Math Core

In the last two units you learned how to compute derivatives. Now the question changes from "what is $f'(x)$ ?" to "what does $f'(x)$ mean?" On the AP exam, a correct number with no interpretation earns only part of the credit, so learning to read a derivative as a sentence about the real world is a skill in its own right.

The derivative is a rate with units

Suppose $V(t)$ is the volume of water in a tank, in gallons, $t$ minutes after a pump is turned on. The derivative is the limit of a difference quotient:

V'(t) = \lim_{h \to 0} \frac{V(t + h) - V(t)}{h}.

The numerator is measured in gallons and the denominator in minutes, so $V'(t)$ is measured in gallons per minute. That is true for every derivative:

\text{units of } f'(x) = \frac{\text{units of } f}{\text{units of } x}.

A second derivative divides by the input's units one more time. If $V$ is in gallons and $t$ in minutes, then $V''(t)$ is in gallons per minute per minute (gallons/min²). It tells you how fast the rate itself is changing.

Reading f'(a) = k in context

If $f(x)$ measures some quantity and $f'(a) = k$ , then at the instant $x = a$ , the quantity $f$ is changing at a rate of $k$ (units of $f$ ) per (unit of $x$ ).

$k > 0$ : the quantity is increasing at that instant.
$k < 0$ : the quantity is decreasing at that instant, at a rate of $|k|$ .

A complete interpretation has four parts. Always include all of them:

What is changing (the quantity $f$ measures, not just "the function").
When: the specific input value, such as "at time $t = 3$ minutes."
How fast: the number, with correct units.
Which way: increasing or decreasing.

For example, if $V'(3) = -12$ , a full answer is: "At time $t = 3$ minutes, the volume of water in the tank is decreasing at a rate of 12 gallons per minute."

Instantaneous, not average

A derivative describes one instant. It does not say that the tank loses 12 gallons over the next minute; the rate might change during that minute. The statement " $V'(3) = -12$ " says that if the rate stayed the same, the tank would lose about 12 gallons in the next minute. That "about" is the idea behind linear approximation, which comes later in this unit.

Compare this with the average rate of change over an interval:

\frac{V(b) - V(a)}{b - a},

which is also in gallons per minute but describes the whole interval $[a, b]$ , not a single moment.

Common mistake

Don't confuse $f(a)$ , $f'(a)$ and the average rate of change. " $V(3) = 200$ " says there are 200 gallons at $t = 3$ . " $V'(3) = -12$ " says the amount is changing by $-12$ gallons per minute at $t = 3$ . Mixing them up (for example, writing "the tank has $-12$ gallons") is the most common interpretation error on the exam.

Estimating a derivative from a table

AP free-response questions often give a function only as a table of values. You can't differentiate a table, but you can estimate $f'(c)$ with the slope of a secant line between the two data points closest to $c$ that surround it:

f'(c) \approx \frac{f(b) - f(a)}{b - a}, \qquad a < c < b.

Show the difference quotient itself, with the numbers from the table, and then give units. Graders look for that setup.

Worked examples

Worked example: Interpreting a derivative

A cup of tea cools in a room. Let $H(t)$ be its temperature in degrees Fahrenheit $t$ minutes after it is poured, and suppose $H'(4) = -3.5$ . Interpret this value in context.

Solution. The units of $H'(t)$ are degrees Fahrenheit per minute. So: at time $t = 4$ minutes, the temperature of the tea is decreasing at a rate of 3.5 degrees Fahrenheit per minute.

Worked example: Marginal cost

A workshop's cost to build $x$ chairs is $C(x) = 0.02x^2 + 5x + 300$ dollars. Find $C'(100)$ and explain what it means.

Solution. $C'(x) = 0.04x + 5$ , so $C'(100) = 0.04(100) + 5 = 9$ .

The units are dollars per chair. When 100 chairs have been built, the cost is increasing at a rate of $9 per chair. Economists call this the marginal cost: building the 101st chair costs about $9.

Worked example: Estimating from a table

The temperature $T(t)$ of a lake, in degrees Fahrenheit, is recorded at selected times $t$ (hours after sunrise).

$t$ (hours)	0	4	8	12
$T(t)$ (°F)	70	62	57	54

Estimate $T'(6)$ and interpret it.

Solution. The data points surrounding $t = 6$ are $t = 4$ and $t = 8$ :

T'(6) \approx \frac{T(8) - T(4)}{8 - 4} = \frac{57 - 62}{4} = -1.25 \text{ °F per hour}.

At $t = 6$ hours, the lake's temperature is decreasing at a rate of about 1.25 degrees Fahrenheit per hour.

Worked example: A second derivative in context

A town's population is modeled by $P(t) = 500e^{0.03t}$ people, $t$ years after 2010. Find $P'(10)$ and $P''(10)$ and interpret both.

Solution. $P'(t) = 15e^{0.03t}$ and $P''(t) = 0.45e^{0.03t}$ .

$P'(10) = 15e^{0.3} \approx 20.248$ people per year: in 2020 the population is increasing at about 20.2 people per year.

$P''(10) = 0.45e^{0.3} \approx 0.607$ people per year per year: in 2020 the growth rate is itself increasing, by about 0.61 people per year each year. The population is growing faster and faster.

Tip

Before you write an interpretation, say the units out loud: "degrees per minute," "people per year per year." If your sentence doesn't use those units, it is probably describing the wrong thing.

Practice

Practice 1

The height of a sunflower is $h(t)$ centimeters, where $t$ is measured in days. Which statement is the best interpretation of $h'(10) = 0.8$ ?

Practice 2

The area of a circle is $A(r) = \pi r^2$ square inches, where $r$ is the radius in inches. Find $A'(5)$ , the rate of change of area with respect to radius when $r = 5$ inches. Give an exact answer (in square inches per inch).

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 3

The amount of water in a reservoir, $W(t)$ , in millions of gallons, is measured at selected times $t$ in days.

$t$ (days)	0	2	5	9
$W(t)$ (million gallons)	120	116	109	101

Use the data to estimate $W'(7)$ , in millions of gallons per day.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 4

A company's revenue from selling $x$ lamps is $R(x) = 40x - 0.05x^2$ dollars. Find $R'(300)$ , in dollars per lamp.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 5

The temperature of an oven is $F(t)$ degrees Fahrenheit, $t$ minutes after it is switched on. What are the units of $F''(t)$ ?

Practice 6

(Calculator, part a.) The temperature of a bowl of soup is modeled by $H(t) = 70 + 110e^{-0.1t}$ degrees Fahrenheit, where $t$ is in minutes. Find $H'(5)$ in degrees Fahrenheit per minute. Round to two decimal places.

Enter a number. Fractions like 3/4 and sqrt(2) are OK.

Practice 7

(Part b.) For the soup in part (a), $H''(5) = 1.1e^{-0.5} \approx 0.667$ . Which statement is correct at $t = 5$ ?