Interpreting the derivative: rates and units

Warm-up

Three quick questions from earlier lessons.

Discover

$T(m)$ is the temperature of a cup of coffee, in °F, $m$ minutes after it is poured. $T'(10) = -2$ . What are the units of $-2$ ?

°F per minute
°F

Key idea

A derivative is a rate, with units.

If

f(x)

is measured in output units and

x

in input units, then

f'(a)

is in output units per input unit.

f'(a) = r

means: at

x = a

,

f

changes by about

r

for each extra unit of

x

.

Worked example

Interpret $T'(10) = -2$ .

Ten minutes after it is poured…

Input 10 minutes, output a rate in °F per minute.

…the coffee is cooling by about 2 °F per minute

Negative: the temperature goes down.

So

T(11) \approx T(10) - 2

A rate lets you estimate a small change.

Your turn, guided

Your turn: $s(t)$ meters, $t$ seconds. $s(4) = 30$ and $s(4.5) = 34$ .

s'(4) \approx \dfrac{34 - 30}{0.5}

4 meters in 0.5 seconds.

s'(4) \approx 8

m/s

4 ÷ 0.5.

Practice ★

$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(2) = 8$ . What does this mean?
1. At $t = 2$ min, the volume is increasing at about 8 liters per minute.
2. After 2 minutes, the tank holds 8 liters.
3. In the first 2 minutes, 8 liters flowed in.
4. It takes 8 minutes to add 2 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(7) = 11$ . What does this mean?
1. At $t = 7$ min, the volume is increasing at about 11 liters per minute.
2. After 7 minutes, the tank holds 11 liters.
3. In the first 7 minutes, 11 liters flowed in.
4. It takes 11 minutes to add 7 liters.

$C(q)$ is the cost, in dollars, of making $q$ bikes. What are the units of $C'(125)$ ?
1. dollars per bike
2. dollars
3. bikes per dollar

Watch out!

Practice ★★

A car's position is $s(t)$ meters after $t$ seconds. $s(9) = 83$ and $s(9.5) = 88$ . Estimate $s'(9)$ , in m/s.
A car's position is $s(t)$ meters after $t$ seconds. $s(3) = 31$ and $s(3.5) = 37$ . Estimate $s'(3)$ , in m/s.
A car's position is $s(t)$ meters after $t$ seconds. $s(7) = 49$ and $s(7.5) = 53$ . Estimate $s'(7)$ , in m/s.

Exit ticket

Three questions to check you have got it.

$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(2) = 8$ . What does this mean?
1. At $t = 2$ min, the volume is increasing at about 8 liters per minute.
2. After 2 minutes, the tank holds 8 liters.
3. In the first 2 minutes, 8 liters flowed in.
4. It takes 8 minutes to add 2 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(7) = 11$ . What does this mean?
1. At $t = 7$ min, the volume is increasing at about 11 liters per minute.
2. After 7 minutes, the tank holds 11 liters.
3. In the first 7 minutes, 11 liters flowed in.
4. It takes 11 minutes to add 7 liters.
$V(t)$ is the volume of water in a tank, in liters, $t$ minutes after a tap is opened. $V'(3) = 19$ . What does this mean?
1. At $t = 3$ min, the volume is increasing at about 19 liters per minute.
2. After 3 minutes, the tank holds 19 liters.
3. In the first 3 minutes, 19 liters flowed in.
4. It takes 19 minutes to add 3 liters.

Summary card

Reading a derivative

Units: output units per input unit.
$f'(a) = r$ : at $x = a$ , $f$ changes by about $r$ per extra unit of $x$ .

Watch out: $f'(a)$ is a rate at one moment, not a value and not a total change.

About this lesson