Mister Exam
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-
How to use it?
- Graphing y =:
- y=x^3-6x^2+3
- (x-2)*(x-4)/x^2+5*x+4
- (3x^3+3x^2-6x)/(2x^2+6x+4)
- sqrt(x)/tan(x)
- Limit of the function:
-
sqrt(x)/tan(x)
-
Identical expressions
- sqrt(x)/tan(x)
- square root of (x) divide by tangent of (x)
- √(x)/tan(x)
- sqrtx/tanx
- sqrt(x) divide by tan(x)
Graphing y = sqrt(x)/tan(x)
The solution
You have entered
[src]
___
\/ x
f(x) = ------
tan(x)
$$f{\left(x \right)} = \frac{\sqrt{x}}{\tan{\left(x \right)}}$$
f = sqrt(x)/tan(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:
$$\frac{\sqrt{x}}{\tan{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -1.5707963267949$$
$$x_{2} = -64.4026493985908$$
$$x_{3} = 76.9690200129499$$
$$x_{4} = -23.5619449019235$$
$$x_{5} = -58.1194640914112$$
$$x_{6} = 61.261056745001$$
$$x_{7} = 80.1106126665397$$
$$x_{8} = -48.6946861306418$$
$$x_{9} = -29.845130209103$$
$$x_{10} = -4.71238898038469$$
$$x_{11} = -86.3937979737193$$
$$x_{12} = -36.1283155162826$$
$$x_{13} = -98.9601685880785$$
$$x_{14} = 1.5707963267949$$
$$x_{15} = -39.2699081698724$$
$$x_{16} = 73.8274273593601$$
$$x_{17} = -92.6769832808989$$
$$x_{18} = 42.4115008234622$$
$$x_{19} = 67.5442420521806$$
$$x_{20} = -32.9867228626928$$
$$x_{21} = 14.1371669411541$$
$$x_{22} = 4.71238898038469$$
$$x_{23} = 32.9867228626928$$
$$x_{24} = -10.9955742875643$$
$$x_{25} = 70.6858347057703$$
$$x_{26} = 36.1283155162826$$
$$x_{27} = 20.4203522483337$$
$$x_{28} = -70.6858347057703$$
$$x_{29} = -26.7035375555132$$
$$x_{30} = 10.9955742875643$$
$$x_{31} = 23.5619449019235$$
$$x_{32} = 45.553093477052$$
$$x_{33} = 83.2522053201295$$
$$x_{34} = -67.5442420521806$$
$$x_{35} = -89.5353906273091$$
$$x_{36} = -54.9778714378214$$
$$x_{37} = 95.8185759344887$$
$$x_{38} = -17.2787595947439$$
$$x_{39} = 26.7035375555132$$
$$x_{40} = 17.2787595947439$$
$$x_{41} = -42.4115008234622$$
$$x_{42} = 54.9778714378214$$
$$x_{43} = -7.85398163397448$$
$$x_{44} = 48.6946861306418$$
$$x_{45} = -51.8362787842316$$
$$x_{46} = 89.5353906273091$$
$$x_{47} = 92.6769832808989$$
$$x_{48} = 58.1194640914112$$
$$x_{49} = -80.1106126665397$$
$$x_{50} = -73.8274273593601$$
$$x_{51} = 86.3937979737193$$
$$x_{52} = -76.9690200129499$$
$$x_{53} = 51.8362787842316$$
$$x_{54} = 39.2699081698724$$
$$x_{55} = -20.4203522483337$$
$$x_{56} = 64.4026493985908$$
$$x_{57} = -83.2522053201295$$
$$x_{58} = 98.9601685880785$$
$$x_{59} = 7.85398163397448$$
$$x_{60} = -95.8185759344887$$
$$x_{61} = -14.1371669411541$$
$$x_{62} = 29.845130209103$$
$$x_{63} = -45.553093477052$$
$$x_{64} = -61.261056745001$$
so we need to solve the equation:
$$\frac{\sqrt{x}}{\tan{\left(x \right)}} = 0$$
Solve this equation
The points of intersection with the axis X:
Numerical solution
$$x_{1} = -1.5707963267949$$
$$x_{2} = -64.4026493985908$$
$$x_{3} = 76.9690200129499$$
$$x_{4} = -23.5619449019235$$
$$x_{5} = -58.1194640914112$$
$$x_{6} = 61.261056745001$$
$$x_{7} = 80.1106126665397$$
$$x_{8} = -48.6946861306418$$
$$x_{9} = -29.845130209103$$
$$x_{10} = -4.71238898038469$$
$$x_{11} = -86.3937979737193$$
$$x_{12} = -36.1283155162826$$
$$x_{13} = -98.9601685880785$$
$$x_{14} = 1.5707963267949$$
$$x_{15} = -39.2699081698724$$
$$x_{16} = 73.8274273593601$$
$$x_{17} = -92.6769832808989$$
$$x_{18} = 42.4115008234622$$
$$x_{19} = 67.5442420521806$$
$$x_{20} = -32.9867228626928$$
$$x_{21} = 14.1371669411541$$
$$x_{22} = 4.71238898038469$$
$$x_{23} = 32.9867228626928$$
$$x_{24} = -10.9955742875643$$
$$x_{25} = 70.6858347057703$$
$$x_{26} = 36.1283155162826$$
$$x_{27} = 20.4203522483337$$
$$x_{28} = -70.6858347057703$$
$$x_{29} = -26.7035375555132$$
$$x_{30} = 10.9955742875643$$
$$x_{31} = 23.5619449019235$$
$$x_{32} = 45.553093477052$$
$$x_{33} = 83.2522053201295$$
$$x_{34} = -67.5442420521806$$
$$x_{35} = -89.5353906273091$$
$$x_{36} = -54.9778714378214$$
$$x_{37} = 95.8185759344887$$
$$x_{38} = -17.2787595947439$$
$$x_{39} = 26.7035375555132$$
$$x_{40} = 17.2787595947439$$
$$x_{41} = -42.4115008234622$$
$$x_{42} = 54.9778714378214$$
$$x_{43} = -7.85398163397448$$
$$x_{44} = 48.6946861306418$$
$$x_{45} = -51.8362787842316$$
$$x_{46} = 89.5353906273091$$
$$x_{47} = 92.6769832808989$$
$$x_{48} = 58.1194640914112$$
$$x_{49} = -80.1106126665397$$
$$x_{50} = -73.8274273593601$$
$$x_{51} = 86.3937979737193$$
$$x_{52} = -76.9690200129499$$
$$x_{53} = 51.8362787842316$$
$$x_{54} = 39.2699081698724$$
$$x_{55} = -20.4203522483337$$
$$x_{56} = 64.4026493985908$$
$$x_{57} = -83.2522053201295$$
$$x_{58} = 98.9601685880785$$
$$x_{59} = 7.85398163397448$$
$$x_{60} = -95.8185759344887$$
$$x_{61} = -14.1371669411541$$
$$x_{62} = 29.845130209103$$
$$x_{63} = -45.553093477052$$
$$x_{64} = -61.261056745001$$
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
$$\frac{\sqrt{x} \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} + \frac{1}{2 \sqrt{x} \tan{\left(x \right)}} = 0$$
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
$$\frac{\sqrt{x} \left(- \tan^{2}{\left(x \right)} - 1\right)}{\tan^{2}{\left(x \right)}} + \frac{1}{2 \sqrt{x} \tan{\left(x \right)}} = 0$$
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
$$\frac{2 \sqrt{x} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x} \tan{\left(x \right)}} - \frac{1}{4 x^{\frac{3}{2}}}}{\tan{\left(x \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 23.5407035002745$$
$$x_{2} = -86.3880101011813$$
$$x_{3} = -58.1108597415507$$
$$x_{4} = 17.2497696527847$$
$$x_{5} = 36.1144702082836$$
$$x_{6} = 98.9551157707129$$
$$x_{7} = -17.2497696527847$$
$$x_{8} = 86.3880101011813$$
$$x_{9} = -83.2461990037822$$
$$x_{10} = -36.1144702082836$$
$$x_{11} = -42.39970801623$$
$$x_{12} = 39.2571712992709$$
$$x_{13} = 67.5368386175974$$
$$x_{14} = 1.11791403207435$$
$$x_{15} = -7.7897512567741$$
$$x_{16} = 32.9715576965888$$
$$x_{17} = 92.6715878578012$$
$$x_{18} = -89.5298058659662$$
$$x_{19} = 14.101702831878$$
$$x_{20} = 76.9625232987661$$
$$x_{21} = 26.6847992011637$$
$$x_{22} = -67.5368386175974$$
$$x_{23} = 70.6787603857481$$
$$x_{24} = -64.3948847286922$$
$$x_{25} = 80.104370769356$$
$$x_{26} = -70.6787603857481$$
$$x_{27} = -51.8266310849037$$
$$x_{28} = -14.101702831878$$
$$x_{29} = -48.6844157231755$$
$$x_{30} = 61.2528937747202$$
$$x_{31} = -98.9551157707129$$
$$x_{32} = 64.3948847286922$$
$$x_{33} = 58.1108597415507$$
$$x_{34} = 54.9687752392829$$
$$x_{35} = -32.9715576965888$$
$$x_{36} = -39.2571712992709$$
$$x_{37} = -23.5407035002745$$
$$x_{38} = -95.8133574317394$$
$$x_{39} = 89.5298058659662$$
$$x_{40} = 10.949895994345$$
$$x_{41} = -92.6715878578012$$
$$x_{42} = -20.3958349886934$$
$$x_{43} = 95.8133574317394$$
$$x_{44} = 29.8283668577611$$
$$x_{45} = 20.3958349886934$$
$$x_{46} = 7.7897512567741$$
$$x_{47} = -29.8283668577611$$
$$x_{48} = -73.8206541354127$$
$$x_{49} = 42.39970801623$$
$$x_{50} = 45.5421144075225$$
$$x_{51} = -76.9625232987661$$
$$x_{52} = -1.11791403207435$$
$$x_{53} = 4.60357245433596$$
$$x_{54} = -4.60357245433596$$
$$x_{55} = -26.6847992011637$$
$$x_{56} = 83.2461990037822$$
$$x_{57} = 51.8266310849037$$
$$x_{58} = -80.104370769356$$
$$x_{59} = -54.9687752392829$$
$$x_{60} = 73.8206541354127$$
$$x_{61} = 48.6844157231755$$
$$x_{62} = -45.5421144075225$$
$$x_{63} = -10.949895994345$$
$$x_{64} = -61.2528937747202$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \sqrt{x} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x} \tan{\left(x \right)}} - \frac{1}{4 x^{\frac{3}{2}}}}{\tan{\left(x \right)}}\right) = - \infty i$$
$$\lim_{x \to 0^+}\left(\frac{2 \sqrt{x} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x} \tan{\left(x \right)}} - \frac{1}{4 x^{\frac{3}{2}}}}{\tan{\left(x \right)}}\right) = \infty$$
- 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
$$\left(-\infty, 1.11791403207435\right]$$
Convex at the intervals
$$\left[98.9551157707129, \infty\right)$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$\frac{2 \sqrt{x} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x} \tan{\left(x \right)}} - \frac{1}{4 x^{\frac{3}{2}}}}{\tan{\left(x \right)}} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 23.5407035002745$$
$$x_{2} = -86.3880101011813$$
$$x_{3} = -58.1108597415507$$
$$x_{4} = 17.2497696527847$$
$$x_{5} = 36.1144702082836$$
$$x_{6} = 98.9551157707129$$
$$x_{7} = -17.2497696527847$$
$$x_{8} = 86.3880101011813$$
$$x_{9} = -83.2461990037822$$
$$x_{10} = -36.1144702082836$$
$$x_{11} = -42.39970801623$$
$$x_{12} = 39.2571712992709$$
$$x_{13} = 67.5368386175974$$
$$x_{14} = 1.11791403207435$$
$$x_{15} = -7.7897512567741$$
$$x_{16} = 32.9715576965888$$
$$x_{17} = 92.6715878578012$$
$$x_{18} = -89.5298058659662$$
$$x_{19} = 14.101702831878$$
$$x_{20} = 76.9625232987661$$
$$x_{21} = 26.6847992011637$$
$$x_{22} = -67.5368386175974$$
$$x_{23} = 70.6787603857481$$
$$x_{24} = -64.3948847286922$$
$$x_{25} = 80.104370769356$$
$$x_{26} = -70.6787603857481$$
$$x_{27} = -51.8266310849037$$
$$x_{28} = -14.101702831878$$
$$x_{29} = -48.6844157231755$$
$$x_{30} = 61.2528937747202$$
$$x_{31} = -98.9551157707129$$
$$x_{32} = 64.3948847286922$$
$$x_{33} = 58.1108597415507$$
$$x_{34} = 54.9687752392829$$
$$x_{35} = -32.9715576965888$$
$$x_{36} = -39.2571712992709$$
$$x_{37} = -23.5407035002745$$
$$x_{38} = -95.8133574317394$$
$$x_{39} = 89.5298058659662$$
$$x_{40} = 10.949895994345$$
$$x_{41} = -92.6715878578012$$
$$x_{42} = -20.3958349886934$$
$$x_{43} = 95.8133574317394$$
$$x_{44} = 29.8283668577611$$
$$x_{45} = 20.3958349886934$$
$$x_{46} = 7.7897512567741$$
$$x_{47} = -29.8283668577611$$
$$x_{48} = -73.8206541354127$$
$$x_{49} = 42.39970801623$$
$$x_{50} = 45.5421144075225$$
$$x_{51} = -76.9625232987661$$
$$x_{52} = -1.11791403207435$$
$$x_{53} = 4.60357245433596$$
$$x_{54} = -4.60357245433596$$
$$x_{55} = -26.6847992011637$$
$$x_{56} = 83.2461990037822$$
$$x_{57} = 51.8266310849037$$
$$x_{58} = -80.104370769356$$
$$x_{59} = -54.9687752392829$$
$$x_{60} = 73.8206541354127$$
$$x_{61} = 48.6844157231755$$
$$x_{62} = -45.5421144075225$$
$$x_{63} = -10.949895994345$$
$$x_{64} = -61.2528937747202$$
You also need to calculate the limits of y '' for arguments seeking to indeterminate points of a function:
Points where there is an indetermination:
$$x_{1} = 0$$
$$\lim_{x \to 0^-}\left(\frac{2 \sqrt{x} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x} \tan{\left(x \right)}} - \frac{1}{4 x^{\frac{3}{2}}}}{\tan{\left(x \right)}}\right) = - \infty i$$
$$\lim_{x \to 0^+}\left(\frac{2 \sqrt{x} \left(\frac{\tan^{2}{\left(x \right)} + 1}{\tan^{2}{\left(x \right)}} - 1\right) \left(\tan^{2}{\left(x \right)} + 1\right) - \frac{\tan^{2}{\left(x \right)} + 1}{\sqrt{x} \tan{\left(x \right)}} - \frac{1}{4 x^{\frac{3}{2}}}}{\tan{\left(x \right)}}\right) = \infty$$
- 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
$$\left(-\infty, 1.11791403207435\right]$$
Convex at the intervals
$$\left[98.9551157707129, \infty\right)$$
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
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \lim_{x \to -\infty}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
True
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \lim_{x \to \infty}\left(\frac{\sqrt{x}}{\tan{\left(x \right)}}\right)$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(x)/tan(x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{1}{\sqrt{x} \tan{\left(x \right)}}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{1}{\sqrt{x} \tan{\left(x \right)}}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{1}{\sqrt{x} \tan{\left(x \right)}}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{1}{\sqrt{x} \tan{\left(x \right)}}\right)$$
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:
$$\frac{\sqrt{x}}{\tan{\left(x \right)}} = - \frac{\sqrt{- x}}{\tan{\left(x \right)}}$$
- No
$$\frac{\sqrt{x}}{\tan{\left(x \right)}} = \frac{\sqrt{- x}}{\tan{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd
So, check:
$$\frac{\sqrt{x}}{\tan{\left(x \right)}} = - \frac{\sqrt{- x}}{\tan{\left(x \right)}}$$
- No
$$\frac{\sqrt{x}}{\tan{\left(x \right)}} = \frac{\sqrt{- x}}{\tan{\left(x \right)}}$$
- No
so, the function
not is
neither even, nor odd