Chapter 2.3, Problem 66AYU

70. $f\left(x\right)=-4x+1$

(a) Find the average rate of change from 2 to 5.

(b) Find an equation of the secant line containing $\left(2,f\left(2\right)\right)$ and $\left(5,f\left(5\right)\right)$.

To determine

To find: The following values using the function $f\left(x\right)=-4x+1$,

a. Average rate of change of a function $f\left(x\right)$ from 2 to 5.

Answer to Problem 66AYU

a. Average rate of change of a function $f\left(x\right)$ from 2 to 5 $=-4$.

Explanation of Solution

Given:

The function $f\left(x\right)=-4x+1$.

Calculation:

It is asked to find the average rate of change of $f\left(x\right)=-4x+1$ the values from 2 to 5. Also the equation of the secant line containing $\left(2,f\left(2\right)\right)$ and $\left(5,f\left(5\right)\right)$.

a. The average rate of change of a function $f\left(x\right)$ from 2 to 5 is,

$\frac{\text{Δ}y}{\text{Δ}x}=\frac{f\left(5\right)-f\left(2\right)}{5-2}=\frac{-19+7}{3}=-4$

To determine

To find: The following values using the function $f\left(x\right)=-4x+1$,

b. An equation of the secant line containing $\left(2,f\left(2\right)\right)$ and $\left(5,f\left(5\right)\right)$.

Answer to Problem 66AYU

b. An equation of the secant line containing $\left(2,f\left(2\right)\right)$ and $\left(5,f\left(5\right)\right)$ is $y=15-4x$.

Explanation of Solution

Given:

The function $f\left(x\right)=-4x+1$.

Calculation:

It is asked to find the average rate of change of $f\left(x\right)=-4x+1$ the values from 2 to 5. Also the equation of the secant line containing $\left(2,f\left(2\right)\right)$ and $\left(5,f\left(5\right)\right)$.

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 $\left(2,f\left(2\right)\right)$ and $\left(5,f\left(5\right)\right)$ becomes,

$m_{sec}=\frac{f\left(5\right)-f\left(2\right)}{5-2}=-4$

One of the coordinate is $\left(2,f\left(2\right)\right)=\left(2, -7\right)$.

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

$y+7=-4\left(x-2\right)$

$y=15-4x$