Mathematics

Problem

Consider the function

f(x)=\sin x

on the closed interval

[0,\pi]

. Answer the following questions. a. (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. b. (ii) Calculate the average rate of change of

f

[0,\pi]

. c. (iii) Show that there exists a number

c

(0,\pi)

where the instantaneous rate of change of

f

equals the average rate found in part (ii). Determine one such value of

c

and justify why it lies inside the interval.

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