Mister Exam
Graphing y = tan(tan(2*x))
The solution
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f(x) = tan(tan(2*x))
$$f{\left(x \right)} = \tan{\left(\tan{\left(2 x \right)} \right)}$$
f = tan(tan(2*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:
$$\tan{\left(\tan{\left(2 x \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -97.3893722612836$$
$$x_{2} = -43.9822971502571$$
$$x_{3} = -29.845130209103$$
$$x_{4} = -84.0058153926878$$
$$x_{5} = -40.0235182424307$$
$$x_{6} = 56.5486677646163$$
$$x_{7} = 26.0722239276738$$
$$x_{8} = 28.2743338823081$$
$$x_{9} = 29.845130209103$$
$$x_{10} = -81.6814089933346$$
$$x_{11} = -15.707963267949$$
$$x_{12} = 72.2566310325652$$
$$x_{13} = 95.8185759344887$$
$$x_{14} = 68.2507246204339$$
$$x_{15} = 76.9690200129499$$
$$x_{16} = -53.4070751110265$$
$$x_{17} = -85.687315405466$$
$$x_{18} = 40.1081602314601$$
$$x_{19} = 100.530964914873$$
$$x_{20} = -9.42477796076938$$
$$x_{21} = 81.6814089933346$$
$$x_{22} = 70.6858347057703$$
$$x_{23} = -21.9911485751286$$
$$x_{24} = -25.7640548565578$$
$$x_{25} = -14.1371669411541$$
$$x_{26} = 42.4115008234622$$
$$x_{27} = -72.2566310325652$$
$$x_{28} = 10.2630300223571$$
$$x_{29} = -55.7433857219175$$
$$x_{30} = -47.7552034316864$$
$$x_{31} = 15.707963267949$$
$$x_{32} = 50.2654824574367$$
$$x_{33} = -28.2743338823081$$
$$x_{34} = -6.28318530717959$$
$$x_{35} = -105.243353895258$$
$$x_{36} = -17.2787595947439$$
$$x_{37} = 62.2005394439564$$
$$x_{38} = -80.1106126665397$$
$$x_{39} = -87.9645943005142$$
$$x_{40} = 0$$
$$x_{41} = 20.4203522483337$$
$$x_{42} = 18.2182422936993$$
$$x_{43} = -65.9734457253857$$
$$x_{44} = -73.8274273593601$$
$$x_{45} = 6.28318530717959$$
$$x_{46} = -67.5442420521806$$
$$x_{47} = -36.1283155162826$$
$$x_{48} = 94.2477796076938$$
$$x_{49} = 65.9734457253857$$
$$x_{50} = -11.8598880461058$$
$$x_{51} = 36.1283155162826$$
$$x_{52} = 32.9867228626928$$
$$x_{53} = 21.9911485751286$$
$$x_{54} = 54.1135576792799$$
$$x_{55} = 78.5398163397448$$
$$x_{56} = -89.5353906273091$$
$$x_{57} = -62.0993088065887$$
$$x_{58} = 7.85398163397448$$
$$x_{59} = 14.1371669411541$$
$$x_{60} = 86.3937979737193$$
$$x_{61} = 3.95877890782638$$
$$x_{62} = 87.9645943005142$$
$$x_{63} = 12.5663706143592$$
$$x_{64} = -3.14159265358979$$
$$x_{65} = -63.6437522155381$$
$$x_{66} = 51.8362787842316$$
$$x_{67} = -37.6991118430775$$
$$x_{68} = 46.1844071048915$$
$$x_{69} = -51.8362787842316$$
$$x_{70} = 90.4748733262645$$
$$x_{71} = -1.5707963267949$$
$$x_{72} = 59.6902604182061$$
$$x_{73} = -23.5619449019235$$
$$x_{74} = 64.4026493985908$$
$$x_{75} = 43.9822971502571$$
$$x_{76} = -50.2654824574367$$
$$x_{77} = 37.6991118430775$$
$$x_{78} = -95.8185759344887$$
$$x_{79} = 80.1106126665397$$
$$x_{80} = 73.8274273593601$$
$$x_{81} = -59.6902604182061$$
$$x_{82} = -7.85398163397448$$
$$x_{83} = -77.7809191566922$$
$$x_{84} = -75.398223686155$$
$$x_{85} = -69.7463520068149$$
$$x_{86} = 23.5619449019235$$
$$x_{87} = 92.6769832808989$$
$$x_{88} = -91.7375005819435$$
$$x_{89} = -45.553093477052$$
$$x_{90} = -33.6180364905323$$
$$x_{91} = 83.2522053201295$$
$$x_{92} = -19.5560384897921$$
$$x_{93} = 1.5707963267949$$
$$x_{94} = -58.1194640914112$$
$$x_{95} = 58.1194640914112$$
$$x_{96} = -94.2477796076938$$
$$x_{97} = 48.0633725028023$$
$$x_{98} = 98.9601685880785$$
$$x_{99} = 34.5575191894877$$
$$x_{100} = -31.4159265358979$$
so we need to solve the equation:
$$\tan{\left(\tan{\left(2 x \right)} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -97.3893722612836$$
$$x_{2} = -43.9822971502571$$
$$x_{3} = -29.845130209103$$
$$x_{4} = -84.0058153926878$$
$$x_{5} = -40.0235182424307$$
$$x_{6} = 56.5486677646163$$
$$x_{7} = 26.0722239276738$$
$$x_{8} = 28.2743338823081$$
$$x_{9} = 29.845130209103$$
$$x_{10} = -81.6814089933346$$
$$x_{11} = -15.707963267949$$
$$x_{12} = 72.2566310325652$$
$$x_{13} = 95.8185759344887$$
$$x_{14} = 68.2507246204339$$
$$x_{15} = 76.9690200129499$$
$$x_{16} = -53.4070751110265$$
$$x_{17} = -85.687315405466$$
$$x_{18} = 40.1081602314601$$
$$x_{19} = 100.530964914873$$
$$x_{20} = -9.42477796076938$$
$$x_{21} = 81.6814089933346$$
$$x_{22} = 70.6858347057703$$
$$x_{23} = -21.9911485751286$$
$$x_{24} = -25.7640548565578$$
$$x_{25} = -14.1371669411541$$
$$x_{26} = 42.4115008234622$$
$$x_{27} = -72.2566310325652$$
$$x_{28} = 10.2630300223571$$
$$x_{29} = -55.7433857219175$$
$$x_{30} = -47.7552034316864$$
$$x_{31} = 15.707963267949$$
$$x_{32} = 50.2654824574367$$
$$x_{33} = -28.2743338823081$$
$$x_{34} = -6.28318530717959$$
$$x_{35} = -105.243353895258$$
$$x_{36} = -17.2787595947439$$
$$x_{37} = 62.2005394439564$$
$$x_{38} = -80.1106126665397$$
$$x_{39} = -87.9645943005142$$
$$x_{40} = 0$$
$$x_{41} = 20.4203522483337$$
$$x_{42} = 18.2182422936993$$
$$x_{43} = -65.9734457253857$$
$$x_{44} = -73.8274273593601$$
$$x_{45} = 6.28318530717959$$
$$x_{46} = -67.5442420521806$$
$$x_{47} = -36.1283155162826$$
$$x_{48} = 94.2477796076938$$
$$x_{49} = 65.9734457253857$$
$$x_{50} = -11.8598880461058$$
$$x_{51} = 36.1283155162826$$
$$x_{52} = 32.9867228626928$$
$$x_{53} = 21.9911485751286$$
$$x_{54} = 54.1135576792799$$
$$x_{55} = 78.5398163397448$$
$$x_{56} = -89.5353906273091$$
$$x_{57} = -62.0993088065887$$
$$x_{58} = 7.85398163397448$$
$$x_{59} = 14.1371669411541$$
$$x_{60} = 86.3937979737193$$
$$x_{61} = 3.95877890782638$$
$$x_{62} = 87.9645943005142$$
$$x_{63} = 12.5663706143592$$
$$x_{64} = -3.14159265358979$$
$$x_{65} = -63.6437522155381$$
$$x_{66} = 51.8362787842316$$
$$x_{67} = -37.6991118430775$$
$$x_{68} = 46.1844071048915$$
$$x_{69} = -51.8362787842316$$
$$x_{70} = 90.4748733262645$$
$$x_{71} = -1.5707963267949$$
$$x_{72} = 59.6902604182061$$
$$x_{73} = -23.5619449019235$$
$$x_{74} = 64.4026493985908$$
$$x_{75} = 43.9822971502571$$
$$x_{76} = -50.2654824574367$$
$$x_{77} = 37.6991118430775$$
$$x_{78} = -95.8185759344887$$
$$x_{79} = 80.1106126665397$$
$$x_{80} = 73.8274273593601$$
$$x_{81} = -59.6902604182061$$
$$x_{82} = -7.85398163397448$$
$$x_{83} = -77.7809191566922$$
$$x_{84} = -75.398223686155$$
$$x_{85} = -69.7463520068149$$
$$x_{86} = 23.5619449019235$$
$$x_{87} = 92.6769832808989$$
$$x_{88} = -91.7375005819435$$
$$x_{89} = -45.553093477052$$
$$x_{90} = -33.6180364905323$$
$$x_{91} = 83.2522053201295$$
$$x_{92} = -19.5560384897921$$
$$x_{93} = 1.5707963267949$$
$$x_{94} = -58.1194640914112$$
$$x_{95} = 58.1194640914112$$
$$x_{96} = -94.2477796076938$$
$$x_{97} = 48.0633725028023$$
$$x_{98} = 98.9601685880785$$
$$x_{99} = 34.5575191894877$$
$$x_{100} = -31.4159265358979$$
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
$$\left(2 \tan^{2}{\left(2 x \right)} + 2\right) \left(\tan^{2}{\left(\tan{\left(2 x \right)} \right)} + 1\right) = 0$$
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
$$\left(2 \tan^{2}{\left(2 x \right)} + 2\right) \left(\tan^{2}{\left(\tan{\left(2 x \right)} \right)} + 1\right) = 0$$
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
$$8 \left(\left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(\tan{\left(2 x \right)} \right)} + \tan{\left(2 x \right)}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\tan^{2}{\left(\tan{\left(2 x \right)} \right)} + 1\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -97.3893722612836$$
$$x_{2} = -43.9822971502571$$
$$x_{3} = -29.845130209103$$
$$x_{4} = 56.5486677646163$$
$$x_{5} = 62.2158522491989$$
$$x_{6} = 28.2743338823081$$
$$x_{7} = 29.845130209103$$
$$x_{8} = -81.6814089933346$$
$$x_{9} = -15.707963267949$$
$$x_{10} = 72.2566310325652$$
$$x_{11} = 95.8185759344887$$
$$x_{12} = 45.553093477052$$
$$x_{13} = -55.9326669420193$$
$$x_{14} = -53.4070751110265$$
$$x_{15} = 18.2335550989418$$
$$x_{16} = -11.9503697917622$$
$$x_{17} = 100.530964914873$$
$$x_{18} = -9.42477796076938$$
$$x_{19} = 81.6814089933346$$
$$x_{20} = 70.6858347057703$$
$$x_{21} = -21.9911485751286$$
$$x_{22} = -14.1371669411541$$
$$x_{23} = 32.0319273584949$$
$$x_{24} = 42.4115008234622$$
$$x_{25} = -72.2566310325652$$
$$x_{26} = 68.2487278775419$$
$$x_{27} = 15.707963267949$$
$$x_{28} = 50.2654824574367$$
$$x_{29} = -28.2743338823081$$
$$x_{30} = -91.722187776701$$
$$x_{31} = 84.2070008243274$$
$$x_{32} = -6.28318530717959$$
$$x_{33} = -17.2787595947439$$
$$x_{34} = -80.1106126665397$$
$$x_{35} = -87.9645943005142$$
$$x_{36} = 76.014224508752$$
$$x_{37} = 48.6946861306418$$
$$x_{38} = -20.4203522483337$$
$$x_{39} = 0$$
$$x_{40} = 20.4203522483337$$
$$x_{41} = 89.5353906273091$$
$$x_{42} = 98.0053730838806$$
$$x_{43} = -65.9734457253857$$
$$x_{44} = -73.8274273593601$$
$$x_{45} = -39.8859089924694$$
$$x_{46} = 6.28318530717959$$
$$x_{47} = -67.5442420521806$$
$$x_{48} = -36.1283155162826$$
$$x_{49} = 94.2477796076938$$
$$x_{50} = -61.261056745001$$
$$x_{51} = 65.9734457253857$$
$$x_{52} = 36.1283155162826$$
$$x_{53} = 21.9911485751286$$
$$x_{54} = -33.9415183668907$$
$$x_{55} = 78.5398163397448$$
$$x_{56} = 40.2247036740703$$
$$x_{57} = -69.7310392015724$$
$$x_{58} = -47.7398906264439$$
$$x_{59} = 4.71238898038469$$
$$x_{60} = -3.75759347618678$$
$$x_{61} = -89.5353906273091$$
$$x_{62} = 7.85398163397448$$
$$x_{63} = 14.1371669411541$$
$$x_{64} = 26.0875367329163$$
$$x_{65} = 86.3937979737193$$
$$x_{66} = 87.9645943005142$$
$$x_{67} = 12.5663706143592$$
$$x_{68} = -25.7487420513153$$
$$x_{69} = 51.8362787842316$$
$$x_{70} = -37.6991118430775$$
$$x_{71} = -51.8362787842316$$
$$x_{72} = -1.5707963267949$$
$$x_{73} = 59.6902604182061$$
$$x_{74} = -83.8682061427265$$
$$x_{75} = -23.5619449019235$$
$$x_{76} = -99.9149640922764$$
$$x_{77} = 64.4026493985908$$
$$x_{78} = 43.9822971502571$$
$$x_{79} = -50.2654824574367$$
$$x_{80} = 37.6991118430775$$
$$x_{81} = -95.8185759344887$$
$$x_{82} = -64.4026493985908$$
$$x_{83} = 80.1106126665397$$
$$x_{84} = 73.8274273593601$$
$$x_{85} = -59.6902604182061$$
$$x_{86} = -7.85398163397448$$
$$x_{87} = -75.398223686155$$
$$x_{88} = -86.3937979737193$$
$$x_{89} = 23.5619449019235$$
$$x_{90} = 92.6769832808989$$
$$x_{91} = 9.42477796076938$$
$$x_{92} = 54.0230759336235$$
$$x_{93} = -45.553093477052$$
$$x_{94} = 1.5707963267949$$
$$x_{95} = -58.1194640914112$$
$$x_{96} = 58.1194640914112$$
$$x_{97} = -94.2477796076938$$
$$x_{98} = -77.9238155171478$$
$$x_{99} = -42.4115008234622$$
$$x_{100} = 34.5575191894877$$
$$x_{101} = -31.4159265358979$$
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
$$\left[100.530964914873, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -99.9149640922764\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$8 \left(\left(\tan^{2}{\left(2 x \right)} + 1\right) \tan{\left(\tan{\left(2 x \right)} \right)} + \tan{\left(2 x \right)}\right) \left(\tan^{2}{\left(2 x \right)} + 1\right) \left(\tan^{2}{\left(\tan{\left(2 x \right)} \right)} + 1\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -97.3893722612836$$
$$x_{2} = -43.9822971502571$$
$$x_{3} = -29.845130209103$$
$$x_{4} = 56.5486677646163$$
$$x_{5} = 62.2158522491989$$
$$x_{6} = 28.2743338823081$$
$$x_{7} = 29.845130209103$$
$$x_{8} = -81.6814089933346$$
$$x_{9} = -15.707963267949$$
$$x_{10} = 72.2566310325652$$
$$x_{11} = 95.8185759344887$$
$$x_{12} = 45.553093477052$$
$$x_{13} = -55.9326669420193$$
$$x_{14} = -53.4070751110265$$
$$x_{15} = 18.2335550989418$$
$$x_{16} = -11.9503697917622$$
$$x_{17} = 100.530964914873$$
$$x_{18} = -9.42477796076938$$
$$x_{19} = 81.6814089933346$$
$$x_{20} = 70.6858347057703$$
$$x_{21} = -21.9911485751286$$
$$x_{22} = -14.1371669411541$$
$$x_{23} = 32.0319273584949$$
$$x_{24} = 42.4115008234622$$
$$x_{25} = -72.2566310325652$$
$$x_{26} = 68.2487278775419$$
$$x_{27} = 15.707963267949$$
$$x_{28} = 50.2654824574367$$
$$x_{29} = -28.2743338823081$$
$$x_{30} = -91.722187776701$$
$$x_{31} = 84.2070008243274$$
$$x_{32} = -6.28318530717959$$
$$x_{33} = -17.2787595947439$$
$$x_{34} = -80.1106126665397$$
$$x_{35} = -87.9645943005142$$
$$x_{36} = 76.014224508752$$
$$x_{37} = 48.6946861306418$$
$$x_{38} = -20.4203522483337$$
$$x_{39} = 0$$
$$x_{40} = 20.4203522483337$$
$$x_{41} = 89.5353906273091$$
$$x_{42} = 98.0053730838806$$
$$x_{43} = -65.9734457253857$$
$$x_{44} = -73.8274273593601$$
$$x_{45} = -39.8859089924694$$
$$x_{46} = 6.28318530717959$$
$$x_{47} = -67.5442420521806$$
$$x_{48} = -36.1283155162826$$
$$x_{49} = 94.2477796076938$$
$$x_{50} = -61.261056745001$$
$$x_{51} = 65.9734457253857$$
$$x_{52} = 36.1283155162826$$
$$x_{53} = 21.9911485751286$$
$$x_{54} = -33.9415183668907$$
$$x_{55} = 78.5398163397448$$
$$x_{56} = 40.2247036740703$$
$$x_{57} = -69.7310392015724$$
$$x_{58} = -47.7398906264439$$
$$x_{59} = 4.71238898038469$$
$$x_{60} = -3.75759347618678$$
$$x_{61} = -89.5353906273091$$
$$x_{62} = 7.85398163397448$$
$$x_{63} = 14.1371669411541$$
$$x_{64} = 26.0875367329163$$
$$x_{65} = 86.3937979737193$$
$$x_{66} = 87.9645943005142$$
$$x_{67} = 12.5663706143592$$
$$x_{68} = -25.7487420513153$$
$$x_{69} = 51.8362787842316$$
$$x_{70} = -37.6991118430775$$
$$x_{71} = -51.8362787842316$$
$$x_{72} = -1.5707963267949$$
$$x_{73} = 59.6902604182061$$
$$x_{74} = -83.8682061427265$$
$$x_{75} = -23.5619449019235$$
$$x_{76} = -99.9149640922764$$
$$x_{77} = 64.4026493985908$$
$$x_{78} = 43.9822971502571$$
$$x_{79} = -50.2654824574367$$
$$x_{80} = 37.6991118430775$$
$$x_{81} = -95.8185759344887$$
$$x_{82} = -64.4026493985908$$
$$x_{83} = 80.1106126665397$$
$$x_{84} = 73.8274273593601$$
$$x_{85} = -59.6902604182061$$
$$x_{86} = -7.85398163397448$$
$$x_{87} = -75.398223686155$$
$$x_{88} = -86.3937979737193$$
$$x_{89} = 23.5619449019235$$
$$x_{90} = 92.6769832808989$$
$$x_{91} = 9.42477796076938$$
$$x_{92} = 54.0230759336235$$
$$x_{93} = -45.553093477052$$
$$x_{94} = 1.5707963267949$$
$$x_{95} = -58.1194640914112$$
$$x_{96} = 58.1194640914112$$
$$x_{97} = -94.2477796076938$$
$$x_{98} = -77.9238155171478$$
$$x_{99} = -42.4115008234622$$
$$x_{100} = 34.5575191894877$$
$$x_{101} = -31.4159265358979$$
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
$$\left[100.530964914873, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -99.9149640922764\right]$$
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:
$$y = \lim_{x \to -\infty} \tan{\left(\tan{\left(2 x \right)} \right)}$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \tan{\left(\tan{\left(2 x \right)} \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty} \tan{\left(\tan{\left(2 x \right)} \right)}$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty} \tan{\left(\tan{\left(2 x \right)} \right)}$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(tan(2*x)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(\tan{\left(2 x \right)} \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\tan{\left(2 x \right)} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(\tan{\left(2 x \right)} \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(\tan{\left(2 x \right)} \right)}}{x}\right)$$
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:
$$\tan{\left(\tan{\left(2 x \right)} \right)} = - \tan{\left(\tan{\left(2 x \right)} \right)}$$
- No
$$\tan{\left(\tan{\left(2 x \right)} \right)} = \tan{\left(\tan{\left(2 x \right)} \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\tan{\left(\tan{\left(2 x \right)} \right)} = - \tan{\left(\tan{\left(2 x \right)} \right)}$$
- No
$$\tan{\left(\tan{\left(2 x \right)} \right)} = \tan{\left(\tan{\left(2 x \right)} \right)}$$
- Yes
so, the function
is
odd