Mister Exam
Graphing y = (2sin^(2)x)+2sin(2x)+1
The solution
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2 f(x) = 2*sin (x) + 2*sin(2*x) + 1
$$f{\left(x \right)} = \left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1$$
f = 2*sin(x)^2 + 2*sin(2*x) + 1
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:
$$\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{4}$$
$$x_{2} = - i \log{\left(\left(- i\right)^{\frac{5}{2}} \right)}$$
Numerical solution
$$x_{1} = -79.3252145031423$$
$$x_{2} = -85.6083998103219$$
$$x_{3} = -50.5872330118333$$
$$x_{4} = 84.037603483527$$
$$x_{5} = -1608.81718919237$$
$$x_{6} = 24.3473430653209$$
$$x_{7} = -6.60493586157623$$
$$x_{8} = -7.06858347057703$$
$$x_{9} = 52.621676947629$$
$$x_{10} = 46.3384916404494$$
$$x_{11} = 36.9137136796801$$
$$x_{12} = -88.2863448549109$$
$$x_{13} = -25.9181393921158$$
$$x_{14} = -31.7376770902946$$
$$x_{15} = -94.5695301620904$$
$$x_{16} = -19.6349540849362$$
$$x_{17} = 21.6693980207319$$
$$x_{18} = 68.329640215578$$
$$x_{19} = -60.0120109726027$$
$$x_{20} = 96.6039740978861$$
$$x_{21} = -91.8915851175014$$
$$x_{22} = -13.3517687777566$$
$$x_{23} = -44.3040477046537$$
$$x_{24} = -73.0420291959627$$
$$x_{25} = -72.5783815869619$$
$$x_{26} = -66.2951962797823$$
$$x_{27} = 78.2180657853482$$
$$x_{28} = 27.9525833279115$$
$$x_{29} = 12.2446200599625$$
$$x_{30} = -29.0597320457056$$
$$x_{31} = -47.9092879672443$$
$$x_{32} = 100.209214360477$$
$$x_{33} = -95.0331777710912$$
$$x_{34} = 80.8960108299372$$
$$x_{35} = 81.359658438938$$
$$x_{36} = -638.528706842126$$
$$x_{37} = 34.2357686350911$$
$$x_{38} = -57.3340659280137$$
$$x_{39} = 8.63937979737193$$
$$x_{40} = 58.9048622548086$$
$$x_{41} = 40.0553063332699$$
$$x_{42} = -69.9004365423729$$
$$x_{43} = 37.3773612886809$$
$$x_{44} = 87.6428437461176$$
$$x_{45} = 59.3685098638094$$
$$x_{46} = 90.3207887907066$$
$$x_{47} = -69.4367889333721$$
$$x_{48} = 74.6128255227576$$
$$x_{49} = -16.0297138223456$$
$$x_{50} = 5.96143475278294$$
$$x_{51} = 62.0464549083984$$
$$x_{52} = 43.6605465958605$$
$$x_{53} = -85.1447522013211$$
$$x_{54} = 15.3862127135523$$
$$x_{55} = -0.321750554396642$$
$$x_{56} = -3.92699081698724$$
$$x_{57} = -75.7199742405517$$
$$x_{58} = -22.3128991295252$$
$$x_{59} = -28.5960844367048$$
$$x_{60} = -51.0508806208341$$
$$x_{61} = -82.0031595477313$$
$$x_{62} = 65.651695170989$$
$$x_{63} = -97.7111228156802$$
$$x_{64} = 71.9348804781686$$
$$x_{65} = -63.6172512351933$$
$$x_{66} = 93.9260290532972$$
$$x_{67} = -38.0208623974742$$
$$x_{68} = 14.9225651045515$$
$$x_{69} = 2.35619449019234$$
$$x_{70} = 18.0641577581413$$
$$x_{71} = 49.9437319030401$$
$$x_{72} = -9.74652851516602$$
$$x_{73} = -41.6261026600648$$
$$x_{74} = 30.6305283725005$$
$$x_{75} = -35.3429173528852$$
$$x_{76} = -53.7288256654231$$
$$x_{77} = 56.2269172102196$$
so we need to solve the equation:
$$\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{4}$$
$$x_{2} = - i \log{\left(\left(- i\right)^{\frac{5}{2}} \right)}$$
Numerical solution
$$x_{1} = -79.3252145031423$$
$$x_{2} = -85.6083998103219$$
$$x_{3} = -50.5872330118333$$
$$x_{4} = 84.037603483527$$
$$x_{5} = -1608.81718919237$$
$$x_{6} = 24.3473430653209$$
$$x_{7} = -6.60493586157623$$
$$x_{8} = -7.06858347057703$$
$$x_{9} = 52.621676947629$$
$$x_{10} = 46.3384916404494$$
$$x_{11} = 36.9137136796801$$
$$x_{12} = -88.2863448549109$$
$$x_{13} = -25.9181393921158$$
$$x_{14} = -31.7376770902946$$
$$x_{15} = -94.5695301620904$$
$$x_{16} = -19.6349540849362$$
$$x_{17} = 21.6693980207319$$
$$x_{18} = 68.329640215578$$
$$x_{19} = -60.0120109726027$$
$$x_{20} = 96.6039740978861$$
$$x_{21} = -91.8915851175014$$
$$x_{22} = -13.3517687777566$$
$$x_{23} = -44.3040477046537$$
$$x_{24} = -73.0420291959627$$
$$x_{25} = -72.5783815869619$$
$$x_{26} = -66.2951962797823$$
$$x_{27} = 78.2180657853482$$
$$x_{28} = 27.9525833279115$$
$$x_{29} = 12.2446200599625$$
$$x_{30} = -29.0597320457056$$
$$x_{31} = -47.9092879672443$$
$$x_{32} = 100.209214360477$$
$$x_{33} = -95.0331777710912$$
$$x_{34} = 80.8960108299372$$
$$x_{35} = 81.359658438938$$
$$x_{36} = -638.528706842126$$
$$x_{37} = 34.2357686350911$$
$$x_{38} = -57.3340659280137$$
$$x_{39} = 8.63937979737193$$
$$x_{40} = 58.9048622548086$$
$$x_{41} = 40.0553063332699$$
$$x_{42} = -69.9004365423729$$
$$x_{43} = 37.3773612886809$$
$$x_{44} = 87.6428437461176$$
$$x_{45} = 59.3685098638094$$
$$x_{46} = 90.3207887907066$$
$$x_{47} = -69.4367889333721$$
$$x_{48} = 74.6128255227576$$
$$x_{49} = -16.0297138223456$$
$$x_{50} = 5.96143475278294$$
$$x_{51} = 62.0464549083984$$
$$x_{52} = 43.6605465958605$$
$$x_{53} = -85.1447522013211$$
$$x_{54} = 15.3862127135523$$
$$x_{55} = -0.321750554396642$$
$$x_{56} = -3.92699081698724$$
$$x_{57} = -75.7199742405517$$
$$x_{58} = -22.3128991295252$$
$$x_{59} = -28.5960844367048$$
$$x_{60} = -51.0508806208341$$
$$x_{61} = -82.0031595477313$$
$$x_{62} = 65.651695170989$$
$$x_{63} = -97.7111228156802$$
$$x_{64} = 71.9348804781686$$
$$x_{65} = -63.6172512351933$$
$$x_{66} = 93.9260290532972$$
$$x_{67} = -38.0208623974742$$
$$x_{68} = 14.9225651045515$$
$$x_{69} = 2.35619449019234$$
$$x_{70} = 18.0641577581413$$
$$x_{71} = 49.9437319030401$$
$$x_{72} = -9.74652851516602$$
$$x_{73} = -41.6261026600648$$
$$x_{74} = 30.6305283725005$$
$$x_{75} = -35.3429173528852$$
$$x_{76} = -53.7288256654231$$
$$x_{77} = 56.2269172102196$$
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
$$4 \sin{\left(x \right)} \cos{\left(x \right)} + 4 \cos{\left(2 x \right)} = 0$$
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
$$4 \sin{\left(x \right)} \cos{\left(x \right)} + 4 \cos{\left(2 x \right)} = 0$$
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
$$4 \left(- \sin^{2}{\left(x \right)} - 2 \sin{\left(2 x \right)} + \cos^{2}{\left(x \right)}\right) = 0$$
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
$$4 \left(- \sin^{2}{\left(x \right)} - 2 \sin{\left(2 x \right)} + \cos^{2}{\left(x \right)}\right) = 0$$
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
$$\lim_{x \to -\infty}\left(\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1\right) = \left\langle -1, 5\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 5\right\rangle$$
$$\lim_{x \to \infty}\left(\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1\right) = \left\langle -1, 5\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 5\right\rangle$$
$$\lim_{x \to -\infty}\left(\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1\right) = \left\langle -1, 5\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -1, 5\right\rangle$$
$$\lim_{x \to \infty}\left(\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1\right) = \left\langle -1, 5\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -1, 5\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*sin(x)^2 + 2*sin(2*x) + 1, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1}{x}\right) = 0$$
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:
$$\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1 = 2 \sin^{2}{\left(x \right)} - 2 \sin{\left(2 x \right)} + 1$$
- No
$$\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1 = - 2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)} - 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1 = 2 \sin^{2}{\left(x \right)} - 2 \sin{\left(2 x \right)} + 1$$
- No
$$\left(2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)}\right) + 1 = - 2 \sin^{2}{\left(x \right)} + 2 \sin{\left(2 x \right)} - 1$$
- No
so, the function
not is
neither even, nor odd