Chapter 2, Problem 43RE

To determine

To find: The average rate of change of function.

Answer to Problem 43RE

The average rate of change of function is $\frac{2}{\pi }$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=1+\sin x$ and the interval is $\left[0,\frac{\pi }{2}\right]$ .

Formula used: The formula for average rate of change is given by,

  $\frac{\Delta y}{\Delta t}=\underset{h\to 0}{\lim }\frac{f\left(a+h\right)-f\left(a\right)}{h}$

Calculation:

The average rate of change of the function can be calculated as:

  $\begin{array}{c}\text{avg rate change}=\frac{f\left(\frac{\pi }{2}\right)-f\left(0\right)}{\frac{\pi }{2}-0}\\ =\frac{1+\sin \frac{\pi }{2}-1-\sin 0}{\frac{\pi }{2}}\\ =\frac{2\left(1+1-1\right)}{\pi }\\ =\frac{2}{\pi }\end{array}$

Therefore, the average rate of change of the function is $\frac{2}{\pi }$ .