Sr Examen

Otras calculadoras

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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       sin(x)                      2*x         
f(x) = ------ + (cos(x) + sin(x))*e    + cos(x)
         2
f(x)=((sin⁡(x)+cos⁡(x))e2x+sin⁡(x)2)+cos⁡(x)f{\left(x \right)} = \left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} + \frac{\sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)} 
f = (sin(x) + cos(x))*exp(2*x) + sin(x)/2 + cos(x)
Gráfico de la función
02468-8-6-4-2-1010-1000000000500000000
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:
((sin⁡(x)+cos⁡(x))e2x+sin⁡(x)2)+cos⁡(x)=0\left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} + \frac{\sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−32.523075253692x_{1} = -32.523075253692
x2=−23.0982972929226x_{2} = -23.0982972929226
x3=−16.8151119857431x_{3} = -16.8151119857431
x4=−41.9478532144614x_{4} = -41.9478532144614
x5=−85.9301503647185x_{5} = -85.9301503647185
x6=−60.7974091360002x_{6} = -60.7974091360002
x7=−63.93900178959x_{7} = -63.93900178959
x8=−98.4965209790777x_{8} = -98.4965209790777
x9=−76.5053724039491x_{9} = -76.5053724039491
x10=−95.3549283254879x_{10} = -95.3549283254879
x11=−92.2133356718981x_{11} = -92.2133356718981
x12=−4.24865978556446x_{12} = -4.24865978556446
x13=−79.6469650575389x_{13} = -79.6469650575389
x14=−67.0805944431797x_{14} = -67.0805944431797
x15=−51.3726311752308x_{15} = -51.3726311752308
x16=−89.0717430183083x_{16} = -89.0717430183083
x17=5.49778294985435x_{17} = 5.49778294985435
x18=−73.3637797503593x_{18} = -73.3637797503593
x19=11.7809724509471x_{19} = 11.7809724509471
x20=−70.2221870967695x_{20} = -70.2221870967695
x21=−1.06561515086674x_{21} = -1.06561515086674
x22=8.63937978953998x_{22} = 8.63937978953998
x23=−13.6735193321527x_{23} = -13.6735193321527
x24=−45.0894458680512x_{24} = -45.0894458680512
x25=−54.5142238288206x_{25} = -54.5142238288206
x26=−29.3814826001022x_{26} = -29.3814826001022
x27=14.9225651045515x_{27} = 14.9225651045515
x28=−26.2398899465124x_{28} = -26.2398899465124
x29=−35.6646679072818x_{29} = -35.6646679072818
x30=−19.9567046393328x_{30} = -19.9567046393328
x31=−82.7885577111287x_{31} = -82.7885577111287
x32=2.35395374874329x_{32} = 2.35395374874329
x33=−10.5319266782789x_{33} = -10.5319266782789
x34=−243.009783044208x_{34} = -243.009783044208
x35=−7.39033387260442x_{35} = -7.39033387260442
x36=−104.779706286257x_{36} = -104.779706286257
x37=−57.6558164824104x_{37} = -57.6558164824104
x38=−38.8062605608716x_{38} = -38.8062605608716
x39=−48.231038521641x_{39} = -48.231038521641
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 sin(x)/2 + (cos(x) + sin(x))*exp(2*x) + cos(x).
(sin⁡(0)2+(sin⁡(0)+cos⁡(0))e0⋅2)+cos⁡(0)\left(\frac{\sin{\left(0 \right)}}{2} + \left(\sin{\left(0 \right)} + \cos{\left(0 \right)}\right) e^{0 \cdot 2}\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))e2x+2(sin⁡(x)+cos⁡(x))e2x−sin⁡(x)+cos⁡(x)2=0\left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} + 2 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} - \sin{\left(x \right)} + \frac{\cos{\left(x \right)}}{2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−46.6602421948461x_{1} = -46.6602421948461
x2=−24.6690936197175x_{2} = -24.6690936197175
x3=−62.3682054627951x_{3} = -62.3682054627951
x4=−49.8018348484359x_{4} = -49.8018348484359
x5=−40.3770568876665x_{5} = -40.3770568876665
x6=−68.6513907699746x_{6} = -68.6513907699746
x7=−65.5097981163849x_{7} = -65.5097981163849
x8=−71.7929834235644x_{8} = -71.7929834235644
x9=−34.0938715804869x_{9} = -34.0938715804869
x10=−43.5186495412563x_{10} = -43.5186495412563
x11=−90.6425393451032x_{11} = -90.6425393451032
x12=−18.385908312538x_{12} = -18.385908312538
x13=−21.5275009661277x_{13} = -21.5275009661277
x14=−87.5009466915134x_{14} = -87.5009466915134
x15=−12.1027230052723x_{15} = -12.1027230052723
x16=−27.8106862733073x_{16} = -27.8106862733073
x17=−93.784131998693x_{17} = -93.784131998693
x18=−96.9257246522828x_{18} = -96.9257246522828
x19=14.4589174955506x_{19} = 14.4589174955506
x20=5.03412469313861x_{20} = 5.03412469313861
x21=11.3173248419092x_{21} = 11.3173248419092
x22=−15.244315658948x_{22} = -15.244315658948
x23=−30.9522789268971x_{23} = -30.9522789268971
x24=−5.81951301553248x_{24} = -5.81951301553248
x25=8.17573216065597x_{25} = 8.17573216065597
x26=−59.2266128092053x_{26} = -59.2266128092053
x27=−2.66439227655508x_{27} = -2.66439227655508
x28=−81.2177613843338x_{28} = -81.2177613843338
x29=−37.2354642340767x_{29} = -37.2354642340767
x30=−84.3593540379236x_{30} = -84.3593540379236
x31=−56.0850201556155x_{31} = -56.0850201556155
x32=1.88447974266388x_{32} = 1.88447974266388
x33=−100.067317305873x_{33} = -100.067317305873
x34=−78.076168730744x_{34} = -78.076168730744
x35=−52.9434275020257x_{35} = -52.9434275020257
x36=−8.96113030567725x_{36} = -8.96113030567725
x37=−74.9345760771542x_{37} = -74.9345760771542
Signos de extremos en los puntos:
(-46.66024219484609, -1.11803398874989)

(-24.66909361971754, 1.11803398874989)

(-62.36820546279506, 1.11803398874989)

(-49.80183484843589, 1.11803398874989)

(-40.377056887666505, -1.11803398874989)

(-68.65139076997464, 1.11803398874989)

(-65.50979811638486, -1.11803398874989)

(-71.79298342356444, -1.11803398874989)

(-34.09387158048692, -1.11803398874989)

(-43.5186495412563, 1.11803398874989)

(-90.6425393451032, -1.11803398874989)

(-18.385908312537953, 1.11803398874989)

(-21.527500966127747, -1.11803398874989)

(-87.5009466915134, 1.11803398874989)

(-12.102723005272294, 1.11803398879114)

(-27.810686273307333, -1.11803398874989)

(-93.78413199869298, 1.11803398874989)

(-96.92572465228278, -1.11803398874989)

(14.458917495550615, 2290267239515.66)

(5.034124693138614, -14915.3222875833)

(11.317324841909162, -4276942910.26027)

(-15.244315658947999, -1.11803398874997)

(-30.952278926897126, 1.11803398874989)

(-5.81951301553248, 1.11804581542154)

(8.17573216065597, 7986946.10951418)

(-59.226612809205264, -1.11803398874989)

(-2.6643922765550845, -1.12446701441265)

(-81.21776138433381, 1.11803398874989)

(-37.235464234076716, 1.11803398874989)

(-84.35935403792361, -1.11803398874989)

(-56.085020155615474, 1.11803398874989)

(1.8844797426638766, 28.0157690849564)

(-100.06731730587258, 1.11803398874989)

(-78.07616873074403, -1.11803398874989)

(-52.94342750202568, -1.11803398874989)

(-8.961130305677255, -1.11803401083489)

(-74.93457607715423, 1.11803398874989)


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=−46.6602421948461x_{1} = -46.6602421948461
x2=−40.3770568876665x_{2} = -40.3770568876665
x3=−65.5097981163849x_{3} = -65.5097981163849
x4=−71.7929834235644x_{4} = -71.7929834235644
x5=−34.0938715804869x_{5} = -34.0938715804869
x6=−90.6425393451032x_{6} = -90.6425393451032
x7=−21.5275009661277x_{7} = -21.5275009661277
x8=−27.8106862733073x_{8} = -27.8106862733073
x9=−96.9257246522828x_{9} = -96.9257246522828
x10=5.03412469313861x_{10} = 5.03412469313861
x11=11.3173248419092x_{11} = 11.3173248419092
x12=−15.244315658948x_{12} = -15.244315658948
x13=−59.2266128092053x_{13} = -59.2266128092053
x14=−2.66439227655508x_{14} = -2.66439227655508
x15=−84.3593540379236x_{15} = -84.3593540379236
x16=−78.076168730744x_{16} = -78.076168730744
x17=−52.9434275020257x_{17} = -52.9434275020257
x18=−8.96113030567725x_{18} = -8.96113030567725
Puntos máximos de la función:
x18=−24.6690936197175x_{18} = -24.6690936197175
x18=−62.3682054627951x_{18} = -62.3682054627951
x18=−49.8018348484359x_{18} = -49.8018348484359
x18=−68.6513907699746x_{18} = -68.6513907699746
x18=−43.5186495412563x_{18} = -43.5186495412563
x18=−18.385908312538x_{18} = -18.385908312538
x18=−87.5009466915134x_{18} = -87.5009466915134
x18=−12.1027230052723x_{18} = -12.1027230052723
x18=−93.784131998693x_{18} = -93.784131998693
x18=14.4589174955506x_{18} = 14.4589174955506
x18=−30.9522789268971x_{18} = -30.9522789268971
x18=−5.81951301553248x_{18} = -5.81951301553248
x18=8.17573216065597x_{18} = 8.17573216065597
x18=−81.2177613843338x_{18} = -81.2177613843338
x18=−37.2354642340767x_{18} = -37.2354642340767
x18=−56.0850201556155x_{18} = -56.0850201556155
x18=1.88447974266388x_{18} = 1.88447974266388
x18=−100.067317305873x_{18} = -100.067317305873
x18=−74.9345760771542x_{18} = -74.9345760771542
Decrece en los intervalos
[11.3173248419092,∞)\left[11.3173248419092, \infty\right)
Crece en los intervalos
(−∞,−96.9257246522828]\left(-\infty, -96.9257246522828\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
−4(sin⁡(x)−cos⁡(x))e2x+3(sin⁡(x)+cos⁡(x))e2x−sin⁡(x)2−cos⁡(x)=0- 4 \left(\sin{\left(x \right)} - \cos{\left(x \right)}\right) e^{2 x} + 3 \left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} - \frac{\sin{\left(x \right)}}{2} - \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−10.5319266760026x_{1} = -10.5319266760026
x2=−95.3549283254879x_{2} = -95.3549283254879
x3=−4.24800517515229x_{3} = -4.24800517515229
x4=10.8536772329265x_{4} = 10.8536772329265
x5=−67.0805944431797x_{5} = -67.0805944431797
x6=−63.93900178959x_{6} = -63.93900178959
x7=1.42363948641901x_{7} = 1.42363948641901
x8=−104.779706286257x_{8} = -104.779706286257
x9=−73.3637797503593x_{9} = -73.3637797503593
x10=−16.815111985743x_{10} = -16.815111985743
x11=7.71208456135628x_{11} = 7.71208456135628
x12=−38.8062605608716x_{12} = -38.8062605608716
x13=−41.9478532144614x_{13} = -41.9478532144614
x14=4.5704822790956x_{14} = 4.5704822790956
x15=−76.5053724039491x_{15} = -76.5053724039491
x16=−57.6558164824104x_{16} = -57.6558164824104
x17=−54.5142238288206x_{17} = -54.5142238288206
x18=−26.2398899465124x_{18} = -26.2398899465124
x19=−35.6646679072818x_{19} = -35.6646679072818
x20=−32.523075253692x_{20} = -32.523075253692
x21=−82.7885577111287x_{21} = -82.7885577111287
x22=−51.3726311752308x_{22} = -51.3726311752308
x23=−92.2133356718981x_{23} = -92.2133356718981
x24=−2030.5760029368x_{24} = -2030.5760029368
x25=−45.0894458680512x_{25} = -45.0894458680512
x26=−98.4965209790777x_{26} = -98.4965209790777
x27=−70.2221870967695x_{27} = -70.2221870967695
x28=13.9952698865498x_{28} = 13.9952698865498
x29=−29.3814826001022x_{29} = -29.3814826001022
x30=−13.6735193321485x_{30} = -13.6735193321485
x31=−19.9567046393328x_{31} = -19.9567046393328
x32=−85.9301503647185x_{32} = -85.9301503647185
x33=−7.39033265364367x_{33} = -7.39033265364367
x34=−60.7974091360002x_{34} = -60.7974091360002
x35=−48.231038521641x_{35} = -48.231038521641
x36=−23.0982972929226x_{36} = -23.0982972929226
x37=−89.0717430183083x_{37} = -89.0717430183083
x38=−79.6469650575389x_{38} = -79.6469650575389
x39=−243.009783044208x_{39} = -243.009783044208

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
[10.8536772329265,∞)\left[10.8536772329265, \infty\right)
Convexa en los intervalos
(−∞,−243.009783044208]\left(-\infty, -243.009783044208\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((sin⁡(x)+cos⁡(x))e2x+sin⁡(x)2)+cos⁡(x))=⟨−32,32⟩\lim_{x \to -\infty}\left(\left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} + \frac{\sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)}\right) = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−32,32⟩y = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(((sin⁡(x)+cos⁡(x))e2x+sin⁡(x)2)+cos⁡(x))y = \lim_{x \to \infty}\left(\left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} + \frac{\sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función sin(x)/2 + (cos(x) + sin(x))*exp(2*x) + cos(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((sin⁡(x)+cos⁡(x))e2x+sin⁡(x)2)+cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} + \frac{\sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)}}{x}\right) = 0
Tomamos como el límite
es decir,
la inclinada coincide con la asíntota horizontal a la derecha
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞(((sin⁡(x)+cos⁡(x))e2x+sin⁡(x)2)+cos⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} + \frac{\sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)}}{x}\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:
((sin⁡(x)+cos⁡(x))e2x+sin⁡(x)2)+cos⁡(x)=(−sin⁡(x)+cos⁡(x))e−2x−sin⁡(x)2+cos⁡(x)\left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} + \frac{\sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)} = \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- 2 x} - \frac{\sin{\left(x \right)}}{2} + \cos{\left(x \right)}
- No
((sin⁡(x)+cos⁡(x))e2x+sin⁡(x)2)+cos⁡(x)=−(−sin⁡(x)+cos⁡(x))e−2x+sin⁡(x)2−cos⁡(x)\left(\left(\sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{2 x} + \frac{\sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)} = - \left(- \sin{\left(x \right)} + \cos{\left(x \right)}\right) e^{- 2 x} + \frac{\sin{\left(x \right)}}{2} - \cos{\left(x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор