Consider the function on the closed interval

MATHEMATICS · Multi-part · CBSE · Mathematics · Applications of Derivatives · Applications of Derivatives

Problem

Consider the function f(x)=sin⁡xf(x)=\sin x on the closed interval [0,π][0,\pi]. Answer the following questions. (i) Define the average rate of change of a real-valued function on a closed interval and the instantaneous rate of change at a point. (ii) Calculate the average rate of change of ff on [0,π][0,\pi]. (iii) Show that there exists a number cc in (0,π)(0,\pi) where the instantaneous rate of change of ff equals the average rate found in part (ii). Determine one such value of cc and justify why it lies inside the interval.

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