Mister Exam
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-
How to use it?
- Graphing y =:
- x³-6x²+4
- y=tg3x
- y=ctgx-1
- x*sin(x)^2
- Derivative of:
-
tg3x
- Integral of d{x}:
-
tg3x
-
Identical expressions
- y=tg3x
- y equally tg3x
-
Similar expressions
- ctg(3*x-(4*pi)/3)
- ctg(3x)
- tg(3*x)
Graphing y = y=tg3x
The solution
You have entered
[src]
f(x) = tan(3*x)
$$f{\left(x \right)} = \tan{\left(3 x \right)}$$
f = tan(3*x)
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:
$$\tan{\left(3 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -4.18879020478639$$
$$x_{2} = 34.5575191894877$$
$$x_{3} = 32.4631240870945$$
$$x_{4} = -96.342174710087$$
$$x_{5} = -43.9822971502571$$
$$x_{6} = 28.2743338823081$$
$$x_{7} = 39.7935069454707$$
$$x_{8} = 8.37758040957278$$
$$x_{9} = 59.6902604182061$$
$$x_{10} = 56.5486677646163$$
$$x_{11} = 100.530964914873$$
$$x_{12} = -46.0766922526503$$
$$x_{13} = 63.8790506229925$$
$$x_{14} = 41.8879020478639$$
$$x_{15} = 15.707963267949$$
$$x_{16} = -63.8790506229925$$
$$x_{17} = 48.1710873550435$$
$$x_{18} = -15.707963267949$$
$$x_{19} = -37.6991118430775$$
$$x_{20} = -41.8879020478639$$
$$x_{21} = -72.2566310325652$$
$$x_{22} = 30.3687289847013$$
$$x_{23} = 36.6519142918809$$
$$x_{24} = 17.8023583703422$$
$$x_{25} = 65.9734457253857$$
$$x_{26} = -79.5870138909414$$
$$x_{27} = 12.5663706143592$$
$$x_{28} = -48.1710873550435$$
$$x_{29} = 85.870199198121$$
$$x_{30} = -19.8967534727354$$
$$x_{31} = 24.0855436775217$$
$$x_{32} = 21.9911485751286$$
$$x_{33} = -13.6135681655558$$
$$x_{34} = -6.28318530717959$$
$$x_{35} = -65.9734457253857$$
$$x_{36} = 4.18879020478639$$
$$x_{37} = -74.3510261349584$$
$$x_{38} = 81.6814089933346$$
$$x_{39} = -92.1533845053006$$
$$x_{40} = -17.8023583703422$$
$$x_{41} = 78.5398163397448$$
$$x_{42} = -2.0943951023932$$
$$x_{43} = 98.4365698124802$$
$$x_{44} = -35.6047167406843$$
$$x_{45} = 14.6607657167524$$
$$x_{46} = -24.0855436775217$$
$$x_{47} = -97.3893722612836$$
$$x_{48} = 37.6991118430775$$
$$x_{49} = 54.4542726622231$$
$$x_{50} = 26.1799387799149$$
$$x_{51} = -21.9911485751286$$
$$x_{52} = 10.471975511966$$
$$x_{53} = -50.2654824574367$$
$$x_{54} = -94.2477796076938$$
$$x_{55} = 83.7758040957278$$
$$x_{56} = 19.8967534727354$$
$$x_{57} = -26.1799387799149$$
$$x_{58} = -53.4070751110265$$
$$x_{59} = 96.342174710087$$
$$x_{60} = -90.0589894029074$$
$$x_{61} = -59.6902604182061$$
$$x_{62} = -57.5958653158129$$
$$x_{63} = -55.5014702134197$$
$$x_{64} = -52.3598775598299$$
$$x_{65} = -99.4837673636768$$
$$x_{66} = -87.9645943005142$$
$$x_{67} = -28.2743338823081$$
$$x_{68} = 61.7846555205993$$
$$x_{69} = -77.4926187885482$$
$$x_{70} = -70.162235930172$$
$$x_{71} = -61.7846555205993$$
$$x_{72} = 6.28318530717959$$
$$x_{73} = 46.0766922526503$$
$$x_{74} = 0$$
$$x_{75} = -85.870199198121$$
$$x_{76} = 87.9645943005142$$
$$x_{77} = -7.33038285837618$$
$$x_{78} = 43.9822971502571$$
$$x_{79} = 52.3598775598299$$
$$x_{80} = -68.0678408277789$$
$$x_{81} = -75.398223686155$$
$$x_{82} = 58.6430628670095$$
$$x_{83} = -9.42477796076938$$
$$x_{84} = -33.5103216382911$$
$$x_{85} = 80.634211442138$$
$$x_{86} = 2.0943951023932$$
$$x_{87} = -30.3687289847013$$
$$x_{88} = -31.4159265358979$$
$$x_{89} = 72.2566310325652$$
$$x_{90} = 90.0589894029074$$
$$x_{91} = -11.5191730631626$$
$$x_{92} = 94.2477796076938$$
$$x_{93} = 76.4454212373516$$
$$x_{94} = -83.7758040957278$$
$$x_{95} = 68.0678408277789$$
$$x_{96} = -81.6814089933346$$
$$x_{97} = 92.1533845053006$$
$$x_{98} = -39.7935069454707$$
$$x_{99} = 50.2654824574367$$
$$x_{100} = 74.3510261349584$$
$$x_{101} = 70.162235930172$$
so we need to solve the equation:
$$\tan{\left(3 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -4.18879020478639$$
$$x_{2} = 34.5575191894877$$
$$x_{3} = 32.4631240870945$$
$$x_{4} = -96.342174710087$$
$$x_{5} = -43.9822971502571$$
$$x_{6} = 28.2743338823081$$
$$x_{7} = 39.7935069454707$$
$$x_{8} = 8.37758040957278$$
$$x_{9} = 59.6902604182061$$
$$x_{10} = 56.5486677646163$$
$$x_{11} = 100.530964914873$$
$$x_{12} = -46.0766922526503$$
$$x_{13} = 63.8790506229925$$
$$x_{14} = 41.8879020478639$$
$$x_{15} = 15.707963267949$$
$$x_{16} = -63.8790506229925$$
$$x_{17} = 48.1710873550435$$
$$x_{18} = -15.707963267949$$
$$x_{19} = -37.6991118430775$$
$$x_{20} = -41.8879020478639$$
$$x_{21} = -72.2566310325652$$
$$x_{22} = 30.3687289847013$$
$$x_{23} = 36.6519142918809$$
$$x_{24} = 17.8023583703422$$
$$x_{25} = 65.9734457253857$$
$$x_{26} = -79.5870138909414$$
$$x_{27} = 12.5663706143592$$
$$x_{28} = -48.1710873550435$$
$$x_{29} = 85.870199198121$$
$$x_{30} = -19.8967534727354$$
$$x_{31} = 24.0855436775217$$
$$x_{32} = 21.9911485751286$$
$$x_{33} = -13.6135681655558$$
$$x_{34} = -6.28318530717959$$
$$x_{35} = -65.9734457253857$$
$$x_{36} = 4.18879020478639$$
$$x_{37} = -74.3510261349584$$
$$x_{38} = 81.6814089933346$$
$$x_{39} = -92.1533845053006$$
$$x_{40} = -17.8023583703422$$
$$x_{41} = 78.5398163397448$$
$$x_{42} = -2.0943951023932$$
$$x_{43} = 98.4365698124802$$
$$x_{44} = -35.6047167406843$$
$$x_{45} = 14.6607657167524$$
$$x_{46} = -24.0855436775217$$
$$x_{47} = -97.3893722612836$$
$$x_{48} = 37.6991118430775$$
$$x_{49} = 54.4542726622231$$
$$x_{50} = 26.1799387799149$$
$$x_{51} = -21.9911485751286$$
$$x_{52} = 10.471975511966$$
$$x_{53} = -50.2654824574367$$
$$x_{54} = -94.2477796076938$$
$$x_{55} = 83.7758040957278$$
$$x_{56} = 19.8967534727354$$
$$x_{57} = -26.1799387799149$$
$$x_{58} = -53.4070751110265$$
$$x_{59} = 96.342174710087$$
$$x_{60} = -90.0589894029074$$
$$x_{61} = -59.6902604182061$$
$$x_{62} = -57.5958653158129$$
$$x_{63} = -55.5014702134197$$
$$x_{64} = -52.3598775598299$$
$$x_{65} = -99.4837673636768$$
$$x_{66} = -87.9645943005142$$
$$x_{67} = -28.2743338823081$$
$$x_{68} = 61.7846555205993$$
$$x_{69} = -77.4926187885482$$
$$x_{70} = -70.162235930172$$
$$x_{71} = -61.7846555205993$$
$$x_{72} = 6.28318530717959$$
$$x_{73} = 46.0766922526503$$
$$x_{74} = 0$$
$$x_{75} = -85.870199198121$$
$$x_{76} = 87.9645943005142$$
$$x_{77} = -7.33038285837618$$
$$x_{78} = 43.9822971502571$$
$$x_{79} = 52.3598775598299$$
$$x_{80} = -68.0678408277789$$
$$x_{81} = -75.398223686155$$
$$x_{82} = 58.6430628670095$$
$$x_{83} = -9.42477796076938$$
$$x_{84} = -33.5103216382911$$
$$x_{85} = 80.634211442138$$
$$x_{86} = 2.0943951023932$$
$$x_{87} = -30.3687289847013$$
$$x_{88} = -31.4159265358979$$
$$x_{89} = 72.2566310325652$$
$$x_{90} = 90.0589894029074$$
$$x_{91} = -11.5191730631626$$
$$x_{92} = 94.2477796076938$$
$$x_{93} = 76.4454212373516$$
$$x_{94} = -83.7758040957278$$
$$x_{95} = 68.0678408277789$$
$$x_{96} = -81.6814089933346$$
$$x_{97} = 92.1533845053006$$
$$x_{98} = -39.7935069454707$$
$$x_{99} = 50.2654824574367$$
$$x_{100} = 74.3510261349584$$
$$x_{101} = 70.162235930172$$
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
$$\lim_{x \to -\infty} \tan{\left(3 x \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \tan{\left(3 x \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty} \tan{\left(3 x \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \tan{\left(3 x \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(3*x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(3 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(3 x \right)}}{x}\right)$$
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:
$$\tan{\left(3 x \right)} = - \tan{\left(3 x \right)}$$
- No
$$\tan{\left(3 x \right)} = \tan{\left(3 x \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\tan{\left(3 x \right)} = - \tan{\left(3 x \right)}$$
- No
$$\tan{\left(3 x \right)} = \tan{\left(3 x \right)}$$
- Yes
so, the function
is
odd