Mister Exam
Graphing y = abs(cos(x*2)*sin(x))
The solution
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f(x) = |cos(x*2)*sin(x)|
$$f{\left(x \right)} = \left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right|$$
f = Abs(sin(x)*cos(2*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:
$$\left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{4}$$
$$x_{3} = \frac{3 \pi}{4}$$
$$x_{4} = \pi$$
Numerical solution
$$x_{1} = -59.6902604182061$$
$$x_{2} = -55.7632696012188$$
$$x_{3} = 2.35619449019234$$
$$x_{4} = -21.9911485751286$$
$$x_{5} = 19.6349540849362$$
$$x_{6} = 5.49778714378214$$
$$x_{7} = -75.398223686155$$
$$x_{8} = -46.3384916404494$$
$$x_{9} = 29.0597320457056$$
$$x_{10} = 50.2654824574367$$
$$x_{11} = 96.6039740978861$$
$$x_{12} = 91.106186954104$$
$$x_{13} = -41.6261026600648$$
$$x_{14} = 157.865030842887$$
$$x_{15} = 25.9181393921158$$
$$x_{16} = 91.8915851175014$$
$$x_{17} = 0$$
$$x_{18} = -19.6349540849362$$
$$x_{19} = 43.9822971502571$$
$$x_{20} = -43.9822971502571$$
$$x_{21} = -85.6083998103219$$
$$x_{22} = 24.3473430653209$$
$$x_{23} = 76.1836218495525$$
$$x_{24} = 101.316363078271$$
$$x_{25} = -65.9734457253857$$
$$x_{26} = -99.7455667514759$$
$$x_{27} = -87.9645943005142$$
$$x_{28} = 40.8407044966673$$
$$x_{29} = 18.8495559215388$$
$$x_{30} = 46.3384916404494$$
$$x_{31} = -11.7809724509617$$
$$x_{32} = 87.9645943005142$$
$$x_{33} = -81.6814089933346$$
$$x_{34} = 72.2566310325652$$
$$x_{35} = -18.0641577581413$$
$$x_{36} = -79.3252145031423$$
$$x_{37} = -31.4159265358979$$
$$x_{38} = -62.0464549083984$$
$$x_{39} = 36.9137136796801$$
$$x_{40} = 74.6128255227576$$
$$x_{41} = -13.3517687777566$$
$$x_{42} = -24.3473430653209$$
$$x_{43} = 28.2743338823081$$
$$x_{44} = 69.9004365423729$$
$$x_{45} = 47.9092879672443$$
$$x_{46} = -63.6172512351933$$
$$x_{47} = -62.8318530717959$$
$$x_{48} = 54.1924732744239$$
$$x_{49} = -53.4070751110265$$
$$x_{50} = 94.2477796076938$$
$$x_{51} = 21.9911485751286$$
$$x_{52} = -68.329640215578$$
$$x_{53} = -15.707963267949$$
$$x_{54} = 65.9734457253857$$
$$x_{55} = -58.9048622548086$$
$$x_{56} = -84.037603483527$$
$$x_{57} = 191.637151868977$$
$$x_{58} = -97.3893722612836$$
$$x_{59} = -69.9004365423729$$
$$x_{60} = 41.6261026600648$$
$$x_{61} = -25.1327412287183$$
$$x_{62} = 90.3207887907066$$
$$x_{63} = 6.28318530717959$$
$$x_{64} = 3.92699081698724$$
$$x_{65} = 32.2013246992954$$
$$x_{66} = -106.814150222053$$
$$x_{67} = -2.35619449019234$$
$$x_{68} = -90.3207887907066$$
$$x_{69} = -77.7544181763474$$
$$x_{70} = 98.174770424681$$
$$x_{71} = 69.1150383789755$$
$$x_{72} = 52.621676947629$$
$$x_{73} = -40.0553063332699$$
$$x_{74} = -37.6991118430775$$
$$x_{75} = 68.329640215578$$
$$x_{76} = -33.7721210260903$$
$$x_{77} = 10.2101761241668$$
$$x_{78} = -47.1238898038469$$
so we need to solve the equation:
$$\left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right| = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
$$x_{2} = \frac{\pi}{4}$$
$$x_{3} = \frac{3 \pi}{4}$$
$$x_{4} = \pi$$
Numerical solution
$$x_{1} = -59.6902604182061$$
$$x_{2} = -55.7632696012188$$
$$x_{3} = 2.35619449019234$$
$$x_{4} = -21.9911485751286$$
$$x_{5} = 19.6349540849362$$
$$x_{6} = 5.49778714378214$$
$$x_{7} = -75.398223686155$$
$$x_{8} = -46.3384916404494$$
$$x_{9} = 29.0597320457056$$
$$x_{10} = 50.2654824574367$$
$$x_{11} = 96.6039740978861$$
$$x_{12} = 91.106186954104$$
$$x_{13} = -41.6261026600648$$
$$x_{14} = 157.865030842887$$
$$x_{15} = 25.9181393921158$$
$$x_{16} = 91.8915851175014$$
$$x_{17} = 0$$
$$x_{18} = -19.6349540849362$$
$$x_{19} = 43.9822971502571$$
$$x_{20} = -43.9822971502571$$
$$x_{21} = -85.6083998103219$$
$$x_{22} = 24.3473430653209$$
$$x_{23} = 76.1836218495525$$
$$x_{24} = 101.316363078271$$
$$x_{25} = -65.9734457253857$$
$$x_{26} = -99.7455667514759$$
$$x_{27} = -87.9645943005142$$
$$x_{28} = 40.8407044966673$$
$$x_{29} = 18.8495559215388$$
$$x_{30} = 46.3384916404494$$
$$x_{31} = -11.7809724509617$$
$$x_{32} = 87.9645943005142$$
$$x_{33} = -81.6814089933346$$
$$x_{34} = 72.2566310325652$$
$$x_{35} = -18.0641577581413$$
$$x_{36} = -79.3252145031423$$
$$x_{37} = -31.4159265358979$$
$$x_{38} = -62.0464549083984$$
$$x_{39} = 36.9137136796801$$
$$x_{40} = 74.6128255227576$$
$$x_{41} = -13.3517687777566$$
$$x_{42} = -24.3473430653209$$
$$x_{43} = 28.2743338823081$$
$$x_{44} = 69.9004365423729$$
$$x_{45} = 47.9092879672443$$
$$x_{46} = -63.6172512351933$$
$$x_{47} = -62.8318530717959$$
$$x_{48} = 54.1924732744239$$
$$x_{49} = -53.4070751110265$$
$$x_{50} = 94.2477796076938$$
$$x_{51} = 21.9911485751286$$
$$x_{52} = -68.329640215578$$
$$x_{53} = -15.707963267949$$
$$x_{54} = 65.9734457253857$$
$$x_{55} = -58.9048622548086$$
$$x_{56} = -84.037603483527$$
$$x_{57} = 191.637151868977$$
$$x_{58} = -97.3893722612836$$
$$x_{59} = -69.9004365423729$$
$$x_{60} = 41.6261026600648$$
$$x_{61} = -25.1327412287183$$
$$x_{62} = 90.3207887907066$$
$$x_{63} = 6.28318530717959$$
$$x_{64} = 3.92699081698724$$
$$x_{65} = 32.2013246992954$$
$$x_{66} = -106.814150222053$$
$$x_{67} = -2.35619449019234$$
$$x_{68} = -90.3207887907066$$
$$x_{69} = -77.7544181763474$$
$$x_{70} = 98.174770424681$$
$$x_{71} = 69.1150383789755$$
$$x_{72} = 52.621676947629$$
$$x_{73} = -40.0553063332699$$
$$x_{74} = -37.6991118430775$$
$$x_{75} = 68.329640215578$$
$$x_{76} = -33.7721210260903$$
$$x_{77} = 10.2101761241668$$
$$x_{78} = -47.1238898038469$$
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
$$2 \left(2 \sin{\left(x \right)} \sin{\left(2 x \right)} - \cos{\left(x \right)} \cos{\left(2 x \right)}\right)^{2} \delta\left(\sin{\left(x \right)} \cos{\left(2 x \right)}\right) - \left(5 \sin{\left(x \right)} \cos{\left(2 x \right)} + 4 \sin{\left(2 x \right)} \cos{\left(x \right)}\right) \operatorname{sign}{\left(\sin{\left(x \right)} \cos{\left(2 x \right)} \right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 \left(2 \sin{\left(x \right)} \sin{\left(2 x \right)} - \cos{\left(x \right)} \cos{\left(2 x \right)}\right)^{2} \delta\left(\sin{\left(x \right)} \cos{\left(2 x \right)}\right) - \left(5 \sin{\left(x \right)} \cos{\left(2 x \right)} + 4 \sin{\left(2 x \right)} \cos{\left(x \right)}\right) \operatorname{sign}{\left(\sin{\left(x \right)} \cos{\left(2 x \right)} \right)} = 0$$
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right| = \left|{\left\langle -1, 1\right\rangle}\right|$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to \infty} \left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right| = \left|{\left\langle -1, 1\right\rangle}\right|$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to -\infty} \left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right| = \left|{\left\langle -1, 1\right\rangle}\right|$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left|{\left\langle -1, 1\right\rangle}\right|$$
$$\lim_{x \to \infty} \left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right| = \left|{\left\langle -1, 1\right\rangle}\right|$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left|{\left\langle -1, 1\right\rangle}\right|$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of Abs(cos(x*2)*sin(x)), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right|}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right|}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right|}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right|}{x}\right) = 0$$
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:
$$\left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right| = \left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right|$$
- No
$$\left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right| = - \left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right|$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right| = \left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right|$$
- No
$$\left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right| = - \left|{\sin{\left(x \right)} \cos{\left(2 x \right)}}\right|$$
- No
so, the function
not is
neither even, nor odd