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

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

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
f(x) = (-1 + 2*x)*sin(x) + (-1 + 2*log(cos(x)))*cos(x)
f(x)=(2x−1)sin⁡(x)+(2log⁡(cos⁡(x))−1)cos⁡(x)f{\left(x \right)} = \left(2 x - 1\right) \sin{\left(x \right)} + \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\right) \cos{\left(x \right)} 
f = (2*x - 1)*sin(x) + (2*log(cos(x)) - 1)*cos(x)
Gráfico de la función
02468-8-6-4-2-1010-5050
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:
(2x−1)sin⁡(x)+(2log⁡(cos⁡(x))−1)cos⁡(x)=0\left(2 x - 1\right) \sin{\left(x \right)} + \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\right) \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Puntos de cruce con el eje X:

Solución numérica
x1=62.8398739645683x_{1} = 62.8398739645683
x2=−6.35637018729289x_{2} = -6.35637018729289
x3=−25.1522376444577x_{3} = -25.1522376444577
x4=37.7125497878719x_{4} = 37.7125497878719
x5=−62.8397473371473x_{5} = -62.8397473371473
x6=−37.7121981638125x_{6} = -37.7121981638125
x7=12.6077136364616x_{7} = 12.6077136364616
x8=94.2531128650743x_{8} = 94.2531128650743
x9=43.9937940561679x_{9} = 43.9937940561679
x10=100.535963200605x_{10} = 100.535963200605
x11=−31.4315875721432x_{11} = -31.4315875721432
x12=−0.560803089410854x_{12} = -0.560803089410854
x13=56.5575876368736x_{13} = 56.5575876368736
x14=69.122324895401x_{14} = 69.122324895401
x15=−18.8753733408446x_{15} = -18.8753733408446
x16=50.2755282300483x_{16} = 50.2755282300483
x17=−56.5574313136906x_{17} = -56.5574313136906
x18=−100.535913731043x_{18} = -100.535913731043
x19=81.6875677274511x_{19} = 81.6875677274511
x20=87.9703106503065x_{20} = 87.9703106503065
x21=−87.9702460384676x_{21} = -87.9702460384676
x22=−12.6045623515965x_{22} = -12.6045623515965
x23=−94.2530565802804x_{23} = -94.2530565802804
x24=31.4320937932335x_{24} = 31.4320937932335
x25=−43.9935356839531x_{25} = -43.9935356839531
x26=−69.1222202411893x_{26} = -69.1222202411893
x27=18.8767776121694x_{27} = 18.8767776121694
x28=−50.2753303958452x_{28} = -50.2753303958452
x29=1.13465350173012x_{29} = 1.13465350173012
x30=25.1530282768851x_{30} = 25.1530282768851
x31=−75.404811074455x_{31} = -75.404811074455
x32=−81.6874927941712x_{32} = -81.6874927941712
x33=75.4048990152687x_{33} = 75.4048990152687
x34=6.36879706683389x_{34} = 6.36879706683389
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 + 2*x)*sin(x) + (-1 + 2*log(cos(x)))*cos(x).
(−1+2log⁡(cos⁡(0)))cos⁡(0)+(−1+0⋅2)sin⁡(0)\left(-1 + 2 \log{\left(\cos{\left(0 \right)} \right)}\right) \cos{\left(0 \right)} + \left(-1 + 0 \cdot 2\right) \sin{\left(0 \right)}
Resultado:
f(0)=−1f{\left(0 \right)} = -1
Punto:
(0, -1)
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
(2x−1)cos⁡(x)−(2log⁡(cos⁡(x))−1)sin⁡(x)=0\left(2 x - 1\right) \cos{\left(x \right)} - \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\right) \sin{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=11.1923851684872x_{1} = 11.1923851684872
x2=−98.9980706522146x_{2} = -98.9980706522146
x3=55.038401154341x_{3} = 55.038401154341
x4=−80.1553167277427x_{4} = -80.1553167277427
x5=−11.1809835436224x_{5} = -11.1809835436224
x6=−67.5952476455873x_{6} = -67.5952476455873
x7=−92.7168859456839x_{7} = -92.7168859456839
x8=−17.4165779471253x_{8} = -17.4165779471253
x9=17.4222474171951x_{9} = 17.4222474171951
x10=98.9983741473088x_{10} = 98.9983741473088
x11=86.4363349953052x_{11} = 86.4363349953052
x12=−61.3160269646839x_{12} = -61.3160269646839
x13=−55.0375642637124x_{13} = -55.0375642637124
x14=23.6765689520874x_{14} = 23.6765689520874
x15=−36.2100137852466x_{15} = -36.2100137852466
x16=−48.7601129098911x_{16} = -48.7601129098911
x17=−5.01383357965488x_{17} = -5.01383357965488
x18=−29.9390384094334x_{18} = -29.9390384094334
x19=−73.8750521835207x_{19} = -73.8750521835207
x20=29.9413760959307x_{20} = 29.9413760959307
x21=0.318280570583234x_{21} = 0.318280570583234
x22=67.5958357764642x_{22} = 67.5958357764642
x23=−42.4840575004816x_{23} = -42.4840575004816
x24=42.4853572957077x_{24} = 42.4853572957077
x25=5.05092038437637x_{25} = 5.05092038437637
x26=36.2117151991956x_{26} = 36.2117151991956
x27=73.8755567303519x_{27} = 73.8755567303519
x28=−23.6731239795525x_{28} = -23.6731239795525
x29=48.7611418014431x_{29} = 48.7611418014431
x30=−86.4359506343332x_{30} = -86.4359506343332
x31=92.7172261668224x_{31} = 92.7172261668224
x32=80.1557548358302x_{32} = 80.1557548358302
x33=61.31672245894x_{33} = 61.31672245894
Signos de extremos en los puntos:
(11.192385168487238, -21.8057261059614)

(-98.99807065221458, -199.139162503015)

(55.03840115434104, -109.276928144792)

(-80.15531672774274, -161.471953649099)

(-11.18098354362239, -23.7693527072617)

(-67.5952476455873, -136.367842119426)

(-92.71688594568388, -186.582292460034)

(-17.41657794712531, -36.1761757384812)

(17.42224741719514, -34.195916949177)

(98.99837414730884, -197.140610394632)

(86.4363349953052, -172.028280355938)

(-61.31602696468392, -123.819080435662)

(-55.03756426371237, -111.273316069569)

(23.676568952087397, -46.6593232370661)

(-36.210013785246616, -73.6657357811634)

(-48.76011290989109, -98.7314685427634)

(-5.01383357965488, -11.5484009067772)

(-29.939038409433376, -61.1475009977644)

(-73.87505218352074, -148.918955768372)

(29.94137609593075, -59.1565322392085)

(0.3182805705832339, -1.16139158186211)

(67.59583577646421, -134.370473098101)

(-42.484057500481576, -86.1949007766244)

(42.485357295707665, -84.2002570967299)

(5.050920384376369, -9.64951433101058)

(36.211715199195645, -71.6725454043035)

(73.87555673035195, -146.921247474119)

(-23.67312397955255, -48.6465934897612)

(48.76114180144308, -96.7358149067473)

(-86.43595063433324, -174.026487580461)

(92.71722616682241, -184.583898039479)

(80.15575483583017, -159.473971341137)

(61.31672245894002, -121.822139593797)


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=11.1923851684872x_{1} = 11.1923851684872
x2=−98.9980706522146x_{2} = -98.9980706522146
x3=55.038401154341x_{3} = 55.038401154341
x4=−80.1553167277427x_{4} = -80.1553167277427
x5=−11.1809835436224x_{5} = -11.1809835436224
x6=−67.5952476455873x_{6} = -67.5952476455873
x7=−92.7168859456839x_{7} = -92.7168859456839
x8=−17.4165779471253x_{8} = -17.4165779471253
x9=17.4222474171951x_{9} = 17.4222474171951
x10=98.9983741473088x_{10} = 98.9983741473088
x11=86.4363349953052x_{11} = 86.4363349953052
x12=−61.3160269646839x_{12} = -61.3160269646839
x13=−55.0375642637124x_{13} = -55.0375642637124
x14=23.6765689520874x_{14} = 23.6765689520874
x15=−36.2100137852466x_{15} = -36.2100137852466
x16=−48.7601129098911x_{16} = -48.7601129098911
x17=−5.01383357965488x_{17} = -5.01383357965488
x18=−29.9390384094334x_{18} = -29.9390384094334
x19=−73.8750521835207x_{19} = -73.8750521835207
x20=29.9413760959307x_{20} = 29.9413760959307
x21=0.318280570583234x_{21} = 0.318280570583234
x22=67.5958357764642x_{22} = 67.5958357764642
x23=−42.4840575004816x_{23} = -42.4840575004816
x24=42.4853572957077x_{24} = 42.4853572957077
x25=5.05092038437637x_{25} = 5.05092038437637
x26=36.2117151991956x_{26} = 36.2117151991956
x27=73.8755567303519x_{27} = 73.8755567303519
x28=−23.6731239795525x_{28} = -23.6731239795525
x29=48.7611418014431x_{29} = 48.7611418014431
x30=−86.4359506343332x_{30} = -86.4359506343332
x31=92.7172261668224x_{31} = 92.7172261668224
x32=80.1557548358302x_{32} = 80.1557548358302
x33=61.31672245894x_{33} = 61.31672245894
La función no tiene puntos máximos
Decrece en los intervalos
[98.9983741473088,∞)\left[98.9983741473088, \infty\right)
Crece en los intervalos
(−∞,−98.9980706522146]\left(-\infty, -98.9980706522146\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
−(2x−1)sin⁡(x)−(2log⁡(cos⁡(x))−1)cos⁡(x)+2sin⁡2(x)cos⁡(x)+2cos⁡(x)=0- \left(2 x - 1\right) \sin{\left(x \right)} - \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\right) \cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)}} + 2 \cos{\left(x \right)} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=25.193635912389x_{1} = 25.193635912389
x2=−12.6811720339471x_{2} = -12.6811720339471
x3=87.9817440973062x_{3} = 87.9817440973062
x4=100.545960271683x_{4} = 100.545960271683
x5=−6.50439884791303x_{5} = -6.50439884791303
x6=50.2956238346493x_{6} = 50.2956238346493
x7=62.8559178155627x_{7} = 62.8559178155627
x8=−37.7383797852679x_{8} = -37.7383797852679
x9=−44.016018436621x_{9} = -44.016018436621
x10=−50.2950300967502x_{10} = -50.2950300967502
x11=−69.1365854481381x_{11} = -69.1365854481381
x12=−18.9270774861546x_{12} = -18.9270774861546
x13=56.5754302225125x_{13} = 56.5754302225125
x14=−25.1912602350172x_{14} = -25.1912602350172
x15=−7.72595060570768x_{15} = -7.72595060570768
x16=69.136899476556x_{16} = 69.136899476556
x17=12.6906873803376x_{17} = 12.6906873803376
x18=−94.2636111134011x_{18} = -94.2636111134011
x19=44.0167939548267x_{19} = 44.0167939548267
x20=18.931302298166x_{20} = 18.931302298166
x21=−31.4629250479627x_{21} = -31.4629250479627
x22=−75.4179869949679x_{22} = -75.4179869949679
x23=81.6998861304508x_{23} = 81.6998861304508
x24=−81.699661296896x_{24} = -81.699661296896
x25=94.2637799868012x_{25} = 94.2637799868012
x26=−87.9815502367254x_{26} = -87.9815502367254
x27=75.4182508638508x_{27} = 75.4182508638508
x28=37.739435401867x_{28} = 37.739435401867
x29=31.4644452565135x_{29} = 31.4644452565135
x30=−56.5749611061152x_{30} = -56.5749611061152
x31=6.54273414364566x_{31} = 6.54273414364566
x32=−100.545811848304x_{32} = -100.545811848304
x33=−62.8555378369724x_{33} = -62.8555378369724

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
[−12.6811720339471,−7.72595060570768]∪[−6.50439884791303,∞)\left[-12.6811720339471, -7.72595060570768\right] \cup \left[-6.50439884791303, \infty\right)
Convexa en los intervalos
(−∞,−100.545811848304]\left(-\infty, -100.545811848304\right]
Asíntotas horizontales
Hallemos las asíntotas horizontales mediante los límites de esta función con x->+oo y x->-oo
lim⁡x→−∞((2x−1)sin⁡(x)+(2log⁡(cos⁡(x))−1)cos⁡(x))=⟨−∞,∞⟩\lim_{x \to -\infty}\left(\left(2 x - 1\right) \sin{\left(x \right)} + \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\right) \cos{\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→∞((2x−1)sin⁡(x)+(2log⁡(cos⁡(x))−1)cos⁡(x))=⟨−∞,∞⟩\lim_{x \to \infty}\left(\left(2 x - 1\right) \sin{\left(x \right)} + \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\right) \cos{\left(x \right)}\right) = \left\langle -\infty, \infty\right\rangle
Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=⟨−∞,∞⟩y = \left\langle -\infty, \infty\right\rangle
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (-1 + 2*x)*sin(x) + (-1 + 2*log(cos(x)))*cos(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→−∞((2x−1)sin⁡(x)+(2log⁡(cos⁡(x))−1)cos⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\left(2 x - 1\right) \sin{\left(x \right)} + \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\right) \cos{\left(x \right)}}{x}\right)
True

Tomamos como el límite
es decir,
ecuación de la asíntota inclinada a la derecha:
y=xlim⁡x→∞((2x−1)sin⁡(x)+(2log⁡(cos⁡(x))−1)cos⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\left(2 x - 1\right) \sin{\left(x \right)} + \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\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:
(2x−1)sin⁡(x)+(2log⁡(cos⁡(x))−1)cos⁡(x)=−(−2x−1)sin⁡(x)+(2log⁡(cos⁡(x))−1)cos⁡(x)\left(2 x - 1\right) \sin{\left(x \right)} + \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\right) \cos{\left(x \right)} = - \left(- 2 x - 1\right) \sin{\left(x \right)} + \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\right) \cos{\left(x \right)}
- No
(2x−1)sin⁡(x)+(2log⁡(cos⁡(x))−1)cos⁡(x)=(−2x−1)sin⁡(x)−(2log⁡(cos⁡(x))−1)cos⁡(x)\left(2 x - 1\right) \sin{\left(x \right)} + \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\right) \cos{\left(x \right)} = \left(- 2 x - 1\right) \sin{\left(x \right)} - \left(2 \log{\left(\cos{\left(x \right)} \right)} - 1\right) \cos{\left(x \right)}
- No
es decir, función
no es
par ni impar
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