§1.6 The second derivative

Introduction

Increasing or decreasing

The Second Derivative

Concavity

Summary

A car's position $s(t)$ (in thousands of feet) rises ever more steeply on $[0,1]$, rises ever less steeply on $[1,2]$ to a stop at $s=4$, stays horizontal on $[2,3]$, and repeats that rise-brake-stop pattern on later intervals, never backing up. What is the car doing?

For that position graph, on which intervals is $s(t)$ increasing, and on which is it decreasing?

For the same car, the position graph bends upward (concave up) on $0 \lt t \lt 1$, bends downward on $1 \lt t \lt 2$, and is horizontal on $2 \lt t \lt 3$. What does the velocity $v(t) = s'(t)$ do on these intervals?

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

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

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

If a potato's temperature $F(t)$ is measured in degrees Fahrenheit with $t$ in minutes, then $F'(20) = 3.645$ carries units of degrees Fahrenheit per minute.

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.

For the potato, $F(30) = 167.6$ degrees, $F'(30) \approx 2.065$ degrees per minute, and $F''(30) \approx -0.1065$. What does the triple of values say about the minute from $t = 30$ to $t = 31$?

On the interval $t = 10$ to $t = 40$, the potato's temperature is increasing at a decreasing rate.

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 function $g$ has a derivative $g'$ whose graph is always positive, with $g'(2) \approx 1.4$, and $g'$ rises and falls in waves, changing direction $9$ times on $-3 \lt x \lt 3$. What follows?

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.

Which everyday shape describes a function that is increasing on $-3 \lt x \lt 3$, concave up on $-3 \lt x \lt 0$, and concave down on $0 \lt x \lt 3$?