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Plotting a function graph/ xtanx-1/3

Graphing y = xtanx-1/3

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = x*tan(x) - 1/3
f(x)=xtan⁡(x)−13f{\left(x \right)} = x \tan{\left(x \right)} - \frac{1}{3} 
f = x*tan(x) - 1/3
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)−13=0x \tan{\left(x \right)} - \frac{1}{3} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=40.8488644771026x_{1} = 40.8488644771026
x2=75.402644368808x_{2} = 75.402644368808
x3=−84.8269311963367x_{3} = -84.8269311963367
x4=−31.426532886749x_{4} = -31.426532886749
x5=65.978497833395x_{5} = 65.978497833395
x6=−69.1198608821607x_{6} = -69.1198608821607
x7=−91.1098455251857x_{7} = -91.1098455251857
x8=97.3927948145887x_{8} = 97.3927948145887
x9=31.426532886749x_{9} = 31.426532886749
x10=87.9683835225577x_{10} = 87.9683835225577
x11=−100.534280521352x_{11} = -100.534280521352
x12=−81.6854896627962x_{12} = -81.6854896627962
x13=22.0062945981151x_{13} = 22.0062945981151
x14=18.8672214087353x_{14} = 18.8672214087353
x15=0.54716075726033x_{15} = 0.54716075726033
x16=−72.2612438921163x_{16} = -72.2612438921163
x17=−34.5671619539785x_{17} = -34.5671619539785
x18=34.5671619539785x_{18} = 34.5671619539785
x19=91.1098455251857x_{19} = 91.1098455251857
x20=6.33574836234573x_{20} = 6.33574836234573
x21=−97.3927948145887x_{21} = -97.3927948145887
x22=−37.7079514813517x_{22} = -37.7079514813517
x23=−12.5928345144433x_{23} = -12.5928345144433
x24=78.5440602167642x_{24} = 78.5440602167642
x25=15.7291521688357x_{25} = 15.7291521688357
x26=56.5545617095679x_{26} = 56.5545617095679
x27=−94.2513162367499x_{27} = -94.2513162367499
x28=−6.33574836234573x_{28} = -6.33574836234573
x29=28.2861176805762x_{29} = 28.2861176805762
x30=−59.6958442217916x_{30} = -59.6958442217916
x31=81.6854896627962x_{31} = 81.6854896627962
x32=69.1198608821607x_{32} = 69.1198608821607
x33=−50.2721129415958x_{33} = -50.2721129415958
x34=−87.9683835225577x_{34} = -87.9683835225577
x35=84.8269311963367x_{35} = 84.8269311963367
x36=50.2721129415958x_{36} = 50.2721129415958
x37=43.9898745065796x_{37} = 43.9898745065796
x38=25.145996373039x_{38} = 25.145996373039
x39=−53.4133156711182x_{39} = -53.4133156711182
x40=−43.9898745065796x_{40} = -43.9898745065796
x41=−47.1309621775118x_{41} = -47.1309621775118
x42=−25.145996373039x_{42} = -25.145996373039
x43=−28.2861176805762x_{43} = -28.2861176805762
x44=12.5928345144433x_{44} = 12.5928345144433
x45=53.4133156711182x_{45} = 53.4133156711182
x46=−22.0062945981151x_{46} = -22.0062945981151
x47=−40.8488644771026x_{47} = -40.8488644771026
x48=−15.7291521688357x_{48} = -15.7291521688357
x49=3.24398748493294x_{49} = 3.24398748493294
x50=59.6958442217916x_{50} = 59.6958442217916
x51=100.534280521352x_{51} = 100.534280521352
x52=37.7079514813517x_{52} = 37.7079514813517
x53=47.1309621775118x_{53} = 47.1309621775118
x54=−65.978497833395x_{54} = -65.978497833395
x55=−9.45999947251134x_{55} = -9.45999947251134
x56=9.45999947251134x_{56} = 9.45999947251134
x57=−62.83715773895x_{57} = -62.83715773895
x58=−56.5545617095679x_{58} = -56.5545617095679
x59=72.2612438921163x_{59} = 72.2612438921163
x60=62.83715773895x_{60} = 62.83715773895
x61=−3.24398748493294x_{61} = -3.24398748493294
x62=94.2513162367499x_{62} = 94.2513162367499
x63=−18.8672214087353x_{63} = -18.8672214087353
x64=−78.5440602167642x_{64} = -78.5440602167642
x65=−75.402644368808x_{65} = -75.402644368808
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) - 1/3.
−13+0tan⁡(0)- \frac{1}{3} + 0 \tan{\left(0 \right)}
The result:
f(0)=−13f{\left(0 \right)} = - \frac{1}{3}
The point:
(0, -1/3)
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, -0.333333333333333)

(-4.4704381302316267e-13, -0.333333333333333)

(0, -1/3)

(3.6245359934199923e-17, -0.333333333333333)


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=−69.100567727981x_{1} = -69.100567727981
x2=−12.4864543952238x_{2} = -12.4864543952238
x3=81.6691650818489x_{3} = 81.6691650818489
x4=−2.79838604578389x_{4} = -2.79838604578389
x5=91.0952098694071x_{5} = 91.0952098694071
x6=78.5270825679419x_{6} = 78.5270825679419
x7=−72.2427897046973x_{7} = -72.2427897046973
x8=15.644128370333x_{8} = 15.644128370333
x9=−43.9595528888955x_{9} = -43.9595528888955
x10=50.2455828375744x_{10} = 50.2455828375744
x11=−62.8159348889734x_{11} = -62.8159348889734
x12=6.12125046689807x_{12} = 6.12125046689807
x13=−84.811211299318x_{13} = -84.811211299318
x14=62.8159348889734x_{14} = 62.8159348889734
x15=56.5309801938186x_{15} = 56.5309801938186
x16=−28.2389365752603x_{16} = -28.2389365752603
x17=−56.5309801938186x_{17} = -56.5309801938186
x18=−25.0929104121121x_{18} = -25.0929104121121
x19=−59.6735041304405x_{19} = -59.6735041304405
x20=69.100567727981x_{20} = 69.100567727981
x21=−47.1026627703624x_{21} = -47.1026627703624
x22=72.2427897046973x_{22} = 72.2427897046973
x23=−40.8162093266346x_{23} = -40.8162093266346
x24=47.1026627703624x_{24} = 47.1026627703624
x25=−78.5270825679419x_{25} = -78.5270825679419
x26=97.3791034786112x_{26} = 97.3791034786112
x27=−31.3840740178899x_{27} = -31.3840740178899
x28=−37.672573565113x_{28} = -37.672573565113
x29=18.7964043662102x_{29} = 18.7964043662102
x30=−75.3849592185347x_{30} = -75.3849592185347
x31=40.8162093266346x_{31} = 40.8162093266346
x32=−34.5285657554621x_{32} = -34.5285657554621
x33=−53.3883466217256x_{33} = -53.3883466217256
x34=34.5285657554621x_{34} = 34.5285657554621
x35=37.672573565113x_{35} = 37.672573565113
x36=−100.521017074687x_{36} = -100.521017074687
x37=−6.12125046689807x_{37} = -6.12125046689807
x38=−65.9582857893902x_{38} = -65.9582857893902
x39=59.6735041304405x_{39} = 59.6735041304405
x40=−91.0952098694071x_{40} = -91.0952098694071
x41=75.3849592185347x_{41} = 75.3849592185347
x42=65.9582857893902x_{42} = 65.9582857893902
x43=84.811211299318x_{43} = 84.811211299318
x44=−87.9532251106725x_{44} = -87.9532251106725
x45=100.521017074687x_{45} = 100.521017074687
x46=25.0929104121121x_{46} = 25.0929104121121
x47=94.2371684817036x_{47} = 94.2371684817036
x48=−94.2371684817036x_{48} = -94.2371684817036
x49=2.79838604578389x_{49} = 2.79838604578389
x50=28.2389365752603x_{50} = 28.2389365752603
x51=−21.945612879981x_{51} = -21.945612879981
x52=53.3883466217256x_{52} = 53.3883466217256
x53=9.31786646179107x_{53} = 9.31786646179107
x54=−50.2455828375744x_{54} = -50.2455828375744
x55=43.9595528888955x_{55} = 43.9595528888955
x56=−18.7964043662102x_{56} = -18.7964043662102
x57=−81.6691650818489x_{57} = -81.6691650818489
x58=−9.31786646179107x_{58} = -9.31786646179107
x59=87.9532251106725x_{59} = 87.9532251106725
x60=12.4864543952238x_{60} = 12.4864543952238
x61=−97.3791034786112x_{61} = -97.3791034786112
x62=31.3840740178899x_{62} = 31.3840740178899
x63=−15.644128370333x_{63} = -15.644128370333
x64=21.945612879981x_{64} = 21.945612879981

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)−13)y = \lim_{x \to -\infty}\left(x \tan{\left(x \right)} - \frac{1}{3}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(xtan⁡(x)−13)y = \lim_{x \to \infty}\left(x \tan{\left(x \right)} - \frac{1}{3}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of x*tan(x) - 1/3, divided by x at x->+oo and x ->-oo
True

Let's take the limit
so,
inclined asymptote equation on the left:
y=xlim⁡x→−∞(xtan⁡(x)−13x)y = x \lim_{x \to -\infty}\left(\frac{x \tan{\left(x \right)} - \frac{1}{3}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(xtan⁡(x)−13x)y = x \lim_{x \to \infty}\left(\frac{x \tan{\left(x \right)} - \frac{1}{3}}{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:
xtan⁡(x)−13=xtan⁡(x)−13x \tan{\left(x \right)} - \frac{1}{3} = x \tan{\left(x \right)} - \frac{1}{3}
- Yes
xtan⁡(x)−13=−xtan⁡(x)+13x \tan{\left(x \right)} - \frac{1}{3} = - x \tan{\left(x \right)} + \frac{1}{3}
- No
so, the function
is
even
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