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Plotting a function graph/ 4*cos(x)

Graphing y = 4*cos(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 4*cos(x)
f(x)=4cos⁡(x)f{\left(x \right)} = 4 \cos{\left(x \right)} 
f = 4*cos(x)
The graph of the function
02468-8-6-4-2-1010-1010
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:
4cos⁡(x)=04 \cos{\left(x \right)} = 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=−67.5442420521806x_{3} = -67.5442420521806
x4=−70.6858347057703x_{4} = -70.6858347057703
x5=64.4026493985908x_{5} = 64.4026493985908
x6=−36.1283155162826x_{6} = -36.1283155162826
x7=−92.6769832808989x_{7} = -92.6769832808989
x8=−61.261056745001x_{8} = -61.261056745001
x9=−76.9690200129499x_{9} = -76.9690200129499
x10=−98.9601685880785x_{10} = -98.9601685880785
x11=−95.8185759344887x_{11} = -95.8185759344887
x12=29.845130209103x_{12} = 29.845130209103
x13=80.1106126665397x_{13} = 80.1106126665397
x14=−64.4026493985908x_{14} = -64.4026493985908
x15=36.1283155162826x_{15} = 36.1283155162826
x16=73.8274273593601x_{16} = 73.8274273593601
x17=−2266.65909956504x_{17} = -2266.65909956504
x18=32.9867228626928x_{18} = 32.9867228626928
x19=−4.71238898038469x_{19} = -4.71238898038469
x20=−39.2699081698724x_{20} = -39.2699081698724
x21=26.7035375555132x_{21} = 26.7035375555132
x22=−7.85398163397448x_{22} = -7.85398163397448
x23=95.8185759344887x_{23} = 95.8185759344887
x24=−17.2787595947439x_{24} = -17.2787595947439
x25=−10.9955742875643x_{25} = -10.9955742875643
x26=98.9601685880785x_{26} = 98.9601685880785
x27=−86.3937979737193x_{27} = -86.3937979737193
x28=92.6769832808989x_{28} = 92.6769832808989
x29=−48.6946861306418x_{29} = -48.6946861306418
x30=54.9778714378214x_{30} = 54.9778714378214
x31=45.553093477052x_{31} = 45.553093477052
x32=23.5619449019235x_{32} = 23.5619449019235
x33=76.9690200129499x_{33} = 76.9690200129499
x34=−89.5353906273091x_{34} = -89.5353906273091
x35=4.71238898038469x_{35} = 4.71238898038469
x36=−26.7035375555132x_{36} = -26.7035375555132
x37=−80.1106126665397x_{37} = -80.1106126665397
x38=7.85398163397448x_{38} = 7.85398163397448
x39=14.1371669411541x_{39} = 14.1371669411541
x40=86.3937979737193x_{40} = 86.3937979737193
x41=−45.553093477052x_{41} = -45.553093477052
x42=−83.2522053201295x_{42} = -83.2522053201295
x43=70.6858347057703x_{43} = 70.6858347057703
x44=83.2522053201295x_{44} = 83.2522053201295
x45=48.6946861306418x_{45} = 48.6946861306418
x46=−20.4203522483337x_{46} = -20.4203522483337
x47=51.8362787842316x_{47} = 51.8362787842316
x48=10.9955742875643x_{48} = 10.9955742875643
x49=20.4203522483337x_{49} = 20.4203522483337
x50=1.5707963267949x_{50} = 1.5707963267949
x51=89.5353906273091x_{51} = 89.5353906273091
x52=17.2787595947439x_{52} = 17.2787595947439
x53=58.1194640914112x_{53} = 58.1194640914112
x54=61.261056745001x_{54} = 61.261056745001
x55=−32.9867228626928x_{55} = -32.9867228626928
x56=−51.8362787842316x_{56} = -51.8362787842316
x57=−14.1371669411541x_{57} = -14.1371669411541
x58=−58.1194640914112x_{58} = -58.1194640914112
x59=−42.4115008234622x_{59} = -42.4115008234622
x60=−54.9778714378214x_{60} = -54.9778714378214
x61=−1.5707963267949x_{61} = -1.5707963267949
x62=42.4115008234622x_{62} = 42.4115008234622
x63=39.2699081698724x_{63} = 39.2699081698724
x64=67.5442420521806x_{64} = 67.5442420521806
x65=−23.5619449019235x_{65} = -23.5619449019235
x66=−168.075206967054x_{66} = -168.075206967054
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 4*cos(x).
4cos⁡(0)4 \cos{\left(0 \right)}
The result:
f(0)=4f{\left(0 \right)} = 4
The point:
(0, 4)
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
−4sin⁡(x)=0- 4 \sin{\left(x \right)} = 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, 4)

(pi, -4)


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
−4cos⁡(x)=0- 4 \cos{\left(x \right)} = 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→−∞(4cos⁡(x))=⟨−4,4⟩\lim_{x \to -\infty}\left(4 \cos{\left(x \right)}\right) = \left\langle -4, 4\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−4,4⟩y = \left\langle -4, 4\right\rangle
lim⁡x→∞(4cos⁡(x))=⟨−4,4⟩\lim_{x \to \infty}\left(4 \cos{\left(x \right)}\right) = \left\langle -4, 4\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−4,4⟩y = \left\langle -4, 4\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 4*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(4cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{4 \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(4cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{4 \cos{\left(x \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:
4cos⁡(x)=4cos⁡(x)4 \cos{\left(x \right)} = 4 \cos{\left(x \right)}
- Yes
4cos⁡(x)=−4cos⁡(x)4 \cos{\left(x \right)} = - 4 \cos{\left(x \right)}
- No
so, the function
is
even
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