Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • y=x^2+3x-1
  • |x|+x
  • x+x
  • -x*x
  • Derivative of:
  • tan(x)/x^3 tan(x)/x^3
  • Limit of the function:
  • tan(x)/x^3 tan(x)/x^3
  • Identical expressions

  • tan(x)/x^ three
  • tangent of (x) divide by x cubed
  • tangent of (x) divide by x to the power of three
  • tan(x)/x3
  • tanx/x3
  • tan(x)/x³
  • tan(x)/x to the power of 3
  • tanx/x^3
  • tan(x) divide by x^3

Graphing y = tan(x)/x^3

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src]
       tan(x)
f(x) = ------
          3  
         x   
$$f{\left(x \right)} = \frac{\tan{\left(x \right)}}{x^{3}}$$
f = tan(x)/x^3
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
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:
$$\frac{\tan{\left(x \right)}}{x^{3}} = 0$$
Solve this equation
The points of intersection with the axis X:

Numerical solution
$$x_{1} = 12.5663706143592$$
$$x_{2} = 53.4070751110265$$
$$x_{3} = -97.3893722612836$$
$$x_{4} = 37.6991118430775$$
$$x_{5} = 97.3893722612836$$
$$x_{6} = 78.5398163397448$$
$$x_{7} = -59.6902604182061$$
$$x_{8} = -65.9734457253857$$
$$x_{9} = -31.4159265358979$$
$$x_{10} = -50.2654824574367$$
$$x_{11} = -21.9911485751286$$
$$x_{12} = 6.28318530717959$$
$$x_{13} = -34.5575191894877$$
$$x_{14} = -94.2477796076938$$
$$x_{15} = -69.1150383789755$$
$$x_{16} = -15.707963267949$$
$$x_{17} = 21.9911485751286$$
$$x_{18} = 69.1150383789755$$
$$x_{19} = 62.8318530717959$$
$$x_{20} = 50.2654824574367$$
$$x_{21} = 81.6814089933346$$
$$x_{22} = 100.530964914873$$
$$x_{23} = -40.8407044966673$$
$$x_{24} = 9.42477796076938$$
$$x_{25} = -87.9645943005142$$
$$x_{26} = 34.5575191894877$$
$$x_{27} = 65.9734457253857$$
$$x_{28} = -62.8318530717959$$
$$x_{29} = -18.8495559215388$$
$$x_{30} = -28.2743338823081$$
$$x_{31} = -56.5486677646163$$
$$x_{32} = -53.4070751110265$$
$$x_{33} = -37.6991118430775$$
$$x_{34} = -25.1327412287183$$
$$x_{35} = -100.530964914873$$
$$x_{36} = -9.42477796076938$$
$$x_{37} = 40.8407044966673$$
$$x_{38} = -91.106186954104$$
$$x_{39} = -75.398223686155$$
$$x_{40} = 18.8495559215388$$
$$x_{41} = 87.9645943005142$$
$$x_{42} = 59.6902604182061$$
$$x_{43} = -6.28318530717959$$
$$x_{44} = 25.1327412287183$$
$$x_{45} = 47.1238898038469$$
$$x_{46} = 91.106186954104$$
$$x_{47} = 28.2743338823081$$
$$x_{48} = 56.5486677646163$$
$$x_{49} = -43.9822971502571$$
$$x_{50} = -47.1238898038469$$
$$x_{51} = -3.14159265358979$$
$$x_{52} = 31.4159265358979$$
$$x_{53} = 94.2477796076938$$
$$x_{54} = -12.5663706143592$$
$$x_{55} = 75.398223686155$$
$$x_{56} = -72.2566310325652$$
$$x_{57} = -84.8230016469244$$
$$x_{58} = 84.8230016469244$$
$$x_{59} = 72.2566310325652$$
$$x_{60} = -81.6814089933346$$
$$x_{61} = 43.9822971502571$$
$$x_{62} = -78.5398163397448$$
$$x_{63} = 15.707963267949$$
$$x_{64} = 3.14159265358979$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(x)/x^3.
$$\frac{\tan{\left(0 \right)}}{0^{3}}$$
The result:
$$f{\left(0 \right)} = \text{NaN}$$
- the solutions of the equation d'not exist
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
$$\frac{\tan^{2}{\left(x \right)} + 1}{x^{3}} - \frac{3 \tan{\left(x \right)}}{x^{4}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -594339.741656121$$
$$x_{2} = 25945.8066003249$$
The values of the extrema at the points:
(-594339.7416561211, 3.8288337974316e-18)

(25945.80660032492, -3.97707611760823e-14)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
The function has no maxima
Decreasing at the entire real axis
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
$$\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \tan{\left(x \right)}}{x^{2}}\right)}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 84.8583106190521$$
$$x_{2} = -66.0187938781021$$
$$x_{3} = -69.1583357737765$$
$$x_{4} = -62.8794552929189$$
$$x_{5} = -94.2795677093343$$
$$x_{6} = -56.6015217996642$$
$$x_{7} = -91.1390680936179$$
$$x_{8} = -25.2499214706009$$
$$x_{9} = -50.3248847443636$$
$$x_{10} = 3.69358314705748$$
$$x_{11} = 94.2795677093343$$
$$x_{12} = 19.0036683009869$$
$$x_{13} = 34.6434756494872$$
$$x_{14} = -84.8583106190521$$
$$x_{15} = 53.4630130760851$$
$$x_{16} = 12.7892677205421$$
$$x_{17} = 44.0500884228484$$
$$x_{18} = -15.8904615094706$$
$$x_{19} = -6.66644065689776$$
$$x_{20} = 37.7780284480501$$
$$x_{21} = 15.8904615094706$$
$$x_{22} = 97.420137568151$$
$$x_{23} = -78.5779395044499$$
$$x_{24} = 28.3788902944365$$
$$x_{25} = 9.70876728081698$$
$$x_{26} = -40.9136395730782$$
$$x_{27} = -59.7403517227637$$
$$x_{28} = -72.2980547201869$$
$$x_{29} = 22.1243392482168$$
$$x_{30} = -37.7780284480501$$
$$x_{31} = -28.3788902944365$$
$$x_{32} = -12.7892677205421$$
$$x_{33} = 59.7403517227637$$
$$x_{34} = 72.2980547201869$$
$$x_{35} = 91.1390680936179$$
$$x_{36} = -53.4630130760851$$
$$x_{37} = 87.9986462029283$$
$$x_{38} = -9.70876728081698$$
$$x_{39} = -3.69358314705748$$
$$x_{40} = -44.0500884228484$$
$$x_{41} = 66.0187938781021$$
$$x_{42} = 78.5779395044499$$
$$x_{43} = 6.66644065689776$$
$$x_{44} = 47.1872114097454$$
$$x_{45} = 25.2499214706009$$
$$x_{46} = 75.4379287840465$$
$$x_{47} = 40.9136395730782$$
$$x_{48} = -22.1243392482168$$
$$x_{49} = -47.1872114097454$$
$$x_{50} = -81.718071233071$$
$$x_{51} = -100.560771117475$$
$$x_{52} = -87.9986462029283$$
$$x_{53} = -75.4379287840465$$
$$x_{54} = 56.6015217996642$$
$$x_{55} = 50.3248847443636$$
$$x_{56} = 69.1583357737765$$
$$x_{57} = 100.560771117475$$
$$x_{58} = 31.5102856697475$$
$$x_{59} = -97.420137568151$$
$$x_{60} = 62.8794552929189$$
$$x_{61} = 81.718071233071$$
$$x_{62} = -34.6434756494872$$
$$x_{63} = -19.0036683009869$$
$$x_{64} = -31.5102856697475$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$

$$\lim_{x \to 0^-}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \tan{\left(x \right)}}{x^{2}}\right)}{x^{3}}\right) = \infty$$
$$\lim_{x \to 0^+}\left(\frac{2 \left(\left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} - \frac{3 \left(\tan^{2}{\left(x \right)} + 1\right)}{x} + \frac{6 \tan{\left(x \right)}}{x^{2}}\right)}{x^{3}}\right) = \infty$$
- limits are equal, then skip the corresponding 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
$$\left[100.560771117475, \infty\right)$$
Convex at the intervals
$$\left[-3.69358314705748, 3.69358314705748\right]$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x^{3}}\right)$$
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x^{3}}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x)/x^3, divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)}}{x x^{3}}\right)$$
True

Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)}}{x x^{3}}\right)$$
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:
$$\frac{\tan{\left(x \right)}}{x^{3}} = \frac{\tan{\left(x \right)}}{x^{3}}$$
- No
$$\frac{\tan{\left(x \right)}}{x^{3}} = - \frac{\tan{\left(x \right)}}{x^{3}}$$
- No
so, the function
not is
neither even, nor odd