Chapter 2.3, Problem 68AYU

72. $g\left(x\right)=x^2+1$

(a) Find the average rate of change from -1 to 2.

(b) Find an equation of the secant line containing $\left(-1,g\left(-1\right)\right)$ and $\left(2,g\left(2\right)\right)$.

(c) Using a graphing utility, draw the graph of g and the secant line obtained in part (b) on the same screen.

To determine

To find: The following values using the function $g\left(x\right)=x^2+1$.

a. Average rate of change of a function $g\left(x\right)$ from $-1$ to 2.

Answer to Problem 68AYU

a. Average rate of change of a function $g\left(x\right)$ from $-1$ to 2 is 1.

Explanation of Solution

Given:

The function $g\left(x\right)=x^2+1$.

Calculation:

It is asked to find the average rate of change of $g\left(x\right)=x^2+1$ the values from $-1$ to 2. Also the equation of the secant line containing $\left(-1,g\left(-1\right)\right)$ and $\left(2,g\left(2\right)\right)$.

a. The average rate of change of $g\left(x\right)$ from $-1$ to 2 is,

$\frac{\text{Δ}y}{\text{Δ}x}=\frac{g\left(2\right)-g\left(-1\right)}{2-\left(-1\right)}=\frac{5-2}{3}=1$

To determine

To find: The following values using the function $g\left(x\right)=x^2+1$.

b. An equation of the secant line containing $\left(-1,g\left(-1\right)\right)$ and $\left(2,g\left(2\right)\right)$.

Answer to Problem 68AYU

b. An equation of the secant line containing $\left(-1,g\left(-1\right)\right)$ and $\left(2,g\left(2\right)\right)$ is $y=x+3$.

Explanation of Solution

Given:

The function $g\left(x\right)=x^2+1$.

Calculation:

It is asked to find the average rate of change of $g\left(x\right)=x^2+1$ the values from $-1$ to 2. Also the equation of the secant line containing $\left(-1,g\left(-1\right)\right)$ and $\left(2,g\left(2\right)\right)$.

b. The line containing the two points $\left(a,f\left(a\right)\right)$ and $\left(b,f\left(b\right)\right)$ are called the secant line; its slope is $m_{sec}=\frac{f\left(b\right)-f\left(a\right)}{b-a}=\frac{f\left(a+h\right)-f\left(a\right)}{h}$.

An equation of the secant line containing $\left(-1,g\left(-1\right)\right)$ and $\left(2,g\left(2\right)\right)$ becomes,

$m_{sec}=\frac{g\left(2\right)-g\left(-1\right)}{2-\left(-1\right)}=1$

One of the coordinate is $\left(-1,g\left(-1\right)\right)=\left(-1, 2\right)$.

Equation containing a point and slope is given by $y-y_1=m\left(x-x_1\right)$.

$y-2=1\left(x+1\right)$

$y=x+3$

To determine

To find: The following values using the function $g\left(x\right)=x^2+1$.

c. Draw the graph of $g$ and the secant line obtained in part (b) on the same screen using graphing utility.

Answer to Problem 68AYU

c.

Explanation of Solution

Given:

The function $g\left(x\right)=x^2+1$.

Calculation:

It is asked to find the average rate of change of $g\left(x\right)=x^2+1$ the values from $-1$ to 2. Also the equation of the secant line containing $\left(-1,g\left(-1\right)\right)$ and $\left(2,g\left(2\right)\right)$.

c. Graph: