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

Elleanor Mckenzie

Answered question

2021-03-05

find curvature of the plane curve

y = \ln (\cos x)

Answer & Explanation

tafzijdeq

Skilled2021-03-06Added 92 answers

The given function is
$f (x) = \ln (\cos (x))$
We know that the formula for the curvature is
$k (x) = | \frac{\frac{f^{″} x}{{(1 + {(f^{'} (x))}^{2})}^{3}}}{2} |$
Now note that
$f' (x) = (\frac{d}{d x}) \ln (\cos (x)) = (\frac{1}{\cos (x)}) \times \frac{d}{d x} (\cos (x)) = - \frac{\sin x}{\cos x} = - \tan x$
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
$k (x) = ∣ \frac{(- \sec^{2}) x}{\frac{{(1 + {(- \tan x)}^{2})}^{3}}{2}} |$

$= | \frac{\sec^{2} x}{\frac{(1 + \tan^{2} x)^{2}}{2}} | = | \frac{\sec^{2} x}{\frac{(\sec^{2} x)^{3}}{2}} | = | \frac{\sec^{2} x}{\sec^{3} x} | = | \cos x |$
Hence the final answer is
$κ (x) =∣ \cos (x) ∣$

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