Question

Find the average rate of change between the following points. A. x= -6 to x=4B. x= -2 to x=2

Answer 1

We have a funtion f(x) in the picture and watn to know the average rate of change in some interval of x, so:

$\begin{gathered} \text{The rate of change of a function is its derivative:} \\ \frac{df}{dx} \\ \text{The average of th rate betwe}en_{}\text{ x1 and x2 is:} \\ \text{average of rate=}\frac{1}{(x_2-x_1)}\int ^{x_2}_{x_1}\frac{df}{dx}dx \\ \text{average of rate=}\frac{1}{(x_2-x_1)}(f(x_2)-f(x_2)) \\ \text{average of rate=}\frac{(y_2-y_1)}{(x_2-x_1)} \end{gathered}$

In case a), (x1, y1) = (-6, 1) and (x2, y2) = (4, 3), so:

$\text{average of rate=}\frac{3-1}{4-(-6)}=\frac{2}{10}=0.2$

In case b), (x1, y1) = (-2, -3) and (x2, y2) = (2, -1), so:

$\text{average of rate=}\frac{(-1-(-3))}{(2-(-2))}=\frac{2}{4}=\frac{1}{2}=0.5$