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Trazado el gráfico de la función/ -(0.1*e^(0.1*x)*(sin⁡(x)+cos⁡(x)))/(2*(1-e^(0.1*x)*sin⁡(x)))

Gráfico de la función y = -(0.1*e^(0.1*x)*(sin⁡(x)+cos⁡(x)))/(2*(1-e^(0.1*x)*sin⁡(x)))

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
          x                    
          --                   
          10                   
         E                     
       - ---*(sin(x) + cos(x)) 
          10                   
f(x) = ------------------------
            /     x        \   
            |     --       |   
            |     10       |   
          2*\1 - E  *sin(x)/
f(x)=(−1)ex1010(sin⁡(x)+cos⁡(x))2(−ex10sin⁡(x)+1)f{\left(x \right)} = \frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} 
f = (-E^(x/10)/10*(sin(x) + cos(x)))/((2*(-E^(x/10)*sin(x) + 1)))
Gráfico de la función
0.001.000.100.200.300.400.500.600.700.800.900-2
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:
(−1)ex1010(sin⁡(x)+cos⁡(x))2(−ex10sin⁡(x)+1)=0\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución analítica
x1=−π4x_{1} = - \frac{\pi}{4}
x2=3π4x_{2} = \frac{3 \pi}{4}
Solución numérica
x1=93.4623814442964x_{1} = 93.4623814442964
x2=21.2057504117311x_{2} = 21.2057504117311
x3=43.1968989868597x_{3} = 43.1968989868597
x4=90.3207887907066x_{4} = 90.3207887907066
x5=−10.2101761241668x_{5} = -10.2101761241668
x6=−19.6349540849362x_{6} = -19.6349540849362
x7=71.4712328691678x_{7} = 71.4712328691678
x8=−44.7676953136546x_{8} = -44.7676953136546
x9=−104.457955731861x_{9} = -104.457955731861
x10=11.7809724509617x_{10} = 11.7809724509617
x11=−41.6261026600648x_{11} = -41.6261026600648
x12=18.0641577581413x_{12} = 18.0641577581413
x13=8.63937979737193x_{13} = 8.63937979737193
x14=−66.7588438887831x_{14} = -66.7588438887831
x15=−60.4756585816035x_{15} = -60.4756585816035
x16=27.4889357189107x_{16} = 27.4889357189107
x17=74.6128255227576x_{17} = 74.6128255227576
x18=−29.0597320457056x_{18} = -29.0597320457056
x19=40.0553063332699x_{19} = 40.0553063332699
x20=−85.6083998103219x_{20} = -85.6083998103219
x21=33.7721210260903x_{21} = 33.7721210260903
x22=−47.9092879672443x_{22} = -47.9092879672443
x23=−57.3340659280137x_{23} = -57.3340659280137
x24=84.037603483527x_{24} = 84.037603483527
x25=−54.1924732744239x_{25} = -54.1924732744239
x26=−76.1836218495525x_{26} = -76.1836218495525
x27=−32.2013246992954x_{27} = -32.2013246992954
x28=−73.0420291959627x_{28} = -73.0420291959627
x29=30.6305283725005x_{29} = 30.6305283725005
x30=−82.4668071567321x_{30} = -82.4668071567321
x31=−38.484510006475x_{31} = -38.484510006475
x32=96.6039740978861x_{32} = 96.6039740978861
x33=−22.776546738526x_{33} = -22.776546738526
x34=−79.3252145031423x_{34} = -79.3252145031423
x35=2.35619449019234x_{35} = 2.35619449019234
x36=68.329640215578x_{36} = 68.329640215578
x37=65.1880475619882x_{37} = 65.1880475619882
x38=−13.3517687777566x_{38} = -13.3517687777566
x39=−88.7499924639117x_{39} = -88.7499924639117
x40=80.8960108299372x_{40} = 80.8960108299372
x41=−51.0508806208341x_{41} = -51.0508806208341
x42=36.9137136796801x_{42} = 36.9137136796801
x43=−69.9004365423729x_{43} = -69.9004365423729
x44=−7.06858347057703x_{44} = -7.06858347057703
x45=−16.4933614313464x_{45} = -16.4933614313464
x46=−3.92699081698724x_{46} = -3.92699081698724
x47=62.0464549083984x_{47} = 62.0464549083984
x48=49.4800842940392x_{48} = 49.4800842940392
x49=−98.174770424681x_{49} = -98.174770424681
x50=46.3384916404494x_{50} = 46.3384916404494
x51=−25.9181393921158x_{51} = -25.9181393921158
x52=52.621676947629x_{52} = 52.621676947629
x53=24.3473430653209x_{53} = 24.3473430653209
x54=−0.785398163397448x_{54} = -0.785398163397448
x55=5.49778714378214x_{55} = 5.49778714378214
x56=−63.6172512351933x_{56} = -63.6172512351933
x57=99.7455667514759x_{57} = 99.7455667514759
x58=58.9048622548086x_{58} = 58.9048622548086
x59=14.9225651045515x_{59} = 14.9225651045515
x60=−91.8915851175014x_{60} = -91.8915851175014
x61=55.7632696012188x_{61} = 55.7632696012188
x62=−95.0331777710912x_{62} = -95.0331777710912
x63=−35.3429173528852x_{63} = -35.3429173528852
x64=77.7544181763474x_{64} = 77.7544181763474
x65=87.1791961371168x_{65} = 87.1791961371168
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 (-E^(x/10)/10*(sin(x) + cos(x)))/((2*(1 - E^(x/10)*sin(x)))).
(−1)e01010(sin⁡(0)+cos⁡(0))2(−e010sin⁡(0)+1)\frac{\left(-1\right) \frac{e^{\frac{0}{10}}}{10} \left(\sin{\left(0 \right)} + \cos{\left(0 \right)}\right)}{2 \left(- e^{\frac{0}{10}} \sin{\left(0 \right)} + 1\right)}
Resultado:
f(0)=−120f{\left(0 \right)} = - \frac{1}{20}
Punto:
(0, -1/20)
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
12(−ex10sin⁡(x)+1)(−(−sin⁡(x)+cos⁡(x))ex1010−(sin⁡(x)+cos⁡(x))ex10100)−(ex10sin⁡(x)5+2ex10cos⁡(x))(sin⁡(x)+cos⁡(x))ex1040(−ex10sin⁡(x)+1)2=0\frac{1}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} \left(- \frac{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{10} - \frac{\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{100}\right) - \frac{\left(\frac{e^{\frac{x}{10}} \sin{\left(x \right)}}{5} + 2 e^{\frac{x}{10}} \cos{\left(x \right)}\right) \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{40 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)^{2}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−1924.82277051613x_{1} = -1924.82277051613
x2=−124.778636646949x_{2} = -124.778636646949
x3=−2330.50523719648x_{3} = -2330.50523719648
x4=−83.9380940296237x_{4} = -83.9380940296237
x5=−2.81619635397987x_{5} = -2.81619635397987
x6=−30.4975236705277x_{6} = -30.4975236705277
x7=−36.7962906872222x_{7} = -36.7962906872222
x8=−71.3721235521918x_{8} = -71.3721235521918
x9=1.90312690149253x_{9} = 1.90312690149253
x10=−582.0968365869x_{10} = -582.0968365869
x11=−74.5127482905657x_{11} = -74.5127482905657
x12=−486.062167657715x_{12} = -486.062167657715
x13=−61.9453505282283x_{13} = -61.9453505282283
x14=−11.4556169574971x_{14} = -11.4556169574971
x15=−153.052973368617x_{15} = -153.052973368617
x16=−822.325556649958x_{16} = -822.325556649958
x17=−49.375369247982x_{17} = -49.375369247982
x18=−898.224266525647x_{18} = -898.224266525647
x19=−27.4345573572623x_{19} = -27.4345573572623
x20=−39.9685654523554x_{20} = -39.9685654523554
x21=−52.5256909712287x_{21} = -52.5256909712287
x22=−80.7961242092005x_{22} = -80.7961242092005
x23=−55.6609093999258x_{23} = -55.6609093999258
x24=−77.6550479304676x_{24} = -77.6550479304676
x25=−435.796312033093x_{25} = -435.796312033093
x26=−2733.12508743085x_{26} = -2733.12508743085
x27=−567.743203483864x_{27} = -567.743203483864
x28=−14.9808472121154x_{28} = -14.9808472121154
x29=−39.968565452453x_{29} = -39.968565452453
x30=−90.2212050693967x_{30} = -90.2212050693967
x31=2.82358414056662x_{31} = 2.82358414056662
x32=−4047.08933720441x_{32} = -4047.08933720441
x33=−3830.03441994575x_{33} = -3830.03441994575
x34=−21.1907151682264x_{34} = -21.1907151682264
x35=−43.0877668109453x_{35} = -43.0877668109453
x36=−24.1849742273843x_{36} = -24.1849742273843
x37=−21.1907151682264x_{37} = -21.1907151682264
x38=−33.6966592867304x_{38} = -33.6966592867304
x39=−234.734382203391x_{39} = -234.734382203391
x40=−93.3626507569302x_{40} = -93.3626507569302
x41=−5821.90700008095x_{41} = -5821.90700008095
x42=−102.787514924946x_{42} = -102.787514924946
x43=−87.0794112023455x_{43} = -87.0794112023455
x44=−68.2292056709602x_{44} = -68.2292056709602
x45=−12204.9931764696x_{45} = -12204.9931764696
x46=−65.0894273082213x_{46} = -65.0894273082213
x47=−99.6458650042135x_{47} = -99.6458650042135
x48=−131.061823204732x_{48} = -131.061823204732
x49=−8.83480155558271x_{49} = -8.83480155558271
x50=−476.63701652976x_{50} = -476.63701652976
x51=−1074.39547911081x_{51} = -1074.39547911081
x52=−46.2457238206076x_{52} = -46.2457238206076
x53=−564.547138764174x_{53} = -564.547138764174
x54=−442.080574029362x_{54} = -442.080574029362
x55=−17.8461051872707x_{55} = -17.8461051872707
x56=−96.5043507553482x_{56} = -96.5043507553482
x57=−58.8071586004732x_{57} = -58.8071586004732
Signos de extremos en los puntos:
(-1924.8227705161255, 1.7691438401015e-85)

(-124.77863664694901, -2.68076129189499e-7)

(-2330.505237196484, -4.22135311569299e-103)

(-83.93809402962373, 1.59173249107843e-5)

(-2.8161963539798656, 0.0385179368983429)

(-30.497523670527663, -0.00344988864624691)

(-36.79629068722217, -0.00180771879571231)

(-71.37212355219184, 5.5902289213802e-5)

(1.9031269014925303, 0.261012330751754)

(-582.0968365869004, -4.32301071762198e-28)

(-74.51274829056572, -4.08746729062026e-5)

(-486.06216765771495, 5.46926244404683e-23)

(-61.94535052822834, -0.000143779294596033)

(-11.455616957497138, -0.0298024142051202)

(-153.05297336861673, 1.58605436357488e-8)

(-822.3255566499581, -1.36867144921588e-37)

(-49.37536924798202, -0.000507203209118152)

(-898.2242665256471, -6.0251928423631e-41)

(-27.434557357262268, 0.00433563189151737)

(-39.968565452355406, 0.00127634792817982)

(-52.52569097122874, 0.000366918027567138)

(-80.79612420920053, -2.18015769450153e-5)

(-55.660909399925835, -0.000269881932289126)

(-77.65504793046756, 2.98317600251334e-5)

(-435.79631203309265, 8.33548665036684e-21)

(-2733.1250874308507, -1.060741158431e-120)

(-567.743203483864, 1.55078029557284e-26)

(-14.980847212115394, 0.0137393797338713)

(-39.96856545245299, 0.00127634792817982)

(-90.22120506939667, 8.49239687877201e-6)

(2.8235841405666235, 0.0721894678915088)

(-4047.089337204406, -8.2215084778855e-179)

(-3830.034419945751, 1.12522718794921e-168)

(-21.190715168226397, 0.00782017337166575)

(-43.08776681094528, -0.00095542252796214)

(-24.184974227384313, -0.00669881624888215)

(-21.1907151682264, 0.00782017337166574)

(-33.69665928673042, 0.00236392962632975)

(-234.73438220339085, 4.4971733134714e-12)

(-93.36265075693017, -6.203872366296e-6)

(-5821.907000080949, 2.51175383826673e-255)

(-102.78751492494612, 2.41717872746826e-6)

(-87.0794112023455, -1.16295803774142e-5)

(-68.22920567096024, -7.66479316941045e-5)

(-12204.993176469627, 4.78805184243152e-532)

(-65.08942730822133, 0.000104730111883607)

(-99.6458650042135, -3.30958665751948e-6)

(-131.0618232047325, -1.43015225686589e-7)

(-8.834801555582711, 0.0233108239607265)

(-476.63701652976, -1.40359463750799e-22)

(-1074.3954791108067, -1.12431696078796e-48)

(-46.245723820607644, 0.000685364558211034)

(-564.547138764174, -2.11999310241646e-26)

(-442.08057402936174, 4.44688025828588e-21)

(-17.84610518727068, -0.0134996580421051)

(-96.50435075534824, 4.5307900389751e-6)

(-58.8071586004732, 0.000196114702958014)


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=−124.778636646949x_{1} = -124.778636646949
x2=−30.4975236705277x_{2} = -30.4975236705277
x3=−36.7962906872222x_{3} = -36.7962906872222
x4=1.90312690149253x_{4} = 1.90312690149253
x5=−74.5127482905657x_{5} = -74.5127482905657
x6=−61.9453505282283x_{6} = -61.9453505282283
x7=−11.4556169574971x_{7} = -11.4556169574971
x8=−49.375369247982x_{8} = -49.375369247982
x9=−80.7961242092005x_{9} = -80.7961242092005
x10=−55.6609093999258x_{10} = -55.6609093999258
x11=−43.0877668109453x_{11} = -43.0877668109453
x12=−24.1849742273843x_{12} = -24.1849742273843
x13=−93.3626507569302x_{13} = -93.3626507569302
x14=−87.0794112023455x_{14} = -87.0794112023455
x15=−68.2292056709602x_{15} = -68.2292056709602
x16=−99.6458650042135x_{16} = -99.6458650042135
x17=−131.061823204732x_{17} = -131.061823204732
x18=−476.63701652976x_{18} = -476.63701652976
x19=−17.8461051872707x_{19} = -17.8461051872707
Puntos máximos de la función:
x19=−83.9380940296237x_{19} = -83.9380940296237
x19=−2.81619635397987x_{19} = -2.81619635397987
x19=−71.3721235521918x_{19} = -71.3721235521918
x19=−153.052973368617x_{19} = -153.052973368617
x19=−27.4345573572623x_{19} = -27.4345573572623
x19=−39.9685654523554x_{19} = -39.9685654523554
x19=−52.5256909712287x_{19} = -52.5256909712287
x19=−77.6550479304676x_{19} = -77.6550479304676
x19=−435.796312033093x_{19} = -435.796312033093
x19=−567.743203483864x_{19} = -567.743203483864
x19=−14.9808472121154x_{19} = -14.9808472121154
x19=−39.968565452453x_{19} = -39.968565452453
x19=−90.2212050693967x_{19} = -90.2212050693967
x19=2.82358414056662x_{19} = 2.82358414056662
x19=−21.1907151682264x_{19} = -21.1907151682264
x19=−21.1907151682264x_{19} = -21.1907151682264
x19=−33.6966592867304x_{19} = -33.6966592867304
x19=−234.734382203391x_{19} = -234.734382203391
x19=−102.787514924946x_{19} = -102.787514924946
x19=−65.0894273082213x_{19} = -65.0894273082213
x19=−8.83480155558271x_{19} = -8.83480155558271
x19=−46.2457238206076x_{19} = -46.2457238206076
x19=−96.5043507553482x_{19} = -96.5043507553482
x19=−58.8071586004732x_{19} = -58.8071586004732
Decrece en los intervalos
[1.90312690149253,∞)\left[1.90312690149253, \infty\right)
Crece en los intervalos
(−∞,−476.63701652976]\left(-\infty, -476.63701652976\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((−1)ex1010(sin⁡(x)+cos⁡(x))2(−ex10sin⁡(x)+1))=0\lim_{x \to -\infty}\left(\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)}\right) = 0
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=0y = 0
lim⁡x→∞((−1)ex1010(sin⁡(x)+cos⁡(x))2(−ex10sin⁡(x)+1))=⟨−2,2⟩lim⁡x→0+(e110x20e110xsin⁡(1x)−20)\lim_{x \to \infty}\left(\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)}\right) = \left\langle -2, 2\right\rangle \lim_{x \to 0^+}\left(\frac{e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−2,2⟩lim⁡x→0+(e110x20e110xsin⁡(1x)−20)y = \left\langle -2, 2\right\rangle \lim_{x \to 0^+}\left(\frac{e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-E^(x/10)/10*(sin(x) + cos(x)))/((2*(1 - E^(x/10)*sin(x)))), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(−12(−ex10sin⁡(x)+1)(sin⁡(x)+cos⁡(x))ex1010x)=0\lim_{x \to -\infty}\left(- \frac{\frac{1}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{10 x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
lim⁡x→∞(−12(−ex10sin⁡(x)+1)(sin⁡(x)+cos⁡(x))ex1010x)=⟨−2,2⟩lim⁡x→0+(xe110x20e110xsin⁡(1x)−20)\lim_{x \to \infty}\left(- \frac{\frac{1}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{\frac{x}{10}}}{10 x}\right) = \left\langle -2, 2\right\rangle \lim_{x \to 0^+}\left(\frac{x e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)
Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=⟨−2,2⟩xlim⁡x→0+(xe110x20e110xsin⁡(1x)−20)y = \left\langle -2, 2\right\rangle x \lim_{x \to 0^+}\left(\frac{x e^{\frac{1}{10 x}}}{20 e^{\frac{1}{10 x}} \sin{\left(\frac{1}{x} \right)} - 20}\right)
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:
(−1)ex1010(sin⁡(x)+cos⁡(x))2(−ex10sin⁡(x)+1)=−(−sin⁡(x)+cos⁡(x))e−x1010(2+2e−x10sin⁡(x))\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} = - \frac{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- \frac{x}{10}}}{10 \left(2 + 2 e^{- \frac{x}{10}} \sin{\left(x \right)}\right)}
- No
(−1)ex1010(sin⁡(x)+cos⁡(x))2(−ex10sin⁡(x)+1)=(−sin⁡(x)+cos⁡(x))e−x1010(2+2e−x10sin⁡(x))\frac{\left(-1\right) \frac{e^{\frac{x}{10}}}{10} \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right)}{2 \left(- e^{\frac{x}{10}} \sin{\left(x \right)} + 1\right)} = \frac{\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- \frac{x}{10}}}{10 \left(2 + 2 e^{- \frac{x}{10}} \sin{\left(x \right)}\right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор