Chapter 2.3, Problem 64AYU

a)

To determine

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

Answer to Problem 64AYU

Average rate of change is $-2$ .

Explanation of Solution

Given:

A function is given as $h\left(x\right)=x^2-2x+3$ .

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.

  $h\left(x\right)=x^2-2x+3$ …… (1)

Consider the given $x$ values.

  $x=-1$ and $x=1$

Now, substitute $x$ by $-1$ in equation (1).

  $\begin{array}{c}h\left(-1\right)={\left(-1\right)}^2-2\left(-1\right)+3\\ h\left(-1\right)=1+2+3\\ h\left(-1\right)=6\end{array}$ …… (2)

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

  $\begin{array}{c}h\left(1\right)={\left(1\right)}^2-2\left(1\right)+3\\ h\left(1\right)=1-2+3\\ h\left(1\right)=2\end{array}$ …… (3)

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

  $\begin{array}{c}Avgh\left(x\right)=\frac{h\left(b\right)-h\left(a\right)}{b-a}\\ =\frac{h\left(1\right)-h\left(-1\right)}{1+1}\\ =\frac{2-6}{2}\\ =\frac{-4}{2}\\ =-2\end{array}$

Hence, the average rate change of the function is $-2$ .

b)

To determine

To find: The average rate of change of the given function between $x=0$ and $x=2$ .

Answer to Problem 64AYU

Average rate of change is $0$ .

Explanation of Solution

Given:

A function is given as $h\left(x\right)=x^2-2x+3$ .

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.

  $h\left(x\right)=x^2-2x+3$ …… (1)

Consider the given $x$ values.

  $x=0$ and $x=2$

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

  $\begin{array}{c}h\left(2\right)={\left(2\right)}^2-2\left(2\right)+3\\ h\left(2\right)=4-4+3\\ h\left(2\right)=3\end{array}$ …… (4)

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

  $\begin{array}{c}h\left(0\right)={\left(0\right)}^2-2\left(0\right)+3\\ h\left(0\right)=3\\ h\left(0\right)=3\end{array}$ …… (5)

Substitute equation (4) and (5) in the formula of average rate of change and simplify as shown:

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

Hence, the average rate change of the function is $0$ .

c)

To determine

To find: The average rate of change of the given function between $x=2$ and $x=5$ .

Answer to Problem 64AYU

Average rate of change is $5$ .

Explanation of Solution

Given:

A function is given as $h\left(x\right)=x^2-2x+3$ .

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.

  $h\left(x\right)=x^2-2x+3$ …… (1)

Consider the given $x$ values.

  $x=2$ and $x=5$

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

  $\begin{array}{c}h\left(2\right)={\left(2\right)}^2-2\left(2\right)+3\\ h\left(2\right)=4-4+3\\ h\left(2\right)=3\end{array}$ …… (4)

Now, substitute $x$ by $5$ in equation (1).

  $\begin{array}{c}h\left(5\right)={\left(5\right)}^2-2\left(5\right)+3\\ h\left(5\right)=25-10+3\\ h\left(5\right)=18\end{array}$ …… (6)

Substitute equation (6) and (4) in the formula of average rate of change and simplify as shown:

  $\begin{array}{c}Avgh\left(x\right)=\frac{h\left(b\right)-h\left(a\right)}{b-a}\\ =\frac{h\left(5\right)-h\left(2\right)}{5-2}\\ =\frac{18-3}{3}\\ =5\end{array}$

Hence, the average rate change of the function is $5$ .