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Plotting a function graph/ 1.5cosx

Graphing y = 1.5cosx

v

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=14.1371669411541x_{1} = 14.1371669411541
x2=73.8274273593601x_{2} = 73.8274273593601
x3=−92.6769832808989x_{3} = -92.6769832808989
x4=−48.6946861306418x_{4} = -48.6946861306418
x5=89.5353906273091x_{5} = 89.5353906273091
x6=−2266.65909956504x_{6} = -2266.65909956504
x7=−23.5619449019235x_{7} = -23.5619449019235
x8=−86.3937979737193x_{8} = -86.3937979737193
x9=39.2699081698724x_{9} = 39.2699081698724
x10=−17.2787595947439x_{10} = -17.2787595947439
x11=20.4203522483337x_{11} = 20.4203522483337
x12=−26.7035375555132x_{12} = -26.7035375555132
x13=61.261056745001x_{13} = 61.261056745001
x14=42.4115008234622x_{14} = 42.4115008234622
x15=−64.4026493985908x_{15} = -64.4026493985908
x16=−83.2522053201295x_{16} = -83.2522053201295
x17=−4.71238898038469x_{17} = -4.71238898038469
x18=−42.4115008234622x_{18} = -42.4115008234622
x19=−29.845130209103x_{19} = -29.845130209103
x20=17.2787595947439x_{20} = 17.2787595947439
x21=51.8362787842316x_{21} = 51.8362787842316
x22=1.5707963267949x_{22} = 1.5707963267949
x23=−67.5442420521806x_{23} = -67.5442420521806
x24=−36.1283155162826x_{24} = -36.1283155162826
x25=45.553093477052x_{25} = 45.553093477052
x26=−80.1106126665397x_{26} = -80.1106126665397
x27=86.3937979737193x_{27} = 86.3937979737193
x28=−73.8274273593601x_{28} = -73.8274273593601
x29=32.9867228626928x_{29} = 32.9867228626928
x30=64.4026493985908x_{30} = 64.4026493985908
x31=−1.5707963267949x_{31} = -1.5707963267949
x32=95.8185759344887x_{32} = 95.8185759344887
x33=−20.4203522483337x_{33} = -20.4203522483337
x34=−10.9955742875643x_{34} = -10.9955742875643
x35=−98.9601685880785x_{35} = -98.9601685880785
x36=92.6769832808989x_{36} = 92.6769832808989
x37=36.1283155162826x_{37} = 36.1283155162826
x38=−32.9867228626928x_{38} = -32.9867228626928
x39=−39.2699081698724x_{39} = -39.2699081698724
x40=−58.1194640914112x_{40} = -58.1194640914112
x41=−61.261056745001x_{41} = -61.261056745001
x42=4.71238898038469x_{42} = 4.71238898038469
x43=−76.9690200129499x_{43} = -76.9690200129499
x44=−95.8185759344887x_{44} = -95.8185759344887
x45=48.6946861306418x_{45} = 48.6946861306418
x46=−51.8362787842316x_{46} = -51.8362787842316
x47=23.5619449019235x_{47} = 23.5619449019235
x48=67.5442420521806x_{48} = 67.5442420521806
x49=−14.1371669411541x_{49} = -14.1371669411541
x50=76.9690200129499x_{50} = 76.9690200129499
x51=98.9601685880785x_{51} = 98.9601685880785
x52=80.1106126665397x_{52} = 80.1106126665397
x53=−7.85398163397448x_{53} = -7.85398163397448
x54=7.85398163397448x_{54} = 7.85398163397448
x55=−387.986692718339x_{55} = -387.986692718339
x56=58.1194640914112x_{56} = 58.1194640914112
x57=−45.553093477052x_{57} = -45.553093477052
x58=83.2522053201295x_{58} = 83.2522053201295
x59=54.9778714378214x_{59} = 54.9778714378214
x60=26.7035375555132x_{60} = 26.7035375555132
x61=−89.5353906273091x_{61} = -89.5353906273091
x62=10.9955742875643x_{62} = 10.9955742875643
x63=−70.6858347057703x_{63} = -70.6858347057703
x64=70.6858347057703x_{64} = 70.6858347057703
x65=−54.9778714378214x_{65} = -54.9778714378214
x66=29.845130209103x_{66} = 29.845130209103
x67=−3626.96871856942x_{67} = -3626.96871856942
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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