Mister Exam
Graphing y = 2tgx+1
The solution
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f(x) = 2*tan(x) + 1
$$f{\left(x \right)} = 2 \tan{\left(x \right)} + 1$$
f = 2*tan(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:
$$2 \tan{\left(x \right)} + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \operatorname{atan}{\left(\frac{1}{2} \right)}$$
Numerical solution
$$x_{1} = -79.0034639487456$$
$$x_{2} = 65.5097981163849$$
$$x_{3} = -22.4547961841294$$
$$x_{4} = 78.076168730744$$
$$x_{5} = -35.0211667984885$$
$$x_{6} = -88.428241909515$$
$$x_{7} = -91.5698345631048$$
$$x_{8} = 15.2443156589482$$
$$x_{9} = 62.3682054627951$$
$$x_{10} = 56.0850201556155$$
$$x_{11} = -0.463647609000806$$
$$x_{12} = 74.9345760771542$$
$$x_{13} = -16.1716108769498$$
$$x_{14} = -6.74683291618039$$
$$x_{15} = -25.5963888377192$$
$$x_{16} = 46.6602421948461$$
$$x_{17} = -47.5875374128477$$
$$x_{18} = 8.96113035176857$$
$$x_{19} = -63.2955006807967$$
$$x_{20} = -19.3132035305396$$
$$x_{21} = -69.5786859879763$$
$$x_{22} = -53.8707227200273$$
$$x_{23} = -41.3043521056681$$
$$x_{24} = 21.5275009661277$$
$$x_{25} = -75.8618712951559$$
$$x_{26} = 37.2354642340767$$
$$x_{27} = -60.1539080272069$$
$$x_{28} = 24.6690936197175$$
$$x_{29} = 81.2177613843338$$
$$x_{30} = 49.8018348484359$$
$$x_{31} = -97.8530198702844$$
$$x_{32} = -44.4459447592579$$
$$x_{33} = 27.8106862733073$$
$$x_{34} = -31.8795741448987$$
$$x_{35} = 40.3770568876665$$
$$x_{36} = 18.385908312538$$
$$x_{37} = -3.6052402625906$$
$$x_{38} = -9.88842556977019$$
$$x_{39} = 30.9522789268971$$
$$x_{40} = 87.5009466915134$$
$$x_{41} = -100.994612523874$$
$$x_{42} = 34.0938715804869$$
$$x_{43} = 71.7929834235644$$
$$x_{44} = -85.2866492559252$$
$$x_{45} = 59.2266128092053$$
$$x_{46} = -38.1627594520783$$
$$x_{47} = -50.7291300664375$$
$$x_{48} = 96.9257246522828$$
$$x_{49} = -28.7379814913089$$
$$x_{50} = -13.03001822336$$
$$x_{51} = 84.3593540379236$$
$$x_{52} = 43.5186495412563$$
$$x_{53} = -82.1450566023354$$
$$x_{54} = 68.6513907699746$$
$$x_{55} = 12.1027230053584$$
$$x_{56} = -57.0123153736171$$
$$x_{57} = 2.67794504458899$$
$$x_{58} = 90.6425393451032$$
$$x_{59} = -66.4370933343865$$
$$x_{60} = -94.7114272166946$$
$$x_{61} = 100.067317305873$$
$$x_{62} = 93.784131998693$$
$$x_{63} = 5.81953769817878$$
$$x_{64} = 52.9434275020257$$
$$x_{65} = -72.7202786415661$$
so we need to solve the equation:
$$2 \tan{\left(x \right)} + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \operatorname{atan}{\left(\frac{1}{2} \right)}$$
Numerical solution
$$x_{1} = -79.0034639487456$$
$$x_{2} = 65.5097981163849$$
$$x_{3} = -22.4547961841294$$
$$x_{4} = 78.076168730744$$
$$x_{5} = -35.0211667984885$$
$$x_{6} = -88.428241909515$$
$$x_{7} = -91.5698345631048$$
$$x_{8} = 15.2443156589482$$
$$x_{9} = 62.3682054627951$$
$$x_{10} = 56.0850201556155$$
$$x_{11} = -0.463647609000806$$
$$x_{12} = 74.9345760771542$$
$$x_{13} = -16.1716108769498$$
$$x_{14} = -6.74683291618039$$
$$x_{15} = -25.5963888377192$$
$$x_{16} = 46.6602421948461$$
$$x_{17} = -47.5875374128477$$
$$x_{18} = 8.96113035176857$$
$$x_{19} = -63.2955006807967$$
$$x_{20} = -19.3132035305396$$
$$x_{21} = -69.5786859879763$$
$$x_{22} = -53.8707227200273$$
$$x_{23} = -41.3043521056681$$
$$x_{24} = 21.5275009661277$$
$$x_{25} = -75.8618712951559$$
$$x_{26} = 37.2354642340767$$
$$x_{27} = -60.1539080272069$$
$$x_{28} = 24.6690936197175$$
$$x_{29} = 81.2177613843338$$
$$x_{30} = 49.8018348484359$$
$$x_{31} = -97.8530198702844$$
$$x_{32} = -44.4459447592579$$
$$x_{33} = 27.8106862733073$$
$$x_{34} = -31.8795741448987$$
$$x_{35} = 40.3770568876665$$
$$x_{36} = 18.385908312538$$
$$x_{37} = -3.6052402625906$$
$$x_{38} = -9.88842556977019$$
$$x_{39} = 30.9522789268971$$
$$x_{40} = 87.5009466915134$$
$$x_{41} = -100.994612523874$$
$$x_{42} = 34.0938715804869$$
$$x_{43} = 71.7929834235644$$
$$x_{44} = -85.2866492559252$$
$$x_{45} = 59.2266128092053$$
$$x_{46} = -38.1627594520783$$
$$x_{47} = -50.7291300664375$$
$$x_{48} = 96.9257246522828$$
$$x_{49} = -28.7379814913089$$
$$x_{50} = -13.03001822336$$
$$x_{51} = 84.3593540379236$$
$$x_{52} = 43.5186495412563$$
$$x_{53} = -82.1450566023354$$
$$x_{54} = 68.6513907699746$$
$$x_{55} = 12.1027230053584$$
$$x_{56} = -57.0123153736171$$
$$x_{57} = 2.67794504458899$$
$$x_{58} = 90.6425393451032$$
$$x_{59} = -66.4370933343865$$
$$x_{60} = -94.7114272166946$$
$$x_{61} = 100.067317305873$$
$$x_{62} = 93.784131998693$$
$$x_{63} = 5.81953769817878$$
$$x_{64} = 52.9434275020257$$
$$x_{65} = -72.7202786415661$$
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
$$2 \tan^{2}{\left(x \right)} + 2 = 0$$
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
$$2 \tan^{2}{\left(x \right)} + 2 = 0$$
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
$$4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$4 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
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:
$$y = \lim_{x \to -\infty}\left(2 \tan{\left(x \right)} + 1\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(2 \tan{\left(x \right)} + 1\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(2 \tan{\left(x \right)} + 1\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(2 \tan{\left(x \right)} + 1\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*tan(x) + 1, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{2 \tan{\left(x \right)} + 1}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{2 \tan{\left(x \right)} + 1}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{2 \tan{\left(x \right)} + 1}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{2 \tan{\left(x \right)} + 1}{x}\right)$$
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:
$$2 \tan{\left(x \right)} + 1 = 1 - 2 \tan{\left(x \right)}$$
- No
$$2 \tan{\left(x \right)} + 1 = 2 \tan{\left(x \right)} - 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$2 \tan{\left(x \right)} + 1 = 1 - 2 \tan{\left(x \right)}$$
- No
$$2 \tan{\left(x \right)} + 1 = 2 \tan{\left(x \right)} - 1$$
- No
so, the function
not is
neither even, nor odd