Chapter 2.3, Problem 69AYU

a)

To determine

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

Answer to Problem 69AYU

Average rate of change is $6$ .

Explanation of Solution

Given:

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

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.

  $h\left(x\right)=x^2-2x$ …… (1)

Consider the given $x$ values.

  $x=2$ and $x=4$

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

  $\begin{array}{c}h\left(2\right)={\left(2\right)}^2+2\left(2\right)\\ h\left(2\right)=4+4\\ h\left(2\right)=8\end{array}$ …… (2)

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

  $\begin{array}{c}h\left(4\right)={\left(4\right)}^2+2\left(2\right)\\ h\left(4\right)=16+4\\ h\left(4\right)=20\end{array}$ …… (3)

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

  $\begin{array}{c}Avgh\left(x\right)=\frac{h\left(b\right)-h\left(a\right)}{b-a}\\ =\frac{h\left(4\right)-h\left(2\right)}{4-2}\\ =\frac{20-8}{2}\\ =\frac{12}{2}\\ =6\end{array}$

Hence, the average rate change of the function is $6$ .

b)

To determine

To find: The equation for the secant line to the given curve joining the point $\left(2,h\left(2\right)\right)$ and $\left(4,h\left(4\right)\right)$ .

Answer to Problem 69AYU

The equation of the secant to the curve is $y-6x+8=0$ .

Explanation of Solution

Given:

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

Formula used:

Slope of the secant line to a curve $y=f\left(x\right)$ joining two points $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right)$ is

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

Equation of secant line of a curve joining two points $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right)$ is

  $y-f\left(a\right)=m\left(x-a\right)$$$

Calculation:

First calculate the slope of the secant line as shown:

Consider the given function.

  $h\left(x\right)=x^2-2x$ …… (1)

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

  $\begin{array}{c}h\left(2\right)={\left(2\right)}^2+2\left(2\right)\\ h\left(2\right)=4+4\\ h\left(2\right)=8\end{array}$ …… (2)

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

  $\begin{array}{c}h\left(4\right)={\left(4\right)}^2+2\left(2\right)\\ h\left(4\right)=16+4\\ h\left(4\right)=20\end{array}$ …… (3)

Slope of the secant line

  $\begin{array}{c}Slope\left(m\right)=\frac{h\left(b\right)-h\left(a\right)}{b-a}\\ =\frac{h\left(4\right)-h\left(2\right)}{4-2}\\ =\frac{20-8}{2}\\ =\frac{12}{2}\\ =6\end{array}$

Therefore, equation of the secant to the curve at $\left(2,h\left(2\right)\right)$ is

  $\begin{array}{l}y-g\left(a\right)=m\left(x-a\right)\\ y-h\left(2\right)=6\left(x-2\right)\\ y-8=6x-12\\ y-6x+8=0\end{array}$

Therefore, the equation of the secant to the curve is $y-6x+8=0$ .