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

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 2*cos(x)*sin(x)
f(x)=sin⁡(x)2cos⁡(x)f{\left(x \right)} = \sin{\left(x \right)} 2 \cos{\left(x \right)} 
f = sin(x)*(2*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:
sin⁡(x)2cos⁡(x)=0\sin{\left(x \right)} 2 \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=73.8274273593601x_{1} = 73.8274273593601
x2=34.5575191894877x_{2} = 34.5575191894877
x3=14.1371669411541x_{3} = 14.1371669411541
x4=58.1194640914112x_{4} = 58.1194640914112
x5=92.6769832808989x_{5} = 92.6769832808989
x6=−43.9822971502571x_{6} = -43.9822971502571
x7=28.2743338823081x_{7} = 28.2743338823081
x8=36.1283155162826x_{8} = 36.1283155162826
x9=−7.85398163397448x_{9} = -7.85398163397448
x10=−58.1194640914112x_{10} = -58.1194640914112
x11=−48.6946861306418x_{11} = -48.6946861306418
x12=86.3937979737193x_{12} = 86.3937979737193
x13=51.8362787842316x_{13} = 51.8362787842316
x14=−42.4115008234622x_{14} = -42.4115008234622
x15=−89.5353906273091x_{15} = -89.5353906273091
x16=59.6902604182061x_{16} = 59.6902604182061
x17=−40.8407044966673x_{17} = -40.8407044966673
x18=1.5707963267949x_{18} = 1.5707963267949
x19=56.5486677646163x_{19} = 56.5486677646163
x20=100.530964914873x_{20} = 100.530964914873
x21=15.707963267949x_{21} = 15.707963267949
x22=−15.707963267949x_{22} = -15.707963267949
x23=−37.6991118430775x_{23} = -37.6991118430775
x24=−72.2566310325652x_{24} = -72.2566310325652
x25=65.9734457253857x_{25} = 65.9734457253857
x26=−36.1283155162826x_{26} = -36.1283155162826
x27=12.5663706143592x_{27} = 12.5663706143592
x28=21.9911485751286x_{28} = 21.9911485751286
x29=−6.28318530717959x_{29} = -6.28318530717959
x30=−65.9734457253857x_{30} = -65.9734457253857
x31=81.6814089933346x_{31} = 81.6814089933346
x32=−14.1371669411541x_{32} = -14.1371669411541
x33=−64.4026493985908x_{33} = -64.4026493985908
x34=80.1106126665397x_{34} = 80.1106126665397
x35=95.8185759344887x_{35} = 95.8185759344887
x36=78.5398163397448x_{36} = 78.5398163397448
x37=45.553093477052x_{37} = 45.553093477052
x38=−17.2787595947439x_{38} = -17.2787595947439
x39=4.71238898038469x_{39} = 4.71238898038469
x40=20.4203522483337x_{40} = 20.4203522483337
x41=−23.5619449019235x_{41} = -23.5619449019235
x42=−51.8362787842316x_{42} = -51.8362787842316
x43=−29.845130209103x_{43} = -29.845130209103
x44=7.85398163397448x_{44} = 7.85398163397448
x45=−95.8185759344887x_{45} = -95.8185759344887
x46=−97.3893722612836x_{46} = -97.3893722612836
x47=−39.2699081698724x_{47} = -39.2699081698724
x48=37.6991118430775x_{48} = 37.6991118430775
x49=−21.9911485751286x_{49} = -21.9911485751286
x50=−50.2654824574367x_{50} = -50.2654824574367
x51=−94.2477796076938x_{51} = -94.2477796076938
x52=−53.4070751110265x_{52} = -53.4070751110265
x53=23.5619449019235x_{53} = 23.5619449019235
x54=−86.3937979737193x_{54} = -86.3937979737193
x55=−61.261056745001x_{55} = -61.261056745001
x56=−67.5442420521806x_{56} = -67.5442420521806
x57=−59.6902604182061x_{57} = -59.6902604182061
x58=113.097335529233x_{58} = 113.097335529233
x59=−45.553093477052x_{59} = -45.553093477052
x60=−87.9645943005142x_{60} = -87.9645943005142
x61=−83.2522053201295x_{61} = -83.2522053201295
x62=−28.2743338823081x_{62} = -28.2743338823081
x63=42.4115008234622x_{63} = 42.4115008234622
x64=67.5442420521806x_{64} = 67.5442420521806
x65=6.28318530717959x_{65} = 6.28318530717959
x66=−73.8274273593601x_{66} = -73.8274273593601
x67=0x_{67} = 0
x68=−1.5707963267949x_{68} = -1.5707963267949
x69=87.9645943005142x_{69} = 87.9645943005142
x70=43.9822971502571x_{70} = 43.9822971502571
x71=−483.805268652828x_{71} = -483.805268652828
x72=70.6858347057703x_{72} = 70.6858347057703
x73=−75.398223686155x_{73} = -75.398223686155
x74=−9.42477796076938x_{74} = -9.42477796076938
x75=−119.380520836412x_{75} = -119.380520836412
x76=−31.4159265358979x_{76} = -31.4159265358979
x77=31.4159265358979x_{77} = 31.4159265358979
x78=72.2566310325652x_{78} = 72.2566310325652
x79=26.7035375555132x_{79} = 26.7035375555132
x80=94.2477796076938x_{80} = 94.2477796076938
x81=590.619418874881x_{81} = 590.619418874881
x82=48.6946861306418x_{82} = 48.6946861306418
x83=−81.6814089933346x_{83} = -81.6814089933346
x84=−20.4203522483337x_{84} = -20.4203522483337
x85=−80.1106126665397x_{85} = -80.1106126665397
x86=64.4026493985908x_{86} = 64.4026493985908
x87=29.845130209103x_{87} = 29.845130209103
x88=89.5353906273091x_{88} = 89.5353906273091
x89=50.2654824574367x_{89} = 50.2654824574367
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (2*cos(x))*sin(x).
sin⁡(0)2cos⁡(0)\sin{\left(0 \right)} 2 \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→−∞(sin⁡(x)2cos⁡(x))=⟨−2,2⟩\lim_{x \to -\infty}\left(\sin{\left(x \right)} 2 \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→∞(sin⁡(x)2cos⁡(x))=⟨−2,2⟩\lim_{x \to \infty}\left(\sin{\left(x \right)} 2 \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*cos(x))*sin(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:
sin⁡(x)2cos⁡(x)=−2sin⁡(x)cos⁡(x)\sin{\left(x \right)} 2 \cos{\left(x \right)} = - 2 \sin{\left(x \right)} \cos{\left(x \right)}
- No
sin⁡(x)2cos⁡(x)=2sin⁡(x)cos⁡(x)\sin{\left(x \right)} 2 \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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