Mister Exam
Graphing y = cos^5(x)*sin(2x)
The solution
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5 f(x) = cos (x)*sin(2*x)
$$f{\left(x \right)} = \sin{\left(2 x \right)} \cos^{5}{\left(x \right)}$$
f = sin(2*x)*cos(x)^5
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:
$$\sin{\left(2 x \right)} \cos^{5}{\left(x \right)} = 0$$
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
$$x_{1} = -1.56311407308359$$
$$x_{2} = -36.1247796422959$$
$$x_{3} = 29.8493694072738$$
$$x_{4} = -58.1161660043884$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = 0$$
$$x_{7} = -65.9734457253857$$
$$x_{8} = -51.8340849947567$$
$$x_{9} = 21.9911485751286$$
$$x_{10} = -21.9911485751286$$
$$x_{11} = 6.28318530717959$$
$$x_{12} = -15.707963267949$$
$$x_{13} = 83.2603901610201$$
$$x_{14} = -67.5491842058323$$
$$x_{15} = -76.9633618963966$$
$$x_{16} = 50.2654824574367$$
$$x_{17} = 58.123586000812$$
$$x_{18} = -87.9645943005142$$
$$x_{19} = -95.8168830935879$$
$$x_{20} = -14.1333948786472$$
$$x_{21} = -1.57512579782855$$
$$x_{22} = 65.9734457253857$$
$$x_{23} = 73.8321260601189$$
$$x_{24} = -83.2493538856472$$
$$x_{25} = -89.5405344525944$$
$$x_{26} = -95.8108139103479$$
$$x_{27} = 14.1406276946085$$
$$x_{28} = 51.8407487312802$$
$$x_{29} = 80.1182869798115$$
$$x_{30} = -29.8426107104596$$
$$x_{31} = 95.8235013185802$$
$$x_{32} = 7.85798817257468$$
$$x_{33} = 64.3989431463717$$
$$x_{34} = -37.6991118430775$$
$$x_{35} = -80.1075538414439$$
$$x_{36} = 42.4075857436484$$
$$x_{37} = -23.5664798348119$$
$$x_{38} = -7.85102998717213$$
$$x_{39} = -73.825498286505$$
$$x_{40} = -32.9866360585369$$
$$x_{41} = -54.9746108621408$$
$$x_{42} = 86.3903015646994$$
$$x_{43} = 28.2743338823081$$
$$x_{44} = 20.4162294115111$$
$$x_{45} = -43.9822971502571$$
$$x_{46} = 36.132143284935$$
$$x_{47} = -45.5578326523853$$
$$x_{48} = 72.2566310325652$$
$$x_{49} = 94.2477796076938$$
$$x_{50} = -17.2708783142353$$
$$x_{51} = -10.9997033924722$$
$$x_{52} = -81.6814089933346$$
$$x_{53} = 43.9822971502571$$
$$x_{54} = -98.9526897100225$$
$$x_{55} = 87.9645943005142$$
so we need to solve the equation:
$$\sin{\left(2 x \right)} \cos^{5}{\left(x \right)} = 0$$
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
$$x_{1} = -1.56311407308359$$
$$x_{2} = -36.1247796422959$$
$$x_{3} = 29.8493694072738$$
$$x_{4} = -58.1161660043884$$
$$x_{5} = -59.6902604182061$$
$$x_{6} = 0$$
$$x_{7} = -65.9734457253857$$
$$x_{8} = -51.8340849947567$$
$$x_{9} = 21.9911485751286$$
$$x_{10} = -21.9911485751286$$
$$x_{11} = 6.28318530717959$$
$$x_{12} = -15.707963267949$$
$$x_{13} = 83.2603901610201$$
$$x_{14} = -67.5491842058323$$
$$x_{15} = -76.9633618963966$$
$$x_{16} = 50.2654824574367$$
$$x_{17} = 58.123586000812$$
$$x_{18} = -87.9645943005142$$
$$x_{19} = -95.8168830935879$$
$$x_{20} = -14.1333948786472$$
$$x_{21} = -1.57512579782855$$
$$x_{22} = 65.9734457253857$$
$$x_{23} = 73.8321260601189$$
$$x_{24} = -83.2493538856472$$
$$x_{25} = -89.5405344525944$$
$$x_{26} = -95.8108139103479$$
$$x_{27} = 14.1406276946085$$
$$x_{28} = 51.8407487312802$$
$$x_{29} = 80.1182869798115$$
$$x_{30} = -29.8426107104596$$
$$x_{31} = 95.8235013185802$$
$$x_{32} = 7.85798817257468$$
$$x_{33} = 64.3989431463717$$
$$x_{34} = -37.6991118430775$$
$$x_{35} = -80.1075538414439$$
$$x_{36} = 42.4075857436484$$
$$x_{37} = -23.5664798348119$$
$$x_{38} = -7.85102998717213$$
$$x_{39} = -73.825498286505$$
$$x_{40} = -32.9866360585369$$
$$x_{41} = -54.9746108621408$$
$$x_{42} = 86.3903015646994$$
$$x_{43} = 28.2743338823081$$
$$x_{44} = 20.4162294115111$$
$$x_{45} = -43.9822971502571$$
$$x_{46} = 36.132143284935$$
$$x_{47} = -45.5578326523853$$
$$x_{48} = 72.2566310325652$$
$$x_{49} = 94.2477796076938$$
$$x_{50} = -17.2708783142353$$
$$x_{51} = -10.9997033924722$$
$$x_{52} = -81.6814089933346$$
$$x_{53} = 43.9822971502571$$
$$x_{54} = -98.9526897100225$$
$$x_{55} = 87.9645943005142$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} \cos^{5}{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} \cos^{5}{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} \cos^{5}{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} \cos^{5}{\left(x \right)}\right) = \left\langle -1, 1\right\rangle$$
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)^5*sin(2*x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} \cos^{5}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} \cos^{5}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} \cos^{5}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} \cos^{5}{\left(x \right)}}{x}\right) = 0$$
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:
$$\sin{\left(2 x \right)} \cos^{5}{\left(x \right)} = - \sin{\left(2 x \right)} \cos^{5}{\left(x \right)}$$
- No
$$\sin{\left(2 x \right)} \cos^{5}{\left(x \right)} = \sin{\left(2 x \right)} \cos^{5}{\left(x \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\sin{\left(2 x \right)} \cos^{5}{\left(x \right)} = - \sin{\left(2 x \right)} \cos^{5}{\left(x \right)}$$
- No
$$\sin{\left(2 x \right)} \cos^{5}{\left(x \right)} = \sin{\left(2 x \right)} \cos^{5}{\left(x \right)}$$
- Yes
so, the function
is
odd