Answer
Average rate of change for interval [1,2]: $\frac{1}{2}$
Instantaneous rate of change for endpoint 1: $1$
Instantaneous rate of change for endpoint 2: $\frac{1}{4}$
Work Step by Step
Average rate of change is given by the formula: $$\frac{f(x_2) - f(x_1)}{x_2 - x_1}$$ for points $(x_1, f(x_1))$ and $(x_2, f(x_2))$. In this case, for endpoints $[1,2]$: $$f(x) = -\frac{1}{x}$$ $$f(1) =-\frac{1}{(1)} = -1$$ $$f(2) = -\frac{1}{(2)} = -\frac{1}{2}$$ giving an average rate of change: $$AverageRate = \frac{-0.5 - (-1)}{2 - 1} = \frac{1}{2}$$ The instantaneous rate of change is given by the first derivative of the function: $$f(x) = -\frac{1}{x} = x^{-1}$$ $$f'(x) = x^{-1-1} =x^{-2} = \frac{1}{x^{2}}$$ which, for the endpoints of the interval [1,2] are as follows: $$f'(2) = \frac{1}{(2)^2} = \frac{1}{4}$$ $$f'(1) = \frac{1}{(1)^{2}} = 1$$
