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

Graphing y = 3*cos(x)

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = 3*cos(x)
f(x)=3cos⁡(x)f{\left(x \right)} = 3 \cos{\left(x \right)} 
f = 3*cos(x)
The graph of the function
0-30-20-101020304050607080905-5
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)=03 \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=54.9778714378214x_{1} = 54.9778714378214
x2=95.8185759344887x_{2} = 95.8185759344887
x3=−39.2699081698724x_{3} = -39.2699081698724
x4=−29.845130209103x_{4} = -29.845130209103
x5=−54.9778714378214x_{5} = -54.9778714378214
x6=−73.8274273593601x_{6} = -73.8274273593601
x7=−7.85398163397448x_{7} = -7.85398163397448
x8=92.6769832808989x_{8} = 92.6769832808989
x9=80.1106126665397x_{9} = 80.1106126665397
x10=86.3937979737193x_{10} = 86.3937979737193
x11=−76.9690200129499x_{11} = -76.9690200129499
x12=−45.553093477052x_{12} = -45.553093477052
x13=61.261056745001x_{13} = 61.261056745001
x14=−48.6946861306418x_{14} = -48.6946861306418
x15=32.9867228626928x_{15} = 32.9867228626928
x16=−3626.96871856942x_{16} = -3626.96871856942
x17=20.4203522483337x_{17} = 20.4203522483337
x18=23.5619449019235x_{18} = 23.5619449019235
x19=−86.3937979737193x_{19} = -86.3937979737193
x20=−23.5619449019235x_{20} = -23.5619449019235
x21=−17.2787595947439x_{21} = -17.2787595947439
x22=−387.986692718339x_{22} = -387.986692718339
x23=−67.5442420521806x_{23} = -67.5442420521806
x24=−89.5353906273091x_{24} = -89.5353906273091
x25=−32.9867228626928x_{25} = -32.9867228626928
x26=64.4026493985908x_{26} = 64.4026493985908
x27=4.71238898038469x_{27} = 4.71238898038469
x28=−10.9955742875643x_{28} = -10.9955742875643
x29=−20.4203522483337x_{29} = -20.4203522483337
x30=−80.1106126665397x_{30} = -80.1106126665397
x31=−64.4026493985908x_{31} = -64.4026493985908
x32=−14.1371669411541x_{32} = -14.1371669411541
x33=−26.7035375555132x_{33} = -26.7035375555132
x34=10.9955742875643x_{34} = 10.9955742875643
x35=58.1194640914112x_{35} = 58.1194640914112
x36=−83.2522053201295x_{36} = -83.2522053201295
x37=26.7035375555132x_{37} = 26.7035375555132
x38=−70.6858347057703x_{38} = -70.6858347057703
x39=48.6946861306418x_{39} = 48.6946861306418
x40=−42.4115008234622x_{40} = -42.4115008234622
x41=−2266.65909956504x_{41} = -2266.65909956504
x42=70.6858347057703x_{42} = 70.6858347057703
x43=−92.6769832808989x_{43} = -92.6769832808989
x44=7.85398163397448x_{44} = 7.85398163397448
x45=−51.8362787842316x_{45} = -51.8362787842316
x46=98.9601685880785x_{46} = 98.9601685880785
x47=42.4115008234622x_{47} = 42.4115008234622
x48=51.8362787842316x_{48} = 51.8362787842316
x49=−58.1194640914112x_{49} = -58.1194640914112
x50=−61.261056745001x_{50} = -61.261056745001
x51=39.2699081698724x_{51} = 39.2699081698724
x52=45.553093477052x_{52} = 45.553093477052
x53=29.845130209103x_{53} = 29.845130209103
x54=−4.71238898038469x_{54} = -4.71238898038469
x55=17.2787595947439x_{55} = 17.2787595947439
x56=89.5353906273091x_{56} = 89.5353906273091
x57=1.5707963267949x_{57} = 1.5707963267949
x58=83.2522053201295x_{58} = 83.2522053201295
x59=−36.1283155162826x_{59} = -36.1283155162826
x60=−95.8185759344887x_{60} = -95.8185759344887
x61=36.1283155162826x_{61} = 36.1283155162826
x62=67.5442420521806x_{62} = 67.5442420521806
x63=73.8274273593601x_{63} = 73.8274273593601
x64=76.9690200129499x_{64} = 76.9690200129499
x65=−1.5707963267949x_{65} = -1.5707963267949
x66=−98.9601685880785x_{66} = -98.9601685880785
x67=14.1371669411541x_{67} = 14.1371669411541
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).
3cos⁡(0)3 \cos{\left(0 \right)}
The result:
f(0)=3f{\left(0 \right)} = 3
The point:
(0, 3)
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)=0- 3 \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, 3)

(pi, -3)


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)=0- 3 \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→−∞(3cos⁡(x))=⟨−3,3⟩\lim_{x \to -\infty}\left(3 \cos{\left(x \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
lim⁡x→∞(3cos⁡(x))=⟨−3,3⟩\lim_{x \to \infty}\left(3 \cos{\left(x \right)}\right) = \left\langle -3, 3\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−3,3⟩y = \left\langle -3, 3\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 3*cos(x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(3cos⁡(x)x)=0\lim_{x \to -\infty}\left(\frac{3 \cos{\left(x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(3cos⁡(x)x)=0\lim_{x \to \infty}\left(\frac{3 \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:
3cos⁡(x)=3cos⁡(x)3 \cos{\left(x \right)} = 3 \cos{\left(x \right)}
- Yes
3cos⁡(x)=−3cos⁡(x)3 \cos{\left(x \right)} = - 3 \cos{\left(x \right)}
- No
so, the function
is
even
The graph
Graphing y = 3*cos(x)
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