Mister Exam
Graphing y = (2tan(x)-sin(2x))(1-cos(3x))
The solution
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f(x) = (2*tan(x) - sin(2*x))*(1 - cos(3*x))
$$f{\left(x \right)} = \left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right)$$
f = (1 - cos(3*x))*(-sin(2*x) + 2*tan(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:
$$\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right) = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 78.5397617781518$$
$$x_{2} = -9.4248341105532$$
$$x_{3} = 65.9734547876023$$
$$x_{4} = -33.510321669889$$
$$x_{5} = 48.1710875317851$$
$$x_{6} = 21.9911516407816$$
$$x_{7} = 94.2478033786606$$
$$x_{8} = -79.5870140821134$$
$$x_{9} = 43.9825183594456$$
$$x_{10} = -87.9650360077111$$
$$x_{11} = 6.28283015343438$$
$$x_{12} = 34.5574635725452$$
$$x_{13} = -92.1533846177873$$
$$x_{14} = 96.3421746230425$$
$$x_{15} = 4.18879030222116$$
$$x_{16} = 8.37758027594926$$
$$x_{17} = 72.2566292959236$$
$$x_{18} = 10.4719754149671$$
$$x_{19} = -43.982518352659$$
$$x_{20} = -39.7935068925054$$
$$x_{21} = -85.8701991481314$$
$$x_{22} = 87.9650364881714$$
$$x_{23} = -46.0766921494917$$
$$x_{24} = -37.6996561012896$$
$$x_{25} = 15.7080214499647$$
$$x_{26} = -72.2565749460944$$
$$x_{27} = -35.604716901487$$
$$x_{28} = 59.6903189708094$$
$$x_{29} = -15.7079741088947$$
$$x_{30} = -81.6821417701444$$
$$x_{31} = -21.9911516394019$$
$$x_{32} = -59.6902756308872$$
$$x_{33} = 28.2743275447995$$
$$x_{34} = -65.9734546760391$$
$$x_{35} = -97.389427219391$$
$$x_{36} = -90.0589893262434$$
$$x_{37} = 54.4542728076009$$
$$x_{38} = 41.8879022116746$$
$$x_{39} = -2.09439497036349$$
$$x_{40} = 50.2653167153794$$
$$x_{41} = -0.00231765937064296$$
$$x_{42} = 52.3598774506673$$
$$x_{43} = 46.0766923327432$$
$$x_{44} = -83.7758040705397$$
$$x_{45} = 0$$
$$x_{46} = -53.4071311214517$$
$$x_{47} = -28.2742765405245$$
$$x_{48} = 85.8701993972745$$
so we need to solve the equation:
$$\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right) = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 78.5397617781518$$
$$x_{2} = -9.4248341105532$$
$$x_{3} = 65.9734547876023$$
$$x_{4} = -33.510321669889$$
$$x_{5} = 48.1710875317851$$
$$x_{6} = 21.9911516407816$$
$$x_{7} = 94.2478033786606$$
$$x_{8} = -79.5870140821134$$
$$x_{9} = 43.9825183594456$$
$$x_{10} = -87.9650360077111$$
$$x_{11} = 6.28283015343438$$
$$x_{12} = 34.5574635725452$$
$$x_{13} = -92.1533846177873$$
$$x_{14} = 96.3421746230425$$
$$x_{15} = 4.18879030222116$$
$$x_{16} = 8.37758027594926$$
$$x_{17} = 72.2566292959236$$
$$x_{18} = 10.4719754149671$$
$$x_{19} = -43.982518352659$$
$$x_{20} = -39.7935068925054$$
$$x_{21} = -85.8701991481314$$
$$x_{22} = 87.9650364881714$$
$$x_{23} = -46.0766921494917$$
$$x_{24} = -37.6996561012896$$
$$x_{25} = 15.7080214499647$$
$$x_{26} = -72.2565749460944$$
$$x_{27} = -35.604716901487$$
$$x_{28} = 59.6903189708094$$
$$x_{29} = -15.7079741088947$$
$$x_{30} = -81.6821417701444$$
$$x_{31} = -21.9911516394019$$
$$x_{32} = -59.6902756308872$$
$$x_{33} = 28.2743275447995$$
$$x_{34} = -65.9734546760391$$
$$x_{35} = -97.389427219391$$
$$x_{36} = -90.0589893262434$$
$$x_{37} = 54.4542728076009$$
$$x_{38} = 41.8879022116746$$
$$x_{39} = -2.09439497036349$$
$$x_{40} = 50.2653167153794$$
$$x_{41} = -0.00231765937064296$$
$$x_{42} = 52.3598774506673$$
$$x_{43} = 46.0766923327432$$
$$x_{44} = -83.7758040705397$$
$$x_{45} = 0$$
$$x_{46} = -53.4071311214517$$
$$x_{47} = -28.2742765405245$$
$$x_{48} = 85.8701993972745$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right)\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right)\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right)\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right)\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (2*tan(x) - sin(2*x))*(1 - cos(3*x)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right)}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right)}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right)}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right)}{x}\right)$$
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:
$$\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right) = \left(1 - \cos{\left(3 x \right)}\right) \left(\sin{\left(2 x \right)} - 2 \tan{\left(x \right)}\right)$$
- No
$$\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right) = - \left(1 - \cos{\left(3 x \right)}\right) \left(\sin{\left(2 x \right)} - 2 \tan{\left(x \right)}\right)$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right) = \left(1 - \cos{\left(3 x \right)}\right) \left(\sin{\left(2 x \right)} - 2 \tan{\left(x \right)}\right)$$
- No
$$\left(1 - \cos{\left(3 x \right)}\right) \left(- \sin{\left(2 x \right)} + 2 \tan{\left(x \right)}\right) = - \left(1 - \cos{\left(3 x \right)}\right) \left(\sin{\left(2 x \right)} - 2 \tan{\left(x \right)}\right)$$
- No
so, the function
not is
neither even, nor odd