Mister Exam
Graphing y = arctg(tg(x/sqrt(2)))/sqrt(2)
The solution
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$$f{\left(x \right)} = \frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{2}}$$
f = atan(tan(x/sqrt(2)))/sqrt(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:
$$\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{2}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -35.5430635052669$$
$$x_{2} = 84.414775825009$$
$$x_{3} = 0$$
$$x_{4} = 48.871712319742$$
$$x_{5} = -93.3005417013257$$
$$x_{6} = 39.9859464434253$$
$$x_{7} = 26.6572976289502$$
$$x_{8} = -39.9859464434253$$
$$x_{9} = 8.88576587631673$$
$$x_{10} = -31.1001805671086$$
$$x_{11} = -88.8576587631673$$
$$x_{12} = 22.2144146907918$$
$$x_{13} = -22.2144146907918$$
$$x_{14} = -13.3286488144751$$
$$x_{15} = -8.88576587631673$$
$$x_{16} = 62.2003611342171$$
$$x_{17} = -26.6572976289502$$
$$x_{18} = 13.3286488144751$$
$$x_{19} = -53.3145952579004$$
$$x_{20} = 97.7434246394841$$
$$x_{21} = -48.871712319742$$
$$x_{22} = 88.8576587631673$$
$$x_{23} = -4.44288293815837$$
$$x_{24} = -75.5290099486922$$
$$x_{25} = 93.3005417013257$$
$$x_{26} = 44.4288293815837$$
$$x_{27} = -79.9718928868506$$
$$x_{28} = 57.7574781960588$$
$$x_{29} = -66.6432440723755$$
$$x_{30} = -84.414775825009$$
$$x_{31} = 35.5430635052669$$
$$x_{32} = 31.1001805671086$$
$$x_{33} = -71.0861270105339$$
$$x_{34} = -97.7434246394841$$
$$x_{35} = 102.186307577642$$
$$x_{36} = 71.0861270105339$$
$$x_{37} = 75.5290099486922$$
$$x_{38} = 4.44288293815837$$
$$x_{39} = 17.7715317526335$$
$$x_{40} = 53.3145952579004$$
$$x_{41} = 66.6432440723755$$
$$x_{42} = -57.7574781960588$$
$$x_{43} = -62.2003611342171$$
$$x_{44} = 79.9718928868506$$
$$x_{45} = -44.4288293815837$$
$$x_{46} = -17.7715317526335$$
so we need to solve the equation:
$$\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{2}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -35.5430635052669$$
$$x_{2} = 84.414775825009$$
$$x_{3} = 0$$
$$x_{4} = 48.871712319742$$
$$x_{5} = -93.3005417013257$$
$$x_{6} = 39.9859464434253$$
$$x_{7} = 26.6572976289502$$
$$x_{8} = -39.9859464434253$$
$$x_{9} = 8.88576587631673$$
$$x_{10} = -31.1001805671086$$
$$x_{11} = -88.8576587631673$$
$$x_{12} = 22.2144146907918$$
$$x_{13} = -22.2144146907918$$
$$x_{14} = -13.3286488144751$$
$$x_{15} = -8.88576587631673$$
$$x_{16} = 62.2003611342171$$
$$x_{17} = -26.6572976289502$$
$$x_{18} = 13.3286488144751$$
$$x_{19} = -53.3145952579004$$
$$x_{20} = 97.7434246394841$$
$$x_{21} = -48.871712319742$$
$$x_{22} = 88.8576587631673$$
$$x_{23} = -4.44288293815837$$
$$x_{24} = -75.5290099486922$$
$$x_{25} = 93.3005417013257$$
$$x_{26} = 44.4288293815837$$
$$x_{27} = -79.9718928868506$$
$$x_{28} = 57.7574781960588$$
$$x_{29} = -66.6432440723755$$
$$x_{30} = -84.414775825009$$
$$x_{31} = 35.5430635052669$$
$$x_{32} = 31.1001805671086$$
$$x_{33} = -71.0861270105339$$
$$x_{34} = -97.7434246394841$$
$$x_{35} = 102.186307577642$$
$$x_{36} = 71.0861270105339$$
$$x_{37} = 75.5290099486922$$
$$x_{38} = 4.44288293815837$$
$$x_{39} = 17.7715317526335$$
$$x_{40} = 53.3145952579004$$
$$x_{41} = 66.6432440723755$$
$$x_{42} = -57.7574781960588$$
$$x_{43} = -62.2003611342171$$
$$x_{44} = 79.9718928868506$$
$$x_{45} = -44.4288293815837$$
$$x_{46} = -17.7715317526335$$
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
$$\frac{\frac{\sqrt{2}}{2} \sqrt{2}}{2} = 0$$
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
$$\frac{\frac{\sqrt{2}}{2} \sqrt{2}}{2} = 0$$
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
$$0 = 0$$
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
$$0 = 0$$
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
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{2}}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{2}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{2}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{2}}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of atan(tan(x/sqrt(2)))/sqrt(2), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{\sqrt{2}}{2} \operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{\sqrt{2}}{2} \operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\frac{\sqrt{2}}{2} \operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\frac{\sqrt{2}}{2} \operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{x}\right)$$
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:
$$\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{2}} = - \frac{\sqrt{2}}{2} \operatorname{atan}{\left(\tan{\left(\frac{\sqrt{2}}{2} x \right)} \right)}$$
- No
$$\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \operatorname{atan}{\left(\tan{\left(\frac{\sqrt{2}}{2} x \right)} \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{2}} = - \frac{\sqrt{2}}{2} \operatorname{atan}{\left(\tan{\left(\frac{\sqrt{2}}{2} x \right)} \right)}$$
- No
$$\frac{\operatorname{atan}{\left(\tan{\left(\frac{x}{\sqrt{2}} \right)} \right)}}{\sqrt{2}} = \frac{\sqrt{2}}{2} \operatorname{atan}{\left(\tan{\left(\frac{\sqrt{2}}{2} x \right)} \right)}$$
- No
so, the function
not is
neither even, nor odd