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Plotting a function graph/ (1/2)cos(x/3)

Graphing y = (1/2)cos(x/3)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /x\
       cos|-|
          \3/
f(x) = ------
         2
f(x)=cos⁡(x3)2f{\left(x \right)} = \frac{\cos{\left(\frac{x}{3} \right)}}{2} 
f = cos(x/3)/2
The graph of the function
02468-8-6-4-2-10101-1
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:
cos⁡(x3)2=0\frac{\cos{\left(\frac{x}{3} \right)}}{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=3π2x_{1} = \frac{3 \pi}{2}
x2=9π2x_{2} = \frac{9 \pi}{2}
Numerical solution
x1=4.71238898038469x_{1} = 4.71238898038469
x2=−89.5353906273091x_{2} = -89.5353906273091
x3=70.6858347057703x_{3} = 70.6858347057703
x4=−98.9601685880785x_{4} = -98.9601685880785
x5=−23.5619449019235x_{5} = -23.5619449019235
x6=23.5619449019235x_{6} = 23.5619449019235
x7=61.261056745001x_{7} = 61.261056745001
x8=5659.57916544201x_{8} = 5659.57916544201
x9=−32.9867228626928x_{9} = -32.9867228626928
x10=−51.8362787842316x_{10} = -51.8362787842316
x11=−80.1106126665397x_{11} = -80.1106126665397
x12=98.9601685880785x_{12} = 98.9601685880785
x13=51.8362787842316x_{13} = 51.8362787842316
x14=−249.756615960389x_{14} = -249.756615960389
x15=2653.07499595658x_{15} = 2653.07499595658
x16=−4.71238898038469x_{16} = -4.71238898038469
x17=−14.1371669411541x_{17} = -14.1371669411541
x18=−70.6858347057703x_{18} = -70.6858347057703
x19=14.1371669411541x_{19} = 14.1371669411541
x20=89.5353906273091x_{20} = 89.5353906273091
x21=80.1106126665397x_{21} = 80.1106126665397
x22=−61.261056745001x_{22} = -61.261056745001
x23=−4867.89781673738x_{23} = -4867.89781673738
x24=−42.4115008234622x_{24} = -42.4115008234622
x25=32.9867228626928x_{25} = 32.9867228626928
x26=155.508836352695x_{26} = 155.508836352695
x27=42.4115008234622x_{27} = 42.4115008234622
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x/3)/2.
cos⁡(03)2\frac{\cos{\left(\frac{0}{3} \right)}}{2}
The result:
f(0)=12f{\left(0 \right)} = \frac{1}{2}
The point:
(0, 1/2)
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
−sin⁡(x3)6=0- \frac{\sin{\left(\frac{x}{3} \right)}}{6} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=3πx_{2} = 3 \pi
The values of the extrema at the points:
(0, 1/2)

(3*pi, -1/2)


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=3πx_{1} = 3 \pi
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[3π,∞)\left(-\infty, 0\right] \cup \left[3 \pi, \infty\right)
Increasing at intervals
[0,3π]\left[0, 3 \pi\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
−cos⁡(x3)18=0- \frac{\cos{\left(\frac{x}{3} \right)}}{18} = 0
Solve this equation
The roots of this equation
x1=3π2x_{1} = \frac{3 \pi}{2}
x2=9π2x_{2} = \frac{9 \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
[3π2,9π2]\left[\frac{3 \pi}{2}, \frac{9 \pi}{2}\right]
Convex at the intervals
(−∞,3π2]∪[9π2,∞)\left(-\infty, \frac{3 \pi}{2}\right] \cup \left[\frac{9 \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
lim⁡x→−∞(cos⁡(x3)2)=⟨−12,12⟩\lim_{x \to -\infty}\left(\frac{\cos{\left(\frac{x}{3} \right)}}{2}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−12,12⟩y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
lim⁡x→∞(cos⁡(x3)2)=⟨−12,12⟩\lim_{x \to \infty}\left(\frac{\cos{\left(\frac{x}{3} \right)}}{2}\right) = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−12,12⟩y = \left\langle - \frac{1}{2}, \frac{1}{2}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x/3)/2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x3)2x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\frac{x}{3} \right)}}{2 x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x3)2x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(\frac{x}{3} \right)}}{2 x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
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:
cos⁡(x3)2=cos⁡(x3)2\frac{\cos{\left(\frac{x}{3} \right)}}{2} = \frac{\cos{\left(\frac{x}{3} \right)}}{2}
- No
cos⁡(x3)2=−cos⁡(x3)2\frac{\cos{\left(\frac{x}{3} \right)}}{2} = - \frac{\cos{\left(\frac{x}{3} \right)}}{2}
- No
so, the function
not is
neither even, nor odd
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