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Chapter 10: Problem 41

Find the curvature and radius of curvature of the plane curve at the given value of \(x\). $$ y=\sqrt{a^{2}-x^{2}}, \quad x=0 $$

Short Answer

Expert verified
After evaluating the derivations and getting expressions for \(y'\), \(y''\), the curvature \(k\) can be found, and subsequently the radius of curvature \(R\) is calculated.

Step by step solution

01

Derive y

Differentiate \(y=\sqrt{a^2 - x^2}\) with respect to x to find \(y'\). This involves using the chain rule of differentiation.
02

Derive y'

Differentiate \(y'\) with respect to x to find \(y''\). This also involves the chain rule of differentiation.
03

Find the curvature (k)

Substitute \(y'\) and \(y''\) values into the curvature formula \(k = |y''| /(1 + (y')^2)^{3/2}\) to find k.
04

Find the radius of curvature (R)

Find the radius of curvature using the formula \(R = 1 / k\).

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