What is \varphi^{\prime}(a) equal to ?

Consider the following for the next two (02) items that follow : Let \varphi(a)=\int_{a}^{a+100\pi}|\sin x|\,dx

  1. A. 0
  2. B. \pi
  3. C. 100
  4. D. 200

Correct Answer: A. 0

Explanation

Since \varphi(a) = 200, a constant independent of a, its derivative with respect to a is zero. Alternatively, using Leibniz's rule, \varphi'(a) = |\sin(a+100\pi)| \cdot 1 - |\sin a| \cdot 1 = |\sin a| - |\sin a| = 0.

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