Mister Exam
Other calculators
- Graph of implicit function
- Derivative Step by Step
- Integral Step by Step
- Graph of parametric function
- Graph in Polar Coordinates
- Surface defined parametrically
- Surface defined by equation
- Curve in 3D space defined parametrically
- Area between lines
- The intersection points of functions
-
How to use it?
- Graphing y =:
- 8/(16-x^2)
- -5x+7
- 5/2x-1
- 3*x^4+4*x^3+1
- Integral of d{x}:
- x*cos(1/x)
- Limit of the function:
-
x*cos(1/x)
-
Identical expressions
- x*cos(one /x)
- x multiply by co sinus of e of (1 divide by x)
- x multiply by co sinus of e of (one divide by x)
- xcos(1/x)
- xcos1/x
- x*cos(1 divide by x)
-
Similar expressions
- x*cos(1/(x+pi))
- ((e^x)*cos(1/x))
- xcos(1/(x-1))
Graphing y = x*cos(1/x)
The solution
You have entered
[src]
/1\
f(x) = x*cos|-|
\x/
$$f{\left(x \right)} = x \cos{\left(\frac{1}{x} \right)}$$
f = x*cos(1/x)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{1} = 0$$
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:
$$x \cos{\left(\frac{1}{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{2}{\pi}$$
$$x_{2} = \frac{2}{\pi}$$
Numerical solution
$$x_{1} = 0.00478661482983144$$
$$x_{2} = 0.0235785100876882$$
$$x_{3} = 0.0303152272555991$$
$$x_{4} = -0.0155273115211605$$
$$x_{5} = 0.010436389710944$$
$$x_{6} = 0.00144358225026662$$
$$x_{7} = 0.00385830165071261$$
$$x_{8} = -0.0909456817667973$$
$$x_{9} = 0.0276791205377209$$
$$x_{10} = 0.0172059397937184$$
$$x_{11} = 0.00191177108819094$$
$$x_{12} = 0.0707355302630646$$
$$x_{13} = 0.0335063038088201$$
$$x_{14} = -0.00509295817894065$$
$$x_{15} = -0.0254647908947033$$
$$x_{16} = -0.0129922402523996$$
$$x_{17} = -0.127323954473516$$
$$x_{18} = -0.0192915082535631$$
$$x_{19} = 0.0374482219039754$$
$$x_{20} = 0.00553582410754419$$
$$x_{21} = 0.636619772367581$$
$$x_{22} = -4.56260139301642 \cdot 10^{-5}$$
$$x_{23} = -0.00848826363156775$$
$$x_{24} = 0.00848826363156775$$
$$x_{25} = -0.00116383870633927$$
$$x_{26} = -0.00367988307726926$$
$$x_{27} = 0.0120116938182563$$
$$x_{28} = 5.45658500357917 \cdot 10^{-5}$$
$$x_{29} = 0.127323954473516$$
$$x_{30} = -7.44671625181403 \cdot 10^{-5}$$
$$x_{31} = -0.010436389710944$$
$$x_{32} = 0.00190035752945547$$
$$x_{33} = -0.00326471678137221$$
$$x_{34} = 0.0124827406346585$$
$$x_{35} = -0.0205361216892768$$
$$x_{36} = 0.00115121116160503$$
$$x_{37} = -0.0578745247606892$$
$$x_{38} = -7.90537405150356 \cdot 10^{-5}$$
$$x_{39} = -0.0335063038088201$$
$$x_{40} = -0.636619772367581$$
$$x_{41} = -0.0303152272555991$$
$$x_{42} = 8.63916097662616 \cdot 10^{-5}$$
$$x_{43} = 0.00236661625415458$$
$$x_{44} = -0.000567903454386781$$
$$x_{45} = -0.0489707517205832$$
$$x_{46} = 0.0909456817667973$$
so we need to solve the equation:
$$x \cos{\left(\frac{1}{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = - \frac{2}{\pi}$$
$$x_{2} = \frac{2}{\pi}$$
Numerical solution
$$x_{1} = 0.00478661482983144$$
$$x_{2} = 0.0235785100876882$$
$$x_{3} = 0.0303152272555991$$
$$x_{4} = -0.0155273115211605$$
$$x_{5} = 0.010436389710944$$
$$x_{6} = 0.00144358225026662$$
$$x_{7} = 0.00385830165071261$$
$$x_{8} = -0.0909456817667973$$
$$x_{9} = 0.0276791205377209$$
$$x_{10} = 0.0172059397937184$$
$$x_{11} = 0.00191177108819094$$
$$x_{12} = 0.0707355302630646$$
$$x_{13} = 0.0335063038088201$$
$$x_{14} = -0.00509295817894065$$
$$x_{15} = -0.0254647908947033$$
$$x_{16} = -0.0129922402523996$$
$$x_{17} = -0.127323954473516$$
$$x_{18} = -0.0192915082535631$$
$$x_{19} = 0.0374482219039754$$
$$x_{20} = 0.00553582410754419$$
$$x_{21} = 0.636619772367581$$
$$x_{22} = -4.56260139301642 \cdot 10^{-5}$$
$$x_{23} = -0.00848826363156775$$
$$x_{24} = 0.00848826363156775$$
$$x_{25} = -0.00116383870633927$$
$$x_{26} = -0.00367988307726926$$
$$x_{27} = 0.0120116938182563$$
$$x_{28} = 5.45658500357917 \cdot 10^{-5}$$
$$x_{29} = 0.127323954473516$$
$$x_{30} = -7.44671625181403 \cdot 10^{-5}$$
$$x_{31} = -0.010436389710944$$
$$x_{32} = 0.00190035752945547$$
$$x_{33} = -0.00326471678137221$$
$$x_{34} = 0.0124827406346585$$
$$x_{35} = -0.0205361216892768$$
$$x_{36} = 0.00115121116160503$$
$$x_{37} = -0.0578745247606892$$
$$x_{38} = -7.90537405150356 \cdot 10^{-5}$$
$$x_{39} = -0.0335063038088201$$
$$x_{40} = -0.636619772367581$$
$$x_{41} = -0.0303152272555991$$
$$x_{42} = 8.63916097662616 \cdot 10^{-5}$$
$$x_{43} = 0.00236661625415458$$
$$x_{44} = -0.000567903454386781$$
$$x_{45} = -0.0489707517205832$$
$$x_{46} = 0.0909456817667973$$
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
$$\cos{\left(\frac{1}{x} \right)} + \frac{\sin{\left(\frac{1}{x} \right)}}{x} = 0$$
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
$$\cos{\left(\frac{1}{x} \right)} + \frac{\sin{\left(\frac{1}{x} \right)}}{x} = 0$$
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
$$- \frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{2}{3 \pi}$$
$$x_{2} = \frac{2}{\pi}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}}\right) = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to 0^+}\left(- \frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}}\right) = \left\langle -\infty, \infty\right\rangle$$
- 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
$$\left[\frac{2}{3 \pi}, \frac{2}{\pi}\right]$$
Convex at the intervals
$$\left(-\infty, \frac{2}{3 \pi}\right] \cup \left[\frac{2}{\pi}, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$- \frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = \frac{2}{3 \pi}$$
$$x_{2} = \frac{2}{\pi}$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}}\right) = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to 0^+}\left(- \frac{\cos{\left(\frac{1}{x} \right)}}{x^{3}}\right) = \left\langle -\infty, \infty\right\rangle$$
- 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
$$\left[\frac{2}{3 \pi}, \frac{2}{\pi}\right]$$
Convex at the intervals
$$\left(-\infty, \frac{2}{3 \pi}\right] \cup \left[\frac{2}{\pi}, \infty\right)$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(x \cos{\left(\frac{1}{x} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x \cos{\left(\frac{1}{x} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(x \cos{\left(\frac{1}{x} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(x \cos{\left(\frac{1}{x} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x*cos(1/x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty} \cos{\left(\frac{1}{x} \right)} = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty} \cos{\left(\frac{1}{x} \right)} = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty} \cos{\left(\frac{1}{x} \right)} = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty} \cos{\left(\frac{1}{x} \right)} = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
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:
$$x \cos{\left(\frac{1}{x} \right)} = - x \cos{\left(\frac{1}{x} \right)}$$
- No
$$x \cos{\left(\frac{1}{x} \right)} = x \cos{\left(\frac{1}{x} \right)}$$
- Yes
so, the function
is
odd
So, check:
$$x \cos{\left(\frac{1}{x} \right)} = - x \cos{\left(\frac{1}{x} \right)}$$
- No
$$x \cos{\left(\frac{1}{x} \right)} = x \cos{\left(\frac{1}{x} \right)}$$
- Yes
so, the function
is
odd