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Plotting a function graph/ x*tan(x)

Graphing y = x*tan(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = x*tan(x)
f(x)=xtan⁡(x)f{\left(x \right)} = x \tan{\left(x \right)} 
f = x*tan(x)
The graph of the function
02468-8-6-4-2-1010-500500
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:
xtan⁡(x)=0x \tan{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=72.2566310325652x_{1} = 72.2566310325652
x2=−59.6902604182061x_{2} = -59.6902604182061
x3=3.14159265358979x_{3} = 3.14159265358979
x4=−43.9822971502571x_{4} = -43.9822971502571
x5=81.6814089933346x_{5} = 81.6814089933346
x6=−100.530964914873x_{6} = -100.530964914873
x7=28.2743338823081x_{7} = 28.2743338823081
x8=65.9734457253857x_{8} = 65.9734457253857
x9=−31.4159265358979x_{9} = -31.4159265358979
x10=−9.42477796076938x_{10} = -9.42477796076938
x11=40.8407044966673x_{11} = 40.8407044966673
x12=56.5486677646163x_{12} = 56.5486677646163
x13=−56.5486677646163x_{13} = -56.5486677646163
x14=12.5663706143592x_{14} = 12.5663706143592
x15=43.9822971502571x_{15} = 43.9822971502571
x16=100.530964914873x_{16} = 100.530964914873
x17=−3.14159265358979x_{17} = -3.14159265358979
x18=−15.707963267949x_{18} = -15.707963267949
x19=59.6902604182061x_{19} = 59.6902604182061
x20=6.28318530717959x_{20} = 6.28318530717959
x21=9.42477796076938x_{21} = 9.42477796076938
x22=−53.4070751110265x_{22} = -53.4070751110265
x23=−47.1238898038469x_{23} = -47.1238898038469
x24=−87.9645943005142x_{24} = -87.9645943005142
x25=69.1150383789755x_{25} = 69.1150383789755
x26=21.9911485751286x_{26} = 21.9911485751286
x27=87.9645943005142x_{27} = 87.9645943005142
x28=18.8495559215388x_{28} = 18.8495559215388
x29=−84.8230016469244x_{29} = -84.8230016469244
x30=−72.2566310325652x_{30} = -72.2566310325652
x31=25.1327412287183x_{31} = 25.1327412287183
x32=37.6991118430775x_{32} = 37.6991118430775
x33=−25.1327412287183x_{33} = -25.1327412287183
x34=0x_{34} = 0
x35=50.2654824574367x_{35} = 50.2654824574367
x36=−6.28318530717959x_{36} = -6.28318530717959
x37=−65.9734457253857x_{37} = -65.9734457253857
x38=−21.9911485751286x_{38} = -21.9911485751286
x39=−62.8318530717959x_{39} = -62.8318530717959
x40=75.398223686155x_{40} = 75.398223686155
x41=84.8230016469244x_{41} = 84.8230016469244
x42=53.4070751110265x_{42} = 53.4070751110265
x43=34.5575191894877x_{43} = 34.5575191894877
x44=−28.2743338823081x_{44} = -28.2743338823081
x45=15.707963267949x_{45} = 15.707963267949
x46=−91.106186954104x_{46} = -91.106186954104
x47=47.1238898038469x_{47} = 47.1238898038469
x48=97.3893722612836x_{48} = 97.3893722612836
x49=−69.1150383789755x_{49} = -69.1150383789755
x50=94.2477796076938x_{50} = 94.2477796076938
x51=−18.8495559215388x_{51} = -18.8495559215388
x52=−50.2654824574367x_{52} = -50.2654824574367
x53=−37.6991118430775x_{53} = -37.6991118430775
x54=−81.6814089933346x_{54} = -81.6814089933346
x55=62.8318530717959x_{55} = 62.8318530717959
x56=78.5398163397448x_{56} = 78.5398163397448
x57=31.4159265358979x_{57} = 31.4159265358979
x58=−78.5398163397448x_{58} = -78.5398163397448
x59=−40.8407044966673x_{59} = -40.8407044966673
x60=−97.3893722612836x_{60} = -97.3893722612836
x61=−75.398223686155x_{61} = -75.398223686155
x62=91.106186954104x_{62} = 91.106186954104
x63=−12.5663706143592x_{63} = -12.5663706143592
x64=−94.2477796076938x_{64} = -94.2477796076938
x65=−34.5575191894877x_{65} = -34.5575191894877
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to x*tan(x).
0tan⁡(0)0 \tan{\left(0 \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
x(tan⁡2(x)+1)+tan⁡(x)=0x \left(\tan^{2}{\left(x \right)} + 1\right) + \tan{\left(x \right)} = 0
Solve this equation
The roots of this equation
x1=3.46683696838538⋅10−18x_{1} = 3.46683696838538 \cdot 10^{-18}
x2=−4.47043813023163⋅10−13x_{2} = -4.47043813023163 \cdot 10^{-13}
x3=0x_{3} = 0
x4=3.62453599341999⋅10−17x_{4} = 3.62453599341999 \cdot 10^{-17}
The values of the extrema at the points:
(3.4668369683853792e-18, 1.20189585653635e-35)

(-4.4704381302316267e-13, 1.99848170762288e-25)

(0, 0)

(3.6245359934199923e-17, 1.31372611675971e-33)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
x1=3.46683696838538⋅10−18x_{1} = 3.46683696838538 \cdot 10^{-18}
x2=−4.47043813023163⋅10−13x_{2} = -4.47043813023163 \cdot 10^{-13}
x3=0x_{3} = 0
x4=3.62453599341999⋅10−17x_{4} = 3.62453599341999 \cdot 10^{-17}
The function has no maxima
Decreasing at intervals
[3.62453599341999⋅10−17,∞)\left[3.62453599341999 \cdot 10^{-17}, \infty\right)
Increasing at intervals
(−∞,−4.47043813023163⋅10−13]\left(-\infty, -4.47043813023163 \cdot 10^{-13}\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
2(x(tan⁡2(x)+1)tan⁡(x)+tan⁡2(x)+1)=02 \left(x \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)} + \tan^{2}{\left(x \right)} + 1\right) = 0
Solve this equation
The roots of this equation
x1=100.521017074687x_{1} = 100.521017074687
x2=−72.2427897046973x_{2} = -72.2427897046973
x3=21.945612879981x_{3} = 21.945612879981
x4=91.0952098694071x_{4} = 91.0952098694071
x5=−28.2389365752603x_{5} = -28.2389365752603
x6=−50.2455828375744x_{6} = -50.2455828375744
x7=−53.3883466217256x_{7} = -53.3883466217256
x8=53.3883466217256x_{8} = 53.3883466217256
x9=56.5309801938186x_{9} = 56.5309801938186
x10=−56.5309801938186x_{10} = -56.5309801938186
x11=87.9532251106725x_{11} = 87.9532251106725
x12=97.3791034786112x_{12} = 97.3791034786112
x13=62.8159348889734x_{13} = 62.8159348889734
x14=−62.8159348889734x_{14} = -62.8159348889734
x15=40.8162093266346x_{15} = 40.8162093266346
x16=9.31786646179107x_{16} = 9.31786646179107
x17=−65.9582857893902x_{17} = -65.9582857893902
x18=−37.672573565113x_{18} = -37.672573565113
x19=69.100567727981x_{19} = 69.100567727981
x20=−75.3849592185347x_{20} = -75.3849592185347
x21=6.12125046689807x_{21} = 6.12125046689807
x22=−87.9532251106725x_{22} = -87.9532251106725
x23=84.811211299318x_{23} = 84.811211299318
x24=12.4864543952238x_{24} = 12.4864543952238
x25=−12.4864543952238x_{25} = -12.4864543952238
x26=−69.100567727981x_{26} = -69.100567727981
x27=−78.5270825679419x_{27} = -78.5270825679419
x28=−43.9595528888955x_{28} = -43.9595528888955
x29=−84.811211299318x_{29} = -84.811211299318
x30=37.672573565113x_{30} = 37.672573565113
x31=18.7964043662102x_{31} = 18.7964043662102
x32=59.6735041304405x_{32} = 59.6735041304405
x33=−91.0952098694071x_{33} = -91.0952098694071
x34=81.6691650818489x_{34} = 81.6691650818489
x35=−97.3791034786112x_{35} = -97.3791034786112
x36=−40.8162093266346x_{36} = -40.8162093266346
x37=47.1026627703624x_{37} = 47.1026627703624
x38=−25.0929104121121x_{38} = -25.0929104121121
x39=78.5270825679419x_{39} = 78.5270825679419
x40=−31.3840740178899x_{40} = -31.3840740178899
x41=94.2371684817036x_{41} = 94.2371684817036
x42=25.0929104121121x_{42} = 25.0929104121121
x43=−100.521017074687x_{43} = -100.521017074687
x44=2.79838604578389x_{44} = 2.79838604578389
x45=65.9582857893902x_{45} = 65.9582857893902
x46=34.5285657554621x_{46} = 34.5285657554621
x47=−9.31786646179107x_{47} = -9.31786646179107
x48=50.2455828375744x_{48} = 50.2455828375744
x49=−34.5285657554621x_{49} = -34.5285657554621
x50=−18.7964043662102x_{50} = -18.7964043662102
x51=75.3849592185347x_{51} = 75.3849592185347
x52=−21.945612879981x_{52} = -21.945612879981
x53=15.644128370333x_{53} = 15.644128370333
x54=43.9595528888955x_{54} = 43.9595528888955
x55=31.3840740178899x_{55} = 31.3840740178899
x56=−15.644128370333x_{56} = -15.644128370333
x57=−47.1026627703624x_{57} = -47.1026627703624
x58=−94.2371684817036x_{58} = -94.2371684817036
x59=−81.6691650818489x_{59} = -81.6691650818489
x60=−6.12125046689807x_{60} = -6.12125046689807
x61=28.2389365752603x_{61} = 28.2389365752603
x62=−2.79838604578389x_{62} = -2.79838604578389
x63=72.2427897046973x_{63} = 72.2427897046973
x64=−59.6735041304405x_{64} = -59.6735041304405

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
[100.521017074687,∞)\left[100.521017074687, \infty\right)
Convex at the intervals
[−2.79838604578389,2.79838604578389]\left[-2.79838604578389, 2.79838604578389\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
True

Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=lim⁡x→−∞(xtan⁡(x))y = \lim_{x \to -\infty}\left(x \tan{\left(x \right)}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(xtan⁡(x))y = \lim_{x \to \infty}\left(x \tan{\left(x \right)}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x*tan(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞tan⁡(x)=⟨−∞,∞⟩\lim_{x \to -\infty} \tan{\left(x \right)} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the left:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
lim⁡x→∞tan⁡(x)=⟨−∞,∞⟩\lim_{x \to \infty} \tan{\left(x \right)} = \left\langle -\infty, \infty\right\rangle
Let's take the limit
so,
inclined asymptote equation on the right:
y=⟨−∞,∞⟩xy = \left\langle -\infty, \infty\right\rangle x
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:
xtan⁡(x)=xtan⁡(x)x \tan{\left(x \right)} = x \tan{\left(x \right)}
- Yes
xtan⁡(x)=−xtan⁡(x)x \tan{\left(x \right)} = - x \tan{\left(x \right)}
- No
so, the function
is
even
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