Mister Exam

Other calculators

Plotting a function graph/ cos(x/2)

Graphing y = cos(x/2)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
          /x\
f(x) = cos|-|
          \2/
f(x)=cos⁡(x2)f{\left(x \right)} = \cos{\left(\frac{x}{2} \right)} 
f = cos(x/2)
The graph of the function
02468-8-6-4-2-10102-2
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⁡(x2)=0\cos{\left(\frac{x}{2} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=πx_{1} = \pi
x2=3πx_{2} = 3 \pi
Numerical solution
x1=40.8407044966673x_{1} = 40.8407044966673
x2=53.4070751110265x_{2} = 53.4070751110265
x3=97.3893722612836x_{3} = 97.3893722612836
x4=34.5575191894877x_{4} = 34.5575191894877
x5=47.1238898038469x_{5} = 47.1238898038469
x6=−97.3893722612836x_{6} = -97.3893722612836
x7=−21.9911485751286x_{7} = -21.9911485751286
x8=3.14159265358979x_{8} = 3.14159265358979
x9=65.9734457253857x_{9} = 65.9734457253857
x10=−53.4070751110265x_{10} = -53.4070751110265
x11=−9.42477796076938x_{11} = -9.42477796076938
x12=−34.5575191894877x_{12} = -34.5575191894877
x13=−9591.28237140964x_{13} = -9591.28237140964
x14=21.9911485751286x_{14} = 21.9911485751286
x15=−47.1238898038469x_{15} = -47.1238898038469
x16=28.2743338823081x_{16} = 28.2743338823081
x17=−3.14159265358979x_{17} = -3.14159265358979
x18=−65.9734457253857x_{18} = -65.9734457253857
x19=72.2566310325652x_{19} = 72.2566310325652
x20=−59.6902604182061x_{20} = -59.6902604182061
x21=7517042.68028432x_{21} = 7517042.68028432
x22=−91.106186954104x_{22} = -91.106186954104
x23=59.6902604182061x_{23} = 59.6902604182061
x24=−40.8407044966673x_{24} = -40.8407044966673
x25=91.106186954104x_{25} = 91.106186954104
x26=78.5398163397448x_{26} = 78.5398163397448
x27=84.8230016469244x_{27} = 84.8230016469244
x28=9.42477796076938x_{28} = 9.42477796076938
x29=−160.221225333079x_{29} = -160.221225333079
x30=−84.8230016469244x_{30} = -84.8230016469244
x31=−78.5398163397448x_{31} = -78.5398163397448
x32=15.707963267949x_{32} = 15.707963267949
x33=−28.2743338823081x_{33} = -28.2743338823081
x34=−15.707963267949x_{34} = -15.707963267949
x35=−72.2566310325652x_{35} = -72.2566310325652
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(x/2).
cos⁡(02)\cos{\left(\frac{0}{2} \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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⁡(x2)2=0- \frac{\sin{\left(\frac{x}{2} \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=2πx_{2} = 2 \pi
The values of the extrema at the points:
(0, 1)

(2*pi, -1)


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=2πx_{1} = 2 \pi
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[2π,∞)\left(-\infty, 0\right] \cup \left[2 \pi, \infty\right)
Increasing at intervals
[0,2π]\left[0, 2 \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⁡(x2)4=0- \frac{\cos{\left(\frac{x}{2} \right)}}{4} = 0
Solve this equation
The roots of this equation
x1=πx_{1} = \pi
x2=3πx_{2} = 3 \pi

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
[π,3π]\left[\pi, 3 \pi\right]
Convex at the intervals
(−∞,π]∪[3π,∞)\left(-\infty, \pi\right] \cup \left[3 \pi, \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⁡(x2)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\left(\frac{x}{2} \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞cos⁡(x2)=⟨−1,1⟩\lim_{x \to \infty} \cos{\left(\frac{x}{2} \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(x/2), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(x2)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(\frac{x}{2} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(x2)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(\frac{x}{2} \right)}}{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⁡(x2)=cos⁡(x2)\cos{\left(\frac{x}{2} \right)} = \cos{\left(\frac{x}{2} \right)}
- No
cos⁡(x2)=−cos⁡(x2)\cos{\left(\frac{x}{2} \right)} = - \cos{\left(\frac{x}{2} \right)}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator