Mister Exam
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-
How to use it?
- Graphing y =:
- x/(4-x^2)
- -x^4+x^2+5
- x^4-5x^2+6
- x^4-6x^2+1
- Integral of d{x}:
-
x/(x^4-16)
-
Identical expressions
- x/(x^ four - sixteen)
- x divide by (x to the power of 4 minus 16)
- x divide by (x to the power of four minus sixteen)
- x/(x4-16)
- x/x4-16
- x/(x⁴-16)
- x/x^4-16
- x divide by (x^4-16)
-
Similar expressions
- x/(x^4+16)
Graphing y = x/(x^4-16)
The solution
You have entered
[src]
x
f(x) = -------
4
x - 16
$$f{\left(x \right)} = \frac{x}{x^{4} - 16}$$
f = x/(x^4 - 16)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = -2$$
$$x_{2} = 2$$
$$x_{1} = -2$$
$$x_{2} = 2$$
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:
$$\frac{x}{x^{4} - 16} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -30448.1852635265$$
$$x_{2} = 13627.3156894593$$
$$x_{3} = -37229.0125605144$$
$$x_{4} = 16170.1475026008$$
$$x_{5} = -9257.90753976151$$
$$x_{6} = -23667.3580679472$$
$$x_{7} = 31427.021528788$$
$$x_{8} = 14474.9286237899$$
$$x_{9} = -42314.6338027785$$
$$x_{10} = 33969.831693598$$
$$x_{11} = 39055.4525011759$$
$$x_{12} = 42445.8667509803$$
$$x_{13} = -35533.8056321403$$
$$x_{14} = -32143.391993459$$
$$x_{15} = 27189.0047552326$$
$$x_{16} = -40619.4266472391$$
$$x_{17} = -20276.9425445156$$
$$x_{18} = -18581.7330944549$$
$$x_{19} = 30579.418167128$$
$$x_{20} = -33838.5987785747$$
$$x_{21} = -27057.7718541128$$
$$x_{22} = 21255.7797208243$$
$$x_{23} = 11084.4482824334$$
$$x_{24} = 35665.038553917$$
$$x_{25} = -15191.3052003182$$
$$x_{26} = -10953.2106823887$$
$$x_{27} = 40750.6595892069$$
$$x_{28} = -38924.2195657489$$
$$x_{29} = -16886.5212863653$$
$$x_{30} = -32990.9953780887$$
$$x_{31} = 26341.4013776652$$
$$x_{32} = 22103.3836362641$$
$$x_{33} = -10105.5682307074$$
$$x_{34} = 38207.8489855768$$
$$x_{35} = -21972.1506746415$$
$$x_{36} = -25362.5650615$$
$$x_{37} = -38076.6160535142$$
$$x_{38} = -28752.97856445$$
$$x_{39} = -22819.7544381535$$
$$x_{40} = 39903.0560358041$$
$$x_{41} = -21124.5467305885$$
$$x_{42} = 32274.6249022224$$
$$x_{43} = 12779.699148658$$
$$x_{44} = 19560.5711165887$$
$$x_{45} = 22950.9873788779$$
$$x_{46} = 28036.608114504$$
$$x_{47} = -13496.0811048387$$
$$x_{48} = -34686.2021962782$$
$$x_{49} = -27905.3752142785$$
$$x_{50} = 41598.2631610587$$
$$x_{51} = 11932.0774603584$$
$$x_{52} = 15322.5389900482$$
$$x_{53} = 0$$
$$x_{54} = 24646.1945150236$$
$$x_{55} = 36512.6420119861$$
$$x_{56} = -36381.40908677$$
$$x_{57} = -12648.4639053896$$
$$x_{58} = 9389.15057513904$$
$$x_{59} = 20408.175573558$$
$$x_{60} = 23798.5909936427$$
$$x_{61} = -11800.8412650999$$
$$x_{62} = 37360.2454891671$$
$$x_{63} = -29600.5819122437$$
$$x_{64} = 25493.7979695406$$
$$x_{65} = -41467.0302159327$$
$$x_{66} = -17734.1275771308$$
$$x_{67} = -14343.6945028541$$
$$x_{68} = 10236.80795557$$
$$x_{69} = -24514.9615998732$$
$$x_{70} = -26210.1684740975$$
$$x_{71} = -19429.3380351854$$
$$x_{72} = 28884.2114649697$$
$$x_{73} = 34817.4351146439$$
$$x_{74} = -39771.8230970718$$
$$x_{75} = 17017.7546618872$$
$$x_{76} = -16038.9139523032$$
$$x_{77} = 33122.2282898894$$
$$x_{78} = -31295.7886227931$$
$$x_{79} = 29731.8148139625$$
$$x_{80} = 18712.966246377$$
$$x_{81} = 17865.3608241084$$
so we need to solve the equation:
$$\frac{x}{x^{4} - 16} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = -30448.1852635265$$
$$x_{2} = 13627.3156894593$$
$$x_{3} = -37229.0125605144$$
$$x_{4} = 16170.1475026008$$
$$x_{5} = -9257.90753976151$$
$$x_{6} = -23667.3580679472$$
$$x_{7} = 31427.021528788$$
$$x_{8} = 14474.9286237899$$
$$x_{9} = -42314.6338027785$$
$$x_{10} = 33969.831693598$$
$$x_{11} = 39055.4525011759$$
$$x_{12} = 42445.8667509803$$
$$x_{13} = -35533.8056321403$$
$$x_{14} = -32143.391993459$$
$$x_{15} = 27189.0047552326$$
$$x_{16} = -40619.4266472391$$
$$x_{17} = -20276.9425445156$$
$$x_{18} = -18581.7330944549$$
$$x_{19} = 30579.418167128$$
$$x_{20} = -33838.5987785747$$
$$x_{21} = -27057.7718541128$$
$$x_{22} = 21255.7797208243$$
$$x_{23} = 11084.4482824334$$
$$x_{24} = 35665.038553917$$
$$x_{25} = -15191.3052003182$$
$$x_{26} = -10953.2106823887$$
$$x_{27} = 40750.6595892069$$
$$x_{28} = -38924.2195657489$$
$$x_{29} = -16886.5212863653$$
$$x_{30} = -32990.9953780887$$
$$x_{31} = 26341.4013776652$$
$$x_{32} = 22103.3836362641$$
$$x_{33} = -10105.5682307074$$
$$x_{34} = 38207.8489855768$$
$$x_{35} = -21972.1506746415$$
$$x_{36} = -25362.5650615$$
$$x_{37} = -38076.6160535142$$
$$x_{38} = -28752.97856445$$
$$x_{39} = -22819.7544381535$$
$$x_{40} = 39903.0560358041$$
$$x_{41} = -21124.5467305885$$
$$x_{42} = 32274.6249022224$$
$$x_{43} = 12779.699148658$$
$$x_{44} = 19560.5711165887$$
$$x_{45} = 22950.9873788779$$
$$x_{46} = 28036.608114504$$
$$x_{47} = -13496.0811048387$$
$$x_{48} = -34686.2021962782$$
$$x_{49} = -27905.3752142785$$
$$x_{50} = 41598.2631610587$$
$$x_{51} = 11932.0774603584$$
$$x_{52} = 15322.5389900482$$
$$x_{53} = 0$$
$$x_{54} = 24646.1945150236$$
$$x_{55} = 36512.6420119861$$
$$x_{56} = -36381.40908677$$
$$x_{57} = -12648.4639053896$$
$$x_{58} = 9389.15057513904$$
$$x_{59} = 20408.175573558$$
$$x_{60} = 23798.5909936427$$
$$x_{61} = -11800.8412650999$$
$$x_{62} = 37360.2454891671$$
$$x_{63} = -29600.5819122437$$
$$x_{64} = 25493.7979695406$$
$$x_{65} = -41467.0302159327$$
$$x_{66} = -17734.1275771308$$
$$x_{67} = -14343.6945028541$$
$$x_{68} = 10236.80795557$$
$$x_{69} = -24514.9615998732$$
$$x_{70} = -26210.1684740975$$
$$x_{71} = -19429.3380351854$$
$$x_{72} = 28884.2114649697$$
$$x_{73} = 34817.4351146439$$
$$x_{74} = -39771.8230970718$$
$$x_{75} = 17017.7546618872$$
$$x_{76} = -16038.9139523032$$
$$x_{77} = 33122.2282898894$$
$$x_{78} = -31295.7886227931$$
$$x_{79} = 29731.8148139625$$
$$x_{80} = 18712.966246377$$
$$x_{81} = 17865.3608241084$$
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
$$- \frac{4 x^{4}}{\left(x^{4} - 16\right)^{2}} + \frac{1}{x^{4} - 16} = 0$$
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
$$- \frac{4 x^{4}}{\left(x^{4} - 16\right)^{2}} + \frac{1}{x^{4} - 16} = 0$$
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
$$\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}} = 0$$
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:
$$x_{1} = -2$$
$$x_{2} = 2$$
$$\lim_{x \to -2^-}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = -\infty$$
$$\lim_{x \to -2^+}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = -2$$
- is an inflection point
$$\lim_{x \to 2^-}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 2$$
- 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
$$\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}} = 0$$
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:
$$x_{1} = -2$$
$$x_{2} = 2$$
$$\lim_{x \to -2^-}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = -\infty$$
$$\lim_{x \to -2^+}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{1} = -2$$
- is an inflection point
$$\lim_{x \to 2^-}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = -\infty$$
$$\lim_{x \to 2^+}\left(\frac{4 x^{3} \left(\frac{8 x^{4}}{x^{4} - 16} - 5\right)}{\left(x^{4} - 16\right)^{2}}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 2$$
- 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:
$$x_{1} = -2$$
$$x_{2} = 2$$
$$x_{1} = -2$$
$$x_{2} = 2$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x}{x^{4} - 16}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{x}{x^{4} - 16}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{x}{x^{4} - 16}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{x}{x^{4} - 16}\right) = 0$$
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 - 16), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \frac{1}{x^{4} - 16} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{x^{4} - 16} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty} \frac{1}{x^{4} - 16} = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty} \frac{1}{x^{4} - 16} = 0$$
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:
$$\frac{x}{x^{4} - 16} = - \frac{x}{x^{4} - 16}$$
- No
$$\frac{x}{x^{4} - 16} = \frac{x}{x^{4} - 16}$$
- Yes
so, the function
is
odd
So, check:
$$\frac{x}{x^{4} - 16} = - \frac{x}{x^{4} - 16}$$
- No
$$\frac{x}{x^{4} - 16} = \frac{x}{x^{4} - 16}$$
- Yes
so, the function
is
odd