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  • Graphing y =:
  • x^3-3x+1 x^3-3x+1
  • x+1
  • y=-x^2+12
  • y=3x⁴-6x²+2 y=3x⁴-6x²+2

  • Identical expressions

  • ctg^2x/ one -sinx
  • ctg squared x divide by 1 minus sinus of x
  • ctg squared x divide by one minus sinus of x
  • ctg2x/1-sinx
  • ctg²x/1-sinx
  • ctg to the power of 2x/1-sinx
  • ctg^2x divide by 1-sinx

  • Similar expressions

  • ctg^2x/1+sinx
Plotting a function graph/ ctg^2x/1-sinx

Graphing y = ctg^2x/1-sinx

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          2            
       cot (x)         
f(x) = ------- - sin(x)
          1
f(x)=−sin⁡(x)+cot⁡2(x)1f{\left(x \right)} = - \sin{\left(x \right)} + \frac{\cot^{2}{\left(x \right)}}{1} 
f = -sin(x) + cot(x)^2/1
The graph of the function
02468-8-6-4-2-1010-500000500000
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)+cot⁡2(x)1=0- \sin{\left(x \right)} + \frac{\cot^{2}{\left(x \right)}}{1} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=90.2507193772162x_{1} = 90.2507193772162
x2=−35.4129867663755x_{2} = -35.4129867663755
x3=−30.5604589590101x_{3} = -30.5604589590101
x4=−16.5634308448368x_{4} = -16.5634308448368
x5=−91.9616545309918x_{5} = -91.9616545309918
x6=346.430659471765x_{6} = 346.430659471765
x7=−437.536846425869x_{7} = -437.536846425869
x8=−79.3952839166326x_{8} = -79.3952839166326
x9=14.8524956910612x_{9} = 14.8524956910612
x10=−357.286094932349x_{10} = -357.286094932349
x11=−55.6932001877285x_{11} = -55.6932001877285
x12=44.8377647271449x_{12} = 44.8377647271449
x13=88.820061877402x_{13} = 88.820061877402
x14=−41.6961720735551x_{14} = -41.6961720735551
x15=−457.817059847222x_{15} = -457.817059847222
x16=−99.6754973379856x_{16} = -99.6754973379856
x17=38.5545794199653x_{17} = 38.5545794199653
x18=25.9882088056062x_{18} = 25.9882088056062
x19=32.2713941127857x_{19} = 32.2713941127857
x20=−54.2625426879143x_{20} = -54.2625426879143
x21=−17.994088344651x_{21} = -17.994088344651
x22=−11.7109030374714x_{22} = -11.7109030374714
x23=46.2684222269591x_{23} = 46.2684222269591
x24=−1763.28894624076x_{24} = -1763.28894624076
x25=−98.2448398381714x_{25} = -98.2448398381714
x26=−24.2772736518305x_{26} = -24.2772736518305
x27=115.383460605935x_{27} = 115.383460605935
x28=51.1209500343245x_{28} = 51.1209500343245
x29=71.4011634556774x_{29} = 71.4011634556774
x30=−87.1091267236264x_{30} = -87.1091267236264
x31=69.9705059558633x_{31} = 69.9705059558633
x32=96.5339046843958x_{32} = 96.5339046843958
x33=19.7050234984266x_{33} = 19.7050234984266
x34=52.5516075341387x_{34} = 52.5516075341387
x35=−61.9763854949081x_{35} = -61.9763854949081
x36=27.4188663054203x_{36} = 27.4188663054203
x37=33.7020516125999x_{37} = 33.7020516125999
x38=13.421838191247x_{38} = 13.421838191247
x39=−1059.57219183665x_{39} = -1059.57219183665
x40=−10.2802455376572x_{40} = -10.2802455376572
x41=−3.9970602304776x_{41} = -3.9970602304776
x42=127.949831220294x_{42} = 127.949831220294
x43=77.684348762857x_{43} = 77.684348762857
x44=−73.1120986094531x_{44} = -73.1120986094531
x45=2.28612507670199x_{45} = 2.28612507670199
x46=8.56931038388157x_{46} = 8.56931038388157
x47=−85.6784692238122x_{47} = -85.6784692238122
x48=−93.392312030806x_{48} = -93.392312030806
x49=0.855467576887808x_{49} = 0.855467576887808
x50=58.8347928413183x_{50} = 58.8347928413183
x51=63.6873206486837x_{51} = 63.6873206486837
x52=82.5368765702224x_{52} = 82.5368765702224
x53=−399.837734582792x_{53} = -399.837734582792
x54=−43.1268295733693x_{54} = -43.1268295733693
x55=−68.2595708020876x_{55} = -68.2595708020876
x56=39.9852369197795x_{56} = 39.9852369197795
x57=76.2536912630428x_{57} = 76.2536912630428
x58=−49.4100148805489x_{58} = -49.4100148805489
x59=83.9675340700366x_{59} = 83.9675340700366
x60=−5.42771773029178x_{60} = -5.42771773029178
x61=−80.8259414164468x_{61} = -80.8259414164468
x62=21.1356809982407x_{62} = 21.1356809982407
x63=−60.5457279950939x_{63} = -60.5457279950939
x64=−47.9793573807347x_{64} = -47.9793573807347
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
(−2cot⁡2(x)−2)cot⁡(x)−cos⁡(x)=0\left(- 2 \cot^{2}{\left(x \right)} - 2\right) \cot{\left(x \right)} - \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
The values of the extrema at the points:
 -pi     
(----, 1)
  2      

 pi     
(--, -1)
 2


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=−π2x_{1} = - \frac{\pi}{2}
x2=π2x_{2} = \frac{\pi}{2}
The function has no maxima
Decreasing at intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Increasing at intervals
(−∞,−π2]\left(-\infty, - \frac{\pi}{2}\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2(cot⁡2(x)+1)2+4(cot⁡2(x)+1)cot⁡2(x)+sin⁡(x)=02 \left(\cot^{2}{\left(x \right)} + 1\right)^{2} + 4 \left(\cot^{2}{\left(x \right)} + 1\right) \cot^{2}{\left(x \right)} + \sin{\left(x \right)} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
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)+cot⁡2(x)1)y = \lim_{x \to -\infty}\left(- \sin{\left(x \right)} + \frac{\cot^{2}{\left(x \right)}}{1}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(−sin⁡(x)+cot⁡2(x)1)y = \lim_{x \to \infty}\left(- \sin{\left(x \right)} + \frac{\cot^{2}{\left(x \right)}}{1}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(x)^2/1 - sin(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)+cot⁡2(x)1x)y = x \lim_{x \to -\infty}\left(\frac{- \sin{\left(x \right)} + \frac{\cot^{2}{\left(x \right)}}{1}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(−sin⁡(x)+cot⁡2(x)1x)y = x \lim_{x \to \infty}\left(\frac{- \sin{\left(x \right)} + \frac{\cot^{2}{\left(x \right)}}{1}}{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)+cot⁡2(x)1=sin⁡(x)+cot⁡2(x)1- \sin{\left(x \right)} + \frac{\cot^{2}{\left(x \right)}}{1} = \sin{\left(x \right)} + \frac{\cot^{2}{\left(x \right)}}{1}
- No
−sin⁡(x)+cot⁡2(x)1=−sin⁡(x)−cot⁡2(x)1- \sin{\left(x \right)} + \frac{\cot^{2}{\left(x \right)}}{1} = - \sin{\left(x \right)} - \frac{\cot^{2}{\left(x \right)}}{1}
- No
so, the function
not is
neither even, nor odd
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