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  • Graphing y =:
  • (x-1)/(x+5)
  • 3x^4-4x^3
  • 3x^2-12x
  • -2x^2+x+1
  • Limit of the function:
  • cos(x)^cot(2*x) cos(x)^cot(2*x)

  • Identical expressions

  • cos(x)^cot(two *x)
  • co sinus of e of (x) to the power of cotangent of (2 multiply by x)
  • co sinus of e of (x) to the power of cotangent of (two multiply by x)
  • cos(x)cot(2*x)
  • cosxcot2*x
  • cos(x)^cot(2x)
  • cos(x)cot(2x)
  • cosxcot2x
  • cosx^cot2x

  • Similar expressions

  • cosx^cot(2*x)
Plotting a function graph/ cos(x)^cot(2*x)

Graphing y = cos(x)^cot(2*x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          cot(2*x)   
f(x) = cos        (x)
f(x)=cos⁡cot⁡(2x)(x)f{\left(x \right)} = \cos^{\cot{\left(2 x \right)}}{\left(x \right)} 
f = cos(x)^cot(2*x)
The graph of the function
02468-8-6-4-2-101005e72
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:
cos⁡cot⁡(2x)(x)=0\cos^{\cot{\left(2 x \right)}}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=−7.80647576424025x_{1} = -7.80647576424025
x2=29.8961611029878x_{2} = 29.8961611029878
x3=80.1625278498171x_{3} = 80.1625278498171
x4=67.5958622856422x_{4} = 67.5958622856422
x5=36.1783104859298x_{5} = 36.1783104859298
x6=−95.7787409002634x_{6} = -95.7787409002634
x7=−58.068874593333x_{7} = -58.068874593333
x8=−14.0845791396593x_{8} = -14.0845791396593
x9=−51.7920506100739x_{9} = -51.7920506100739
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)^cot(2*x).
cos⁡cot⁡(0⋅2)(0)\cos^{\cot{\left(0 \cdot 2 \right)}}{\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
((−2cot⁡2(2x)−2)log⁡(cos⁡(x))−sin⁡(x)cot⁡(2x)cos⁡(x))cos⁡cot⁡(2x)(x)=0\left(\left(- 2 \cot^{2}{\left(2 x \right)} - 2\right) \log{\left(\cos{\left(x \right)} \right)} - \frac{\sin{\left(x \right)} \cot{\left(2 x \right)}}{\cos{\left(x \right)}}\right) \cos^{\cot{\left(2 x \right)}}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−87.4788029226985x_{1} = -87.4788029226985
x2=−12.0805792365435x_{2} = -12.0805792365435
x3=24.6469498509027x_{3} = 24.6469498509027
x4=−18.3637645437231x_{4} = -18.3637645437231
x5=−31.9017179137136x_{5} = -31.9017179137136
x6=75.8840150639707x_{6} = 75.8840150639707
x7=93.7619882298781x_{7} = 93.7619882298781
x8=44.4680885280728x_{8} = 44.4680885280728
x9=56.0628763868006x_{9} = 56.0628763868006
x10=−69.6008297567911x_{10} = -69.6008297567911
x11=−25.618532606534x_{11} = -25.618532606534
x12=−38.1849032208932x_{12} = -38.1849032208932
x13=−82.1672003711503x_{13} = -82.1672003711503
x14=−56.0628763868006x_{14} = -56.0628763868006
x15=62.3460616939802x_{15} = 62.3460616939802
x16=82.1672003711503x_{16} = 82.1672003711503
x17=12.0805792365435x_{17} = 12.0805792365435
x18=18.3637645437231x_{18} = 18.3637645437231
x19=−75.8840150639707x_{19} = -75.8840150639707
x20=100.045173537058x_{20} = 100.045173537058
x21=−63.3176444496115x_{21} = -63.3176444496115
x22=−19.3353472993544x_{22} = -19.3353472993544
x23=−49.779691079621x_{23} = -49.779691079621
x24=−93.7619882298781x_{24} = -93.7619882298781
x25=69.6008297567911x_{25} = 69.6008297567911
x26=38.1849032208932x_{26} = 38.1849032208932
x27=19.3353472993544x_{27} = 19.3353472993544
x28=68.6292470011598x_{28} = 68.6292470011598
x29=−5.79739392936392x_{29} = -5.79739392936392
x30=−68.6292470011598x_{30} = -68.6292470011598
x31=−24.6469498509027x_{31} = -24.6469498509027
x32=49.779691079621x_{32} = 49.779691079621
x33=−43.4965057724414x_{33} = -43.4965057724414
x34=31.9017179137136x_{34} = 31.9017179137136
x35=5.79739392936392x_{35} = 5.79739392936392
x36=25.618532606534x_{36} = 25.618532606534
x37=88.4503856783299x_{37} = 88.4503856783299
x38=63.3176444496115x_{38} = 63.3176444496115
x39=−100.045173537058x_{39} = -100.045173537058
x40=−62.3460616939802x_{40} = -62.3460616939802
The values of the extrema at the points:
(-87.47880292269855, 0.919454783241476)

(-12.08057923654351, 0.919454783241477)

(24.646949850902683, 1.08760106339821)

(-18.363764543723097, 0.919454783241477)

(-31.901717913713597, 1.08760106339821)

(75.8840150639707, 0.919454783241477)

(93.76198822987813, 1.08760106339821)

(44.46808852807277, 0.919454783241477)

(56.06287638680062, 1.08760106339821)

(-69.60082975679111, 1.08760106339821)

(-25.618532606534007, 1.08760106339821)

(-38.18490322089318, 1.08760106339821)

(-82.16720037115029, 1.08760106339821)

(-56.06287638680062, 0.919454783241476)

(62.346061693980204, 1.08760106339821)

(82.16720037115029, 0.919454783241477)

(12.08057923654351, 1.08760106339821)

(18.363764543723097, 1.08760106339821)

(-75.8840150639707, 1.08760106339821)

(100.04517353705772, 1.08760106339821)

(-63.31764444961153, 1.08760106339821)

(-19.33534729935442, 1.08760106339821)

(-49.77969107962103, 0.919454783241476)

(-93.76198822987813, 0.919454783241476)

(69.60082975679111, 0.919454783241476)

(38.18490322089318, 0.919454783241477)

(19.33534729935442, 0.919454783241477)

(68.62924700115978, 1.08760106339821)

(-5.797393929363924, 0.919454783241477)

(-68.62924700115978, 0.919454783241476)

(-24.646949850902683, 0.919454783241477)

(49.77969107962103, 1.08760106339821)

(-43.496505772441445, 0.919454783241476)

(31.901717913713597, 0.919454783241477)

(5.797393929363924, 1.08760106339821)

(25.618532606534007, 0.919454783241477)

(88.45038567832988, 0.919454783241476)

(63.31764444961153, 0.919454783241477)

(-100.04517353705772, 0.919454783241477)

(-62.346061693980204, 0.919454783241476)


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=−87.4788029226985x_{1} = -87.4788029226985
x2=−12.0805792365435x_{2} = -12.0805792365435
x3=−18.3637645437231x_{3} = -18.3637645437231
x4=75.8840150639707x_{4} = 75.8840150639707
x5=44.4680885280728x_{5} = 44.4680885280728
x6=−56.0628763868006x_{6} = -56.0628763868006
x7=82.1672003711503x_{7} = 82.1672003711503
x8=−49.779691079621x_{8} = -49.779691079621
x9=−93.7619882298781x_{9} = -93.7619882298781
x10=69.6008297567911x_{10} = 69.6008297567911
x11=38.1849032208932x_{11} = 38.1849032208932
x12=19.3353472993544x_{12} = 19.3353472993544
x13=−5.79739392936392x_{13} = -5.79739392936392
x14=−68.6292470011598x_{14} = -68.6292470011598
x15=−24.6469498509027x_{15} = -24.6469498509027
x16=−43.4965057724414x_{16} = -43.4965057724414
x17=31.9017179137136x_{17} = 31.9017179137136
x18=25.618532606534x_{18} = 25.618532606534
x19=88.4503856783299x_{19} = 88.4503856783299
x20=63.3176444496115x_{20} = 63.3176444496115
x21=−100.045173537058x_{21} = -100.045173537058
x22=−62.3460616939802x_{22} = -62.3460616939802
Maxima of the function at points:
x22=24.6469498509027x_{22} = 24.6469498509027
x22=−31.9017179137136x_{22} = -31.9017179137136
x22=93.7619882298781x_{22} = 93.7619882298781
x22=56.0628763868006x_{22} = 56.0628763868006
x22=−69.6008297567911x_{22} = -69.6008297567911
x22=−25.618532606534x_{22} = -25.618532606534
x22=−38.1849032208932x_{22} = -38.1849032208932
x22=−82.1672003711503x_{22} = -82.1672003711503
x22=62.3460616939802x_{22} = 62.3460616939802
x22=12.0805792365435x_{22} = 12.0805792365435
x22=18.3637645437231x_{22} = 18.3637645437231
x22=−75.8840150639707x_{22} = -75.8840150639707
x22=100.045173537058x_{22} = 100.045173537058
x22=−63.3176444496115x_{22} = -63.3176444496115
x22=−19.3353472993544x_{22} = -19.3353472993544
x22=68.6292470011598x_{22} = 68.6292470011598
x22=49.779691079621x_{22} = 49.779691079621
x22=5.79739392936392x_{22} = 5.79739392936392
Decreasing at intervals
[88.4503856783299,∞)\left[88.4503856783299, \infty\right)
Increasing at intervals
(−∞,−100.045173537058]\left(-\infty, -100.045173537058\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→−∞cos⁡cot⁡(2x)(x)y = \lim_{x \to -\infty} \cos^{\cot{\left(2 x \right)}}{\left(x \right)}
True

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

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