Final answer:
The average rate of change of the function f(x) = -2x² - 7x - 10 from x₁ = -0.5 to x₂ = 6.4 is calculated by finding the difference of f(x₂) and f(x₁) and dividing by the difference of x₂ and x₁, resulting in an average rate of change of approximately -18.8.
Step-by-step explanation:
The average rate of change of a function f(x) over the interval [x₁, x₂] is calculated by f(x₂) - f(x₁) divided by x₂ - x₁. For the given function f(x) = -2x² - 7x - 10, the values of f(x) at x₁ = -0.5 and x₂ = 6.4 are:
- f(-0.5) = -2(-0.5)² - 7(-0.5) - 10 = -2(0.25) + 3.5 - 10 = -0.5 + 3.5 - 10 = -7
- f(6.4) = -2(6.4)² - 7(6.4) - 10 = -2(40.96) - 44.8 - 10 = -81.92 - 44.8 - 10 = -136.72
Therefore, the average rate of change from x₁ = -0.5 to x₂ = 6.4 is:
(f(6.4) - f(-0.5)) / (6.4 - (-0.5)) = (-136.72 - (-7)) / (6.4 + 0.5) = (-136.72 + 7) / 6.9 = -129.72 / 6.9 ≈ -18.8
So, the average rate of change of the function from x₁ = -0.5 to x₂ = 6.4 is approximately -18.8.