Mister Exam
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-
How to use it?
- Graphing y =:
- (x^2+5)/(x-3)
- -x^4+2x^2+3
- x^3-x^2-x+2
-
x^2+x-2
- Derivative of:
-
tan(5*x)
- Limit of the function:
-
tan(5*x)
-
Identical expressions
- tan(five *x)
- tangent of (5 multiply by x)
- tangent of (five multiply by x)
- tan(5x)
- tan5x
-
Similar expressions
- tan(5*x^3-3*x^2)
- atan(5^x+1)^3
Graphing y = tan(5*x)
The solution
You have entered
[src]
f(x) = tan(5*x)
$$f{\left(x \right)} = \tan{\left(5 x \right)}$$
f = tan(5*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:
$$\tan{\left(5 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 65.9734457253857$$
$$x_{2} = -13.8230076757951$$
$$x_{3} = 26.3893782901543$$
$$x_{4} = -21.9911485751286$$
$$x_{5} = 21.9911485751286$$
$$x_{6} = 96.1327351998477$$
$$x_{7} = 42.0973415581032$$
$$x_{8} = 4.39822971502571$$
$$x_{9} = -76.026542216873$$
$$x_{10} = -15.707963267949$$
$$x_{11} = 33.9292006587698$$
$$x_{12} = 16.3362817986669$$
$$x_{13} = 48.3805268652828$$
$$x_{14} = -74.1415866247191$$
$$x_{15} = 99.9026463841554$$
$$x_{16} = -64.0884901332318$$
$$x_{17} = -20.1061929829747$$
$$x_{18} = 6.28318530717959$$
$$x_{19} = -61.5752160103599$$
$$x_{20} = 74.1415866247191$$
$$x_{21} = -32.0442450666159$$
$$x_{22} = -30.159289474462$$
$$x_{23} = -77.9114978090269$$
$$x_{24} = -33.9292006587698$$
$$x_{25} = 86.0796387083603$$
$$x_{26} = 14.451326206513$$
$$x_{27} = 18.2212373908208$$
$$x_{28} = 28.2743338823081$$
$$x_{29} = -99.9026463841554$$
$$x_{30} = -71.6283125018473$$
$$x_{31} = 8.16814089933346$$
$$x_{32} = 58.4336233567702$$
$$x_{33} = 89.8495498926681$$
$$x_{34} = -10.0530964914873$$
$$x_{35} = 23.8761041672824$$
$$x_{36} = 11.9380520836412$$
$$x_{37} = -39.5840674352314$$
$$x_{38} = -57.8053048260522$$
$$x_{39} = 38.3274303737955$$
$$x_{40} = 55.9203492338983$$
$$x_{41} = -5.65486677646163$$
$$x_{42} = -49.6371639267187$$
$$x_{43} = -27.6460153515902$$
$$x_{44} = 72.2566310325652$$
$$x_{45} = 92.3628240155399$$
$$x_{46} = 45.867252742411$$
$$x_{47} = -87.9645943005142$$
$$x_{48} = -69.7433569096934$$
$$x_{49} = -1.88495559215388$$
$$x_{50} = -93.6194610769758$$
$$x_{51} = -37.6991118430775$$
$$x_{52} = -43.9822971502571$$
$$x_{53} = 40.2123859659494$$
$$x_{54} = -55.9203492338983$$
$$x_{55} = 36.4424747816416$$
$$x_{56} = 98.0176907920015$$
$$x_{57} = -47.7522083345649$$
$$x_{58} = 62.2035345410779$$
$$x_{59} = -79.7964534011807$$
$$x_{60} = -11.9380520836412$$
$$x_{61} = -23.8761041672824$$
$$x_{62} = 20.1061929829747$$
$$x_{63} = -45.867252742411$$
$$x_{64} = -52.1504380495906$$
$$x_{65} = -25.7610597594363$$
$$x_{66} = -17.5929188601028$$
$$x_{67} = 76.026542216873$$
$$x_{68} = -81.6814089933346$$
$$x_{69} = 64.0884901332318$$
$$x_{70} = -3.76991118430775$$
$$x_{71} = -65.9734457253857$$
$$x_{72} = 54.0353936417444$$
$$x_{73} = 0$$
$$x_{74} = 80.4247719318987$$
$$x_{75} = -35.8141562509236$$
$$x_{76} = -98.0176907920015$$
$$x_{77} = 1.88495559215388$$
$$x_{78} = 60.318578948924$$
$$x_{79} = -83.5663645854885$$
$$x_{80} = 43.9822971502571$$
$$x_{81} = 67.8584013175395$$
$$x_{82} = -54.0353936417444$$
$$x_{83} = 84.1946831162065$$
$$x_{84} = -86.0796387083603$$
$$x_{85} = -42.0973415581032$$
$$x_{86} = -67.8584013175395$$
$$x_{87} = 50.2654824574367$$
$$x_{88} = 94.2477796076938$$
$$x_{89} = 82.3097275240526$$
$$x_{90} = -59.6902604182061$$
$$x_{91} = 52.1504380495906$$
$$x_{92} = 70.3716754404114$$
$$x_{93} = -96.1327351998477$$
$$x_{94} = -7.5398223686155$$
$$x_{95} = 10.0530964914873$$
$$x_{96} = -89.8495498926681$$
$$x_{97} = 30.159289474462$$
$$x_{98} = 87.9645943005142$$
$$x_{99} = 32.0442450666159$$
$$x_{100} = 77.9114978090269$$
$$x_{101} = -91.734505484822$$
so we need to solve the equation:
$$\tan{\left(5 x \right)} = 0$$
Solve this equation
The points of intersection with the axis X:
Analytical solution
$$x_{1} = 0$$
Numerical solution
$$x_{1} = 65.9734457253857$$
$$x_{2} = -13.8230076757951$$
$$x_{3} = 26.3893782901543$$
$$x_{4} = -21.9911485751286$$
$$x_{5} = 21.9911485751286$$
$$x_{6} = 96.1327351998477$$
$$x_{7} = 42.0973415581032$$
$$x_{8} = 4.39822971502571$$
$$x_{9} = -76.026542216873$$
$$x_{10} = -15.707963267949$$
$$x_{11} = 33.9292006587698$$
$$x_{12} = 16.3362817986669$$
$$x_{13} = 48.3805268652828$$
$$x_{14} = -74.1415866247191$$
$$x_{15} = 99.9026463841554$$
$$x_{16} = -64.0884901332318$$
$$x_{17} = -20.1061929829747$$
$$x_{18} = 6.28318530717959$$
$$x_{19} = -61.5752160103599$$
$$x_{20} = 74.1415866247191$$
$$x_{21} = -32.0442450666159$$
$$x_{22} = -30.159289474462$$
$$x_{23} = -77.9114978090269$$
$$x_{24} = -33.9292006587698$$
$$x_{25} = 86.0796387083603$$
$$x_{26} = 14.451326206513$$
$$x_{27} = 18.2212373908208$$
$$x_{28} = 28.2743338823081$$
$$x_{29} = -99.9026463841554$$
$$x_{30} = -71.6283125018473$$
$$x_{31} = 8.16814089933346$$
$$x_{32} = 58.4336233567702$$
$$x_{33} = 89.8495498926681$$
$$x_{34} = -10.0530964914873$$
$$x_{35} = 23.8761041672824$$
$$x_{36} = 11.9380520836412$$
$$x_{37} = -39.5840674352314$$
$$x_{38} = -57.8053048260522$$
$$x_{39} = 38.3274303737955$$
$$x_{40} = 55.9203492338983$$
$$x_{41} = -5.65486677646163$$
$$x_{42} = -49.6371639267187$$
$$x_{43} = -27.6460153515902$$
$$x_{44} = 72.2566310325652$$
$$x_{45} = 92.3628240155399$$
$$x_{46} = 45.867252742411$$
$$x_{47} = -87.9645943005142$$
$$x_{48} = -69.7433569096934$$
$$x_{49} = -1.88495559215388$$
$$x_{50} = -93.6194610769758$$
$$x_{51} = -37.6991118430775$$
$$x_{52} = -43.9822971502571$$
$$x_{53} = 40.2123859659494$$
$$x_{54} = -55.9203492338983$$
$$x_{55} = 36.4424747816416$$
$$x_{56} = 98.0176907920015$$
$$x_{57} = -47.7522083345649$$
$$x_{58} = 62.2035345410779$$
$$x_{59} = -79.7964534011807$$
$$x_{60} = -11.9380520836412$$
$$x_{61} = -23.8761041672824$$
$$x_{62} = 20.1061929829747$$
$$x_{63} = -45.867252742411$$
$$x_{64} = -52.1504380495906$$
$$x_{65} = -25.7610597594363$$
$$x_{66} = -17.5929188601028$$
$$x_{67} = 76.026542216873$$
$$x_{68} = -81.6814089933346$$
$$x_{69} = 64.0884901332318$$
$$x_{70} = -3.76991118430775$$
$$x_{71} = -65.9734457253857$$
$$x_{72} = 54.0353936417444$$
$$x_{73} = 0$$
$$x_{74} = 80.4247719318987$$
$$x_{75} = -35.8141562509236$$
$$x_{76} = -98.0176907920015$$
$$x_{77} = 1.88495559215388$$
$$x_{78} = 60.318578948924$$
$$x_{79} = -83.5663645854885$$
$$x_{80} = 43.9822971502571$$
$$x_{81} = 67.8584013175395$$
$$x_{82} = -54.0353936417444$$
$$x_{83} = 84.1946831162065$$
$$x_{84} = -86.0796387083603$$
$$x_{85} = -42.0973415581032$$
$$x_{86} = -67.8584013175395$$
$$x_{87} = 50.2654824574367$$
$$x_{88} = 94.2477796076938$$
$$x_{89} = 82.3097275240526$$
$$x_{90} = -59.6902604182061$$
$$x_{91} = 52.1504380495906$$
$$x_{92} = 70.3716754404114$$
$$x_{93} = -96.1327351998477$$
$$x_{94} = -7.5398223686155$$
$$x_{95} = 10.0530964914873$$
$$x_{96} = -89.8495498926681$$
$$x_{97} = 30.159289474462$$
$$x_{98} = 87.9645943005142$$
$$x_{99} = 32.0442450666159$$
$$x_{100} = 77.9114978090269$$
$$x_{101} = -91.734505484822$$
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
$$5 \tan^{2}{\left(5 x \right)} + 5 = 0$$
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
$$5 \tan^{2}{\left(5 x \right)} + 5 = 0$$
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
$$50 \left(\tan^{2}{\left(5 x \right)} + 1\right) \tan{\left(5 x \right)} = 0$$
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
$$50 \left(\tan^{2}{\left(5 x \right)} + 1\right) \tan{\left(5 x \right)} = 0$$
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
$$\lim_{x \to -\infty} \tan{\left(5 x \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \tan{\left(5 x \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the right:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to -\infty} \tan{\left(5 x \right)} = \left\langle -\infty, \infty\right\rangle$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = \left\langle -\infty, \infty\right\rangle$$
$$\lim_{x \to \infty} \tan{\left(5 x \right)} = \left\langle -\infty, \infty\right\rangle$$
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(5*x), divided by x at x->+oo and x ->-oo
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(5 x \right)}}{x}\right)$$
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(5 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the left:
$$y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(5 x \right)}}{x}\right)$$
True
Let's take the limit
so,
inclined asymptote equation on the right:
$$y = x \lim_{x \to \infty}\left(\frac{\tan{\left(5 x \right)}}{x}\right)$$
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:
$$\tan{\left(5 x \right)} = - \tan{\left(5 x \right)}$$
- No
$$\tan{\left(5 x \right)} = \tan{\left(5 x \right)}$$
- Yes
so, the function
is
odd
So, check:
$$\tan{\left(5 x \right)} = - \tan{\left(5 x \right)}$$
- No
$$\tan{\left(5 x \right)} = \tan{\left(5 x \right)}$$
- Yes
so, the function
is
odd