Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ (18*x*(x-1)*(2**x-x**2+x-2)+45*x*(x-1)*(x-2)*(x-3)+124*x*(x-1)*(x-2)+4*(2**x-1)*(2**x-2)+12*x**2+12*x-12*x*2**x)/(3*2**(2*x+6))

Gráfico de la función y = (18*x*(x-1)*(2**x-x**2+x-2)+45*x*(x-1)*(x-2)*(x-3)+124*x*(x-1)*(x-2)+4*(2**x-1)*(2**x-2)+12*x**2+12*x-12*x*2**x)/(3*2**(2*x+6))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                    / x    2        \                                                            / x    \ / x    \       2                x
       18*x*(x - 1)*\2  - x  + x - 2/ + 45*x*(x - 1)*(x - 2)*(x - 3) + 124*x*(x - 1)*(x - 2) + 4*\2  - 1/*\2  - 2/ + 12*x  + 12*x - 12*x*2 
f(x) = ------------------------------------------------------------------------------------------------------------------------------------
                                                                       2*x + 6                                                             
                                                                    3*2
f(x)=−2x12x+(12x+(12x2+((2x−2)4(2x−1)+(124x(x−1)(x−2)+(18x(x−1)((x+(2x−x2))−2)+45x(x−1)(x−2)(x−3))))))3⋅22x+6f{\left(x \right)} = \frac{- 2^{x} 12 x + \left(12 x + \left(12 x^{2} + \left(\left(2^{x} - 2\right) 4 \left(2^{x} - 1\right) + \left(124 x \left(x - 1\right) \left(x - 2\right) + \left(18 x \left(x - 1\right) \left(\left(x + \left(2^{x} - x^{2}\right)\right) - 2\right) + 45 x \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)\right)\right)\right)\right)\right)}{3 \cdot 2^{2 x + 6}} 
f = (-2^x*12*x + 12*x + 12*x^2 + (2^x - 2)*(4*(2^x - 1)) + ((124*x)*(x - 1))*(x - 2) + ((18*x)*(x - 1))*(x + 2^x - x^2 - 2) + (((45*x)*(x - 1))*(x - 2))*(x - 3))/((3*2^(2*x + 6)))
Gráfico de la función
0.05.00.51.01.52.02.53.03.54.04.50.2-0.1
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:
−2x12x+(12x+(12x2+((2x−2)4(2x−1)+(124x(x−1)(x−2)+(18x(x−1)((x+(2x−x2))−2)+45x(x−1)(x−2)(x−3))))))3⋅22x+6=0\frac{- 2^{x} 12 x + \left(12 x + \left(12 x^{2} + \left(\left(2^{x} - 2\right) 4 \left(2^{x} - 1\right) + \left(124 x \left(x - 1\right) \left(x - 2\right) + \left(18 x \left(x - 1\right) \left(\left(x + \left(2^{x} - x^{2}\right)\right) - 2\right) + 45 x \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)\right)\right)\right)\right)\right)}{3 \cdot 2^{2 x + 6}} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=2x_{1} = 2
x2=0x_{2} = 0
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 (((18*x)*(x - 1))*(2^x - x^2 + x - 2) + (((45*x)*(x - 1))*(x - 2))*(x - 3) + ((124*x)*(x - 1))*(x - 2) + (4*(2^x - 1))*(2^x - 2) + 12*x^2 + 12*x - 12*x*2^x)/((3*2^(2*x + 6))).
((((−2+20)4(−1+20)+(((−1)0⋅18(−2+(−02+20))+(−3)(−2)(−1)0⋅45)+(−2)(−1)0⋅124))+12⋅02)+0⋅12)−0⋅12⋅203⋅20⋅2+6\frac{\left(\left(\left(\left(-2 + 2^{0}\right) 4 \left(-1 + 2^{0}\right) + \left(\left(\left(-1\right) 0 \cdot 18 \left(-2 + \left(- 0^{2} + 2^{0}\right)\right) + \left(-3\right) \left(-2\right) \left(-1\right) 0 \cdot 45\right) + \left(-2\right) \left(-1\right) 0 \cdot 124\right)\right) + 12 \cdot 0^{2}\right) + 0 \cdot 12\right) - 0 \cdot 12 \cdot 2^{0}}{3 \cdot 2^{0 \cdot 2 + 6}}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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
−2⋅2−4x−12⋅22x+6(−2x12x+(12x+(12x2+((2x−2)4(2x−1)+(124x(x−1)(x−2)+(18x(x−1)((x+(2x−x2))−2)+45x(x−1)(x−2)(x−3)))))))log⁡(2)3+2−2x−63(−12⋅2xxlog⁡(2)+4⋅2x(2x−2)log⁡(2)+4⋅2x(2x−1)log⁡(2)−12⋅2x+18x(x−1)(2xlog⁡(2)−2x+1)+24x+124x(x−1)+45x(x−1)(x−2)+(x−3)(45x(x−1)+(x−2)(90x−45))+(x−2)(248x−124)+(36x−18)((x+(2x−x2))−2)+12)=0- \frac{2 \cdot 2^{- 4 x - 12} \cdot 2^{2 x + 6} \left(- 2^{x} 12 x + \left(12 x + \left(12 x^{2} + \left(\left(2^{x} - 2\right) 4 \left(2^{x} - 1\right) + \left(124 x \left(x - 1\right) \left(x - 2\right) + \left(18 x \left(x - 1\right) \left(\left(x + \left(2^{x} - x^{2}\right)\right) - 2\right) + 45 x \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)\right)\right)\right)\right)\right)\right) \log{\left(2 \right)}}{3} + \frac{2^{- 2 x - 6}}{3} \left(- 12 \cdot 2^{x} x \log{\left(2 \right)} + 4 \cdot 2^{x} \left(2^{x} - 2\right) \log{\left(2 \right)} + 4 \cdot 2^{x} \left(2^{x} - 1\right) \log{\left(2 \right)} - 12 \cdot 2^{x} + 18 x \left(x - 1\right) \left(2^{x} \log{\left(2 \right)} - 2 x + 1\right) + 24 x + 124 x \left(x - 1\right) + 45 x \left(x - 1\right) \left(x - 2\right) + \left(x - 3\right) \left(45 x \left(x - 1\right) + \left(x - 2\right) \left(90 x - 45\right)\right) + \left(x - 2\right) \left(248 x - 124\right) + \left(36 x - 18\right) \left(\left(x + \left(2^{x} - x^{2}\right)\right) - 2\right) + 12\right) = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=62.7706728546357x_{1} = 62.7706728546357
x2=72.2880718064998x_{2} = 72.2880718064998
x3=55.4079170815051x_{3} = 55.4079170815051
x4=46.9437611133111x_{4} = 46.9437611133111
x5=106.675369004527x_{5} = 106.675369004527
x6=108.202538262165x_{6} = 108.202538262165
x7=104.349029856478x_{7} = 104.349029856478
x8=53.6303611924966x_{8} = 53.6303611924966
x9=106.614683653747x_{9} = 106.614683653747
x10=48.5472772193572x_{10} = 48.5472772193572
x11=64.6538774532576x_{11} = 64.6538774532576
x12=104.711833825778x_{12} = 104.711833825778
x13=76.1486634229241x_{13} = 76.1486634229241
x14=102.830662484033x_{14} = 102.830662484033
x15=60.9013245046264x_{15} = 60.9013245046264
x16=0.0454029670062296x_{16} = 0.0454029670062296
x17=59.0485681882441x_{17} = 59.0485681882441
x18=74.215506646003x_{18} = 74.215506646003
x19=48.687320139481x_{19} = 48.687320139481
x20=85.8802278492015x_{20} = 85.8802278492015
x21=104.92763575097x_{21} = 104.92763575097
x22=101.844774994901x_{22} = 101.844774994901
x23=57.215918666273x_{23} = 57.215918666273
x24=91.7577341619084x_{24} = 91.7577341619084
x25=102.301359192866x_{25} = 102.301359192866
x26=50.1946694626465x_{26} = 50.1946694626465
x27=99.5506499561975x_{27} = 99.5506499561975
x28=83.9266560032107x_{28} = 83.9266560032107
x29=104.16303853973x_{29} = 104.16303853973
x30=4.35594280968514x_{30} = 4.35594280968514
x31=66.5487832531211x_{31} = 66.5487832531211
x32=80.0295928057729x_{32} = 80.0295928057729
x33=97.6990816107479x_{33} = 97.6990816107479
x34=81.9763212601759x_{34} = 81.9763212601759
x35=1.5078759302553x_{35} = 1.5078759302553
x36=93.720646342157x_{36} = 93.720646342157
x37=107.666244343416x_{37} = 107.666244343416
x38=95.6875306444994x_{38} = 95.6875306444994
x39=106.792097107438x_{39} = 106.792097107438
x40=68.4536697633449x_{40} = 68.4536697633449
x41=104.234356096635x_{41} = 104.234356096635
x42=104.088932637854x_{42} = 104.088932637854
x43=89.7959728271257x_{43} = 89.7959728271257
x44=103.152318150569x_{44} = 103.152318150569
x45=106.3328125x_{45} = 106.3328125
x46=70.3671455330397x_{46} = 70.3671455330397
x47=51.8902490796584x_{47} = 51.8902490796584
x48=78.0868795572327x_{48} = 78.0868795572327
x49=87.8366961544431x_{49} = 87.8366961544431
Signos de extremos en los puntos:
(62.77067285463568, 0.0208333333333334)

(72.28807180649983, 0.0208333333333333)

(55.40791708150509, 0.0208333333333392)

(46.943761113311126, 0.020833333334805)

(106.67536900452744, 0.0208333333333333)

(108.20253826216509, 0.0208333333333333)

(104.34902985647803, 0.0208333333333333)

(53.63036119249658, 0.0208333333333521)

(106.61468365374684, 0.0208333333333333)

(48.54727721935724, 0.0208333333338519)

(64.65387745325764, 0.0208333333333334)

(104.71183382577816, 0.0208333333333333)

(76.14866342292414, 0.0208333333333333)

(102.8306624840334, 0.0208333333333333)

(60.90132450462639, 0.0208333333333335)

(0.045402967006229644, -0.000752516371450277)

(59.04856818824414, 0.0208333333333339)

(74.21550664600296, 0.0208333333333333)

(48.68732013948095, 0.0208333333338067)

(85.88022784920155, 0.0208333333333333)

(104.92763575096998, 0.0208333333333333)

(101.84477499490102, 0.0208333333333333)

(57.21591866627299, 0.0208333333333351)

(91.75773416190836, 0.0208333333333333)

(102.3013591928658, 0.0208333333333333)

(50.194669462646544, 0.0208333333335105)

(99.55064995619746, 0.0208333333333333)

(83.9266560032107, 0.0208333333333333)

(104.16303853972975, 0.0208333333333333)

(4.355942809685139, 0.0998218286750877)

(66.54878325312114, 0.0208333333333333)

(80.02959280577288, 0.0208333333333333)

(97.69908161074788, 0.0208333333333333)

(81.97632126017587, 0.0208333333333333)

(1.5078759302553, -0.0130124211612551)

(93.720646342157, 0.0208333333333333)

(107.66624434341605, 0.0208333333333333)

(95.68753064449945, 0.0208333333333333)

(106.79209710743801, 0.0208333333333333)

(68.4536697633449, 0.0208333333333333)

(104.23435609663515, 0.0208333333333333)

(104.08893263785407, 0.0208333333333333)

(89.79597282712574, 0.0208333333333333)

(103.1523181505693, 0.0208333333333333)

(106.3328125, 0.0208333333333333)

(70.36714553303973, 0.0208333333333333)

(51.89024907965835, 0.0208333333333919)

(78.08687955723272, 0.0208333333333333)

(87.83669615444315, 0.0208333333333333)


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=55.4079170815051x_{1} = 55.4079170815051
x2=0.0454029670062296x_{2} = 0.0454029670062296
x3=59.0485681882441x_{3} = 59.0485681882441
x4=85.8802278492015x_{4} = 85.8802278492015
x5=50.1946694626465x_{5} = 50.1946694626465
x6=99.5506499561975x_{6} = 99.5506499561975
x7=97.6990816107479x_{7} = 97.6990816107479
x8=1.5078759302553x_{8} = 1.5078759302553
x9=93.720646342157x_{9} = 93.720646342157
Puntos máximos de la función:
x9=62.7706728546357x_{9} = 62.7706728546357
x9=72.2880718064998x_{9} = 72.2880718064998
x9=108.202538262165x_{9} = 108.202538262165
x9=104.349029856478x_{9} = 104.349029856478
x9=53.6303611924966x_{9} = 53.6303611924966
x9=64.6538774532576x_{9} = 64.6538774532576
x9=104.711833825778x_{9} = 104.711833825778
x9=60.9013245046264x_{9} = 60.9013245046264
x9=91.7577341619084x_{9} = 91.7577341619084
x9=102.301359192866x_{9} = 102.301359192866
x9=4.35594280968514x_{9} = 4.35594280968514
x9=66.5487832531211x_{9} = 66.5487832531211
x9=80.0295928057729x_{9} = 80.0295928057729
x9=81.9763212601759x_{9} = 81.9763212601759
x9=104.088932637854x_{9} = 104.088932637854
x9=51.8902490796584x_{9} = 51.8902490796584
x9=78.0868795572327x_{9} = 78.0868795572327
Decrece en los intervalos
[99.5506499561975,∞)\left[99.5506499561975, \infty\right)
Crece en los intervalos
(−∞,0.0454029670062296]\left(-\infty, 0.0454029670062296\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(−2x12x+(12x+(12x2+((2x−2)4(2x−1)+(124x(x−1)(x−2)+(18x(x−1)((x+(2x−x2))−2)+45x(x−1)(x−2)(x−3))))))3⋅22x+6)=∞\lim_{x \to -\infty}\left(\frac{- 2^{x} 12 x + \left(12 x + \left(12 x^{2} + \left(\left(2^{x} - 2\right) 4 \left(2^{x} - 1\right) + \left(124 x \left(x - 1\right) \left(x - 2\right) + \left(18 x \left(x - 1\right) \left(\left(x + \left(2^{x} - x^{2}\right)\right) - 2\right) + 45 x \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)\right)\right)\right)\right)\right)}{3 \cdot 2^{2 x + 6}}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota horizontal a la izquierda
lim⁡x→∞(−2x12x+(12x+(12x2+((2x−2)4(2x−1)+(124x(x−1)(x−2)+(18x(x−1)((x+(2x−x2))−2)+45x(x−1)(x−2)(x−3))))))3⋅22x+6)=148\lim_{x \to \infty}\left(\frac{- 2^{x} 12 x + \left(12 x + \left(12 x^{2} + \left(\left(2^{x} - 2\right) 4 \left(2^{x} - 1\right) + \left(124 x \left(x - 1\right) \left(x - 2\right) + \left(18 x \left(x - 1\right) \left(\left(x + \left(2^{x} - x^{2}\right)\right) - 2\right) + 45 x \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)\right)\right)\right)\right)\right)}{3 \cdot 2^{2 x + 6}}\right) = \frac{1}{48}
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=148y = \frac{1}{48}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (((18*x)*(x - 1))*(2^x - x^2 + x - 2) + (((45*x)*(x - 1))*(x - 2))*(x - 3) + ((124*x)*(x - 1))*(x - 2) + (4*(2^x - 1))*(2^x - 2) + 12*x^2 + 12*x - 12*x*2^x)/((3*2^(2*x + 6))), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(2−2x−63(−2x12x+(12x+(12x2+((2x−2)4(2x−1)+(124x(x−1)(x−2)+(18x(x−1)((x+(2x−x2))−2)+45x(x−1)(x−2)(x−3)))))))x)=−∞\lim_{x \to -\infty}\left(\frac{\frac{2^{- 2 x - 6}}{3} \left(- 2^{x} 12 x + \left(12 x + \left(12 x^{2} + \left(\left(2^{x} - 2\right) 4 \left(2^{x} - 1\right) + \left(124 x \left(x - 1\right) \left(x - 2\right) + \left(18 x \left(x - 1\right) \left(\left(x + \left(2^{x} - x^{2}\right)\right) - 2\right) + 45 x \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)\right)\right)\right)\right)\right)\right)}{x}\right) = -\infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la izquierda
lim⁡x→∞(2−2x−63(−2x12x+(12x+(12x2+((2x−2)4(2x−1)+(124x(x−1)(x−2)+(18x(x−1)((x+(2x−x2))−2)+45x(x−1)(x−2)(x−3)))))))x)=0\lim_{x \to \infty}\left(\frac{\frac{2^{- 2 x - 6}}{3} \left(- 2^{x} 12 x + \left(12 x + \left(12 x^{2} + \left(\left(2^{x} - 2\right) 4 \left(2^{x} - 1\right) + \left(124 x \left(x - 1\right) \left(x - 2\right) + \left(18 x \left(x - 1\right) \left(\left(x + \left(2^{x} - x^{2}\right)\right) - 2\right) + 45 x \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)\right)\right)\right)\right)\right)\right)}{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:
−2x12x+(12x+(12x2+((2x−2)4(2x−1)+(124x(x−1)(x−2)+(18x(x−1)((x+(2x−x2))−2)+45x(x−1)(x−2)(x−3))))))3⋅22x+6=22x−6(12x2−45x(−x−3)(−x−2)(−x−1)−124x(−x−2)(−x−1)−18x(−x−1)(−x2−x−2+2−x)−12x+(−4+4⋅2−x)(−2+2−x)+12⋅2−xx)3\frac{- 2^{x} 12 x + \left(12 x + \left(12 x^{2} + \left(\left(2^{x} - 2\right) 4 \left(2^{x} - 1\right) + \left(124 x \left(x - 1\right) \left(x - 2\right) + \left(18 x \left(x - 1\right) \left(\left(x + \left(2^{x} - x^{2}\right)\right) - 2\right) + 45 x \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)\right)\right)\right)\right)\right)}{3 \cdot 2^{2 x + 6}} = \frac{2^{2 x - 6} \left(12 x^{2} - 45 x \left(- x - 3\right) \left(- x - 2\right) \left(- x - 1\right) - 124 x \left(- x - 2\right) \left(- x - 1\right) - 18 x \left(- x - 1\right) \left(- x^{2} - x - 2 + 2^{- x}\right) - 12 x + \left(-4 + 4 \cdot 2^{- x}\right) \left(-2 + 2^{- x}\right) + 12 \cdot 2^{- x} x\right)}{3}
- No
−2x12x+(12x+(12x2+((2x−2)4(2x−1)+(124x(x−1)(x−2)+(18x(x−1)((x+(2x−x2))−2)+45x(x−1)(x−2)(x−3))))))3⋅22x+6=−22x−6(12x2−45x(−x−3)(−x−2)(−x−1)−124x(−x−2)(−x−1)−18x(−x−1)(−x2−x−2+2−x)−12x+(−4+4⋅2−x)(−2+2−x)+12⋅2−xx)3\frac{- 2^{x} 12 x + \left(12 x + \left(12 x^{2} + \left(\left(2^{x} - 2\right) 4 \left(2^{x} - 1\right) + \left(124 x \left(x - 1\right) \left(x - 2\right) + \left(18 x \left(x - 1\right) \left(\left(x + \left(2^{x} - x^{2}\right)\right) - 2\right) + 45 x \left(x - 1\right) \left(x - 2\right) \left(x - 3\right)\right)\right)\right)\right)\right)}{3 \cdot 2^{2 x + 6}} = - \frac{2^{2 x - 6} \left(12 x^{2} - 45 x \left(- x - 3\right) \left(- x - 2\right) \left(- x - 1\right) - 124 x \left(- x - 2\right) \left(- x - 1\right) - 18 x \left(- x - 1\right) \left(- x^{2} - x - 2 + 2^{- x}\right) - 12 x + \left(-4 + 4 \cdot 2^{- x}\right) \left(-2 + 2^{- x}\right) + 12 \cdot 2^{- x} x\right)}{3}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор