From the story. A reviewer of the memoir noted that the boys' teachers handed Homer books on calculus as well as rocketry. Calculus is the language of anything that changes, and a rocket's speed changes every instant. (More in October Sky.)
If height is a function of time, $h(t)$, then velocity is how fast it changes: the derivative. Going the other way, if you know the acceleration, the change in velocity is the area under its graph: the integral.
Key idea
$$v(t) = \frac{dh}{dt}, \qquad a(t) = \frac{dv}{dt}, \qquad v(t) = v(0) + \int_0^t a\,dt, \qquad h(t) = h(0) + \int_0^t v\,dt.$$ At apogee $v = h'(t) = 0$.
Worked example: The top of the coast
After burnout, $h(t) = 120 + 45t - 4.9t^2$. Find the greatest height.
- $h'(t) = 45 - 9.8t = 0$ at $t = 4.59\ \text{s}$.
- $h(4.59) = 120 + 45(4.59) - 4.9(4.59)^2 \approx 223\ \text{m}$.
Answer: $\approx 223\ \text{m}$
Common mistake
Forgetting the starting value after integrating: $\int a\,dt$ gives the change in velocity, not the velocity itself.