Mister Exam
Graphing y = arccos3x/x
The solution
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acos(3*x)
f(x) = ---------
x
$$f{\left(x \right)} = \frac{\operatorname{acos}{\left(3 x \right)}}{x}$$
f = acos(3*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:
$$\frac{\operatorname{acos}{\left(3 x \right)}}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{3}$$
Numerical solution
$$x_{1} = 0.333333333333333$$
so we need to solve the equation:
$$\frac{\operatorname{acos}{\left(3 x \right)}}{x} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = \frac{1}{3}$$
Numerical solution
$$x_{1} = 0.333333333333333$$
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
$$- \frac{3}{x \sqrt{1 - 9 x^{2}}} - \frac{\operatorname{acos}{\left(3 x \right)}}{x^{2}} = 0$$
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
$$- \frac{3}{x \sqrt{1 - 9 x^{2}}} - \frac{\operatorname{acos}{\left(3 x \right)}}{x^{2}} = 0$$
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
$$- \frac{27}{\left(1 - 9 x^{2}\right)^{\frac{3}{2}}} + \frac{6}{x^{2} \sqrt{1 - 9 x^{2}}} + \frac{2 \operatorname{acos}{\left(3 x \right)}}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 25781.0971941094$$
$$x_{2} = 36379.3139617626$$
$$x_{3} = 54210.1767017913$$
$$x_{4} = 37433.8020008722$$
$$x_{5} = 27909.572766442$$
$$x_{6} = 52121.5542642327$$
$$x_{7} = 44793.728362934$$
$$x_{8} = 40592.4516681336$$
$$x_{9} = 46890.3771163784$$
$$x_{10} = 42694.4756621236$$
$$x_{11} = 45842.3684869966$$
$$x_{12} = 30033.2229968973$$
$$x_{13} = 41643.8212967933$$
$$x_{14} = 28971.9716543895$$
$$x_{15} = 43744.4375233888$$
$$x_{16} = 39540.3426275751$$
$$x_{17} = 48984.5718298356$$
$$x_{18} = 53166.1261621269$$
$$x_{19} = 38487.4685116018$$
$$x_{20} = 26845.9688890402$$
$$x_{21} = 33210.5981987743$$
$$x_{22} = 51076.4474786614$$
$$x_{23} = 50030.7916258103$$
$$x_{24} = 31093.3794928415$$
$$x_{25} = 34267.7465959127$$
$$x_{26} = 47937.7724682061$$
$$x_{27} = 32152.4896396722$$
$$x_{28} = 35323.9732675495$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{27}{\left(1 - 9 x^{2}\right)^{\frac{3}{2}}} + \frac{6}{x^{2} \sqrt{1 - 9 x^{2}}} + \frac{2 \operatorname{acos}{\left(3 x \right)}}{x^{3}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(- \frac{27}{\left(1 - 9 x^{2}\right)^{\frac{3}{2}}} + \frac{6}{x^{2} \sqrt{1 - 9 x^{2}}} + \frac{2 \operatorname{acos}{\left(3 x \right)}}{x^{3}}\right) = \infty$$
- 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:
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
$$- \frac{27}{\left(1 - 9 x^{2}\right)^{\frac{3}{2}}} + \frac{6}{x^{2} \sqrt{1 - 9 x^{2}}} + \frac{2 \operatorname{acos}{\left(3 x \right)}}{x^{3}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 25781.0971941094$$
$$x_{2} = 36379.3139617626$$
$$x_{3} = 54210.1767017913$$
$$x_{4} = 37433.8020008722$$
$$x_{5} = 27909.572766442$$
$$x_{6} = 52121.5542642327$$
$$x_{7} = 44793.728362934$$
$$x_{8} = 40592.4516681336$$
$$x_{9} = 46890.3771163784$$
$$x_{10} = 42694.4756621236$$
$$x_{11} = 45842.3684869966$$
$$x_{12} = 30033.2229968973$$
$$x_{13} = 41643.8212967933$$
$$x_{14} = 28971.9716543895$$
$$x_{15} = 43744.4375233888$$
$$x_{16} = 39540.3426275751$$
$$x_{17} = 48984.5718298356$$
$$x_{18} = 53166.1261621269$$
$$x_{19} = 38487.4685116018$$
$$x_{20} = 26845.9688890402$$
$$x_{21} = 33210.5981987743$$
$$x_{22} = 51076.4474786614$$
$$x_{23} = 50030.7916258103$$
$$x_{24} = 31093.3794928415$$
$$x_{25} = 34267.7465959127$$
$$x_{26} = 47937.7724682061$$
$$x_{27} = 32152.4896396722$$
$$x_{28} = 35323.9732675495$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(- \frac{27}{\left(1 - 9 x^{2}\right)^{\frac{3}{2}}} + \frac{6}{x^{2} \sqrt{1 - 9 x^{2}}} + \frac{2 \operatorname{acos}{\left(3 x \right)}}{x^{3}}\right) = -\infty$$
$$\lim_{x \to 0^+}\left(- \frac{27}{\left(1 - 9 x^{2}\right)^{\frac{3}{2}}} + \frac{6}{x^{2} \sqrt{1 - 9 x^{2}}} + \frac{2 \operatorname{acos}{\left(3 x \right)}}{x^{3}}\right) = \infty$$
- 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:
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
$$\lim_{x \to -\infty}\left(\frac{\operatorname{acos}{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{acos}{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = 0$$
$$\lim_{x \to -\infty}\left(\frac{\operatorname{acos}{\left(3 x \right)}}{x}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\operatorname{acos}{\left(3 x \right)}}{x}\right) = 0$$
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 acos(3*x)/x, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\operatorname{acos}{\left(3 x \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{acos}{\left(3 x \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
$$\lim_{x \to -\infty}\left(\frac{\operatorname{acos}{\left(3 x \right)}}{x^{2}}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\operatorname{acos}{\left(3 x \right)}}{x^{2}}\right) = 0$$
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:
$$\frac{\operatorname{acos}{\left(3 x \right)}}{x} = - \frac{\operatorname{acos}{\left(- 3 x \right)}}{x}$$
- No
$$\frac{\operatorname{acos}{\left(3 x \right)}}{x} = \frac{\operatorname{acos}{\left(- 3 x \right)}}{x}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\operatorname{acos}{\left(3 x \right)}}{x} = - \frac{\operatorname{acos}{\left(- 3 x \right)}}{x}$$
- No
$$\frac{\operatorname{acos}{\left(3 x \right)}}{x} = \frac{\operatorname{acos}{\left(- 3 x \right)}}{x}$$
- No
so, the function
not is
neither even, nor odd