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-exp(2*x)-exp(4*x)-4*sin(x)/85+33*cos(x)/85
Trazado el gráfico de la función/ -exp(2*x)-exp(4*x)-4*sin(x)/85+33*cos(x)/85

Gráfico de la función y = -exp(2*x)-exp(4*x)-4*sin(x)/85+33*cos(x)/85

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          2*x    4*x   4*sin(x)   33*cos(x)
f(x) = - e    - e    - -------- + ---------
                          85          85
f(x)=((−e4x−e2x)−4sin⁡(x)85)+33cos⁡(x)85f{\left(x \right)} = \left(\left(- e^{4 x} - e^{2 x}\right) - \frac{4 \sin{\left(x \right)}}{85}\right) + \frac{33 \cos{\left(x \right)}}{85} 
f = -exp(4*x) - exp(2*x) - 4*sin(x)/85 + (33*cos(x))/85
Gráfico de la función
02468-8-6-4-2-1010-250000000000000000250000000000000000
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:
((−e4x−e2x)−4sin⁡(x)85)+33cos⁡(x)85=0\left(\left(- e^{4 x} - e^{2 x}\right) - \frac{4 \sin{\left(x \right)}}{85}\right) + \frac{33 \cos{\left(x \right)}}{85} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−1.57749577307796x_{1} = -1.57749577307796
x2=−67.6648657207607x_{2} = -67.6648657207607
x3=−14.2577906097331x_{3} = -14.2577906097331
x4=−42.5321244920423x_{4} = -42.5321244920423
x5=−99.0807922566586x_{5} = -99.0807922566586
x6=−61.3816804135811x_{6} = -61.3816804135811
x7=−29.9657538776831x_{7} = -29.9657538776831
x8=−26.8241612240933x_{8} = -26.8241612240933
x9=−23.6825685705036x_{9} = -23.6825685705036
x10=−77.08964368153x_{10} = -77.08964368153
x11=−95.9391996030688x_{11} = -95.9391996030688
x12=−45.6737171456321x_{12} = -45.6737171456321
x13=−7.97460499980473x_{13} = -7.97460499980473
x14=−33.1073465312729x_{14} = -33.1073465312729
x15=−58.2400877599913x_{15} = -58.2400877599913
x16=−4.83317472659795x_{16} = -4.83317472659795
x17=−86.5144216422994x_{17} = -86.5144216422994
x18=−51.9569024528117x_{18} = -51.9569024528117
x19=−20.5409759169138x_{19} = -20.5409759169138
x20=−48.8153097992219x_{20} = -48.8153097992219
x21=−0.664799719576466x_{21} = -0.664799719576466
x22=−55.0984951064015x_{22} = -55.0984951064015
x23=−73.9480510279402x_{23} = -73.9480510279402
x24=−64.5232730671709x_{24} = -64.5232730671709
x25=−11.1161979567097x_{25} = -11.1161979567097
x26=−36.2489391848627x_{26} = -36.2489391848627
x27=−89.6560142958892x_{27} = -89.6560142958892
x28=−39.3905318384525x_{28} = -39.3905318384525
x29=−70.8064583743505x_{29} = -70.8064583743505
x30=−17.399383263324x_{30} = -17.399383263324
x31=−80.2312363351198x_{31} = -80.2312363351198
x32=−670.850655210001x_{32} = -670.850655210001
x33=−83.3728289887096x_{33} = -83.3728289887096
x34=−92.797606949479x_{34} = -92.797606949479
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 -exp(2*x) - exp(4*x) - 4*sin(x)/85 + (33*cos(x))/85.
((−e0⋅2−e0⋅4)−4sin⁡(0)85)+33cos⁡(0)85\left(\left(- e^{0 \cdot 2} - e^{0 \cdot 4}\right) - \frac{4 \sin{\left(0 \right)}}{85}\right) + \frac{33 \cos{\left(0 \right)}}{85}
Resultado:
f(0)=−13785f{\left(0 \right)} = - \frac{137}{85}
Punto:
(0, -137/85)
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
−4e4x−2e2x−33sin⁡(x)85−4cos⁡(x)85=0- 4 e^{4 x} - 2 e^{2 x} - \frac{33 \sin{\left(x \right)}}{85} - \frac{4 \cos{\left(x \right)}}{85} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−1.04131672607427x_{1} = -1.04131672607427
x2=−97.5099959298637x_{2} = -97.5099959298637
x3=−31.536550204478x_{3} = -31.536550204478
x4=−40.9613281652474x_{4} = -40.9613281652474
x5=−94.3684032762739x_{5} = -94.3684032762739
x6=−25.2533648972985x_{6} = -25.2533648972985
x7=−91.2268106226841x_{7} = -91.2268106226841
x8=−18.9701795901189x_{8} = -18.9701795901189
x9=−3.25457486376707x_{9} = -3.25457486376707
x10=−44.1029208188372x_{10} = -44.1029208188372
x11=−9.54540160318349x_{11} = -9.54540160318349
x12=−6.40382298711638x_{12} = -6.40382298711638
x13=−72.3772547011453x_{13} = -72.3772547011453
x14=−100.651588583453x_{14} = -100.651588583453
x15=−62.952476740376x_{15} = -62.952476740376
x16=−69.2356620475556x_{16} = -69.2356620475556
x17=−12.6869942829881x_{17} = -12.6869942829881
x18=−47.244513472427x_{18} = -47.244513472427
x19=−75.5188473547351x_{19} = -75.5188473547351
x20=−28.3949575508882x_{20} = -28.3949575508882
x21=−81.8020326619147x_{21} = -81.8020326619147
x22=−59.8108840867862x_{22} = -59.8108840867862
x23=−34.6781428580678x_{23} = -34.6781428580678
x24=−56.6692914331964x_{24} = -56.6692914331964
x25=−66.0940693939658x_{25} = -66.0940693939658
x26=−53.5276987796066x_{26} = -53.5276987796066
x27=−78.6604400083249x_{27} = -78.6604400083249
x28=−88.0852179690943x_{28} = -88.0852179690943
x29=−50.3861061260168x_{29} = -50.3861061260168
x30=−22.1117722437087x_{30} = -22.1117722437087
x31=−84.9436253155045x_{31} = -84.9436253155045
x32=−15.828586936529x_{32} = -15.828586936529
x33=−37.8197355116576x_{33} = -37.8197355116576
Signos de extremos en los puntos:
(-1.041316726074265, 0.0965793821534147)

(-97.50999592986369, -0.391076944437521)

(-31.536550204478036, 0.391076944437521)

(-40.961328165247416, -0.391076944437521)

(-94.3684032762739, 0.391076944437521)

(-25.25336489729845, 0.391076944437522)

(-91.22681062268411, -0.391076944437521)

(-18.970179590118864, 0.391076944437521)

(-3.254574863767073, -0.392557491850165)

(-44.102920818837205, 0.391076944437521)

(-9.545401603183485, -0.39107694955398)

(-6.403822987116385, 0.391074204647358)

(-72.37725470114535, -0.391076944437521)

(-100.65158858345349, 0.391076944437521)

(-62.952476740375964, 0.391076944437521)

(-69.23566204755555, 0.391076944437521)

(-12.68699428298814, 0.391076944427967)

(-47.244513472427, -0.391076944437521)

(-75.51884735473514, 0.391076944437521)

(-28.394957550888243, -0.391076944437522)

(-81.80203266191472, 0.391076944437521)

(-59.810884086786174, -0.391076944437521)

(-34.67814285806783, -0.391076944437521)

(-56.66929143319638, 0.391076944437521)

(-66.09406939396577, -0.391076944437521)

(-53.52769877960659, -0.391076944437521)

(-78.66044000832494, -0.391076944437521)

(-88.08521796909432, 0.391076944437521)

(-50.38610612601679, 0.391076944437521)

(-22.111772243708657, -0.391076944437522)

(-84.94362531550452, -0.391076944437521)

(-15.828586936528978, -0.391076944437539)

(-37.81973551165762, 0.391076944437521)


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=−97.5099959298637x_{1} = -97.5099959298637
x2=−40.9613281652474x_{2} = -40.9613281652474
x3=−91.2268106226841x_{3} = -91.2268106226841
x4=−3.25457486376707x_{4} = -3.25457486376707
x5=−9.54540160318349x_{5} = -9.54540160318349
x6=−72.3772547011453x_{6} = -72.3772547011453
x7=−47.244513472427x_{7} = -47.244513472427
x8=−28.3949575508882x_{8} = -28.3949575508882
x9=−59.8108840867862x_{9} = -59.8108840867862
x10=−34.6781428580678x_{10} = -34.6781428580678
x11=−66.0940693939658x_{11} = -66.0940693939658
x12=−53.5276987796066x_{12} = -53.5276987796066
x13=−78.6604400083249x_{13} = -78.6604400083249
x14=−22.1117722437087x_{14} = -22.1117722437087
x15=−84.9436253155045x_{15} = -84.9436253155045
x16=−15.828586936529x_{16} = -15.828586936529
Puntos máximos de la función:
x16=−1.04131672607427x_{16} = -1.04131672607427
x16=−31.536550204478x_{16} = -31.536550204478
x16=−94.3684032762739x_{16} = -94.3684032762739
x16=−25.2533648972985x_{16} = -25.2533648972985
x16=−18.9701795901189x_{16} = -18.9701795901189
x16=−44.1029208188372x_{16} = -44.1029208188372
x16=−6.40382298711638x_{16} = -6.40382298711638
x16=−100.651588583453x_{16} = -100.651588583453
x16=−62.952476740376x_{16} = -62.952476740376
x16=−69.2356620475556x_{16} = -69.2356620475556
x16=−12.6869942829881x_{16} = -12.6869942829881
x16=−75.5188473547351x_{16} = -75.5188473547351
x16=−81.8020326619147x_{16} = -81.8020326619147
x16=−56.6692914331964x_{16} = -56.6692914331964
x16=−88.0852179690943x_{16} = -88.0852179690943
x16=−50.3861061260168x_{16} = -50.3861061260168
x16=−37.8197355116576x_{16} = -37.8197355116576
Decrece en los intervalos
[−3.25457486376707,∞)\left[-3.25457486376707, \infty\right)
Crece en los intervalos
(−∞,−97.5099959298637]\left(-\infty, -97.5099959298637\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
−16e4x−4e2x+4sin⁡(x)85−33cos⁡(x)85=0- 16 e^{4 x} - 4 e^{2 x} + \frac{4 \sin{\left(x \right)}}{85} - \frac{33 \cos{\left(x \right)}}{85} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−67.6648657207607x_{1} = -67.6648657207607
x2=−7.97460651355077x_{2} = -7.97460651355077
x3=−42.5321244920423x_{3} = -42.5321244920423
x4=−11.1161979538829x_{4} = -11.1161979538829
x5=−17.399383263324x_{5} = -17.399383263324
x6=−99.0807922566586x_{6} = -99.0807922566586
x7=−61.3816804135811x_{7} = -61.3816804135811
x8=−29.9657538776831x_{8} = -29.9657538776831
x9=−26.8241612240933x_{9} = -26.8241612240933
x10=−23.6825685705036x_{10} = -23.6825685705036
x11=−77.08964368153x_{11} = -77.08964368153
x12=−95.9391996030688x_{12} = -95.9391996030688
x13=−45.6737171456321x_{13} = -45.6737171456321
x14=−33.1073465312729x_{14} = -33.1073465312729
x15=−58.2400877599913x_{15} = -58.2400877599913
x16=−86.5144216422994x_{16} = -86.5144216422994
x17=−51.9569024528117x_{17} = -51.9569024528117
x18=−20.5409759169138x_{18} = -20.5409759169138
x19=−48.8153097992219x_{19} = -48.8153097992219
x20=−55.0984951064015x_{20} = -55.0984951064015
x21=−73.9480510279402x_{21} = -73.9480510279402
x22=−64.5232730671709x_{22} = -64.5232730671709
x23=−36.2489391848627x_{23} = -36.2489391848627
x24=−4.83236316151132x_{24} = -4.83236316151132
x25=−89.6560142958892x_{25} = -89.6560142958892
x26=−39.3905318384525x_{26} = -39.3905318384525
x27=−14.2577906097384x_{27} = -14.2577906097384
x28=−70.8064583743505x_{28} = -70.8064583743505
x29=−80.2312363351198x_{29} = -80.2312363351198
x30=−1.92818727666857x_{30} = -1.92818727666857
x31=−670.850655210001x_{31} = -670.850655210001
x32=−83.3728289887096x_{32} = -83.3728289887096
x33=−92.797606949479x_{33} = -92.797606949479

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
[−4.83236316151132,∞)\left[-4.83236316151132, \infty\right)
Convexa en los intervalos
(−∞,−670.850655210001]\left(-\infty, -670.850655210001\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((−e4x−e2x)−4sin⁡(x)85)+33cos⁡(x)85)=⟨−3785,3785⟩\lim_{x \to -\infty}\left(\left(\left(- e^{4 x} - e^{2 x}\right) - \frac{4 \sin{\left(x \right)}}{85}\right) + \frac{33 \cos{\left(x \right)}}{85}\right) = \left\langle - \frac{37}{85}, \frac{37}{85}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−3785,3785⟩y = \left\langle - \frac{37}{85}, \frac{37}{85}\right\rangle
lim⁡x→∞(((−e4x−e2x)−4sin⁡(x)85)+33cos⁡(x)85)=−∞\lim_{x \to \infty}\left(\left(\left(- e^{4 x} - e^{2 x}\right) - \frac{4 \sin{\left(x \right)}}{85}\right) + \frac{33 \cos{\left(x \right)}}{85}\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 -exp(2*x) - exp(4*x) - 4*sin(x)/85 + (33*cos(x))/85, dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((−e4x−e2x)−4sin⁡(x)85)+33cos⁡(x)85x)=0\lim_{x \to -\infty}\left(\frac{\left(\left(- e^{4 x} - e^{2 x}\right) - \frac{4 \sin{\left(x \right)}}{85}\right) + \frac{33 \cos{\left(x \right)}}{85}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(((−e4x−e2x)−4sin⁡(x)85)+33cos⁡(x)85x)=−∞\lim_{x \to \infty}\left(\frac{\left(\left(- e^{4 x} - e^{2 x}\right) - \frac{4 \sin{\left(x \right)}}{85}\right) + \frac{33 \cos{\left(x \right)}}{85}}{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:
((−e4x−e2x)−4sin⁡(x)85)+33cos⁡(x)85=4sin⁡(x)85+33cos⁡(x)85−e−2x−e−4x\left(\left(- e^{4 x} - e^{2 x}\right) - \frac{4 \sin{\left(x \right)}}{85}\right) + \frac{33 \cos{\left(x \right)}}{85} = \frac{4 \sin{\left(x \right)}}{85} + \frac{33 \cos{\left(x \right)}}{85} - e^{- 2 x} - e^{- 4 x}
- No
((−e4x−e2x)−4sin⁡(x)85)+33cos⁡(x)85=−4sin⁡(x)85−33cos⁡(x)85+e−2x+e−4x\left(\left(- e^{4 x} - e^{2 x}\right) - \frac{4 \sin{\left(x \right)}}{85}\right) + \frac{33 \cos{\left(x \right)}}{85} = - \frac{4 \sin{\left(x \right)}}{85} - \frac{33 \cos{\left(x \right)}}{85} + e^{- 2 x} + e^{- 4 x}
- No
es decir, función
no es
par ni impar
Gráfico
Gráfico de la función y = -exp(2*x)-exp(4*x)-4*sin(x)/85+33*cos(x)/85
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