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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                         -x         
f(x) = 2*cos(x) + (cos(2*x) + sin(2*x))*e   + sin(x)
f(x)=((sin⁡(2x)+cos⁡(2x))e−x+2cos⁡(x))+sin⁡(x)f{\left(x \right)} = \left(\left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{- x} + 2 \cos{\left(x \right)}\right) + \sin{\left(x \right)} 
f = (sin(2*x) + cos(2*x))*exp(-x) + 2*cos(x) + sin(x)
Gráfico de la función
02468-8-6-4-2-1010-2000020000
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⁡(2x)+cos⁡(2x))e−x+2cos⁡(x))+sin⁡(x)=0\left(\left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) 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=39.7335557788732x_{1} = 39.7335557788732
x2=1.94385309268742x_{2} = 1.94385309268742
x3=5.17955820869071x_{3} = 5.17955820869071
x4=8.31747633578368x_{4} = 8.31747633578368
x5=49.1583337396426x_{5} = 49.1583337396426
x6=61.7247043540018x_{6} = 61.7247043540018
x7=−23.9546439835986x_{7} = -23.9546439835986
x8=86.8574455827201x_{8} = 86.8574455827201
x9=30.3087778181039x_{9} = 30.3087778181039
x10=−25.5254403104213x_{10} = -25.5254403104213
x11=83.7158529291303x_{11} = 83.7158529291303
x12=11.4592285029292x_{12} = 11.4592285029292
x13=74.2910749683609x_{13} = 74.2910749683609
x14=89.9990382363099x_{14} = 89.9990382363099
x15=77.4326676219507x_{15} = 77.4326676219507
x16=99.4238161970793x_{16} = 99.4238161970793
x17=96.2822235434895x_{17} = 96.2822235434895
x18=−2.04631508411971x_{18} = -2.04631508411971
x19=36.5919631252834x_{19} = 36.5919631252834
x20=17.7424072160818x_{20} = 17.7424072160818
x21=27.1671851645131x_{21} = 27.1671851645131
x22=14.6008142646655x_{22} = 14.6008142646655
x23=68.0078896611814x_{23} = 68.0078896611814
x24=42.875148432463x_{24} = 42.875148432463
x25=−30.2378292908017x_{25} = -30.2378292908017
x26=−8.24683726494595x_{26} = -8.24683726494595
x27=52.2999263932324x_{27} = 52.2999263932324
x28=−11.3882666040089x_{28} = -11.3882666040089
x29=−0.606207697652873x_{29} = -0.606207697652873
x30=55.4415190468222x_{30} = 55.4415190468222
x31=−27.096236637213x_{31} = -27.096236637213
x32=−3.518661885146x_{32} = -3.518661885146
x33=−17.6714586638089x_{33} = -17.6714586638089
x34=71.1494823147711x_{34} = 71.1494823147711
x35=80.5742602755405x_{35} = 80.5742602755405
x36=−16.1006622969385x_{36} = -16.1006622969385
x37=24.0255925109473x_{37} = 24.0255925109473
x38=−9.81744881542807x_{38} = -9.81744881542807
x39=93.1406308898997x_{39} = 93.1406308898997
x40=64.8662970075916x_{40} = 64.8662970075916
x41=58.583111700412x_{41} = 58.583111700412
x42=−5.10146359536733x_{42} = -5.10146359536733
x43=46.0167410860528x_{43} = 46.0167410860528
x44=−19.2422550055153x_{44} = -19.2422550055153
x45=20.8839998568013x_{45} = 20.8839998568013
x46=33.4503704716936x_{46} = 33.4503704716936
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 2*cos(x) + (cos(2*x) + sin(2*x))*exp(-x) + sin(x).
sin⁡(0)+((sin⁡(0⋅2)+cos⁡(0⋅2))e−0+2cos⁡(0))\sin{\left(0 \right)} + \left(\left(\sin{\left(0 \cdot 2 \right)} + \cos{\left(0 \cdot 2 \right)}\right) e^{- 0} + 2 \cos{\left(0 \right)}\right)
Resultado:
f(0)=3f{\left(0 \right)} = 3
Punto:
(0, 3)
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
(−2sin⁡(2x)+2cos⁡(2x))e−x−(sin⁡(2x)+cos⁡(2x))e−x−2sin⁡(x)+cos⁡(x)=0\left(- 2 \sin{\left(2 x \right)} + 2 \cos{\left(2 x \right)}\right) e^{- x} - \left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{- x} - 2 \sin{\left(x \right)} + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=44.4459447592579x_{1} = 44.4459447592579
x2=3.62801422609082x_{2} = 3.62801422609082
x3=−15.5470880094102x_{3} = -15.5470880094102
x4=66.4370933343865x_{4} = 66.4370933343865
x5=69.5786859879763x_{5} = 69.5786859879763
x6=28.7379814913092x_{6} = 28.7379814913092
x7=−28.1134586051099x_{7} = -28.1134586051099
x8=0.255701375162867x_{8} = 0.255701375162867
x9=72.7202786415661x_{9} = 72.7202786415661
x10=38.1627594520783x_{10} = 38.1627594520783
x11=31.8795741448987x_{11} = 31.8795741448987
x12=25.596388837713x_{12} = 25.596388837713
x13=−17.117884305128x_{13} = -17.117884305128
x14=88.428241909515x_{14} = 88.428241909515
x15=9.88846643296898x_{15} = 9.88846643296898
x16=−21.8302732979651x_{16} = -21.8302732979651
x17=91.5698345631048x_{17} = 91.5698345631048
x18=19.3132035272422x_{18} = 19.3132035272422
x19=63.2955006807967x_{19} = 63.2955006807967
x20=−13.9762919513053x_{20} = -13.9762919513053
x21=13.0300164576432x_{21} = 13.0300164576432
x22=57.0123153736171x_{22} = 57.0123153736171
x23=−31.2550512586996x_{23} = -31.2550512586996
x24=−29.6842549319047x_{24} = -29.6842549319047
x25=−7.69326019908088x_{25} = -7.69326019908088
x26=22.4547961842719x_{26} = 22.4547961842719
x27=47.5875374128477x_{27} = 47.5875374128477
x28=−23.4010696247019x_{28} = -23.4010696247019
x29=94.7114272166946x_{29} = 94.7114272166946
x30=16.1716109532536x_{30} = 16.1716109532536
x31=97.8530198702844x_{31} = 97.8530198702844
x32=35.0211667984885x_{32} = 35.0211667984885
x33=−20.2594769716719x_{33} = -20.2594769716719
x34=100.994612523874x_{34} = 100.994612523874
x35=−9.26391267581802x_{35} = -9.26391267581802
x36=75.8618712951559x_{36} = 75.8618712951559
x37=79.0034639487456x_{37} = 79.0034639487456
x38=53.8707227200273x_{38} = 53.8707227200273
x39=−1.484665633778x_{39} = -1.484665633778
x40=82.1450566023354x_{40} = 82.1450566023354
x41=6.74588907568972x_{41} = 6.74588907568972
x42=85.2866492559252x_{42} = 85.2866492559252
x43=41.3043521056681x_{43} = 41.3043521056681
x44=−6.12207892521095x_{44} = -6.12207892521095
x45=50.7291300664375x_{45} = 50.7291300664375
x46=60.1539080272069x_{46} = 60.1539080272069
Signos de extremos en los puntos:
(44.44594475925791, 2.23606797749979)

(3.6280142260908157, -2.1985722071687)

(-15.547088009410185, 7146187.15002582)

(66.43709333438646, -2.23606797749979)

(69.57868598797626, 2.23606797749979)

(28.73798149130921, -2.23606797749933)

(-28.113458605109884, 2049179161217.76)

(0.255701375162867, 3.24217432825651)

(72.72027864156605, -2.23606797749979)

(38.162759452078326, 2.23606797749979)

(31.879574144898726, 2.23606797749981)

(25.596388837712993, 2.2360679775105)

(-17.117884305128037, -34376581.2458983)

(88.42824190951502, 2.23606797749979)

(9.888466432968976, -2.23599691704751)

(-21.83027329796508, 3826724728.45321)

(91.5698345631048, -2.23606797749979)

(19.31320352724218, 2.23606798323448)

(63.29550068079667, 2.23606797749979)

(-13.976291951305344, -1485547.46768316)

(13.030016457643212, 2.23607104838452)

(57.012315373617085, 2.23606797749979)

(-31.25505125869961, 47419425119288.2)

(-29.684254931904672, -9857530004593.2)

(-7.6932601990808775, -2774.84046322185)

(22.454796184271853, -2.23606797725197)

(47.5875374128477, -2.23606797749979)

(-23.40106962470194, -18408372759.0055)

(94.7114272166946, 2.23606797749979)

(16.171610953253577, -2.23606784479499)

(97.8530198702844, -2.23606797749979)

(35.02116679848853, -2.23606797749979)

(-20.259476971671944, -795497916.472251)

(100.9946125238742, 2.23606797749979)

(-9.263912675818018, 13342.9648847032)

(75.86187129515585, 2.23606797749979)

(79.00346394874563, -2.23606797749979)

(53.87072272002729, -2.23606797749979)

(-1.4846656337780049, -5.92893219327708)

(82.14505660233543, 2.23606797749979)

(6.745889075689718, 2.23771340654961)

(85.28664925592523, -2.23606797749979)

(41.304352105668116, -2.23606797749979)

(-6.1220789252109515, 578.828461909566)

(50.7291300664375, 2.23606797749979)

(60.153908027206874, -2.23606797749979)


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=3.62801422609082x_{1} = 3.62801422609082
x2=66.4370933343865x_{2} = 66.4370933343865
x3=28.7379814913092x_{3} = 28.7379814913092
x4=72.7202786415661x_{4} = 72.7202786415661
x5=−17.117884305128x_{5} = -17.117884305128
x6=9.88846643296898x_{6} = 9.88846643296898
x7=91.5698345631048x_{7} = 91.5698345631048
x8=−13.9762919513053x_{8} = -13.9762919513053
x9=−29.6842549319047x_{9} = -29.6842549319047
x10=−7.69326019908088x_{10} = -7.69326019908088
x11=22.4547961842719x_{11} = 22.4547961842719
x12=47.5875374128477x_{12} = 47.5875374128477
x13=−23.4010696247019x_{13} = -23.4010696247019
x14=16.1716109532536x_{14} = 16.1716109532536
x15=97.8530198702844x_{15} = 97.8530198702844
x16=35.0211667984885x_{16} = 35.0211667984885
x17=−20.2594769716719x_{17} = -20.2594769716719
x18=79.0034639487456x_{18} = 79.0034639487456
x19=53.8707227200273x_{19} = 53.8707227200273
x20=−1.484665633778x_{20} = -1.484665633778
x21=85.2866492559252x_{21} = 85.2866492559252
x22=41.3043521056681x_{22} = 41.3043521056681
x23=60.1539080272069x_{23} = 60.1539080272069
Puntos máximos de la función:
x23=44.4459447592579x_{23} = 44.4459447592579
x23=−15.5470880094102x_{23} = -15.5470880094102
x23=69.5786859879763x_{23} = 69.5786859879763
x23=−28.1134586051099x_{23} = -28.1134586051099
x23=0.255701375162867x_{23} = 0.255701375162867
x23=38.1627594520783x_{23} = 38.1627594520783
x23=31.8795741448987x_{23} = 31.8795741448987
x23=25.596388837713x_{23} = 25.596388837713
x23=88.428241909515x_{23} = 88.428241909515
x23=−21.8302732979651x_{23} = -21.8302732979651
x23=19.3132035272422x_{23} = 19.3132035272422
x23=63.2955006807967x_{23} = 63.2955006807967
x23=13.0300164576432x_{23} = 13.0300164576432
x23=57.0123153736171x_{23} = 57.0123153736171
x23=−31.2550512586996x_{23} = -31.2550512586996
x23=94.7114272166946x_{23} = 94.7114272166946
x23=100.994612523874x_{23} = 100.994612523874
x23=−9.26391267581802x_{23} = -9.26391267581802
x23=75.8618712951559x_{23} = 75.8618712951559
x23=82.1450566023354x_{23} = 82.1450566023354
x23=6.74588907568972x_{23} = 6.74588907568972
x23=−6.12207892521095x_{23} = -6.12207892521095
x23=50.7291300664375x_{23} = 50.7291300664375
Decrece en los intervalos
[97.8530198702844,∞)\left[97.8530198702844, \infty\right)
Crece en los intervalos
(−∞,−29.6842549319047]\left(-\infty, -29.6842549319047\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⁡(2x)−cos⁡(2x))e−x−3(sin⁡(2x)+cos⁡(2x))e−x−sin⁡(x)−2cos⁡(x)=04 \left(\sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- x} - 3 \left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{- x} - \sin{\left(x \right)} - 2 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=39.7335557788732x_{1} = 39.7335557788732
x2=−25.9890879194181x_{2} = -25.9890879194181
x3=8.31725732812605x_{3} = 8.31725732812605
x4=30.3087778181039x_{4} = 30.3087778181039
x5=−19.7059026123468x_{5} = -19.7059026123468
x6=49.1583337396426x_{6} = 49.1583337396426
x7=−29.1306805730077x_{7} = -29.1306805730077
x8=5.18429956854269x_{8} = 5.18429956854269
x9=−5.56815103833414x_{9} = -5.56815103833414
x10=61.7247043540018x_{10} = 61.7247043540018
x11=86.8574455827201x_{11} = 86.8574455827201
x12=17.7424072337062x_{12} = 17.7424072337062
x13=83.7158529291303x_{13} = 83.7158529291303
x14=−18.1351062834055x_{14} = -18.1351062834055
x15=20.8839998560397x_{15} = 20.8839998560397
x16=74.2910749683609x_{16} = 74.2910749683609
x17=27.1671851645116x_{17} = 27.1671851645116
x18=−10.2811233065204x_{18} = -10.2811233065204
x19=89.9990382363099x_{19} = 89.9990382363099
x20=−7.13956311596467x_{20} = -7.13956311596467
x21=77.4326676219507x_{21} = 77.4326676219507
x22=−13.4227173631788x_{22} = -13.4227173631788
x23=99.4238161970793x_{23} = 99.4238161970793
x24=96.2822235434895x_{24} = 96.2822235434895
x25=−21.2766989391213x_{25} = -21.2766989391213
x26=36.5919631252834x_{26} = 36.5919631252834
x27=−11.8519198869694x_{27} = -11.8519198869694
x28=11.4592379395324x_{28} = 11.4592379395324
x29=171.680447229645x_{29} = 171.680447229645
x30=68.0078896611814x_{30} = 68.0078896611814
x31=1.1565873366713x_{31} = 1.1565873366713
x32=42.875148432463x_{32} = 42.875148432463
x33=52.2999263932324x_{33} = 52.2999263932324
x34=−27.5598842462129x_{34} = -27.5598842462129
x35=24.0255925109802x_{35} = 24.0255925109802
x36=55.4415190468222x_{36} = 55.4415190468222
x37=−3.99721658632344x_{37} = -3.99721658632344
x38=71.1494823147711x_{38} = 71.1494823147711
x39=14.6008138568215x_{39} = 14.6008138568215
x40=80.5742602755405x_{40} = 80.5742602755405
x41=93.1406308898997x_{41} = 93.1406308898997
x42=64.8662970075916x_{42} = 64.8662970075916
x43=58.583111700412x_{43} = 58.583111700412
x44=46.0167410860528x_{44} = 46.0167410860528
x45=33.4503704716936x_{45} = 33.4503704716936

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
[171.680447229645,∞)\left[171.680447229645, \infty\right)
Convexa en los intervalos
(−∞,−27.5598842462129]\left(-\infty, -27.5598842462129\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞(((sin⁡(2x)+cos⁡(2x))e−x+2cos⁡(x))+sin⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(\left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{- x} + 2 \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
lim⁡x→∞(((sin⁡(2x)+cos⁡(2x))e−x+2cos⁡(x))+sin⁡(x))=⟨−3,3⟩\lim_{x \to \infty}\left(\left(\left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{- x} + 2 \cos{\left(x \right)}\right) + \sin{\left(x \right)}\right) = \left\langle -3, 3\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función 2*cos(x) + (cos(2*x) + sin(2*x))*exp(-x) + sin(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→−∞(((sin⁡(2x)+cos⁡(2x))e−x+2cos⁡(x))+sin⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\left(\left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{- x} + 2 \cos{\left(x \right)}\right) + \sin{\left(x \right)}}{x}\right)
lim⁡x→∞(((sin⁡(2x)+cos⁡(2x))e−x+2cos⁡(x))+sin⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\left(\left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) 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 izquierda
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⁡(2x)+cos⁡(2x))e−x+2cos⁡(x))+sin⁡(x)=(−sin⁡(2x)+cos⁡(2x))ex−sin⁡(x)+2cos⁡(x)\left(\left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{- x} + 2 \cos{\left(x \right)}\right) + \sin{\left(x \right)} = \left(- \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{x} - \sin{\left(x \right)} + 2 \cos{\left(x \right)}
- No
((sin⁡(2x)+cos⁡(2x))e−x+2cos⁡(x))+sin⁡(x)=−(−sin⁡(2x)+cos⁡(2x))ex+sin⁡(x)−2cos⁡(x)\left(\left(\sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{- x} + 2 \cos{\left(x \right)}\right) + \sin{\left(x \right)} = - \left(- \sin{\left(2 x \right)} + \cos{\left(2 x \right)}\right) e^{x} + \sin{\left(x \right)} - 2 \cos{\left(x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор