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  • Graphing y =:
  • x^3-2*x^2+x+3 x^3-2*x^2+x+3
  • (x^3+4)/(x^2)
  • x^3+3x+1
  • -x^3+4x^2-4x

  • Identical expressions

  • tg(two x)^(ctg(x/2))
  • tg(2x) to the power of (ctg(x divide by 2))
  • tg(two x) to the power of (ctg(x divide by 2))
  • tg(2x)(ctg(x/2))
  • tg2xctgx/2
  • tg2x^ctgx/2
  • tg(2x)^(ctg(x divide by 2))
Plotting a function graph/ tg(2x)^(ctg(x/2))

Graphing y = tg(2x)^(ctg(x/2))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                    /x\
                 cot|-|
                    \2/
f(x) = (tan(2*x))
f(x)=tan⁡cot⁡(x2)(2x)f{\left(x \right)} = \tan^{\cot{\left(\frac{x}{2} \right)}}{\left(2 x \right)} 
f = tan(2*x)^cot(x/2)
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(2*x)^cot(x/2).
tan⁡cot⁡(02)(0⋅2)\tan^{\cot{\left(\frac{0}{2} \right)}}{\left(0 \cdot 2 \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
((2tan⁡2(2x)+2)cot⁡(x2)tan⁡(2x)+(−cot⁡2(x2)2−12)log⁡(tan⁡(2x)))tan⁡cot⁡(x2)(2x)=0\left(\frac{\left(2 \tan^{2}{\left(2 x \right)} + 2\right) \cot{\left(\frac{x}{2} \right)}}{\tan{\left(2 x \right)}} + \left(- \frac{\cot^{2}{\left(\frac{x}{2} \right)}}{2} - \frac{1}{2}\right) \log{\left(\tan{\left(2 x \right)} \right)}\right) \tan^{\cot{\left(\frac{x}{2} \right)}}{\left(2 x \right)} = 0
Solve this equation
The roots of this equation
x1=−13.3517687777664x_{1} = -13.3517687777664
x2=−19.6349540849381x_{2} = -19.6349540849381
x3=94.25x_{3} = 94.25
x4=−57.3340659280251x_{4} = -57.3340659280251
x5=24.3473430653192x_{5} = 24.3473430653192
x6=68.3296402155766x_{6} = 68.3296402155766
x7=3.30656871038136x_{7} = 3.30656871038136
x8=44x_{8} = 44
x9=−28.1093578255166x_{9} = -28.1093578255166
x10=74.6128255227505x_{10} = 74.6128255227505
x11=34.7224952462793x_{11} = 34.7224952462793
x12=41.0056805534589x_{12} = 41.0056805534589
x13=−53.2420990542349x_{13} = -53.2420990542349
x14=30.6305283724922x_{14} = 30.6305283724922
x15=−78.3748402829533x_{15} = -78.3748402829533
x16=9.58975401756095x_{16} = 9.58975401756095
x17=66.1384217821772x_{17} = 66.1384217821772
x18=84.987977703716x_{18} = 84.987977703716
x19=−84.6580255901329x_{19} = -84.6580255901329
x20=−9.25980190397781x_{20} = -9.25980190397781
x21=22.1561246319201x_{21} = 22.1561246319201
x22=78.7047923965364x_{22} = 78.7047923965364
x23=59.8552364749976x_{23} = 59.8552364749976
x24=88x_{24} = 88
x25=−72.0916549757737x_{25} = -72.0916549757737
x26=−97.224396204492x_{26} = -97.224396204492
x27=−63.6172512351954x_{27} = -63.6172512351954
x28=97.5543483180752x_{28} = 97.5543483180752
x29=15.8729393247405x_{29} = 15.8729393247405
x30=−21.826172518337x_{30} = -21.826172518337
x31=−65.8084696685941x_{31} = -65.8084696685941
x32=−34.3925431326962x_{32} = -34.3925431326962
x33=91.2711630108956x_{33} = 91.2711630108956
The values of the extrema at the points:
(-13.351768777766381, 1.39603369953625e-26)

(-19.634954084938123, 2.73793446490225e-28)

(94.25, 9.38077002596721e-2120)

(-57.33406592802512, 2.02798575047659e-26)

(24.347343065319205, 2.0331559982005e-28)

(68.32964021557656, 1.38119639835943e-28)

(3.3065687103813612, 1.09263531182907)

(44, 1.27246687910515e-164)

(-28.10935782551657, 1.09263531182907)

(74.61282552275046, 6.55133765320612e-27)

(34.722495246279294, 1.09263531182907)

(41.00568055345888, 1.09263531182907)

(-53.24209905423492, 1.09263531182907)

(30.630528372492172, 9.47410787394105e-27)

(-78.37484028295326, 1.09263531182907)

(9.589754017560947, 1.09263531182907)

(66.13842178217723, 1.09263531182907)

(84.98797770371598, 1.09263531182907)

(-84.65802559013285, 1.09263531182907)

(-9.259801903977811, 1.09263531182907)

(22.15612463192012, 1.09263531182907)

(78.7047923965364, 1.09263531182907)

(59.85523647499764, 1.09263531182907)

(88, 1.23699807188943e-65)

(-72.09165497577368, 1.09263531182907)

(-97.22439620449202, 1.09263531182907)

(-63.61725123519544, 3.52560915332702e-28)

(97.55434831807516, 1.09263531182907)

(15.872939324740534, 1.09263531182907)

(-21.826172518336985, 1.09263531182907)

(-65.80846966859409, 1.09263531182907)

(-34.392543132696154, 1.09263531182907)

(91.27116301089558, 1.09263531182907)


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=−13.3517687777664x_{1} = -13.3517687777664
x2=−19.6349540849381x_{2} = -19.6349540849381
x3=−57.3340659280251x_{3} = -57.3340659280251
x4=24.3473430653192x_{4} = 24.3473430653192
x5=68.3296402155766x_{5} = 68.3296402155766
x6=74.6128255227505x_{6} = 74.6128255227505
x7=30.6305283724922x_{7} = 30.6305283724922
x8=−63.6172512351954x_{8} = -63.6172512351954
Maxima of the function at points:
x8=3.30656871038136x_{8} = 3.30656871038136
x8=−28.1093578255166x_{8} = -28.1093578255166
x8=34.7224952462793x_{8} = 34.7224952462793
x8=41.0056805534589x_{8} = 41.0056805534589
x8=−53.2420990542349x_{8} = -53.2420990542349
x8=−78.3748402829533x_{8} = -78.3748402829533
x8=9.58975401756095x_{8} = 9.58975401756095
x8=66.1384217821772x_{8} = 66.1384217821772
x8=84.987977703716x_{8} = 84.987977703716
x8=−84.6580255901329x_{8} = -84.6580255901329
x8=−9.25980190397781x_{8} = -9.25980190397781
x8=22.1561246319201x_{8} = 22.1561246319201
x8=78.7047923965364x_{8} = 78.7047923965364
x8=59.8552364749976x_{8} = 59.8552364749976
x8=−72.0916549757737x_{8} = -72.0916549757737
x8=−97.224396204492x_{8} = -97.224396204492
x8=97.5543483180752x_{8} = 97.5543483180752
x8=15.8729393247405x_{8} = 15.8729393247405
x8=−21.826172518337x_{8} = -21.826172518337
x8=−65.8084696685941x_{8} = -65.8084696685941
x8=−34.3925431326962x_{8} = -34.3925431326962
x8=91.2711630108956x_{8} = 91.2711630108956
Decreasing at intervals
[74.6128255227505,∞)\left[74.6128255227505, \infty\right)
Increasing at intervals
(−∞,−63.6172512351954]\left(-\infty, -63.6172512351954\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→−∞tan⁡cot⁡(x2)(2x)y = \lim_{x \to -\infty} \tan^{\cot{\left(\frac{x}{2} \right)}}{\left(2 x \right)}
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞tan⁡cot⁡(x2)(2x)y = \lim_{x \to \infty} \tan^{\cot{\left(\frac{x}{2} \right)}}{\left(2 x \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(2*x)^cot(x/2), 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→−∞(tan⁡cot⁡(x2)(2x)x)y = x \lim_{x \to -\infty}\left(\frac{\tan^{\cot{\left(\frac{x}{2} \right)}}{\left(2 x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡cot⁡(x2)(2x)x)y = x \lim_{x \to \infty}\left(\frac{\tan^{\cot{\left(\frac{x}{2} \right)}}{\left(2 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:
tan⁡cot⁡(x2)(2x)=(−tan⁡(2x))−cot⁡(x2)\tan^{\cot{\left(\frac{x}{2} \right)}}{\left(2 x \right)} = \left(- \tan{\left(2 x \right)}\right)^{- \cot{\left(\frac{x}{2} \right)}}
- No
tan⁡cot⁡(x2)(2x)=−(−tan⁡(2x))−cot⁡(x2)\tan^{\cot{\left(\frac{x}{2} \right)}}{\left(2 x \right)} = - \left(- \tan{\left(2 x \right)}\right)^{- \cot{\left(\frac{x}{2} \right)}}
- No
so, the function
not is
neither even, nor odd
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