Mister Exam
Graphing y = -2tg(x-1)+3
The solution
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f(x) = -2*tan(x - 1) + 3
$$f{\left(x \right)} = 3 - 2 \tan{\left(x - 1 \right)}$$
f = 3 - 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:
$$3 - 2 \tan{\left(x - 1 \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \operatorname{atan}{\left(\frac{3}{2} \right)} + 1$$
Numerical solution
$$x_{1} = 74.2394247558126$$
$$x_{2} = -1.15879893034246$$
$$x_{3} = -79.6986152700873$$
$$x_{4} = 93.0889806773513$$
$$x_{5} = -92.2649858844465$$
$$x_{6} = 49.1066835270942$$
$$x_{7} = 8.26597903042692$$
$$x_{8} = -38.85791077342$$
$$x_{9} = 55.3898688342738$$
$$x_{10} = 1.98279372324733$$
$$x_{11} = -32.5747254662404$$
$$x_{12} = -95.4065785380363$$
$$x_{13} = 71.0978321022228$$
$$x_{14} = 58.5314614878636$$
$$x_{15} = -63.9906520021383$$
$$x_{16} = -70.2738373093179$$
$$x_{17} = -82.8402079236771$$
$$x_{18} = -10.5835768911118$$
$$x_{19} = -85.9818005772669$$
$$x_{20} = -89.1233932308567$$
$$x_{21} = 17.6907569911963$$
$$x_{22} = 89.9473880237615$$
$$x_{23} = 33.3987202591453$$
$$x_{24} = -60.8490593485485$$
$$x_{25} = -57.7074666949587$$
$$x_{26} = -48.2826887341894$$
$$x_{27} = 45.9650908735044$$
$$x_{28} = -98.5481711916261$$
$$x_{29} = 77.3810174094024$$
$$x_{30} = -76.5570226164975$$
$$x_{31} = 99.3721659845309$$
$$x_{32} = -35.7163181198302$$
$$x_{33} = 5.12438637683712$$
$$x_{34} = 64.8146467950432$$
$$x_{35} = 11.4075716840167$$
$$x_{36} = 23.9739422983759$$
$$x_{37} = -45.1410960805996$$
$$x_{38} = -67.1322446557281$$
$$x_{39} = 86.8057953701718$$
$$x_{40} = -16.8667621982914$$
$$x_{41} = -51.4242813877792$$
$$x_{42} = -29.4331328126506$$
$$x_{43} = -41.9995034270098$$
$$x_{44} = 42.8234982199146$$
$$x_{45} = 36.5403129127351$$
$$x_{46} = -13.7251695447016$$
$$x_{47} = 61.6730541414534$$
$$x_{48} = 14.5491643376065$$
$$x_{49} = -26.2915401590608$$
$$x_{50} = 30.2571276055555$$
$$x_{51} = -54.565874041369$$
$$x_{52} = -73.4154299629077$$
$$x_{53} = -4.30039158393226$$
$$x_{54} = 102.513758638121$$
$$x_{55} = 52.248276180684$$
$$x_{56} = -7.44198423752205$$
$$x_{57} = -23.149947505471$$
$$x_{58} = 67.956239448633$$
$$x_{59} = 80.5226100629922$$
$$x_{60} = 39.6819055663249$$
$$x_{61} = 83.664202716582$$
$$x_{62} = -20.0083548518812$$
$$x_{63} = 96.2305733309411$$
$$x_{64} = 27.1155349519657$$
$$x_{65} = 20.8323496447861$$
so we need to solve the equation:
$$3 - 2 \tan{\left(x - 1 \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \operatorname{atan}{\left(\frac{3}{2} \right)} + 1$$
Numerical solution
$$x_{1} = 74.2394247558126$$
$$x_{2} = -1.15879893034246$$
$$x_{3} = -79.6986152700873$$
$$x_{4} = 93.0889806773513$$
$$x_{5} = -92.2649858844465$$
$$x_{6} = 49.1066835270942$$
$$x_{7} = 8.26597903042692$$
$$x_{8} = -38.85791077342$$
$$x_{9} = 55.3898688342738$$
$$x_{10} = 1.98279372324733$$
$$x_{11} = -32.5747254662404$$
$$x_{12} = -95.4065785380363$$
$$x_{13} = 71.0978321022228$$
$$x_{14} = 58.5314614878636$$
$$x_{15} = -63.9906520021383$$
$$x_{16} = -70.2738373093179$$
$$x_{17} = -82.8402079236771$$
$$x_{18} = -10.5835768911118$$
$$x_{19} = -85.9818005772669$$
$$x_{20} = -89.1233932308567$$
$$x_{21} = 17.6907569911963$$
$$x_{22} = 89.9473880237615$$
$$x_{23} = 33.3987202591453$$
$$x_{24} = -60.8490593485485$$
$$x_{25} = -57.7074666949587$$
$$x_{26} = -48.2826887341894$$
$$x_{27} = 45.9650908735044$$
$$x_{28} = -98.5481711916261$$
$$x_{29} = 77.3810174094024$$
$$x_{30} = -76.5570226164975$$
$$x_{31} = 99.3721659845309$$
$$x_{32} = -35.7163181198302$$
$$x_{33} = 5.12438637683712$$
$$x_{34} = 64.8146467950432$$
$$x_{35} = 11.4075716840167$$
$$x_{36} = 23.9739422983759$$
$$x_{37} = -45.1410960805996$$
$$x_{38} = -67.1322446557281$$
$$x_{39} = 86.8057953701718$$
$$x_{40} = -16.8667621982914$$
$$x_{41} = -51.4242813877792$$
$$x_{42} = -29.4331328126506$$
$$x_{43} = -41.9995034270098$$
$$x_{44} = 42.8234982199146$$
$$x_{45} = 36.5403129127351$$
$$x_{46} = -13.7251695447016$$
$$x_{47} = 61.6730541414534$$
$$x_{48} = 14.5491643376065$$
$$x_{49} = -26.2915401590608$$
$$x_{50} = 30.2571276055555$$
$$x_{51} = -54.565874041369$$
$$x_{52} = -73.4154299629077$$
$$x_{53} = -4.30039158393226$$
$$x_{54} = 102.513758638121$$
$$x_{55} = 52.248276180684$$
$$x_{56} = -7.44198423752205$$
$$x_{57} = -23.149947505471$$
$$x_{58} = 67.956239448633$$
$$x_{59} = 80.5226100629922$$
$$x_{60} = 39.6819055663249$$
$$x_{61} = 83.664202716582$$
$$x_{62} = -20.0083548518812$$
$$x_{63} = 96.2305733309411$$
$$x_{64} = 27.1155349519657$$
$$x_{65} = 20.8323496447861$$
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
$$- 2 \tan^{2}{\left(x - 1 \right)} - 2 = 0$$
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
$$- 2 \tan^{2}{\left(x - 1 \right)} - 2 = 0$$
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
$$- 4 \left(\tan^{2}{\left(x - 1 \right)} + 1\right) \tan{\left(x - 1 \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 1$$
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
$$\left(-\infty, 1\right]$$
Convex at the intervals
$$\left[1, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- 4 \left(\tan^{2}{\left(x - 1 \right)} + 1\right) \tan{\left(x - 1 \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 1$$
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
$$\left(-\infty, 1\right]$$
Convex at the intervals
$$\left[1, \infty\right)$$
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:
$$y = \lim_{x \to -\infty}\left(3 - 2 \tan{\left(x - 1 \right)}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(3 - 2 \tan{\left(x - 1 \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(3 - 2 \tan{\left(x - 1 \right)}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(3 - 2 \tan{\left(x - 1 \right)}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of -2*tan(x - 1) + 3, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{3 - 2 \tan{\left(x - 1 \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{3 - 2 \tan{\left(x - 1 \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{3 - 2 \tan{\left(x - 1 \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{3 - 2 \tan{\left(x - 1 \right)}}{x}\right)$$
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:
$$3 - 2 \tan{\left(x - 1 \right)} = 2 \tan{\left(x + 1 \right)} + 3$$
- No
$$3 - 2 \tan{\left(x - 1 \right)} = - 2 \tan{\left(x + 1 \right)} - 3$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$3 - 2 \tan{\left(x - 1 \right)} = 2 \tan{\left(x + 1 \right)} + 3$$
- No
$$3 - 2 \tan{\left(x - 1 \right)} = - 2 \tan{\left(x + 1 \right)} - 3$$
- No
so, the function
not is
neither even, nor odd