Mister Exam
Graphing y = y=tgx-2
The solution
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f(x) = tan(x) - 2
$$f{\left(x \right)} = \tan{\left(x \right)} - 2$$
f = tan(x) - 2
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:
$$\tan{\left(x \right)} - 2 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \operatorname{atan}{\left(2 \right)}$$
Numerical solution
$$x_{1} = -61.7247043540018$$
$$x_{2} = -17.7424072037447$$
$$x_{3} = 67.0805944431797$$
$$x_{4} = -46.0167410860528$$
$$x_{5} = -52.2999263932324$$
$$x_{6} = -55.4415190468222$$
$$x_{7} = -1258.67150537171$$
$$x_{8} = -74.2910749683609$$
$$x_{9} = -30.3087778181038$$
$$x_{10} = -24.0255925109243$$
$$x_{11} = 101.638113632667$$
$$x_{12} = -96.2822235434895$$
$$x_{13} = 76.5053724039491$$
$$x_{14} = -39.7335557788732$$
$$x_{15} = 10.5319266785635$$
$$x_{16} = 63.93900178959$$
$$x_{17} = -8.31762924297529$$
$$x_{18} = 51.3726311752308$$
$$x_{19} = -33.4503704716936$$
$$x_{20} = 54.5142238288206$$
$$x_{21} = 92.2133356718981$$
$$x_{22} = 41.9478532144614$$
$$x_{23} = -49.1583337396426$$
$$x_{24} = -36.5919631252834$$
$$x_{25} = -102.565408850669$$
$$x_{26} = 23.0982972929226$$
$$x_{27} = -99.4238161970793$$
$$x_{28} = 35.6646679072818$$
$$x_{29} = 19.9567046393328$$
$$x_{30} = 4.24874137138388$$
$$x_{31} = 98.4965209790777$$
$$x_{32} = 85.9301503647185$$
$$x_{33} = -89.9990382363099$$
$$x_{34} = 48.231038521641$$
$$x_{35} = 73.3637797503593$$
$$x_{36} = 70.2221870967695$$
$$x_{37} = 1.10714871779409$$
$$x_{38} = -27.167185164514$$
$$x_{39} = 16.8151119857431$$
$$x_{40} = 32.523075253692$$
$$x_{41} = 13.6735193321533$$
$$x_{42} = 79.6469650575389$$
$$x_{43} = -11.4592218965651$$
$$x_{44} = -77.4326676219507$$
$$x_{45} = 26.2398899465124$$
$$x_{46} = 60.7974091360002$$
$$x_{47} = -68.0078896611814$$
$$x_{48} = 95.3549283254879$$
$$x_{49} = -2.0344439357957$$
$$x_{50} = -83.7158529291303$$
so we need to solve the equation:
$$\tan{\left(x \right)} - 2 = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \operatorname{atan}{\left(2 \right)}$$
Numerical solution
$$x_{1} = -61.7247043540018$$
$$x_{2} = -17.7424072037447$$
$$x_{3} = 67.0805944431797$$
$$x_{4} = -46.0167410860528$$
$$x_{5} = -52.2999263932324$$
$$x_{6} = -55.4415190468222$$
$$x_{7} = -1258.67150537171$$
$$x_{8} = -74.2910749683609$$
$$x_{9} = -30.3087778181038$$
$$x_{10} = -24.0255925109243$$
$$x_{11} = 101.638113632667$$
$$x_{12} = -96.2822235434895$$
$$x_{13} = 76.5053724039491$$
$$x_{14} = -39.7335557788732$$
$$x_{15} = 10.5319266785635$$
$$x_{16} = 63.93900178959$$
$$x_{17} = -8.31762924297529$$
$$x_{18} = 51.3726311752308$$
$$x_{19} = -33.4503704716936$$
$$x_{20} = 54.5142238288206$$
$$x_{21} = 92.2133356718981$$
$$x_{22} = 41.9478532144614$$
$$x_{23} = -49.1583337396426$$
$$x_{24} = -36.5919631252834$$
$$x_{25} = -102.565408850669$$
$$x_{26} = 23.0982972929226$$
$$x_{27} = -99.4238161970793$$
$$x_{28} = 35.6646679072818$$
$$x_{29} = 19.9567046393328$$
$$x_{30} = 4.24874137138388$$
$$x_{31} = 98.4965209790777$$
$$x_{32} = 85.9301503647185$$
$$x_{33} = -89.9990382363099$$
$$x_{34} = 48.231038521641$$
$$x_{35} = 73.3637797503593$$
$$x_{36} = 70.2221870967695$$
$$x_{37} = 1.10714871779409$$
$$x_{38} = -27.167185164514$$
$$x_{39} = 16.8151119857431$$
$$x_{40} = 32.523075253692$$
$$x_{41} = 13.6735193321533$$
$$x_{42} = 79.6469650575389$$
$$x_{43} = -11.4592218965651$$
$$x_{44} = -77.4326676219507$$
$$x_{45} = 26.2398899465124$$
$$x_{46} = 60.7974091360002$$
$$x_{47} = -68.0078896611814$$
$$x_{48} = 95.3549283254879$$
$$x_{49} = -2.0344439357957$$
$$x_{50} = -83.7158529291303$$
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
$$\tan^{2}{\left(x \right)} + 1 = 0$$
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
$$\tan^{2}{\left(x \right)} + 1 = 0$$
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
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0$$
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
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} = 0$$
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:
$$y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} - 2\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(x \right)} - 2\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} - 2\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\tan{\left(x \right)} - 2\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x) - 2, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} - 2}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} - 2}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} - 2}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} - 2}{x}\right)$$
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:
$$\tan{\left(x \right)} - 2 = - \tan{\left(x \right)} - 2$$
- No
$$\tan{\left(x \right)} - 2 = \tan{\left(x \right)} + 2$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\tan{\left(x \right)} - 2 = - \tan{\left(x \right)} - 2$$
- No
$$\tan{\left(x \right)} - 2 = \tan{\left(x \right)} + 2$$
- No
so, the function
not is
neither even, nor odd