Mister Exam
Graphing y = sin(1/x)/(x-pi)
The solution
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/1\
sin|-|
\x/
f(x) = ------
x - pi
$$f{\left(x \right)} = \frac{\sin{\left(\frac{1}{x} \right)}}{x - \pi}$$
f = sin(1/x)/(x - pi)
The graph of the function
The domain of the function
The points at which the function is not precisely defined:
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
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:
$$\frac{\sin{\left(\frac{1}{x} \right)}}{x - \pi} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{\pi}$$
Numerical solution
$$x_{1} = 0.318309886183791$$
so we need to solve the equation:
$$\frac{\sin{\left(\frac{1}{x} \right)}}{x - \pi} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{\pi}$$
Numerical solution
$$x_{1} = 0.318309886183791$$
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
$$- \frac{\sin{\left(\frac{1}{x} \right)}}{\left(x - \pi\right)^{2}} - \frac{\cos{\left(\frac{1}{x} \right)}}{x^{2} \left(x - \pi\right)} = 0$$
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
$$- \frac{\sin{\left(\frac{1}{x} \right)}}{\left(x - \pi\right)^{2}} - \frac{\cos{\left(\frac{1}{x} \right)}}{x^{2} \left(x - \pi\right)} = 0$$
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
$$\frac{\frac{2 \sin{\left(\frac{1}{x} \right)}}{\left(x - \pi\right)^{2}} + \frac{2 \cos{\left(\frac{1}{x} \right)}}{x^{2} \left(x - \pi\right)} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x^{3}}}{x - \pi} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 2672.59914142838$$
$$x_{2} = 9875.86055130205$$
$$x_{3} = -9527.21291879904$$
$$x_{4} = 1795.66481584954$$
$$x_{5} = -7127.5636937748$$
$$x_{6} = 9657.73244039986$$
$$x_{7} = 8567.04750495924$$
$$x_{8} = -4727.03519067431$$
$$x_{9} = -9963.46765765076$$
$$x_{10} = 5730.6688594219$$
$$x_{11} = -8218.37638218406$$
$$x_{12} = -7563.9046250385$$
$$x_{13} = -2103.85757747776$$
$$x_{14} = 10312.1094035261$$
$$x_{15} = 3984.1609499801$$
$$x_{16} = 3110.0820040762$$
$$x_{17} = -1884.49781064975$$
$$x_{18} = 7912.59219570683$$
$$x_{19} = 9003.33121848942$$
$$x_{20} = 10093.9861524262$$
$$x_{21} = 2891.39153275004$$
$$x_{22} = -6254.79868617554$$
$$x_{23} = -7345.73711349166$$
$$x_{24} = -4508.71387076107$$
$$x_{25} = 4202.56336684401$$
$$x_{26} = -6691.19683866279$$
$$x_{27} = 2015.27806635047$$
$$x_{28} = -2322.98364195377$$
$$x_{29} = 2453.67751658293$$
$$x_{30} = 6167.12296517052$$
$$x_{31} = -1664.81794261702$$
$$x_{32} = -2541.93401946594$$
$$x_{33} = 9221.46797442319$$
$$x_{34} = 1575.62328638749$$
$$x_{35} = 4420.93281466911$$
$$x_{36} = 7258.09258415983$$
$$x_{37} = 5294.16278090678$$
$$x_{38} = -4071.99175234101$$
$$x_{39} = 9439.60164515275$$
$$x_{40} = 6603.53544126308$$
$$x_{41} = -10181.5913738514$$
$$x_{42} = -7782.06672155466$$
$$x_{43} = -3635.13246059397$$
$$x_{44} = -11054.0652312485$$
$$x_{45} = -8436.52469271616$$
$$x_{46} = -3853.58211306336$$
$$x_{47} = -2979.45767137699$$
$$x_{48} = 4639.2739666018$$
$$x_{49} = -8000.22384279978$$
$$x_{50} = 4857.59065292343$$
$$x_{51} = -10835.9496655612$$
$$x_{52} = 6385.33387132668$$
$$x_{53} = -9745.34156017184$$
$$x_{54} = -5381.87838463543$$
$$x_{55} = -3198.08109678725$$
$$x_{56} = 2234.58865242645$$
$$x_{57} = 8130.74835775075$$
$$x_{58} = -6909.38381101966$$
$$x_{59} = -9309.08155576863$$
$$x_{60} = 7039.91405776322$$
$$x_{61} = -6036.58577380052$$
$$x_{62} = -6473.00206648509$$
$$x_{63} = 3765.71980294301$$
$$x_{64} = 8785.19114673091$$
$$x_{65} = -8654.66909205839$$
$$x_{66} = -5600.1269454978$$
$$x_{67} = 10748.3494296442$$
$$x_{68} = -9090.94727636317$$
$$x_{69} = 8348.90001232911$$
$$x_{70} = -4945.33463032787$$
$$x_{71} = -10617.8322464235$$
$$x_{72} = 3547.23274074674$$
$$x_{73} = -4290.36738412697$$
$$x_{74} = 10530.2304510805$$
$$x_{75} = 7694.43114135566$$
$$x_{76} = 6821.72857426908$$
$$x_{77} = -5818.36226837651$$
$$x_{78} = 5512.4230916673$$
$$x_{79} = 3328.69068622548$$
$$x_{80} = -3416.63528954843$$
$$x_{81} = 5948.90169104348$$
$$x_{82} = -2760.74905976284$$
$$x_{83} = 5075.88604278917$$
$$x_{84} = -8872.8098668416$$
$$x_{85} = 7476.26476477424$$
$$x_{86} = -10399.7128578208$$
$$x_{87} = -5163.61492826338$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
$$\lim_{x \to 3.14159265358979^-}\left(\frac{\frac{2 \sin{\left(\frac{1}{x} \right)}}{\left(x - \pi\right)^{2}} + \frac{2 \cos{\left(\frac{1}{x} \right)}}{x^{2} \left(x - \pi\right)} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x^{3}}}{x - \pi}\right) = - \frac{1 \left(- 3.57618238781008 \pi^{2} - 0.111889909866023 \pi^{4} - 5.66584098743708 + 0.00588168894525661 \pi^{5} + 7.16097918087282 \pi + 0.888969963637469 \pi^{3}\right)}{- 62.8318530717959 \pi^{3} - 1.90985931710274 \pi^{5} - 186.037660081799 \pi + 97.4090910340024 + 0.101321183642338 \pi^{6} + 148.04406601634 \pi^{2} + 15 \pi^{4}}$$
$$\lim_{x \to 3.14159265358979^+}\left(\frac{\frac{2 \sin{\left(\frac{1}{x} \right)}}{\left(x - \pi\right)^{2}} + \frac{2 \cos{\left(\frac{1}{x} \right)}}{x^{2} \left(x - \pi\right)} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x^{3}}}{x - \pi}\right) = - \frac{1 \left(- 3.57618238781008 \pi^{2} - 0.111889909866023 \pi^{4} - 5.66584098743708 + 0.00588168894525661 \pi^{5} + 7.16097918087282 \pi + 0.888969963637469 \pi^{3}\right)}{- 62.8318530717959 \pi^{3} - 1.90985931710274 \pi^{5} - 186.037660081799 \pi + 97.4090910340024 + 0.101321183642338 \pi^{6} + 148.04406601634 \pi^{2} + 15 \pi^{4}}$$
- 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:
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
$$\frac{\frac{2 \sin{\left(\frac{1}{x} \right)}}{\left(x - \pi\right)^{2}} + \frac{2 \cos{\left(\frac{1}{x} \right)}}{x^{2} \left(x - \pi\right)} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x^{3}}}{x - \pi} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 2672.59914142838$$
$$x_{2} = 9875.86055130205$$
$$x_{3} = -9527.21291879904$$
$$x_{4} = 1795.66481584954$$
$$x_{5} = -7127.5636937748$$
$$x_{6} = 9657.73244039986$$
$$x_{7} = 8567.04750495924$$
$$x_{8} = -4727.03519067431$$
$$x_{9} = -9963.46765765076$$
$$x_{10} = 5730.6688594219$$
$$x_{11} = -8218.37638218406$$
$$x_{12} = -7563.9046250385$$
$$x_{13} = -2103.85757747776$$
$$x_{14} = 10312.1094035261$$
$$x_{15} = 3984.1609499801$$
$$x_{16} = 3110.0820040762$$
$$x_{17} = -1884.49781064975$$
$$x_{18} = 7912.59219570683$$
$$x_{19} = 9003.33121848942$$
$$x_{20} = 10093.9861524262$$
$$x_{21} = 2891.39153275004$$
$$x_{22} = -6254.79868617554$$
$$x_{23} = -7345.73711349166$$
$$x_{24} = -4508.71387076107$$
$$x_{25} = 4202.56336684401$$
$$x_{26} = -6691.19683866279$$
$$x_{27} = 2015.27806635047$$
$$x_{28} = -2322.98364195377$$
$$x_{29} = 2453.67751658293$$
$$x_{30} = 6167.12296517052$$
$$x_{31} = -1664.81794261702$$
$$x_{32} = -2541.93401946594$$
$$x_{33} = 9221.46797442319$$
$$x_{34} = 1575.62328638749$$
$$x_{35} = 4420.93281466911$$
$$x_{36} = 7258.09258415983$$
$$x_{37} = 5294.16278090678$$
$$x_{38} = -4071.99175234101$$
$$x_{39} = 9439.60164515275$$
$$x_{40} = 6603.53544126308$$
$$x_{41} = -10181.5913738514$$
$$x_{42} = -7782.06672155466$$
$$x_{43} = -3635.13246059397$$
$$x_{44} = -11054.0652312485$$
$$x_{45} = -8436.52469271616$$
$$x_{46} = -3853.58211306336$$
$$x_{47} = -2979.45767137699$$
$$x_{48} = 4639.2739666018$$
$$x_{49} = -8000.22384279978$$
$$x_{50} = 4857.59065292343$$
$$x_{51} = -10835.9496655612$$
$$x_{52} = 6385.33387132668$$
$$x_{53} = -9745.34156017184$$
$$x_{54} = -5381.87838463543$$
$$x_{55} = -3198.08109678725$$
$$x_{56} = 2234.58865242645$$
$$x_{57} = 8130.74835775075$$
$$x_{58} = -6909.38381101966$$
$$x_{59} = -9309.08155576863$$
$$x_{60} = 7039.91405776322$$
$$x_{61} = -6036.58577380052$$
$$x_{62} = -6473.00206648509$$
$$x_{63} = 3765.71980294301$$
$$x_{64} = 8785.19114673091$$
$$x_{65} = -8654.66909205839$$
$$x_{66} = -5600.1269454978$$
$$x_{67} = 10748.3494296442$$
$$x_{68} = -9090.94727636317$$
$$x_{69} = 8348.90001232911$$
$$x_{70} = -4945.33463032787$$
$$x_{71} = -10617.8322464235$$
$$x_{72} = 3547.23274074674$$
$$x_{73} = -4290.36738412697$$
$$x_{74} = 10530.2304510805$$
$$x_{75} = 7694.43114135566$$
$$x_{76} = 6821.72857426908$$
$$x_{77} = -5818.36226837651$$
$$x_{78} = 5512.4230916673$$
$$x_{79} = 3328.69068622548$$
$$x_{80} = -3416.63528954843$$
$$x_{81} = 5948.90169104348$$
$$x_{82} = -2760.74905976284$$
$$x_{83} = 5075.88604278917$$
$$x_{84} = -8872.8098668416$$
$$x_{85} = 7476.26476477424$$
$$x_{86} = -10399.7128578208$$
$$x_{87} = -5163.61492826338$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
True
True
- the limits are not equal, so
$$x_{1} = 0$$
- is an inflection point
$$\lim_{x \to 3.14159265358979^-}\left(\frac{\frac{2 \sin{\left(\frac{1}{x} \right)}}{\left(x - \pi\right)^{2}} + \frac{2 \cos{\left(\frac{1}{x} \right)}}{x^{2} \left(x - \pi\right)} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x^{3}}}{x - \pi}\right) = - \frac{1 \left(- 3.57618238781008 \pi^{2} - 0.111889909866023 \pi^{4} - 5.66584098743708 + 0.00588168894525661 \pi^{5} + 7.16097918087282 \pi + 0.888969963637469 \pi^{3}\right)}{- 62.8318530717959 \pi^{3} - 1.90985931710274 \pi^{5} - 186.037660081799 \pi + 97.4090910340024 + 0.101321183642338 \pi^{6} + 148.04406601634 \pi^{2} + 15 \pi^{4}}$$
$$\lim_{x \to 3.14159265358979^+}\left(\frac{\frac{2 \sin{\left(\frac{1}{x} \right)}}{\left(x - \pi\right)^{2}} + \frac{2 \cos{\left(\frac{1}{x} \right)}}{x^{2} \left(x - \pi\right)} + \frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x^{3}}}{x - \pi}\right) = - \frac{1 \left(- 3.57618238781008 \pi^{2} - 0.111889909866023 \pi^{4} - 5.66584098743708 + 0.00588168894525661 \pi^{5} + 7.16097918087282 \pi + 0.888969963637469 \pi^{3}\right)}{- 62.8318530717959 \pi^{3} - 1.90985931710274 \pi^{5} - 186.037660081799 \pi + 97.4090910340024 + 0.101321183642338 \pi^{6} + 148.04406601634 \pi^{2} + 15 \pi^{4}}$$
- 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:
Have no bends at the whole real axis
Vertical asymptotes
Have:
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
$$x_{1} = 0$$
$$x_{2} = 3.14159265358979$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x - \pi}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x - \pi}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x - \pi}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x - \pi}\right) = 0$$
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 sin(1/x)/(x - pi), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x \left(x - \pi\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x \left(x - \pi\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x \left(x - \pi\right)}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\sin{\left(\frac{1}{x} \right)}}{x \left(x - \pi\right)}\right) = 0$$
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:
$$\frac{\sin{\left(\frac{1}{x} \right)}}{x - \pi} = - \frac{\sin{\left(\frac{1}{x} \right)}}{- x - \pi}$$
- No
$$\frac{\sin{\left(\frac{1}{x} \right)}}{x - \pi} = \frac{\sin{\left(\frac{1}{x} \right)}}{- x - \pi}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\sin{\left(\frac{1}{x} \right)}}{x - \pi} = - \frac{\sin{\left(\frac{1}{x} \right)}}{- x - \pi}$$
- No
$$\frac{\sin{\left(\frac{1}{x} \right)}}{x - \pi} = \frac{\sin{\left(\frac{1}{x} \right)}}{- x - \pi}$$
- No
so, the function
not is
neither even, nor odd