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

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

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cos(x)*sin(x*2)
f(x)=sin⁡(2x)cos⁡(x)f{\left(x \right)} = \sin{\left(2 x \right)} \cos{\left(x \right)} 
f = sin(2*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:
sin⁡(2x)cos⁡(x)=0\sin{\left(2 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=12.5663706143592x_{1} = 12.5663706143592
x2=4.71238883532779x_{2} = 4.71238883532779
x3=23.5619450555027x_{3} = 23.5619450555027
x4=−92.6769836764771x_{4} = -92.6769836764771
x5=78.5398163397448x_{5} = 78.5398163397448
x6=−70.6858349962623x_{6} = -70.6858349962623
x7=−39.2699083096144x_{7} = -39.2699083096144
x8=−65.9734457253857x_{8} = -65.9734457253857
x9=−73.8274272804402x_{9} = -73.8274272804402
x10=−15.707963267949x_{10} = -15.707963267949
x11=50.2654824574367x_{11} = 50.2654824574367
x12=81.6814089933346x_{12} = 81.6814089933346
x13=−61.261056881309x_{13} = -61.261056881309
x14=−75.398223686155x_{14} = -75.398223686155
x15=51.8362788934209x_{15} = 51.8362788934209
x16=7.85398173541774x_{16} = 7.85398173541774
x17=80.1106131546315x_{17} = 80.1106131546315
x18=56.5486677646163x_{18} = 56.5486677646163
x19=−23.561945003804x_{19} = -23.561945003804
x20=−64.4026492408158x_{20} = -64.4026492408158
x21=−58.1194640027517x_{21} = -58.1194640027517
x22=48.6946859820148x_{22} = 48.6946859820148
x23=95.8185760508519x_{23} = 95.8185760508519
x24=15.707963267949x_{24} = 15.707963267949
x25=67.5442422018325x_{25} = 67.5442422018325
x26=−45.5530935824522x_{26} = -45.5530935824522
x27=−95.8185758682892x_{27} = -95.8185758682892
x28=−59.6902604182061x_{28} = -59.6902604182061
x29=−83.2522054524035x_{29} = -83.2522054524035
x30=64.4026493118058x_{30} = 64.4026493118058
x31=21.9911485751286x_{31} = 21.9911485751286
x32=6.28318530717959x_{32} = 6.28318530717959
x33=−87.9645943005142x_{33} = -87.9645943005142
x34=9.42477796076938x_{34} = 9.42477796076938
x35=−28.2743338823081x_{35} = -28.2743338823081
x36=−86.3937978155375x_{36} = -86.3937978155375
x37=−9.42477796076938x_{37} = -9.42477796076938
x38=86.3937978909611x_{38} = 86.3937978909611
x39=59.6902604182061x_{39} = 59.6902604182061
x40=−80.1106125824842x_{40} = -80.1106125824842
x41=14.1371670924752x_{41} = 14.1371670924752
x42=28.2743338823081x_{42} = 28.2743338823081
x43=73.8274274722061x_{43} = 73.8274274722061
x44=36.1283160593477x_{44} = 36.1283160593477
x45=−67.5442421609972x_{45} = -67.5442421609972
x46=94.2477796076938x_{46} = 94.2477796076938
x47=−89.5353907394375x_{47} = -89.5353907394375
x48=−72.2566310325652x_{48} = -72.2566310325652
x49=−97.3893722612836x_{49} = -97.3893722612836
x50=37.6991118430775x_{50} = 37.6991118430775
x51=89.5353907744432x_{51} = 89.5353907744432
x52=−14.13716684381x_{52} = -14.13716684381
x53=−50.2654824574367x_{53} = -50.2654824574367
x54=−94.2477796076938x_{54} = -94.2477796076938
x55=−7.85398150264842x_{55} = -7.85398150264842
x56=−42.4115006663339x_{56} = -42.4115006663339
x57=−36.128315423197x_{57} = -36.128315423197
x58=26.7035374084741x_{58} = 26.7035374084741
x59=−37.6991118430775x_{59} = -37.6991118430775
x60=−17.278759737384x_{60} = -17.278759737384
x61=−1.57079642505341x_{61} = -1.57079642505341
x62=70.6858345559153x_{62} = 70.6858345559153
x63=−95.818575585294x_{63} = -95.818575585294
x64=42.4115007327518x_{64} = 42.4115007327518
x65=92.6769831301454x_{65} = 92.6769831301454
x66=20.4203521537986x_{66} = 20.4203521537986
x67=−81.6814089933346x_{67} = -81.6814089933346
x68=43.9822971502571x_{68} = 43.9822971502571
x69=45.5530936288414x_{69} = 45.5530936288414
x70=−31.4159265358979x_{70} = -31.4159265358979
x71=0x_{71} = 0
x72=−21.9911485751286x_{72} = -21.9911485751286
x73=100.530964914873x_{73} = 100.530964914873
x74=34.5575191894877x_{74} = 34.5575191894877
x75=−29.8451300981866x_{75} = -29.8451300981866
x76=65.9734457253857x_{76} = 65.9734457253857
x77=1.57079648184495x_{77} = 1.57079648184495
x78=−20.4203520921076x_{78} = -20.4203520921076
x79=7.85398164444075x_{79} = 7.85398164444075
x80=−53.4070751110265x_{80} = -53.4070751110265
x81=−6.28318530717959x_{81} = -6.28318530717959
x82=−43.9822971502571x_{82} = -43.9822971502571
x83=29.8451303144929x_{83} = 29.8451303144929
x84=−51.8362786906154x_{84} = -51.8362786906154
x85=72.2566310325652x_{85} = 72.2566310325652
x86=87.9645943005142x_{86} = 87.9645943005142
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x)*sin(x*2).
sin⁡(0⋅2)cos⁡(0)\sin{\left(0 \cdot 2 \right)} \cos{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
−(4sin⁡(x)cos⁡(2x)+5sin⁡(2x)cos⁡(x))=0- (4 \sin{\left(x \right)} \cos{\left(2 x \right)} + 5 \sin{\left(2 x \right)} \cos{\left(x \right)}) = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
x3=i(log⁡(9)−log⁡(−5−214i))2x_{3} = \frac{i \left(\log{\left(9 \right)} - \log{\left(-5 - 2 \sqrt{14} i \right)}\right)}{2}
x4=i(log⁡(9)−log⁡(−5+214i))2x_{4} = \frac{i \left(\log{\left(9 \right)} - \log{\left(-5 + 2 \sqrt{14} i \right)}\right)}{2}
x5=−ilog⁡(−−5−214i3)x_{5} = - i \log{\left(- \frac{\sqrt{-5 - 2 \sqrt{14} i}}{3} \right)}
x6=−ilog⁡(−−5+214i3)x_{6} = - i \log{\left(- \frac{\sqrt{-5 + 2 \sqrt{14} i}}{3} \right)}

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
[π,∞)\left[\pi, \infty\right)
Convex at the intervals
(−∞,−π2+atan⁡(2145)2]\left(-\infty, - \frac{\pi}{2} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{14}}{5} \right)}}{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⁡(2x)cos⁡(x))=⟨−1,1⟩\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞(sin⁡(2x)cos⁡(x))=⟨−1,1⟩\lim_{x \to \infty}\left(\sin{\left(2 x \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x)*sin(x*2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(sin⁡(2x)cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{\sin{\left(2 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→∞(sin⁡(2x)cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{\sin{\left(2 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⁡(2x)cos⁡(x)=−sin⁡(2x)cos⁡(x)\sin{\left(2 x \right)} \cos{\left(x \right)} = - \sin{\left(2 x \right)} \cos{\left(x \right)}
- No
sin⁡(2x)cos⁡(x)=sin⁡(2x)cos⁡(x)\sin{\left(2 x \right)} \cos{\left(x \right)} = \sin{\left(2 x \right)} \cos{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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