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:} \\ (df)/(dx) \\ \text{The average of th rate betwe}en_{}\text{ x1 and x2 is:} \\ \text{average of rate=}(1)/((x_2-x_1))\int ^(x_2)_(x_1)(df)/(dx)dx \\ \text{average of rate=}(1)/((x_2-x_1))(f(x_2)-f(x_2)) \\ \text{average of rate=}((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=}(3-1)/(4-(-6))=(2)/(10)=0.2`

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

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