Question

For the curve \(y=a \,log \,sec(\frac{x}{a})\) what is the value of \( \frac{ds}{dx}\)? (where φ is the angle made by tangent to the curve with x-axis)?

(a) cos φ

(b) sec φ

(c) tan φ

(d) cot φ

I got this question at a job interview.

Enquiry is from Derivative of Arc Length topic in section Differential Calculus of Engineering Mathematics

Answer 1

The correct choice is (b) sec φ

Explanation: w.k.t \(\frac{ds}{dx} = \sqrt{1+(\frac{dy}{dx})^2}\)

\(\frac{dy}{dx} = a \frac{tan \frac{x}{a}.sec\frac{x}{a}}{sec\frac{x}{a}} \frac{1}{a} = tan \frac{x}{a} \)

substituting we get

\(\frac{ds}{dx} = \sqrt{1+(tan \frac{x}{a})^2}=sec\frac{x}{a}, \,but \,w.k.t\, \frac{dy}{dx} = tan φ = tan\frac{x}{a} \,thus\, φ=\frac{x}{a}\)

\(\frac{ds}{dx} = sec⁡φ.\)