§1.6 The second derivative

Introduction

Increasing or decreasing

The Second Derivative

Concavity

Summary

An object has position $s(t)$, velocity $v(t) = s'(t)$, and acceleration $a(t)$. Which formula gives $a(t)$ from $s$?

A twice differentiable function satisfies $f(2) = -3$, $f'(2) = 1.5$, and $f''(2) = -0.25$. Near $x = 2$ the graph of $f$ is

A car's position function has $v(t) \gt 0$ and $a(t) \lt 0$ on some interval. What is the car doing there?

For a potato's temperature, central differences gave $F'(20) \approx 3.645$ and $F'(40) \approx 1.515$ degrees per minute. Estimate $F''(30)$ with a central difference, rounded to four decimals, in degrees per minute per minute.

A bungee jumper's height satisfies $h(4) = 55.5$ and $h(5) = 69.8$ feet. Estimate $h'(4.5)$ in feet per second using a central difference over $[4, 5]$, rounded to one decimal.

If $s'$ is increasing on an interval, then $s'' \gt 0$ on that interval.