## Calculus: Early Transcendentals 8th Edition

(a) $$m_{ave}=\frac{f(x_2)-f(x_1)}{x_2-x_1}.$$ (b) $$m_{inst}=\lim_{x\to x_1}\frac{f(x)-f(x_1)}{x-x_1}.$$
(a) By definition, the average rate of change is obtained by dividing the total change of the value of $f$ over the interval with the length of that interval so simply $$m_{ave}=\frac{f(x_2)-f(x_1)}{x_2-x_1}.$$ (b) The instantaneous rate of change would be the ratio of the total change in the value of the function over an interval and the length of that interval but in the limit where this length tends to zero, i.e. when the endpoints tend to each other. This gives $$m_{inst}=\lim_{x\to x_1}\frac{f(x)-f(x_1)}{x-x_1}.$$