Chapter 2.4, Problem 2E

(a)

To determine

To find: The average rate of change of the function $f\left(x\right)=\sqrt{4x+1}$ over the interval $\left[0,2\right]$.

Answer to Problem 2E

The average rate of change of the function $f\left(x\right)=\sqrt{4x+1}$ over the interval $\left[0,2\right]$ is $1$.

Explanation of Solution

Given information:

The function is $f\left(x\right)=\sqrt{4x+1}$ and the interval is $\left[0,2\right]$.

Calculation:

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

The average rate of change of function $f\left(x\right)=\sqrt{4x+1}$ over the interval $\left[0,2\right]$ is:

  $\frac{f\left(2\right)-f\left(0\right)}{2-0}=\frac{f\left(2\right)-f\left(0\right)}{2}$

The function is given as $f\left(x\right)=\sqrt{4x+1}$.

Substitute $2$ for $x$ in the function.

  $\begin{array}{c}f\left(2\right)=\sqrt{4\times 2+1}\\ =\sqrt{9}\\ =3\end{array}$

Substitute $0$ for $x$ in the function.

  $\begin{array}{c}f\left(0\right)=\sqrt{4\times 0+1}\\ =\sqrt{1}\\ =1\end{array}$

Substitute $3$ for $f\left(2\right)$ and $1$ for $f\left(0\right)$ in the formula.

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

Therefore, the average rate of change of $f\left(x\right)=\sqrt{4x+1}$ over the interval $\left[0,2\right]$ is $1$.

(b)

To determine

To find: The average rate of change of the function $f\left(x\right)=\sqrt{4x+1}$ over the interval $\left[10,12\right]$.

Answer to Problem 2E

The average rate of change of the function $f\left(x\right)=\sqrt{4x+1}$ over the interval $\left[10,12\right]$ is $0.30$.

Explanation of Solution

Given information:

The function is $f\left(x\right)=\sqrt{4x+1}$ and the interval is $\left[10,12\right]$.

Calculation:

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

The average rate of change of function $f\left(x\right)=\sqrt{4x+1}$ over the interval $\left[10,12\right]$ is:

  $\frac{f\left(12\right)-f\left(10\right)}{12-10}=\frac{f\left(12\right)-f\left(10\right)}{2}$

The function is given as $f\left(x\right)=\sqrt{4x+1}$.

Substitute $12$ for $x$ in the function.

  $\begin{array}{c}f\left(12\right)=\sqrt{4\times 12+1}\\ =\sqrt{49}\\ =7\end{array}$

Substitute $10$ for $x$ in the function.

  $\begin{array}{c}f\left(10\right)=\sqrt{4\times 10+1}\\ =\sqrt{41}\\ \approx 6.40\end{array}$

Substitute $7$ for $f\left(12\right)$ and $6.40$ for $f\left(10\right)$ in the formula.

  $\begin{array}{c}\frac{f\left(12\right)-f\left(10\right)}{2}=\frac{7-6.40}{2}\\ =\frac{0.60}{2}\\ =0.30\end{array}$

Therefore, the average rate of change of the function $f\left(x\right)=\sqrt{4x+1}$ over the interval $\left[10,12\right]$ is $0.30$.