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Trazado el gráfico de la función/ sh(x)+ch(x)+sin(x)+cos(x)

Gráfico de la función y = sh(x)+ch(x)+sin(x)+cos(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = sinh(x) + cosh(x) + sin(x) + cos(x)
f(x)=((sinh⁡(x)+cosh⁡(x))+sin⁡(x))+cos⁡(x)f{\left(x \right)} = \left(\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) + \sin{\left(x \right)}\right) + \cos{\left(x \right)} 
f = sinh(x) + cosh(x) + sin(x) + cos(x)
Gráfico de la función
02468-8-6-4-2-1010-2500025000
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:
((sinh⁡(x)+cosh⁡(x))+sin⁡(x))+cos⁡(x)=0\left(\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) + \sin{\left(x \right)}\right) + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−98x_{1} = -98
x2=−44.7676953115161x_{2} = -44.7676953115161
x3=−68x_{3} = -68
x4=−54.1925387985637x_{4} = -54.1925387985637
x5=−74x_{5} = -74
x6=−3.91286004365428x_{6} = -3.91286004365428
x7=−38.4845100064698x_{7} = -38.4845100064698
x8=−80x_{8} = -80
x9=−29.0597320457054x_{9} = -29.0597320457054
x10=−64.8x_{10} = -64.8
x11=−47.9092881324719x_{11} = -47.9092881324719
x12=−76x_{12} = -76
x13=−82x_{13} = -82
x14=−10.2101501062122x_{14} = -10.2101501062122
x15=−22.7765467384353x_{15} = -22.7765467384353
x16=−7.06918516629265x_{16} = -7.06918516629265
x17=−96x_{17} = -96
x18=−100x_{18} = -100
x19=−41.6261026602281x_{19} = -41.6261026602281
x20=−51.0508792796484x_{20} = -51.0508792796484
x21=−78x_{21} = -78
x22=−64x_{22} = -64
x23=−35.3429173528853x_{23} = -35.3429173528853
x24=−16.4933613827606x_{24} = -16.4933613827606
x25=−1.03841563726656x_{25} = -1.03841563726656
x26=−32.2013246992954x_{26} = -32.2013246992954
x27=−51.0508787223699x_{27} = -51.0508787223699
x28=−57.333242035742x_{28} = -57.333242035742
x29=−66x_{29} = -66
x30=−94x_{30} = -94
x31=−90x_{31} = -90
x32=−25.9181393921197x_{32} = -25.9181393921197
x33=−38.4845100064697x_{33} = -38.4845100064697
x34=−19.6349540870358x_{34} = -19.6349540870358
x35=−13.3517699020639x_{35} = -13.3517699020639
x36=−72x_{36} = -72
x37=−92x_{37} = -92
x38=−44.7676953119769x_{38} = -44.7676953119769
x39=−86x_{39} = -86
x40=−88x_{40} = -88
x41=−84x_{41} = -84
x42=−70x_{42} = -70
x43=−60.5x_{43} = -60.5
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 sinh(x) + cosh(x) + sin(x) + cos(x).
(sin⁡(0)+(sinh⁡(0)+cosh⁡(0)))+cos⁡(0)\left(\sin{\left(0 \right)} + \left(\sinh{\left(0 \right)} + \cosh{\left(0 \right)}\right)\right) + \cos{\left(0 \right)}
Resultado:
f(0)=2f{\left(0 \right)} = 2
Punto:
(0, 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
−sin⁡(x)+cos⁡(x)+sinh⁡(x)+cosh⁡(x)=0- \sin{\left(x \right)} + \cos{\left(x \right)} + \sinh{\left(x \right)} + \cosh{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−98x_{1} = -98
x2=−40.0553063332566x_{2} = -40.0553063332566
x3=−68x_{3} = -68
x4=−27.4889357189115x_{4} = -27.4889357189115
x5=−30.6305283725005x_{5} = -30.6305283725005
x6=−74x_{6} = -74
x7=−14.9225653382722x_{7} = -14.9225653382722
x8=−18.0641577480413x_{8} = -18.0641577480413
x9=−36.913713679678x_{9} = -36.913713679678
x10=−36.9137136796813x_{10} = -36.9137136796813
x11=−80x_{11} = -80
x12=−49.4800837705626x_{12} = -49.4800837705626
x13=−76x_{13} = -76
x14=−82x_{14} = -82
x15=−21.2057504121676x_{15} = -21.2057504121676
x16=−96x_{16} = -96
x17=−100x_{17} = -100
x18=−62x_{18} = -62
x19=−78x_{19} = -78
x20=−30.6305283725004x_{20} = -30.6305283725004
x21=−64x_{21} = -64
x22=−24.347343065302x_{22} = -24.347343065302
x23=−58.9089362661039x_{23} = -58.9089362661039
x24=−52.6216786429799x_{24} = -52.6216786429799
x25=−43.1968989874661x_{25} = -43.1968989874661
x26=−58.903371320038x_{26} = -58.903371320038
x27=−46.3384916334098x_{27} = -46.3384916334098
x28=−66x_{28} = -66
x29=−8.63950493723709x_{29} = -8.63950493723709
x30=−55.7632311977716x_{30} = -55.7632311977716
x31=−94x_{31} = -94
x32=−90x_{32} = -90
x33=−52.6216698294126x_{33} = -52.6216698294126
x34=−90.3525333679292x_{34} = -90.3525333679292
x35=−72x_{35} = -72
x36=−92x_{36} = -92
x37=−2.41916566066641x_{37} = -2.41916566066641
x38=−11.7809670424718x_{38} = -11.7809670424718
x39=−33.7721210260903x_{39} = -33.7721210260903
x40=−86x_{40} = -86
x41=−88x_{41} = -88
x42=−84x_{42} = -84
x43=−70x_{43} = -70
x44=−5.49488252961338x_{44} = -5.49488252961338
x45=−46.3384916508493x_{45} = -46.3384916508493
Signos de extremos en los puntos:
(-98, -0.245906373301036)

(-40.05530633325657, -1.41421356237309)

(-68, 1.33807070318533)

(-27.488935718911506, -1.41421356237309)

(-30.630528372500454, 1.41421356237309)

(-74, 1.15686360229902)

(-14.922565338272241, -1.41421323175354)

(-18.06415774804132, 1.41421357727426)

(-36.91371367967796, 1.41421356237309)

(-36.913713679681294, 1.41421356237309)

(-80, 0.883501410084328)

(-49.48008377056263, 1.4142135623729)

(-76, 0.258223694209377)

(-82, 0.636448915449458)

(-21.205750412167564, -1.41421356237309)

(-96, -1.16401819472543)

(-100, 1.36868451339744)

(-62, 1.41268785897281)

(-78, -1.37178154923252)

(-30.630528372500446, 1.41421356237309)

(-64, -0.528168807767241)

(-24.347343065302038, 1.4142135623731)

(-58.90893626610386, -1.41420182613642)

(-52.62167864297992, -1.41421356237106)

(-43.196898987466135, 1.41421356237309)

(-58.903371320037984, -1.41421199055528)

(-46.338491633409795, -1.41421356237309)

(-66, -0.973096301942383)

(-8.639504937237092, -1.41403657680579)

(-55.76323119777159, 1.41421356133024)

(-94, 1.21471135213764)

(-90, -1.34207027972973)

(-52.621669829412575, -1.41421356233727)

(-90.35253336792924, -1.413501057849)

(-72, -1.22107395103592)

(-92, 0.153021621705466)

(-2.419165660666407, -1.32241471076109)

(-11.780967042471811, 1.41422121110827)

(-33.772121026090346, -1.41421356237309)

(-86, 0.539760002054318)

(-88, 0.963974980961464)

(-84, -1.41321381566063)

(-70, -0.140571478471589)

(-5.494882529613383, 1.41831533564538)

(-46.338491650849335, -1.41421356237309)


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=−40.0553063332566x_{1} = -40.0553063332566
x2=−27.4889357189115x_{2} = -27.4889357189115
x3=−21.2057504121676x_{3} = -21.2057504121676
x4=−46.3384916334098x_{4} = -46.3384916334098
x5=−8.63950493723709x_{5} = -8.63950493723709
x6=−2.41916566066641x_{6} = -2.41916566066641
x7=−33.7721210260903x_{7} = -33.7721210260903
x8=−46.3384916508493x_{8} = -46.3384916508493
Puntos máximos de la función:
x8=−30.6305283725005x_{8} = -30.6305283725005
x8=−18.0641577480413x_{8} = -18.0641577480413
x8=−36.913713679678x_{8} = -36.913713679678
x8=−36.9137136796813x_{8} = -36.9137136796813
x8=−30.6305283725004x_{8} = -30.6305283725004
x8=−24.347343065302x_{8} = -24.347343065302
x8=−43.1968989874661x_{8} = -43.1968989874661
x8=−11.7809670424718x_{8} = -11.7809670424718
x8=−5.49488252961338x_{8} = -5.49488252961338
Decrece en los intervalos
[−2.41916566066641,∞)\left[-2.41916566066641, \infty\right)
Crece en los intervalos
(−∞,−46.3384916508493]\left(-\infty, -46.3384916508493\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
−sin⁡(x)−cos⁡(x)+sinh⁡(x)+cosh⁡(x)=0- \sin{\left(x \right)} - \cos{\left(x \right)} + \sinh{\left(x \right)} + \cosh{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−98x_{1} = -98
x2=−44.7676953115161x_{2} = -44.7676953115161
x3=−68x_{3} = -68
x4=−54.1925387985637x_{4} = -54.1925387985637
x5=−25.9181393921119x_{5} = -25.9181393921119
x6=0x_{6} = 0
x7=−74x_{7} = -74
x8=−38.4845100064698x_{8} = -38.4845100064698
x9=−80x_{9} = -80
x10=−64.8x_{10} = -64.8
x11=−47.9092881324719x_{11} = -47.9092881324719
x12=−76x_{12} = -76
x13=−82x_{13} = -82
x14=−19.6349540828366x_{14} = -19.6349540828366
x15=−96x_{15} = -96
x16=−100x_{16} = -100
x17=−41.6261026602281x_{17} = -41.6261026602281
x18=−51.0508792796484x_{18} = -51.0508792796484
x19=−29.0597320457058x_{19} = -29.0597320457058
x20=−78x_{20} = -78
x21=−64x_{21} = -64
x22=−3.94073313569291x_{22} = -3.94073313569291
x23=−16.4933614799322x_{23} = -16.4933614799322
x24=−35.3429173528853x_{24} = -35.3429173528853
x25=−32.2013246992954x_{25} = -32.2013246992954
x26=−51.0508787223699x_{26} = -51.0508787223699
x27=−57.333242035742x_{27} = -57.333242035742
x28=−10.2102021407676x_{28} = -10.2102021407676
x29=−66x_{29} = -66
x30=−22.7765467386167x_{30} = -22.7765467386167
x31=−94x_{31} = -94
x32=−90x_{32} = -90
x33=−13.3517676534468x_{33} = -13.3517676534468
x34=−38.4845100064697x_{34} = -38.4845100064697
x35=−72x_{35} = -72
x36=−7.06798104991329x_{36} = -7.06798104991329
x37=−92x_{37} = -92
x38=−44.7676953119769x_{38} = -44.7676953119769
x39=−86x_{39} = -86
x40=−88x_{40} = -88
x41=−84x_{41} = -84
x42=−70x_{42} = -70
x43=−60.5x_{43} = -60.5

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
(−∞,−19.6349540828366]∪[−3.94073313569291,∞)\left(-\infty, -19.6349540828366\right] \cup \left[-3.94073313569291, \infty\right)
Convexa en los intervalos
[−7.06798104991329,−3.94073313569291]\left[-7.06798104991329, -3.94073313569291\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞(((sinh⁡(x)+cosh⁡(x))+sin⁡(x))+cos⁡(x))y = \lim_{x \to -\infty}\left(\left(\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) + \sin{\left(x \right)}\right) + \cos{\left(x \right)}\right)
lim⁡x→∞(((sinh⁡(x)+cosh⁡(x))+sin⁡(x))+cos⁡(x))=∞\lim_{x \to \infty}\left(\left(\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) + \sin{\left(x \right)}\right) + \cos{\left(x \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 sinh(x) + cosh(x) + sin(x) + cos(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→−∞(((sinh⁡(x)+cosh⁡(x))+sin⁡(x))+cos⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\left(\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) + \sin{\left(x \right)}\right) + \cos{\left(x \right)}}{x}\right)
lim⁡x→∞(((sinh⁡(x)+cosh⁡(x))+sin⁡(x))+cos⁡(x)x)=∞\lim_{x \to \infty}\left(\frac{\left(\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) + \sin{\left(x \right)}\right) + \cos{\left(x \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:
((sinh⁡(x)+cosh⁡(x))+sin⁡(x))+cos⁡(x)=−sin⁡(x)+cos⁡(x)−sinh⁡(x)+cosh⁡(x)\left(\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) + \sin{\left(x \right)}\right) + \cos{\left(x \right)} = - \sin{\left(x \right)} + \cos{\left(x \right)} - \sinh{\left(x \right)} + \cosh{\left(x \right)}
- No
((sinh⁡(x)+cosh⁡(x))+sin⁡(x))+cos⁡(x)=sin⁡(x)−cos⁡(x)+sinh⁡(x)−cosh⁡(x)\left(\left(\sinh{\left(x \right)} + \cosh{\left(x \right)}\right) + \sin{\left(x \right)}\right) + \cos{\left(x \right)} = \sin{\left(x \right)} - \cos{\left(x \right)} + \sinh{\left(x \right)} - \cosh{\left(x \right)}
- No
es decir, función
no es
par ni impar
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