Sr Examen

Otras calculadoras

Trazado el gráfico de la función/ -1+(-1-x)*cos(2*x)+(-1-x)*exp(x)+(-1-x)*sin(2*x)

Gráfico de la función y = -1+(-1-x)*cos(2*x)+(-1-x)*exp(x)+(-1-x)*sin(2*x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                                          x                    
f(x) = -1 + (-1 - x)*cos(2*x) + (-1 - x)*e  + (-1 - x)*sin(2*x)
f(x)=(−x−1)sin⁡(2x)+((−x−1)ex+((−x−1)cos⁡(2x)−1))f{\left(x \right)} = \left(- x - 1\right) \sin{\left(2 x \right)} + \left(\left(- x - 1\right) e^{x} + \left(\left(- x - 1\right) \cos{\left(2 x \right)} - 1\right)\right) 
f = (-x - 1)*sin(2*x) + (-x - 1)*exp(x) + (-x - 1)*cos(2*x) - 1
Gráfico de la función
02468-8-6-4-2-1010-250000250000
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:
(−x−1)sin⁡(2x)+((−x−1)ex+((−x−1)cos⁡(2x)−1))=0\left(- x - 1\right) \sin{\left(2 x \right)} + \left(\left(- x - 1\right) e^{x} + \left(\left(- x - 1\right) \cos{\left(2 x \right)} - 1\right)\right) = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=−66.3607354388259x_{1} = -66.3607354388259
x2=−44.3668432507387x_{2} = -44.3668432507387
x3=−56.9350458931008x_{3} = -56.9350458931008
x4=−60.0769747345276x_{4} = -60.0769747345276
x5=−8.29513250323377x_{5} = -8.29513250323377
x6=−52.2358785891094x_{6} = -52.2358785891094
x7=−74.2249548344183x_{7} = -74.2249548344183
x8=−11.4222186132624x_{8} = -11.4222186132624
x9=−69.5025761997543x_{9} = -69.5025761997543
x10=−55.3770725862686x_{10} = -55.3770725862686
x11=−2.22562330179759x_{11} = -2.22562330179759
x12=−30.2499177978201x_{12} = -30.2499177978201
x13=−3.39690076296571x_{13} = -3.39690076296571
x14=−31.7971445381997x_{14} = -31.7971445381997
x15=−99.3564623134632x_{15} = -99.3564623134632
x16=−19.2228484792639x_{16} = -19.2228484792639
x17=−82.069746918288x_{17} = -82.069746918288
x18=−89.9320652954826x_{18} = -89.9320652954826
x19=−91.4949791119948x_{19} = -91.4949791119948
x20=−88.3532459395788x_{20} = -88.3532459395788
x21=−9.777172453449x_{21} = -9.777172453449
x22=−80.5077585853495x_{22} = -80.5077585853495
x23=−83.6491822143175x_{23} = -83.6491822143175
x24=−47.5089867630885x_{24} = -47.5089867630885
x25=−96.2149882618326x_{25} = -96.2149882618326
x26=−25.511014042766x_{26} = -25.511014042766
x27=−41.2246136467735x_{27} = -41.2246136467735
x28=−75.7861951753108x_{28} = -75.7861951753108
x29=−17.6926451981512x_{29} = -17.6926451981512
x30=−58.5183101252021x_{30} = -58.5183101252021
x31=−39.6717501813015x_{31} = -39.6717501813015
x32=−45.9536577245603x_{32} = -45.9536577245603
x33=−23.9700383529872x_{33} = -23.9700383529872
x34=−67.94222270284x_{34} = -67.94222270284
x35=−97.7784180849175x_{35} = -97.7784180849175
x36=−53.7930770274444x_{36} = -53.7930770274444
x37=−77.3663488621854x_{37} = -77.3663488621854
x38=−22.3672981637581x_{38} = -22.3672981637581
x39=−38.0822760510374x_{39} = -38.0822760510374
x40=−61.6595844422668x_{40} = -61.6595844422668
x41=−33.3903382082879x_{41} = -33.3903382082879
x42=−16.0772042492352x_{42} = -16.0772042492352
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 -1 + (-1 - x)*cos(2*x) + (-1 - x)*exp(x) + (-1 - x)*sin(2*x).
((−1+(−1−0)cos⁡(0⋅2))+(−1−0)e0)+(−1−0)sin⁡(0⋅2)\left(\left(-1 + \left(-1 - 0\right) \cos{\left(0 \cdot 2 \right)}\right) + \left(-1 - 0\right) e^{0}\right) + \left(-1 - 0\right) \sin{\left(0 \cdot 2 \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
(−x−1)ex−2(−x−1)sin⁡(2x)+2(−x−1)cos⁡(2x)−ex−sin⁡(2x)−cos⁡(2x)=0\left(- x - 1\right) e^{x} - 2 \left(- x - 1\right) \sin{\left(2 x \right)} + 2 \left(- x - 1\right) \cos{\left(2 x \right)} - e^{x} - \sin{\left(2 x \right)} - \cos{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−20.0407798646334x_{1} = -20.0407798646334
x2=−27.8909305442173x_{2} = -27.8909305442173
x3=−48.307271452037x_{3} = -48.307271452037
x4=−70.2967432346177x_{4} = -70.2967432346177
x5=−42.0248953008302x_{5} = -42.0248953008302
x6=−86.004039894791x_{6} = -86.004039894791
x7=−100.140787478269x_{7} = -100.140787478269
x8=−71.8674596045071x_{8} = -71.8674596045071
x9=−18.4711622278058x_{9} = -18.4711622278058
x10=−95.4285243331501x_{10} = -95.4285243331501
x11=−64.0139176111499x_{11} = -64.0139176111499
x12=−65.5846174571317x_{12} = -65.5846174571317
x13=−15.332699711118x_{13} = -15.332699711118
x14=−7.49975246074226x_{14} = -7.49975246074226
x15=−4.39433925448196x_{15} = -4.39433925448196
x16=−12.1959853260224x_{16} = -12.1959853260224
x17=−34.1723559328195x_{17} = -34.1723559328195
x18=−93.8577727894039x_{18} = -93.8577727894039
x19=−92.2870227867059x_{19} = -92.2870227867059
x20=−5.94054718820995x_{20} = -5.94054718820995
x21=−62.4432226972853x_{21} = -62.4432226972853
x22=−59.301849260036x_{22} = -59.301849260036
x23=−40.4543415140698x_{23} = -40.4543415140698
x24=−37.3132968568621x_{24} = -37.3132968568621
x25=−51.4485350855633x_{25} = -51.4485350855633
x26=−21.6105768091782x_{26} = -21.6105768091782
x27=−81.2918235134642x_{27} = -81.2918235134642
x28=−29.4612141078121x_{28} = -29.4612141078121
x29=−43.595466968147x_{29} = -43.595466968147
x30=−57.731171644018x_{30} = -57.731171644018
x31=−0.782507279185871x_{31} = -0.782507279185871
x32=−87.5747828636033x_{32} = -87.5747828636033
x33=−26.3207105314339x_{33} = -26.3207105314339
x34=−49.8778979835774x_{34} = -49.8778979835774
x35=−78.1503576383583x_{35} = -78.1503576383583
x36=−35.742811670968x_{36} = -35.742811670968
x37=−73.4381794415606x_{37} = -73.4381794415606
x38=−56.1605007874078x_{38} = -56.1605007874078
x39=−1.53112226827525x_{39} = -1.53112226827525
x40=−84.4332989349059x_{40} = -84.4332989349059
x41=−13.7640442903819x_{41} = -13.7640442903819
x42=−79.7210893111989x_{42} = -79.7210893111989
Signos de extremos en los puntos:
(-20.040779864633414, -27.9184497732393)

(-27.890930544217294, 37.0229465437808)

(-48.30727145203698, -67.8988484229824)

(-70.29674323461768, -98.9978432001354)

(-42.024895300830245, -59.0136547990851)

(-86.004039894791, -121.211786502694)

(-100.14078747826939, 139.204463182668)

(-71.8674596045071, 99.2192281275463)

(-18.47116222780578, 23.6978427534278)

(-95.42852433315014, -134.540227757372)

(-64.01391761114989, -90.1123316768659)

(-65.58461745713174, 90.3337049184686)

(-15.33269971111799, 19.2571789544453)

(-7.499752460742258, -10.1612484827881)

(-4.394339254481964, -5.70501005712696)

(-12.195985326022399, 14.8178062726271)

(-34.172355932819535, 45.907467527421)

(-93.85777278940385, 130.318817955819)

(-92.28702278670593, -130.097409244031)

(-5.940547188209952, 5.96499445190665)

(-62.44322269728526, 85.8909619233463)

(-59.30184926003602, 81.4482340080829)

(-40.454341514069796, 54.7923848648336)

(-37.31329685686212, 50.3498895033352)

(-51.448535085563286, -72.3414986783194)

(-21.61057680917818, 28.1391840551561)

(-81.29182351346417, 112.547584147014)

(-29.46121410781208, -41.2440252863291)

(-43.595466968147015, 59.2349373815048)

(-57.731171644017955, -81.2268764881315)

(-0.7825072791858715, -0.883218708073241)

(-87.57478286360335, 121.433190240379)

(-26.320710531433942, -36.8019127773401)

(-49.87789798357739, 68.1201698124549)

(-78.15035763835829, 108.104790859307)

(-35.74281167096796, -50.1286680986204)

(-73.43817944156058, -103.440615506918)

(-56.16050078740785, 77.0055237487324)

(-1.5311222682752508, -1.45667256335889)

(-84.4332989349059, 116.990384185657)

(-13.764044290381873, -19.037237201122)

(-79.72108931119887, -112.326186608784)


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=−20.0407798646334x_{1} = -20.0407798646334
x2=−48.307271452037x_{2} = -48.307271452037
x3=−70.2967432346177x_{3} = -70.2967432346177
x4=−42.0248953008302x_{4} = -42.0248953008302
x5=−86.004039894791x_{5} = -86.004039894791
x6=−95.4285243331501x_{6} = -95.4285243331501
x7=−64.0139176111499x_{7} = -64.0139176111499
x8=−7.49975246074226x_{8} = -7.49975246074226
x9=−4.39433925448196x_{9} = -4.39433925448196
x10=−92.2870227867059x_{10} = -92.2870227867059
x11=−51.4485350855633x_{11} = -51.4485350855633
x12=−29.4612141078121x_{12} = -29.4612141078121
x13=−57.731171644018x_{13} = -57.731171644018
x14=−26.3207105314339x_{14} = -26.3207105314339
x15=−35.742811670968x_{15} = -35.742811670968
x16=−73.4381794415606x_{16} = -73.4381794415606
x17=−1.53112226827525x_{17} = -1.53112226827525
x18=−13.7640442903819x_{18} = -13.7640442903819
x19=−79.7210893111989x_{19} = -79.7210893111989
Puntos máximos de la función:
x19=−27.8909305442173x_{19} = -27.8909305442173
x19=−100.140787478269x_{19} = -100.140787478269
x19=−71.8674596045071x_{19} = -71.8674596045071
x19=−18.4711622278058x_{19} = -18.4711622278058
x19=−65.5846174571317x_{19} = -65.5846174571317
x19=−15.332699711118x_{19} = -15.332699711118
x19=−12.1959853260224x_{19} = -12.1959853260224
x19=−34.1723559328195x_{19} = -34.1723559328195
x19=−93.8577727894039x_{19} = -93.8577727894039
x19=−5.94054718820995x_{19} = -5.94054718820995
x19=−62.4432226972853x_{19} = -62.4432226972853
x19=−59.301849260036x_{19} = -59.301849260036
x19=−40.4543415140698x_{19} = -40.4543415140698
x19=−37.3132968568621x_{19} = -37.3132968568621
x19=−21.6105768091782x_{19} = -21.6105768091782
x19=−81.2918235134642x_{19} = -81.2918235134642
x19=−43.595466968147x_{19} = -43.595466968147
x19=−0.782507279185871x_{19} = -0.782507279185871
x19=−87.5747828636033x_{19} = -87.5747828636033
x19=−49.8778979835774x_{19} = -49.8778979835774
x19=−78.1503576383583x_{19} = -78.1503576383583
x19=−56.1605007874078x_{19} = -56.1605007874078
x19=−84.4332989349059x_{19} = -84.4332989349059
Decrece en los intervalos
[−1.53112226827525,∞)\left[-1.53112226827525, \infty\right)
Crece en los intervalos
(−∞,−95.4285243331501]\left(-\infty, -95.4285243331501\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
−(x+1)ex+4(x+1)sin⁡(2x)+4(x+1)cos⁡(2x)−2ex+4sin⁡(2x)−4cos⁡(2x)=0- \left(x + 1\right) e^{x} + 4 \left(x + 1\right) \sin{\left(2 x \right)} + 4 \left(x + 1\right) \cos{\left(2 x \right)} - 2 e^{x} + 4 \sin{\left(2 x \right)} - 4 \cos{\left(2 x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−22.4071873314378x_{1} = -22.4071873314378
x2=−30.2549137847745x_{2} = -30.2549137847745
x3=−45.9569124852419x_{3} = -45.9569124852419
x4=−9.87358382341354x_{4} = -9.87358382341354
x5=−44.3865185090471x_{5} = -44.3865185090471
x6=−82.0802744903927x_{6} = -82.0802744903927
x7=−52.2387348697591x_{7} = -52.2387348697591
x8=−61.6619974728184x_{8} = -61.6619974728184
x9=−41.2458246715886x_{9} = -41.2458246715886
x10=−66.3737925354503x_{10} = -66.3737925354503
x11=−107.211556750303x_{11} = -107.211556750303
x12=−19.2695955950552x_{12} = -19.2695955950552
x13=−31.8248406147759x_{13} = -31.8248406147759
x14=−80.5095999654955x_{14} = -80.5095999654955
x15=−16.133653330794x_{15} = -16.133653330794
x16=−55.3797640800198x_{16} = -55.3797640800198
x17=0.938840959135837x_{17} = 0.938840959135837
x18=−91.5044104036412x_{18} = -91.5044104036412
x19=−75.7976070754479x_{19} = -75.7976070754479
x20=−5.22162669823754x_{20} = -5.22162669823754
x21=−96.2165260124045x_{21} = -96.2165260124045
x22=−3.71048266493864x_{22} = -3.71048266493864
x23=−69.5150346109041x_{23} = -69.5150346109041
x24=−83.6509536436357x_{24} = -83.6509536436357
x25=−99.3579509677552x_{25} = -99.3579509677552
x26=−38.1052828339243x_{26} = -38.1052828339243
x27=−60.0914201575762x_{27} = -60.0914201575762
x28=−97.7872371295516x_{28} = -97.7872371295516
x29=−8.3146312926029x_{29} = -8.3146312926029
x30=−63.2325858381921x_{30} = -63.2325858381921
x31=−88.3630163778857x_{31} = -88.3630163778857
x32=−53.8092411011661x_{32} = -53.8092411011661
x33=−33.3948515959935x_{33} = -33.3948515959935
x34=−47.5273336025315x_{34} = -47.5273336025315
x35=−20.8382338719616x_{35} = -20.8382338719616
x36=−11.4360392006347x_{36} = -11.4360392006347
x37=−2.2894197897413x_{37} = -2.2894197897413
x38=−39.6755324419894x_{38} = -39.6755324419894
x39=−67.9444094619968x_{39} = -67.9444094619968
x40=0.183856620870702x_{40} = 0.183856620870702
x41=−89.9337116370698x_{41} = -89.9337116370698
x42=−25.5457991361971x_{42} = -25.5457991361971
x43=−17.7013606592206x_{43} = -17.7013606592206
x44=−77.3682659426012x_{44} = -77.3682659426012
x45=−23.9763917261188x_{45} = -23.9763917261188
x46=−74.2269541034648x_{46} = -74.2269541034648

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
[0.183856620870702,∞)\left[0.183856620870702, \infty\right)
Convexa en los intervalos
(−∞,−99.3579509677552]\left(-\infty, -99.3579509677552\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la izquierda:
y=lim⁡x→−∞((−x−1)sin⁡(2x)+((−x−1)ex+((−x−1)cos⁡(2x)−1)))y = \lim_{x \to -\infty}\left(\left(- x - 1\right) \sin{\left(2 x \right)} + \left(\left(- x - 1\right) e^{x} + \left(\left(- x - 1\right) \cos{\left(2 x \right)} - 1\right)\right)\right)
lim⁡x→∞((−x−1)sin⁡(2x)+((−x−1)ex+((−x−1)cos⁡(2x)−1)))=−∞\lim_{x \to \infty}\left(\left(- x - 1\right) \sin{\left(2 x \right)} + \left(\left(- x - 1\right) e^{x} + \left(\left(- x - 1\right) \cos{\left(2 x \right)} - 1\right)\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 -1 + (-1 - x)*cos(2*x) + (-1 - x)*exp(x) + (-1 - x)*sin(2*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→−∞((−x−1)sin⁡(2x)+((−x−1)ex+((−x−1)cos⁡(2x)−1))x)y = x \lim_{x \to -\infty}\left(\frac{\left(- x - 1\right) \sin{\left(2 x \right)} + \left(\left(- x - 1\right) e^{x} + \left(\left(- x - 1\right) \cos{\left(2 x \right)} - 1\right)\right)}{x}\right)
lim⁡x→∞((−x−1)sin⁡(2x)+((−x−1)ex+((−x−1)cos⁡(2x)−1))x)=−∞\lim_{x \to \infty}\left(\frac{\left(- x - 1\right) \sin{\left(2 x \right)} + \left(\left(- x - 1\right) e^{x} + \left(\left(- x - 1\right) \cos{\left(2 x \right)} - 1\right)\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:
(−x−1)sin⁡(2x)+((−x−1)ex+((−x−1)cos⁡(2x)−1))=−(x−1)sin⁡(2x)+(x−1)cos⁡(2x)+(x−1)e−x−1\left(- x - 1\right) \sin{\left(2 x \right)} + \left(\left(- x - 1\right) e^{x} + \left(\left(- x - 1\right) \cos{\left(2 x \right)} - 1\right)\right) = - \left(x - 1\right) \sin{\left(2 x \right)} + \left(x - 1\right) \cos{\left(2 x \right)} + \left(x - 1\right) e^{- x} - 1
- No
(−x−1)sin⁡(2x)+((−x−1)ex+((−x−1)cos⁡(2x)−1))=(x−1)sin⁡(2x)−(x−1)cos⁡(2x)−(x−1)e−x+1\left(- x - 1\right) \sin{\left(2 x \right)} + \left(\left(- x - 1\right) e^{x} + \left(\left(- x - 1\right) \cos{\left(2 x \right)} - 1\right)\right) = \left(x - 1\right) \sin{\left(2 x \right)} - \left(x - 1\right) \cos{\left(2 x \right)} - \left(x - 1\right) e^{- x} + 1
- No
es decir, función
no es
par ni impar
    © Sr Examen – Калькулятор