_{15} = -68.9420813553217
x16=47.1238898038469x_{16} = 47.1238898038469
x17=−57.2892398235793x_{17} = -57.2892398235793
x18=−68.0747882615809x_{18} = -68.0747882615809
x19=−30.9793085211899x_{19} = -30.9793085211899
x20=40.8407044966673x_{20} = 40.8407044966673
x21=−98x_{21} = -98
x22=12.2402395425543x_{22} = 12.2402395425543
x23=−36.1613567108591x_{23} = -36.1613567108591
x24=−59.4326804957891x_{24} = -59.4326804957891
x25=−39.776101681403x_{25} = -39.776101681403
x26=5.56233920333362x_{26} = 5.56233920333362
x27=97.3893722612836x_{27} = 97.3893722612836
x28=75.3451377702063x_{28} = 75.3451377702063
x29=−76x_{29} = -76
x30=53.4070751110265x_{30} = 53.4070751110265
x31=−69.5362836500951x_{31} = -69.5362836500951
x32=21.99114857541x_{32} = 21.99114857541
x33=−88x_{33} = -88
x34=−53.5394968770946x_{34} = -53.5394968770946
x35=−82x_{35} = -82
x36=43.891178329751x_{36} = 43.891178329751
x37=−80x_{37} = -80
x38=31.2881254789911x_{38} = 31.2881254789911
x39=−47.6743358674202x_{39} = -47.6743358674202
x40=94.2053207535601x_{40} = 94.2053207535601
x41=3.18368261799658x_{41} = 3.18368261799658
x42=−66.3036782532999x_{42} = -66.3036782532999
x43=−34.2379542328948x_{43} = -34.2379542328948
x44=−227.832981568342x_{44} = -227.832981568342
x45=−74x_{45} = -74
x46=−67.1432355423531x_{46} = -67.1432355423531
x47=−92x_{47} = -92
x48=−45.531456569077x_{48} = -45.531456569077
x49=28.2743338823087x_{49} = 28.2743338823087
x50=−89.75x_{50} = -89.75
x51=−37.6609370705441x_{51} = -37.6609370705441
x52=−51.3838378084674x_{52} = -51.3838378084674
x53=−68x_{53} = -68
x54=−41.8603414722947x_{54} = -41.8603414722947
x55=34.5575191894877x_{55} = 34.5575191894877
x56=−49.434019319435x_{56} = -49.434019319435
x57=−94x_{57} = -94
x58=50.1857891154357x_{58} = 50.1857891154357
x59=69.0571193521472x_{59} = 69.0571193521472
x60=−180.322097543255x_{60} = -180.322097543255
x61=−31.9581111950784x_{61} = -31.9581111950784
x62=18.6351058599102x_{62} = 18.6351058599102
x63=−63.2165186697298x_{63} = -63.2165186697298
x64=87.9190998881179x_{64} = 87.9190998881179
x65=−70x_{65} = -70
x66=37.5927331737907x_{66} = 37.5927331737907
x67=−86x_{67} = -86
x68=59.6902604182061x_{68} = 59.6902604182061
x69=72.2566310325652x_{69} = 72.2566310325652
x70=−69.0003462863994x_{70} = -69.0003462863994
x71=65.9734457253857x_{71} = 65.9734457253857
x72=−96x_{72} = -96
x73=1027.30079772386x_{73} = 1027.30079772386
x74=56.4778509119883x_{74} = 56.4778509119883
x75=9.42485865447416x_{75} = 9.42485865447416
x76=−55.3794557131517x_{76} = -55.3794557131517
x77=−1.904069216847x_{77} = -1.904069216847
x78=84.8230016469244x_{78} = 84.8230016469244
x79=−66.7546885271059x_{79} = -66.7546885271059
x80=15.7079634186507x_{80} = 15.7079634186507
x81=24.9726498832111x_{81} = 24.9726498832111

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:
No tiene corvaduras en todo el eje numérico
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞asin⁡((x−e−x)+sin⁡(x))=∞i\lim_{x \to -\infty} \operatorname{asin}{\left(\left(x - e^{- x}\right) + \sin{\left(x \right)} \right)} = \infty i
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞asin⁡((x−e−x)+sin⁡(x))=−∞i\lim_{x \to \infty} \operatorname{asin}{\left(\left(x - e^{- x}\right) + \sin{\left(x \right)} \right)} = - \infty i
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 asin(x - exp(-x) + sin(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→−∞(asin⁡((x−e−x)+sin⁡(x))x)y = x \lim_{x \to -\infty}\left(\frac{\operatorname{asin}{\left(\left(x - e^{- x}\right) + \sin{\left(x \right)} \right)}}{x}\right)
True

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