University Calculus: Early Transcendentals (3rd Edition)

Published by Pearson
ISBN 10: 0321999584
ISBN 13: 978-0-32199-958-0

Chapter 2 - Questions to Guide Your Review - Page 110: 2


To find the rate of change of function $g(t)$ at $t=t_0$, the following limit must be calculated: $$\lim_{h\to0}\frac{\Delta y}{\Delta x}=\lim_{h\to0}\frac{g(t_0+h)-g(t_0)}{h}$$

Work Step by Step

To find the rate of change of the function $y=g(t)$ at $t=t_0$, a limit of average rate of change must be calculated. Recall the average rate of change formula for $y=g(t)$ over the interval from $t=t_0$ to $t=t_1$ and $t_1-t_0=h$: $$\frac{\Delta y}{\Delta x}=\frac{g(t_1)-g(t_0)}{t_1-t_0}=\frac{g(t_0+h)-g(t_0)}{h}$$ To calculate the limit of average rate of change for $g(t)$ at $t=t_0$, we calculate the limit of the function of the rate of change above as $h\to0$: $$\lim_{h\to0}\frac{\Delta y}{\Delta x}=\lim_{h\to0}\frac{g(t_0+h)-g(t_0)}{h}$$
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