Let $s(x) = \sin x$.

1. Find the exact value of the average rate of change of $s(x)$ from $x = \frac { \pi } { 4 }$ to $x = \frac { \pi } { 2 }$.

   Step (a):

   $\frac{s\left(\frac{\pi}{2}\right)-s\left(\frac{\pi}{4}\right)}{\left(\frac{\pi}{2}\right)-\left(\frac{\pi}{4}\right)}$

2. Write the expression for the average rate of change of $s(x)$ from $x =\frac { \pi } { 4 }$ to $x =\frac { \pi } { 4 }+ h$.

   Step (b):

   $\frac{s\left(\frac{\pi}{4}+h\right)-s\left(\frac{\pi}{4}\right)}{\left(\frac{\pi}{4}+h\right)-\left(\frac{\pi}{4}\right)}$

3. Use the fact that $\sin\left(A + B\right) = \sin A \cos B + \cos A \sin B$ to show that the expression in part (b) can be written as $\frac { 1 } { \sqrt { 2 } } ( \frac { \operatorname { cos } ( h ) + \operatorname { sin } ( h ) - 1 } { h } )$.

   Step (c):

   $m=\frac{\text{sin}\left ( \frac{\pi }{4}+h \right )-\text{sin}\left ( \frac{\pi }{4} \right )}{h}=\frac{\text{sin}\left ( \frac{\pi }{4}\right )\text{cos}(h)+\cos\left ( \frac{\pi }{4} \right )\text{sin}(h)-\text{sin}\left ( \frac{\pi }{4} \right )}{h}$

   Substitute values for the trigonometric expressions and simplify.