We are given the following function:
[tex]f(x)=x^3-2x+6[/tex]
And we are asked to determine the average rate of change between points -8 and -4. To do this we will use the following formula for the average rate of change "R" of a function f(x) between points "a" and "b":
[tex]R=\frac{f(b)-f(a)}{b-a}[/tex]
Replacing the points a = -8 and b = -4 in the formula we get:
[tex]R=\frac{f(-4)-f(-8)}{-4-(-8)}[/tex]
Now we need to determine the value of the function at x = -4. To do this we will replace the value of -4 in the given function:
[tex]f(-4)=(-4)^3-2(-4)+6[/tex]
Solving the operations:
[tex]f(-4)=-50[/tex]
Now we replace the value of x = -8 in the function:
[tex]f(-8)=(-8)^3-2(-8)+6[/tex]
Solving the operations:
[tex]f(-8)=-490[/tex]
Now we replace the values of the function in the formula for the average rate of change:
[tex]R=\frac{-50-(-490)}{-4-(-8)}[/tex]
Solving the operations:
[tex]R=\frac{440}{4}[/tex]
Solving the fraction:
[tex]R=110[/tex]
Therefore, the average rate of change is 110.
The same procedure can be used to solve for parts b and c.
