AP Calculus ABAP Exams

AP Calculus AB Unit 7 Review: The Most-Tested Differential Equation Questions

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Math Instructor · 10+ Years Experience

October 24, 2026·10 min read

Key takeaways

Unit weight: only 14 of 315 official multiple choice questions (4.4%), but 9.8% of the free-response points.
Top types: separable differential equations with an initial condition, slope fields, and exponential growth and decay.
Practice: a full separation of variables by hand. It was on the free response of 12 of the 15 exams since 2012, worth 4 to 6 of the question’s 9 points.

Unit 7 is small in multiple choice and big in free response. College Board’s Course and Exam Description (the CED) gives Unit 7 5–10% of the multiple choice, and the official exams give it less: 25 of the 719 in-scope multiple choice questions across all 17 official exams from 1969 to 2018 (3.5%), and 14 of 315 on the 7 official practice exams from 2012–2018 (4.4%), 1 to 3 per exam. On the 42-question section that starts in May 2027, that rate is about 2 questions.

The free response is another story. Unit 7 earned 9.8% of the points on the 2012–2026 exams, and 13 of the 86 free-response questions since 2012 had a Unit 7 part, on 13 of the 15 exams, every one in the no-calculator part. One kind of question carries most of that: solving a separable differential equation with an initial condition was on 12 of the 15 exams, worth 4 to 6 of the question’s 9 points. The usual Unit 7 free-response question is a package:

Sketch the slope field at a handful of points (8 questions since 2012), or read one (1).
Use the differential equation: a tangent-line approximation from $\frac{dy}{dx}$ (7 of the 13 questions), sometimes with whether it over- or underestimates (4), or $\frac{d^2y}{dx^2}$ found from $\frac{dy}{dx}$ (5).
Solve it by separating variables (12 questions), usually the last part.

The 4 most-tested Unit 7 question types

Ranked by share of the whole exam. The four add up to about 6.6% of the exam, out of 7.5% for all of Unit 7, and the first one alone is 4.6%. Two smaller types follow them.

1. Separable differential equation with an initial condition

Separate the variables, integrate both sides, add $C$ once, use the initial condition to find $C$ , then solve for $y$ . The free response scores each of those steps, and a question may also ask where the solution is defined.

Topic 7.74.6% of the exam6 of 7 practice exams (6 questions)12 FRQs since 2012, on 12 of the 15 exams

Try it

Find the particular solution

y = f(x)

\frac{dy}{dx} = 2xy^2

with

f(0) = \frac{1}{3}

, and the interval where it is defined.

Solution

$\frac{dy}{y^2} = 2x\,dx$ , so $-\frac{1}{y} = x^2 + C$ . At $\left(0, \frac{1}{3}\right)$ , $-3 = C$ , so $y = \frac{1}{3 - x^2}$ . The denominator is 0 at $x = \pm\sqrt{3}$ , and the solution through $x = 0$ is defined on $-\sqrt{3} < x < \sqrt{3}$ .

Common mistake

adding the constant at the wrong step. For

\frac{dy}{dx} = 2xy

with

y(0) = 3

\ln|y| = x^2 + C

gives

y = Ke^{x^2}

, so the constant multiplies:

y = 3e^{x^2}

. Writing

y = e^{x^2} + 2

meets the initial condition but does not solve the equation.

Practice: Separable Differential Equations worksheet · AP Unit 7 worksheet

2. Slope fields

Each segment has the slope $\frac{dy}{dx}$ at its point. Match a field to an equation by where the segments are horizontal, the sign of the slope off those lines, and whether the slopes change along rows, columns or both.

Topic 7.30.7% of the exam4 of 7 practice exams (4 questions)1 FRQ since 2012 (2026), besides the sketches in type 4

Try it

Which differential equation has the slope field shown?

$\frac{dy}{dx} = x - y$
$\frac{dy}{dx} = y - x$
$\frac{dy}{dx} = x + y$
$\frac{dy}{dx} = xy$

Solution

(A) $\frac{dy}{dx} = x - y$ . It is 0 on $y = x$ , 1 at $(1, 0)$ below the line and $-1$ at $(0, 1)$ above it. (B) has the signs reversed, (C) is 0 on $y = -x$ , and (D) is 0 on the axes.

Common mistake

matching by one feature only. Two equations can both be 0 on the same line; check the sign of the slope at a point off it too.

Practice: Slope Fields worksheet · AP Unit 7 worksheet

3. Exponential growth and decay ( $dy/dt = ky$ )

“The rate of change is proportional to the amount” means $\frac{dy}{dt} = ky$ , so $y = y_0 e^{kt}$ . A doubling time or half-life $T$ gives $k = \pm\frac{\ln 2}{T}$ . This is the Unit 7 type the exam puts in the calculator section.

Topic 7.80.7% of the exam2 of 7 practice exams (2 questions)1 FRQ since 2012 (2021)

Try it

A population grows according to

\frac{dP}{dt} = kP

, with

P(0) = 500

, and it doubles every 6 years. Find

k

and

P(10)

Solution

$500e^{6k} = 1000$ gives $k = \frac{\ln 2}{6} \approx 0.116$ , and $P(10) = 500 \cdot 2^{10/6} \approx 1587.401$ .

Common mistake

using the rate as

k

, or the doubling time as a time constant:

k = \frac{1}{6}

gives

500e^{10/6} \approx 2647.245

. From

\frac{dy}{dt} = ky

k

is the rate divided by the amount, and a doubling time

T

gives

k = \frac{\ln 2}{T}

Practice: Half-Life and Doubling Time worksheet · AP Unit 7 worksheet

4. Sketch a slope field or a solution curve on it (free response)

At each point $(a, b)$ , put $x = a$ and $y = b$ into $\frac{dy}{dx}$ and draw a short segment with that slope. It is usually part (a) of the separable-equation question.

Topic 7.30.6% of the examno multiple choice questions8 FRQs since 2012, 7 of them in the same question as a separable equation

Try it

Sketch the slope field of

\frac{dy}{dx} = x(y - 1)

at the six points with

x = -1

, 0, 1 and

y = 0

, 2.

Solution

on $y = 0$ the slopes are 1, 0, $-1$ ; on $y = 2$ they are $-1$ , 0, 1. The segments at $x = 0$ are horizontal, and the solution through $(0, 1)$ is the horizontal line $y = 1$ .

Common mistake

drawing a segment with the slope of the wrong point, such as swapping

x

and

y

. A solution curve follows the segments and never crosses them.

Practice: Slope Fields worksheet · AP Unit 7 worksheet

Two more Unit 7 types, lower priority

5. Verify a solution of a differential equation

Substitute the function and its derivatives into the whole equation; both sides must agree for every $x$ .

Topic 7.20.3% of the exam1 of 7 practice exams (1 question)1 FRQ since 2012 (2015)

Try it

For which values of

r

y = e^{rx}

a solution of

y'' - y' - 6y = 0

Solution

substituting gives $(r^2 - r - 6)e^{rx} = 0$ , and $e^{rx}$ is never 0, so $r^2 - r - 6 = 0$ : $r = 3$ or $r = -2$ .

Common mistake

checking only part of the equation. Every term has to be substituted, and the equation has to hold for every

x

, not at one point.

Practice: Verifying Solutions worksheet · AP Unit 7 worksheet

6. Write a differential equation from words

“The rate of change of $Q$ ” is $\frac{dQ}{dt}$ ; “proportional to” is $k$ times; “inversely proportional to” is $k$ divided by.

Topic 7.10.2% of the exam1 of 7 practice exams (1 question)no FRQs since 2012 (the last was in 2002)

Try it

A cup of coffee cools in a room at 20°C. The rate of change of its temperature

T

is proportional to the difference between

T

and the room temperature. Write a differential equation, with

k > 0

Solution

$\frac{dT}{dt} = -k(T - 20)$ . While the coffee is hotter than the room, $T - 20 > 0$ , and the minus sign makes the rate negative.

Common mistake

writing the quantity instead of its rate (

T = -k(T - 20)

), or leaving out the room temperature (

\frac{dT}{dt} = -kT

What’s rising and fading in Unit 7

We compare each type’s share of recent exams (multiple choice 2012–2018, free response 2012–2026) with its share across all years: Rising is at least double, Fading under half. No Unit 7 type is Rising, but the unit as a whole grew.

Up: separable equations with an initial condition. 4.81% of recent exams against 2.94% across all years, short of double. Slope fields (topic 7.3) are up too: 1.42% against 0.91%.
Fading: exponential growth and decay (0.38% against 0.78%, confirmed by teacher material). Its last official multiple choice question was in 2017, and its only recent free-response part, in 2021, asked for the long-run value of a solution.
Down a little on the newest free response. Unit 7 went from 11.1% of the free-response points in 2012–2018 to 8.5% in 2019–2026: the separable equation from 8.5% to 7.0%, the slope-field sketch from 1.9% to 1.0%. One question is about 2.3% of a window’s points, so that is about one question, and the separable equation was still on 6 of the 8 exams from 2019 to 2026.

In the CED, rarely asked in multiple choice

Reasoning using slope fields (topic 7.4) and general solutions by separation of variables (topic 7.6) have never been the main topic of an official multiple choice question. Reasoning about a solution from its slope field is folded into the slope-field questions above, and the free response asks for $\frac{d^2y}{dx^2}$ from $\frac{dy}{dx}$ instead.
Writing a differential equation (topic 7.1) has 3 official multiple choice questions, and verifying a solution (7.2) 2.
Euler’s method (topic 7.5) and logistic models (topic 7.9) are BC only.

What students over- and under-practice in Unit 7

We tagged 10,263 multiple choice questions from teacher worksheets, review packets and prep books (reposted official questions and non-AB content removed). Unit 7 is 4.15% of them, against 3.48% of the 719 in-scope official multiple choice questions from 1969–2018, so practice gives the unit a little more than the multiple choice does. Per type:

Question type	Practice vs official share	Ratio
Slope fields	1.38% vs 0.83%	1.7× over
Separable equation with an initial condition	1.53% vs 1.11%	1.4× over
Exponential growth and decay	0.94% vs 0.83%	1.1×, about even
Verify a solution	0.14% vs 0.28%	0.5× under
Write a differential equation	0.17% vs 0.42%	0.4× under

Official counts per type are small (2 to 8 questions), so read the ratios as directions.

Against the whole exam, separable equations are badly under-practiced. Multiple choice alone makes them look over-practiced, but most of their weight is on the free response: practice gives them 1.53% of its questions, about a third of their 4.59% share of the exam.
The slope-field sketch has no multiple choice version, so a multiple choice diet never practices drawing one, though it was on 8 free-response sections since 2012.
Exponential models get about their share while the exam asks them less and less.

Calculator note

Unit 7 is almost all no-calculator. Of the 14 Unit 7 multiple choice questions on the 2012–2018 practice exams, 13 were in the no-calculator part; the one calculator question was an exponential model. Across all 17 official exams, 4 of the 25 were in the calculator section, 3 of them exponential models.

On free response, all 13 questions with a Unit 7 part since 2012 were in the no-calculator part. Separate the variables, integrate and solve for $y$ by hand. Keep the calculator for exponential models, where $k$ or an amount is given to three decimal places.

How we counted

Every number here comes from the public official AP Calculus AB exams: 760 multiple choice questions (1969–2018) and 416 free-response questions (1969–2026), each tagged to its CED topic and question type. Every example problem is our own.

Practice Unit 7

Our free AP Calculus AB Unit 7 worksheet has 20 original multiple choice questions on differential equations, weighted by the exam: 11 on separable equations with an initial condition (the solution, a value of it, the interval where it is defined, the first error in a student’s work, and contexts such as cooling, growth toward a limit and a tank filling with salt), 3 on slope fields (one of them a sketch to choose), 2 on exponential models, and 1 each on writing and verifying a differential equation, on the general solution of a separable equation, and on reasoning about solutions straight from the equation. Part A has 18 questions without a calculator and Part B has 2 with one, both exponential models that need a graphing calculator, with answers to three decimal places. It comes with an answer key and a printable PDF.

For free response, the AP Calculus AB Unit 7 FRQ set has 4 multi-part questions in the exam’s form, each scored out of 9 points, with a scoring guide and a PDF. Both make good practice before the Unit 7 Progress Check in AP Classroom.

Common Questions

The CED gives Unit 7 5–10% of the multiple choice, and the official exams gave it less: 14 of 315 multiple choice questions (4.4%) on the 7 official practice exams from 2012–2018, 1 to 3 per exam. The free response gives it far more: 9.8% of the points on the 2012–2026 exams, with a Unit 7 question on 13 of the 15 exams.

Almost. Solving a separable differential equation with an initial condition was on the free response of 12 of the 15 exams from 2012 to 2026 (not 2015, 2020 or 2025), worth 4 to 6 of the question's 9 points, always without a calculator. Each step earns credit: separating, both antiderivatives, the constant from the initial condition, and solving for y.

Both ways. Multiple choice asks which equation matches a field, or which curve could be a solution on it: 6 official questions, 4 of them on the 2012–2018 practice exams. The free response asks you to sketch the field at a few given points, usually part (a) of the separable-equation question: 8 questions since 2012.

Rarely. Of the 14 Unit 7 multiple choice questions on the 2012–2018 practice exams, 13 were in the no-calculator part, and every free-response question with a Unit 7 part since 2012 was in the no-calculator part. The exception is exponential growth and decay, where a calculator gives k or an amount to three decimal places.

No. Euler's method (topic 7.5) and logistic models (topic 7.9) are BC only. AB Unit 7 covers writing and verifying differential equations, slope fields, separation of variables and exponential models, dy/dt = ky.

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Practice this unit: free AP Calculus AB Unit 7 worksheet

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AP Calculus AB Unit 7 Review: The Most-Tested Differential Equation Questions

The 4 most-tested Unit 7 question types

1. Separable differential equation with an initial condition

2. Slope fields

3. Exponential growth and decay (dy/dt=kydy/dt = kydy/dt=ky)

4. Sketch a slope field or a solution curve on it (free response)

Two more Unit 7 types, lower priority

5. Verify a solution of a differential equation

6. Write a differential equation from words

What’s rising and fading in Unit 7

In the CED, rarely asked in multiple choice

What students over- and under-practice in Unit 7

Calculator note

How we counted

Practice Unit 7

Common Questions

How much of the AP Calculus AB exam is Unit 7?+

Is there a separable differential equation on every AP Calculus AB exam?+

How are slope fields tested on AP Calculus AB?+

Do I need a calculator for Unit 7 questions?+

Are Euler's method and logistic growth on AP Calculus AB?+

Practice this unit: free AP Calculus AB Unit 7 worksheet

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3. Exponential growth and decay ( $dy/dt = ky$ )