57. $f(x)=5x - 2$ (a) find the average rate of change from 1 to 3. (b) find an equation of the secant line…

Explanation:

Step1: Calculate f(1) and f(3)

First, find f(1): [ \begin{align*} f(1)&=5\times1 - 2\ &=3 \end{align*} ] Then, find f(3): [ \begin{align*} f(3)&=5\times3 - 2\ &=13 \end{align*} ]

Step2: Calculate the average rate of change

The formula for the average rate of change of a function (y = f(x)) from (x=a) to (x = b) is (\frac{f(b)-f(a)}{b - a}). Here (a = 1), (b=3), (f(1)=3) and (f(3)=13). [ \begin{align*} \text{Average rate of change}&=\frac{f(3)-f(1)}{3 - 1}\ &=\frac{13 - 3}{2}\ &=5 \end{align*} ]

Step3: Find the equation of the secant line

The slope - intercept form of a line is (y=mx + c), where (m) is the slope and (c) is the y - intercept. The slope (m) of the secant line passing through ((x_1,y_1)=(1,3)) and ((x_2,y_2)=(3,13)) is the average rate of change, so (m = 5). Using the point - slope form (y - y_1=m(x - x_1)) with ((x_1,y_1)=(1,3)) and (m = 5): [ \begin{align*} y-3&=5(x - 1)\ y-3&=5x-5\ y&=5x-2 \end{align*} ]

Answer:

(a) 5 (b) (y = 5x-2)