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Plotting a function graph/ cos5x

Graphing y = cos5x

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = cos(5*x)
f(x)=cos⁡(5x)f{\left(x \right)} = \cos{\left(5 x \right)} 
f = cos(5*x)
The graph of the function
02468-8-6-4-2-10102-2
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:
cos⁡(5x)=0\cos{\left(5 x \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π10x_{1} = \frac{\pi}{10}
x2=3π10x_{2} = \frac{3 \pi}{10}
Numerical solution
x1=93.9336203423348x_{1} = 93.9336203423348
x2=−61.8893752757189x_{2} = -61.8893752757189
x3=95.1902574037707x_{3} = 95.1902574037707
x4=−48.0663675999238x_{4} = -48.0663675999238
x5=39.8982267005904x_{5} = 39.8982267005904
x6=14.1371669411541x_{6} = 14.1371669411541
x7=−63.7743308678728x_{7} = -63.7743308678728
x8=−36.1283155162826x_{8} = -36.1283155162826
x9=60.0044196835651x_{9} = 60.0044196835651
x10=24.1902634326414x_{10} = 24.1902634326414
x11=83.8805238508475x_{11} = 83.8805238508475
x12=−70.0575161750524x_{12} = -70.0575161750524
x13=72.5707902979242x_{13} = 72.5707902979242
x14=−21.6769893097696x_{14} = -21.6769893097696
x15=16.0221225333079x_{15} = 16.0221225333079
x16=−27.9601746169492x_{16} = -27.9601746169492
x17=−31.7300858012569x_{17} = -31.7300858012569
x18=−49.9513231920777x_{18} = -49.9513231920777
x19=68.1725605828985x_{19} = 68.1725605828985
x20=−51.8362787842316x_{20} = -51.8362787842316
x21=−14.1371669411541x_{21} = -14.1371669411541
x22=66.2876049907446x_{22} = 66.2876049907446
x23=−9.73893722612836x_{23} = -9.73893722612836
x24=88.9070720965912x_{24} = 88.9070720965912
x25=78.2256570743859x_{25} = 78.2256570743859
x26=27.9601746169492x_{26} = 27.9601746169492
x27=22.3053078404875x_{27} = 22.3053078404875
x28=90.1637091580271x_{28} = 90.1637091580271
x29=12.2522113490002x_{29} = 12.2522113490002
x30=70.0575161750524x_{30} = 70.0575161750524
x31=32.3584043319749x_{31} = 32.3584043319749
x32=2.19911485751286x_{32} = 2.19911485751286
x33=−5.96902604182061x_{33} = -5.96902604182061
x34=48.0663675999238x_{34} = 48.0663675999238
x35=26.0752190247953x_{35} = 26.0752190247953
x36=−80.1106126665397x_{36} = -80.1106126665397
x37=−29.845130209103x_{37} = -29.845130209103
x38=−60.0044196835651x_{38} = -60.0044196835651
x39=−69.4291976443344x_{39} = -69.4291976443344
x40=−81.9955682586936x_{40} = -81.9955682586936
x41=4.08407044966673x_{41} = 4.08407044966673
x42=98.3318500573605x_{42} = 98.3318500573605
x43=−4.08407044966673x_{43} = -4.08407044966673
x44=56.2345084992573x_{44} = 56.2345084992573
x45=−76.9690200129499x_{45} = -76.9690200129499
x46=44.2964564156161x_{46} = 44.2964564156161
x47=−71.9424717672063x_{47} = -71.9424717672063
x48=61.8893752757189x_{48} = 61.8893752757189
x49=88.2787535658732x_{49} = 88.2787535658732
x50=−19.7920337176157x_{50} = -19.7920337176157
x51=−87.6504350351552x_{51} = -87.6504350351552
x52=−39.8982267005904x_{52} = -39.8982267005904
x53=36.1283155162826x_{53} = 36.1283155162826
x54=−75.712382951514x_{54} = -75.712382951514
x55=−93.9336203423348x_{55} = -93.9336203423348
x56=80.1106126665397x_{56} = 80.1106126665397
x57=58.1194640914112x_{57} = 58.1194640914112
x58=−38.0132711084365x_{58} = -38.0132711084365
x59=−102.730079772386x_{59} = -102.730079772386
x60=−58.1194640914112x_{60} = -58.1194640914112
x61=54.3495529071034x_{61} = 54.3495529071034
x62=100.216805649514x_{62} = 100.216805649514
x63=−16.0221225333079x_{63} = -16.0221225333079
x64=92.0486647501809x_{64} = 92.0486647501809
x65=71.9424717672063x_{65} = 71.9424717672063
x66=−53.7212343763855x_{66} = -53.7212343763855
x67=−95.8185759344887x_{67} = -95.8185759344887
x68=34.2433599241287x_{68} = 34.2433599241287
x69=−43.6681378848981x_{69} = -43.6681378848981
x70=−103.358398303104x_{70} = -103.358398303104
x71=−53.0929158456675x_{71} = -53.0929158456675
x72=27.3318560862312x_{72} = 27.3318560862312
x73=−41.7831822927443x_{73} = -41.7831822927443
x74=46.18141200777x_{74} = 46.18141200777
x75=−17.9070781254618x_{75} = -17.9070781254618
x76=7.22566310325652x_{76} = 7.22566310325652
x77=65.0309679293087x_{77} = 65.0309679293087
x78=0.314159265358979x_{78} = 0.314159265358979
x79=5.96902604182061x_{79} = 5.96902604182061
x80=38.0132711084365x_{80} = 38.0132711084365
x81=−83.8805238508475x_{81} = -83.8805238508475
x82=−97.7035315266426x_{82} = -97.7035315266426
x83=21.0486707790516x_{83} = 21.0486707790516
x84=10.3672557568463x_{84} = 10.3672557568463
x85=76.340701482232x_{85} = 76.340701482232
x86=−73.8274273593601x_{86} = -73.8274273593601
x87=−65.6592864600267x_{87} = -65.6592864600267
x88=−11.6238928182822x_{88} = -11.6238928182822
x89=−92.0486647501809x_{89} = -92.0486647501809
x90=81.9955682586936x_{90} = 81.9955682586936
x91=−26.0752190247953x_{91} = -26.0752190247953
x92=−85.7654794430014x_{92} = -85.7654794430014
x93=17.9070781254618x_{93} = 17.9070781254618
x94=−7.85398163397448x_{94} = -7.85398163397448
x95=49.9513231920777x_{95} = 49.9513231920777
x96=−33.6150413934108x_{96} = -33.6150413934108
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to cos(5*x).
cos⁡(0⋅5)\cos{\left(0 \cdot 5 \right)}
The result:
f(0)=1f{\left(0 \right)} = 1
The point:
(0, 1)
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⁡(5x)=0- 5 \sin{\left(5 x \right)} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π5x_{2} = \frac{\pi}{5}
The values of the extrema at the points:
(0, 1)

 pi     
(--, -1)
 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=π5x_{1} = \frac{\pi}{5}
Maxima of the function at points:
x1=0x_{1} = 0
Decreasing at intervals
(−∞,0]∪[π5,∞)\left(-\infty, 0\right] \cup \left[\frac{\pi}{5}, \infty\right)
Increasing at intervals
[0,π5]\left[0, \frac{\pi}{5}\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
−25cos⁡(5x)=0- 25 \cos{\left(5 x \right)} = 0
Solve this equation
The roots of this equation
x1=π10x_{1} = \frac{\pi}{10}
x2=3π10x_{2} = \frac{3 \pi}{10}

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
[π10,3π10]\left[\frac{\pi}{10}, \frac{3 \pi}{10}\right]
Convex at the intervals
(−∞,π10]∪[3π10,∞)\left(-\infty, \frac{\pi}{10}\right] \cup \left[\frac{3 \pi}{10}, \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→−∞cos⁡(5x)=⟨−1,1⟩\lim_{x \to -\infty} \cos{\left(5 x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
lim⁡x→∞cos⁡(5x)=⟨−1,1⟩\lim_{x \to \infty} \cos{\left(5 x \right)} = \left\langle -1, 1\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−1,1⟩y = \left\langle -1, 1\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(5*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(5x)x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(5 x \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(5x)x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(5 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:
cos⁡(5x)=cos⁡(5x)\cos{\left(5 x \right)} = \cos{\left(5 x \right)}
- Yes
cos⁡(5x)=−cos⁡(5x)\cos{\left(5 x \right)} = - \cos{\left(5 x \right)}
- No
so, the function
is
even
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