Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ -4-cos(x)-sin(x)+2^x/(1+log(2)^2)

Gráfico de la función y = -4-cos(x)-sin(x)+2^x/(1+log(2)^2)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                    x    
                                   2     
f(x) = -4 - cos(x) - sin(x) + -----------
                                     2   
                              1 + log (2)
f(x)=2xlog⁡(2)2+1+((−cos⁡(x)−4)−sin⁡(x))f{\left(x \right)} = \frac{2^{x}}{\log{\left(2 \right)}^{2} + 1} + \left(\left(- \cos{\left(x \right)} - 4\right) - \sin{\left(x \right)}\right) 
f = 2^x/(log(2)^2 + 1) - cos(x) - 4 - sin(x)
Gráfico de la función
02468-8-6-4-2-1010-10001000
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:
2xlog⁡(2)2+1+((−cos⁡(x)−4)−sin⁡(x))=0\frac{2^{x}}{\log{\left(2 \right)}^{2} + 1} + \left(\left(- \cos{\left(x \right)} - 4\right) - \sin{\left(x \right)}\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=2.49413836922879x_{1} = 2.49413836922879
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 -4 - cos(x) - sin(x) + 2^x/(1 + log(2)^2).
((−4−cos⁡(0))−sin⁡(0))+20log⁡(2)2+1\left(\left(-4 - \cos{\left(0 \right)}\right) - \sin{\left(0 \right)}\right) + \frac{2^{0}}{\log{\left(2 \right)}^{2} + 1}
Resultado:
f(0)=−5+1log⁡(2)2+1f{\left(0 \right)} = -5 + \frac{1}{\log{\left(2 \right)}^{2} + 1}
Punto:
(0, -5 + 1/(1 + log(2)^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
2xlog⁡(2)log⁡(2)2+1+sin⁡(x)−cos⁡(x)=0\frac{2^{x} \log{\left(2 \right)}}{\log{\left(2 \right)}^{2} + 1} + \sin{\left(x \right)} - \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−46.3384916404494x_{1} = -46.3384916404494
x2=−30.6305283726996x_{2} = -30.6305283726996
x3=−68.329640215578x_{3} = -68.329640215578
x4=−84.037603483527x_{4} = -84.037603483527
x5=−55.7632696012188x_{5} = -55.7632696012188
x6=−93.4623814442964x_{6} = -93.4623814442964
x7=−71.4712328691678x_{7} = -71.4712328691678
x8=−65.1880475619882x_{8} = -65.1880475619882
x9=−96.6039740978861x_{9} = -96.6039740978861
x10=−18.064158966128x_{10} = -18.064158966128
x11=−14.9225544440083x_{11} = -14.9225544440083
x12=−27.4889357171531x_{12} = -27.4889357171531
x13=−11.78106652319x_{13} = -11.78106652319
x14=−21.2057502748483x_{14} = -21.2057502748483
x15=−49.4800842940392x_{15} = -49.4800842940392
x16=−24.3473430808317x_{16} = -24.3473430808317
x17=−8.63854907961704x_{17} = -8.63854907961704
x18=−62.0464549083984x_{18} = -62.0464549083984
x19=−87.1791961371168x_{19} = -87.1791961371168
x20=−228.550865548657x_{20} = -228.550865548657
x21=−90.3207887907066x_{21} = -90.3207887907066
x22=−52.621676947629x_{22} = -52.621676947629
x23=−5.5050771238138x_{23} = -5.5050771238138
x24=−40.0553063332696x_{24} = -40.0553063332696
x25=0.34983867063653x_{25} = 0.34983867063653
x26=−58.9048622548086x_{26} = -58.9048622548086
x27=−36.9137136796826x_{27} = -36.9137136796826
x28=0.34983867063725x_{28} = 0.34983867063725
x29=−80.8960108299372x_{29} = -80.8960108299372
x30=−43.1968989868597x_{30} = -43.1968989868597
x31=−77.7544181763474x_{31} = -77.7544181763474
x32=−99.7455667514759x_{32} = -99.7455667514759
x33=−2.28837093437763x_{33} = -2.28837093437763
x34=−33.7721210260677x_{34} = -33.7721210260677
x35=−74.6128255227576x_{35} = -74.6128255227576
Signos de extremos en los puntos:
                                         1.12389066087535e-14 
(-46.33849164044945, -2.58578643762691 + --------------------)
                                                    2         
                                             1 + log (2)      

                                         6.01578320347801e-10 
(-30.630528372699647, -5.4142135623731 + --------------------)
                                                    2         
                                             1 + log (2)      

                                       2.69605471782059e-21 
(-68.329640215578, -5.41421356237309 + --------------------)
                                                  2         
                                           1 + log (2)      

                                         5.03686821093774e-26 
(-84.03760348352696, -2.58578643762691 + --------------------)
                                                    2         
                                             1 + log (2)      

                                         1.63524636999539e-17 
(-55.76326960121883, -5.41421356237309 + --------------------)
                                                    2         
                                             1 + log (2)      

                                         7.32857807686204e-29 
(-93.46238144429635, -5.41421356237309 + --------------------)
                                                    2         
                                             1 + log (2)      

                                         3.05502718607258e-22 
(-71.47123286916779, -2.58578643762691 + --------------------)
                                                    2         
                                             1 + log (2)      

                                         2.37926231053509e-20 
(-65.18804756198821, -2.58578643762691 + --------------------)
                                                    2         
                                             1 + log (2)      

                                         8.30435862895539e-30 
(-96.60397409788614, -2.58578643762691 + --------------------)
                                                    2         
                                             1 + log (2)      

                                          3.64876845622627e-6 
(-18.064158966127973, -5.41421356237206 + -------------------)
                                                     2        
                                              1 + log (2)     

                                          3.22005656236401e-5 
(-14.922554444008298, -2.58578643770727 + -------------------)
                                                     2        
                                              1 + log (2)     

                                          5.30891534551978e-9 
(-27.488935717153083, -2.58578643762691 + -------------------)
                                                     2        
                                              1 + log (2)     

                                          0.000284148649063999 
(-11.781066523189953, -5.41421355611549 + --------------------)
                                                     2         
                                              1 + log (2)      

                                          4.13459606253903e-7 
(-21.205750274848292, -2.58578643762692 + -------------------)
                                                     2        
                                              1 + log (2)     

                                          1.27353369367919e-15 
(-49.480084294039244, -5.41421356237309 + --------------------)
                                                     2         
                                              1 + log (2)      

                                          4.68510596492499e-8 
(-24.347343080831735, -5.41421356237309 + -------------------)
                                                     2        
                                              1 + log (2)     

                                       0.00250921345706745 
(-8.63854907961704, -2.5857869255956 + -------------------)
                                                  2        
                                           1 + log (2)     

                                         2.09969371352705e-19 
(-62.04645490839842, -5.41421356237309 + --------------------)
                                                    2         
                                             1 + log (2)      

                                         5.70751372936474e-27 
(-87.17919613711676, -5.41421356237309 + --------------------)
                                                    2         
                                             1 + log (2)      

                                         1.5824643497876e-69 
(-228.55086554865747, -2.5857864376269 + -------------------)
                                                    2        
                                             1 + log (2)     

                                        6.46745390323051e-28 
(-90.32078879070656, -2.5857864376269 + --------------------)
                                                   2         
                                            1 + log (2)      

                                          1.44310129570163e-16 
(-52.621676947629034, -2.58578643762691 + --------------------)
                                                     2         
                                              1 + log (2)      

                                         0.0220194596479633 
(-5.505077123813801, -5.41417598419189 + ------------------)
                                                   2        
                                            1 + log (2)     

                                         8.75288672642304e-13 
(-40.05530633326958, -2.58578643762691 + --------------------)
                                                    2         
                                             1 + log (2)      

                                         1.27441810755427 
(0.34983867063652996, -5.2821742746995 + ----------------)
                                                  2       
                                           1 + log (2)    

                                         1.85297504655278e-18 
(-58.90486225480862, -2.58578643762691 + --------------------)
                                                    2         
                                             1 + log (2)      

                                         7.72440312834413e-12 
(-36.91371367968263, -5.41421356237309 + --------------------)
                                                    2         
                                             1 + log (2)      

                                         1.27441810755491 
(0.3498386706372505, -5.28217427469993 + ----------------)
                                                  2       
                                           1 + log (2)    

                                         4.44502502794306e-25 
(-80.89601082993718, -5.41421356237309 + --------------------)
                                                    2         
                                             1 + log (2)      

                                          9.91831016228282e-14 
(-43.196898986859686, -5.41421356237309 + --------------------)
                                                     2         
                                              1 + log (2)      

                                         3.92272473123968e-24 
(-77.75441817634739, -2.58578643762691 + --------------------)
                                                    2         
                                             1 + log (2)      

                                        9.41006174936374e-31 
(-99.74556675147593, -5.4142135623731 + --------------------)
                                                   2         
                                            1 + log (2)      

                                         0.204706535167316 
(-2.288370934377629, -2.58903790668087 + -----------------)
                                                   2       
                                            1 + log (2)    

                                         6.8167686336223e-11 
(-33.77212102606771, -2.58578643762691 + -------------------)
                                                    2        
                                             1 + log (2)     

                                         3.4617958774914e-23 
(-74.61282552275759, -5.41421356237309 + -------------------)
                                                    2        
                                             1 + log (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=−30.6305283726996x_{1} = -30.6305283726996
x2=−68.329640215578x_{2} = -68.329640215578
x3=−55.7632696012188x_{3} = -55.7632696012188
x4=−93.4623814442964x_{4} = -93.4623814442964
x5=−18.064158966128x_{5} = -18.064158966128
x6=−11.78106652319x_{6} = -11.78106652319
x7=−49.4800842940392x_{7} = -49.4800842940392
x8=−24.3473430808317x_{8} = -24.3473430808317
x9=−62.0464549083984x_{9} = -62.0464549083984
x10=−87.1791961371168x_{10} = -87.1791961371168
x11=−5.5050771238138x_{11} = -5.5050771238138
x12=0.34983867063653x_{12} = 0.34983867063653
x13=−36.9137136796826x_{13} = -36.9137136796826
x14=0.34983867063725x_{14} = 0.34983867063725
x15=−80.8960108299372x_{15} = -80.8960108299372
x16=−43.1968989868597x_{16} = -43.1968989868597
x17=−99.7455667514759x_{17} = -99.7455667514759
x18=−74.6128255227576x_{18} = -74.6128255227576
Puntos máximos de la función:
x18=−46.3384916404494x_{18} = -46.3384916404494
x18=−84.037603483527x_{18} = -84.037603483527
x18=−71.4712328691678x_{18} = -71.4712328691678
x18=−65.1880475619882x_{18} = -65.1880475619882
x18=−96.6039740978861x_{18} = -96.6039740978861
x18=−14.9225544440083x_{18} = -14.9225544440083
x18=−27.4889357171531x_{18} = -27.4889357171531
x18=−21.2057502748483x_{18} = -21.2057502748483
x18=−8.63854907961704x_{18} = -8.63854907961704
x18=−228.550865548657x_{18} = -228.550865548657
x18=−90.3207887907066x_{18} = -90.3207887907066
x18=−52.621676947629x_{18} = -52.621676947629
x18=−40.0553063332696x_{18} = -40.0553063332696
x18=−58.9048622548086x_{18} = -58.9048622548086
x18=−77.7544181763474x_{18} = -77.7544181763474
x18=−2.28837093437763x_{18} = -2.28837093437763
x18=−33.7721210260677x_{18} = -33.7721210260677
Decrece en los intervalos
[0.34983867063725,∞)\left[0.34983867063725, \infty\right)
Crece en los intervalos
(−∞,−99.7455667514759]\left(-\infty, -99.7455667514759\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
2xlog⁡(2)2log⁡(2)2+1+sin⁡(x)+cos⁡(x)=0\frac{2^{x} \log{\left(2 \right)}^{2}}{\log{\left(2 \right)}^{2} + 1} + \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−79.3252145031423x_{1} = -79.3252145031423
x2=−73.0420291959627x_{2} = -73.0420291959627
x3=−66.7588438887831x_{3} = -66.7588438887831
x4=−85.6083998103219x_{4} = -85.6083998103219
x5=−82.4668071567321x_{5} = -82.4668071567321
x6=−22.7765467065873x_{6} = -22.7765467065873
x7=−69.9004365423729x_{7} = -69.9004365423729
x8=−44.7676953136546x_{8} = -44.7676953136546
x9=−35.3429173528799x_{9} = -35.3429173528799
x10=−51.0508806208341x_{10} = -51.0508806208341
x11=−32.2013246993419x_{11} = -32.2013246993419
x12=−16.493358943947x_{12} = -16.493358943947
x13=−60.4756585816035x_{13} = -60.4756585816035
x14=−41.6261026600647x_{14} = -41.6261026600647
x15=−19.6349543667947x_{15} = -19.6349543667947
x16=−10.2099823792365x_{16} = -10.2099823792365
x17=−7.07029101430206x_{17} = -7.07029101430206
x18=−0.907999379078339x_{18} = -0.907999379078339
x19=−107.59954838545x_{19} = -107.59954838545
x20=−63.6172512351933x_{20} = -63.6172512351933
x21=−54.1924732744239x_{21} = -54.1924732744239
x22=−3.91174305409906x_{22} = -3.91174305409906
x23=−47.9092879672443x_{23} = -47.9092879672443
x24=−25.9181393957349x_{24} = -25.9181393957349
x25=−13.3517907286293x_{25} = -13.3517907286293
x26=−88.7499924639117x_{26} = -88.7499924639117
x27=−95.0331777710912x_{27} = -95.0331777710912
x28=−76.1836218495525x_{28} = -76.1836218495525
x29=−38.4845100064756x_{29} = -38.4845100064756
x30=−29.0597320452955x_{30} = -29.0597320452955
x31=−57.3340659280137x_{31} = -57.3340659280137
x32=−98.174770424681x_{32} = -98.174770424681
x33=−91.8915851175014x_{33} = -91.8915851175014

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
[−0.907999379078339,∞)\left[-0.907999379078339, \infty\right)
Convexa en los intervalos
(−∞,−107.59954838545]\left(-\infty, -107.59954838545\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(2xlog⁡(2)2+1+((−cos⁡(x)−4)−sin⁡(x)))=⟨−6,−2⟩\lim_{x \to -\infty}\left(\frac{2^{x}}{\log{\left(2 \right)}^{2} + 1} + \left(\left(- \cos{\left(x \right)} - 4\right) - \sin{\left(x \right)}\right)\right) = \left\langle -6, -2\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−6,−2⟩y = \left\langle -6, -2\right\rangle
lim⁡x→∞(2xlog⁡(2)2+1+((−cos⁡(x)−4)−sin⁡(x)))=∞\lim_{x \to \infty}\left(\frac{2^{x}}{\log{\left(2 \right)}^{2} + 1} + \left(\left(- \cos{\left(x \right)} - 4\right) - \sin{\left(x \right)}\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 -4 - cos(x) - sin(x) + 2^x/(1 + log(2)^2), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(2xlog⁡(2)2+1+((−cos⁡(x)−4)−sin⁡(x))x)=0\lim_{x \to -\infty}\left(\frac{\frac{2^{x}}{\log{\left(2 \right)}^{2} + 1} + \left(\left(- \cos{\left(x \right)} - 4\right) - \sin{\left(x \right)}\right)}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(2xlog⁡(2)2+1+((−cos⁡(x)−4)−sin⁡(x))x)=∞\lim_{x \to \infty}\left(\frac{\frac{2^{x}}{\log{\left(2 \right)}^{2} + 1} + \left(\left(- \cos{\left(x \right)} - 4\right) - \sin{\left(x \right)}\right)}{x}\right) = \infty
Tomamos como el límite
es decir,
no hay asíntota inclinada a la derecha
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:
2xlog⁡(2)2+1+((−cos⁡(x)−4)−sin⁡(x))=sin⁡(x)−cos⁡(x)−4+2−xlog⁡(2)2+1\frac{2^{x}}{\log{\left(2 \right)}^{2} + 1} + \left(\left(- \cos{\left(x \right)} - 4\right) - \sin{\left(x \right)}\right) = \sin{\left(x \right)} - \cos{\left(x \right)} - 4 + \frac{2^{- x}}{\log{\left(2 \right)}^{2} + 1}
- No
2xlog⁡(2)2+1+((−cos⁡(x)−4)−sin⁡(x))=−sin⁡(x)+cos⁡(x)+4−2−xlog⁡(2)2+1\frac{2^{x}}{\log{\left(2 \right)}^{2} + 1} + \left(\left(- \cos{\left(x \right)} - 4\right) - \sin{\left(x \right)}\right) = - \sin{\left(x \right)} + \cos{\left(x \right)} + 4 - \frac{2^{- x}}{\log{\left(2 \right)}^{2} + 1}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор