Mister Exam
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-
How to use it?
- Graphing y =:
- x^4/4-x^3/3-x^2
- x^3-3x-7
- x^3-3x-2
- x^3+3x-2
- Derivative of:
-
tan(4*x)
- Limit of the function:
-
tan(4*x)
-
Identical expressions
- tan(four *x)
- tangent of (4 multiply by x)
- tangent of (four multiply by x)
- tan(4x)
- tan4x
-
Similar expressions
- pi-arctan(4x)
Graphing y = tan(4*x)
The solution
You have entered
[src]
f(x) = tan(4*x)
$$f{\left(x \right)} = \tan{\left(4 x \right)}$$
f = tan(4*x)
The graph of the function
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:
$$\tan{\left(4 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 65.9734457253857$$
$$x_{2} = 2.35619449019234$$
$$x_{3} = -23.5619449019235$$
$$x_{4} = -29.845130209103$$
$$x_{5} = -21.9911485751286$$
$$x_{6} = -11.7809724509617$$
$$x_{7} = 11.7809724509617$$
$$x_{8} = 21.9911485751286$$
$$x_{9} = 46.3384916404494$$
$$x_{10} = -15.707963267949$$
$$x_{11} = 42.4115008234622$$
$$x_{12} = 40.0553063332699$$
$$x_{13} = -32.2013246992954$$
$$x_{14} = -25.9181393921158$$
$$x_{15} = 36.1283155162826$$
$$x_{16} = 33.7721210260903$$
$$x_{17} = 55.7632696012188$$
$$x_{18} = 6.28318530717959$$
$$x_{19} = -80.1106126665397$$
$$x_{20} = 86.3937979737193$$
$$x_{21} = -18.0641577581413$$
$$x_{22} = 64.4026493985908$$
$$x_{23} = -69.9004365423729$$
$$x_{24} = -95.8185759344887$$
$$x_{25} = 28.2743338823081$$
$$x_{26} = 18.0641577581413$$
$$x_{27} = -85.6083998103219$$
$$x_{28} = -1.5707963267949$$
$$x_{29} = 47.9092879672443$$
$$x_{30} = 76.1836218495525$$
$$x_{31} = -62.0464549083984$$
$$x_{32} = 73.8274273593601$$
$$x_{33} = 32.2013246992954$$
$$x_{34} = 62.0464549083984$$
$$x_{35} = -33.7721210260903$$
$$x_{36} = -63.6172512351933$$
$$x_{37} = -10.2101761241668$$
$$x_{38} = -47.9092879672443$$
$$x_{39} = 10.2101761241668$$
$$x_{40} = -5.49778714378214$$
$$x_{41} = 72.2566310325652$$
$$x_{42} = 24.3473430653209$$
$$x_{43} = -51.8362787842316$$
$$x_{44} = 58.1194640914112$$
$$x_{45} = -76.1836218495525$$
$$x_{46} = -93.4623814442964$$
$$x_{47} = -73.8274273593601$$
$$x_{48} = 51.8362787842316$$
$$x_{49} = 78.5398163397448$$
$$x_{50} = -87.9645943005142$$
$$x_{51} = -37.6991118430775$$
$$x_{52} = -43.9822971502571$$
$$x_{53} = 82.4668071567321$$
$$x_{54} = 29.845130209103$$
$$x_{55} = -45.553093477052$$
$$x_{56} = 98.174770424681$$
$$x_{57} = -54.1924732744239$$
$$x_{58} = 80.1106126665397$$
$$x_{59} = -58.1194640914112$$
$$x_{60} = -98.174770424681$$
$$x_{61} = 38.484510006475$$
$$x_{62} = -36.1283155162826$$
$$x_{63} = -81.6814089933346$$
$$x_{64} = -49.4800842940392$$
$$x_{65} = -65.9734457253857$$
$$x_{66} = 0$$
$$x_{67} = -27.4889357189107$$
$$x_{68} = -84.037603483527$$
$$x_{69} = -41.6261026600648$$
$$x_{70} = -67.5442420521806$$
$$x_{71} = -91.8915851175014$$
$$x_{72} = 43.9822971502571$$
$$x_{73} = 60.4756585816035$$
$$x_{74} = 25.9181393921158$$
$$x_{75} = 100.530964914873$$
$$x_{76} = -7.85398163397448$$
$$x_{77} = -55.7632696012188$$
$$x_{78} = -19.6349540849362$$
$$x_{79} = 91.8915851175014$$
$$x_{80} = 7.85398163397448$$
$$x_{81} = -14.1371669411541$$
$$x_{82} = 50.2654824574367$$
$$x_{83} = 94.2477796076938$$
$$x_{84} = -3.92699081698724$$
$$x_{85} = -77.7544181763474$$
$$x_{86} = -59.6902604182061$$
$$x_{87} = 90.3207887907066$$
$$x_{88} = -40.0553063332699$$
$$x_{89} = 3.92699081698724$$
$$x_{90} = 14.1371669411541$$
$$x_{91} = 69.9004365423729$$
$$x_{92} = 20.4203522483337$$
$$x_{93} = 54.1924732744239$$
$$x_{94} = 95.8185759344887$$
$$x_{95} = -89.5353906273091$$
$$x_{96} = 87.9645943005142$$
$$x_{97} = 84.037603483527$$
$$x_{98} = 68.329640215578$$
$$x_{99} = -71.4712328691678$$
$$x_{100} = -99.7455667514759$$
$$x_{101} = 16.4933614313464$$
so we need to solve the equation:
$$\tan{\left(4 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 65.9734457253857$$
$$x_{2} = 2.35619449019234$$
$$x_{3} = -23.5619449019235$$
$$x_{4} = -29.845130209103$$
$$x_{5} = -21.9911485751286$$
$$x_{6} = -11.7809724509617$$
$$x_{7} = 11.7809724509617$$
$$x_{8} = 21.9911485751286$$
$$x_{9} = 46.3384916404494$$
$$x_{10} = -15.707963267949$$
$$x_{11} = 42.4115008234622$$
$$x_{12} = 40.0553063332699$$
$$x_{13} = -32.2013246992954$$
$$x_{14} = -25.9181393921158$$
$$x_{15} = 36.1283155162826$$
$$x_{16} = 33.7721210260903$$
$$x_{17} = 55.7632696012188$$
$$x_{18} = 6.28318530717959$$
$$x_{19} = -80.1106126665397$$
$$x_{20} = 86.3937979737193$$
$$x_{21} = -18.0641577581413$$
$$x_{22} = 64.4026493985908$$
$$x_{23} = -69.9004365423729$$
$$x_{24} = -95.8185759344887$$
$$x_{25} = 28.2743338823081$$
$$x_{26} = 18.0641577581413$$
$$x_{27} = -85.6083998103219$$
$$x_{28} = -1.5707963267949$$
$$x_{29} = 47.9092879672443$$
$$x_{30} = 76.1836218495525$$
$$x_{31} = -62.0464549083984$$
$$x_{32} = 73.8274273593601$$
$$x_{33} = 32.2013246992954$$
$$x_{34} = 62.0464549083984$$
$$x_{35} = -33.7721210260903$$
$$x_{36} = -63.6172512351933$$
$$x_{37} = -10.2101761241668$$
$$x_{38} = -47.9092879672443$$
$$x_{39} = 10.2101761241668$$
$$x_{40} = -5.49778714378214$$
$$x_{41} = 72.2566310325652$$
$$x_{42} = 24.3473430653209$$
$$x_{43} = -51.8362787842316$$
$$x_{44} = 58.1194640914112$$
$$x_{45} = -76.1836218495525$$
$$x_{46} = -93.4623814442964$$
$$x_{47} = -73.8274273593601$$
$$x_{48} = 51.8362787842316$$
$$x_{49} = 78.5398163397448$$
$$x_{50} = -87.9645943005142$$
$$x_{51} = -37.6991118430775$$
$$x_{52} = -43.9822971502571$$
$$x_{53} = 82.4668071567321$$
$$x_{54} = 29.845130209103$$
$$x_{55} = -45.553093477052$$
$$x_{56} = 98.174770424681$$
$$x_{57} = -54.1924732744239$$
$$x_{58} = 80.1106126665397$$
$$x_{59} = -58.1194640914112$$
$$x_{60} = -98.174770424681$$
$$x_{61} = 38.484510006475$$
$$x_{62} = -36.1283155162826$$
$$x_{63} = -81.6814089933346$$
$$x_{64} = -49.4800842940392$$
$$x_{65} = -65.9734457253857$$
$$x_{66} = 0$$
$$x_{67} = -27.4889357189107$$
$$x_{68} = -84.037603483527$$
$$x_{69} = -41.6261026600648$$
$$x_{70} = -67.5442420521806$$
$$x_{71} = -91.8915851175014$$
$$x_{72} = 43.9822971502571$$
$$x_{73} = 60.4756585816035$$
$$x_{74} = 25.9181393921158$$
$$x_{75} = 100.530964914873$$
$$x_{76} = -7.85398163397448$$
$$x_{77} = -55.7632696012188$$
$$x_{78} = -19.6349540849362$$
$$x_{79} = 91.8915851175014$$
$$x_{80} = 7.85398163397448$$
$$x_{81} = -14.1371669411541$$
$$x_{82} = 50.2654824574367$$
$$x_{83} = 94.2477796076938$$
$$x_{84} = -3.92699081698724$$
$$x_{85} = -77.7544181763474$$
$$x_{86} = -59.6902604182061$$
$$x_{87} = 90.3207887907066$$
$$x_{88} = -40.0553063332699$$
$$x_{89} = 3.92699081698724$$
$$x_{90} = 14.1371669411541$$
$$x_{91} = 69.9004365423729$$
$$x_{92} = 20.4203522483337$$
$$x_{93} = 54.1924732744239$$
$$x_{94} = 95.8185759344887$$
$$x_{95} = -89.5353906273091$$
$$x_{96} = 87.9645943005142$$
$$x_{97} = 84.037603483527$$
$$x_{98} = 68.329640215578$$
$$x_{99} = -71.4712328691678$$
$$x_{100} = -99.7455667514759$$
$$x_{101} = 16.4933614313464$$
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
$$4 \tan^{2}{\left(4 x \right)} + 4 = 0$$
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
$$4 \tan^{2}{\left(4 x \right)} + 4 = 0$$
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
$$32 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(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
$$32 \left(\tan^{2}{\left(4 x \right)} + 1\right) \tan{\left(4 x \right)} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = 0$$
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
$$\left[0, \infty\right)$$
Convex at the intervals
$$\left(-\infty, 0\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty} \tan{\left(4 x \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \tan{\left(4 x \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty} \tan{\left(4 x \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \tan{\left(4 x \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(4*x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(4 x \right)}}{x}\right)$$
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:
$$\tan{\left(4 x \right)} = - \tan{\left(4 x \right)}$$
- No
$$\tan{\left(4 x \right)} = \tan{\left(4 x \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\tan{\left(4 x \right)} = - \tan{\left(4 x \right)}$$
- No
$$\tan{\left(4 x \right)} = \tan{\left(4 x \right)}$$
- Yes
so, the function
is
odd