Mister Exam
Graphing y = (x-1)*cosx/x-1
The solution
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(x - 1)*cos(x)
f(x) = -------------- - 1
x
$$f{\left(x \right)} = -1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}$$
f = -1 + ((x - 1)*cos(x))/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:
$$-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -3908.11864147039$$
$$x_{2} = -81.5255757034701$$
$$x_{3} = -37.4706068235463$$
$$x_{4} = -87.8143902428644$$
$$x_{5} = -50.4629398821162$$
$$x_{6} = -81.8369485363712$$
$$x_{7} = -6.79528546545564$$
$$x_{8} = -88.1145447112556$$
$$x_{9} = -31.6640081143412$$
$$x_{10} = -94.1026353004828$$
$$x_{11} = -63.0088484909748$$
$$x_{12} = -56.7350583803468$$
$$x_{13} = -50.0672590311525$$
$$x_{14} = -75.5600271487453$$
$$x_{15} = -43.7705436902596$$
$$x_{16} = -1.03674878997284$$
$$x_{17} = -69.2839281169777$$
$$x_{18} = -12.174233004382$$
$$x_{19} = -150.681555978239$$
$$x_{20} = -24.8537031333593$$
$$x_{21} = -18.528148876584$$
$$x_{22} = -37.9262696941923$$
$$x_{23} = -100.390400893298$$
$$x_{24} = -19.1657971772164$$
$$x_{25} = -56.3616697214293$$
$$x_{26} = -62.6543642083992$$
$$x_{27} = -68.9457398660007$$
$$x_{28} = -44.193054743776$$
$$x_{29} = -25.4088121091366$$
$$x_{30} = -31.165921585635$$
$$x_{31} = -94.3927026839426$$
$$x_{32} = -12.9473476761254$$
$$x_{33} = -75.2360760574743$$
$$x_{34} = -5830.81448414035$$
$$x_{35} = -5.73110630283233$$
so we need to solve the equation:
$$-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -3908.11864147039$$
$$x_{2} = -81.5255757034701$$
$$x_{3} = -37.4706068235463$$
$$x_{4} = -87.8143902428644$$
$$x_{5} = -50.4629398821162$$
$$x_{6} = -81.8369485363712$$
$$x_{7} = -6.79528546545564$$
$$x_{8} = -88.1145447112556$$
$$x_{9} = -31.6640081143412$$
$$x_{10} = -94.1026353004828$$
$$x_{11} = -63.0088484909748$$
$$x_{12} = -56.7350583803468$$
$$x_{13} = -50.0672590311525$$
$$x_{14} = -75.5600271487453$$
$$x_{15} = -43.7705436902596$$
$$x_{16} = -1.03674878997284$$
$$x_{17} = -69.2839281169777$$
$$x_{18} = -12.174233004382$$
$$x_{19} = -150.681555978239$$
$$x_{20} = -24.8537031333593$$
$$x_{21} = -18.528148876584$$
$$x_{22} = -37.9262696941923$$
$$x_{23} = -100.390400893298$$
$$x_{24} = -19.1657971772164$$
$$x_{25} = -56.3616697214293$$
$$x_{26} = -62.6543642083992$$
$$x_{27} = -68.9457398660007$$
$$x_{28} = -44.193054743776$$
$$x_{29} = -25.4088121091366$$
$$x_{30} = -31.165921585635$$
$$x_{31} = -94.3927026839426$$
$$x_{32} = -12.9473476761254$$
$$x_{33} = -75.2360760574743$$
$$x_{34} = -5830.81448414035$$
$$x_{35} = -5.73110630283233$$
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
$$\frac{- \left(x - 1\right) \sin{\left(x \right)} + \cos{\left(x \right)}}{x} - \frac{\left(x - 1\right) \cos{\left(x \right)}}{x^{2}} = 0$$
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
$$\frac{- \left(x - 1\right) \sin{\left(x \right)} + \cos{\left(x \right)}}{x} - \frac{\left(x - 1\right) \cos{\left(x \right)}}{x^{2}} = 0$$
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
$$\frac{- \left(x - 1\right) \cos{\left(x \right)} - 2 \sin{\left(x \right)} + \frac{\left(x - 1\right) \sin{\left(x \right)}}{x} + \frac{\left(x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(x - 1\right) \cos{\left(x \right)}}{x^{2}}}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 29.8474528425252$$
$$x_{2} = -14.1278030885708$$
$$x_{3} = 4.81830169068692$$
$$x_{4} = 32.9886180185199$$
$$x_{5} = -80.11030486752$$
$$x_{6} = -865.508773397218$$
$$x_{7} = -58.1188820004811$$
$$x_{8} = 76.9693620488425$$
$$x_{9} = 54.9785453606757$$
$$x_{10} = 64.4031391874027$$
$$x_{11} = -76.9686867419196$$
$$x_{12} = 73.8277993317264$$
$$x_{13} = 61.2615984927811$$
$$x_{14} = -32.9849386326173$$
$$x_{15} = -230.907022689846$$
$$x_{16} = -98.9599664047991$$
$$x_{17} = 67.5446870137729$$
$$x_{18} = -92.6767529095687$$
$$x_{19} = 83.252497386865$$
$$x_{20} = -29.8429571923873$$
$$x_{21} = -17.2724203975121$$
$$x_{22} = 17.2858609706125$$
$$x_{23} = -39.2686433436629$$
$$x_{24} = 14.1479101581108$$
$$x_{25} = -70.6854400018162$$
$$x_{26} = 23.565705260759$$
$$x_{27} = 80.1109282396762$$
$$x_{28} = 39.2712388323218$$
$$x_{29} = -42.4104144616263$$
$$x_{30} = 98.9603748966698$$
$$x_{31} = -64.4021745660819$$
$$x_{32} = 86.394069065896$$
$$x_{33} = 2.1151044019306$$
$$x_{34} = -7.82491949822255$$
$$x_{35} = 58.1200665270198$$
$$x_{36} = 70.6862407240204$$
$$x_{37} = -67.543810056952$$
$$x_{38} = 48.6955472336803$$
$$x_{39} = -23.5584875548962$$
$$x_{40} = 95.8187960668258$$
$$x_{41} = -86.3935330805343$$
$$x_{42} = 45.5540788615421$$
$$x_{43} = -10.9803508603381$$
$$x_{44} = 51.8370377167161$$
$$x_{45} = -45.5521503065062$$
$$x_{46} = 11.0136769372905$$
$$x_{47} = -36.1268243364932$$
$$x_{48} = -95.8183603465263$$
$$x_{49} = 20.4253916500379$$
$$x_{50} = -48.6938595973699$$
$$x_{51} = -89.5351438981654$$
$$x_{52} = -61.2605323722968$$
$$x_{53} = 92.6772186743975$$
$$x_{54} = -51.8355485164882$$
$$x_{55} = 42.4126394726376$$
$$x_{56} = -4.63467435481449$$
$$x_{57} = 1198.517598738$$
$$x_{58} = -73.8270653183329$$
$$x_{59} = -26.7008332529908$$
$$x_{60} = -20.4157768960442$$
$$x_{61} = 36.1298911934334$$
$$x_{62} = 26.7064504389974$$
$$x_{63} = -54.9772215461932$$
$$x_{64} = 89.5356429258046$$
$$x_{65} = 7.89057914345282$$
$$x_{66} = -83.2519201806158$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \left(x - 1\right) \cos{\left(x \right)} - 2 \sin{\left(x \right)} + \frac{\left(x - 1\right) \sin{\left(x \right)}}{x} + \frac{\left(x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(x - 1\right) \cos{\left(x \right)}}{x^{2}}}{x}\right) = \infty$$
$$\lim_{x \to 0^+}\left(\frac{- \left(x - 1\right) \cos{\left(x \right)} - 2 \sin{\left(x \right)} + \frac{\left(x - 1\right) \sin{\left(x \right)}}{x} + \frac{\left(x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(x - 1\right) \cos{\left(x \right)}}{x^{2}}}{x}\right) = -\infty$$
- 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:
Concave at the intervals
$$\left[95.8187960668258, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -865.508773397218\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{- \left(x - 1\right) \cos{\left(x \right)} - 2 \sin{\left(x \right)} + \frac{\left(x - 1\right) \sin{\left(x \right)}}{x} + \frac{\left(x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(x - 1\right) \cos{\left(x \right)}}{x^{2}}}{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 29.8474528425252$$
$$x_{2} = -14.1278030885708$$
$$x_{3} = 4.81830169068692$$
$$x_{4} = 32.9886180185199$$
$$x_{5} = -80.11030486752$$
$$x_{6} = -865.508773397218$$
$$x_{7} = -58.1188820004811$$
$$x_{8} = 76.9693620488425$$
$$x_{9} = 54.9785453606757$$
$$x_{10} = 64.4031391874027$$
$$x_{11} = -76.9686867419196$$
$$x_{12} = 73.8277993317264$$
$$x_{13} = 61.2615984927811$$
$$x_{14} = -32.9849386326173$$
$$x_{15} = -230.907022689846$$
$$x_{16} = -98.9599664047991$$
$$x_{17} = 67.5446870137729$$
$$x_{18} = -92.6767529095687$$
$$x_{19} = 83.252497386865$$
$$x_{20} = -29.8429571923873$$
$$x_{21} = -17.2724203975121$$
$$x_{22} = 17.2858609706125$$
$$x_{23} = -39.2686433436629$$
$$x_{24} = 14.1479101581108$$
$$x_{25} = -70.6854400018162$$
$$x_{26} = 23.565705260759$$
$$x_{27} = 80.1109282396762$$
$$x_{28} = 39.2712388323218$$
$$x_{29} = -42.4104144616263$$
$$x_{30} = 98.9603748966698$$
$$x_{31} = -64.4021745660819$$
$$x_{32} = 86.394069065896$$
$$x_{33} = 2.1151044019306$$
$$x_{34} = -7.82491949822255$$
$$x_{35} = 58.1200665270198$$
$$x_{36} = 70.6862407240204$$
$$x_{37} = -67.543810056952$$
$$x_{38} = 48.6955472336803$$
$$x_{39} = -23.5584875548962$$
$$x_{40} = 95.8187960668258$$
$$x_{41} = -86.3935330805343$$
$$x_{42} = 45.5540788615421$$
$$x_{43} = -10.9803508603381$$
$$x_{44} = 51.8370377167161$$
$$x_{45} = -45.5521503065062$$
$$x_{46} = 11.0136769372905$$
$$x_{47} = -36.1268243364932$$
$$x_{48} = -95.8183603465263$$
$$x_{49} = 20.4253916500379$$
$$x_{50} = -48.6938595973699$$
$$x_{51} = -89.5351438981654$$
$$x_{52} = -61.2605323722968$$
$$x_{53} = 92.6772186743975$$
$$x_{54} = -51.8355485164882$$
$$x_{55} = 42.4126394726376$$
$$x_{56} = -4.63467435481449$$
$$x_{57} = 1198.517598738$$
$$x_{58} = -73.8270653183329$$
$$x_{59} = -26.7008332529908$$
$$x_{60} = -20.4157768960442$$
$$x_{61} = 36.1298911934334$$
$$x_{62} = 26.7064504389974$$
$$x_{63} = -54.9772215461932$$
$$x_{64} = 89.5356429258046$$
$$x_{65} = 7.89057914345282$$
$$x_{66} = -83.2519201806158$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{- \left(x - 1\right) \cos{\left(x \right)} - 2 \sin{\left(x \right)} + \frac{\left(x - 1\right) \sin{\left(x \right)}}{x} + \frac{\left(x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(x - 1\right) \cos{\left(x \right)}}{x^{2}}}{x}\right) = \infty$$
$$\lim_{x \to 0^+}\left(\frac{- \left(x - 1\right) \cos{\left(x \right)} - 2 \sin{\left(x \right)} + \frac{\left(x - 1\right) \sin{\left(x \right)}}{x} + \frac{\left(x - 1\right) \sin{\left(x \right)} - \cos{\left(x \right)}}{x} - \frac{\cos{\left(x \right)}}{x} + \frac{2 \left(x - 1\right) \cos{\left(x \right)}}{x^{2}}}{x}\right) = -\infty$$
- 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:
Concave at the intervals
$$\left[95.8187960668258, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -865.508773397218\right]$$
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
$$\lim_{x \to -\infty}\left(-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}\right) = \left\langle -2, 0\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 0\right\rangle$$
$$\lim_{x \to \infty}\left(-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}\right) = \left\langle -2, 0\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -2, 0\right\rangle$$
$$\lim_{x \to -\infty}\left(-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}\right) = \left\langle -2, 0\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -2, 0\right\rangle$$
$$\lim_{x \to \infty}\left(-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}\right) = \left\langle -2, 0\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -2, 0\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of ((x - 1)*cos(x))/x - 1, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x}}{x}\right) = 0$$
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:
$$-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x} = -1 - \frac{\left(- x - 1\right) \cos{\left(x \right)}}{x}$$
- No
$$-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x} = 1 + \frac{\left(- x - 1\right) \cos{\left(x \right)}}{x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x} = -1 - \frac{\left(- x - 1\right) \cos{\left(x \right)}}{x}$$
- No
$$-1 + \frac{\left(x - 1\right) \cos{\left(x \right)}}{x} = 1 + \frac{\left(- x - 1\right) \cos{\left(x \right)}}{x}$$
- No
so, the function
not is
neither even, nor odd