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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
        x                  
       e                   
f(x) = -- - cos(x) + sin(x)
       2
f(x)=(ex2−cos⁡(x))+sin⁡(x)f{\left(x \right)} = \left(\frac{e^{x}}{2} - \cos{\left(x \right)}\right) + \sin{\left(x \right)} 
f = exp(x)/2 - cos(x) + sin(x)
Gráfico de la función
02468-8-6-4-2-101020000-10000
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:
(ex2−cos⁡(x))+sin⁡(x)=0\left(\frac{e^{x}}{2} - \cos{\left(x \right)}\right) + \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−8.63931721569265x_{1} = -8.63931721569265
x2=−65.1880475619882x_{2} = -65.1880475619882
x3=0.292130748452471x_{3} = 0.292130748452471
x4=0.292130748452456x_{4} = 0.292130748452456
x5=−14.9225649876911x_{5} = -14.9225649876911
x6=−62.0464549083984x_{6} = -62.0464549083984
x7=−40.0553063332699x_{7} = -40.0553063332699
x8=−30.6305283725005x_{8} = -30.6305283725005
x9=−228.550865548657x_{9} = -228.550865548657
x10=−21.2057504115129x_{10} = -21.2057504115129
x11=−2.32149445348489x_{11} = -2.32149445348489
x12=−43.1968989868597x_{12} = -43.1968989868597
x13=−33.7721210260903x_{13} = -33.7721210260903
x14=−49.4800842940392x_{14} = -49.4800842940392
x15=−5.49923314463276x_{15} = -5.49923314463276
x16=−58.9048622548086x_{16} = -58.9048622548086
x17=−68.329640215578x_{17} = -68.329640215578
x18=−11.7809751551847x_{18} = -11.7809751551847
x19=−84.037603483527x_{19} = -84.037603483527
x20=−24.3473430653303x_{20} = -24.3473430653303
x21=−18.0641577631913x_{21} = -18.0641577631913
x22=−27.4889357189103x_{22} = -27.4889357189103
x23=−46.3384916404494x_{23} = -46.3384916404494
x24=−96.6039740978861x_{24} = -96.6039740978861
x25=−90.3207887907066x_{25} = -90.3207887907066
x26=−74.6128255227576x_{26} = -74.6128255227576
x27=−55.7632696012188x_{27} = -55.7632696012188
x28=−93.4623814442964x_{28} = -93.4623814442964
x29=−71.4712328691678x_{29} = -71.4712328691678
x30=−52.621676947629x_{30} = -52.621676947629
x31=−99.7455667514759x_{31} = -99.7455667514759
x32=−36.9137136796801x_{32} = -36.9137136796801
x33=−80.8960108299372x_{33} = -80.8960108299372
x34=−87.1791961371168x_{34} = -87.1791961371168
x35=−77.7544181763474x_{35} = -77.7544181763474
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(x)/2 - cos(x) + sin(x).
(−cos⁡(0)+e02)+sin⁡(0)\left(- \cos{\left(0 \right)} + \frac{e^{0}}{2}\right) + \sin{\left(0 \right)}
Resultado:
f(0)=−12f{\left(0 \right)} = - \frac{1}{2}
Punto:
(0, -1/2)
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
ex2+sin⁡(x)+cos⁡(x)=0\frac{e^{x}}{2} + \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−35.3429173528852x_{1} = -35.3429173528852
x2=−51.0508806208341x_{2} = -51.0508806208341
x3=−107.59954838545x_{3} = -107.59954838545
x4=−16.4933614070535x_{4} = -16.4933614070535
x5=−0.925925557386978x_{5} = -0.925925557386978
x6=−10.2101631153588x_{6} = -10.2101631153588
x7=−98.174770424681x_{7} = -98.174770424681
x8=−13.3517693399106x_{8} = -13.3517693399106
x9=−38.484510006475x_{9} = -38.484510006475
x10=−79.3252145031423x_{10} = -79.3252145031423
x11=−88.7499924639117x_{11} = -88.7499924639117
x12=−3.91997570268469x_{12} = -3.91997570268469
x13=−66.7588438887831x_{13} = -66.7588438887831
x14=−63.6172512351933x_{14} = -63.6172512351933
x15=−25.9181393921178x_{15} = -25.9181393921178
x16=−95.0331777710912x_{16} = -95.0331777710912
x17=−32.2013246992954x_{17} = -32.2013246992954
x18=−47.9092879672443x_{18} = -47.9092879672443
x19=−73.0420291959627x_{19} = -73.0420291959627
x20=−41.6261026600648x_{20} = -41.6261026600648
x21=−7.06888440891705x_{21} = -7.06888440891705
x22=−91.8915851175014x_{22} = -91.8915851175014
x23=−82.4668071567321x_{23} = -82.4668071567321
x24=−76.1836218495525x_{24} = -76.1836218495525
x25=−22.7765467384806x_{25} = -22.7765467384806
x26=−60.4756585816035x_{26} = -60.4756585816035
x27=−29.0597320457055x_{27} = -29.0597320457055
x28=−19.634954085986x_{28} = -19.634954085986
x29=−69.9004365423729x_{29} = -69.9004365423729
x30=−57.3340659280137x_{30} = -57.3340659280137
x31=−44.7676953136546x_{31} = -44.7676953136546
x32=−54.1924732744239x_{32} = -54.1924732744239
x33=−85.6083998103219x_{33} = -85.6083998103219
Signos de extremos en los puntos:
(-35.34291735288517, 1.4142135623731)

(-51.05088062083414, -1.41421356237309)

(-107.59954838545042, -1.41421356237309)

(-16.493361407053527, 1.41421359672843)

(-0.9259255573869776, -1.20219033071629)

(-10.210163115358759, 1.41423195948623)

(-98.17477042468104, 1.41421356237309)

(-13.351769339910561, -1.41421276736715)

(-38.48451000647497, -1.41421356237309)

(-79.32521450314228, 1.41421356237309)

(-88.74999246391165, -1.4142135623731)

(-3.919975702684686, 1.42409955291581)

(-66.7588438887831, 1.4142135623731)

(-63.617251235193315, -1.41421356237309)

(-25.918139392117755, -1.41421356237032)

(-95.03317777109125, -1.41421356237309)

(-32.201324699295384, -1.41421356237309)

(-47.909287967244346, 1.41421356237309)

(-73.0420291959627, 1.41421356237309)

(-41.62610266006476, 1.41421356237309)

(-7.068884408917046, -1.4137879072593)

(-91.89158511750145, 1.41421356237309)

(-82.46680715673207, -1.41421356237309)

(-76.18362184955248, -1.41421356237309)

(-22.776546738480636, 1.41421356243725)

(-60.47565858160352, 1.41421356237309)

(-29.059732045705502, 1.41421356237321)

(-19.634954085986, -1.41421356088847)

(-69.9004365423729, -1.4142135623731)

(-57.33406592801373, -1.41421356237309)

(-44.767695313654556, -1.41421356237309)

(-54.19247327442393, 1.41421356237309)

(-85.60839981032187, 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=−51.0508806208341x_{1} = -51.0508806208341
x2=−107.59954838545x_{2} = -107.59954838545
x3=−0.925925557386978x_{3} = -0.925925557386978
x4=−13.3517693399106x_{4} = -13.3517693399106
x5=−38.484510006475x_{5} = -38.484510006475
x6=−88.7499924639117x_{6} = -88.7499924639117
x7=−63.6172512351933x_{7} = -63.6172512351933
x8=−25.9181393921178x_{8} = -25.9181393921178
x9=−95.0331777710912x_{9} = -95.0331777710912
x10=−32.2013246992954x_{10} = -32.2013246992954
x11=−7.06888440891705x_{11} = -7.06888440891705
x12=−82.4668071567321x_{12} = -82.4668071567321
x13=−76.1836218495525x_{13} = -76.1836218495525
x14=−19.634954085986x_{14} = -19.634954085986
x15=−69.9004365423729x_{15} = -69.9004365423729
x16=−57.3340659280137x_{16} = -57.3340659280137
x17=−44.7676953136546x_{17} = -44.7676953136546
Puntos máximos de la función:
x17=−35.3429173528852x_{17} = -35.3429173528852
x17=−16.4933614070535x_{17} = -16.4933614070535
x17=−10.2101631153588x_{17} = -10.2101631153588
x17=−98.174770424681x_{17} = -98.174770424681
x17=−79.3252145031423x_{17} = -79.3252145031423
x17=−3.91997570268469x_{17} = -3.91997570268469
x17=−66.7588438887831x_{17} = -66.7588438887831
x17=−47.9092879672443x_{17} = -47.9092879672443
x17=−73.0420291959627x_{17} = -73.0420291959627
x17=−41.6261026600648x_{17} = -41.6261026600648
x17=−91.8915851175014x_{17} = -91.8915851175014
x17=−22.7765467384806x_{17} = -22.7765467384806
x17=−60.4756585816035x_{17} = -60.4756585816035
x17=−29.0597320457055x_{17} = -29.0597320457055
x17=−54.1924732744239x_{17} = -54.1924732744239
x17=−85.6083998103219x_{17} = -85.6083998103219
Decrece en los intervalos
[−0.925925557386978,∞)\left[-0.925925557386978, \infty\right)
Crece en los intervalos
(−∞,−107.59954838545]\left(-\infty, -107.59954838545\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
ex2−sin⁡(x)+cos⁡(x)=0\frac{e^{x}}{2} - \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−65.1880475619882x_{1} = -65.1880475619882
x2=−18.0641577530913x_{2} = -18.0641577530913
x3=−62.0464549083984x_{3} = -62.0464549083984
x4=−40.0553063332699x_{4} = -40.0553063332699
x5=−228.550865548657x_{5} = -228.550865548657
x6=−8.63944237121926x_{6} = -8.63944237121926
x7=−43.1968989868597x_{7} = -43.1968989868597
x8=−33.7721210260903x_{8} = -33.7721210260903
x9=−49.4800842940392x_{9} = -49.4800842940392
x10=−58.9048622548086x_{10} = -58.9048622548086
x11=−68.329640215578x_{11} = -68.329640215578
x12=−27.4889357189111x_{12} = -27.4889357189111
x13=−84.037603483527x_{13} = -84.037603483527
x14=−30.6305283725005x_{14} = -30.6305283725005
x15=−21.2057504119493x_{15} = -21.2057504119493
x16=−2.3886402485139x_{16} = -2.3886402485139
x17=−5.49633694895682x_{17} = -5.49633694895682
x18=−46.3384916404494x_{18} = -46.3384916404494
x19=−11.7809697467241x_{19} = -11.7809697467241
x20=−96.6039740978861x_{20} = -96.6039740978861
x21=−90.3207887907066x_{21} = -90.3207887907066
x22=−74.6128255227576x_{22} = -74.6128255227576
x23=−55.7632696012188x_{23} = -55.7632696012188
x24=−24.3473430653115x_{24} = -24.3473430653115
x25=−93.4623814442964x_{25} = -93.4623814442964
x26=−71.4712328691678x_{26} = -71.4712328691678
x27=−52.621676947629x_{27} = -52.621676947629
x28=−99.7455667514759x_{28} = -99.7455667514759
x29=−36.9137136796801x_{29} = -36.9137136796801
x30=−80.8960108299372x_{30} = -80.8960108299372
x31=−87.1791961371168x_{31} = -87.1791961371168
x32=−14.9225652214119x_{32} = -14.9225652214119
x33=−77.7544181763474x_{33} = -77.7544181763474

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
[−2.3886402485139,∞)\left[-2.3886402485139, \infty\right)
Convexa en los intervalos
(−∞,−228.550865548657]\left(-\infty, -228.550865548657\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((ex2−cos⁡(x))+sin⁡(x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\left(\frac{e^{x}}{2} - \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) = \left\langle -2, 2\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞((ex2−cos⁡(x))+sin⁡(x))=∞\lim_{x \to \infty}\left(\left(\frac{e^{x}}{2} - \cos{\left(x \right)}\right) + \sin{\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 exp(x)/2 - cos(x) + sin(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞((ex2−cos⁡(x))+sin⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\left(\frac{e^{x}}{2} - \cos{\left(x \right)}\right) + \sin{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞((ex2−cos⁡(x))+sin⁡(x)x)=∞\lim_{x \to \infty}\left(\frac{\left(\frac{e^{x}}{2} - \cos{\left(x \right)}\right) + \sin{\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:
(ex2−cos⁡(x))+sin⁡(x)=−sin⁡(x)−cos⁡(x)+e−x2\left(\frac{e^{x}}{2} - \cos{\left(x \right)}\right) + \sin{\left(x \right)} = - \sin{\left(x \right)} - \cos{\left(x \right)} + \frac{e^{- x}}{2}
- No
(ex2−cos⁡(x))+sin⁡(x)=sin⁡(x)+cos⁡(x)−e−x2\left(\frac{e^{x}}{2} - \cos{\left(x \right)}\right) + \sin{\left(x \right)} = \sin{\left(x \right)} + \cos{\left(x \right)} - \frac{e^{- x}}{2}
- No
es decir, función
no es
par ni impar
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