Mister Exam
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-
How to use it?
- Graphing y =:
- (x+2)^2/(x+1)
- x^6+x^3-2
- x^3-12x+1
- x^2(x-3)
- How do you in partial fractions?:
- x/(x^4-1)
- Integral of d{x}:
-
x/(x^4-1)
-
Identical expressions
- x/(x^ four - one)
- x divide by (x to the power of 4 minus 1)
- x divide by (x to the power of four minus one)
- x/(x4-1)
- x/x4-1
- x/(x⁴-1)
- x/x^4-1
- x divide by (x^4-1)
-
Similar expressions
- x/(x^4+1)
Graphing y = x/(x^4-1)
The solution
You have entered
[src]
x
f(x) = ------
4
x - 1
$$f{\left(x \right)} = \frac{x}{x^{4} - 1}$$
f = x/(x^4 - 1)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{1} = -1$$
$$x_{2} = 1$$
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:
$$\frac{x}{x^{4} - 1} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -31295.7962379693$$
$$x_{2} = -13496.1760483594$$
$$x_{3} = 24646.2101098217$$
$$x_{4} = -10953.3882805875$$
$$x_{5} = 41598.266404313$$
$$x_{6} = 28036.6187081671$$
$$x_{7} = 33969.8376493113$$
$$x_{8} = -16038.9705212803$$
$$x_{9} = -15191.3717757046$$
$$x_{10} = -24514.9774427548$$
$$x_{11} = 39055.4564200716$$
$$x_{12} = -17734.1694257125$$
$$x_{13} = -36381.4139341171$$
$$x_{14} = -32991.0018786856$$
$$x_{15} = 25493.8120599588$$
$$x_{16} = 13627.4079554379$$
$$x_{17} = 42445.8698037899$$
$$x_{18} = -40619.4301301675$$
$$x_{19} = 0$$
$$x_{20} = -21124.571491027$$
$$x_{21} = -38924.2235238324$$
$$x_{22} = 16170.2027251269$$
$$x_{23} = -12648.5792422719$$
$$x_{24} = 19560.6023126174$$
$$x_{25} = 21255.8040320689$$
$$x_{26} = 11932.2149075083$$
$$x_{27} = 32274.6318465866$$
$$x_{28} = 23798.6083148292$$
$$x_{29} = -42314.636883662$$
$$x_{30} = 36512.6468080104$$
$$x_{31} = -20276.9705414765$$
$$x_{32} = -16886.5697578526$$
$$x_{33} = 40750.6630390817$$
$$x_{34} = 12779.8110193159$$
$$x_{35} = 15322.6038938897$$
$$x_{36} = -18581.7694737718$$
$$x_{37} = -25362.579368541$$
$$x_{38} = 22951.0066909733$$
$$x_{39} = -23667.3756746024$$
$$x_{40} = -38076.6202818456$$
$$x_{41} = 26341.414151142$$
$$x_{42} = 39903.05971023$$
$$x_{43} = 10237.0256272847$$
$$x_{44} = -27057.783637116$$
$$x_{45} = -14343.7735909283$$
$$x_{46} = -41467.0334896237$$
$$x_{47} = -26210.1814375691$$
$$x_{48} = 35665.0437000849$$
$$x_{49} = -32143.3990219725$$
$$x_{50} = -27905.3859558973$$
$$x_{51} = -21972.1726788051$$
$$x_{52} = 28884.2211530911$$
$$x_{53} = -22819.7740804181$$
$$x_{54} = -11800.9832801479$$
$$x_{55} = -10105.7943654042$$
$$x_{56} = -19429.3698582394$$
$$x_{57} = 30579.4263316509$$
$$x_{58} = -30448.1935325157$$
$$x_{59} = -34686.2077896141$$
$$x_{60} = 33122.2347146414$$
$$x_{61} = 37360.2499661057$$
$$x_{62} = 29731.8236968773$$
$$x_{63} = 38207.853171119$$
$$x_{64} = -39771.8268074555$$
$$x_{65} = -9258.20163820821$$
$$x_{66} = 18713.0018766498$$
$$x_{67} = 31427.0290503421$$
$$x_{68} = -28752.9883838734$$
$$x_{69} = 27189.0163709003$$
$$x_{70} = -33838.6048028281$$
$$x_{71} = 11084.619736454$$
$$x_{72} = 20408.2030416476$$
$$x_{73} = 17017.8020366885$$
$$x_{74} = 17865.4017704484$$
$$x_{75} = 34817.4406458859$$
$$x_{76} = -35533.8108346963$$
$$x_{77} = -37229.0170842667$$
$$x_{78} = 14475.0056109124$$
$$x_{79} = 22103.405256456$$
$$x_{80} = -29600.5909120854$$
$$x_{81} = 9389.43268708208$$
so we need to solve the equation:
$$\frac{x}{x^{4} - 1} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -31295.7962379693$$
$$x_{2} = -13496.1760483594$$
$$x_{3} = 24646.2101098217$$
$$x_{4} = -10953.3882805875$$
$$x_{5} = 41598.266404313$$
$$x_{6} = 28036.6187081671$$
$$x_{7} = 33969.8376493113$$
$$x_{8} = -16038.9705212803$$
$$x_{9} = -15191.3717757046$$
$$x_{10} = -24514.9774427548$$
$$x_{11} = 39055.4564200716$$
$$x_{12} = -17734.1694257125$$
$$x_{13} = -36381.4139341171$$
$$x_{14} = -32991.0018786856$$
$$x_{15} = 25493.8120599588$$
$$x_{16} = 13627.4079554379$$
$$x_{17} = 42445.8698037899$$
$$x_{18} = -40619.4301301675$$
$$x_{19} = 0$$
$$x_{20} = -21124.571491027$$
$$x_{21} = -38924.2235238324$$
$$x_{22} = 16170.2027251269$$
$$x_{23} = -12648.5792422719$$
$$x_{24} = 19560.6023126174$$
$$x_{25} = 21255.8040320689$$
$$x_{26} = 11932.2149075083$$
$$x_{27} = 32274.6318465866$$
$$x_{28} = 23798.6083148292$$
$$x_{29} = -42314.636883662$$
$$x_{30} = 36512.6468080104$$
$$x_{31} = -20276.9705414765$$
$$x_{32} = -16886.5697578526$$
$$x_{33} = 40750.6630390817$$
$$x_{34} = 12779.8110193159$$
$$x_{35} = 15322.6038938897$$
$$x_{36} = -18581.7694737718$$
$$x_{37} = -25362.579368541$$
$$x_{38} = 22951.0066909733$$
$$x_{39} = -23667.3756746024$$
$$x_{40} = -38076.6202818456$$
$$x_{41} = 26341.414151142$$
$$x_{42} = 39903.05971023$$
$$x_{43} = 10237.0256272847$$
$$x_{44} = -27057.783637116$$
$$x_{45} = -14343.7735909283$$
$$x_{46} = -41467.0334896237$$
$$x_{47} = -26210.1814375691$$
$$x_{48} = 35665.0437000849$$
$$x_{49} = -32143.3990219725$$
$$x_{50} = -27905.3859558973$$
$$x_{51} = -21972.1726788051$$
$$x_{52} = 28884.2211530911$$
$$x_{53} = -22819.7740804181$$
$$x_{54} = -11800.9832801479$$
$$x_{55} = -10105.7943654042$$
$$x_{56} = -19429.3698582394$$
$$x_{57} = 30579.4263316509$$
$$x_{58} = -30448.1935325157$$
$$x_{59} = -34686.2077896141$$
$$x_{60} = 33122.2347146414$$
$$x_{61} = 37360.2499661057$$
$$x_{62} = 29731.8236968773$$
$$x_{63} = 38207.853171119$$
$$x_{64} = -39771.8268074555$$
$$x_{65} = -9258.20163820821$$
$$x_{66} = 18713.0018766498$$
$$x_{67} = 31427.0290503421$$
$$x_{68} = -28752.9883838734$$
$$x_{69} = 27189.0163709003$$
$$x_{70} = -33838.6048028281$$
$$x_{71} = 11084.619736454$$
$$x_{72} = 20408.2030416476$$
$$x_{73} = 17017.8020366885$$
$$x_{74} = 17865.4017704484$$
$$x_{75} = 34817.4406458859$$
$$x_{76} = -35533.8108346963$$
$$x_{77} = -37229.0170842667$$
$$x_{78} = 14475.0056109124$$
$$x_{79} = 22103.405256456$$
$$x_{80} = -29600.5909120854$$
$$x_{81} = 9389.43268708208$$
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
$$- \frac{4 x^{4}}{\left(x^{4} - 1\right)^{2}} + \frac{1}{x^{4} - 1} = 0$$
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
$$- \frac{4 x^{4}}{\left(x^{4} - 1\right)^{2}} + \frac{1}{x^{4} - 1} = 0$$
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
$$\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 1} - 5\right)}{\left(x^{4} - 1\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$\lim_{x \to -1^-}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 1} - 5\right)}{\left(x^{4} - 1\right)^{2}}\right) = -\infty$$
$$\lim_{x \to -1^+}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 1} - 5\right)}{\left(x^{4} - 1\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = -1$$
- is an inflection point
$$\lim_{x \to 1^-}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 1} - 5\right)}{\left(x^{4} - 1\right)^{2}}\right) = -\infty$$
$$\lim_{x \to 1^+}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 1} - 5\right)}{\left(x^{4} - 1\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 1$$
- 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:
Have no bends at the whole real axis
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 1} - 5\right)}{\left(x^{4} - 1\right)^{2}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$\lim_{x \to -1^-}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 1} - 5\right)}{\left(x^{4} - 1\right)^{2}}\right) = -\infty$$
$$\lim_{x \to -1^+}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 1} - 5\right)}{\left(x^{4} - 1\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = -1$$
- is an inflection point
$$\lim_{x \to 1^-}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 1} - 5\right)}{\left(x^{4} - 1\right)^{2}}\right) = -\infty$$
$$\lim_{x \to 1^+}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 1} - 5\right)}{\left(x^{4} - 1\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 1$$
- 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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
$$x_{1} = -1$$
$$x_{2} = 1$$
$$x_{1} = -1$$
$$x_{2} = 1$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x}{x^{4} - 1}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{x}{x^{4} - 1}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{x}{x^{4} - 1}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{x}{x^{4} - 1}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x/(x^4 - 1), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \frac{1}{x^{4} - 1} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{x^{4} - 1} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty} \frac{1}{x^{4} - 1} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{x^{4} - 1} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
$$\frac{x}{x^{4} - 1} = - \frac{x}{x^{4} - 1}$$
- No
$$\frac{x}{x^{4} - 1} = \frac{x}{x^{4} - 1}$$
- Yes
so, the function
is
odd
So, check:
$$\frac{x}{x^{4} - 1} = - \frac{x}{x^{4} - 1}$$
- No
$$\frac{x}{x^{4} - 1} = \frac{x}{x^{4} - 1}$$
- Yes
so, the function
is
odd