평균변화율과 미분계수

(a) Observe that $v(t) = 16-32t$ is a linear function with negative slope $-32$, and thus decreasing. Since the function is of interest from $t = 0$ to $t=2$, the velocity is greatest at the left endpoint, when $t=0$. This greatest velocity is $v(0) = 16$ feet per second. (b) Since $v$ is a linear function, its slope is constant, and hence the slope of every tangent line to the function is the same as the slope of the line the function generates. Thus, $v'(1) = -32$. (c) Because $v$ is measured in feet per second and $t$ in seconds, the units on $v'(1)$ are “feet per second per second.” Knowing that $v'(1) = -32$, this tells us that the ball's velocity is decreasing at a rate of 32 feet per second per second; that is, for whatever the ball's velocity is at $t = 1$, we would expect the velocity to decrease by (exactly) $32$ feet per second over the next one second. (Normally this total decrease would be approximate, not exact, but since the velocity function is linear, we can compute the exact total change.) (d) The instantaneous rate of change of velocity is precisely the acceleration of the ball. Indeed, $v'(t)$ tells us how the ball's velocity changes, and that it does so at a constant rate of $-32$ feet per second per second. This is also consistent with the fact that acceleration due to gravity is constant.