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

Graphing y = 3*cos(x)/2

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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       3*cos(x)
f(x) = --------
          2
f(x)=3cos⁡(x)2f{\left(x \right)} = \frac{3 \cos{\left(x \right)}}{2} 
f = (3*cos(x))/2
The graph of the function
02468-8-6-4-2-10105-5
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:
3cos⁡(x)2=0\frac{3 \cos{\left(x \right)}}{2} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
Numerical solution
x1=−387.986692718339x_{1} = -387.986692718339
x2=−29.845130209103x_{2} = -29.845130209103
x3=−3626.96871856942x_{3} = -3626.96871856942
x4=−67.5442420521806x_{4} = -67.5442420521806
x5=−70.6858347057703x_{5} = -70.6858347057703
x6=64.4026493985908x_{6} = 64.4026493985908
x7=−36.1283155162826x_{7} = -36.1283155162826
x8=−92.6769832808989x_{8} = -92.6769832808989
x9=−61.261056745001x_{9} = -61.261056745001
x10=−76.9690200129499x_{10} = -76.9690200129499
x11=−98.9601685880785x_{11} = -98.9601685880785
x12=−95.8185759344887x_{12} = -95.8185759344887
x13=29.845130209103x_{13} = 29.845130209103
x14=80.1106126665397x_{14} = 80.1106126665397
x15=−64.4026493985908x_{15} = -64.4026493985908
x16=36.1283155162826x_{16} = 36.1283155162826
x17=73.8274273593601x_{17} = 73.8274273593601
x18=−2266.65909956504x_{18} = -2266.65909956504
x19=32.9867228626928x_{19} = 32.9867228626928
x20=−4.71238898038469x_{20} = -4.71238898038469
x21=−39.2699081698724x_{21} = -39.2699081698724
x22=26.7035375555132x_{22} = 26.7035375555132
x23=−7.85398163397448x_{23} = -7.85398163397448
x24=95.8185759344887x_{24} = 95.8185759344887
x25=−17.2787595947439x_{25} = -17.2787595947439
x26=−10.9955742875643x_{26} = -10.9955742875643
x27=98.9601685880785x_{27} = 98.9601685880785
x28=−86.3937979737193x_{28} = -86.3937979737193
x29=92.6769832808989x_{29} = 92.6769832808989
x30=−48.6946861306418x_{30} = -48.6946861306418
x31=54.9778714378214x_{31} = 54.9778714378214
x32=45.553093477052x_{32} = 45.553093477052
x33=23.5619449019235x_{33} = 23.5619449019235
x34=76.9690200129499x_{34} = 76.9690200129499
x35=−89.5353906273091x_{35} = -89.5353906273091
x36=4.71238898038469x_{36} = 4.71238898038469
x37=−26.7035375555132x_{37} = -26.7035375555132
x38=−80.1106126665397x_{38} = -80.1106126665397
x39=7.85398163397448x_{39} = 7.85398163397448
x40=14.1371669411541x_{40} = 14.1371669411541
x41=86.3937979737193x_{41} = 86.3937979737193
x42=−45.553093477052x_{42} = -45.553093477052
x43=−83.2522053201295x_{43} = -83.2522053201295
x44=70.6858347057703x_{44} = 70.6858347057703
x45=83.2522053201295x_{45} = 83.2522053201295
x46=48.6946861306418x_{46} = 48.6946861306418
x47=−20.4203522483337x_{47} = -20.4203522483337
x48=51.8362787842316x_{48} = 51.8362787842316
x49=10.9955742875643x_{49} = 10.9955742875643
x50=20.4203522483337x_{50} = 20.4203522483337
x51=1.5707963267949x_{51} = 1.5707963267949
x52=89.5353906273091x_{52} = 89.5353906273091
x53=17.2787595947439x_{53} = 17.2787595947439
x54=58.1194640914112x_{54} = 58.1194640914112
x55=61.261056745001x_{55} = 61.261056745001
x56=−32.9867228626928x_{56} = -32.9867228626928
x57=−51.8362787842316x_{57} = -51.8362787842316
x58=−14.1371669411541x_{58} = -14.1371669411541
x59=−58.1194640914112x_{59} = -58.1194640914112
x60=−42.4115008234622x_{60} = -42.4115008234622
x61=−54.9778714378214x_{61} = -54.9778714378214
x62=−1.5707963267949x_{62} = -1.5707963267949
x63=42.4115008234622x_{63} = 42.4115008234622
x64=39.2699081698724x_{64} = 39.2699081698724
x65=67.5442420521806x_{65} = 67.5442420521806
x66=−23.5619449019235x_{66} = -23.5619449019235
x67=−73.8274273593601x_{67} = -73.8274273593601
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (3*cos(x))/2.
3cos⁡(0)2\frac{3 \cos{\left(0 \right)}}{2}
The result:
f(0)=32f{\left(0 \right)} = \frac{3}{2}
The point:
(0, 3/2)
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
−3sin⁡(x)2=0- \frac{3 \sin{\left(x \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=πx_{2} = \pi
The values of the extrema at the points:
(0, 3/2)

(pi, -3/2)


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=πx_{1} = \pi
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π,∞)\left(-\infty, 0\right] \cup \left[\pi, \infty\right)
Increasing at intervals
[0,π]\left[0, \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
−3cos⁡(x)2=0- \frac{3 \cos{\left(x \right)}}{2} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \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
[π2,3π2]\left[\frac{\pi}{2}, \frac{3 \pi}{2}\right]
Convex at the intervals
(−∞,π2]∪[3π2,∞)\left(-\infty, \frac{\pi}{2}\right] \cup \left[\frac{3 \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→−∞(3cos⁡(x)2)=⟨−32,32⟩\lim_{x \to -\infty}\left(\frac{3 \cos{\left(x \right)}}{2}\right) = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−32,32⟩y = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
lim⁡x→∞(3cos⁡(x)2)=⟨−32,32⟩\lim_{x \to \infty}\left(\frac{3 \cos{\left(x \right)}}{2}\right) = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−32,32⟩y = \left\langle - \frac{3}{2}, \frac{3}{2}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (3*cos(x))/2, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3cos⁡(x)2x)=0\lim_{x \to -\infty}\left(\frac{3 \cos{\left(x \right)}}{2 x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3cos⁡(x)2x)=0\lim_{x \to \infty}\left(\frac{3 \cos{\left(x \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:
3cos⁡(x)2=3cos⁡(x)2\frac{3 \cos{\left(x \right)}}{2} = \frac{3 \cos{\left(x \right)}}{2}
- Yes
3cos⁡(x)2=−3cos⁡(x)2\frac{3 \cos{\left(x \right)}}{2} = - \frac{3 \cos{\left(x \right)}}{2}
- No
so, the function
is
even
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