Mister Exam

Other calculators

Plotting a function graph/ ctg(3*x-(4*pi)/3)

Graphing y = ctg(3*x-(4*pi)/3)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /      4*pi\
f(x) = cot|3*x - ----|
          \       3  /
f(x)=cot⁡(3x−4π3)f{\left(x \right)} = \cot{\left(3 x - \frac{4 \pi}{3} \right)} 
f = cot(3*x - 4*pi/3)
The graph of the function
02468-8-6-4-2-1010-250250
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:
cot⁡(3x−4π3)=0\cot{\left(3 x - \frac{4 \pi}{3} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=−π18x_{1} = - \frac{\pi}{18}
Numerical solution
x1=63.704517697793x_{1} = 63.704517697793
x2=82.5540736193318x_{2} = 82.5540736193318
x3=−44.1568300754565x_{3} = -44.1568300754565
x4=21.8166156499291x_{4} = 21.8166156499291
x5=−68.2423737529783x_{5} = -68.2423737529783
x6=−35.7792496658838x_{6} = -35.7792496658838
x7=91.9788515801012x_{7} = 91.9788515801012
x8=−17.9768912955416x_{8} = -17.9768912955416
x9=19.7222205475359x_{9} = 19.7222205475359
x10=98.2620368872808x_{10} = 98.2620368872808
x11=74.176493209759x_{11} = 74.176493209759
x12=96.1676417848876x_{12} = 96.1676417848876
x13=85.6956662729216x_{13} = 85.6956662729216
x14=18.6750229963393x_{14} = 18.6750229963393
x15=−95.4695100840898x_{15} = -95.4695100840898
x16=32.2885911618951x_{16} = 32.2885911618951
x17=−31.5904594610974x_{17} = -31.5904594610974
x18=45.9021593274509x_{18} = 45.9021593274509
x19=−39.9680398706701x_{19} = -39.9680398706701
x20=47.9965544298441x_{20} = 47.9965544298441
x21=41.7133691226645x_{21} = 41.7133691226645
x22=−7.50491578357562x_{22} = -7.50491578357562
x23=−64.0535835481919x_{23} = -64.0535835481919
x24=28.0998009571087x_{24} = 28.0998009571087
x25=12.3918376891597x_{25} = 12.3918376891597
x26=−49.3928178314395x_{26} = -49.3928178314395
x27=−5.41052068118242x_{27} = -5.41052068118242
x28=30.1941960595019x_{28} = 30.1941960595019
x29=−93.3751149816966x_{29} = -93.3751149816966
x30=54.2797397370236x_{30} = 54.2797397370236
x31=8.20304748437335x_{31} = 8.20304748437335
x32=56.3741348394168x_{32} = 56.3741348394168
x33=80.4596785169386x_{33} = 80.4596785169386
x34=−97.563905186483x_{34} = -97.563905186483
x35=−4.36332312998582x_{35} = -4.36332312998582
x36=−26.3544717051144x_{36} = -26.3544717051144
x37=−22.165681500328x_{37} = -22.165681500328
x38=−86.0447321233204x_{38} = -86.0447321233204
x39=−66.1479786505851x_{39} = -66.1479786505851
x40=−92.3279174305x_{40} = -92.3279174305
x41=−46.2512251778497x_{41} = -46.2512251778497
x42=69.9877030049726x_{42} = 69.9877030049726
x43=16.5806278939461x_{43} = 16.5806278939461
x44=58.46852994181x_{44} = 58.46852994181
x45=60.5629250442032x_{45} = 60.5629250442032
x46=36.4773813666815x_{46} = 36.4773813666815
x47=1.91986217719376x_{47} = 1.91986217719376
x48=14.4862327915529x_{48} = 14.4862327915529
x49=87.7900613753148x_{49} = 87.7900613753148
x50=−79.7615468161409x_{50} = -79.7615468161409
x51=−37.873644768277x_{51} = -37.873644768277
x52=26.0054058547155x_{52} = 26.0054058547155
x53=−42.0624349730633x_{53} = -42.0624349730633
x54=52.1853446346305x_{54} = 52.1853446346305
x55=−13.7881010907552x_{55} = -13.7881010907552
x56=43.8077642250577x_{56} = 43.8077642250577
x57=−73.4783615089613x_{57} = -73.4783615089613
x58=−20.0712863979348x_{58} = -20.0712863979348
x59=65.7989128001862x_{59} = 65.7989128001862
x60=10.2974425867665x_{60} = 10.2974425867665
x61=84.648468721725x_{61} = 84.648468721725
x62=−99.6583002888762x_{62} = -99.6583002888762
x63=−77.6671517137477x_{63} = -77.6671517137477
x64=−0.174532925199433x_{64} = -0.174532925199433
x65=−59.8647933434055x_{65} = -59.8647933434055
x66=−71.3839664065681x_{66} = -71.3839664065681
x67=78.3652834145454x_{67} = 78.3652834145454
x68=50.0909495322373x_{68} = 50.0909495322373
x69=94.0732466824944x_{69} = 94.0732466824944
x70=−29.4960643587042x_{70} = -29.4960643587042
x71=−11.693705988362x_{71} = -11.693705988362
x72=−48.3456202802429x_{72} = -48.3456202802429
x73=−61.9591884457987x_{73} = -61.9591884457987
x74=76.2708883121522x_{74} = 76.2708883121522
x75=−51.4872129338327x_{75} = -51.4872129338327
x76=−24.2600766027212x_{76} = -24.2600766027212
x77=89.884456477708x_{77} = 89.884456477708
x78=−81.8559419185341x_{78} = -81.8559419185341
x79=38.5717764690747x_{79} = 38.5717764690747
x80=−57.7703982410123x_{80} = -57.7703982410123
x81=40.6661715714679x_{81} = 40.6661715714679
x82=62.6573201465964x_{82} = 62.6573201465964
x83=4.01425727958696x_{83} = 4.01425727958696
x84=100.356431989674x_{84} = 100.356431989674
x85=6.10865238198015x_{85} = 6.10865238198015
x86=23.9110107523223x_{86} = 23.9110107523223
x87=−9.59931088596881x_{87} = -9.59931088596881
x88=−90.2335223281068x_{88} = -90.2335223281068
x89=−75.5727566113545x_{89} = -75.5727566113545
x90=−83.9503370209273x_{90} = -83.9503370209273
x91=−27.401669256311x_{91} = -27.401669256311
x92=−88.1391272257137x_{92} = -88.1391272257137
x93=72.0820981073658x_{93} = 72.0820981073658
x94=−2.26892802759263x_{94} = -2.26892802759263
x95=−15.8824961931484x_{95} = -15.8824961931484
x96=−55.6760031386191x_{96} = -55.6760031386191
x97=−53.5816080362259x_{97} = -53.5816080362259
x98=−33.6848545634906x_{98} = -33.6848545634906
x99=−70.3367688553715x_{99} = -70.3367688553715
x100=67.8933079025794x_{100} = 67.8933079025794
x101=34.3829862642883x_{101} = 34.3829862642883
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cot(3*x - 4*pi/3).
cot⁡(−4π3+0⋅3)\cot{\left(- \frac{4 \pi}{3} + 0 \cdot 3 \right)}
The result:
f(0)=−33f{\left(0 \right)} = - \frac{\sqrt{3}}{3}
The point:
(0, -sqrt(3)/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
−3cot⁡2(3x−4π3)−3=0- 3 \cot^{2}{\left(3 x - \frac{4 \pi}{3} \right)} - 3 = 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
−18(tan⁡2(3x+π6)+1)tan⁡(3x+π6)=0- 18 \left(\tan^{2}{\left(3 x + \frac{\pi}{6} \right)} + 1\right) \tan{\left(3 x + \frac{\pi}{6} \right)} = 0
Solve this equation
The roots of this equation
x1=−π18x_{1} = - \frac{\pi}{18}

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
(−∞,−π18]\left(-\infty, - \frac{\pi}{18}\right]
Convex at the intervals
[−π18,∞)\left[- \frac{\pi}{18}, \infty\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→−∞cot⁡(3x−4π3)y = \lim_{x \to -\infty} \cot{\left(3 x - \frac{4 \pi}{3} \right)}
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞cot⁡(3x−4π3)y = \lim_{x \to \infty} \cot{\left(3 x - \frac{4 \pi}{3} \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cot(3*x - 4*pi/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→−∞(cot⁡(3x−4π3)x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(3 x - \frac{4 \pi}{3} \right)}}{x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(3x−4π3)x)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(3 x - \frac{4 \pi}{3} \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:
cot⁡(3x−4π3)=−cot⁡(3x+4π3)\cot{\left(3 x - \frac{4 \pi}{3} \right)} = - \cot{\left(3 x + \frac{4 \pi}{3} \right)}
- No
cot⁡(3x−4π3)=cot⁡(3x+4π3)\cot{\left(3 x - \frac{4 \pi}{3} \right)} = \cot{\left(3 x + \frac{4 \pi}{3} \right)}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator