Mister Exam
Graphing y = cos(x)*sin(x*2)
The solution
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f(x) = cos(x)*sin(x*2)
$$f{\left(x \right)} = \sin{\left(2 x \right)} \cos{\left(x \right)}$$
f = sin(2*x)*cos(x)
The graph of the function
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:
$$\sin{\left(2 x \right)} \cos{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 12.5663706143592$$
$$x_{2} = 4.71238883532779$$
$$x_{3} = 23.5619450555027$$
$$x_{4} = -92.6769836764771$$
$$x_{5} = 78.5398163397448$$
$$x_{6} = -70.6858349962623$$
$$x_{7} = -39.2699083096144$$
$$x_{8} = -65.9734457253857$$
$$x_{9} = -73.8274272804402$$
$$x_{10} = -15.707963267949$$
$$x_{11} = 50.2654824574367$$
$$x_{12} = 81.6814089933346$$
$$x_{13} = -61.261056881309$$
$$x_{14} = -75.398223686155$$
$$x_{15} = 51.8362788934209$$
$$x_{16} = 7.85398173541774$$
$$x_{17} = 80.1106131546315$$
$$x_{18} = 56.5486677646163$$
$$x_{19} = -23.561945003804$$
$$x_{20} = -64.4026492408158$$
$$x_{21} = -58.1194640027517$$
$$x_{22} = 48.6946859820148$$
$$x_{23} = 95.8185760508519$$
$$x_{24} = 15.707963267949$$
$$x_{25} = 67.5442422018325$$
$$x_{26} = -45.5530935824522$$
$$x_{27} = -95.8185758682892$$
$$x_{28} = -59.6902604182061$$
$$x_{29} = -83.2522054524035$$
$$x_{30} = 64.4026493118058$$
$$x_{31} = 21.9911485751286$$
$$x_{32} = 6.28318530717959$$
$$x_{33} = -87.9645943005142$$
$$x_{34} = 9.42477796076938$$
$$x_{35} = -28.2743338823081$$
$$x_{36} = -86.3937978155375$$
$$x_{37} = -9.42477796076938$$
$$x_{38} = 86.3937978909611$$
$$x_{39} = 59.6902604182061$$
$$x_{40} = -80.1106125824842$$
$$x_{41} = 14.1371670924752$$
$$x_{42} = 28.2743338823081$$
$$x_{43} = 73.8274274722061$$
$$x_{44} = 36.1283160593477$$
$$x_{45} = -67.5442421609972$$
$$x_{46} = 94.2477796076938$$
$$x_{47} = -89.5353907394375$$
$$x_{48} = -72.2566310325652$$
$$x_{49} = -97.3893722612836$$
$$x_{50} = 37.6991118430775$$
$$x_{51} = 89.5353907744432$$
$$x_{52} = -14.13716684381$$
$$x_{53} = -50.2654824574367$$
$$x_{54} = -94.2477796076938$$
$$x_{55} = -7.85398150264842$$
$$x_{56} = -42.4115006663339$$
$$x_{57} = -36.128315423197$$
$$x_{58} = 26.7035374084741$$
$$x_{59} = -37.6991118430775$$
$$x_{60} = -17.278759737384$$
$$x_{61} = -1.57079642505341$$
$$x_{62} = 70.6858345559153$$
$$x_{63} = -95.818575585294$$
$$x_{64} = 42.4115007327518$$
$$x_{65} = 92.6769831301454$$
$$x_{66} = 20.4203521537986$$
$$x_{67} = -81.6814089933346$$
$$x_{68} = 43.9822971502571$$
$$x_{69} = 45.5530936288414$$
$$x_{70} = -31.4159265358979$$
$$x_{71} = 0$$
$$x_{72} = -21.9911485751286$$
$$x_{73} = 100.530964914873$$
$$x_{74} = 34.5575191894877$$
$$x_{75} = -29.8451300981866$$
$$x_{76} = 65.9734457253857$$
$$x_{77} = 1.57079648184495$$
$$x_{78} = -20.4203520921076$$
$$x_{79} = 7.85398164444075$$
$$x_{80} = -53.4070751110265$$
$$x_{81} = -6.28318530717959$$
$$x_{82} = -43.9822971502571$$
$$x_{83} = 29.8451303144929$$
$$x_{84} = -51.8362786906154$$
$$x_{85} = 72.2566310325652$$
$$x_{86} = 87.9645943005142$$
so we need to solve the equation:
$$\sin{\left(2 x \right)} \cos{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = - \frac{\pi}{2}$$
$$x_{3} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 12.5663706143592$$
$$x_{2} = 4.71238883532779$$
$$x_{3} = 23.5619450555027$$
$$x_{4} = -92.6769836764771$$
$$x_{5} = 78.5398163397448$$
$$x_{6} = -70.6858349962623$$
$$x_{7} = -39.2699083096144$$
$$x_{8} = -65.9734457253857$$
$$x_{9} = -73.8274272804402$$
$$x_{10} = -15.707963267949$$
$$x_{11} = 50.2654824574367$$
$$x_{12} = 81.6814089933346$$
$$x_{13} = -61.261056881309$$
$$x_{14} = -75.398223686155$$
$$x_{15} = 51.8362788934209$$
$$x_{16} = 7.85398173541774$$
$$x_{17} = 80.1106131546315$$
$$x_{18} = 56.5486677646163$$
$$x_{19} = -23.561945003804$$
$$x_{20} = -64.4026492408158$$
$$x_{21} = -58.1194640027517$$
$$x_{22} = 48.6946859820148$$
$$x_{23} = 95.8185760508519$$
$$x_{24} = 15.707963267949$$
$$x_{25} = 67.5442422018325$$
$$x_{26} = -45.5530935824522$$
$$x_{27} = -95.8185758682892$$
$$x_{28} = -59.6902604182061$$
$$x_{29} = -83.2522054524035$$
$$x_{30} = 64.4026493118058$$
$$x_{31} = 21.9911485751286$$
$$x_{32} = 6.28318530717959$$
$$x_{33} = -87.9645943005142$$
$$x_{34} = 9.42477796076938$$
$$x_{35} = -28.2743338823081$$
$$x_{36} = -86.3937978155375$$
$$x_{37} = -9.42477796076938$$
$$x_{38} = 86.3937978909611$$
$$x_{39} = 59.6902604182061$$
$$x_{40} = -80.1106125824842$$
$$x_{41} = 14.1371670924752$$
$$x_{42} = 28.2743338823081$$
$$x_{43} = 73.8274274722061$$
$$x_{44} = 36.1283160593477$$
$$x_{45} = -67.5442421609972$$
$$x_{46} = 94.2477796076938$$
$$x_{47} = -89.5353907394375$$
$$x_{48} = -72.2566310325652$$
$$x_{49} = -97.3893722612836$$
$$x_{50} = 37.6991118430775$$
$$x_{51} = 89.5353907744432$$
$$x_{52} = -14.13716684381$$
$$x_{53} = -50.2654824574367$$
$$x_{54} = -94.2477796076938$$
$$x_{55} = -7.85398150264842$$
$$x_{56} = -42.4115006663339$$
$$x_{57} = -36.128315423197$$
$$x_{58} = 26.7035374084741$$
$$x_{59} = -37.6991118430775$$
$$x_{60} = -17.278759737384$$
$$x_{61} = -1.57079642505341$$
$$x_{62} = 70.6858345559153$$
$$x_{63} = -95.818575585294$$
$$x_{64} = 42.4115007327518$$
$$x_{65} = 92.6769831301454$$
$$x_{66} = 20.4203521537986$$
$$x_{67} = -81.6814089933346$$
$$x_{68} = 43.9822971502571$$
$$x_{69} = 45.5530936288414$$
$$x_{70} = -31.4159265358979$$
$$x_{71} = 0$$
$$x_{72} = -21.9911485751286$$
$$x_{73} = 100.530964914873$$
$$x_{74} = 34.5575191894877$$
$$x_{75} = -29.8451300981866$$
$$x_{76} = 65.9734457253857$$
$$x_{77} = 1.57079648184495$$
$$x_{78} = -20.4203520921076$$
$$x_{79} = 7.85398164444075$$
$$x_{80} = -53.4070751110265$$
$$x_{81} = -6.28318530717959$$
$$x_{82} = -43.9822971502571$$
$$x_{83} = 29.8451303144929$$
$$x_{84} = -51.8362786906154$$
$$x_{85} = 72.2566310325652$$
$$x_{86} = 87.9645943005142$$
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$- (4 \sin{\left(x \right)} \cos{\left(2 x \right)} + 5 \sin{\left(2 x \right)} \cos{\left(x \right)}) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
$$x_{2} = \pi$$
$$x_{3} = \frac{i \left(\log{\left(9 \right)} - \log{\left(-5 - 2 \sqrt{14} i \right)}\right)}{2}$$
$$x_{4} = \frac{i \left(\log{\left(9 \right)} - \log{\left(-5 + 2 \sqrt{14} i \right)}\right)}{2}$$
$$x_{5} = - i \log{\left(- \frac{\sqrt{-5 - 2 \sqrt{14} i}}{3} \right)}$$
$$x_{6} = - i \log{\left(- \frac{\sqrt{-5 + 2 \sqrt{14} i}}{3} \right)}$$
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
$$\left[\pi, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{\pi}{2} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{14}}{5} \right)}}{2}\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$- (4 \sin{\left(x \right)} \cos{\left(2 x \right)} + 5 \sin{\left(2 x \right)} \cos{\left(x \right)}) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
$$x_{2} = \pi$$
$$x_{3} = \frac{i \left(\log{\left(9 \right)} - \log{\left(-5 - 2 \sqrt{14} i \right)}\right)}{2}$$
$$x_{4} = \frac{i \left(\log{\left(9 \right)} - \log{\left(-5 + 2 \sqrt{14} i \right)}\right)}{2}$$
$$x_{5} = - i \log{\left(- \frac{\sqrt{-5 - 2 \sqrt{14} i}}{3} \right)}$$
$$x_{6} = - i \log{\left(- \frac{\sqrt{-5 + 2 \sqrt{14} i}}{3} \right)}$$
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
$$\left[\pi, \infty\right)$$
Convex at the intervals
$$\left(-\infty, - \frac{\pi}{2} + \frac{\operatorname{atan}{\left(\frac{2 \sqrt{14}}{5} \right)}}{2}\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} \cos{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
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
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} \cos{\left(x \right)}}{x}\right) = 0$$
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:
$$\sin{\left(2 x \right)} \cos{\left(x \right)} = - \sin{\left(2 x \right)} \cos{\left(x \right)}$$
- No
$$\sin{\left(2 x \right)} \cos{\left(x \right)} = \sin{\left(2 x \right)} \cos{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sin{\left(2 x \right)} \cos{\left(x \right)} = - \sin{\left(2 x \right)} \cos{\left(x \right)}$$
- No
$$\sin{\left(2 x \right)} \cos{\left(x \right)} = \sin{\left(2 x \right)} \cos{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd