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Plotting a function graph/ 2*sin(x)*cos(x)

Graphing y = 2*sin(x)*cos(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*sin(x)*cos(x)
f(x)=2sin⁡(x)cos⁡(x)f{\left(x \right)} = 2 \sin{\left(x \right)} \cos{\left(x \right)} 
f = (2*sin(x))*cos(x)
The graph of the function
02468-8-6-4-2-10102-2
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:
2sin⁡(x)cos⁡(x)=02 \sin{\left(x \right)} \cos{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}
Numerical solution
x1=65.9734457253857x_{1} = 65.9734457253857
x2=−64.4026493985908x_{2} = -64.4026493985908
x3=−23.5619449019235x_{3} = -23.5619449019235
x4=−29.845130209103x_{4} = -29.845130209103
x5=−21.9911485751286x_{5} = -21.9911485751286
x6=21.9911485751286x_{6} = 21.9911485751286
x7=1.5707963267949x_{7} = 1.5707963267949
x8=−15.707963267949x_{8} = -15.707963267949
x9=42.4115008234622x_{9} = 42.4115008234622
x10=4.71238898038469x_{10} = 4.71238898038469
x11=36.1283155162826x_{11} = 36.1283155162826
x12=23.5619449019235x_{12} = 23.5619449019235
x13=6.28318530717959x_{13} = 6.28318530717959
x14=−17.2787595947439x_{14} = -17.2787595947439
x15=26.7035375555132x_{15} = 26.7035375555132
x16=−80.1106126665397x_{16} = -80.1106126665397
x17=86.3937979737193x_{17} = 86.3937979737193
x18=64.4026493985908x_{18} = 64.4026493985908
x19=−83.2522053201295x_{19} = -83.2522053201295
x20=−95.8185759344887x_{20} = -95.8185759344887
x21=28.2743338823081x_{21} = 28.2743338823081
x22=−94.2477796076938x_{22} = -94.2477796076938
x23=−1.5707963267949x_{23} = -1.5707963267949
x24=−48.6946861306418x_{24} = -48.6946861306418
x25=−86.3937979737193x_{25} = -86.3937979737193
x26=73.8274273593601x_{26} = 73.8274273593601
x27=−53.4070751110265x_{27} = -53.4070751110265
x28=−39.2699081698724x_{28} = -39.2699081698724
x29=67.5442420521806x_{29} = 67.5442420521806
x30=70.6858347057703x_{30} = 70.6858347057703
x31=59.6902604182061x_{31} = 59.6902604182061
x32=56.5486677646163x_{32} = 56.5486677646163
x33=−42.4115008234622x_{33} = -42.4115008234622
x34=72.2566310325652x_{34} = 72.2566310325652
x35=−50.2654824574367x_{35} = -50.2654824574367
x36=−51.8362787842316x_{36} = -51.8362787842316
x37=58.1194640914112x_{37} = 58.1194640914112
x38=−73.8274273593601x_{38} = -73.8274273593601
x39=51.8362787842316x_{39} = 51.8362787842316
x40=78.5398163397448x_{40} = 78.5398163397448
x41=−87.9645943005142x_{41} = -87.9645943005142
x42=37.6991118430775x_{42} = 37.6991118430775
x43=−6.28318530717959x_{43} = -6.28318530717959
x44=−37.6991118430775x_{44} = -37.6991118430775
x45=−43.9822971502571x_{45} = -43.9822971502571
x46=29.845130209103x_{46} = 29.845130209103
x47=−45.553093477052x_{47} = -45.553093477052
x48=80.1106126665397x_{48} = 80.1106126665397
x49=−58.1194640914112x_{49} = -58.1194640914112
x50=590.619418874881x_{50} = 590.619418874881
x51=113.097335529233x_{51} = 113.097335529233
x52=−36.1283155162826x_{52} = -36.1283155162826
x53=−72.2566310325652x_{53} = -72.2566310325652
x54=−81.6814089933346x_{54} = -81.6814089933346
x55=−40.8407044966673x_{55} = -40.8407044966673
x56=−65.9734457253857x_{56} = -65.9734457253857
x57=0x_{57} = 0
x58=−28.2743338823081x_{58} = -28.2743338823081
x59=−67.5442420521806x_{59} = -67.5442420521806
x60=31.4159265358979x_{60} = 31.4159265358979
x61=−483.805268652828x_{61} = -483.805268652828
x62=43.9822971502571x_{62} = 43.9822971502571
x63=−119.380520836412x_{63} = -119.380520836412
x64=100.530964914873x_{64} = 100.530964914873
x65=−7.85398163397448x_{65} = -7.85398163397448
x66=48.6946861306418x_{66} = 48.6946861306418
x67=−97.3893722612836x_{67} = -97.3893722612836
x68=89.5353906273091x_{68} = 89.5353906273091
x69=81.6814089933346x_{69} = 81.6814089933346
x70=−75.398223686155x_{70} = -75.398223686155
x71=7.85398163397448x_{71} = 7.85398163397448
x72=−14.1371669411541x_{72} = -14.1371669411541
x73=50.2654824574367x_{73} = 50.2654824574367
x74=94.2477796076938x_{74} = 94.2477796076938
x75=−59.6902604182061x_{75} = -59.6902604182061
x76=12.5663706143592x_{76} = 12.5663706143592
x77=14.1371669411541x_{77} = 14.1371669411541
x78=34.5575191894877x_{78} = 34.5575191894877
x79=20.4203522483337x_{79} = 20.4203522483337
x80=45.553093477052x_{80} = 45.553093477052
x81=95.8185759344887x_{81} = 95.8185759344887
x82=15.707963267949x_{82} = 15.707963267949
x83=−89.5353906273091x_{83} = -89.5353906273091
x84=87.9645943005142x_{84} = 87.9645943005142
x85=92.6769832808989x_{85} = 92.6769832808989
x86=−9.42477796076938x_{86} = -9.42477796076938
x87=−20.4203522483337x_{87} = -20.4203522483337
x88=−31.4159265358979x_{88} = -31.4159265358979
x89=−61.261056745001x_{89} = -61.261056745001
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (2*sin(x))*cos(x).
2sin⁡(0)cos⁡(0)2 \sin{\left(0 \right)} \cos{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
−2sin⁡2(x)+2cos⁡2(x)=0- 2 \sin^{2}{\left(x \right)} + 2 \cos^{2}{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=−π4x_{1} = - \frac{\pi}{4}
x2=π4x_{2} = \frac{\pi}{4}
The values of the extrema at the points:
 -pi      
(----, -1)
  4       

 pi    
(--, 1)
 4


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=−π4x_{1} = - \frac{\pi}{4}
Maxima of the function at points:
x1=π4x_{1} = \frac{\pi}{4}
Decreasing at intervals
[−π4,π4]\left[- \frac{\pi}{4}, \frac{\pi}{4}\right]
Increasing at intervals
(−∞,−π4]∪[π4,∞)\left(-\infty, - \frac{\pi}{4}\right] \cup \left[\frac{\pi}{4}, \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
−8sin⁡(x)cos⁡(x)=0- 8 \sin{\left(x \right)} \cos{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=−π2x_{2} = - \frac{\pi}{2}
x3=π2x_{3} = \frac{\pi}{2}

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
[−π2,0]∪[π2,∞)\left[- \frac{\pi}{2}, 0\right] \cup \left[\frac{\pi}{2}, \infty\right)
Convex at the intervals
(−∞,−π2]∪[0,π2]\left(-\infty, - \frac{\pi}{2}\right] \cup \left[0, \frac{\pi}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(2sin⁡(x)cos⁡(x))=⟨−2,2⟩\lim_{x \to -\infty}\left(2 \sin{\left(x \right)} \cos{\left(x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(2sin⁡(x)cos⁡(x))=⟨−2,2⟩\lim_{x \to \infty}\left(2 \sin{\left(x \right)} \cos{\left(x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (2*sin(x))*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(x)cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(x)cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{2 \sin{\left(x \right)} \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
2sin⁡(x)cos⁡(x)=−2sin⁡(x)cos⁡(x)2 \sin{\left(x \right)} \cos{\left(x \right)} = - 2 \sin{\left(x \right)} \cos{\left(x \right)}
- No
2sin⁡(x)cos⁡(x)=2sin⁡(x)cos⁡(x)2 \sin{\left(x \right)} \cos{\left(x \right)} = 2 \sin{\left(x \right)} \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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