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  • Graphing y =:
  • 3x^4-4x^3
  • 3x^2-12x+1
  • 2x^2+5x+2
  • 2x^2-3x+4

  • Identical expressions

  • one / five *cos(five *x)
  • 1 divide by 5 multiply by co sinus of e of (5 multiply by x)
  • one divide by five multiply by co sinus of e of (five multiply by x)
  • 1/5cos(5x)
  • 1/5cos5x
  • 1 divide by 5*cos(5*x)
Plotting a function graph/ 1/5*cos(5*x)

Graphing y = 1/5*cos(5*x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
       cos(5*x)
f(x) = --------
          5
f(x)=cos⁡(5x)5f{\left(x \right)} = \frac{\cos{\left(5 x \right)}}{5} 
f = cos(5*x)/5
The graph of the function
02468-8-6-4-2-10100.5-0.5
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:
cos⁡(5x)5=0\frac{\cos{\left(5 x \right)}}{5} = 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=−26.0752190247953x_{1} = -26.0752190247953
x2=−81.9955682586936x_{2} = -81.9955682586936
x3=−97.7035315266426x_{3} = -97.7035315266426
x4=92.0486647501809x_{4} = 92.0486647501809
x5=14.1371669411541x_{5} = 14.1371669411541
x6=58.1194640914112x_{6} = 58.1194640914112
x7=24.1902634326414x_{7} = 24.1902634326414
x8=17.9070781254618x_{8} = 17.9070781254618
x9=95.1902574037707x_{9} = 95.1902574037707
x10=38.0132711084365x_{10} = 38.0132711084365
x11=36.1283155162826x_{11} = 36.1283155162826
x12=−7.85398163397448x_{12} = -7.85398163397448
x13=−58.1194640914112x_{13} = -58.1194640914112
x14=−102.730079772386x_{14} = -102.730079772386
x15=46.18141200777x_{15} = 46.18141200777
x16=100.216805649514x_{16} = 100.216805649514
x17=27.3318560862312x_{17} = 27.3318560862312
x18=12.2522113490002x_{18} = 12.2522113490002
x19=−69.4291976443344x_{19} = -69.4291976443344
x20=−11.6238928182822x_{20} = -11.6238928182822
x21=0.314159265358979x_{21} = 0.314159265358979
x22=−75.712382951514x_{22} = -75.712382951514
x23=−36.1283155162826x_{23} = -36.1283155162826
x24=61.8893752757189x_{24} = 61.8893752757189
x25=32.3584043319749x_{25} = 32.3584043319749
x26=−65.6592864600267x_{26} = -65.6592864600267
x27=44.2964564156161x_{27} = 44.2964564156161
x28=−85.7654794430014x_{28} = -85.7654794430014
x29=−4.08407044966673x_{29} = -4.08407044966673
x30=−14.1371669411541x_{30} = -14.1371669411541
x31=80.1106126665397x_{31} = 80.1106126665397
x32=78.2256570743859x_{32} = 78.2256570743859
x33=−61.8893752757189x_{33} = -61.8893752757189
x34=56.2345084992573x_{34} = 56.2345084992573
x35=−93.9336203423348x_{35} = -93.9336203423348
x36=−92.0486647501809x_{36} = -92.0486647501809
x37=54.3495529071034x_{37} = 54.3495529071034
x38=90.1637091580271x_{38} = 90.1637091580271
x39=93.9336203423348x_{39} = 93.9336203423348
x40=72.5707902979242x_{40} = 72.5707902979242
x41=83.8805238508475x_{41} = 83.8805238508475
x42=−51.8362787842316x_{42} = -51.8362787842316
x43=−103.358398303104x_{43} = -103.358398303104
x44=−29.845130209103x_{44} = -29.845130209103
x45=60.0044196835651x_{45} = 60.0044196835651
x46=−39.8982267005904x_{46} = -39.8982267005904
x47=−60.0044196835651x_{47} = -60.0044196835651
x48=5.96902604182061x_{48} = 5.96902604182061
x49=−31.7300858012569x_{49} = -31.7300858012569
x50=7.22566310325652x_{50} = 7.22566310325652
x51=16.0221225333079x_{51} = 16.0221225333079
x52=21.0486707790516x_{52} = 21.0486707790516
x53=76.340701482232x_{53} = 76.340701482232
x54=−95.8185759344887x_{54} = -95.8185759344887
x55=65.0309679293087x_{55} = 65.0309679293087
x56=68.1725605828985x_{56} = 68.1725605828985
x57=70.0575161750524x_{57} = 70.0575161750524
x58=−83.8805238508475x_{58} = -83.8805238508475
x59=22.3053078404875x_{59} = 22.3053078404875
x60=−53.0929158456675x_{60} = -53.0929158456675
x61=66.2876049907446x_{61} = 66.2876049907446
x62=−49.9513231920777x_{62} = -49.9513231920777
x63=−5.96902604182061x_{63} = -5.96902604182061
x64=−43.6681378848981x_{64} = -43.6681378848981
x65=71.9424717672063x_{65} = 71.9424717672063
x66=48.0663675999238x_{66} = 48.0663675999238
x67=−9.73893722612836x_{67} = -9.73893722612836
x68=−63.7743308678728x_{68} = -63.7743308678728
x69=98.3318500573605x_{69} = 98.3318500573605
x70=88.2787535658732x_{70} = 88.2787535658732
x71=−17.9070781254618x_{71} = -17.9070781254618
x72=27.9601746169492x_{72} = 27.9601746169492
x73=34.2433599241287x_{73} = 34.2433599241287
x74=−16.0221225333079x_{74} = -16.0221225333079
x75=−76.9690200129499x_{75} = -76.9690200129499
x76=−73.8274273593601x_{76} = -73.8274273593601
x77=−19.7920337176157x_{77} = -19.7920337176157
x78=88.9070720965912x_{78} = 88.9070720965912
x79=81.9955682586936x_{79} = 81.9955682586936
x80=49.9513231920777x_{80} = 49.9513231920777
x81=−87.6504350351552x_{81} = -87.6504350351552
x82=26.0752190247953x_{82} = 26.0752190247953
x83=−21.6769893097696x_{83} = -21.6769893097696
x84=−71.9424717672063x_{84} = -71.9424717672063
x85=2.19911485751286x_{85} = 2.19911485751286
x86=−27.9601746169492x_{86} = -27.9601746169492
x87=10.3672557568463x_{87} = 10.3672557568463
x88=39.8982267005904x_{88} = 39.8982267005904
x89=−33.6150413934108x_{89} = -33.6150413934108
x90=−70.0575161750524x_{90} = -70.0575161750524
x91=−41.7831822927443x_{91} = -41.7831822927443
x92=4.08407044966673x_{92} = 4.08407044966673
x93=−53.7212343763855x_{93} = -53.7212343763855
x94=−80.1106126665397x_{94} = -80.1106126665397
x95=−38.0132711084365x_{95} = -38.0132711084365
x96=−48.0663675999238x_{96} = -48.0663675999238
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)/5.
cos⁡(0⋅5)5\frac{\cos{\left(0 \cdot 5 \right)}}{5}
The result:
f(0)=15f{\left(0 \right)} = \frac{1}{5}
The point:
(0, 1/5)
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
−sin⁡(5x)=0- \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, 1/5)

 pi       
(--, -1/5)
 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
−5cos⁡(5x)=0- 5 \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)5)=⟨−15,15⟩\lim_{x \to -\infty}\left(\frac{\cos{\left(5 x \right)}}{5}\right) = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−15,15⟩y = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle
lim⁡x→∞(cos⁡(5x)5)=⟨−15,15⟩\lim_{x \to \infty}\left(\frac{\cos{\left(5 x \right)}}{5}\right) = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−15,15⟩y = \left\langle - \frac{1}{5}, \frac{1}{5}\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of cos(5*x)/5, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(cos⁡(5x)5x)=0\lim_{x \to -\infty}\left(\frac{\cos{\left(5 x \right)}}{5 x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(cos⁡(5x)5x)=0\lim_{x \to \infty}\left(\frac{\cos{\left(5 x \right)}}{5 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)5=cos⁡(5x)5\frac{\cos{\left(5 x \right)}}{5} = \frac{\cos{\left(5 x \right)}}{5}
- Yes
cos⁡(5x)5=−cos⁡(5x)5\frac{\cos{\left(5 x \right)}}{5} = - \frac{\cos{\left(5 x \right)}}{5}
- No
so, the function
is
even
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