Calculus (3rd Edition)

a) $-144.7213595; 0$ b) $0$
We are given the function: $h(t)=8\cos (12\pi t)$ a) The average rate of change on $[t_0,t_1]$ is: $\dfrac{Δh}{Δt}=\dfrac{h(t_1)−h(t_0)}{t_1−t_0}$ Compute the average rate of change on $[0,0.1]$: $\dfrac{Δh}{Δt}=\dfrac{h(0.1)−h(0)}{0.1-0}=\dfrac{8\cos(12\pi\cdot 0.1)-8\cos(12\pi\cdot 0)}{0.1}$ $\approx -144.7213595$ Compute the average rate of change on $[3,3.5]$: $\dfrac{Δh}{Δt}=\dfrac{h(3.5)−h(3)}{3.5-3}=\dfrac{8\cos(12\pi\cdot 3.5)-8\cos(12\pi\cdot 3)}{0.5}$ $=0$ b) The instantaneous rate of change is the limit of the average rate of change. In order to estimate the instantaneous rate of change at $t=0$, consider intervals $[t_1,t_0],[t_0,t_1]$ for $t_1$ close to $t_0$, where $t_0=3$ and build the table: From the table we find that the instantaneous rate of of change at $t=3$ is approximately $0$.