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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
       -sin(x)                            x         
f(x) = -------- + (-cos(2*x) - sin(2*x))*e  + cos(x)
          2
f(x)=((−sin⁡(2x)−cos⁡(2x))ex+(−1)sin⁡(x)2)+cos⁡(x)f{\left(x \right)} = \left(\left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{x} + \frac{\left(-1\right) \sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)} 
f = (-sin(2*x) - cos(2*x))*exp(x) + (-sin(x))/2 + cos(x)
Gráfico de la función
02468-8-6-4-2-1010-5000050000
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))ex+(−1)sin⁡(x)2)+cos⁡(x)=0\left(\left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{x} + \frac{\left(-1\right) \sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−99.4238161970793x_{1} = -99.4238161970793
x2=−2.01659565024917x_{2} = -2.01659565024917
x3=−61.7247043540018x_{3} = -61.7247043540018
x4=−49.1583337396426x_{4} = -49.1583337396426
x5=−11.45922378416x_{5} = -11.45922378416
x6=−33.4503704716936x_{6} = -33.4503704716936
x7=−77.4326676219507x_{7} = -77.4326676219507
x8=18.4568568436428x_{8} = 18.4568568436428
x9=26.3108384738146x_{9} = 26.3108384738146
x10=−39.7335557788732x_{10} = -39.7335557788732
x11=−27.1671851645138x_{11} = -27.1671851645138
x12=20.0276531666911x_{12} = 20.0276531666911
x13=−96.2822235434895x_{13} = -96.2822235434895
x14=−80.5742602755405x_{14} = -80.5742602755405
x15=−17.7424072072696x_{15} = -17.7424072072696
x16=29.4524311274043x_{16} = 29.4524311274043
x17=−89.9990382363099x_{17} = -89.9990382363099
x18=−58.583111700412x_{18} = -58.583111700412
x19=−74.2910749683609x_{19} = -74.2910749683609
x20=21.5984494932655x_{20} = 21.5984494932655
x21=−36.5919631252834x_{21} = -36.5919631252834
x22=−24.0255925109308x_{22} = -24.0255925109308
x23=−86.8574455827201x_{23} = -86.8574455827201
x24=13.7444678895398x_{24} = 13.7444678895398
x25=5.89157541591041x_{25} = 5.89157541591041
x26=−46.0167410860528x_{26} = -46.0167410860528
x27=27.8816348006091x_{27} = 27.8816348006091
x28=−93.1406308898997x_{28} = -93.1406308898997
x29=−30.3087778181039x_{29} = -30.3087778181039
x30=−68.0078896611814x_{30} = -68.0078896611814
x31=−14.6008144685866x_{31} = -14.6008144685866
x32=9.0320317535734x_{32} = 9.0320317535734
x33=−71.1494823147711x_{33} = -71.1494823147711
x34=−64.8662970075916x_{34} = -64.8662970075916
x35=−20.8839998571821x_{35} = -20.8839998571821
x36=−83.7158529291303x_{36} = -83.7158529291303
x37=7.46129866565413x_{37} = 7.46129866565413
x38=0x_{38} = 0
x39=−52.2999263932324x_{39} = -52.2999263932324
x40=−5.17706079773958x_{40} = -5.17706079773958
x41=−55.4415190468222x_{41} = -55.4415190468222
x42=−8.31758558858182x_{42} = -8.31758558858182
x43=12.1736735690306x_{43} = 12.1736735690306
x44=15.3152640982505x_{44} = 15.3152640982505
x45=4.31931891092025x_{45} = 4.31931891092025
x46=−42.875148432463x_{46} = -42.875148432463
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(2*x) - sin(2*x))*exp(x) + cos(x).
((−cos⁡(0⋅2)−sin⁡(0⋅2))e0+(−1)sin⁡(0)2)+cos⁡(0)\left(\left(- \cos{\left(0 \cdot 2 \right)} - \sin{\left(0 \cdot 2 \right)}\right) e^{0} + \frac{\left(-1\right) \sin{\left(0 \right)}}{2}\right) + \cos{\left(0 \right)}
Resultado:
f(0)=0f{\left(0 \right)} = 0
Punto:
(0, 0)
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⁡(2x)−cos⁡(2x))ex+(2sin⁡(2x)−2cos⁡(2x))ex−sin⁡(x)−cos⁡(x)2=0\left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{x} + \left(2 \sin{\left(2 x \right)} - 2 \cos{\left(2 x \right)}\right) e^{x} - \sin{\left(x \right)} - \frac{\cos{\left(x \right)}}{2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−44.4459447592579x_{1} = -44.4459447592579
x2=−35.0211667984885x_{2} = -35.0211667984885
x3=−50.7291300664375x_{3} = -50.7291300664375
x4=17.9032824823194x_{4} = 17.9032824823194
x5=−3.53084441183211x_{5} = -3.53084441183211
x6=−41.3043521056681x_{6} = -41.3043521056681
x7=−22.4547961837177x_{7} = -22.4547961837177
x8=13.1908937929858x_{8} = 13.1908937929858
x9=10.0492940799707x_{9} = 10.0492940799707
x10=11.6200979108149x_{10} = 11.6200979108149
x11=−6.74954677059035x_{11} = -6.74954677059035
x12=−75.8618712951559x_{12} = -75.8618712951559
x13=25.7572641149185x_{13} = 25.7572641149185
x14=−53.8707227200273x_{14} = -53.8707227200273
x15=−100.994612523874x_{15} = -100.994612523874
x16=19.474078808284x_{16} = 19.474078808284
x17=−97.8530198702844x_{17} = -97.8530198702844
x18=−82.1450566023354x_{18} = -82.1450566023354
x19=−94.7114272166946x_{19} = -94.7114272166946
x20=2.1859345198368x_{20} = 2.1859345198368
x21=16.3324861415112x_{21} = 16.3324861415112
x22=−31.8795741448988x_{22} = -31.8795741448988
x23=−66.4370933343865x_{23} = -66.4370933343865
x24=27.3280604417123x_{24} = 27.3280604417123
x25=24.1864677881251x_{25} = 24.1864677881251
x26=5.33730609993229x_{26} = 5.33730609993229
x27=−72.7202786415661x_{27} = -72.7202786415661
x28=−57.0123153736171x_{28} = -57.0123153736171
x29=−28.7379814913082x_{29} = -28.7379814913082
x30=−63.2955006807967x_{30} = -63.2955006807967
x31=−60.1539080272069x_{31} = -60.1539080272069
x32=−19.3132035400653x_{32} = -19.3132035400653
x33=−9.88830749640405x_{33} = -9.88830749640405
x34=−25.5963888377369x_{34} = -25.5963888377369
x35=−16.1716106565165x_{35} = -16.1716106565165
x36=−85.2866492559252x_{36} = -85.2866492559252
x37=−88.428241909515x_{37} = -88.428241909515
x38=−47.5875374128477x_{38} = -47.5875374128477
x39=30.4696530953022x_{39} = 30.4696530953022
x40=−79.0034639487456x_{40} = -79.0034639487456
x41=−38.1627594520783x_{41} = -38.1627594520783
x42=8.47848746340932x_{42} = 8.47848746340932
x43=−13.0300233242745x_{43} = -13.0300233242745
x44=3.76248561206274x_{44} = 3.76248561206274
x45=−69.5786859879763x_{45} = -69.5786859879763
x46=−91.5698345631048x_{46} = -91.5698345631048
Signos de extremos en los puntos:
(-44.44594475925791, 1.11803398874989)

(-35.02116679848853, -1.11803398874989)

(-50.7291300664375, 1.11803398874989)

(17.903282482319394, 75397472.3118449)

(-3.530844411832112, -1.11522689459104)

(-41.304352105668116, -1.11803398874989)

(-22.45479618371771, -1.11803398871449)

(13.19089379298581, -677316.908251517)

(10.049294079970668, -29270.0588253816)

(11.620097910814875, 140801.45014009)

(-6.749546770590349, 1.11827305148252)

(-75.86187129515585, 1.11803398874989)

(25.75726411491849, -194221661641.761)

(-53.87072272002729, -1.11803398874989)

(-100.9946125238742, 1.11803398874989)

(19.474078808283977, -362697829.855237)

(-97.8530198702844, -1.11803398874989)

(-82.14505660233543, 1.11803398874989)

(-94.7114272166946, 1.11803398874989)

(2.1859345198367977, 10.3746247928743)

(16.332486141511154, -15673594.915122)

(-31.879574144898772, 1.1180339887499)

(-66.43709333438646, -1.11803398874989)

(27.32806044171226, 934298910222.704)

(24.186467788125146, 40374716741.3286)

(5.337306099932291, 263.927228962533)

(-72.72027864156605, -1.11803398874989)

(-57.012315373617085, 1.11803398874989)

(-28.737981491308176, -1.11803398874983)

(-63.29550068079667, 1.11803398874989)

(-60.153908027206874, -1.11803398874989)

(-19.31320354006535, 1.11803398956914)

(-9.888307496404046, -1.11802384478128)

(-25.596388837736942, 1.11803398875142)

(-16.17161065651647, -1.11803396979209)

(-85.28664925592523, -1.11803398874989)

(-88.42824190951502, 1.11803398874989)

(-47.5875374128477, -1.11803398874989)

(30.469653095302167, 21620323908628.1)

(-79.00346394874563, -1.11803398874989)

(-38.162759452078326, 1.11803398874989)

(8.47848746340932, 6083.549234715)

(-13.030023324274493, 1.11803442746176)

(3.762485612062741, -55.1798718939363)

(-69.57868598797626, 1.11803398874989)

(-91.5698345631048, -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=−35.0211667984885x_{1} = -35.0211667984885
x2=−3.53084441183211x_{2} = -3.53084441183211
x3=−41.3043521056681x_{3} = -41.3043521056681
x4=−22.4547961837177x_{4} = -22.4547961837177
x5=13.1908937929858x_{5} = 13.1908937929858
x6=10.0492940799707x_{6} = 10.0492940799707
x7=25.7572641149185x_{7} = 25.7572641149185
x8=−53.8707227200273x_{8} = -53.8707227200273
x9=19.474078808284x_{9} = 19.474078808284
x10=−97.8530198702844x_{10} = -97.8530198702844
x11=16.3324861415112x_{11} = 16.3324861415112
x12=−66.4370933343865x_{12} = -66.4370933343865
x13=−72.7202786415661x_{13} = -72.7202786415661
x14=−28.7379814913082x_{14} = -28.7379814913082
x15=−60.1539080272069x_{15} = -60.1539080272069
x16=−9.88830749640405x_{16} = -9.88830749640405
x17=−16.1716106565165x_{17} = -16.1716106565165
x18=−85.2866492559252x_{18} = -85.2866492559252
x19=−47.5875374128477x_{19} = -47.5875374128477
x20=−79.0034639487456x_{20} = -79.0034639487456
x21=3.76248561206274x_{21} = 3.76248561206274
x22=−91.5698345631048x_{22} = -91.5698345631048
Puntos máximos de la función:
x22=−44.4459447592579x_{22} = -44.4459447592579
x22=−50.7291300664375x_{22} = -50.7291300664375
x22=17.9032824823194x_{22} = 17.9032824823194
x22=11.6200979108149x_{22} = 11.6200979108149
x22=−6.74954677059035x_{22} = -6.74954677059035
x22=−75.8618712951559x_{22} = -75.8618712951559
x22=−100.994612523874x_{22} = -100.994612523874
x22=−82.1450566023354x_{22} = -82.1450566023354
x22=−94.7114272166946x_{22} = -94.7114272166946
x22=2.1859345198368x_{22} = 2.1859345198368
x22=−31.8795741448988x_{22} = -31.8795741448988
x22=27.3280604417123x_{22} = 27.3280604417123
x22=24.1864677881251x_{22} = 24.1864677881251
x22=5.33730609993229x_{22} = 5.33730609993229
x22=−57.0123153736171x_{22} = -57.0123153736171
x22=−63.2955006807967x_{22} = -63.2955006807967
x22=−19.3132035400653x_{22} = -19.3132035400653
x22=−25.5963888377369x_{22} = -25.5963888377369
x22=−88.428241909515x_{22} = -88.428241909515
x22=30.4696530953022x_{22} = 30.4696530953022
x22=−38.1627594520783x_{22} = -38.1627594520783
x22=8.47848746340932x_{22} = 8.47848746340932
x22=−13.0300233242745x_{22} = -13.0300233242745
x22=−69.5786859879763x_{22} = -69.5786859879763
Decrece en los intervalos
[25.7572641149185,∞)\left[25.7572641149185, \infty\right)
Crece en los intervalos
(−∞,−97.8530198702844]\left(-\infty, -97.8530198702844\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))ex+3(sin⁡(2x)+cos⁡(2x))ex+sin⁡(x)2−cos⁡(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} + \frac{\sin{\left(x \right)}}{2} - \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−64.8662970075916x_{1} = -64.8662970075916
x2=−99.4238161970793x_{2} = -99.4238161970793
x3=−46.0167410860528x_{3} = -46.0167410860528
x4=23.6328934292233x_{4} = 23.6328934292233
x5=22.0620971024128x_{5} = 22.0620971024128
x6=−30.3087778181042x_{6} = -30.3087778181042
x7=−93.1406308898997x_{7} = -93.1406308898997
x8=−61.7247043540018x_{8} = -61.7247043540018
x9=−17.7424073130162x_{9} = -17.7424073130162
x10=−86.8574455827201x_{10} = -86.8574455827201
x11=−58.583111700412x_{11} = -58.583111700412
x12=−1.65547837114724x_{12} = -1.65547837114724
x13=−39.7335557788732x_{13} = -39.7335557788732
x14=1.64958620257498x_{14} = 1.64958620257498
x15=−68.0078896611814x_{15} = -68.0078896611814
x16=−27.1671851645052x_{16} = -27.1671851645052
x17=−49.1583337396426x_{17} = -49.1583337396426
x18=−36.5919631252834x_{18} = -36.5919631252834
x19=9.49572137431647x_{19} = 9.49572137431647
x20=15.7789117857015x_{20} = 15.7789117857015
x21=31.4868750632x_{21} = 31.4868750632
x22=−71.1494823147711x_{22} = -71.1494823147711
x23=7.92494472691859x_{23} = 7.92494472691859
x24=−83.7158529291303x_{24} = -83.7158529291303
x25=−80.5742602755405x_{25} = -80.5742602755405
x26=28.3452824096102x_{26} = 28.3452824096102
x27=17.3497081208705x_{27} = 17.3497081208705
x28=−77.4326676219507x_{28} = -77.4326676219507
x29=−52.2999263932324x_{29} = -52.2999263932324
x30=12.6373193626448x_{30} = 12.6373193626448
x31=14.2081154956564x_{31} = 14.2081154956564
x32=20.4913007756865x_{32} = 20.4913007756865
x33=−20.8839998526124x_{33} = -20.8839998526124
x34=−5.20739217375729x_{34} = -5.20739217375729
x35=−11.4592804109039x_{35} = -11.4592804109039
x36=−14.6008120215359x_{36} = -14.6008120215359
x37=−24.0255925111283x_{37} = -24.0255925111283
x38=−55.4415190468222x_{38} = -55.4415190468222
x39=−33.4503704716936x_{39} = -33.4503704716936
x40=−89.9990382363099x_{40} = -89.9990382363099
x41=−8.31627537174744x_{41} = -8.31627537174744
x42=−42.875148432463x_{42} = -42.875148432463
x43=−96.2822235434895x_{43} = -96.2822235434895
x44=0.12874079676591x_{44} = 0.12874079676591
x45=6.35425214706242x_{45} = 6.35425214706242
x46=−74.2910749683609x_{46} = -74.2910749683609

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
[31.4868750632,∞)\left[31.4868750632, \infty\right)
Convexa en los intervalos
(−∞,−99.4238161970793]\left(-\infty, -99.4238161970793\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))ex+(−1)sin⁡(x)2)+cos⁡(x))=⟨−32,32⟩\lim_{x \to -\infty}\left(\left(\left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{x} + \frac{\left(-1\right) \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⁡(2x)−cos⁡(2x))ex+(−1)sin⁡(x)2)+cos⁡(x))y = \lim_{x \to \infty}\left(\left(\left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{x} + \frac{\left(-1\right) \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(2*x) - sin(2*x))*exp(x) + cos(x), dividida por x con x->+oo y x ->-oo
lim⁡x→−∞(((−sin⁡(2x)−cos⁡(2x))ex+(−1)sin⁡(x)2)+cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\left(\left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{x} + \frac{\left(-1\right) \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⁡(2x)−cos⁡(2x))ex+(−1)sin⁡(x)2)+cos⁡(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} + \frac{\left(-1\right) \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⁡(2x)−cos⁡(2x))ex+(−1)sin⁡(x)2)+cos⁡(x)=(sin⁡(2x)−cos⁡(2x))e−x+sin⁡(x)2+cos⁡(x)\left(\left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{x} + \frac{\left(-1\right) \sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)} = \left(\sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- x} + \frac{\sin{\left(x \right)}}{2} + \cos{\left(x \right)}
- No
((−sin⁡(2x)−cos⁡(2x))ex+(−1)sin⁡(x)2)+cos⁡(x)=−(sin⁡(2x)−cos⁡(2x))e−x−sin⁡(x)2−cos⁡(x)\left(\left(- \sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{x} + \frac{\left(-1\right) \sin{\left(x \right)}}{2}\right) + \cos{\left(x \right)} = - \left(\sin{\left(2 x \right)} - \cos{\left(2 x \right)}\right) e^{- x} - \frac{\sin{\left(x \right)}}{2} - \cos{\left(x \right)}
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор