Sr Examen

Otras calculadoras

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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
             /       x\    
          log\1 + x*E /    
f(x) = --------------------
          /       ________\
          |      /      2 |
       log\x + \/  1 + x  /
f(x)=log⁡(exx+1)log⁡(x+x2+1)f{\left(x \right)} = \frac{\log{\left(e^{x} x + 1 \right)}}{\log{\left(x + \sqrt{x^{2} + 1} \right)}} 
f = log(E^x*x + 1)/log(x + sqrt(x^2 + 1))
Gráfico de la función
02468-8-6-4-2-101005
Dominio de definición de la función
Puntos en los que la función no está definida exactamente:
x1=0x_{1} = 0
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⁡(exx+1)log⁡(x+x2+1)=0\frac{\log{\left(e^{x} x + 1 \right)}}{\log{\left(x + \sqrt{x^{2} + 1} \right)}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−43.3994514654678x_{1} = -43.3994514654678
x2=−33.7002291278265x_{2} = -33.7002291278265
x3=−45.7597410257952x_{3} = -45.7597410257952
x4=−84x_{4} = -84
x5=−51.2941476996224x_{5} = -51.2941476996224
x6=−68x_{6} = -68
x7=−90x_{7} = -90
x8=−46.53087513762x_{8} = -46.53087513762
x9=−38.3836333251876x_{9} = -38.3836333251876
x10=−45.3713261043746x_{10} = -45.3713261043746
x11=−45.3682799020102x_{11} = -45.3682799020102
x12=−98x_{12} = -98
x13=−37.0441473368719x_{13} = -37.0441473368719
x14=−44.5884552810949x_{14} = -44.5884552810949
x15=−40.1081793647041x_{15} = -40.1081793647041
x16=−48.8029498623465x_{16} = -48.8029498623465
x17=−37.4956282276754x_{17} = -37.4956282276754
x18=−32.6632900979411x_{18} = -32.6632900979411
x19=−39.2537917748258x_{19} = -39.2537917748258
x20=−35.1786456692558x_{20} = -35.1786456692558
x21=−387.445030565266x_{21} = -387.445030565266
x22=−34.6946355029704x_{22} = -34.6946355029704
x23=−35.6549717870873x_{23} = -35.6549717870873
x24=−47.2948294456441x_{24} = -47.2948294456441
x25=−65.2185316851804x_{25} = -65.2185316851804
x26=−92x_{26} = -92
x27=−42.592245598174x_{27} = -42.592245598174
x28=−39.4760749477738x_{28} = -39.4760749477738
x29=−86x_{29} = -86
x30=−67.485973353229x_{30} = -67.485973353229
x31=−49.3161717843167x_{31} = -49.3161717843167
x32=−42.1855534362529x_{32} = -42.1855534362529
x33=−100x_{33} = -100
x34=−36.1242900447651x_{34} = -36.1242900447651
x35=−59.2246697567999x_{35} = -59.2246697567999
x36=−33.1877436645797x_{36} = -33.1877436645797
x37=−41.3638095051796x_{37} = -41.3638095051796
x38=−76x_{38} = -76
x39=−42.9963140722135x_{39} = -42.9963140722135
x40=−68.0275704279569x_{40} = -68.0275704279569
x41=−47.3407233500386x_{41} = -47.3407233500386
x42=−48.4282673214477x_{42} = -48.4282673214477
x43=−37.5240176245332x_{43} = -37.5240176245332
x44=−78x_{44} = -78
x45=−31.7343085974711x_{45} = -31.7343085974711
x46=−48.0520533355827x_{46} = -48.0520533355827
x47=−34.2021456592464x_{47} = -34.2021456592464
x48=−57.2397727579464x_{48} = -57.2397727579464
x49=−41.4350316533908x_{49} = -41.4350316533908
x50=−67.528274142588x_{50} = -67.528274142588
x51=−39.6828434480538x_{51} = -39.6828434480538
x52=−36.5871809504796x_{52} = -36.5871809504796
x53=−30.4023040253644x_{53} = -30.4023040253644
x54=−63.1970847942379x_{54} = -63.1970847942379
x55=−30.9980586467113x_{55} = -30.9980586467113
x56=−392.30577478002x_{56} = -392.30577478002
x57=−82x_{57} = -82
x58=−38.820803308266x_{58} = -38.820803308266
x59=−33.6495469060591x_{59} = -33.6495469060591
x60=−31.5710171171308x_{60} = -31.5710171171308
x61=−88x_{61} = -88
x62=−393.262799640241x_{62} = -393.262799640241
x63=−29.7781851770083x_{63} = -29.7781851770083
x64=−41.7761179000335x_{64} = -41.7761179000335
x65=−32.1251230266095x_{65} = -32.1251230266095
x66=−46.9137204075844x_{66} = -46.9137204075844
x67=−72x_{67} = -72
x68=−40.5299999850927x_{68} = -40.5299999850927
x69=−46.146235748999x_{69} = -46.146235748999
x70=−55.2562382064487x_{70} = -55.2562382064487
x71=−66.8449972607667x_{71} = -66.8449972607667
x72=−74x_{72} = -74
x73=−61.2124560959278x_{73} = -61.2124560959278
x74=−35.5808727505208x_{74} = -35.5808727505208
x75=−391.113901380411x_{75} = -391.113901380411
x76=−67.3375706608183x_{76} = -67.3375706608183
x77=−37.9420096728669x_{77} = -37.9420096728669
x78=−53.2742716967352x_{78} = -53.2742716967352
x79=−393.431988661188x_{79} = -393.431988661188
x80=−43.3978697635907x_{80} = -43.3978697635907
x81=−43.7970156806803x_{81} = -43.7970156806803
x82=−47.6742561836685x_{82} = -47.6742561836685
x83=−387.356894030113x_{83} = -387.356894030113
x84=−29.8411235040176x_{84} = -29.8411235040176
x85=−44.9809219917996x_{85} = -44.9809219917996
x86=−44.1938477072055x_{86} = -44.1938477072055
x87=−40.9484877961702x_{87} = -40.9484877961702
x88=−70x_{88} = -70
x89=−96x_{89} = -96
x90=−80x_{90} = -80
x91=−94x_{91} = -94
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(1 + x*E^x)/log(x + sqrt(1 + x^2)).
log⁡(0e0+1)log⁡(02+1)\frac{\log{\left(0 e^{0} + 1 \right)}}{\log{\left(\sqrt{0^{2} + 1} \right)}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
Asíntotas verticales
Hay:
x1=0x_{1} = 0
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(log⁡(exx+1)log⁡(x+x2+1))=0\lim_{x \to -\infty}\left(\frac{\log{\left(e^{x} x + 1 \right)}}{\log{\left(x + \sqrt{x^{2} + 1} \right)}}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞(log⁡(exx+1)log⁡(x+x2+1))=∞\lim_{x \to \infty}\left(\frac{\log{\left(e^{x} x + 1 \right)}}{\log{\left(x + \sqrt{x^{2} + 1} \right)}}\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(1 + x*E^x)/log(x + sqrt(1 + x^2)), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(log⁡(exx+1)xlog⁡(x+x2+1))=0\lim_{x \to -\infty}\left(\frac{\log{\left(e^{x} x + 1 \right)}}{x \log{\left(x + \sqrt{x^{2} + 1} \right)}}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(log⁡(exx+1)xlog⁡(x+x2+1))=0\lim_{x \to \infty}\left(\frac{\log{\left(e^{x} x + 1 \right)}}{x \log{\left(x + \sqrt{x^{2} + 1} \right)}}\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⁡(exx+1)log⁡(x+x2+1)=log⁡(−xe−x+1)log⁡(−x+x2+1)\frac{\log{\left(e^{x} x + 1 \right)}}{\log{\left(x + \sqrt{x^{2} + 1} \right)}} = \frac{\log{\left(- x e^{- x} + 1 \right)}}{\log{\left(- x + \sqrt{x^{2} + 1} \right)}}
- No
log⁡(exx+1)log⁡(x+x2+1)=−log⁡(−xe−x+1)log⁡(−x+x2+1)\frac{\log{\left(e^{x} x + 1 \right)}}{\log{\left(x + \sqrt{x^{2} + 1} \right)}} = - \frac{\log{\left(- x e^{- x} + 1 \right)}}{\log{\left(- x + \sqrt{x^{2} + 1} \right)}}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор