Mister Exam
Graphing y = sin2x-cos^2x
The solution
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2 f(x) = sin(2*x) - cos (x)
$$f{\left(x \right)} = \sin{\left(2 x \right)} - \cos^{2}{\left(x \right)}$$
f = sin(2*x) - cos(x)^2
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:
$$\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 86.3937979737193$$
$$x_{2} = -7.85398163397448$$
$$x_{3} = 1.5707963267949$$
$$x_{4} = -64.4026493985908$$
$$x_{5} = -58.1194640914112$$
$$x_{6} = -20.4203522483337$$
$$x_{7} = 97.8530198702844$$
$$x_{8} = -93.784131998693$$
$$x_{9} = -45.553093477052$$
$$x_{10} = 64.4026493985908$$
$$x_{11} = -29.845130209103$$
$$x_{12} = -87.5009466915134$$
$$x_{13} = 42.4115008234622$$
$$x_{14} = 22.4547961841294$$
$$x_{15} = 50.7291300664375$$
$$x_{16} = 72.7202786415661$$
$$x_{17} = -43.5186495412563$$
$$x_{18} = -78.076168730744$$
$$x_{19} = 95.8185759344887$$
$$x_{20} = 58.1194640914112$$
$$x_{21} = -73.8274273593601$$
$$x_{22} = 7.85398163397448$$
$$x_{23} = 0.463647609000806$$
$$x_{24} = -34.0938715804869$$
$$x_{25} = 89.5353906273091$$
$$x_{26} = 23.5619449019235$$
$$x_{27} = 16.1716108769498$$
$$x_{28} = -92.6769832808989$$
$$x_{29} = -21.5275009661277$$
$$x_{30} = -15.2443156589482$$
$$x_{31} = -81.2177613843338$$
$$x_{32} = -42.4115008234622$$
$$x_{33} = -30.9522789268971$$
$$x_{34} = -100.067317305873$$
$$x_{35} = -67.5442420521806$$
$$x_{36} = 20.4203522483337$$
$$x_{37} = -59.2266128092053$$
$$x_{38} = -56.0850201556155$$
$$x_{39} = -65.5097981163849$$
$$x_{40} = -14.1371669411541$$
$$x_{41} = 66.4370933343865$$
$$x_{42} = 57.0123153736171$$
$$x_{43} = 29.845130209103$$
$$x_{44} = 28.7379814913089$$
$$x_{45} = -5.81953769817878$$
$$x_{46} = 47.5875374128477$$
$$x_{47} = -86.3937979737193$$
$$x_{48} = 80.1106126665397$$
$$x_{49} = -12.1027230053584$$
$$x_{50} = -71.7929834235644$$
$$x_{51} = 44.4459447592579$$
$$x_{52} = -95.8185759344887$$
$$x_{53} = -36.1283155162826$$
$$x_{54} = 6.74683291618039$$
$$x_{55} = 51.8362787842316$$
$$x_{56} = 9.88842556977019$$
$$x_{57} = 31.8795741448987$$
$$x_{58} = 53.8707227200273$$
$$x_{59} = -1.5707963267949$$
$$x_{60} = -37.2354642340767$$
$$x_{61} = 45.553093477052$$
$$x_{62} = -51.8362787842316$$
$$x_{63} = 75.8618712951559$$
$$x_{64} = 36.1283155162826$$
$$x_{65} = 60.1539080272069$$
$$x_{66} = -49.8018348484359$$
$$x_{67} = -23.5619449019235$$
$$x_{68} = -27.8106862733073$$
$$x_{69} = 94.7114272166946$$
$$x_{70} = 73.8274273593601$$
$$x_{71} = 14.1371669411541$$
$$x_{72} = 41.3043521056681$$
$$x_{73} = 180.641577581413$$
$$x_{74} = -89.5353906273091$$
$$x_{75} = 82.1450566023354$$
$$x_{76} = -84.3593540379236$$
$$x_{77} = -80.1106126665397$$
$$x_{78} = 38.1627594520783$$
$$x_{79} = 67.5442420521806$$
$$x_{80} = 88.428241909515$$
so we need to solve the equation:
$$\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{2}$$
$$x_{2} = \frac{\pi}{2}$$
Numerical solution
$$x_{1} = 86.3937979737193$$
$$x_{2} = -7.85398163397448$$
$$x_{3} = 1.5707963267949$$
$$x_{4} = -64.4026493985908$$
$$x_{5} = -58.1194640914112$$
$$x_{6} = -20.4203522483337$$
$$x_{7} = 97.8530198702844$$
$$x_{8} = -93.784131998693$$
$$x_{9} = -45.553093477052$$
$$x_{10} = 64.4026493985908$$
$$x_{11} = -29.845130209103$$
$$x_{12} = -87.5009466915134$$
$$x_{13} = 42.4115008234622$$
$$x_{14} = 22.4547961841294$$
$$x_{15} = 50.7291300664375$$
$$x_{16} = 72.7202786415661$$
$$x_{17} = -43.5186495412563$$
$$x_{18} = -78.076168730744$$
$$x_{19} = 95.8185759344887$$
$$x_{20} = 58.1194640914112$$
$$x_{21} = -73.8274273593601$$
$$x_{22} = 7.85398163397448$$
$$x_{23} = 0.463647609000806$$
$$x_{24} = -34.0938715804869$$
$$x_{25} = 89.5353906273091$$
$$x_{26} = 23.5619449019235$$
$$x_{27} = 16.1716108769498$$
$$x_{28} = -92.6769832808989$$
$$x_{29} = -21.5275009661277$$
$$x_{30} = -15.2443156589482$$
$$x_{31} = -81.2177613843338$$
$$x_{32} = -42.4115008234622$$
$$x_{33} = -30.9522789268971$$
$$x_{34} = -100.067317305873$$
$$x_{35} = -67.5442420521806$$
$$x_{36} = 20.4203522483337$$
$$x_{37} = -59.2266128092053$$
$$x_{38} = -56.0850201556155$$
$$x_{39} = -65.5097981163849$$
$$x_{40} = -14.1371669411541$$
$$x_{41} = 66.4370933343865$$
$$x_{42} = 57.0123153736171$$
$$x_{43} = 29.845130209103$$
$$x_{44} = 28.7379814913089$$
$$x_{45} = -5.81953769817878$$
$$x_{46} = 47.5875374128477$$
$$x_{47} = -86.3937979737193$$
$$x_{48} = 80.1106126665397$$
$$x_{49} = -12.1027230053584$$
$$x_{50} = -71.7929834235644$$
$$x_{51} = 44.4459447592579$$
$$x_{52} = -95.8185759344887$$
$$x_{53} = -36.1283155162826$$
$$x_{54} = 6.74683291618039$$
$$x_{55} = 51.8362787842316$$
$$x_{56} = 9.88842556977019$$
$$x_{57} = 31.8795741448987$$
$$x_{58} = 53.8707227200273$$
$$x_{59} = -1.5707963267949$$
$$x_{60} = -37.2354642340767$$
$$x_{61} = 45.553093477052$$
$$x_{62} = -51.8362787842316$$
$$x_{63} = 75.8618712951559$$
$$x_{64} = 36.1283155162826$$
$$x_{65} = 60.1539080272069$$
$$x_{66} = -49.8018348484359$$
$$x_{67} = -23.5619449019235$$
$$x_{68} = -27.8106862733073$$
$$x_{69} = 94.7114272166946$$
$$x_{70} = 73.8274273593601$$
$$x_{71} = 14.1371669411541$$
$$x_{72} = 41.3043521056681$$
$$x_{73} = 180.641577581413$$
$$x_{74} = -89.5353906273091$$
$$x_{75} = 82.1450566023354$$
$$x_{76} = -84.3593540379236$$
$$x_{77} = -80.1106126665397$$
$$x_{78} = 38.1627594520783$$
$$x_{79} = 67.5442420521806$$
$$x_{80} = 88.428241909515$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$2 \sin{\left(x \right)} \cos{\left(x \right)} + 2 \cos{\left(2 x \right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$2 \sin{\left(x \right)} \cos{\left(x \right)} + 2 \cos{\left(2 x \right)} = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
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
$$2 \left(- \sin^{2}{\left(x \right)} - 2 \sin{\left(2 x \right)} + \cos^{2}{\left(x \right)}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - i \log{\left(- e^{\frac{i \operatorname{atan}{\left(\frac{4}{3} \right)}}{4}} \right)}$$
$$x_{2} = \frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4}$$
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
$$\left(-\infty, - \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4} \right)}} \right)}\right]$$
Convex at the intervals
$$\left[\frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 \left(- \sin^{2}{\left(x \right)} - 2 \sin{\left(2 x \right)} + \cos^{2}{\left(x \right)}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = - i \log{\left(- e^{\frac{i \operatorname{atan}{\left(\frac{4}{3} \right)}}{4}} \right)}$$
$$x_{2} = \frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4}$$
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
$$\left(-\infty, - \pi + \operatorname{atan}{\left(\frac{\sin{\left(\frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4} \right)}}{\cos{\left(\frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4} \right)}} \right)}\right]$$
Convex at the intervals
$$\left[\frac{\operatorname{atan}{\left(\frac{4}{3} \right)}}{4}, \infty\right)$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)}\right) = \left\langle -2, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)}\right) = \left\langle -2, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -2, 1\right\rangle$$
$$\lim_{x \to -\infty}\left(\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)}\right) = \left\langle -2, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 1\right\rangle$$
$$\lim_{x \to \infty}\left(\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)}\right) = \left\langle -2, 1\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -2, 1\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sin(2*x) - cos(x)^2, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)}}{x}\right) = 0$$
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:
$$\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)} = - \sin{\left(2 x \right)} - \cos^{2}{\left(x \right)}$$
- No
$$\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)} = \sin{\left(2 x \right)} + \cos^{2}{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)} = - \sin{\left(2 x \right)} - \cos^{2}{\left(x \right)}$$
- No
$$\sin{\left(2 x \right)} - \cos^{2}{\left(x \right)} = \sin{\left(2 x \right)} + \cos^{2}{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd