Find the average rate of change of the function over the given interval.
-$$h ( t ) = \sin ( 4 t ) , \left[ 0 , \frac { \pi } { 8 } \right]$$
A) $\frac { 4 } { \pi }$
B) $- \frac { 8 } { \pi }$
C) $\frac { \pi } { 8 }$
D) $\frac { 8 } { \pi }$

Question

Quizplus · Accepted Answer

The average rate of change of a function $f(x)$ over the interval $[a,b]$ is given by $\frac{f(b)-f(a)}{b-a}$. Using this formula, we can compute the average rate of change of $h(t) = \sin(4t)$ over $\left[ 0,\frac{\pi}8ight]$ as follows:

\begin{align*}
	ext{average rate of change } &= \frac{h\left(\frac{\pi}{8}ight) - h(0)}{\frac{\pi}{8} -0} \
&= \frac{\sin(4\cdot \frac{\pi}{8}) - \sin(4\cdot 0)}{\frac{\pi}{8}}\
&= \frac{\sin(\frac{\pi}{2})}{\frac{\pi}{8}} \
&= \frac{8}{\pi}.\
\end{align*}

Therefore, the answer is D.

Find the Average Rate of Change of the Function Over h(t)=sin⁡(4t),[0,π8]h ( t ) = \sin ( 4 t ) , \left[ 0 , \frac { \pi } { 8 } \right]h(t)=sin(4t),[0,8π​]

Find the Average Rate of Change of the Function Over $h ( t ) = \sin ( 4 t ) , \left[ 0 , \frac { \pi } { 8 } \right]$