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Plotting a function graph/ ln(2cos(x))

Graphing y = ln(2cos(x))

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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

Analytical solution
x1=π3x_{1} = \frac{\pi}{3}
x2=5π3x_{2} = \frac{5 \pi}{3}
Numerical solution
x1=−19.8967534727354x_{1} = -19.8967534727354
x2=38.7463093942741x_{2} = 38.7463093942741
x3=−233.525053916841x_{3} = -233.525053916841
x4=49.2182849062401x_{4} = 49.2182849062401
x5=−86.9173967493176x_{5} = -86.9173967493176
x6=93.2005820564972x_{6} = 93.2005820564972
x7=−51.3126800086333x_{7} = -51.3126800086333
x8=63.8790506229925x_{8} = 63.8790506229925
x9=5.23598775598299x_{9} = 5.23598775598299
x10=45.0294947014537x_{10} = 45.0294947014537
x11=−61.7846555205993x_{11} = -61.7846555205993
x12=−74.3510261349584x_{12} = -74.3510261349584
x13=30.3687289847013x_{13} = 30.3687289847013
x14=−55.5014702134197x_{14} = -55.5014702134197
x15=26.1799387799149x_{15} = 26.1799387799149
x16=74.3510261349584x_{16} = 74.3510261349584
x17=−24.0855436775217x_{17} = -24.0855436775217
x18=−70.162235930172x_{18} = -70.162235930172
x19=−26.1799387799149x_{19} = -26.1799387799149
x20=13.6135681655558x_{20} = 13.6135681655558
x21=−82.7286065445312x_{21} = -82.7286065445312
x22=32.4631240870945x_{22} = 32.4631240870945
x23=−95.2949771588904x_{23} = -95.2949771588904
x24=89.0117918517108x_{24} = 89.0117918517108
x25=101.57816246607x_{25} = 101.57816246607
x26=−5.23598775598299x_{26} = -5.23598775598299
x27=−80.634211442138x_{27} = -80.634211442138
x28=−17.8023583703422x_{28} = -17.8023583703422
x29=−99.4837673636768x_{29} = -99.4837673636768
x30=57.5958653158129x_{30} = 57.5958653158129
x31=55.5014702134197x_{31} = 55.5014702134197
x32=−68.0678408277789x_{32} = -68.0678408277789
x33=76.4454212373516x_{33} = 76.4454212373516
x34=68.0678408277789x_{34} = 68.0678408277789
x35=24.0855436775217x_{35} = 24.0855436775217
x36=86.9173967493176x_{36} = 86.9173967493176
x37=42.9350995990605x_{37} = 42.9350995990605
x38=−93.2005820564972x_{38} = -93.2005820564972
x39=−63.8790506229925x_{39} = -63.8790506229925
x40=−11.5191730631626x_{40} = -11.5191730631626
x41=36.6519142918809x_{41} = 36.6519142918809
x42=−76.4454212373516x_{42} = -76.4454212373516
x43=11.5191730631626x_{43} = 11.5191730631626
x44=70.162235930172x_{44} = 70.162235930172
x45=−49.2182849062401x_{45} = -49.2182849062401
x46=−57.5958653158129x_{46} = -57.5958653158129
x47=61.7846555205993x_{47} = 61.7846555205993
x48=−32.4631240870945x_{48} = -32.4631240870945
x49=17.8023583703422x_{49} = 17.8023583703422
x50=19.8967534727354x_{50} = 19.8967534727354
x51=80.634211442138x_{51} = 80.634211442138
x52=82.7286065445312x_{52} = 82.7286065445312
x53=−7.33038285837618x_{53} = -7.33038285837618
x54=−101.57816246607x_{54} = -101.57816246607
x55=99.4837673636768x_{55} = 99.4837673636768
x56=−13.6135681655558x_{56} = -13.6135681655558
x57=−30.3687289847013x_{57} = -30.3687289847013
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(2*cos(x)).
log⁡(2cos⁡(0))\log{\left(2 \cos{\left(0 \right)} \right)}
The result:
f(0)=log⁡(2)f{\left(0 \right)} = \log{\left(2 \right)}
The point:
(0, log(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
−sin⁡(x)cos⁡(x)=0- \frac{\sin{\left(x \right)}}{\cos{\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, log(2))

(pi, pi*I + log(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
−(sin⁡2(x)cos⁡2(x)+1)=0- (\frac{\sin^{2}{\left(x \right)}}{\cos^{2}{\left(x \right)}} + 1) = 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⁡(2cos⁡(x))=log⁡(⟨−2,2⟩)\lim_{x \to -\infty} \log{\left(2 \cos{\left(x \right)} \right)} = \log{\left(\left\langle -2, 2\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=log⁡(⟨−2,2⟩)y = \log{\left(\left\langle -2, 2\right\rangle \right)}
lim⁡x→∞log⁡(2cos⁡(x))=log⁡(⟨−2,2⟩)\lim_{x \to \infty} \log{\left(2 \cos{\left(x \right)} \right)} = \log{\left(\left\langle -2, 2\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=log⁡(⟨−2,2⟩)y = \log{\left(\left\langle -2, 2\right\rangle \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(2*cos(x)), divided by x at x->+oo and x ->-oo
lim⁡x→−∞(log⁡(2cos⁡(x))x)=0\lim_{x \to -\infty}\left(\frac{\log{\left(2 \cos{\left(x \right)} \right)}}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(log⁡(2cos⁡(x))x)=0\lim_{x \to \infty}\left(\frac{\log{\left(2 \cos{\left(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⁡(2cos⁡(x))=log⁡(2cos⁡(x))\log{\left(2 \cos{\left(x \right)} \right)} = \log{\left(2 \cos{\left(x \right)} \right)}
- Yes
log⁡(2cos⁡(x))=−log⁡(2cos⁡(x))\log{\left(2 \cos{\left(x \right)} \right)} = - \log{\left(2 \cos{\left(x \right)} \right)}
- No
so, the function
is
even
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