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Plotting a function graph/ tg(x)+arctg(x)

Graphing y = tg(x)+arctg(x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = tan(x) + atan(x)
f(x)=tan⁡(x)+atan⁡(x)f{\left(x \right)} = \tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} 
f = tan(x) + atan(x)
The graph of the function
02468-8-6-4-2-1010-200200
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⁡(x)+atan⁡(x)=0\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Numerical solution
x1=77.5396725417462x_{1} = 77.5396725417462
x2=61.83266643575x_{2} = 61.83266643575
x3=−14.7242522567935x_{3} = -14.7242522567935
x4=−64.9740303869064x_{4} = -64.9740303869064
x5=−30.4216613078004x_{5} = -30.4216613078004
x6=24.1410236736903x_{6} = 24.1410236736903
x7=−99.5299908589546x_{7} = -99.5299908589546
x8=−49.2675048394235x_{8} = -49.2675048394235
x9=−17.8622178450272x_{9} = -17.8622178450272
x10=39.8441394477448x_{10} = 39.8441394477448
x11=−24.1410236736903x_{11} = -24.1410236736903
x12=58.6913271474582x_{12} = 58.6913271474582
x13=49.2675048394235x_{13} = 49.2675048394235
x14=86.9640430028565x_{14} = 86.9640430028565
x15=33.562342086631x_{15} = 33.562342086631
x16=−77.5396725417462x_{16} = -77.5396725417462
x17=−52.408740441672x_{17} = -52.408740441672
x18=−55.5500167379063x_{18} = -55.5500167379063
x19=−27.2811936639259x_{19} = -27.2811936639259
x20=−86.9640430028565x_{20} = -86.9640430028565
x21=0x_{21} = 0
x22=42.9851917481598x_{22} = 42.9851917481598
x23=−33.562342086631x_{23} = -33.562342086631
x24=17.8622178450272x_{24} = 17.8622178450272
x25=−71.2568191529722x_{25} = -71.2568191529722
x26=−90.1055188584242x_{26} = -90.1055188584242
x27=52.408740441672x_{27} = 52.408740441672
x28=46.1263183201969x_{28} = 46.1263183201969
x29=−61.83266643575x_{29} = -61.83266643575
x30=71.2568191529722x_{30} = 71.2568191529722
x31=−46.1263183201969x_{31} = -46.1263183201969
x32=55.5500167379063x_{32} = 55.5500167379063
x33=−58.6913271474582x_{33} = -58.6913271474582
x34=−42.9851917481598x_{34} = -42.9851917481598
x35=−36.703180699745x_{35} = -36.703180699745
x36=−5.33753969228673x_{36} = -5.33753969228673
x37=21.0012872745219x_{37} = 21.0012872745219
x38=11.5883131166121x_{38} = 11.5883131166121
x39=27.2811936639259x_{39} = 27.2811936639259
x40=−93.2470026209387x_{40} = -93.2470026209387
x41=−39.8441394477448x_{41} = -39.8441394477448
x42=30.4216613078004x_{42} = 30.4216613078004
x43=2.2831221774726x_{43} = 2.2831221774726
x44=80.6811187268223x_{44} = 80.6811187268223
x45=99.5299908589546x_{45} = 99.5299908589546
x46=68.1154155674723x_{46} = 68.1154155674723
x47=−83.8225759478241x_{47} = -83.8225759478241
x48=−74.3982387986425x_{48} = -74.3982387986425
x49=5.33753969228673x_{49} = 5.33753969228673
x50=14.7242522567935x_{50} = 14.7242522567935
x51=8.45673516163459x_{51} = 8.45673516163459
x52=64.9740303869064x_{52} = 64.9740303869064
x53=−21.0012872745219x_{53} = -21.0012872745219
x54=74.3982387986425x_{54} = 74.3982387986425
x55=96.3884935138203x_{55} = 96.3884935138203
x56=−96.3884935138203x_{56} = -96.3884935138203
x57=−8.45673516163459x_{57} = -8.45673516163459
x58=83.8225759478241x_{58} = 83.8225759478241
x59=93.2470026209387x_{59} = 93.2470026209387
x60=−80.6811187268223x_{60} = -80.6811187268223
x61=−11.5883131166121x_{61} = -11.5883131166121
x62=−68.1154155674723x_{62} = -68.1154155674723
x63=−2.2831221774726x_{63} = -2.2831221774726
x64=36.703180699745x_{64} = 36.703180699745
x65=90.1055188584242x_{65} = 90.1055188584242
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to tan(x) + atan(x).
tan⁡(0)+atan⁡(0)\tan{\left(0 \right)} + \operatorname{atan}{\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
tan⁡2(x)+1+1x2+1=0\tan^{2}{\left(x \right)} + 1 + \frac{1}{x^{2} + 1} = 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
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(x2+1)2+(tan⁡2(x)+1)tan⁡(x))=02 \left(- \frac{x}{\left(x^{2} + 1\right)^{2}} + \left(\tan^{2}{\left(x \right)} + 1\right) \tan{\left(x \right)}\right) = 0
Solve this equation
The roots of this equation
x1=−91.1061882761657x_{1} = -91.1061882761657
x2=34.5575433799626x_{2} = 34.5575433799626
x3=−97.3893733436485x_{3} = -97.3893733436485
x4=−25.1328040205507x_{4} = -25.1328040205507
x5=−94.2477808019261x_{5} = -94.2477808019261
x6=31.4159587220774x_{6} = 31.4159587220774
x7=0x_{7} = 0
x8=−21.9912422140771x_{8} = -21.9912422140771
x9=78.5398184031738x_{9} = 78.5398184031738
x10=9.42594558462241x_{10} = 9.42594558462241
x11=25.1328040205507x_{11} = 25.1328040205507
x12=−53.4070816709532x_{12} = -53.4070816709532
x13=−9.42594558462241x_{13} = -9.42594558462241
x14=40.8407191588716x_{14} = 40.8407191588716
x15=59.6902651176419x_{15} = 59.6902651176419
x16=3.16758836114823x_{16} = 3.16758836114823
x17=−43.98230889158x_{17} = -43.98230889158
x18=75.3982260183446x_{18} = 75.3982260183446
x19=−75.3982260183446x_{19} = -75.3982260183446
x20=−34.5575433799626x_{20} = -34.5575433799626
x21=37.6991304808952x_{21} = 37.6991304808952
x22=−62.8318571011953x_{22} = -62.8318571011953
x23=91.1061882761657x_{23} = 91.1061882761657
x24=−59.6902651176419x_{24} = -59.6902651176419
x25=94.2477808019261x_{25} = 94.2477808019261
x26=87.9645957693188x_{26} = 87.9645957693188
x27=18.8497043937593x_{27} = 18.8497043937593
x28=−78.5398184031738x_{28} = -78.5398184031738
x29=−87.9645957693188x_{29} = -87.9645957693188
x30=−37.6991304808952x_{30} = -37.6991304808952
x31=−56.548673291256x_{31} = -56.548673291256
x32=53.4070816709532x_{32} = 53.4070816709532
x33=84.8230032850167x_{33} = 84.8230032850167
x34=15.7082191890374x_{34} = 15.7082191890374
x35=97.3893733436485x_{35} = 97.3893733436485
x36=−65.9734492062964x_{36} = -65.9734492062964
x37=−72.2566336822883x_{37} = -72.2566336822883
x38=6.28701317511834x_{38} = 6.28701317511834
x39=−31.4159587220774x_{39} = -31.4159587220774
x40=−47.1238993512507x_{40} = -47.1238993512507
x41=72.2566336822883x_{41} = 72.2566336822883
x42=−18.8497043937593x_{42} = -18.8497043937593
x43=12.5668681635809x_{43} = 12.5668681635809
x44=−12.5668681635809x_{44} = -12.5668681635809
x45=−69.1150414065892x_{45} = -69.1150414065892
x46=100.530965898917x_{46} = 100.530965898917
x47=56.548673291256x_{47} = 56.548673291256
x48=62.8318571011953x_{48} = 62.8318571011953
x49=−28.2743780124173x_{49} = -28.2743780124173
x50=−6.28701317511834x_{50} = -6.28701317511834
x51=−84.8230032850167x_{51} = -84.8230032850167
x52=−3.16758836114823x_{52} = -3.16758836114823
x53=47.1238993512507x_{53} = 47.1238993512507
x54=−40.8407191588716x_{54} = -40.8407191588716
x55=−15.7082191890374x_{55} = -15.7082191890374
x56=65.9734492062964x_{56} = 65.9734492062964
x57=−81.6814108277603x_{57} = -81.6814108277603
x58=28.2743780124173x_{58} = 28.2743780124173
x59=−50.2654903251137x_{59} = -50.2654903251137
x60=21.9912422140771x_{60} = 21.9912422140771
x61=81.6814108277603x_{61} = 81.6814108277603
x62=69.1150414065892x_{62} = 69.1150414065892
x63=50.2654903251137x_{63} = 50.2654903251137
x64=−100.530965898917x_{64} = -100.530965898917
x65=43.98230889158x_{65} = 43.98230889158

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
[100.530965898917,∞)\left[100.530965898917, \infty\right)
Convex at the intervals
(−∞,−100.530965898917]\left(-\infty, -100.530965898917\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→−∞(tan⁡(x)+atan⁡(x))y = \lim_{x \to -\infty}\left(\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(tan⁡(x)+atan⁡(x))y = \lim_{x \to \infty}\left(\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of tan(x) + atan(x), 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→−∞(tan⁡(x)+atan⁡(x)x)y = x \lim_{x \to -\infty}\left(\frac{\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(tan⁡(x)+atan⁡(x)x)y = x \lim_{x \to \infty}\left(\frac{\tan{\left(x \right)} + \operatorname{atan}{\left(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⁡(x)+atan⁡(x)=−tan⁡(x)−atan⁡(x)\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} = - \tan{\left(x \right)} - \operatorname{atan}{\left(x \right)}
- No
tan⁡(x)+atan⁡(x)=tan⁡(x)+atan⁡(x)\tan{\left(x \right)} + \operatorname{atan}{\left(x \right)} = \tan{\left(x \right)} + \operatorname{atan}{\left(x \right)}
- No
so, the function
not is
neither even, nor odd
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