Final answer:
The average rate of change of the function f(x)=-x²-5x+12 over the interval −5≤x≤4 is calculated by evaluating the function at the interval endpoints and using the formula. The result is -4.
Step-by-step explanation:
The average rate of change of a function over a given interval is similar to the slope of the line that connects the end points of the function on that interval. To calculate the average rate of change of the function f(x)=-x²-5x+12 over the interval −5≤x≤4, we need to evaluate the function at the endpoints of the interval and then apply the average rate of change formula: (f(x_2) - f(x_1)) / (x_2 - x_1).
Firstly, let's evaluate f(x) at the endpoints:
- f(-5) = -(-5)² - 5(-5) + 12 = -25 + 25 + 12 = 12
- f(4) = -(4)² - 5(4) + 12 = -16 - 20 + 12 = -24
Now, apply the formula to find the average rate of change:
(f(4) - f(-5)) / (4 - (-5)) = (-24 - 12) / (4 + 5) = -36 / 9 = -4.
Therefore, the average rate of change of the function f(x) over the interval −5≤x≤4 is -4.