Chapter 2.3, Problem 67AYU

a)

To determine

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

Answer to Problem 67AYU

Average rate of change is $1$ .

Explanation of Solution

Given:

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

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.

  $g\left(x\right)=x^2-2$ …… (1)

Consider the given $x$ values.

  $x=-2$ and $x=1$

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

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

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

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

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

  $\begin{array}{c}Avgg\left(x\right)=\frac{g\left(b\right)-g\left(a\right)}{b-a}\\ =\frac{g\left(1\right)-g\left(-2\right)}{1+2}\\ =\frac{2+1}{3}\\ =\frac{3}{3}\\ =1\end{array}$

Hence, the average rate change of the function is $1$ .

b)

To determine

To find: The equation for the secant line to the given curve joining the point $\left(1,g\left(1\right)\right)$ and $\left(-2,g\left(-2\right)\right)$ .

Answer to Problem 67AYU

The equation of the secant to the curve is $y-x+2=0$ .

Explanation of Solution

Given:

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

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.

  $g\left(x\right)=x^2-2$ …… (1)

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

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

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

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

Slope of the secant line

  $\begin{array}{c}Slope\left(m\right)=\frac{g\left(b\right)-g\left(a\right)}{b-a}\\ =\frac{g\left(1\right)-g\left(-2\right)}{1+2}\\ =\frac{2+1}{3}\\ =\frac{3}{3}\\ =1\end{array}$

Therefore, equation of the secant to the curve at $\left(1,g\left(1\right)\right)$ is

  $\begin{array}{l}y-g\left(a\right)=m\left(x-a\right)\\ y-g\left(1\right)=1\left(x-1\right)\\ y+1=x-1\\ y-x+2=0\end{array}$

Therefore, the equation of the secant to the curve is $y-x+2=0$ .