Final answer:
To find the average rate of change of the function − between x = -1 and x = 2, calculate the difference in function values and divide by the change in x. The instantaneous rate of change at x = -1 is found by evaluating the derivative of the function at that point.
Step-by-step explanation:
The question involves finding the average rate of change of the function f(x) = x2 − 3x between two points and finding the instantaneous rate of change at a given point. To find the average rate of change between x=-1 and x=2, we calculate the difference in function values and divide by the change in x:
Calculate f(-1) and f(2).
Compute the average rate of change as [f(2) - f(-1)] / (2 - (-1)).
The instantaneous rate of change at x=-1 is the derivative of the function at x=-1. The derivative, f'(x) = 2x - 3, gives us the slope of the tangent line to the curve at any point x. So we calculate f'(-1) = 2(-1) - 3.
Find the derivative of the function, f'(x).
Plug in x=-1 to find f'(-1).
To find the average rate of change of the function, we need to determine the change in the function's value over the given interval and divide it by the change in the independent variable. In this case, the function is f(x) = x^2 - 3x. So, we calculate f(2) - f(-1) and divide it by 2 - (-1).
To find the instantaneous rate of change at x = -1, we need to find the derivative of the function and evaluate it at x = -1. The derivative of f(x) is f'(x) = 2x - 3. So, we evaluate f'(-1).