Mister Exam

Other calculators

Plotting a function graph/ 2*cos(5*x)

Graphing y = 2*cos(5*x)

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = 2*cos(5*x)
f(x)=2cos⁡(5x)f{\left(x \right)} = 2 \cos{\left(5 x \right)} 
f = 2*cos(5*x)
The graph of the function
02468-8-6-4-2-10105-5
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:
2cos⁡(5x)=02 \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=88.2787535658732x_{1} = 88.2787535658732
x2=4.08407044966673x_{2} = 4.08407044966673
x3=21.0486707790516x_{3} = 21.0486707790516
x4=−85.7654794430014x_{4} = -85.7654794430014
x5=−48.0663675999238x_{5} = -48.0663675999238
x6=−27.9601746169492x_{6} = -27.9601746169492
x7=−29.845130209103x_{7} = -29.845130209103
x8=−26.0752190247953x_{8} = -26.0752190247953
x9=−75.712382951514x_{9} = -75.712382951514
x10=39.8982267005904x_{10} = 39.8982267005904
x11=−36.1283155162826x_{11} = -36.1283155162826
x12=76.340701482232x_{12} = 76.340701482232
x13=−102.730079772386x_{13} = -102.730079772386
x14=−19.7920337176157x_{14} = -19.7920337176157
x15=−16.0221225333079x_{15} = -16.0221225333079
x16=27.9601746169492x_{16} = 27.9601746169492
x17=54.3495529071034x_{17} = 54.3495529071034
x18=−95.8185759344887x_{18} = -95.8185759344887
x19=−38.0132711084365x_{19} = -38.0132711084365
x20=−14.1371669411541x_{20} = -14.1371669411541
x21=46.18141200777x_{21} = 46.18141200777
x22=58.1194640914112x_{22} = 58.1194640914112
x23=−39.8982267005904x_{23} = -39.8982267005904
x24=−53.0929158456675x_{24} = -53.0929158456675
x25=−70.0575161750524x_{25} = -70.0575161750524
x26=12.2522113490002x_{26} = 12.2522113490002
x27=2.19911485751286x_{27} = 2.19911485751286
x28=72.5707902979242x_{28} = 72.5707902979242
x29=5.96902604182061x_{29} = 5.96902604182061
x30=−93.9336203423348x_{30} = -93.9336203423348
x31=92.0486647501809x_{31} = 92.0486647501809
x32=98.3318500573605x_{32} = 98.3318500573605
x33=7.22566310325652x_{33} = 7.22566310325652
x34=22.3053078404875x_{34} = 22.3053078404875
x35=27.3318560862312x_{35} = 27.3318560862312
x36=−69.4291976443344x_{36} = -69.4291976443344
x37=38.0132711084365x_{37} = 38.0132711084365
x38=49.9513231920777x_{38} = 49.9513231920777
x39=−11.6238928182822x_{39} = -11.6238928182822
x40=−80.1106126665397x_{40} = -80.1106126665397
x41=−41.7831822927443x_{41} = -41.7831822927443
x42=36.1283155162826x_{42} = 36.1283155162826
x43=−49.9513231920777x_{43} = -49.9513231920777
x44=44.2964564156161x_{44} = 44.2964564156161
x45=−76.9690200129499x_{45} = -76.9690200129499
x46=93.9336203423348x_{46} = 93.9336203423348
x47=10.3672557568463x_{47} = 10.3672557568463
x48=83.8805238508475x_{48} = 83.8805238508475
x49=60.0044196835651x_{49} = 60.0044196835651
x50=66.2876049907446x_{50} = 66.2876049907446
x51=65.0309679293087x_{51} = 65.0309679293087
x52=95.1902574037707x_{52} = 95.1902574037707
x53=−60.0044196835651x_{53} = -60.0044196835651
x54=−4.08407044966673x_{54} = -4.08407044966673
x55=−9.73893722612836x_{55} = -9.73893722612836
x56=−97.7035315266426x_{56} = -97.7035315266426
x57=26.0752190247953x_{57} = 26.0752190247953
x58=71.9424717672063x_{58} = 71.9424717672063
x59=48.0663675999238x_{59} = 48.0663675999238
x60=−33.6150413934108x_{60} = -33.6150413934108
x61=−61.8893752757189x_{61} = -61.8893752757189
x62=17.9070781254618x_{62} = 17.9070781254618
x63=−21.6769893097696x_{63} = -21.6769893097696
x64=32.3584043319749x_{64} = 32.3584043319749
x65=−81.9955682586936x_{65} = -81.9955682586936
x66=−87.6504350351552x_{66} = -87.6504350351552
x67=−63.7743308678728x_{67} = -63.7743308678728
x68=−58.1194640914112x_{68} = -58.1194640914112
x69=34.2433599241287x_{69} = 34.2433599241287
x70=−31.7300858012569x_{70} = -31.7300858012569
x71=−92.0486647501809x_{71} = -92.0486647501809
x72=24.1902634326414x_{72} = 24.1902634326414
x73=80.1106126665397x_{73} = 80.1106126665397
x74=−5.96902604182061x_{74} = -5.96902604182061
x75=68.1725605828985x_{75} = 68.1725605828985
x76=56.2345084992573x_{76} = 56.2345084992573
x77=14.1371669411541x_{77} = 14.1371669411541
x78=−17.9070781254618x_{78} = -17.9070781254618
x79=−53.7212343763855x_{79} = -53.7212343763855
x80=−65.6592864600267x_{80} = -65.6592864600267
x81=−71.9424717672063x_{81} = -71.9424717672063
x82=61.8893752757189x_{82} = 61.8893752757189
x83=16.0221225333079x_{83} = 16.0221225333079
x84=78.2256570743859x_{84} = 78.2256570743859
x85=90.1637091580271x_{85} = 90.1637091580271
x86=−73.8274273593601x_{86} = -73.8274273593601
x87=88.9070720965912x_{87} = 88.9070720965912
x88=0.314159265358979x_{88} = 0.314159265358979
x89=−43.6681378848981x_{89} = -43.6681378848981
x90=−51.8362787842316x_{90} = -51.8362787842316
x91=−7.85398163397448x_{91} = -7.85398163397448
x92=−103.358398303104x_{92} = -103.358398303104
x93=100.216805649514x_{93} = 100.216805649514
x94=81.9955682586936x_{94} = 81.9955682586936
x95=70.0575161750524x_{95} = 70.0575161750524
x96=−83.8805238508475x_{96} = -83.8805238508475
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to 2*cos(5*x).
2cos⁡(0⋅5)2 \cos{\left(0 \cdot 5 \right)}
The result:
f(0)=2f{\left(0 \right)} = 2
The point:
(0, 2)
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
−10sin⁡(5x)=0- 10 \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, 2)

 pi     
(--, -2)
 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
−50cos⁡(5x)=0- 50 \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→−∞(2cos⁡(5x))=⟨−2,2⟩\lim_{x \to -\infty}\left(2 \cos{\left(5 x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
lim⁡x→∞(2cos⁡(5x))=⟨−2,2⟩\lim_{x \to \infty}\left(2 \cos{\left(5 x \right)}\right) = \left\langle -2, 2\right\rangle
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=⟨−2,2⟩y = \left\langle -2, 2\right\rangle
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of 2*cos(5*x), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2cos⁡(5x)x)=0\lim_{x \to -\infty}\left(\frac{2 \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→∞(2cos⁡(5x)x)=0\lim_{x \to \infty}\left(\frac{2 \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:
2cos⁡(5x)=2cos⁡(5x)2 \cos{\left(5 x \right)} = 2 \cos{\left(5 x \right)}
- Yes
2cos⁡(5x)=−2cos⁡(5x)2 \cos{\left(5 x \right)} = - 2 \cos{\left(5 x \right)}
- No
so, the function
is
even
    © Mister Exam – Calculator