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

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

Analytical solution
x1=π4x_{1} = \frac{\pi}{4}
x2=7π4x_{2} = \frac{7 \pi}{4}
Numerical solution
x1=−68.329640215578x_{1} = -68.329640215578
x2=18.0641577581413x_{2} = 18.0641577581413
x3=25.9181393921158x_{3} = 25.9181393921158
x4=69.9004365423729x_{4} = 69.9004365423729
x5=63.6172512351933x_{5} = 63.6172512351933
x6=43.1968989868597x_{6} = 43.1968989868597
x7=88.7499924639117x_{7} = 88.7499924639117
x8=7.06858347057703x_{8} = 7.06858347057703
x9=82.4668071567321x_{9} = 82.4668071567321
x10=−30.6305283725005x_{10} = -30.6305283725005
x11=80.8960108299372x_{11} = 80.8960108299372
x12=68.329640215578x_{12} = 68.329640215578
x13=−13.3517687777566x_{13} = -13.3517687777566
x14=−62.0464549083984x_{14} = -62.0464549083984
x15=57.3340659280137x_{15} = 57.3340659280137
x16=0.785398163397448x_{16} = 0.785398163397448
x17=−24.3473430653209x_{17} = -24.3473430653209
x18=19.6349540849362x_{18} = 19.6349540849362
x19=−7.06858347057703x_{19} = -7.06858347057703
x20=−49.4800842940392x_{20} = -49.4800842940392
x21=95.0331777710912x_{21} = 95.0331777710912
x22=−63.6172512351933x_{22} = -63.6172512351933
x23=55.7632696012188x_{23} = 55.7632696012188
x24=51.0508806208341x_{24} = 51.0508806208341
x25=−51.0508806208341x_{25} = -51.0508806208341
x26=32.2013246992954x_{26} = 32.2013246992954
x27=−36.9137136796801x_{27} = -36.9137136796801
x28=13.3517687777566x_{28} = 13.3517687777566
x29=−57.3340659280137x_{29} = -57.3340659280137
x30=−32.2013246992954x_{30} = -32.2013246992954
x31=30.6305283725005x_{31} = 30.6305283725005
x32=−82.4668071567321x_{32} = -82.4668071567321
x33=−88.7499924639117x_{33} = -88.7499924639117
x34=76.1836218495525x_{34} = 76.1836218495525
x35=−74.6128255227576x_{35} = -74.6128255227576
x36=−95.0331777710912x_{36} = -95.0331777710912
x37=−0.785398163397448x_{37} = -0.785398163397448
x38=5.49778714378214x_{38} = 5.49778714378214
x39=24.3473430653209x_{39} = 24.3473430653209
x40=−43.1968989868597x_{40} = -43.1968989868597
x41=99.7455667514759x_{41} = 99.7455667514759
x42=−80.8960108299372x_{42} = -80.8960108299372
x43=−19.6349540849362x_{43} = -19.6349540849362
x44=−76.1836218495525x_{44} = -76.1836218495525
x45=−38.484510006475x_{45} = -38.484510006475
x46=−11.7809724509617x_{46} = -11.7809724509617
x47=36.9137136796801x_{47} = 36.9137136796801
x48=62.0464549083984x_{48} = 62.0464549083984
x49=−25.9181393921158x_{49} = -25.9181393921158
x50=−5.49778714378214x_{50} = -5.49778714378214
x51=−55.7632696012188x_{51} = -55.7632696012188
x52=11.7809724509617x_{52} = 11.7809724509617
x53=74.6128255227576x_{53} = 74.6128255227576
x54=−99.7455667514759x_{54} = -99.7455667514759
x55=−18.0641577581413x_{55} = -18.0641577581413
x56=38.484510006475x_{56} = 38.484510006475
x57=−87.1791961371168x_{57} = -87.1791961371168
x58=101.316363078271x_{58} = 101.316363078271
x59=−44.7676953136546x_{59} = -44.7676953136546
x60=−69.9004365423729x_{60} = -69.9004365423729
x61=−101.316363078271x_{61} = -101.316363078271
x62=44.7676953136546x_{62} = 44.7676953136546
x63=87.1791961371168x_{63} = 87.1791961371168
x64=−93.4623814442964x_{64} = -93.4623814442964
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to log(sqrt(2)*cos(x)).
log⁡(2cos⁡(0))\log{\left(\sqrt{2} \cos{\left(0 \right)} \right)}
The result:
f(0)=log⁡(2)f{\left(0 \right)} = \log{\left(\sqrt{2} \right)}
The point:
(0, log(sqrt(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⟨−1,1⟩)\lim_{x \to -\infty} \log{\left(\sqrt{2} \cos{\left(x \right)} \right)} = \log{\left(\sqrt{2} \left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=log⁡(2⟨−1,1⟩)y = \log{\left(\sqrt{2} \left\langle -1, 1\right\rangle \right)}
lim⁡x→∞log⁡(2cos⁡(x))=log⁡(2⟨−1,1⟩)\lim_{x \to \infty} \log{\left(\sqrt{2} \cos{\left(x \right)} \right)} = \log{\left(\sqrt{2} \left\langle -1, 1\right\rangle \right)}
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=log⁡(2⟨−1,1⟩)y = \log{\left(\sqrt{2} \left\langle -1, 1\right\rangle \right)}
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of log(sqrt(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(\sqrt{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(\sqrt{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(\sqrt{2} \cos{\left(x \right)} \right)} = \log{\left(\sqrt{2} \cos{\left(x \right)} \right)}
- Yes
log⁡(2cos⁡(x))=−log⁡(2cos⁡(x))\log{\left(\sqrt{2} \cos{\left(x \right)} \right)} = - \log{\left(\sqrt{2} \cos{\left(x \right)} \right)}
- No
so, the function
is
even
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