Mister Exam
Graphing y = tg2x-ctg3x+1
The solution
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f(x) = tan(2*x) - cot(3*x) + 1
$$f{\left(x \right)} = \left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1$$
f = tan(2*x) - cot(3*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:
$$\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 91.3095296112788$$
$$x_{2} = -36.3120940895226$$
$$x_{3} = -85.2686902599117$$
$$x_{4} = 42.2277222502222$$
$$x_{5} = 13.9533883679141$$
$$x_{6} = -9.8704665737567$$
$$x_{7} = -38.1448004560648$$
$$x_{8} = 27.8286452693208$$
$$x_{9} = 56.102979151629$$
$$x_{10} = -6.07984265000475$$
$$x_{11} = 75.6015663433299$$
$$x_{12} = -91.5518755670913$$
$$x_{13} = -43.7789544930823$$
$$x_{14} = -63.2775416847832$$
$$x_{15} = -89.7191692005491$$
$$x_{16} = 93.8020909947065$$
$$x_{17} = -21.7878059179537$$
$$x_{18} = -19.2952445345261$$
$$x_{19} = -47.5695784168342$$
$$x_{20} = 49.8197938444494$$
$$x_{21} = 31.6192691930728$$
$$x_{22} = 37.9024545002524$$
$$x_{23} = -100.327622257699$$
$$x_{24} = -3.58728126657711$$
$$x_{25} = -12.3630279571843$$
$$x_{26} = 15.9113059251238$$
$$x_{27} = -52.0200573574716$$
$$x_{28} = 98.3120291517831$$
$$x_{29} = -65.7701030682108$$
$$x_{30} = 64.2188708253508$$
$$x_{31} = -78.33647368257$$
$$x_{32} = -82.1270976063219$$
$$x_{33} = -56.3453251074414$$
$$x_{34} = 70.0376952694749$$
$$x_{35} = 53.6104177682013$$
$$x_{36} = 3.34493531076463$$
$$x_{37} = 66.1767883825605$$
$$x_{38} = 20.2365736750937$$
$$x_{39} = 12.1206820013719$$
$$x_{40} = -45.736872050292$$
$$x_{41} = -58.3032426646512$$
$$x_{42} = -14.3209455143941$$
$$x_{43} = -53.8527637240138$$
$$x_{44} = 44.1856398074319$$
$$x_{45} = 88.1679369576891$$
$$x_{46} = 68.6693497659881$$
$$x_{47} = 35.9445369430426$$
$$x_{48} = 25.3360838858932$$
$$x_{49} = -25.5784298417057$$
$$x_{50} = -63.7920330422437$$
$$x_{51} = -96.0023545077287$$
$$x_{52} = -31.8616151488853$$
$$x_{53} = -74.0112059326001$$
$$x_{54} = -30.028908782343$$
$$x_{55} = 59.8936030753809$$
$$x_{56} = 57.9356855181712$$
$$x_{57} = -1.75457490003489$$
$$x_{58} = 79.9268340932997$$
$$x_{59} = 29.661351635863$$
$$x_{60} = 2.69590404060247$$
$$x_{61} = -8.03776020721448$$
$$x_{62} = 51.6525002109916$$
$$x_{63} = 81.8847516505095$$
$$x_{64} = -80.2943912397797$$
$$x_{65} = 100.085276301886$$
$$x_{66} = -75.8439122991424$$
$$x_{67} = 9.62812061794421$$
$$x_{68} = 84.3773130339371$$
$$x_{69} = 71.8109424195779$$
$$x_{70} = 40.39501588368$$
$$x_{71} = -16.1536518809363$$
$$x_{72} = 7.67020306073449$$
$$x_{73} = -87.7612516433394$$
$$x_{74} = -28.0709912251333$$
$$x_{75} = 86.2100194004793$$
$$x_{76} = 73.6436487861201$$
$$x_{77} = -34.3541765323129$$
$$x_{78} = -60.1359490311934$$
$$x_{79} = 95.6347973612487$$
$$x_{80} = 78.0941277267575$$
$$x_{81} = 22.1944912323034$$
$$x_{82} = -67.7280206254206$$
$$x_{83} = 47.3272324610217$$
$$x_{84} = 34.1118305765004$$
$$x_{85} = 62.3861644588085$$
$$x_{86} = -69.5607269919628$$
$$x_{87} = 5.83749669419227$$
$$x_{88} = -23.7457234751634$$
$$x_{89} = -85.7831816173722$$
$$x_{90} = -18.6462132643639$$
$$x_{91} = -50.0621398002619$$
$$x_{92} = 24.687052615731$$
$$x_{93} = 18.4038673085514$$
$$x_{94} = -41.2863931096546$$
$$x_{95} = -94.044436950519$$
$$x_{96} = -97.8350608742709$$
$$x_{97} = -72.0532883753904$$
so we need to solve the equation:
$$\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 91.3095296112788$$
$$x_{2} = -36.3120940895226$$
$$x_{3} = -85.2686902599117$$
$$x_{4} = 42.2277222502222$$
$$x_{5} = 13.9533883679141$$
$$x_{6} = -9.8704665737567$$
$$x_{7} = -38.1448004560648$$
$$x_{8} = 27.8286452693208$$
$$x_{9} = 56.102979151629$$
$$x_{10} = -6.07984265000475$$
$$x_{11} = 75.6015663433299$$
$$x_{12} = -91.5518755670913$$
$$x_{13} = -43.7789544930823$$
$$x_{14} = -63.2775416847832$$
$$x_{15} = -89.7191692005491$$
$$x_{16} = 93.8020909947065$$
$$x_{17} = -21.7878059179537$$
$$x_{18} = -19.2952445345261$$
$$x_{19} = -47.5695784168342$$
$$x_{20} = 49.8197938444494$$
$$x_{21} = 31.6192691930728$$
$$x_{22} = 37.9024545002524$$
$$x_{23} = -100.327622257699$$
$$x_{24} = -3.58728126657711$$
$$x_{25} = -12.3630279571843$$
$$x_{26} = 15.9113059251238$$
$$x_{27} = -52.0200573574716$$
$$x_{28} = 98.3120291517831$$
$$x_{29} = -65.7701030682108$$
$$x_{30} = 64.2188708253508$$
$$x_{31} = -78.33647368257$$
$$x_{32} = -82.1270976063219$$
$$x_{33} = -56.3453251074414$$
$$x_{34} = 70.0376952694749$$
$$x_{35} = 53.6104177682013$$
$$x_{36} = 3.34493531076463$$
$$x_{37} = 66.1767883825605$$
$$x_{38} = 20.2365736750937$$
$$x_{39} = 12.1206820013719$$
$$x_{40} = -45.736872050292$$
$$x_{41} = -58.3032426646512$$
$$x_{42} = -14.3209455143941$$
$$x_{43} = -53.8527637240138$$
$$x_{44} = 44.1856398074319$$
$$x_{45} = 88.1679369576891$$
$$x_{46} = 68.6693497659881$$
$$x_{47} = 35.9445369430426$$
$$x_{48} = 25.3360838858932$$
$$x_{49} = -25.5784298417057$$
$$x_{50} = -63.7920330422437$$
$$x_{51} = -96.0023545077287$$
$$x_{52} = -31.8616151488853$$
$$x_{53} = -74.0112059326001$$
$$x_{54} = -30.028908782343$$
$$x_{55} = 59.8936030753809$$
$$x_{56} = 57.9356855181712$$
$$x_{57} = -1.75457490003489$$
$$x_{58} = 79.9268340932997$$
$$x_{59} = 29.661351635863$$
$$x_{60} = 2.69590404060247$$
$$x_{61} = -8.03776020721448$$
$$x_{62} = 51.6525002109916$$
$$x_{63} = 81.8847516505095$$
$$x_{64} = -80.2943912397797$$
$$x_{65} = 100.085276301886$$
$$x_{66} = -75.8439122991424$$
$$x_{67} = 9.62812061794421$$
$$x_{68} = 84.3773130339371$$
$$x_{69} = 71.8109424195779$$
$$x_{70} = 40.39501588368$$
$$x_{71} = -16.1536518809363$$
$$x_{72} = 7.67020306073449$$
$$x_{73} = -87.7612516433394$$
$$x_{74} = -28.0709912251333$$
$$x_{75} = 86.2100194004793$$
$$x_{76} = 73.6436487861201$$
$$x_{77} = -34.3541765323129$$
$$x_{78} = -60.1359490311934$$
$$x_{79} = 95.6347973612487$$
$$x_{80} = 78.0941277267575$$
$$x_{81} = 22.1944912323034$$
$$x_{82} = -67.7280206254206$$
$$x_{83} = 47.3272324610217$$
$$x_{84} = 34.1118305765004$$
$$x_{85} = 62.3861644588085$$
$$x_{86} = -69.5607269919628$$
$$x_{87} = 5.83749669419227$$
$$x_{88} = -23.7457234751634$$
$$x_{89} = -85.7831816173722$$
$$x_{90} = -18.6462132643639$$
$$x_{91} = -50.0621398002619$$
$$x_{92} = 24.687052615731$$
$$x_{93} = 18.4038673085514$$
$$x_{94} = -41.2863931096546$$
$$x_{95} = -94.044436950519$$
$$x_{96} = -97.8350608742709$$
$$x_{97} = -72.0532883753904$$
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
$$2 \tan^{2}{\left(2 x \right)} + 3 \cot^{2}{\left(3 x \right)} + 5 = 0$$
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
$$2 \tan^{2}{\left(2 x \right)} + 3 \cot^{2}{\left(3 x \right)} + 5 = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
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:
$$y = \lim_{x \to -\infty}\left(\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(2*x) - cot(3*x) + 1, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1}{x}\right)$$
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:
$$\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1 = - \tan{\left(2 x \right)} + \cot{\left(3 x \right)} + 1$$
- No
$$\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1 = \tan{\left(2 x \right)} - \cot{\left(3 x \right)} - 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1 = - \tan{\left(2 x \right)} + \cot{\left(3 x \right)} + 1$$
- No
$$\left(\tan{\left(2 x \right)} - \cot{\left(3 x \right)}\right) + 1 = \tan{\left(2 x \right)} - \cot{\left(3 x \right)} - 1$$
- No
so, the function
not is
neither even, nor odd