What is the average rate of change of ff over the interval -1<= x <= 1?

f(x)=x^2−x−1

That would be (f(1)-f(-1))/(1 - (-1)) = (-1-1)/(1+1) = -1

To find the average rate of change of a function over an interval, we need to calculate the difference between the function values at the endpoints of the interval and divide it by the difference in x-values.

In this case, the function is f(x) = x^2 - x - 1, and the interval is -1 <= x <= 1.

First, let's find the value of f(x) at the endpoints of the interval:

When x = -1:
f(-1) = (-1)^2 - (-1) - 1 = 1 + 1 - 1 = 1

When x = 1:
f(1) = (1)^2 - (1) - 1 = 1 - 1 - 1 = -1

Now, let's calculate the difference in y-values:
Δf = f(1) - f(-1) = -1 - 1 = -2

Next, let's calculate the difference in x-values:
Δx = 1 - (-1) = 1 + 1 = 2

Finally, we can find the average rate of change:
Average rate of change = Δf / Δx = -2 / 2 = -1

Therefore, the average rate of change of f(x) over the interval -1 <= x <= 1 is -1.