Derivatives in context

What you’ll learn

• Interpreting the contextual meaning of $f^{\prime }(x)$.
• Determining units of a derivative using the output-over-input rule.
• Using Desmos to compute derivatives of complicated functions.

Suppose a function $V(t)$ represents the volume of water, in gallons, in a tank at time $t$, where $t$ is in minutes.

Since $V(t)$ represents the amount of water in the tank, an equation such as $V(5)=30$ is interpreted as:

At $t=5$ minutes, the volume of water in the tank is $30$ gallons.

On the other hand, the derivative $V^{\prime }(t)$ represents the instantaneous rate of change.

1. Sign determines direction:

   • If $V^{\prime }(t)>0\to V(t)$ is increasing
   • If $V^{\prime }(t)<0\to V(t)$ is decreasing.
   • If $V^{\prime }(t)=0\to V(t)$ is not changing.

2. Units are a ratio: The units of the derivative, a rate of change, are always

$$\frac{\text{Output units}}{\text{Input units}}$$

AP tip:

For full credit on FRQ questions that ask you to “interpret the meaning of the derivative in context,” your answer must include 4 specific components:

1. Specific input value/time
2. Noun/subject (output)
3. Direction of change (increasing or decreasing)
4. Numerical value with correct units

Use the words “increasing” or “decreasing” based on the sign of the derivative. Do not say “changing at a rate of negative…”

For example, a correct interpretation of the equation $V^{\prime }(5)=12$ satisfies all 4 points above:

At $t=5$ minutes${}^{(1)}$, the volume of water in the tank${}^{(2)}$ is increasing${}^{(3)}$ at a rate of $12$ gallons per minute${}^{(4)}$.

Derivatives with Desmos

For the calculator-active sections of the AP exam, evaluating derivatives using the built-in graphing calculator Desmos is highly efficient.

The temperature of a chemical solution is modeled by the function $H(t)=20+150e^{-0.12t}\sin (0.5t)$, where $t$ is measured in seconds since the experiment began ($t\ge 0$) and $H(t)$ is measured in degrees Celsius.

Find the value of $H^{\prime }(8)$. Using correct units, interpret the meaning of your answer in the context of the problem.

Solution

AP tip:

Unless otherwise specified, final answers on the calculator-allowed FRQ portion must be accurate to 3 decimal places.

Non-time inputs

While most of the contextual questions on the exam use time $t$ as the independent variable, some problems involve other input/output relationships (e.g. see the 2021 AB FRQ #1).

The table below gives values for the weight $W(x)$, in grams, of a 3D-printed plastic cone at select heights $x$, in centimeters. $W$ is a differentiable function.

$x$	$W(x)$
$0$	$10$
$2$	$15$
$4$	$22$
$7$	$35$

a) Find the average rate of change of $W$ over the interval $2\le x\le 7$. Indicate units of measure.

Estimate the value of $W^{\prime }(3)$. Using correct units, interpret the meaning of your answer in the context of the problem.

Solutions

a) Average rate of change of $W$ over $[2,7]$

b) Estimating and interpreting $W^{\prime }(3)$

Interpreting a “rate of a rate”

Some exam questions give a function that is already a rate of change. You can spot these if the prompt explicitly uses the word “rate” or if the units look like gallons per minute or degrees per hour.

In these cases, taking the derivative of the rate gives you the rate of change of the rate itself.

$$\text{Units of }{f}^{\prime \prime }(x)=\frac{\text{Units of }{f}^{\prime }(x)}{\text{Units of }x}$$

For example, if a rate function $R(t)$ is given in gallons per minute, then the units of $R^{\prime }(t)$ will be

$$\frac{\text{gal/min}}{\text{min}}=\frac{\text{gal}}{{\text{min}}^2}$$

Variations in wording such as “gallons per minute per minute” are also acceptable.

The rate at which the population of bacteria in a petri dish grows is modeled by the function $P(t)=200+40e^{-0.15t^2}$, where $P(t)$ is measured in bacteria per hour and $t$ is measured in hours for $0\le t\le 6$.

Find the value of $P^{\prime }(3.5)$ and interpret the meaning of your answer in the context of the problem using correct units.

Solution