Mister Exam

Other calculators

Plotting a function graph/ 1/2ctgx/2

Graphing y = 1/2ctgx/2

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       /cot(x)\
       |------|
       \  2   /
f(x) = --------
          2
f(x)=12cot⁡(x)2f{\left(x \right)} = \frac{\frac{1}{2} \cot{\left(x \right)}}{2} 
f = (cot(x)/2)/2
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:
12cot⁡(x)2=0\frac{\frac{1}{2} \cot{\left(x \right)}}{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
Numerical solution
x1=7.85398163397448x_{1} = 7.85398163397448
x2=−73.8274273593601x_{2} = -73.8274273593601
x3=−54.9778714378214x_{3} = -54.9778714378214
x4=73.8274273593601x_{4} = 73.8274273593601
x5=−26.7035375555132x_{5} = -26.7035375555132
x6=−1.5707963267949x_{6} = -1.5707963267949
x7=−95.8185759344887x_{7} = -95.8185759344887
x8=−39.2699081698724x_{8} = -39.2699081698724
x9=−4.71238898038469x_{9} = -4.71238898038469
x10=14.1371669411541x_{10} = 14.1371669411541
x11=10.9955742875643x_{11} = 10.9955742875643
x12=58.1194640914112x_{12} = 58.1194640914112
x13=70.6858347057703x_{13} = 70.6858347057703
x14=−36.1283155162826x_{14} = -36.1283155162826
x15=54.9778714378214x_{15} = 54.9778714378214
x16=23.5619449019235x_{16} = 23.5619449019235
x17=−92.6769832808989x_{17} = -92.6769832808989
x18=−86.3937979737193x_{18} = -86.3937979737193
x19=−10.9955742875643x_{19} = -10.9955742875643
x20=92.6769832808989x_{20} = 92.6769832808989
x21=39.2699081698724x_{21} = 39.2699081698724
x22=−32.9867228626928x_{22} = -32.9867228626928
x23=98.9601685880785x_{23} = 98.9601685880785
x24=36.1283155162826x_{24} = 36.1283155162826
x25=−7.85398163397448x_{25} = -7.85398163397448
x26=−58.1194640914112x_{26} = -58.1194640914112
x27=−67.5442420521806x_{27} = -67.5442420521806
x28=−61.261056745001x_{28} = -61.261056745001
x29=26.7035375555132x_{29} = 26.7035375555132
x30=86.3937979737193x_{30} = 86.3937979737193
x31=−48.6946861306418x_{31} = -48.6946861306418
x32=51.8362787842316x_{32} = 51.8362787842316
x33=−42.4115008234622x_{33} = -42.4115008234622
x34=−89.5353906273091x_{34} = -89.5353906273091
x35=−98.9601685880785x_{35} = -98.9601685880785
x36=−14.1371669411541x_{36} = -14.1371669411541
x37=80.1106126665397x_{37} = 80.1106126665397
x38=−64.4026493985908x_{38} = -64.4026493985908
x39=95.8185759344887x_{39} = 95.8185759344887
x40=1.5707963267949x_{40} = 1.5707963267949
x41=45.553093477052x_{41} = 45.553093477052
x42=−17.2787595947439x_{42} = -17.2787595947439
x43=4.71238898038469x_{43} = 4.71238898038469
x44=48.6946861306418x_{44} = 48.6946861306418
x45=76.9690200129499x_{45} = 76.9690200129499
x46=−45.553093477052x_{46} = -45.553093477052
x47=20.4203522483337x_{47} = 20.4203522483337
x48=17.2787595947439x_{48} = 17.2787595947439
x49=−83.2522053201295x_{49} = -83.2522053201295
x50=−20.4203522483337x_{50} = -20.4203522483337
x51=−80.1106126665397x_{51} = -80.1106126665397
x52=61.261056745001x_{52} = 61.261056745001
x53=32.9867228626928x_{53} = 32.9867228626928
x54=64.4026493985908x_{54} = 64.4026493985908
x55=−23.5619449019235x_{55} = -23.5619449019235
x56=29.845130209103x_{56} = 29.845130209103
x57=42.4115008234622x_{57} = 42.4115008234622
x58=89.5353906273091x_{58} = 89.5353906273091
x59=−51.8362787842316x_{59} = -51.8362787842316
x60=−70.6858347057703x_{60} = -70.6858347057703
x61=83.2522053201295x_{61} = 83.2522053201295
x62=67.5442420521806x_{62} = 67.5442420521806
x63=−29.845130209103x_{63} = -29.845130209103
x64=−76.9690200129499x_{64} = -76.9690200129499
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (cot(x)/2)/2.
12cot⁡(0)2\frac{\frac{1}{2} \cot{\left(0 \right)}}{2}
The result:
f(0)=∞~f{\left(0 \right)} = \tilde{\infty}
sof doesn't intersect Y
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
−cot⁡2(x)4−14=0- \frac{\cot^{2}{\left(x \right)}}{4} - \frac{1}{4} = 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
(cot⁡2(x)+1)cot⁡(x)2=0\frac{\left(\cot^{2}{\left(x \right)} + 1\right) \cot{\left(x \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}

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
(−∞,π2]\left(-\infty, \frac{\pi}{2}\right]
Convex at the intervals
[π2,∞)\left[\frac{\pi}{2}, \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→−∞(12cot⁡(x)2)y = \lim_{x \to -\infty}\left(\frac{\frac{1}{2} \cot{\left(x \right)}}{2}\right)
True

Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=lim⁡x→∞(12cot⁡(x)2)y = \lim_{x \to \infty}\left(\frac{\frac{1}{2} \cot{\left(x \right)}}{2}\right)
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (cot(x)/2)/2, 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⁡(x)4x)y = x \lim_{x \to -\infty}\left(\frac{\cot{\left(x \right)}}{4 x}\right)
True

Let's take the limit
so,
inclined asymptote equation on the right:
y=xlim⁡x→∞(cot⁡(x)4x)y = x \lim_{x \to \infty}\left(\frac{\cot{\left(x \right)}}{4 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:
12cot⁡(x)2=−cot⁡(x)4\frac{\frac{1}{2} \cot{\left(x \right)}}{2} = - \frac{\cot{\left(x \right)}}{4}
- No
12cot⁡(x)2=cot⁡(x)4\frac{\frac{1}{2} \cot{\left(x \right)}}{2} = \frac{\cot{\left(x \right)}}{4}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator