Chapter 2, Problem 41RE

a)

To determine

To find: The average rate of change of the given function between $x=1$ and $x=2$ .

Answer to Problem 41RE

Average rate of change is $23$ .

Explanation of Solution

Given:

A function is given as $f\left(x\right)=8x^2-x$ .

Formula used:

Average rate of change of a function $f\left(x\right)$ between $x=b$ and $x=a$ is given by

  $Avgf\left(x\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Calculation:

Consider the given function.

  $f\left(x\right)=8x^2-x$ …… (1)

Consider the given $x$ values.

  $x=1$ and $x=2$

Now, replace $x$ by $1$ in equation (1).

  $\begin{array}{c}f\left(1\right)=8{\left(1\right)}^{{}_2}-1\\ =8-1\\ =7\end{array}$ …… (2)

Now, replace $x$ by $2$ in equation (1).

  $\begin{array}{c}f\left(2\right)=8{\left(2\right)}^{{}_2}-2\\ =32-2\\ =30\end{array}$ …… (3)

Substitute equation (2) and (3) in the formula of average rate of change and simplify as shown:

  $\begin{array}{c}Avgf\left(x\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}\\ =\frac{f\left(2\right)-f\left(1\right)}{2-1}\\ =\frac{30-7}{1}\\ =23\end{array}$

Hence, the average rate change of the function is $23$ .

b)

To determine

To find: The average rate of change of the given function between $x=0$ and $x=1$ .

Answer to Problem 41RE

Average rate of change is $7$ .

Explanation of Solution

Given:

A function is given as $f\left(x\right)=8x^2-x$ .

Formula used:

Average rate of change of a function $f\left(x\right)$ between $x=b$ and $x=a$ is given by

  $Avgf\left(x\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Calculation:

Consider the given function.

  $f\left(x\right)=8x^2-x$ …… (1)

Consider the given $x$ values.

  $x=1$ and $x=0$

Now, replace $x$ by $1$ in equation (1).

  $\begin{array}{c}f\left(1\right)=8{\left(1\right)}^{{}_2}-1\\ =8-1\\ =7\end{array}$ …… (2)

Now, replace $x$ by $0$ in equation (1).

  $\begin{array}{c}f\left(0\right)=8{\left(0\right)}^{{}_2}-0\\ =0\end{array}$ …… (4)

Substitute equation (2) and (4) in the formula of average rate of change and simplify as shown:

  $\begin{array}{c}Avgf\left(x\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}\\ =\frac{f\left(1\right)-f\left(0\right)}{1-0}\\ =\frac{7-0}{1}\\ =7\end{array}$

Hence, the average rate change of the function is $7$ .

c)

To determine

To find: The average rate of change of the given function between $x=4$ and $x=2$ .

Answer to Problem 41RE

Average rate of change is $49$ .

Explanation of Solution

Given:

A function is given as $f\left(x\right)=8x^2-x$ .

Formula used:

Average rate of change of a function $f\left(x\right)$ between $x=b$ and $x=a$ is given by

  $Avgf\left(x\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}$

Calculation:

Consider the given function.

  $f\left(x\right)=8x^2-x$ …… (1)

Consider the given $x$ values.

  $x=4$ and $x=2$

Now, replace $x$ by $4$ in equation (1).

  $\begin{array}{c}f\left(4\right)=8{\left(4\right)}^{{}_2}-4\\ =128-4\\ =124\end{array}$ …… (5)

Now, replace $x$ by $2$ in equation (1).

  $\begin{array}{c}f\left(2\right)=8{\left(2\right)}^{{}_2}-2\\ =32-2\\ =30\end{array}$ …… (3)

Substitute equation (5) and (3) in the formula of average rate of change and simplify as shown:

  $\begin{array}{c}Avgf\left(x\right)=\frac{f\left(b\right)-f\left(a\right)}{b-a}\\ =\frac{f\left(4\right)-f\left(2\right)}{4-2}\\ =\frac{128-30}{2}\\ =49\end{array}$

Hence, the average rate change of the function is $49$ .