Mister Exam
Graphing y = tan(x)-1/(4*x)-1
The solution
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f(x) = tan(x) - --- - 1
4*x
$$f{\left(x \right)} = \left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1$$
f = tan(x) - 1/(4*x) - 1
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 95.0344913544903$$
$$x_{2} = 60.4777211923292$$
$$x_{3} = -93.4637206519092$$
$$x_{4} = -77.7560283558673$$
$$x_{5} = 91.8929435473717$$
$$x_{6} = 41.6290963693637$$
$$x_{7} = 82.4683205951513$$
$$x_{8} = 88.7513989106074$$
$$x_{9} = -24.3525024452919$$
$$x_{10} = 57.3362413039713$$
$$x_{11} = -62.0484735262681$$
$$x_{12} = 25.9229381995883$$
$$x_{13} = -80.8975583839916$$
$$x_{14} = 101.317595301642$$
$$x_{15} = -18.0711229382539$$
$$x_{16} = -30.6346254172489$$
$$x_{17} = -27.4935029811728$$
$$x_{18} = -43.1998009073906$$
$$x_{19} = 85.609857792695$$
$$x_{20} = 51.0533230563024$$
$$x_{21} = 32.205191035161$$
$$x_{22} = -14.9310074230454$$
$$x_{23} = -8.65403456872203$$
$$x_{24} = 98.1760420280135$$
$$x_{25} = -84.039093101098$$
$$x_{26} = 13.3610373580831$$
$$x_{27} = -71.4729848439766$$
$$x_{28} = -87.1806319995788$$
$$x_{29} = 79.3267877829935$$
$$x_{30} = 35.3464412999816$$
$$x_{31} = 19.6412779024164$$
$$x_{32} = -46.3411963133816$$
$$x_{33} = -96.6052696988463$$
$$x_{34} = -40.0584365319855$$
$$x_{35} = 44.7704795511379$$
$$x_{36} = 10.2222560339859$$
$$x_{37} = 69.9022215606051$$
$$x_{38} = 76.1852598978093$$
$$x_{39} = -58.9069887533201$$
$$x_{40} = -36.9171111340582$$
$$x_{41} = 7.08591656277011$$
$$x_{42} = 29.0640144525706$$
$$x_{43} = -99.746821495994$$
$$x_{44} = -55.7655161617889$$
$$x_{45} = 38.4877472680578$$
$$x_{46} = -52.6240579382353$$
$$x_{47} = -90.3221746425414$$
$$x_{48} = 66.7607127461121$$
$$x_{49} = -21.2116782555737$$
$$x_{50} = 73.0437375741928$$
$$x_{51} = 63.6192121950275$$
$$x_{52} = 22.7820035307747$$
$$x_{53} = -2.41082493019699$$
$$x_{54} = -68.3314728841616$$
$$x_{55} = -74.6145036099593$$
$$x_{56} = -49.4826168258104$$
$$x_{57} = -11.7916863086823$$
$$x_{58} = -33.775835627013$$
$$x_{59} = 3.95759900591204$$
$$x_{60} = 16.5008796888555$$
$$x_{61} = -65.1899687163157$$
$$x_{62} = -5.52094852985428$$
$$x_{63} = 54.19477445812$$
$$x_{64} = 47.9118901281572$$
so we need to solve the equation:
$$\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1 = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 95.0344913544903$$
$$x_{2} = 60.4777211923292$$
$$x_{3} = -93.4637206519092$$
$$x_{4} = -77.7560283558673$$
$$x_{5} = 91.8929435473717$$
$$x_{6} = 41.6290963693637$$
$$x_{7} = 82.4683205951513$$
$$x_{8} = 88.7513989106074$$
$$x_{9} = -24.3525024452919$$
$$x_{10} = 57.3362413039713$$
$$x_{11} = -62.0484735262681$$
$$x_{12} = 25.9229381995883$$
$$x_{13} = -80.8975583839916$$
$$x_{14} = 101.317595301642$$
$$x_{15} = -18.0711229382539$$
$$x_{16} = -30.6346254172489$$
$$x_{17} = -27.4935029811728$$
$$x_{18} = -43.1998009073906$$
$$x_{19} = 85.609857792695$$
$$x_{20} = 51.0533230563024$$
$$x_{21} = 32.205191035161$$
$$x_{22} = -14.9310074230454$$
$$x_{23} = -8.65403456872203$$
$$x_{24} = 98.1760420280135$$
$$x_{25} = -84.039093101098$$
$$x_{26} = 13.3610373580831$$
$$x_{27} = -71.4729848439766$$
$$x_{28} = -87.1806319995788$$
$$x_{29} = 79.3267877829935$$
$$x_{30} = 35.3464412999816$$
$$x_{31} = 19.6412779024164$$
$$x_{32} = -46.3411963133816$$
$$x_{33} = -96.6052696988463$$
$$x_{34} = -40.0584365319855$$
$$x_{35} = 44.7704795511379$$
$$x_{36} = 10.2222560339859$$
$$x_{37} = 69.9022215606051$$
$$x_{38} = 76.1852598978093$$
$$x_{39} = -58.9069887533201$$
$$x_{40} = -36.9171111340582$$
$$x_{41} = 7.08591656277011$$
$$x_{42} = 29.0640144525706$$
$$x_{43} = -99.746821495994$$
$$x_{44} = -55.7655161617889$$
$$x_{45} = 38.4877472680578$$
$$x_{46} = -52.6240579382353$$
$$x_{47} = -90.3221746425414$$
$$x_{48} = 66.7607127461121$$
$$x_{49} = -21.2116782555737$$
$$x_{50} = 73.0437375741928$$
$$x_{51} = 63.6192121950275$$
$$x_{52} = 22.7820035307747$$
$$x_{53} = -2.41082493019699$$
$$x_{54} = -68.3314728841616$$
$$x_{55} = -74.6145036099593$$
$$x_{56} = -49.4826168258104$$
$$x_{57} = -11.7916863086823$$
$$x_{58} = -33.775835627013$$
$$x_{59} = 3.95759900591204$$
$$x_{60} = 16.5008796888555$$
$$x_{61} = -65.1899687163157$$
$$x_{62} = -5.52094852985428$$
$$x_{63} = 54.19477445812$$
$$x_{64} = 47.9118901281572$$
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
$$\tan^{2}{\left(x \right)} + 1 + \frac{1}{4 x^{2}} = 0$$
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
$$\tan^{2}{\left(x \right)} + 1 + \frac{1}{4 x^{2}} = 0$$
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
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{1}{2 x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 59.6902615937248$$
$$x_{2} = 31.4159345987753$$
$$x_{3} = -12.566496593124$$
$$x_{4} = -21.9911720820024$$
$$x_{5} = 28.2743449424922$$
$$x_{6} = 9.42507655767126$$
$$x_{7} = -6.28419268165633$$
$$x_{8} = -25.1327569765083$$
$$x_{9} = 78.5398168557694$$
$$x_{10} = -15.7080277702229$$
$$x_{11} = 34.5575252472483$$
$$x_{12} = 69.1150391361959$$
$$x_{13} = 84.8230020565613$$
$$x_{14} = -69.1150391361959$$
$$x_{15} = 100.530965160933$$
$$x_{16} = -40.8407081666179$$
$$x_{17} = 56.5486691471408$$
$$x_{18} = -18.8495932494818$$
$$x_{19} = -56.5486691471408$$
$$x_{20} = 40.8407081666179$$
$$x_{21} = -43.9823000886252$$
$$x_{22} = -47.1238921928491$$
$$x_{23} = -53.4070767521588$$
$$x_{24} = -3.14959356380938$$
$$x_{25} = -78.5398168557694$$
$$x_{26} = -75.3982242694076$$
$$x_{27} = 87.9645946678103$$
$$x_{28} = 25.1327569765083$$
$$x_{29} = -87.9645946678103$$
$$x_{30} = 75.3982242694076$$
$$x_{31} = -100.530965160933$$
$$x_{32} = 6.28419268165633$$
$$x_{33} = 65.9734465960134$$
$$x_{34} = -31.4159345987753$$
$$x_{35} = 47.1238921928491$$
$$x_{36} = -34.5575252472483$$
$$x_{37} = -72.2566316952498$$
$$x_{38} = -62.8318540796563$$
$$x_{39} = -50.2654844259139$$
$$x_{40} = 97.3893725319319$$
$$x_{41} = -97.3893725319319$$
$$x_{42} = -91.1061872846991$$
$$x_{43} = 12.566496593124$$
$$x_{44} = -37.6991165090964$$
$$x_{45} = 15.7080277702229$$
$$x_{46} = 50.2654844259139$$
$$x_{47} = 3.14959356380938$$
$$x_{48} = 43.9823000886252$$
$$x_{49} = 94.2477799063191$$
$$x_{50} = 53.4070767521588$$
$$x_{51} = -84.8230020565613$$
$$x_{52} = 72.2566316952498$$
$$x_{53} = -59.6902615937248$$
$$x_{54} = 21.9911720820024$$
$$x_{55} = -65.9734465960134$$
$$x_{56} = -81.6814094520786$$
$$x_{57} = 91.1061872846991$$
$$x_{58} = 18.8495932494818$$
$$x_{59} = 62.8318540796563$$
$$x_{60} = 37.6991165090964$$
$$x_{61} = -28.2743449424922$$
$$x_{62} = 81.6814094520786$$
$$x_{63} = -9.42507655767126$$
$$x_{64} = -94.2477799063191$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{1}{2 x^{3}}\right) = \infty$$
$$\lim_{x \to 0^+}\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{1}{2 x^{3}}\right) = -\infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
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
$$\left[100.530965160933, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -100.530965160933\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{1}{2 x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 59.6902615937248$$
$$x_{2} = 31.4159345987753$$
$$x_{3} = -12.566496593124$$
$$x_{4} = -21.9911720820024$$
$$x_{5} = 28.2743449424922$$
$$x_{6} = 9.42507655767126$$
$$x_{7} = -6.28419268165633$$
$$x_{8} = -25.1327569765083$$
$$x_{9} = 78.5398168557694$$
$$x_{10} = -15.7080277702229$$
$$x_{11} = 34.5575252472483$$
$$x_{12} = 69.1150391361959$$
$$x_{13} = 84.8230020565613$$
$$x_{14} = -69.1150391361959$$
$$x_{15} = 100.530965160933$$
$$x_{16} = -40.8407081666179$$
$$x_{17} = 56.5486691471408$$
$$x_{18} = -18.8495932494818$$
$$x_{19} = -56.5486691471408$$
$$x_{20} = 40.8407081666179$$
$$x_{21} = -43.9823000886252$$
$$x_{22} = -47.1238921928491$$
$$x_{23} = -53.4070767521588$$
$$x_{24} = -3.14959356380938$$
$$x_{25} = -78.5398168557694$$
$$x_{26} = -75.3982242694076$$
$$x_{27} = 87.9645946678103$$
$$x_{28} = 25.1327569765083$$
$$x_{29} = -87.9645946678103$$
$$x_{30} = 75.3982242694076$$
$$x_{31} = -100.530965160933$$
$$x_{32} = 6.28419268165633$$
$$x_{33} = 65.9734465960134$$
$$x_{34} = -31.4159345987753$$
$$x_{35} = 47.1238921928491$$
$$x_{36} = -34.5575252472483$$
$$x_{37} = -72.2566316952498$$
$$x_{38} = -62.8318540796563$$
$$x_{39} = -50.2654844259139$$
$$x_{40} = 97.3893725319319$$
$$x_{41} = -97.3893725319319$$
$$x_{42} = -91.1061872846991$$
$$x_{43} = 12.566496593124$$
$$x_{44} = -37.6991165090964$$
$$x_{45} = 15.7080277702229$$
$$x_{46} = 50.2654844259139$$
$$x_{47} = 3.14959356380938$$
$$x_{48} = 43.9823000886252$$
$$x_{49} = 94.2477799063191$$
$$x_{50} = 53.4070767521588$$
$$x_{51} = -84.8230020565613$$
$$x_{52} = 72.2566316952498$$
$$x_{53} = -59.6902615937248$$
$$x_{54} = 21.9911720820024$$
$$x_{55} = -65.9734465960134$$
$$x_{56} = -81.6814094520786$$
$$x_{57} = 91.1061872846991$$
$$x_{58} = 18.8495932494818$$
$$x_{59} = 62.8318540796563$$
$$x_{60} = 37.6991165090964$$
$$x_{61} = -28.2743449424922$$
$$x_{62} = 81.6814094520786$$
$$x_{63} = -9.42507655767126$$
$$x_{64} = -94.2477799063191$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{1}{2 x^{3}}\right) = \infty$$
$$\lim_{x \to 0^+}\left(2 \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{1}{2 x^{3}}\right) = -\infty$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
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
$$\left[100.530965160933, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -100.530965160933\right]$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$y = \lim_{x \to -\infty}\left(\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x) - 1/(4*x) - 1, divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1}{x}\right)$$
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:
$$\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1 = - \tan{\left(x \right)} - 1 + \frac{1}{4 x}$$
- No
$$\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1 = \tan{\left(x \right)} + 1 - \frac{1}{4 x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1 = - \tan{\left(x \right)} - 1 + \frac{1}{4 x}$$
- No
$$\left(\tan{\left(x \right)} - \frac{1}{4 x}\right) - 1 = \tan{\left(x \right)} + 1 - \frac{1}{4 x}$$
- No
so, the function
not is
neither even, nor odd