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  • Graphing y =:
  • x^4-4x^2+3
  • x^3-5x
  • |x|+2x
  • x^2-x-2

  • Identical expressions

  • sin^ three (ln(tg2x))
  • sinus of cubed (ln(tg2x))
  • sinus of to the power of three (ln(tg2x))
  • sin3(ln(tg2x))
  • sin3lntg2x
  • sin³(ln(tg2x))
  • sin to the power of 3(ln(tg2x))
  • sin^3lntg2x
Plotting a function graph/ sin^3(ln(tg2x))

Graphing y = sin^3(ln(tg2x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          3               
f(x) = sin (log(tan(2*x)))
f(x)=sin⁡3(log⁡(tan⁡(2x)))f{\left(x \right)} = \sin^{3}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)} 
f = sin(log(tan(2*x)))^3
The graph of the function
02468-8-6-4-2-10102-2
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sin(log(tan(2*x)))^3.
sin⁡3(log⁡(tan⁡(0⋅2)))\sin^{3}{\left(\log{\left(\tan{\left(0 \cdot 2 \right)} \right)} \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
3(2tan⁡2(2x)+2)sin⁡2(log⁡(tan⁡(2x)))cos⁡(log⁡(tan⁡(2x)))tan⁡(2x)=0\frac{3 \left(2 \tan^{2}{\left(2 x \right)} + 2\right) \sin^{2}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)} \cos{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}}{\tan{\left(2 x \right)}} = 0
Solve this equation
The roots of this equation
x1=π8x_{1} = \frac{\pi}{8}
x2=atan⁡(e−π2)2x_{2} = \frac{\operatorname{atan}{\left(e^{- \frac{\pi}{2}} \right)}}{2}
x3=atan⁡(eπ2)2x_{3} = \frac{\operatorname{atan}{\left(e^{\frac{\pi}{2}} \right)}}{2}
The values of the extrema at the points:
 pi    
(--, 0)
 8     

     / -pi \     
     | ----|     
     |  2  |     
 atan\e    /     
(-----------, -1)
      2          

     / pi\    
     | --|    
     | 2 |    
 atan\e  /    
(---------, 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=atan⁡(e−π2)2x_{1} = \frac{\operatorname{atan}{\left(e^{- \frac{\pi}{2}} \right)}}{2}
Maxima of the function at points:
x1=atan⁡(eπ2)2x_{1} = \frac{\operatorname{atan}{\left(e^{\frac{\pi}{2}} \right)}}{2}
Decreasing at intervals
[atan⁡(e−π2)2,atan⁡(eπ2)2]\left[\frac{\operatorname{atan}{\left(e^{- \frac{\pi}{2}} \right)}}{2}, \frac{\operatorname{atan}{\left(e^{\frac{\pi}{2}} \right)}}{2}\right]
Increasing at intervals
(−∞,atan⁡(e−π2)2]∪[atan⁡(eπ2)2,∞)\left(-\infty, \frac{\operatorname{atan}{\left(e^{- \frac{\pi}{2}} \right)}}{2}\right] \cup \left[\frac{\operatorname{atan}{\left(e^{\frac{\pi}{2}} \right)}}{2}, \infty\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
12(tan⁡2(2x)+1)(−(tan⁡2(2x)+1)sin⁡2(log⁡(tan⁡(2x)))tan⁡2(2x)−(tan⁡2(2x)+1)sin⁡(log⁡(tan⁡(2x)))cos⁡(log⁡(tan⁡(2x)))tan⁡2(2x)+2(tan⁡2(2x)+1)cos⁡2(log⁡(tan⁡(2x)))tan⁡2(2x)+2sin⁡(log⁡(tan⁡(2x)))cos⁡(log⁡(tan⁡(2x))))sin⁡(log⁡(tan⁡(2x)))=012 \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(- \frac{\left(\tan^{2}{\left(2 x \right)} + 1\right) \sin^{2}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}}{\tan^{2}{\left(2 x \right)}} - \frac{\left(\tan^{2}{\left(2 x \right)} + 1\right) \sin{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)} \cos{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}}{\tan^{2}{\left(2 x \right)}} + \frac{2 \left(\tan^{2}{\left(2 x \right)} + 1\right) \cos^{2}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}}{\tan^{2}{\left(2 x \right)}} + 2 \sin{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)} \cos{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}\right) \sin{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)} = 0
Solve this equation
The roots of this equation
x1=52.2289778659303x_{1} = 52.2289778659303
x2=61.9990653791785x_{2} = 61.9990653791785
x3=1.96349540849362x_{3} = 1.96349540849362
x4=83.990213954307x_{4} = 83.990213954307
x5=−4.31968989868597x_{5} = -4.31968989868597
x6=22.3838476568273x_{6} = 22.3838476568273
x7=−78.1471172580461x_{7} = -78.1471172580461
x8=66.3661448070844x_{8} = 66.3661448070844
x9=94.4116768693319x_{9} = 94.4116768693319
x10=−7.69008437233639x_{10} = -7.69008437233639
x11=80.2745099281778x_{11} = 80.2745099281778
x12=60.0829594999048x_{12} = 60.0829594999048
x13=40.0079168040499x_{13} = 40.0079168040499
x14=82.0741080750334x_{14} = 82.0741080750334
x15=−71.8639319508665x_{15} = -71.8639319508665
x16=−79.717913584841x_{16} = -79.717913584841
x17=76.0197245879144x_{17} = 76.0197245879144
x18=8.24668071567321x_{18} = 8.24668071567321
x19=−51.6723815225935x_{19} = -51.6723815225935
x20=−86.0010988920206x_{20} = -86.0010988920206
x21=−35.7356164345839x_{21} = -35.7356164345839
x22=−93.8550805259951x_{22} = -93.8550805259951
x23=−26.0820366537539x_{23} = -26.0820366537539
x24=−92.0554823791395x_{24} = -92.0554823791395
x25=54.0285760127858x_{25} = 54.0285760127858
x26=98.0108731630429x_{26} = 98.0108731630429
x27=−20.0276531666349x_{27} = -20.0276531666349
x28=36.2922127779207x_{28} = 36.2922127779207
x29=96.2112750161874x_{29} = 96.2112750161874
x30=67.9369411338793x_{30} = 67.9369411338793
x31=10.0462788625287x_{31} = 10.0462788625287
x32=88.3572933822129x_{32} = 88.3572933822129
x33=45.9457925587507x_{33} = 45.9457925587507
x34=23.9546439836222x_{34} = 23.9546439836222
x35=−57.7267650097125x_{35} = -57.7267650097125
x36=−64.009950316892x_{36} = -64.009950316892
x37=−42.0188017417635x_{37} = -42.0188017417635
x38=−40.6768072350292x_{38} = -40.6768072350292
x39=−73.663530097722x_{39} = -73.663530097722
x40=−27.8816348006094x_{40} = -27.8816348006094
x41=14.3010642027922x_{41} = 14.3010642027922
x42=−13.7444678594553x_{42} = -13.7444678594553
x43=30.2378292908018x_{43} = 30.2378292908018
x44=−32.5940237809941x_{44} = -32.5940237809941
x45=−5.89048622548086x_{45} = -5.89048622548086
x46=−21.8272513134905x_{46} = -21.8272513134905
x47=58.2833613530493x_{47} = 58.2833613530493
x48=−29.6812329474649x_{48} = -29.6812329474649
x49=89.9280897090078x_{49} = 89.9280897090078
x50=44.3749962319558x_{50} = 44.3749962319558
x51=−87.8006970388761x_{51} = -87.8006970388761
x52=32.0374274376573x_{52} = 32.0374274376573
x53=74.2201264410589x_{53} = 74.2201264410589
x54=−49.872783375738x_{54} = -49.872783375738
x55=−48.0731852288824x_{55} = -48.0731852288824
x56=16.1006623496477x_{56} = 16.1006623496477
x57=−100.138265833175x_{57} = -100.138265833175
x58=−70.064333804011x_{58} = -70.064333804011
x59=38.0918109247762x_{59} = 38.0918109247762

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
[96.2112750161874,∞)\left[96.2112750161874, \infty\right)
Convex at the intervals
(−∞,−100.138265833175]\left(-\infty, -100.138265833175\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⁡3(log⁡(tan⁡(2x)))y = \lim_{x \to -\infty} \sin^{3}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞sin⁡3(log⁡(tan⁡(2x)))y = \lim_{x \to \infty} \sin^{3}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(log(tan(2*x)))^3, 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⁡3(log⁡(tan⁡(2x)))x)y = x \lim_{x \to -\infty}\left(\frac{\sin^{3}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(sin⁡3(log⁡(tan⁡(2x)))x)y = x \lim_{x \to \infty}\left(\frac{\sin^{3}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \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⁡3(log⁡(tan⁡(2x)))=sin⁡3(log⁡(−tan⁡(2x)))\sin^{3}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)} = \sin^{3}{\left(\log{\left(- \tan{\left(2 x \right)} \right)} \right)}
- No
sin⁡3(log⁡(tan⁡(2x)))=−sin⁡3(log⁡(−tan⁡(2x)))\sin^{3}{\left(\log{\left(\tan{\left(2 x \right)} \right)} \right)} = - \sin^{3}{\left(\log{\left(- \tan{\left(2 x \right)} \right)} \right)}
- No
so, the function
not is
neither even, nor odd
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