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Plotting a function graph/ (1+sin(x))^cot(x)

Graphing y = (1+sin(x))^cot(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [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)
The graph of the function
02468-8-6-4-2-101005
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
(sin⁡(x)+1)cot⁡(x)=0\left(\sin{\left(x \right)} + 1\right)^{\cot{\left(x \right)}} = 0
Solve this equation
Solution is not found,
it's possible that the graph doesn't intersect the axis X
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (1 + sin(x))^cot(x).
(sin⁡(0)+1)cot⁡(0)\left(\sin{\left(0 \right)} + 1\right)^{\cot{\left(0 \right)}}
The result:
f(0)=NaNf{\left(0 \right)} = \text{NaN}
- the solutions of the equation d'not exist
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
((−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
Solve this equation
The roots of this equation
x1=3.89455409802898x_{1} = 3.89455409802898
x2=29.0272953267473x_{2} = 29.0272953267473
x3=74.6452622417159x_{3} = 74.6452622417159
x4=−32.1688879803371x_{4} = -32.1688879803371
x5=−63.584814516235x_{5} = -63.584814516235
x6=−76.1511851305942x_{6} = -76.1511851305942
x7=91.8591483985432x_{7} = 91.8591483985432
x8=−52.6541136665873x_{8} = -52.6541136665873
x9=16.4609247123882x_{9} = 16.4609247123882
x10=−96.6364108168444x_{10} = -96.6364108168444
x11=22.7441100195677x_{11} = 22.7441100195677
x12=−724.954941534803x_{12} = -724.954941534803
x13=−57.3016292090555x_{13} = -57.3016292090555
x14=−46.3709283594077x_{14} = -46.3709283594077
x15=−2.38863120915061x_{15} = -2.38863120915061
x16=−33.8045577450485x_{16} = -33.8045577450485
x17=−19.6025173659779x_{17} = -19.6025173659779
x18=−13.3193320587984x_{18} = -13.3193320587984
x19=98.1423337057228x_{19} = 98.1423337057228
x20=24.3797797842792x_{20} = 24.3797797842792
x21=47.8768512482861x_{21} = 47.8768512482861
x22=62.0788916273567x_{22} = 62.0788916273567
x23=−21.2381871306894x_{23} = -21.2381871306894
x24=−58.9372989737669x_{24} = -58.9372989737669
x25=−77.7868548953056x_{25} = -77.7868548953056
x26=41.5936659411065x_{26} = 41.5936659411065
x27=66.7264071698248x_{27} = 66.7264071698248
x28=68.3620769345363x_{28} = 68.3620769345363
x29=−69.8679998234146x_{29} = -69.8679998234146
x30=−8.67181651633019x_{30} = -8.67181651633019
x31=59248.0488154944x_{31} = 59248.0488154944
x32=−25.8857026731575x_{32} = -25.8857026731575
x33=104.425519012902x_{33} = 104.425519012902
x34=−65.2204842809465x_{34} = -65.2204842809465
x35=−90.3532255096648x_{35} = -90.3532255096648
x36=99.7780034704342x_{36} = 99.7780034704342
x37=−335.397452489669x_{37} = -335.397452489669
x38=73.0095924770044x_{38} = 73.0095924770044
x39=18.0965944770996x_{39} = 18.0965944770996
x40=−14.9550018235098x_{40} = -14.9550018235098
x41=−84.0700402024852x_{41} = -84.0700402024852
x42=−40.0877430522281x_{42} = -40.0877430522281
x43=30.6629650914587x_{43} = 30.6629650914587
x44=−71.5036695881261x_{44} = -71.5036695881261
x45=−82.4343704377738x_{45} = -82.4343704377738
x46=60.4432218626453x_{46} = 60.4432218626453
x47=−27.521372437869x_{47} = -27.521372437869
x48=85.5759630913636x_{48} = 85.5759630913636
x49=10.1777394052086x_{49} = 10.1777394052086
x50=11.81340916992x_{50} = 11.81340916992
x51=79.292777784184x_{51} = 79.292777784184
x52=54.1600365554657x_{52} = 54.1600365554657
x53=55.7957063201771x_{53} = 55.7957063201771
x54=35.3104806339269x_{54} = 35.3104806339269
The values of the extrema at the points:
(3.894554098028981, 0.292697608379478)

(29.02729532674733, 0.292697608379478)

(74.64526224171586, 3.41649528855568)

(-32.16888798033712, 3.41649528855568)

(-63.58481451623505, 3.41649528855568)

(-76.15118513059423, 3.41649528855568)

(91.8591483985432, 0.292697608379478)

(-52.6541136665873, 0.292697608379478)

(16.460924712388152, 0.292697608379477)

(-96.6364108168444, 0.292697608379478)

(22.74411001956774, 0.292697608379478)

(-724.9549415348031, 0.292697608379478)

(-57.30162920905546, 3.41649528855568)

(-46.37092835940771, 0.292697608379478)

(-2.388631209150606, 0.292697608379478)

(-33.80455774504854, 0.292697608379478)

(-19.602517365977945, 3.41649528855568)

(-13.319332058798361, 3.41649528855568)

(98.14233370572278, 0.292697608379478)

(24.379779784279158, 3.41649528855568)

(47.87685124828609, 0.292697608379478)

(62.078891627356676, 3.41649528855568)

(-21.238187130689365, 0.292697608379478)

(-58.937298973766886, 0.292697608379477)

(-77.78685489530564, 0.292697608379478)

(41.5936659411065, 0.292697608379477)

(66.72640716982484, 0.292697608379477)

(68.36207693453626, 3.41649528855568)

(-69.86799982341464, 3.41649528855568)

(-8.671816516330193, 0.292697608379478)

(59248.04881549435, 0.292697608379477)

(-25.88570267315753, 3.41649528855568)

(104.42551901290237, 0.292697608379478)

(-65.22048428094647, 0.292697608379478)

(-90.35322550966481, 0.292697608379478)

(99.7780034704342, 3.41649528855568)

(-335.3974524896687, 0.292697608379478)

(73.00959247700443, 0.292697608379477)

(18.096594477099572, 3.41649528855568)

(-14.955001823509779, 0.292697608379478)

(-84.07004020248523, 0.292697608379478)

(-40.08774305222813, 0.292697608379478)

(30.662965091458744, 3.41649528855568)

(-71.50366958812606, 0.292697608379478)

(-82.43437043777381, 3.41649528855568)

(60.44322186264526, 0.292697608379478)

(-27.52137243786895, 0.292697608379478)

(85.5759630913636, 0.292697608379478)

(10.177739405208568, 0.292697608379478)

(11.813409169919986, 3.41649528855568)

(79.29277778418403, 0.292697608379477)

(54.16003655546567, 0.292697608379478)

(55.79570632017709, 3.41649528855568)

(35.310480633926915, 0.292697608379478)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=3.89455409802898x_{1} = 3.89455409802898
x2=29.0272953267473x_{2} = 29.0272953267473
x3=91.8591483985432x_{3} = 91.8591483985432
x4=−52.6541136665873x_{4} = -52.6541136665873
x5=16.4609247123882x_{5} = 16.4609247123882
x6=−96.6364108168444x_{6} = -96.6364108168444
x7=22.7441100195677x_{7} = 22.7441100195677
x8=−724.954941534803x_{8} = -724.954941534803
x9=−46.3709283594077x_{9} = -46.3709283594077
x10=−2.38863120915061x_{10} = -2.38863120915061
x11=−33.8045577450485x_{11} = -33.8045577450485
x12=98.1423337057228x_{12} = 98.1423337057228
x13=47.8768512482861x_{13} = 47.8768512482861
x14=−21.2381871306894x_{14} = -21.2381871306894
x15=−58.9372989737669x_{15} = -58.9372989737669
x16=−77.7868548953056x_{16} = -77.7868548953056
x17=41.5936659411065x_{17} = 41.5936659411065
x18=66.7264071698248x_{18} = 66.7264071698248
x19=−8.67181651633019x_{19} = -8.67181651633019
x20=59248.0488154944x_{20} = 59248.0488154944
x21=104.425519012902x_{21} = 104.425519012902
x22=−65.2204842809465x_{22} = -65.2204842809465
x23=−90.3532255096648x_{23} = -90.3532255096648
x24=−335.397452489669x_{24} = -335.397452489669
x25=73.0095924770044x_{25} = 73.0095924770044
x26=−14.9550018235098x_{26} = -14.9550018235098
x27=−84.0700402024852x_{27} = -84.0700402024852
x28=−40.0877430522281x_{28} = -40.0877430522281
x29=−71.5036695881261x_{29} = -71.5036695881261
x30=60.4432218626453x_{30} = 60.4432218626453
x31=−27.521372437869x_{31} = -27.521372437869
x32=85.5759630913636x_{32} = 85.5759630913636
x33=10.1777394052086x_{33} = 10.1777394052086
x34=79.292777784184x_{34} = 79.292777784184
x35=54.1600365554657x_{35} = 54.1600365554657
x36=35.3104806339269x_{36} = 35.3104806339269
Maxima of the function at points:
x36=74.6452622417159x_{36} = 74.6452622417159
x36=−32.1688879803371x_{36} = -32.1688879803371
x36=−63.584814516235x_{36} = -63.584814516235
x36=−76.1511851305942x_{36} = -76.1511851305942
x36=−57.3016292090555x_{36} = -57.3016292090555
x36=−19.6025173659779x_{36} = -19.6025173659779
x36=−13.3193320587984x_{36} = -13.3193320587984
x36=24.3797797842792x_{36} = 24.3797797842792
x36=62.0788916273567x_{36} = 62.0788916273567
x36=68.3620769345363x_{36} = 68.3620769345363
x36=−69.8679998234146x_{36} = -69.8679998234146
x36=−25.8857026731575x_{36} = -25.8857026731575
x36=99.7780034704342x_{36} = 99.7780034704342
x36=18.0965944770996x_{36} = 18.0965944770996
x36=30.6629650914587x_{36} = 30.6629650914587
x36=−82.4343704377738x_{36} = -82.4343704377738
x36=11.81340916992x_{36} = 11.81340916992
x36=55.7957063201771x_{36} = 55.7957063201771
Decreasing at intervals
[59248.0488154944,∞)\left[59248.0488154944, \infty\right)
Increasing at intervals
(−∞,−724.954941534803]\left(-\infty, -724.954941534803\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
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

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(sin⁡(x)+1)cot⁡(x)y = \lim_{x \to \infty} \left(\sin{\left(x \right)} + 1\right)^{\cot{\left(x \right)}}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (1 + sin(x))^cot(x), divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
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

Let's take the limit
so,
inclined asymptote equation on the right:
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)
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
(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
so, the function
not is
neither even, nor odd
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