Mister Exam
Graphing y = x(sin(x))/(1-(cos(x)))
The solution
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x*sin(x)
f(x) = ----------
1 - cos(x)
$$f{\left(x \right)} = \frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}}$$
f = (x*sin(x))/(1 - cos(x))
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
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:
$$\frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \pi$$
Numerical solution
$$x_{1} = 72.2566310325652$$
$$x_{2} = -59.6902604182061$$
$$x_{3} = 3.14159265358979$$
$$x_{4} = 28.2743338823081$$
$$x_{5} = 65.9734457253857$$
$$x_{6} = -9.42477796076938$$
$$x_{7} = 40.8407044966673$$
$$x_{8} = -3.14159265358979$$
$$x_{9} = -15.707963267949$$
$$x_{10} = 59.6902604182061$$
$$x_{11} = 9.42477796076938$$
$$x_{12} = -53.4070751110265$$
$$x_{13} = -47.1238898038469$$
$$x_{14} = -84.8230016469244$$
$$x_{15} = 21.9911485751286$$
$$x_{16} = -72.2566310325652$$
$$x_{17} = 34.5575191894877$$
$$x_{18} = -21.9911485751286$$
$$x_{19} = -65.9734457253857$$
$$x_{20} = 53.4070751110265$$
$$x_{21} = 15.707963267949$$
$$x_{22} = -28.2743338823081$$
$$x_{23} = -91.106186954104$$
$$x_{24} = 47.1238898038469$$
$$x_{25} = 97.3893722612836$$
$$x_{26} = 78.5398163397448$$
$$x_{27} = -78.5398163397448$$
$$x_{28} = -40.8407044966673$$
$$x_{29} = -97.3893722612836$$
$$x_{30} = 84.8230016469244$$
$$x_{31} = 91.106186954104$$
$$x_{32} = -34.5575191894877$$
so we need to solve the equation:
$$\frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \pi$$
Numerical solution
$$x_{1} = 72.2566310325652$$
$$x_{2} = -59.6902604182061$$
$$x_{3} = 3.14159265358979$$
$$x_{4} = 28.2743338823081$$
$$x_{5} = 65.9734457253857$$
$$x_{6} = -9.42477796076938$$
$$x_{7} = 40.8407044966673$$
$$x_{8} = -3.14159265358979$$
$$x_{9} = -15.707963267949$$
$$x_{10} = 59.6902604182061$$
$$x_{11} = 9.42477796076938$$
$$x_{12} = -53.4070751110265$$
$$x_{13} = -47.1238898038469$$
$$x_{14} = -84.8230016469244$$
$$x_{15} = 21.9911485751286$$
$$x_{16} = -72.2566310325652$$
$$x_{17} = 34.5575191894877$$
$$x_{18} = -21.9911485751286$$
$$x_{19} = -65.9734457253857$$
$$x_{20} = 53.4070751110265$$
$$x_{21} = 15.707963267949$$
$$x_{22} = -28.2743338823081$$
$$x_{23} = -91.106186954104$$
$$x_{24} = 47.1238898038469$$
$$x_{25} = 97.3893722612836$$
$$x_{26} = 78.5398163397448$$
$$x_{27} = -78.5398163397448$$
$$x_{28} = -40.8407044966673$$
$$x_{29} = -97.3893722612836$$
$$x_{30} = 84.8230016469244$$
$$x_{31} = 91.106186954104$$
$$x_{32} = -34.5575191894877$$
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
$$- \frac{x \sin^{2}{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} + \frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{1 - \cos{\left(x \right)}} = 0$$
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
$$- \frac{x \sin^{2}{\left(x \right)}}{\left(1 - \cos{\left(x \right)}\right)^{2}} + \frac{x \cos{\left(x \right)} + \sin{\left(x \right)}}{1 - \cos{\left(x \right)}} = 0$$
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
$$\frac{x \sin{\left(x \right)} - \frac{x \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - 2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 53.3321085176254$$
$$x_{2} = -91.0622680279826$$
$$x_{3} = -53.3321085176254$$
$$x_{4} = 78.4888647223284$$
$$x_{5} = -40.7426059185751$$
$$x_{6} = 15.4505036738754$$
$$x_{7} = -34.4415105438615$$
$$x_{8} = 8.98681891581813$$
$$x_{9} = -8.98681891581813$$
$$x_{10} = -84.7758271362638$$
$$x_{11} = 91.0622680279826$$
$$x_{12} = -78.4888647223284$$
$$x_{13} = 34.4415105438615$$
$$x_{14} = -59.6231975817859$$
$$x_{15} = 97.3482884639088$$
$$x_{16} = 21.8082433188578$$
$$x_{17} = 59.6231975817859$$
$$x_{18} = 40.7426059185751$$
$$x_{19} = -72.2012444887512$$
$$x_{20} = -65.912778079645$$
$$x_{21} = -97.3482884639088$$
$$x_{22} = 72.2012444887512$$
$$x_{23} = -21.8082433188578$$
$$x_{24} = -47.038904997378$$
$$x_{25} = 28.1323878256629$$
$$x_{26} = 65.912778079645$$
$$x_{27} = 84.7758271362638$$
$$x_{28} = -28.1323878256629$$
$$x_{29} = 47.038904997378$$
$$x_{30} = -15.4505036738754$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
$$\lim_{x \to 0^-}\left(\frac{x \sin{\left(x \right)} - \frac{x \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - 2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) = - \frac{1}{3}$$
$$\lim_{x \to 0^+}\left(\frac{x \sin{\left(x \right)} - \frac{x \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - 2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) = - \frac{1}{3}$$
- limits are equal, then skip the corresponding point
$$\lim_{x \to 6.28318530717959^-}\left(\frac{x \sin{\left(x \right)} - \frac{x \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - 2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) = -\infty$$
$$\lim_{x \to 6.28318530717959^+}\left(\frac{x \sin{\left(x \right)} - \frac{x \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - 2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 6.28318530717959$$
- 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:
Concave at the intervals
$$\left[-8.98681891581813, 8.98681891581813\right]$$
Convex at the intervals
$$\left(-\infty, -97.3482884639088\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{x \sin{\left(x \right)} - \frac{x \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - 2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 53.3321085176254$$
$$x_{2} = -91.0622680279826$$
$$x_{3} = -53.3321085176254$$
$$x_{4} = 78.4888647223284$$
$$x_{5} = -40.7426059185751$$
$$x_{6} = 15.4505036738754$$
$$x_{7} = -34.4415105438615$$
$$x_{8} = 8.98681891581813$$
$$x_{9} = -8.98681891581813$$
$$x_{10} = -84.7758271362638$$
$$x_{11} = 91.0622680279826$$
$$x_{12} = -78.4888647223284$$
$$x_{13} = 34.4415105438615$$
$$x_{14} = -59.6231975817859$$
$$x_{15} = 97.3482884639088$$
$$x_{16} = 21.8082433188578$$
$$x_{17} = 59.6231975817859$$
$$x_{18} = 40.7426059185751$$
$$x_{19} = -72.2012444887512$$
$$x_{20} = -65.912778079645$$
$$x_{21} = -97.3482884639088$$
$$x_{22} = 72.2012444887512$$
$$x_{23} = -21.8082433188578$$
$$x_{24} = -47.038904997378$$
$$x_{25} = 28.1323878256629$$
$$x_{26} = 65.912778079645$$
$$x_{27} = 84.7758271362638$$
$$x_{28} = -28.1323878256629$$
$$x_{29} = 47.038904997378$$
$$x_{30} = -15.4505036738754$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
$$\lim_{x \to 0^-}\left(\frac{x \sin{\left(x \right)} - \frac{x \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - 2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) = - \frac{1}{3}$$
$$\lim_{x \to 0^+}\left(\frac{x \sin{\left(x \right)} - \frac{x \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - 2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) = - \frac{1}{3}$$
- limits are equal, then skip the corresponding point
$$\lim_{x \to 6.28318530717959^-}\left(\frac{x \sin{\left(x \right)} - \frac{x \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - 2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) = -\infty$$
$$\lim_{x \to 6.28318530717959^+}\left(\frac{x \sin{\left(x \right)} - \frac{x \left(\cos{\left(x \right)} + \frac{2 \sin^{2}{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - \frac{2 \left(x \cos{\left(x \right)} + \sin{\left(x \right)}\right) \sin{\left(x \right)}}{\cos{\left(x \right)} - 1} - 2 \cos{\left(x \right)}}{\cos{\left(x \right)} - 1}\right) = \infty$$
- the limits are not equal, so
$$x_{2} = 6.28318530717959$$
- 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:
Concave at the intervals
$$\left[-8.98681891581813, 8.98681891581813\right]$$
Convex at the intervals
$$\left(-\infty, -97.3482884639088\right]$$
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
$$x_{1} = 0$$
$$x_{2} = 6.28318530717959$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty}\left(\frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right) = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty}\left(\frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right) = \left\langle -\infty, \infty\right\rangle$$
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 (x*sin(x))/(1 - cos(x)), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\sin{\left(x \right)}}{1 - \cos{\left(x \right)}}\right)$$
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:
$$\frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}} = \frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}}$$
- Yes
$$\frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}} = - \frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}}$$
- No
so, the function
is
even
So, check:
$$\frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}} = \frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}}$$
- Yes
$$\frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}} = - \frac{x \sin{\left(x \right)}}{1 - \cos{\left(x \right)}}$$
- No
so, the function
is
even