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

Graphing y = log(cos(2*x))

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
f(x) = log(cos(2*x))
f(x)=log⁡(cos⁡(2x))f{\left(x \right)} = \log{\left(\cos{\left(2 x \right)} \right)} 
f = log(cos(2*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:
log⁡(cos⁡(2x))=0\log{\left(\cos{\left(2 x \right)} \right)} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=0x_{1} = 0
x2=πx_{2} = \pi
Numerical solution
x1=69.1150383612121x_{1} = 69.1150383612121
x2=−72.2566308356894x_{2} = -72.2566308356894
x3=37.6991120767477x_{3} = 37.6991120767477
x4=−94.2477794177114x_{4} = -94.2477794177114
x5=62.8318525607898x_{5} = 62.8318525607898
x6=6.28318528407908x_{6} = 6.28318528407908
x7=−34.5575189626208x_{7} = -34.5575189626208
x8=28.2743338651142x_{8} = 28.2743338651142
x9=−50.2654822535294x_{9} = -50.2654822535294
x10=18.8495552720944x_{10} = 18.8495552720944
x11=78.5398161548118x_{11} = 78.5398161548118
x12=−31.4159267547793x_{12} = -31.4159267547793
x13=72.2566310277135x_{13} = 72.2566310277135
x14=−59.6902604582916x_{14} = -59.6902604582916
x15=−65.9734457646558x_{15} = -65.9734457646558
x16=−75.3982236054042x_{16} = -75.3982236054042
x17=−84.8230016623642x_{17} = -84.8230016623642
x18=−69.1150391289658x_{18} = -69.1150391289658
x19=56.5486675731909x_{19} = 56.5486675731909
x20=−34.557518409417x_{20} = -34.557518409417
x21=−3.14159332624222x_{21} = -3.14159332624222
x22=59.6902606605194x_{22} = 59.6902606605194
x23=53.4070752703581x_{23} = 53.4070752703581
x24=84.8230011231247x_{24} = 84.8230011231247
x25=−97.389372502654x_{25} = -97.389372502654
x26=34.5575194141501x_{26} = 34.5575194141501
x27=62.8318524651379x_{27} = 62.8318524651379
x28=34.5575189914319x_{28} = 34.5575189914319
x29=15.7079634939052x_{29} = 15.7079634939052
x30=59.6902606597981x_{30} = 59.6902606597981
x31=−15.7079632968187x_{31} = -15.7079632968187
x32=−100.530964223024x_{32} = -100.530964223024
x33=40.8407038692067x_{33} = 40.8407038692067
x34=−37.6991118776023x_{34} = -37.6991118776023
x35=−18.8495559442882x_{35} = -18.8495559442882
x36=−21.9911485864121x_{36} = -21.9911485864121
x37=87.9645943363558x_{37} = 87.9645943363558
x38=75.3982238314004x_{38} = 75.3982238314004
x39=94.2477796093519x_{39} = 94.2477796093519
x40=25.1327412137936x_{40} = 25.1327412137936
x41=81.6814092224531x_{41} = 81.6814092224531
x42=84.8230010599183x_{42} = 84.8230010599183
x43=−56.5486669960941x_{43} = -56.5486669960941
x44=12.5663704095248x_{44} = 12.5663704095248
x45=−6.28318508874543x_{45} = -6.28318508874543
x46=65.9734457532363x_{46} = 65.9734457532363
x47=9.4247781515942x_{47} = 9.4247781515942
x48=−87.9645943581379x_{48} = -87.9645943581379
x49=−53.4070753372009x_{49} = -53.4070753372009
x50=−81.6814090389034x_{50} = -81.6814090389034
x51=−47.1238905267517x_{51} = -47.1238905267517
x52=91.106186934433x_{52} = 91.106186934433
x53=−56.5486675180935x_{53} = -56.5486675180935
x54=81.681409243074x_{54} = 81.681409243074
x55=−9.42477817254169x_{55} = -9.42477817254169
x56=50.2654824463146x_{56} = 50.2654824463146
x57=−78.5398155749623x_{57} = -78.5398155749623
x58=−28.274333671219x_{58} = -28.274333671219
x59=21.9911485852348x_{59} = 21.9911485852348
x60=18.8495554506519x_{60} = 18.8495554506519
x61=100.530964736304x_{61} = 100.530964736304
x62=−53.4070750099862x_{62} = -53.4070750099862
x63=0x_{63} = 0
x64=−12.566370406077x_{64} = -12.566370406077
x65=43.982297169579x_{65} = 43.982297169579
x66=−62.8318530897596x_{66} = -62.8318530897596
x67=−43.9822971744191x_{67} = -43.9822971744191
x68=40.8407040035115x_{68} = 40.8407040035115
x69=31.4159267104104x_{69} = 31.4159267104104
x70=−9.42477780989129x_{70} = -9.42477780989129
x71=−40.8407045170434x_{71} = -40.8407045170434
x72=47.1238897876436x_{72} = 47.1238897876436
x73=−31.4159264120844x_{73} = -31.4159264120844
x74=−97.3893721997423x_{74} = -97.3893721997423
x75=−78.5398160725979x_{75} = -78.5398160725979
x76=12.5663707984054x_{76} = 12.5663707984054
x77=97.3893723934834x_{77} = 97.3893723934834
x78=−25.1327419258466x_{78} = -25.1327419258466
x79=−75.3982239198207x_{79} = -75.3982239198207
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(cos(2*x)).
log⁡(cos⁡(0⋅2))\log{\left(\cos{\left(0 \cdot 2 \right)} \right)}
The result:
f(0)=0f{\left(0 \right)} = 0
The point:
(0, 0)
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
−2sin⁡(2x)cos⁡(2x)=0- \frac{2 \sin{\left(2 x \right)}}{\cos{\left(2 x \right)}} = 0
Solve this equation
The roots of this equation
x1=0x_{1} = 0
x2=π2x_{2} = \frac{\pi}{2}
The values of the extrema at the points:
(0, 0)

 pi       
(--, pi*I)
 2


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:
The function has no minima
Maxima of the function at points:
x2=0x_{2} = 0
Decreasing at intervals
(−∞,0]\left(-\infty, 0\right]
Increasing at intervals
[0,∞)\left[0, \infty\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
−4(sin⁡2(2x)cos⁡2(2x)+1)=0- 4 \left(\frac{\sin^{2}{\left(2 x \right)}}{\cos^{2}{\left(2 x \right)}} + 1\right) = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞log⁡(cos⁡(2x))=log⁡(⟨−1,1⟩)\lim_{x \to -\infty} \log{\left(\cos{\left(2 x \right)} \right)} = \log{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=log⁡(⟨−1,1⟩)y = \log{\left(\left\langle -1, 1\right\rangle \right)}
lim⁡x→∞log⁡(cos⁡(2x))=log⁡(⟨−1,1⟩)\lim_{x \to \infty} \log{\left(\cos{\left(2 x \right)} \right)} = \log{\left(\left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=log⁡(⟨−1,1⟩)y = \log{\left(\left\langle -1, 1\right\rangle \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(cos(2*x)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(cos⁡(2x))x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(\cos{\left(2 x \right)} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(cos⁡(2x))x)=0\lim_{x \to \infty}\left(\frac{\log{\left(\cos{\left(2 x \right)} \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:
log⁡(cos⁡(2x))=log⁡(cos⁡(2x))\log{\left(\cos{\left(2 x \right)} \right)} = \log{\left(\cos{\left(2 x \right)} \right)}
- Yes
log⁡(cos⁡(2x))=−log⁡(cos⁡(2x))\log{\left(\cos{\left(2 x \right)} \right)} = - \log{\left(\cos{\left(2 x \right)} \right)}
- No
so, the function
is
even
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