Mister Exam
Other calculators
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-
How to use it?
- Graphing y =:
- x^4/4-x^3/3-x^2
- x^3+6x
- x^3-4x^2+4x
- x^3-3x-7
- Integral of d{x}:
-
tan(t)
- Derivative of:
-
tan(t)
-
Identical expressions
- tan(t)
- tangent of (t)
- tant
-
Similar expressions
- tan(tan(x))/sin(sin(x))
- atan(tan(x/2)+1)
- tan(tan(2*x))
- (tan(tan(x))-sin(sin(x)))/(tan(x)-sin(x))
- -tan(1+atan(t^2))
Graphing y = tan(t)
The solution
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis T at f = 0
so we need to solve the equation:
$$\tan{\left(t \right)} = 0$$
Solve this equation
The points of intersection with the axis T:
Analytical solution
$$t_{1} = 0$$
Numerical solution
$$t_{1} = 72.2566310325652$$
$$t_{2} = 94.2477796076938$$
$$t_{3} = -3.14159265358979$$
$$t_{4} = 40.8407044966673$$
$$t_{5} = -91.106186954104$$
$$t_{6} = -15.707963267949$$
$$t_{7} = -37.6991118430775$$
$$t_{8} = -31.4159265358979$$
$$t_{9} = -81.6814089933346$$
$$t_{10} = 6.28318530717959$$
$$t_{11} = 62.8318530717959$$
$$t_{12} = 53.4070751110265$$
$$t_{13} = 81.6814089933346$$
$$t_{14} = 100.530964914873$$
$$t_{15} = 50.2654824574367$$
$$t_{16} = 84.8230016469244$$
$$t_{17} = -47.1238898038469$$
$$t_{18} = -56.5486677646163$$
$$t_{19} = -53.4070751110265$$
$$t_{20} = 3.14159265358979$$
$$t_{21} = 25.1327412287183$$
$$t_{22} = -9.42477796076938$$
$$t_{23} = -87.9645943005142$$
$$t_{24} = 15.707963267949$$
$$t_{25} = 91.106186954104$$
$$t_{26} = 97.3893722612836$$
$$t_{27} = 78.5398163397448$$
$$t_{28} = 9.42477796076938$$
$$t_{29} = -6.28318530717959$$
$$t_{30} = 75.398223686155$$
$$t_{31} = 12.5663706143592$$
$$t_{32} = 56.5486677646163$$
$$t_{33} = -12.5663706143592$$
$$t_{34} = -34.5575191894877$$
$$t_{35} = -28.2743338823081$$
$$t_{36} = 18.8495559215388$$
$$t_{37} = -78.5398163397448$$
$$t_{38} = -94.2477796076938$$
$$t_{39} = -25.1327412287183$$
$$t_{40} = 59.6902604182061$$
$$t_{41} = -21.9911485751286$$
$$t_{42} = -72.2566310325652$$
$$t_{43} = -75.398223686155$$
$$t_{44} = 31.4159265358979$$
$$t_{45} = 43.9822971502571$$
$$t_{46} = 28.2743338823081$$
$$t_{47} = -40.8407044966673$$
$$t_{48} = -50.2654824574367$$
$$t_{49} = 47.1238898038469$$
$$t_{50} = -84.8230016469244$$
$$t_{51} = -97.3893722612836$$
$$t_{52} = -43.9822971502571$$
$$t_{53} = 69.1150383789755$$
$$t_{54} = -18.8495559215388$$
$$t_{55} = 65.9734457253857$$
$$t_{56} = -65.9734457253857$$
$$t_{57} = 34.5575191894877$$
$$t_{58} = 0$$
$$t_{59} = -59.6902604182061$$
$$t_{60} = -62.8318530717959$$
$$t_{61} = 21.9911485751286$$
$$t_{62} = 87.9645943005142$$
$$t_{63} = 37.6991118430775$$
$$t_{64} = -100.530964914873$$
$$t_{65} = -69.1150383789755$$
so we need to solve the equation:
$$\tan{\left(t \right)} = 0$$
Solve this equation
The points of intersection with the axis T:
Analytical solution
$$t_{1} = 0$$
Numerical solution
$$t_{1} = 72.2566310325652$$
$$t_{2} = 94.2477796076938$$
$$t_{3} = -3.14159265358979$$
$$t_{4} = 40.8407044966673$$
$$t_{5} = -91.106186954104$$
$$t_{6} = -15.707963267949$$
$$t_{7} = -37.6991118430775$$
$$t_{8} = -31.4159265358979$$
$$t_{9} = -81.6814089933346$$
$$t_{10} = 6.28318530717959$$
$$t_{11} = 62.8318530717959$$
$$t_{12} = 53.4070751110265$$
$$t_{13} = 81.6814089933346$$
$$t_{14} = 100.530964914873$$
$$t_{15} = 50.2654824574367$$
$$t_{16} = 84.8230016469244$$
$$t_{17} = -47.1238898038469$$
$$t_{18} = -56.5486677646163$$
$$t_{19} = -53.4070751110265$$
$$t_{20} = 3.14159265358979$$
$$t_{21} = 25.1327412287183$$
$$t_{22} = -9.42477796076938$$
$$t_{23} = -87.9645943005142$$
$$t_{24} = 15.707963267949$$
$$t_{25} = 91.106186954104$$
$$t_{26} = 97.3893722612836$$
$$t_{27} = 78.5398163397448$$
$$t_{28} = 9.42477796076938$$
$$t_{29} = -6.28318530717959$$
$$t_{30} = 75.398223686155$$
$$t_{31} = 12.5663706143592$$
$$t_{32} = 56.5486677646163$$
$$t_{33} = -12.5663706143592$$
$$t_{34} = -34.5575191894877$$
$$t_{35} = -28.2743338823081$$
$$t_{36} = 18.8495559215388$$
$$t_{37} = -78.5398163397448$$
$$t_{38} = -94.2477796076938$$
$$t_{39} = -25.1327412287183$$
$$t_{40} = 59.6902604182061$$
$$t_{41} = -21.9911485751286$$
$$t_{42} = -72.2566310325652$$
$$t_{43} = -75.398223686155$$
$$t_{44} = 31.4159265358979$$
$$t_{45} = 43.9822971502571$$
$$t_{46} = 28.2743338823081$$
$$t_{47} = -40.8407044966673$$
$$t_{48} = -50.2654824574367$$
$$t_{49} = 47.1238898038469$$
$$t_{50} = -84.8230016469244$$
$$t_{51} = -97.3893722612836$$
$$t_{52} = -43.9822971502571$$
$$t_{53} = 69.1150383789755$$
$$t_{54} = -18.8495559215388$$
$$t_{55} = 65.9734457253857$$
$$t_{56} = -65.9734457253857$$
$$t_{57} = 34.5575191894877$$
$$t_{58} = 0$$
$$t_{59} = -59.6902604182061$$
$$t_{60} = -62.8318530717959$$
$$t_{61} = 21.9911485751286$$
$$t_{62} = 87.9645943005142$$
$$t_{63} = 37.6991118430775$$
$$t_{64} = -100.530964914873$$
$$t_{65} = -69.1150383789755$$
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d t} f{\left(t \right)} = $$
the first derivative
$$\tan^{2}{\left(t \right)} + 1 = 0$$
Solve this equation
Solutions are not found,
function may have no extrema
$$\frac{d}{d t} f{\left(t \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d t} f{\left(t \right)} = $$
the first derivative
$$\tan^{2}{\left(t \right)} + 1 = 0$$
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
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = $$
the second derivative
$$2 \left(\tan^{2}{\left(t \right)} + 1\right) \tan{\left(t \right)} = 0$$
Solve this equation
The roots of this equation
$$t_{1} = 0$$
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
$$\left(-\infty, 0\right]$$
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d t^{2}} f{\left(t \right)} = $$
the second derivative
$$2 \left(\tan^{2}{\left(t \right)} + 1\right) \tan{\left(t \right)} = 0$$
Solve this equation
The roots of this equation
$$t_{1} = 0$$
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 t->+oo and t->-oo
$$\lim_{t \to -\infty} \tan{\left(t \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{t \to \infty} \tan{\left(t \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{t \to -\infty} \tan{\left(t \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{t \to \infty} \tan{\left(t \right)} = \left\langle -\infty, \infty\right\rangle$$
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(t), divided by t at t->+oo and t ->-oo
$$\lim_{t \to -\infty}\left(\frac{\tan{\left(t \right)}}{t}\right) = \lim_{t \to -\infty}\left(\frac{\tan{\left(t \right)}}{t}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = t \lim_{t \to -\infty}\left(\frac{\tan{\left(t \right)}}{t}\right)$$
$$\lim_{t \to \infty}\left(\frac{\tan{\left(t \right)}}{t}\right) = \lim_{t \to \infty}\left(\frac{\tan{\left(t \right)}}{t}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = t \lim_{t \to \infty}\left(\frac{\tan{\left(t \right)}}{t}\right)$$
$$\lim_{t \to -\infty}\left(\frac{\tan{\left(t \right)}}{t}\right) = \lim_{t \to -\infty}\left(\frac{\tan{\left(t \right)}}{t}\right)$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = t \lim_{t \to -\infty}\left(\frac{\tan{\left(t \right)}}{t}\right)$$
$$\lim_{t \to \infty}\left(\frac{\tan{\left(t \right)}}{t}\right) = \lim_{t \to \infty}\left(\frac{\tan{\left(t \right)}}{t}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = t \lim_{t \to \infty}\left(\frac{\tan{\left(t \right)}}{t}\right)$$
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-t) и f = -f(-t).
So, check:
$$\tan{\left(t \right)} = - \tan{\left(t \right)}$$
- No
$$\tan{\left(t \right)} = \tan{\left(t \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\tan{\left(t \right)} = - \tan{\left(t \right)}$$
- No
$$\tan{\left(t \right)} = \tan{\left(t \right)}$$
- Yes
so, the function
is
odd
The graph