Key to "y=ln⁡(cos⁡x) find curvature of the plane"

Josalynn

Answered question

2020-11-10

y = \ln (\cos x)

find curvature of the plane

Answer & Explanation

insonsipthinye

Skilled2020-11-11Added 83 answers

The given function is
$f (x) = \ln (\cos (x))$
We know that the formula for the curvature is
$(x) = | | | | \frac{f^{″} x}{\frac{(1 + (f^{'} (x))^{2})^{3}}{2}} | | | |$
Now note that
$f^{'} (x) = (\frac{d}{d x}) \ln (\cos (x)) = \frac{1}{\cos (x)} \cdot (\frac{d}{d x}) (\cos (x)) = - (\frac{\sin x}{\cos x}) = - \tan x$
The second derivative is
$f^{″} (x) = \frac{d}{d x} (f^{'} (x)) = \frac{d}{d x} (- \tan x) = - \sec^{2} x$
By pluggin all values in the curveture formula yields us
$k (x) = | | | | | \frac{(- \sec^{2}) x}{\frac{{(1 + {(- \tan x)}^{2})}^{3}}{2}} | | | | | = | | | | | \frac{\sec^{2} x}{\frac{{(1 + \tan^{2} x)}^{3}}{2}} | | | | | = | | | | \frac{\sec^{2} x}{\frac{(\sec^{2} x)^{3}}{2}} | | | | = | \cos x |$
Hence the final answer is
$κ (x) =∣ \cos (x) ∣$

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