Question

For the cartesian curve y=f(x) with ‘s’ as arc length which of the following condition holds good?

(a) \(\frac{ds}{dx} = \sqrt{1+(\frac{dy}{dx})^2}\)

(b) \(\frac{d^2s}{dx^2} = \sqrt{1-(\frac{dy}{dx})^2}\)

(c) \(\frac{dy}{dx} = \sqrt{1+(\frac{ds}{dx})^2}\)

(d) \(\sqrt{(\frac{ds}{dx})^2 + (\frac{dy}{dx})^2} = 1\)

This question was posed to me during an interview for a job.

Origin of the question is Derivative of Arc Length topic in section Differential Calculus of Engineering Mathematics

Answer 1

The correct choice is (a) \(\frac{ds}{dx} = \sqrt{1+(\frac{dy}{dx})^2}\)

To explain: From the below figure we can notice that ds is parallel to tangent to the curve, thus ds segment make an angle of φ with x-axis in positive direction

\(\frac{dy}{ds} = sin⁡φ, \frac{dx}{ds} = cos⁡φ, \frac{ds}{dx} = sec⁡φ, i.e (\frac{dx}{ds})^2 = sec^2 φ = 1+tan^2 φ…(1)\)

but \(tan⁡ φ=\frac{dy}{dx}\) = slope

substituting in (1) we get \( 1+(\frac{dy}{dx})^2 = (\frac{ds}{dx})^2 = \frac{ds}{dx} = \sqrt{1+(\frac{dy}{dx})^2}\).