Question

f(x) = \(\int_{0}^{\pi/2}sin(ax)da\) then the value of f^(100)(0) is?

(a) a^(100) sin(a)

(b) -a^(100) sin(a)

(c) a^(100) cos(a)

(d) 0

I got this question in an interview for job.

This interesting question is from The nth Derivative of Some Elementary Functions topic in division Differential Calculus of Engineering Mathematics

Answer 1

The correct choice is (d) 0

Easy explanation: First solve the integral f(x)=\(\int_{0}^{\pi/2}sin(ax)da\)

Which gives =\([\frac{-cos(ax)}{x}]_{0}^{\pi/2}=\frac{1-cos(\frac{\pi x}{2})}{x}\)

Now expanding into Taylor series yields

\(=(\frac{(\frac{\pi x}{2})^2}{x2!}-\frac{(\frac{\pi x}{2})^4}{x4!}….\infty)\)

Observe that every term in the expansion is odd powered

Hence even derivative at x = 0 has to be 0.