Mister Exam
Graphing y = x-2+sin(1/x)
The solution
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/1\
f(x) = x - 2 + sin|-|
\x/
$$f{\left(x \right)} = \left(x - 2\right) + \sin{\left(\frac{1}{x} \right)}$$
f = x - 2 + sin(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:
$$\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 1.30766271568223$$
so we need to solve the equation:
$$\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = 1.30766271568223$$
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
$$1 - \frac{\cos{\left(\frac{1}{x} \right)}}{x^{2}} = 0$$
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
$$1 - \frac{\cos{\left(\frac{1}{x} \right)}}{x^{2}} = 0$$
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
$$\frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -24517.764514177$$
$$x_{2} = -31297.9789349291$$
$$x_{3} = -22822.7684696829$$
$$x_{4} = -37230.8517263672$$
$$x_{5} = -42316.2509339008$$
$$x_{6} = 33971.8484557919$$
$$x_{7} = 28886.5862895034$$
$$x_{8} = -38925.9782260908$$
$$x_{9} = -13501.2455714192$$
$$x_{10} = -27060.3085181287$$
$$x_{11} = -30450.4370366936$$
$$x_{12} = -23670.2626854463$$
$$x_{13} = -28755.3642709925$$
$$x_{14} = 32276.7483423665$$
$$x_{15} = -34688.1770045647$$
$$x_{16} = -16043.2338667424$$
$$x_{17} = -17738.0242579883$$
$$x_{18} = 23801.4794641224$$
$$x_{19} = -21127.8065465298$$
$$x_{20} = 39904.771377312$$
$$x_{21} = 33124.297012115$$
$$x_{22} = 20411.5519538498$$
$$x_{23} = 25496.4920696083$$
$$x_{24} = -33840.6233739026$$
$$x_{25} = -19432.887686874$$
$$x_{26} = -26212.7880538116$$
$$x_{27} = 17021.8196886792$$
$$x_{28} = -18585.4480940284$$
$$x_{29} = 18716.6547734102$$
$$x_{30} = 22106.496925216$$
$$x_{31} = 42447.4788863631$$
$$x_{32} = -11806.7845665482$$
$$x_{33} = 37362.078192229$$
$$x_{34} = 17869.2283625365$$
$$x_{35} = -15195.8737165349$$
$$x_{36} = 39057.2052525216$$
$$x_{37} = -32145.5241220507$$
$$x_{38} = 21259.0191811792$$
$$x_{39} = 12785.166209279$$
$$x_{40} = 34819.4024707177$$
$$x_{41} = 40752.3390879025$$
$$x_{42} = 15327.0673496263$$
$$x_{43} = 30581.6602478055$$
$$x_{44} = -21975.2827488203$$
$$x_{45} = 16174.4315462694$$
$$x_{46} = -29602.8987100326$$
$$x_{47} = 29734.1213504079$$
$$x_{48} = 13632.4287111505$$
$$x_{49} = 22953.9840195429$$
$$x_{50} = 14479.7313150492$$
$$x_{51} = -16890.6185554691$$
$$x_{52} = 28039.0554132717$$
$$x_{53} = -40621.1115695252$$
$$x_{54} = 27191.5291133465$$
$$x_{55} = -35535.7330502043$$
$$x_{56} = -27907.8340740483$$
$$x_{57} = -25365.273191964$$
$$x_{58} = -38078.4140636161$$
$$x_{59} = 19564.09661214$$
$$x_{60} = -41468.6805452608$$
$$x_{61} = 35666.9588734903$$
$$x_{62} = -14348.5424650725$$
$$x_{63} = 36514.5174977468$$
$$x_{64} = -36383.2913419629$$
$$x_{65} = 26344.0078318137$$
$$x_{66} = 24648.982396646$$
$$x_{67} = 11937.9523841008$$
$$x_{68} = -32993.0723444395$$
$$x_{69} = 31429.2026711056$$
$$x_{70} = 38209.640819107$$
$$x_{71} = 41599.9082873259$$
$$x_{72} = -39773.5440970563$$
$$x_{73} = -12653.9899632038$$
$$x_{74} = -20280.3410578956$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
- the limits are not equal, so
$$x_{1} = 0$$
- 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
$$\frac{2 \cos{\left(\frac{1}{x} \right)} - \frac{\sin{\left(\frac{1}{x} \right)}}{x}}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -24517.764514177$$
$$x_{2} = -31297.9789349291$$
$$x_{3} = -22822.7684696829$$
$$x_{4} = -37230.8517263672$$
$$x_{5} = -42316.2509339008$$
$$x_{6} = 33971.8484557919$$
$$x_{7} = 28886.5862895034$$
$$x_{8} = -38925.9782260908$$
$$x_{9} = -13501.2455714192$$
$$x_{10} = -27060.3085181287$$
$$x_{11} = -30450.4370366936$$
$$x_{12} = -23670.2626854463$$
$$x_{13} = -28755.3642709925$$
$$x_{14} = 32276.7483423665$$
$$x_{15} = -34688.1770045647$$
$$x_{16} = -16043.2338667424$$
$$x_{17} = -17738.0242579883$$
$$x_{18} = 23801.4794641224$$
$$x_{19} = -21127.8065465298$$
$$x_{20} = 39904.771377312$$
$$x_{21} = 33124.297012115$$
$$x_{22} = 20411.5519538498$$
$$x_{23} = 25496.4920696083$$
$$x_{24} = -33840.6233739026$$
$$x_{25} = -19432.887686874$$
$$x_{26} = -26212.7880538116$$
$$x_{27} = 17021.8196886792$$
$$x_{28} = -18585.4480940284$$
$$x_{29} = 18716.6547734102$$
$$x_{30} = 22106.496925216$$
$$x_{31} = 42447.4788863631$$
$$x_{32} = -11806.7845665482$$
$$x_{33} = 37362.078192229$$
$$x_{34} = 17869.2283625365$$
$$x_{35} = -15195.8737165349$$
$$x_{36} = 39057.2052525216$$
$$x_{37} = -32145.5241220507$$
$$x_{38} = 21259.0191811792$$
$$x_{39} = 12785.166209279$$
$$x_{40} = 34819.4024707177$$
$$x_{41} = 40752.3390879025$$
$$x_{42} = 15327.0673496263$$
$$x_{43} = 30581.6602478055$$
$$x_{44} = -21975.2827488203$$
$$x_{45} = 16174.4315462694$$
$$x_{46} = -29602.8987100326$$
$$x_{47} = 29734.1213504079$$
$$x_{48} = 13632.4287111505$$
$$x_{49} = 22953.9840195429$$
$$x_{50} = 14479.7313150492$$
$$x_{51} = -16890.6185554691$$
$$x_{52} = 28039.0554132717$$
$$x_{53} = -40621.1115695252$$
$$x_{54} = 27191.5291133465$$
$$x_{55} = -35535.7330502043$$
$$x_{56} = -27907.8340740483$$
$$x_{57} = -25365.273191964$$
$$x_{58} = -38078.4140636161$$
$$x_{59} = 19564.09661214$$
$$x_{60} = -41468.6805452608$$
$$x_{61} = 35666.9588734903$$
$$x_{62} = -14348.5424650725$$
$$x_{63} = 36514.5174977468$$
$$x_{64} = -36383.2913419629$$
$$x_{65} = 26344.0078318137$$
$$x_{66} = 24648.982396646$$
$$x_{67} = 11937.9523841008$$
$$x_{68} = -32993.0723444395$$
$$x_{69} = 31429.2026711056$$
$$x_{70} = 38209.640819107$$
$$x_{71} = 41599.9082873259$$
$$x_{72} = -39773.5440970563$$
$$x_{73} = -12653.9899632038$$
$$x_{74} = -20280.3410578956$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
True
True
- the limits are not equal, so
$$x_{1} = 0$$
- 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:
$$x_{1} = 0$$
$$x_{1} = 0$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
$$\lim_{x \to -\infty}\left(\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)}\right) = -\infty$$
Let's take the limit
so,
horizontal asymptote on the left doesn’t exist
$$\lim_{x \to \infty}\left(\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)}\right) = \infty$$
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 - 2 + sin(1/x), divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x$$
$$\lim_{x \to -\infty}\left(\frac{\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)}}{x}\right) = 1$$
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x$$
$$\lim_{x \to \infty}\left(\frac{\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)}}{x}\right) = 1$$
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:
$$\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)} = - x - \sin{\left(\frac{1}{x} \right)} - 2$$
- No
$$\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)} = x + \sin{\left(\frac{1}{x} \right)} + 2$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)} = - x - \sin{\left(\frac{1}{x} \right)} - 2$$
- No
$$\left(x - 2\right) + \sin{\left(\frac{1}{x} \right)} = x + \sin{\left(\frac{1}{x} \right)} + 2$$
- No
so, the function
not is
neither even, nor odd