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Plotting a function graph/ 5*cos(x)

Graphing y = 5*cos(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 5*cos(x)
f(x)=5cos⁡(x)f{\left(x \right)} = 5 \cos{\left(x \right)} 
f = 5*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:
5cos⁡(x)=05 \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=48.6946861306418x_{1} = 48.6946861306418
x2=92.6769832808989x_{2} = 92.6769832808989
x3=86.3937979737193x_{3} = 86.3937979737193
x4=−7.85398163397448x_{4} = -7.85398163397448
x5=−86.3937979737193x_{5} = -86.3937979737193
x6=1.5707963267949x_{6} = 1.5707963267949
x7=−64.4026493985908x_{7} = -64.4026493985908
x8=−58.1194640914112x_{8} = -58.1194640914112
x9=−83.2522053201295x_{9} = -83.2522053201295
x10=−54.9778714378214x_{10} = -54.9778714378214
x11=54.9778714378214x_{11} = 54.9778714378214
x12=89.5353906273091x_{12} = 89.5353906273091
x13=−20.4203522483337x_{13} = -20.4203522483337
x14=32.9867228626928x_{14} = 32.9867228626928
x15=−17.2787595947439x_{15} = -17.2787595947439
x16=23.5619449019235x_{16} = 23.5619449019235
x17=−45.553093477052x_{17} = -45.553093477052
x18=64.4026493985908x_{18} = 64.4026493985908
x19=45.553093477052x_{19} = 45.553093477052
x20=83.2522053201295x_{20} = 83.2522053201295
x21=−29.845130209103x_{21} = -29.845130209103
x22=−51.8362787842316x_{22} = -51.8362787842316
x23=80.1106126665397x_{23} = 80.1106126665397
x24=−39.2699081698724x_{24} = -39.2699081698724
x25=−92.6769832808989x_{25} = -92.6769832808989
x26=4.71238898038469x_{26} = 4.71238898038469
x27=70.6858347057703x_{27} = 70.6858347057703
x28=36.1283155162826x_{28} = 36.1283155162826
x29=−70.6858347057703x_{29} = -70.6858347057703
x30=−48.6946861306418x_{30} = -48.6946861306418
x31=42.4115008234622x_{31} = 42.4115008234622
x32=−2266.65909956504x_{32} = -2266.65909956504
x33=−42.4115008234622x_{33} = -42.4115008234622
x34=−67.5442420521806x_{34} = -67.5442420521806
x35=10.9955742875643x_{35} = 10.9955742875643
x36=98.9601685880785x_{36} = 98.9601685880785
x37=−23.5619449019235x_{37} = -23.5619449019235
x38=20.4203522483337x_{38} = 20.4203522483337
x39=−61.261056745001x_{39} = -61.261056745001
x40=−10.9955742875643x_{40} = -10.9955742875643
x41=17.2787595947439x_{41} = 17.2787595947439
x42=−95.8185759344887x_{42} = -95.8185759344887
x43=−36.1283155162826x_{43} = -36.1283155162826
x44=61.261056745001x_{44} = 61.261056745001
x45=73.8274273593601x_{45} = 73.8274273593601
x46=−3639.53508918378x_{46} = -3639.53508918378
x47=14.1371669411541x_{47} = 14.1371669411541
x48=−26.7035375555132x_{48} = -26.7035375555132
x49=51.8362787842316x_{49} = 51.8362787842316
x50=−89.5353906273091x_{50} = -89.5353906273091
x51=39.2699081698724x_{51} = 39.2699081698724
x52=−387.986692718339x_{52} = -387.986692718339
x53=−32.9867228626928x_{53} = -32.9867228626928
x54=−14.1371669411541x_{54} = -14.1371669411541
x55=−4.71238898038469x_{55} = -4.71238898038469
x56=−76.9690200129499x_{56} = -76.9690200129499
x57=95.8185759344887x_{57} = 95.8185759344887
x58=−45375.5934921242x_{58} = -45375.5934921242
x59=76.9690200129499x_{59} = 76.9690200129499
x60=58.1194640914112x_{60} = 58.1194640914112
x61=−80.1106126665397x_{61} = -80.1106126665397
x62=−73.8274273593601x_{62} = -73.8274273593601
x63=7.85398163397448x_{63} = 7.85398163397448
x64=−1.5707963267949x_{64} = -1.5707963267949
x65=29.845130209103x_{65} = 29.845130209103
x66=67.5442420521806x_{66} = 67.5442420521806
x67=26.7035375555132x_{67} = 26.7035375555132
x68=−98.9601685880785x_{68} = -98.9601685880785
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 5*cos(x).
5cos⁡(0)5 \cos{\left(0 \right)}
The result:
f(0)=5f{\left(0 \right)} = 5
The point:
(0, 5)
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
−5sin⁡(x)=0- 5 \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, 5)

(pi, -5)


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
−5cos⁡(x)=0- 5 \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→−∞(5cos⁡(x))=⟨−5,5⟩\lim_{x \to -\infty}\left(5 \cos{\left(x \right)}\right) = \left\langle -5, 5\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−5,5⟩y = \left\langle -5, 5\right\rangle
lim⁡x→∞(5cos⁡(x))=⟨−5,5⟩\lim_{x \to \infty}\left(5 \cos{\left(x \right)}\right) = \left\langle -5, 5\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−5,5⟩y = \left\langle -5, 5\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 5*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(5cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{5 \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(5cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{5 \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:
5cos⁡(x)=5cos⁡(x)5 \cos{\left(x \right)} = 5 \cos{\left(x \right)}
- Yes
5cos⁡(x)=−5cos⁡(x)5 \cos{\left(x \right)} = - 5 \cos{\left(x \right)}
- No
so, the function
is
even
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