Mister Exam
Graphing y = tg(x)+arctg(x)
The solution
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f(x) = tan(x) + atan(x)
$$f{\left(x \right)} = \tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}$$
f = tan(x) + atan(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:
$$\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 77.5396725417462$$
$$x_{2} = 61.83266643575$$
$$x_{3} = -14.7242522567935$$
$$x_{4} = -64.9740303869064$$
$$x_{5} = -30.4216613078004$$
$$x_{6} = 24.1410236736903$$
$$x_{7} = -99.5299908589546$$
$$x_{8} = -49.2675048394235$$
$$x_{9} = -17.8622178450272$$
$$x_{10} = 39.8441394477448$$
$$x_{11} = -24.1410236736903$$
$$x_{12} = 58.6913271474582$$
$$x_{13} = 49.2675048394235$$
$$x_{14} = 86.9640430028565$$
$$x_{15} = 33.562342086631$$
$$x_{16} = -77.5396725417462$$
$$x_{17} = -52.408740441672$$
$$x_{18} = -55.5500167379063$$
$$x_{19} = -27.2811936639259$$
$$x_{20} = -86.9640430028565$$
$$x_{21} = 0$$
$$x_{22} = 42.9851917481598$$
$$x_{23} = -33.562342086631$$
$$x_{24} = 17.8622178450272$$
$$x_{25} = -71.2568191529722$$
$$x_{26} = -90.1055188584242$$
$$x_{27} = 52.408740441672$$
$$x_{28} = 46.1263183201969$$
$$x_{29} = -61.83266643575$$
$$x_{30} = 71.2568191529722$$
$$x_{31} = -46.1263183201969$$
$$x_{32} = 55.5500167379063$$
$$x_{33} = -58.6913271474582$$
$$x_{34} = -42.9851917481598$$
$$x_{35} = -36.703180699745$$
$$x_{36} = -5.33753969228673$$
$$x_{37} = 21.0012872745219$$
$$x_{38} = 11.5883131166121$$
$$x_{39} = 27.2811936639259$$
$$x_{40} = -93.2470026209387$$
$$x_{41} = -39.8441394477448$$
$$x_{42} = 30.4216613078004$$
$$x_{43} = 2.2831221774726$$
$$x_{44} = 80.6811187268223$$
$$x_{45} = 99.5299908589546$$
$$x_{46} = 68.1154155674723$$
$$x_{47} = -83.8225759478241$$
$$x_{48} = -74.3982387986425$$
$$x_{49} = 5.33753969228673$$
$$x_{50} = 14.7242522567935$$
$$x_{51} = 8.45673516163459$$
$$x_{52} = 64.9740303869064$$
$$x_{53} = -21.0012872745219$$
$$x_{54} = 74.3982387986425$$
$$x_{55} = 96.3884935138203$$
$$x_{56} = -96.3884935138203$$
$$x_{57} = -8.45673516163459$$
$$x_{58} = 83.8225759478241$$
$$x_{59} = 93.2470026209387$$
$$x_{60} = -80.6811187268223$$
$$x_{61} = -11.5883131166121$$
$$x_{62} = -68.1154155674723$$
$$x_{63} = -2.2831221774726$$
$$x_{64} = 36.703180699745$$
$$x_{65} = 90.1055188584242$$
so we need to solve the equation:
$$\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 77.5396725417462$$
$$x_{2} = 61.83266643575$$
$$x_{3} = -14.7242522567935$$
$$x_{4} = -64.9740303869064$$
$$x_{5} = -30.4216613078004$$
$$x_{6} = 24.1410236736903$$
$$x_{7} = -99.5299908589546$$
$$x_{8} = -49.2675048394235$$
$$x_{9} = -17.8622178450272$$
$$x_{10} = 39.8441394477448$$
$$x_{11} = -24.1410236736903$$
$$x_{12} = 58.6913271474582$$
$$x_{13} = 49.2675048394235$$
$$x_{14} = 86.9640430028565$$
$$x_{15} = 33.562342086631$$
$$x_{16} = -77.5396725417462$$
$$x_{17} = -52.408740441672$$
$$x_{18} = -55.5500167379063$$
$$x_{19} = -27.2811936639259$$
$$x_{20} = -86.9640430028565$$
$$x_{21} = 0$$
$$x_{22} = 42.9851917481598$$
$$x_{23} = -33.562342086631$$
$$x_{24} = 17.8622178450272$$
$$x_{25} = -71.2568191529722$$
$$x_{26} = -90.1055188584242$$
$$x_{27} = 52.408740441672$$
$$x_{28} = 46.1263183201969$$
$$x_{29} = -61.83266643575$$
$$x_{30} = 71.2568191529722$$
$$x_{31} = -46.1263183201969$$
$$x_{32} = 55.5500167379063$$
$$x_{33} = -58.6913271474582$$
$$x_{34} = -42.9851917481598$$
$$x_{35} = -36.703180699745$$
$$x_{36} = -5.33753969228673$$
$$x_{37} = 21.0012872745219$$
$$x_{38} = 11.5883131166121$$
$$x_{39} = 27.2811936639259$$
$$x_{40} = -93.2470026209387$$
$$x_{41} = -39.8441394477448$$
$$x_{42} = 30.4216613078004$$
$$x_{43} = 2.2831221774726$$
$$x_{44} = 80.6811187268223$$
$$x_{45} = 99.5299908589546$$
$$x_{46} = 68.1154155674723$$
$$x_{47} = -83.8225759478241$$
$$x_{48} = -74.3982387986425$$
$$x_{49} = 5.33753969228673$$
$$x_{50} = 14.7242522567935$$
$$x_{51} = 8.45673516163459$$
$$x_{52} = 64.9740303869064$$
$$x_{53} = -21.0012872745219$$
$$x_{54} = 74.3982387986425$$
$$x_{55} = 96.3884935138203$$
$$x_{56} = -96.3884935138203$$
$$x_{57} = -8.45673516163459$$
$$x_{58} = 83.8225759478241$$
$$x_{59} = 93.2470026209387$$
$$x_{60} = -80.6811187268223$$
$$x_{61} = -11.5883131166121$$
$$x_{62} = -68.1154155674723$$
$$x_{63} = -2.2831221774726$$
$$x_{64} = 36.703180699745$$
$$x_{65} = 90.1055188584242$$
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
$$\tan^{2}{\left(x \right)} + 1 + \frac{1}{x^{2} + 1} = 0$$
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
$$\tan^{2}{\left(x \right)} + 1 + \frac{1}{x^{2} + 1} = 0$$
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
$$2 \left(- \frac{x}{\left(x^{2} + 1\right)^{2}} + \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -91.1061882761657$$
$$x_{2} = 34.5575433799626$$
$$x_{3} = -97.3893733436485$$
$$x_{4} = -25.1328040205507$$
$$x_{5} = -94.2477808019261$$
$$x_{6} = 31.4159587220774$$
$$x_{7} = 0$$
$$x_{8} = -21.9912422140771$$
$$x_{9} = 78.5398184031738$$
$$x_{10} = 9.42594558462241$$
$$x_{11} = 25.1328040205507$$
$$x_{12} = -53.4070816709532$$
$$x_{13} = -9.42594558462241$$
$$x_{14} = 40.8407191588716$$
$$x_{15} = 59.6902651176419$$
$$x_{16} = 3.16758836114823$$
$$x_{17} = -43.98230889158$$
$$x_{18} = 75.3982260183446$$
$$x_{19} = -75.3982260183446$$
$$x_{20} = -34.5575433799626$$
$$x_{21} = 37.6991304808952$$
$$x_{22} = -62.8318571011953$$
$$x_{23} = 91.1061882761657$$
$$x_{24} = -59.6902651176419$$
$$x_{25} = 94.2477808019261$$
$$x_{26} = 87.9645957693188$$
$$x_{27} = 18.8497043937593$$
$$x_{28} = -78.5398184031738$$
$$x_{29} = -87.9645957693188$$
$$x_{30} = -37.6991304808952$$
$$x_{31} = -56.548673291256$$
$$x_{32} = 53.4070816709532$$
$$x_{33} = 84.8230032850167$$
$$x_{34} = 15.7082191890374$$
$$x_{35} = 97.3893733436485$$
$$x_{36} = -65.9734492062964$$
$$x_{37} = -72.2566336822883$$
$$x_{38} = 6.28701317511834$$
$$x_{39} = -31.4159587220774$$
$$x_{40} = -47.1238993512507$$
$$x_{41} = 72.2566336822883$$
$$x_{42} = -18.8497043937593$$
$$x_{43} = 12.5668681635809$$
$$x_{44} = -12.5668681635809$$
$$x_{45} = -69.1150414065892$$
$$x_{46} = 100.530965898917$$
$$x_{47} = 56.548673291256$$
$$x_{48} = 62.8318571011953$$
$$x_{49} = -28.2743780124173$$
$$x_{50} = -6.28701317511834$$
$$x_{51} = -84.8230032850167$$
$$x_{52} = -3.16758836114823$$
$$x_{53} = 47.1238993512507$$
$$x_{54} = -40.8407191588716$$
$$x_{55} = -15.7082191890374$$
$$x_{56} = 65.9734492062964$$
$$x_{57} = -81.6814108277603$$
$$x_{58} = 28.2743780124173$$
$$x_{59} = -50.2654903251137$$
$$x_{60} = 21.9912422140771$$
$$x_{61} = 81.6814108277603$$
$$x_{62} = 69.1150414065892$$
$$x_{63} = 50.2654903251137$$
$$x_{64} = -100.530965898917$$
$$x_{65} = 43.98230889158$$
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
$$\left[100.530965898917, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -100.530965898917\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 \left(- \frac{x}{\left(x^{2} + 1\right)^{2}} + \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}\right) = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -91.1061882761657$$
$$x_{2} = 34.5575433799626$$
$$x_{3} = -97.3893733436485$$
$$x_{4} = -25.1328040205507$$
$$x_{5} = -94.2477808019261$$
$$x_{6} = 31.4159587220774$$
$$x_{7} = 0$$
$$x_{8} = -21.9912422140771$$
$$x_{9} = 78.5398184031738$$
$$x_{10} = 9.42594558462241$$
$$x_{11} = 25.1328040205507$$
$$x_{12} = -53.4070816709532$$
$$x_{13} = -9.42594558462241$$
$$x_{14} = 40.8407191588716$$
$$x_{15} = 59.6902651176419$$
$$x_{16} = 3.16758836114823$$
$$x_{17} = -43.98230889158$$
$$x_{18} = 75.3982260183446$$
$$x_{19} = -75.3982260183446$$
$$x_{20} = -34.5575433799626$$
$$x_{21} = 37.6991304808952$$
$$x_{22} = -62.8318571011953$$
$$x_{23} = 91.1061882761657$$
$$x_{24} = -59.6902651176419$$
$$x_{25} = 94.2477808019261$$
$$x_{26} = 87.9645957693188$$
$$x_{27} = 18.8497043937593$$
$$x_{28} = -78.5398184031738$$
$$x_{29} = -87.9645957693188$$
$$x_{30} = -37.6991304808952$$
$$x_{31} = -56.548673291256$$
$$x_{32} = 53.4070816709532$$
$$x_{33} = 84.8230032850167$$
$$x_{34} = 15.7082191890374$$
$$x_{35} = 97.3893733436485$$
$$x_{36} = -65.9734492062964$$
$$x_{37} = -72.2566336822883$$
$$x_{38} = 6.28701317511834$$
$$x_{39} = -31.4159587220774$$
$$x_{40} = -47.1238993512507$$
$$x_{41} = 72.2566336822883$$
$$x_{42} = -18.8497043937593$$
$$x_{43} = 12.5668681635809$$
$$x_{44} = -12.5668681635809$$
$$x_{45} = -69.1150414065892$$
$$x_{46} = 100.530965898917$$
$$x_{47} = 56.548673291256$$
$$x_{48} = 62.8318571011953$$
$$x_{49} = -28.2743780124173$$
$$x_{50} = -6.28701317511834$$
$$x_{51} = -84.8230032850167$$
$$x_{52} = -3.16758836114823$$
$$x_{53} = 47.1238993512507$$
$$x_{54} = -40.8407191588716$$
$$x_{55} = -15.7082191890374$$
$$x_{56} = 65.9734492062964$$
$$x_{57} = -81.6814108277603$$
$$x_{58} = 28.2743780124173$$
$$x_{59} = -50.2654903251137$$
$$x_{60} = 21.9912422140771$$
$$x_{61} = 81.6814108277603$$
$$x_{62} = 69.1150414065892$$
$$x_{63} = 50.2654903251137$$
$$x_{64} = -100.530965898917$$
$$x_{65} = 43.98230889158$$
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
$$\left[100.530965898917, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -100.530965898917\right]$$
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:
$$y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x) + atan(x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x}\right)$$
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:
$$\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} = - \tan{\left(x \right)} - \operatorname{atan}{\left(x \right)}$$
- No
$$\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} = \tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} = - \tan{\left(x \right)} - \operatorname{atan}{\left(x \right)}$$
- No
$$\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} = \tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}$$
- No
so, the function
not is
neither even, nor odd