Mister Exam
Graphing y = tg3x+1
The solution
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f(x) = tan(3*x) + 1
$$f{\left(x \right)} = \tan{\left(3 x \right)} + 1$$
f = tan(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:
$$\tan{\left(3 x \right)} + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{12}$$
Numerical solution
$$x_{1} = 61.5228561328001$$
$$x_{2} = -72.5184304203644$$
$$x_{3} = -2.35619449019234$$
$$x_{4} = 41.6261026600648$$
$$x_{5} = -84.037603483527$$
$$x_{6} = 80.3724120543389$$
$$x_{7} = -96.6039740978861$$
$$x_{8} = 34.2957198016886$$
$$x_{9} = -55.7632696012188$$
$$x_{10} = -22.2529479629277$$
$$x_{11} = 30.1069295969022$$
$$x_{12} = 93.9859802198946$$
$$x_{13} = 87.7027949127151$$
$$x_{14} = -29.5833308213039$$
$$x_{15} = 39.5317075576716$$
$$x_{16} = -15.9697626557481$$
$$x_{17} = -11.7809724509617$$
$$x_{18} = -9.68657734856853$$
$$x_{19} = 78.2780169519457$$
$$x_{20} = 76.1836218495525$$
$$x_{21} = -31.6777259236971$$
$$x_{22} = 23.8237442897226$$
$$x_{23} = 36.3901149040818$$
$$x_{24} = -77.7544181763474$$
$$x_{25} = -50.5272818452358$$
$$x_{26} = 69.9004365423729$$
$$x_{27} = 32.2013246992954$$
$$x_{28} = -51.5744793964324$$
$$x_{29} = 37.4373124552784$$
$$x_{30} = -48.4328867428426$$
$$x_{31} = 14.3989663289532$$
$$x_{32} = 74.0892267471593$$
$$x_{33} = -75.6600230739542$$
$$x_{34} = 83.5140047079287$$
$$x_{35} = 56.2868683768171$$
$$x_{36} = 63.6172512351933$$
$$x_{37} = -18.0641577581413$$
$$x_{38} = -62.0464549083984$$
$$x_{39} = -73.565627971561$$
$$x_{40} = 15.4461638801498$$
$$x_{41} = -7.59218224617533$$
$$x_{42} = 28.012534494509$$
$$x_{43} = -37.9609112308767$$
$$x_{44} = -97.6511716490827$$
$$x_{45} = 12.30457122656$$
$$x_{46} = 47.9092879672443$$
$$x_{47} = 67.8060414399797$$
$$x_{48} = -26.4417381677141$$
$$x_{49} = -44.2440965380563$$
$$x_{50} = 54.1924732744239$$
$$x_{51} = -64.1408500107916$$
$$x_{52} = -86.1319985859202$$
$$x_{53} = -13.8753675533549$$
$$x_{54} = -0.261799387799149$$
$$x_{55} = 19.6349540849362$$
$$x_{56} = 85.6083998103219$$
$$x_{57} = -59.9520598060052$$
$$x_{58} = -66.2352451131848$$
$$x_{59} = -28.5361332701073$$
$$x_{60} = -68.329640215578$$
$$x_{61} = -70.4240353179712$$
$$x_{62} = 21.7293491873294$$
$$x_{63} = -20.1585528605345$$
$$x_{64} = 3.92699081698724$$
$$x_{65} = -79.8488132787406$$
$$x_{66} = -4.45058959258554$$
$$x_{67} = -35.8665161284835$$
$$x_{68} = 10.2101761241668$$
$$x_{69} = 59.4284610304069$$
$$x_{70} = 45.8148928648512$$
$$x_{71} = 100.269165527074$$
$$x_{72} = 8.11578102177363$$
$$x_{73} = -88.2263936883134$$
$$x_{74} = -33.7721210260903$$
$$x_{75} = 89.7971900151083$$
$$x_{76} = -92.4151838930998$$
$$x_{77} = 43.720497762458$$
$$x_{78} = 17.540558982543$$
$$x_{79} = 71.9948316447661$$
$$x_{80} = -57.857664703612$$
$$x_{81} = -46.3384916404494$$
$$x_{82} = -53.6688744988256$$
$$x_{83} = 96.0803753222878$$
$$x_{84} = 25.9181393921158$$
$$x_{85} = -24.3473430653209$$
$$x_{86} = 91.8915851175014$$
$$x_{87} = -90.3207887907066$$
$$x_{88} = 52.0980781720307$$
$$x_{89} = -40.0553063332699$$
$$x_{90} = 98.174770424681$$
$$x_{91} = 58.3812634792103$$
$$x_{92} = -81.9432083811338$$
$$x_{93} = 81.4196096055355$$
$$x_{94} = -94.5095789954929$$
$$x_{95} = 1.83259571459405$$
$$x_{96} = -42.1497014356631$$
$$x_{97} = -6.54498469497874$$
$$x_{98} = 65.7116463375865$$
$$x_{99} = 50.0036830696375$$
$$x_{100} = 6.02138591938044$$
$$x_{101} = -99.7455667514759$$
so we need to solve the equation:
$$\tan{\left(3 x \right)} + 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{\pi}{12}$$
Numerical solution
$$x_{1} = 61.5228561328001$$
$$x_{2} = -72.5184304203644$$
$$x_{3} = -2.35619449019234$$
$$x_{4} = 41.6261026600648$$
$$x_{5} = -84.037603483527$$
$$x_{6} = 80.3724120543389$$
$$x_{7} = -96.6039740978861$$
$$x_{8} = 34.2957198016886$$
$$x_{9} = -55.7632696012188$$
$$x_{10} = -22.2529479629277$$
$$x_{11} = 30.1069295969022$$
$$x_{12} = 93.9859802198946$$
$$x_{13} = 87.7027949127151$$
$$x_{14} = -29.5833308213039$$
$$x_{15} = 39.5317075576716$$
$$x_{16} = -15.9697626557481$$
$$x_{17} = -11.7809724509617$$
$$x_{18} = -9.68657734856853$$
$$x_{19} = 78.2780169519457$$
$$x_{20} = 76.1836218495525$$
$$x_{21} = -31.6777259236971$$
$$x_{22} = 23.8237442897226$$
$$x_{23} = 36.3901149040818$$
$$x_{24} = -77.7544181763474$$
$$x_{25} = -50.5272818452358$$
$$x_{26} = 69.9004365423729$$
$$x_{27} = 32.2013246992954$$
$$x_{28} = -51.5744793964324$$
$$x_{29} = 37.4373124552784$$
$$x_{30} = -48.4328867428426$$
$$x_{31} = 14.3989663289532$$
$$x_{32} = 74.0892267471593$$
$$x_{33} = -75.6600230739542$$
$$x_{34} = 83.5140047079287$$
$$x_{35} = 56.2868683768171$$
$$x_{36} = 63.6172512351933$$
$$x_{37} = -18.0641577581413$$
$$x_{38} = -62.0464549083984$$
$$x_{39} = -73.565627971561$$
$$x_{40} = 15.4461638801498$$
$$x_{41} = -7.59218224617533$$
$$x_{42} = 28.012534494509$$
$$x_{43} = -37.9609112308767$$
$$x_{44} = -97.6511716490827$$
$$x_{45} = 12.30457122656$$
$$x_{46} = 47.9092879672443$$
$$x_{47} = 67.8060414399797$$
$$x_{48} = -26.4417381677141$$
$$x_{49} = -44.2440965380563$$
$$x_{50} = 54.1924732744239$$
$$x_{51} = -64.1408500107916$$
$$x_{52} = -86.1319985859202$$
$$x_{53} = -13.8753675533549$$
$$x_{54} = -0.261799387799149$$
$$x_{55} = 19.6349540849362$$
$$x_{56} = 85.6083998103219$$
$$x_{57} = -59.9520598060052$$
$$x_{58} = -66.2352451131848$$
$$x_{59} = -28.5361332701073$$
$$x_{60} = -68.329640215578$$
$$x_{61} = -70.4240353179712$$
$$x_{62} = 21.7293491873294$$
$$x_{63} = -20.1585528605345$$
$$x_{64} = 3.92699081698724$$
$$x_{65} = -79.8488132787406$$
$$x_{66} = -4.45058959258554$$
$$x_{67} = -35.8665161284835$$
$$x_{68} = 10.2101761241668$$
$$x_{69} = 59.4284610304069$$
$$x_{70} = 45.8148928648512$$
$$x_{71} = 100.269165527074$$
$$x_{72} = 8.11578102177363$$
$$x_{73} = -88.2263936883134$$
$$x_{74} = -33.7721210260903$$
$$x_{75} = 89.7971900151083$$
$$x_{76} = -92.4151838930998$$
$$x_{77} = 43.720497762458$$
$$x_{78} = 17.540558982543$$
$$x_{79} = 71.9948316447661$$
$$x_{80} = -57.857664703612$$
$$x_{81} = -46.3384916404494$$
$$x_{82} = -53.6688744988256$$
$$x_{83} = 96.0803753222878$$
$$x_{84} = 25.9181393921158$$
$$x_{85} = -24.3473430653209$$
$$x_{86} = 91.8915851175014$$
$$x_{87} = -90.3207887907066$$
$$x_{88} = 52.0980781720307$$
$$x_{89} = -40.0553063332699$$
$$x_{90} = 98.174770424681$$
$$x_{91} = 58.3812634792103$$
$$x_{92} = -81.9432083811338$$
$$x_{93} = 81.4196096055355$$
$$x_{94} = -94.5095789954929$$
$$x_{95} = 1.83259571459405$$
$$x_{96} = -42.1497014356631$$
$$x_{97} = -6.54498469497874$$
$$x_{98} = 65.7116463375865$$
$$x_{99} = 50.0036830696375$$
$$x_{100} = 6.02138591938044$$
$$x_{101} = -99.7455667514759$$
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
$$3 \tan^{2}{\left(3 x \right)} + 3 = 0$$
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
$$3 \tan^{2}{\left(3 x \right)} + 3 = 0$$
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
$$18 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} = 0$$
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
$$18 \left(\tan^{2}{\left(3 x \right)} + 1\right) \tan{\left(3 x \right)} = 0$$
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:
$$y = \lim_{x \to -\infty}\left(\tan{\left(3 x \right)} + 1\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(3 x \right)} + 1\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\tan{\left(3 x \right)} + 1\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(3 x \right)} + 1\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(3*x) + 1, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)} + 1}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)} + 1}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)} + 1}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)} + 1}{x}\right)$$
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:
$$\tan{\left(3 x \right)} + 1 = 1 - \tan{\left(3 x \right)}$$
- No
$$\tan{\left(3 x \right)} + 1 = \tan{\left(3 x \right)} - 1$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(3 x \right)} + 1 = 1 - \tan{\left(3 x \right)}$$
- No
$$\tan{\left(3 x \right)} + 1 = \tan{\left(3 x \right)} - 1$$
- No
so, the function
not is
neither even, nor odd