Chapter 2.4, Problem 1QQ

To determine

Tofind:The correct value for the average rate of $f\left(x\right)=\sqrt{x+1}$ over the interval $\left(0,3\right)$ .

Answer to Problem 1QQ

The correct choice is (D) such that the average rate of $f\left(x\right)=\sqrt{x+1}$ over the interval $\left(0,3\right)$ is $\frac{1}{3}$ .

Explanation of Solution

Given information:The function is $f\left(x\right)=\sqrt{x+1}$ and the interval is $\left(0,3\right)$ .

The formula for the average rate of $f\left(x\right)$ over the interval $\left(a,b\right)$ is $\frac{f\left(b\right)-f\left(a\right)}{b-a}$ .

The value of $f\left(x\right)=\sqrt{x+1}$ at $x=3$ is:

  $\begin{array}{c}f\left(3\right)=\sqrt{3+1}\\ =\sqrt{4}\\ =2\end{array}$

So, the value of $f\left(3\right)$ is $2$ .

The value of $f\left(x\right)=\sqrt{x+1}$ at $x=0$ is:

  $\begin{array}{c}f\left(0\right)=\sqrt{0+1}\\ =\sqrt{1}\\ =1\end{array}$

So, the value of $f\left(0\right)$ is $1$ .

Now, the average rate can be calculated as:

  $\begin{array}{c}\frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{f\left(3\right)-f\left(0\right)}{3-0}\\ =\frac{2-1}{3}\\ =\frac{1}{3}\end{array}$

Therefore, the correct choice is (D) such that the average rate of $f\left(x\right)=\sqrt{x+1}$ over the interval $\left(0,3\right)$ is $\frac{1}{3}$ .