Sr Examen

Otras calculadoras

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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

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

Solución numérica
x1=88.7519560218585x_{1} = 88.7519560218585
x2=63.6199966221486x_{2} = 63.6199966221486
x3=−99.7472953527304x_{3} = -99.7472953527304
x4=−63.6199966221486x_{4} = -63.6199966221486
x5=24.3543156967572x_{5} = 24.3543156967572
x6=11.7950708789993x_{6} = 11.7950708789993
x7=−13.365241885338x_{7} = -13.365241885338
x8=−68.3321577671221x_{8} = -68.3321577671221
x9=−24.3543156967572x_{9} = -24.3543156967572
x10=−57.3371148034946x_{10} = -57.3371148034946
x11=−55.7663494057152x_{11} = -55.7663494057152
x12=19.6440069748152x_{12} = 19.6440069748152
x13=18.073488825764x_{13} = 18.073488825764
x14=74.6151324910722x_{14} = 74.6151324910722
x15=69.9029333888055x_{15} = 69.9029333888055
x16=−19.6440069748152x_{16} = -19.6440069748152
x17=62.0492253482485x_{17} = 62.0492253482485
x18=76.1859114196687x_{18} = 76.1859114196687
x19=55.7663494057152x_{19} = 55.7663494057152
x20=−49.4835512354109x_{20} = -49.4835512354109
x21=32.2067902406885x_{21} = 32.2067902406885
x22=−38.4890716436065x_{22} = -38.4890716436065
x23=−76.1859114196687x_{23} = -76.1859114196687
x24=5.52657917248173x_{24} = 5.52657917248173
x25=−18.073488825764x_{25} = -18.073488825764
x26=44.7716095805499x_{26} = 44.7716095805499
x27=−82.4689212262039x_{27} = -82.4689212262039
x28=−1.09517824356037x_{28} = -1.09517824356037
x29=−32.2067902406885x_{29} = -32.2067902406885
x30=38.4890716436065x_{30} = 38.4890716436065
x31=25.9249554978121x_{31} = 25.9249554978121
x32=99.7472953527304x_{32} = 99.7472953527304
x33=−5.52657917248173x_{33} = -5.52657917248173
x34=68.3321577671221x_{34} = 68.3321577671221
x35=−93.4642256324517x_{35} = -93.4642256324517
x36=−11.7950708789993x_{36} = -11.7950708789993
x37=13.365241885338x_{37} = 13.365241885338
x38=49.4835512354109x_{38} = 49.4835512354109
x39=0x_{39} = 0
x40=−25.9249554978121x_{40} = -25.9249554978121
x41=30.6360941054098x_{41} = 30.6360941054098
x42=−74.6151324910722x_{42} = -74.6151324910722
x43=−62.0492253482485x_{43} = -62.0492253482485
x44=82.4689212262039x_{44} = 82.4689212262039
x45=−69.9029333888055x_{45} = -69.9029333888055
x46=−7.09488666894283x_{46} = -7.09488666894283
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 (log(cos(x))*sin(2*x))/2 - x*cos(2*x)/2.
log⁡(cos⁡(0))sin⁡(0⋅2)2−0cos⁡(0⋅2)2\frac{\log{\left(\cos{\left(0 \right)} \right)} \sin{\left(0 \cdot 2 \right)}}{2} - \frac{0 \cos{\left(0 \cdot 2 \right)}}{2}
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
xsin⁡(2x)+log⁡(cos⁡(x))cos⁡(2x)−sin⁡(x)sin⁡(2x)2cos⁡(x)−cos⁡(2x)2=0x \sin{\left(2 x \right)} + \log{\left(\cos{\left(x \right)} \right)} \cos{\left(2 x \right)} - \frac{\sin{\left(x \right)} \sin{\left(2 x \right)}}{2 \cos{\left(x \right)}} - \frac{\cos{\left(2 x \right)}}{2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=−73.8497629783191x_{1} = -73.8497629783191
x2=−36.1664669726414x_{2} = -36.1664669726414
x3=4.85535400289239x_{3} = 4.85535400289239
x4=37.705742618707x_{4} = 37.705742618707
x5=81.6844695984197x_{5} = 81.6844695984197
x6=−67.5681487106479x_{6} = -67.5681487106479
x7=92.6957272072728x_{7} = 92.6957272072728
x8=56.5530885336565x_{8} = 56.5530885336565
x9=50.2704557623692x_{9} = 50.2704557623692
x10=−87.9674362995073x_{10} = -87.9674362995073
x11=86.4135889345361x_{11} = 86.4135889345361
x12=−31.4238831075354x_{12} = -31.4238831075354
x13=23.61378868006x_{13} = 23.61378868006
x14=55.0058122331746x_{14} = 55.0058122331746
x15=42.4454109582606x_{15} = 42.4454109582606
x16=12.5862466409933x_{16} = 12.5862466409933
x17=6.32282805574228x_{17} = 6.32282805574228
x18=−61.2868049125343x_{18} = -61.2868049125343
x19=94.2504321465305x_{19} = 94.2504321465305
x20=87.9674362995073x_{20} = 87.9674362995073
x21=29.8889485733125x_{21} = 29.8889485733125
x22=−80.1315903728764x_{22} = -80.1315903728764
x23=−29.8889485733125x_{23} = -29.8889485733125
x24=−17.3430209611095x_{24} = -17.3430209611095
x25=−81.6844695984197x_{25} = -81.6844695984197
x26=−42.4454109582606x_{26} = -42.4454109582606
x27=−6.32282805574228x_{27} = -6.32282805574228
x28=36.1664669726414x_{28} = 36.1664669726414
x29=100.533451674976x_{29} = 100.533451674976
x30=0.606974603888332x_{30} = 0.606974603888332
x31=−50.2704557623692x_{31} = -50.2704557623692
x32=43.987980826915x_{32} = 43.987980826915
x33=67.5681487106479x_{33} = 67.5681487106479
x34=−37.705742618707x_{34} = -37.705742618707
x35=80.1315903728764x_{35} = 80.1315903728764
x36=−86.4135889345361x_{36} = -86.4135889345361
x37=−75.4015393290825x_{37} = -75.4015393290825
x38=−94.2504321465305x_{38} = -94.2504321465305
x39=−43.987980826915x_{39} = -43.987980826915
x40=48.7252881019068x_{40} = 48.7252881019068
x41=73.8497629783191x_{41} = 73.8497629783191
x42=−23.61378868006x_{42} = -23.61378868006
Signos de extremos en los puntos:
(-73.84976297831912, -36.9729295514706)

(-36.16646697264143, -18.1551158356313)

(4.855354002892385, 2.60390881873152)

(37.705742618706985, -18.8512136640367)

(81.6844695984197, -40.8414696527169)

(-67.56814871064788, -33.8346907310889)

(92.69572720727275, 46.3898262730079)

(56.553088533656485, -28.2754390889677)

(50.27045576236922, -25.1339845754532)

(-87.96743629950727, 43.9830076538312)

(86.41358893453605, 43.2505638866455)

(-31.42388310753541, 15.7099524948124)

(23.61378868006, 11.8966636969236)

(55.00581223317458, 27.5598893998792)

(42.44541095826064, 21.2885881662263)

(12.5862466409933, -6.28815562281456)

(6.322828055742285, -3.15151373312277)

(-61.286804912534265, -30.6969640745736)

(94.25043214653046, -47.1245529416666)

(87.96743629950727, -43.9830076538312)

(29.88894857331252, 15.0240122000754)

(-80.13159037287639, -40.1115791214216)

(-29.88894857331252, -15.0240122000754)

(-17.34302096110948, -8.77593391046638)

(-81.6844695984197, 40.8414696527169)

(-42.44541095826064, -21.2885881662263)

(-6.322828055742285, 3.15151373312277)

(36.16646697264143, 18.1551158356313)

(100.53345167497645, -50.2661041500255)

(0.606974603888332, -0.198202936057562)

(-50.27045576236922, 25.1339845754532)

(43.987980826914956, -21.9925695248946)

(67.56814871064788, 33.8346907310889)

(-37.705742618706985, 18.8512136640367)

(80.13159037287639, 40.1115791214216)

(-86.41358893453605, -43.2505638866455)

(-75.40153932908247, 37.6999407598845)

(-94.25043214653046, 47.1245529416666)

(-43.987980826914956, 21.9925695248946)

(48.72528810190678, 24.4236657883981)

(73.84976297831912, 36.9729295514706)

(-23.61378868006, -11.8966636969236)


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=−73.8497629783191x_{1} = -73.8497629783191
x2=−36.1664669726414x_{2} = -36.1664669726414
x3=37.705742618707x_{3} = 37.705742618707
x4=81.6844695984197x_{4} = 81.6844695984197
x5=−67.5681487106479x_{5} = -67.5681487106479
x6=56.5530885336565x_{6} = 56.5530885336565
x7=50.2704557623692x_{7} = 50.2704557623692
x8=12.5862466409933x_{8} = 12.5862466409933
x9=6.32282805574228x_{9} = 6.32282805574228
x10=−61.2868049125343x_{10} = -61.2868049125343
x11=94.2504321465305x_{11} = 94.2504321465305
x12=87.9674362995073x_{12} = 87.9674362995073
x13=−80.1315903728764x_{13} = -80.1315903728764
x14=−29.8889485733125x_{14} = -29.8889485733125
x15=−17.3430209611095x_{15} = -17.3430209611095
x16=−42.4454109582606x_{16} = -42.4454109582606
x17=100.533451674976x_{17} = 100.533451674976
x18=0.606974603888332x_{18} = 0.606974603888332
x19=43.987980826915x_{19} = 43.987980826915
x20=−86.4135889345361x_{20} = -86.4135889345361
x21=−23.61378868006x_{21} = -23.61378868006
Puntos máximos de la función:
x21=4.85535400289239x_{21} = 4.85535400289239
x21=92.6957272072728x_{21} = 92.6957272072728
x21=−87.9674362995073x_{21} = -87.9674362995073
x21=86.4135889345361x_{21} = 86.4135889345361
x21=−31.4238831075354x_{21} = -31.4238831075354
x21=23.61378868006x_{21} = 23.61378868006
x21=55.0058122331746x_{21} = 55.0058122331746
x21=42.4454109582606x_{21} = 42.4454109582606
x21=29.8889485733125x_{21} = 29.8889485733125
x21=−81.6844695984197x_{21} = -81.6844695984197
x21=−6.32282805574228x_{21} = -6.32282805574228
x21=36.1664669726414x_{21} = 36.1664669726414
x21=−50.2704557623692x_{21} = -50.2704557623692
x21=67.5681487106479x_{21} = 67.5681487106479
x21=−37.705742618707x_{21} = -37.705742618707
x21=80.1315903728764x_{21} = 80.1315903728764
x21=−75.4015393290825x_{21} = -75.4015393290825
x21=−94.2504321465305x_{21} = -94.2504321465305
x21=−43.987980826915x_{21} = -43.987980826915
x21=48.7252881019068x_{21} = 48.7252881019068
x21=73.8497629783191x_{21} = 73.8497629783191
Decrece en los intervalos
[100.533451674976,∞)\left[100.533451674976, \infty\right)
Crece en los intervalos
(−∞,−86.4135889345361]\left(-\infty, -86.4135889345361\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
2xcos⁡(2x)−2log⁡(cos⁡(x))sin⁡(2x)−sin⁡2(x)sin⁡(2x)2cos⁡2(x)−2sin⁡(x)cos⁡(2x)cos⁡(x)+3sin⁡(2x)2=02 x \cos{\left(2 x \right)} - 2 \log{\left(\cos{\left(x \right)} \right)} \sin{\left(2 x \right)} - \frac{\sin^{2}{\left(x \right)} \sin{\left(2 x \right)}}{2 \cos^{2}{\left(x \right)}} - \frac{2 \sin{\left(x \right)} \cos{\left(2 x \right)}}{\cos{\left(x \right)}} + \frac{3 \sin{\left(2 x \right)}}{2} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=82.4720029556833x_{1} = 82.4720029556833
x2=−55.7707265878262x_{2} = -55.7707265878262
x3=−69.9065799652927x_{3} = -69.9065799652927
x4=−5.56286224935976x_{4} = -5.56286224935976
x5=18.0863589993736x_{5} = 18.0863589993736
x6=62.0531687465299x_{6} = 62.0531687465299
x7=99.7497682823529x_{7} = 99.7497682823529
x8=−11.8140836229578x_{8} = -11.8140836229578
x9=38.4958021257939x_{9} = 38.4958021257939
x10=57.3415797376218x_{10} = 57.3415797376218
x11=−25.9351258754077x_{11} = -25.9351258754077
x12=−57.3415797376218x_{12} = -57.3415797376218
x13=88.7548162292528x_{13} = 88.7548162292528
x14=69.9065799652927x_{14} = 69.9065799652927
x15=−32.2148906822907x_{15} = -32.2148906822907
x16=32.2148906822907x_{16} = 32.2148906822907
x17=0x_{17} = 0
x18=−18.0863589993736x_{18} = -18.0863589993736
x19=−24.3640416913461x_{19} = -24.3640416913461
x20=−13.3860342558761x_{20} = -13.3860342558761
x21=25.9351258754077x_{21} = 25.9351258754077
x22=55.7707265878262x_{22} = 55.7707265878262
x23=74.6184235765065x_{23} = 74.6184235765065
x24=−82.4720029556833x_{24} = -82.4720029556833
x25=−51.0593376668002x_{25} = -51.0593376668002
x26=−19.6576674023468x_{26} = -19.6576674023468
x27=−7.13832336762488x_{27} = -7.13832336762488
x28=49.4884694202809x_{28} = 49.4884694202809
x29=−99.7497682823529x_{29} = -99.7497682823529
x30=−76.1892518639602x_{30} = -76.1892518639602
x31=44.7773664051067x_{31} = 44.7773664051067
x32=30.643910181686x_{32} = 30.643910181686
x33=−93.4668624391249x_{33} = -93.4668624391249
x34=5.56286224935976x_{34} = 5.56286224935976
x35=13.3860342558761x_{35} = 13.3860342558761
x36=19.6576674023468x_{36} = 19.6576674023468
x37=76.1892518639602x_{37} = 76.1892518639602
x38=−68.3357456020097x_{38} = -68.3357456020097
x39=24.3640416913461x_{39} = 24.3640416913461
x40=11.8140836229578x_{40} = 11.8140836229578
x41=−62.0531687465299x_{41} = -62.0531687465299
x42=63.6240110985299x_{42} = 63.6240110985299
x43=−80.901179396507x_{43} = -80.901179396507
x44=−38.4958021257939x_{44} = -38.4958021257939
x45=68.3357456020097x_{45} = 68.3357456020097
x46=−63.6240110985299x_{46} = -63.6240110985299
x47=−49.4884694202809x_{47} = -49.4884694202809

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
[99.7497682823529,∞)\left[99.7497682823529, \infty\right)
Convexa en los intervalos
(−∞,−99.7497682823529]\left(-\infty, -99.7497682823529\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→−∞(−xcos⁡(2x)2+log⁡(cos⁡(x))sin⁡(2x)2)y = \lim_{x \to -\infty}\left(- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\log{\left(\cos{\left(x \right)} \right)} \sin{\left(2 x \right)}}{2}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(−xcos⁡(2x)2+log⁡(cos⁡(x))sin⁡(2x)2)y = \lim_{x \to \infty}\left(- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\log{\left(\cos{\left(x \right)} \right)} \sin{\left(2 x \right)}}{2}\right)
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (log(cos(x))*sin(2*x))/2 - x*cos(2*x)/2, 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→−∞(−xcos⁡(2x)2+log⁡(cos⁡(x))sin⁡(2x)2x)y = x \lim_{x \to -\infty}\left(\frac{- \frac{x \cos{\left(2 x \right)}}{2} + \frac{\log{\left(\cos{\left(x \right)} \right)} \sin{\left(2 x \right)}}{2}}{x}\right)
True

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