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Trazado el gráfico de la función/ (1+sin(x))^cot(x)

Gráfico de la función y = (1+sin(x))^cot(x)

v

Gráfico:

interior superior

Puntos de intersección:

mostrar?

Definida a trozos:

Solución

Ha introducido [src] 
                   cot(x)
f(x) = (1 + sin(x))
f(x)=(sin⁡(x)+1)cot⁡(x)f{\left(x \right)} = \left(\sin{\left(x \right)} + 1\right)^{\cot{\left(x \right)}} 
f = (sin(x) + 1)^cot(x)
Gráfico de la función
02468-8-6-4-2-101005
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⁡(x)+1)cot⁡(x)=0\left(\sin{\left(x \right)} + 1\right)^{\cot{\left(x \right)}} = 0
Resolvermos esta ecuación
Solución no hallada,
puede ser que el gráfico no cruce el eje X
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 + sin(x))^cot(x).
(sin⁡(0)+1)cot⁡(0)\left(\sin{\left(0 \right)} + 1\right)^{\cot{\left(0 \right)}}
Resultado:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- no hay soluciones de la ecuación
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
((−cot⁡2(x)−1)log⁡(sin⁡(x)+1)+cos⁡(x)cot⁡(x)sin⁡(x)+1)(sin⁡(x)+1)cot⁡(x)=0\left(\left(- \cot^{2}{\left(x \right)} - 1\right) \log{\left(\sin{\left(x \right)} + 1 \right)} + \frac{\cos{\left(x \right)} \cot{\left(x \right)}}{\sin{\left(x \right)} + 1}\right) \left(\sin{\left(x \right)} + 1\right)^{\cot{\left(x \right)}} = 0
Resolvermos esta ecuación
Raíces de esta ecuación
x1=59248.0488154944x_{1} = 59248.0488154944
x2=60.4432218626453x_{2} = 60.4432218626453
x3=−96.6364108168444x_{3} = -96.6364108168444
x4=3.89455409802898x_{4} = 3.89455409802898
x5=85.5759630913636x_{5} = 85.5759630913636
x6=−40.0877430522281x_{6} = -40.0877430522281
x7=−46.3709283594077x_{7} = -46.3709283594077
x8=22.7441100195677x_{8} = 22.7441100195677
x9=−25.8857026731575x_{9} = -25.8857026731575
x10=16.4609247123882x_{10} = 16.4609247123882
x11=−71.5036695881261x_{11} = -71.5036695881261
x12=−90.3532255096648x_{12} = -90.3532255096648
x13=55.7957063201771x_{13} = 55.7957063201771
x14=74.6452622417159x_{14} = 74.6452622417159
x15=29.0272953267473x_{15} = 29.0272953267473
x16=−8.67181651633019x_{16} = -8.67181651633019
x17=−13.3193320587984x_{17} = -13.3193320587984
x18=68.3620769345363x_{18} = 68.3620769345363
x19=−77.7868548953056x_{19} = -77.7868548953056
x20=−724.954941534803x_{20} = -724.954941534803
x21=−63.584814516235x_{21} = -63.584814516235
x22=−65.2204842809465x_{22} = -65.2204842809465
x23=11.81340916992x_{23} = 11.81340916992
x24=−57.3016292090555x_{24} = -57.3016292090555
x25=10.1777394052086x_{25} = 10.1777394052086
x26=−33.8045577450485x_{26} = -33.8045577450485
x27=24.3797797842792x_{27} = 24.3797797842792
x28=−84.0700402024852x_{28} = -84.0700402024852
x29=54.1600365554657x_{29} = 54.1600365554657
x30=−58.9372989737669x_{30} = -58.9372989737669
x31=−76.1511851305942x_{31} = -76.1511851305942
x32=62.0788916273567x_{32} = 62.0788916273567
x33=73.0095924770044x_{33} = 73.0095924770044
x34=104.425519012902x_{34} = 104.425519012902
x35=−14.9550018235098x_{35} = -14.9550018235098
x36=−2.38863120915061x_{36} = -2.38863120915061
x37=−21.2381871306894x_{37} = -21.2381871306894
x38=−19.6025173659779x_{38} = -19.6025173659779
x39=79.292777784184x_{39} = 79.292777784184
x40=−69.8679998234146x_{40} = -69.8679998234146
x41=18.0965944770996x_{41} = 18.0965944770996
x42=−32.1688879803371x_{42} = -32.1688879803371
x43=−82.4343704377738x_{43} = -82.4343704377738
x44=35.3104806339269x_{44} = 35.3104806339269
x45=91.8591483985432x_{45} = 91.8591483985432
x46=47.8768512482861x_{46} = 47.8768512482861
x47=41.5936659411065x_{47} = 41.5936659411065
x48=−52.6541136665873x_{48} = -52.6541136665873
x49=−335.397452489669x_{49} = -335.397452489669
x50=99.7780034704342x_{50} = 99.7780034704342
x51=30.6629650914587x_{51} = 30.6629650914587
x52=66.7264071698248x_{52} = 66.7264071698248
x53=98.1423337057228x_{53} = 98.1423337057228
x54=−27.521372437869x_{54} = -27.521372437869
Signos de extremos en los puntos:
(59248.04881549435, 0.292697608379477)

(60.44322186264526, 0.292697608379478)

(-96.6364108168444, 0.292697608379478)

(3.894554098028981, 0.292697608379478)

(85.5759630913636, 0.292697608379478)

(-40.08774305222813, 0.292697608379478)

(-46.37092835940771, 0.292697608379478)

(22.74411001956774, 0.292697608379478)

(-25.88570267315753, 3.41649528855568)

(16.460924712388152, 0.292697608379477)

(-71.50366958812606, 0.292697608379478)

(-90.35322550966481, 0.292697608379478)

(55.79570632017709, 3.41649528855568)

(74.64526224171586, 3.41649528855568)

(29.02729532674733, 0.292697608379478)

(-8.671816516330193, 0.292697608379478)

(-13.319332058798361, 3.41649528855568)

(68.36207693453626, 3.41649528855568)

(-77.78685489530564, 0.292697608379478)

(-724.9549415348031, 0.292697608379478)

(-63.58481451623505, 3.41649528855568)

(-65.22048428094647, 0.292697608379478)

(11.813409169919986, 3.41649528855568)

(-57.30162920905546, 3.41649528855568)

(10.177739405208568, 0.292697608379478)

(-33.80455774504854, 0.292697608379478)

(24.379779784279158, 3.41649528855568)

(-84.07004020248523, 0.292697608379478)

(54.16003655546567, 0.292697608379478)

(-58.937298973766886, 0.292697608379477)

(-76.15118513059423, 3.41649528855568)

(62.078891627356676, 3.41649528855568)

(73.00959247700443, 0.292697608379477)

(104.42551901290237, 0.292697608379478)

(-14.955001823509779, 0.292697608379478)

(-2.388631209150606, 0.292697608379478)

(-21.238187130689365, 0.292697608379478)

(-19.602517365977945, 3.41649528855568)

(79.29277778418403, 0.292697608379477)

(-69.86799982341464, 3.41649528855568)

(18.096594477099572, 3.41649528855568)

(-32.16888798033712, 3.41649528855568)

(-82.43437043777381, 3.41649528855568)

(35.310480633926915, 0.292697608379478)

(91.8591483985432, 0.292697608379478)

(47.87685124828609, 0.292697608379478)

(41.5936659411065, 0.292697608379477)

(-52.6541136665873, 0.292697608379478)

(-335.3974524896687, 0.292697608379478)

(99.7780034704342, 3.41649528855568)

(30.662965091458744, 3.41649528855568)

(66.72640716982484, 0.292697608379477)

(98.14233370572278, 0.292697608379478)

(-27.52137243786895, 0.292697608379478)


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=59248.0488154944x_{1} = 59248.0488154944
x2=60.4432218626453x_{2} = 60.4432218626453
x3=−96.6364108168444x_{3} = -96.6364108168444
x4=3.89455409802898x_{4} = 3.89455409802898
x5=85.5759630913636x_{5} = 85.5759630913636
x6=−40.0877430522281x_{6} = -40.0877430522281
x7=−46.3709283594077x_{7} = -46.3709283594077
x8=22.7441100195677x_{8} = 22.7441100195677
x9=16.4609247123882x_{9} = 16.4609247123882
x10=−71.5036695881261x_{10} = -71.5036695881261
x11=−90.3532255096648x_{11} = -90.3532255096648
x12=29.0272953267473x_{12} = 29.0272953267473
x13=−8.67181651633019x_{13} = -8.67181651633019
x14=−77.7868548953056x_{14} = -77.7868548953056
x15=−724.954941534803x_{15} = -724.954941534803
x16=−65.2204842809465x_{16} = -65.2204842809465
x17=10.1777394052086x_{17} = 10.1777394052086
x18=−33.8045577450485x_{18} = -33.8045577450485
x19=−84.0700402024852x_{19} = -84.0700402024852
x20=54.1600365554657x_{20} = 54.1600365554657
x21=−58.9372989737669x_{21} = -58.9372989737669
x22=73.0095924770044x_{22} = 73.0095924770044
x23=104.425519012902x_{23} = 104.425519012902
x24=−14.9550018235098x_{24} = -14.9550018235098
x25=−2.38863120915061x_{25} = -2.38863120915061
x26=−21.2381871306894x_{26} = -21.2381871306894
x27=79.292777784184x_{27} = 79.292777784184
x28=35.3104806339269x_{28} = 35.3104806339269
x29=91.8591483985432x_{29} = 91.8591483985432
x30=47.8768512482861x_{30} = 47.8768512482861
x31=41.5936659411065x_{31} = 41.5936659411065
x32=−52.6541136665873x_{32} = -52.6541136665873
x33=−335.397452489669x_{33} = -335.397452489669
x34=66.7264071698248x_{34} = 66.7264071698248
x35=98.1423337057228x_{35} = 98.1423337057228
x36=−27.521372437869x_{36} = -27.521372437869
Puntos máximos de la función:
x36=−25.8857026731575x_{36} = -25.8857026731575
x36=55.7957063201771x_{36} = 55.7957063201771
x36=74.6452622417159x_{36} = 74.6452622417159
x36=−13.3193320587984x_{36} = -13.3193320587984
x36=68.3620769345363x_{36} = 68.3620769345363
x36=−63.584814516235x_{36} = -63.584814516235
x36=11.81340916992x_{36} = 11.81340916992
x36=−57.3016292090555x_{36} = -57.3016292090555
x36=24.3797797842792x_{36} = 24.3797797842792
x36=−76.1511851305942x_{36} = -76.1511851305942
x36=62.0788916273567x_{36} = 62.0788916273567
x36=−19.6025173659779x_{36} = -19.6025173659779
x36=−69.8679998234146x_{36} = -69.8679998234146
x36=18.0965944770996x_{36} = 18.0965944770996
x36=−32.1688879803371x_{36} = -32.1688879803371
x36=−82.4343704377738x_{36} = -82.4343704377738
x36=99.7780034704342x_{36} = 99.7780034704342
x36=30.6629650914587x_{36} = 30.6629650914587
Decrece en los intervalos
[59248.0488154944,∞)\left[59248.0488154944, \infty\right)
Crece en los intervalos
(−∞,−724.954941534803]\left(-\infty, -724.954941534803\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→−∞(sin⁡(x)+1)cot⁡(x)y = \lim_{x \to -\infty} \left(\sin{\left(x \right)} + 1\right)^{\cot{\left(x \right)}}
True

Tomamos como el límite
es decir,
ecuación de la asíntota horizontal a la derecha:
y=lim⁡x→∞(sin⁡(x)+1)cot⁡(x)y = \lim_{x \to \infty} \left(\sin{\left(x \right)} + 1\right)^{\cot{\left(x \right)}}
Asíntotas inclinadas
Se puede hallar la asíntota inclinada calculando el límite de la función (1 + sin(x))^cot(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⁡(x)+1)cot⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\left(\sin{\left(x \right)} + 1\right)^{\cot{\left(x \right)}}}{x}\right)
True

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