AP Calculus ABAP Exams

AP Calculus AB Unit 4 Review: The Most-Tested Motion and Related Rates Questions

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Math Instructor · 10+ Years Experience

October 24, 2026·11 min read

Key takeaways

Unit weight: 28 of 315 official multiple choice questions (8.9%), but 19.5% of the free-response points, with a Unit 4 part on all 15 exams since 2012.
Top types: tangent-line approximation, particle motion (at rest, changing direction, speeding up) and related rates with a given equation.
Practice: particle motion. A particle-motion part was on 13 of the 15 exams since 2012.

Unit 4 is under a tenth of the AP Calculus AB multiple choice but about a fifth of the free-response points. College Board’s Course and Exam Description (the CED) gives Unit 4 10–15% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 28 of 315 questions (8.9%), between 3 and 6 per exam, below that range. On the 42-question section that starts in May 2027, that rate is about 4 questions.

The free response is where Unit 4 counts. It earned 19.5% of the points on the 2012–2026 exams, and 49 of the 86 free-response questions since 2012 had a Unit 4 part, on every one of the 15 exams. The common ones:

Particle motion (13 questions, on 13 of the 15 exams): velocity and acceleration, when the particle is at rest or changes direction, and whether it is speeding up.
A rate in context (17 questions): the value of a rate at a time, what a derivative means with units, and whether a rate is increasing or decreasing. 11 of them allowed a calculator.
Tangent-line approximation (11 questions), sometimes followed by “is it an over- or underestimate?”
Related rates with a given equation (8 questions, 2016–2025), often a particle moving along an implicit curve.
L’Hospital’s rule (5 questions, 2018–2023).

The 10 most-tested Unit 4 question types

Ranked by share of the whole exam. The ten add up to about 11.5% of the exam, out of 13.2% for all of Unit 4. Four smaller types follow them.

1. Tangent-line (linear) approximation

Near $x = a$ , $f(x) \approx f(a) + f'(a)(x - a)$ : the tangent line stands in for the function.

Topic 4.61.9% of the exam3 of 7 practice exams (4 questions)10 FRQs since 2012

Try it

Use the line tangent to

f(x) = \sqrt{x}

x = 9

to approximate

\sqrt{9.3}

Solution

$f(9) = 3$ and $f'(9) = \frac{1}{2 \cdot 3} = \frac{1}{6}$ , so $\sqrt{9.3} \approx 3 + \frac{1}{6}(0.3) = 3.05$ .

Common mistake

giving only the change

f'(a)(x - a)

(here 0.05) and forgetting to add

f(a)

, or using the slope at the point being approximated instead of at the point of tangency.

Practice: Linear Approximation worksheet · AP Unit 4 worksheet

2. Particle at rest or changing direction

At rest means $v(t) = 0$ . Changing direction means $v$ changes sign, which is stronger.

Topic 4.21.8% of the exam3 of 7 practice exams8 FRQs since 2012

Try it

A particle moves with velocity

v(t) = (t - 1)(t - 3)^2

for

t \ge 0

. When is it at rest, and when does it change direction?

Solution

it is at rest at $t = 1$ and $t = 3$ . It changes direction only at $t = 1$ : the squared factor keeps $v$ positive on both sides of $t = 3$ .

Common mistake

counting every zero of

v

as a change of direction, or looking for zeros of

a(t)

x(t)

instead of

v(t)

Practice: Motion Along a Line worksheet · AP Unit 4 worksheet

3. Velocity and acceleration from position or velocity

$v(t) = x'(t)$ and $a(t) = v'(t) = x''(t)$ . On the calculator section the model is usually too messy to differentiate by hand.

Topic 4.21.6% of the exam4 of 7 practice exams11 FRQs since 2012

Try it

A particle’s position is

x(t) = t^3 - 6t^2 + 9t

. Find its acceleration at each time its velocity is 0.

Solution

$v(t) = 3t^2 - 12t + 9 = 3(t - 1)(t - 3)$ and $a(t) = 6t - 12$ , so $a(1) = -6$ and $a(3) = 6$ .

Common mistake

giving the position or the wrong derivative (velocity for acceleration). On the calculator, a degree-mode setting turns a correct method into a wrong answer.

Practice: Motion Along a Line worksheet · AP Unit 4 worksheet

4. Related rates with a given equation

The equation relating the quantities is given. Differentiate it with respect to $t$ , then substitute the values at the instant.

Topic 4.41.5% of the exam3 of 7 practice exams, all 3 of its official multiple choice questions8 FRQs since 2012

Try it

x

and

y

are functions of

t

with

x^2 y = 12

. At the instant when

x = 2

y = 3

and

\frac{dx}{dt} = 4

, find

\frac{dy}{dt}

Solution

$2xy \cdot \frac{dx}{dt} + x^2 \cdot \frac{dy}{dt} = 0$ , so $2(2)(3)(4) + 4 \cdot \frac{dy}{dt} = 0$ and $\frac{dy}{dt} = -12$ .

Common mistake

substituting the values before differentiating. A quantity that is changing has to stay a variable until after you differentiate, or its rate disappears.

5. Speeding up or slowing down

The speed $|v|$ increases when $v$ and $a$ have the same sign, and decreases when their signs differ.

Topic 4.21.2% of the exam2 of 7 practice exams11 FRQs since 2012

Try it

v(t) = t^2 - 4t + 3

. Is the particle speeding up or slowing down at

t = 2.5

Solution

$v(2.5) = -0.75$ and $a(2.5) = 2(2.5) - 4 = 1$ . The signs differ, so it is slowing down.

Common mistake

deciding from the sign of

a

alone (

a > 0

, so “speeding up”) or from the sign of

v

alone.

Practice: Motion Along a Line worksheet · AP Unit 4 worksheet

6. L’Hospital’s rule

For a $\frac{0}{0}$ or $\frac{\infty}{\infty}$ form, the limit of $\frac{f}{g}$ equals the limit of $\frac{f'}{g'}$ . Check the form first, every time.

Topic 4.70.9% of the exam2 of 7 practice exams5 FRQs since 2012, all from 2018 to 2023

Try it

Find

\lim\limits_{x \to 0} \frac{e^{3x} - 1 - 3x}{x^2}

Solution

$\frac{0}{0}$ , so take $\frac{3e^{3x} - 3}{2x}$ , which is still $\frac{0}{0}$ . Once more: $\frac{9e^{3x}}{2} \to \frac{9}{2}$ .

Common mistake

applying the rule to a limit that is not

\frac{0}{0}

\frac{\infty}{\infty}

, using the quotient rule instead of differentiating the top and bottom separately, or stopping while the form is still

\frac{0}{0}

. On free response, saying why the rule applies is worth a point.

Practice: L'Hôpital's Rule worksheet · AP Unit 4 worksheet

7. Rate of change in an applied context

The rate at an instant is the derivative there. When a quantity has a rate in and a rate out, it changes at the rate in minus the rate out.

Topic 4.30.8% of the exam2 of 7 practice exams7 FRQs since 2012

Try it

Water enters a tank at

E(t) = 50 + 6t

gallons per minute and leaves at

L(t) = 8t + t^2

gallons per minute. At

t = 4

, is the amount of water increasing or decreasing, and how fast?

Solution

$E(4) - L(4) = 74 - 48 = 26$ , so it is increasing at 26 gallons per minute.

Common mistake

using only the rate in (74), or giving the amount or an average rate instead of the rate at that instant.

Practice: Instantaneous Rate of Change worksheet · AP Unit 4 worksheet

8. Related rates: circles and spheres

Write the formula for the quantity asked (area, volume, circumference), differentiate with respect to $t$ , then substitute.

Topic 4.50.6% of the exam1 of 7 practice examsno FRQs since 2012 (the last was in 2009)

Try it

The radius of a sphere grows at 0.5 cm per second. How fast is its volume growing when the radius is 6 cm?

Solution

$\frac{dV}{dt} = 4\pi r^2 \cdot \frac{dr}{dt} = 4\pi(36)(0.5) = 72\pi$ cubic cm per second.

Common mistake

using a given diameter or circumference as the radius, or differentiating the formula for the wrong measure, such as surface area for volume.

9. Related rates: other geometry

Triangles, cylinders and other shapes where more than one dimension changes. Every changing dimension brings its own rate.

Topic 4.50.6% of the exam1 of 7 practice exams1 FRQ since 2012 (2019)

Try it

A triangle’s base grows at 2 cm per second while its height shrinks at 1 cm per second. How fast is its area changing when the base is 10 cm and the height is 6 cm?

Solution

$A = \frac{bh}{2}$ , so $\frac{dA}{dt} = \frac{b'h + bh'}{2} = \frac{2 \cdot 6 + 10 \cdot (-1)}{2} = 1$ square cm per second.

Common mistake

dropping one of the rates in a product (here 6, from the base alone), or giving a shrinking dimension a positive rate.

10. Interpret a derivative in context

$f'(a)$ is how fast $f$ is changing at the instant $a$ , in units of $f$ per unit of the input.

Topic 4.10.5% of the exam4 of 7 practice exams8 FRQs since 2012

Try it

W(t)

is the number of gallons of water in a tank

t

minutes after noon, and

W'(5) = -3

. What does that mean?

Solution

at 12:05, the amount of water in the tank is decreasing at 3 gallons per minute.

Common mistake

reading

W'(5)

as an amount (“there are 3 gallons”), as a change over the first 5 minutes, or with the wrong units.

Practice: Instantaneous Rate of Change worksheet · AP Unit 4 worksheet

Four more Unit 4 types, lower priority

11. Related rates: ladders and right triangles

The Pythagorean theorem ties the sides together: $x^2 + y^2 = L^2$ , so $x \cdot \frac{dx}{dt} + y \cdot \frac{dy}{dt} = 0$ .

Topic 4.50.5% of the exam1 of 7 practice exams1 FRQ since 2012 (2014)

Try it

A 13-foot ladder leans against a wall. Its bottom slides away from the wall at 2 feet per second. How fast is the top sliding down when the bottom is 5 feet from the wall?

Solution

the top is 12 feet up, and $5(2) + 12 \cdot \frac{dy}{dt} = 0$ gives $\frac{dy}{dt} = -\frac{5}{6}$ . The top slides down at $\frac{5}{6}$ foot per second.

Common mistake

inverting the ratio (

-\frac{12}{5}(2) = -\frac{24}{5}

) or dropping the minus sign of a shrinking length.

12. Is the quantity increasing or decreasing at time $t$ ? (free response)

A rate $R$ is increasing where $R' > 0$ . The sign of $R$ itself says whether the amount is increasing, which is a different question.

Topic 4.30.5% of the examno multiple choice questions8 FRQs since 2012, 6 of them on the calculator section

Try it

A rate is

R(t) = 20 + 12t - t^3

. Is the rate increasing or decreasing at

t = 3

Solution

$R'(t) = 12 - 3t^2$ , so $R'(3) = -15 < 0$ and the rate is decreasing, even though $R(3) = 29$ is positive.

Common mistake

answering from the sign of

R(3)

. The free response wants the sign of

R'(3)

as the reason.

Practice: Instantaneous Rate of Change worksheet · AP Unit 4 worksheet

13. Related rates: cones and similar triangles

Use similar triangles to write the radius in terms of the height, then substitute before you differentiate.

Topic 4.50.4% of the exam1 of 7 practice exams1 FRQ since 2012 (2022)

Try it

Water pours at 3 cubic feet per minute into a tank shaped like an upside-down cone, 10 feet tall with a top radius of 5 feet. When the water is 4 feet deep, at what rate is its depth growing?

Solution

$r = \frac{h}{2}$ , so $V = \frac{\pi h^3}{12}$ and $\frac{dV}{dt} = \frac{\pi h^2}{4} \cdot \frac{dh}{dt}$ . Then $3 = 4\pi \cdot \frac{dh}{dt}$ , and $\frac{dh}{dt} = \frac{3}{4\pi}$ feet per minute.

Common mistake

holding the radius of the water surface constant, or setting up the similar triangles with the wrong sides.

14. Is the tangent-line approximation an over- or underestimate? (free response)

Concave up, the graph lies above its tangent lines, so the tangent line underestimates. Concave down, it overestimates.

Topic 4.60.3% of the examno multiple choice questions4 FRQs since 2012 (2017, 2022, 2023, 2026)

Try it

\frac{dH}{dt} = -0.5(H - 20)

with

H(0) = 100

. Does the tangent line at

t = 0

over- or underestimate

H(1)

Solution

$\frac{d^2H}{dt^2} = -0.5 \cdot \frac{dH}{dt} = 0.25(H - 20)$ , which is positive while $H > 20$ . The graph is concave up, so the tangent line gives an underestimate.

Common mistake

deciding from the first derivative (“

H

is decreasing, so it is an overestimate”), or getting the conclusion backward.

Practice: Linear Approximation worksheet · AP Unit 4 worksheet

What’s rising and fading in Unit 4

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. Unit 4 splits cleanly: speeding up, rates in context, related rates from an equation and L’Hospital’s rule are rising, and the geometry problems are fading.

Rising: speeding up or slowing down. 1.73% of recent exams against 0.63% across all years. Both of its official multiple choice questions are from 2012–2018, and it was in 11 FRQs since 2012. Teacher material written since 2019 confirms the rise.
Rising: related rates with a given equation. 1.98% against 0.95%. All 3 of its official multiple choice questions are on the 2012–2018 practice exams, and its share of free-response points went from 2.4% (2012–2018) to 3.6% (2019–2026).
Rising: L’Hospital’s rule. 1.17% against 0.40%. It was on the free response in 2018, 2019 (twice), 2021 and 2023, and its free-response share went from 0.8% to 2.6%.
Rising: rates in context. The rate of change in an applied context (1.17% against 0.47%) and the free-response question whether a rate is increasing or decreasing (0.72% against 0.30%).
Fading: geometric related rates. Circles and spheres 0.16% against 0.96%, cones 0.16% against 0.75%, other shapes 0.29% against 0.88%. Circles and spheres had 8 official multiple choice questions, only 1 of them from 2012–2018 (2013), and no free-response part since 2009.

Tangent-line approximation is steady, but less on the newest free response. It is 2.20% of recent exams against 1.23% across all years, yet its free-response points went from 4.2% (2012–2018) to 2.1% (2019–2026). Unit 4 as a whole went from 19.8% to 19.2% of the free-response points, about the same. One question is about 2.3% of a window’s points, so read each of these moves as a question or less.

In the CED, rarely asked in multiple choice

Rates in applied contexts (topic 4.3). Two official multiple choice questions ever, both from 2012–2018, but 14 free-response questions since 1998.
Related rates from a given equation (topic 4.4) and L’Hospital’s rule (topic 4.7). Three official multiple choice questions each, but 13 and 5 free-response questions since 1998.
The two free-response-only types, whether a rate is increasing or decreasing and whether an approximation is an over- or underestimate, have never been a multiple choice question.

So a Unit 4 study plan built only from multiple choice undercounts the topics that carry the free-response points.

What students over- and under-practice in Unit 4

We tagged 10,263 multiple choice questions from teacher worksheets, review packets and prep books (reposted official questions and non-AB content removed). Unit 4 is 9.1% of them, against 7.9% of the 719 in-scope official multiple choice questions from 1969–2018. Per type:

Question type	Practice vs official share	Ratio
Related rates: other geometry	0.60% vs 0.28%	2.2× over
L’Hospital’s rule	0.84% vs 0.42%	2.0× over
Related rates: cones and similar triangles	0.55% vs 0.28%	2.0× over
Speeding up or slowing down	0.42% vs 0.28%	1.5× over
Velocity and acceleration	1.83% vs 1.39%	1.3× over
Related rates: ladders and right triangles	0.70% vs 0.56%	1.3× over
Tangent-line approximation	1.42% vs 1.25%	1.1×, even
Rate of change in an applied context	0.30% vs 0.28%	1.1×, even
Related rates with a given equation	0.44% vs 0.42%	1.0×, even
Particle at rest or changing direction	1.00% vs 0.97%	1.0×, even
Related rates: circles and spheres	0.79% vs 1.11%	0.7× under
Interpret a derivative in context	0.20% vs 0.70%	0.3× under

Official counts per type are small (2 to 10 questions), so read the ratios as directions.

Practice leans on the fading geometry. Cones and the other related-rates shapes get about twice their official share, on types that are fading.
L’Hospital’s rule is over-practiced in multiple choice but rising on the free response, where you also have to say why the rule applies. Practice writing that sentence.
Interpreting a derivative is the biggest gap: under a third of its official share, though it was on 4 of the 7 practice exams and in 8 FRQs since 2012.
The free-response-only types get no multiple choice practice at all, because there is none to copy. Each is worth one or two points with a short reason.

Calculator note

Unit 4 is the first unit where the calculator does real work. Of the 28 Unit 4 multiple choice questions on the 2012–2018 practice exams, 16 were in the no-calculator part and 12 in the calculator part. The calculator part had all 4 derivative interpretations, 3 of the 4 velocity and acceleration questions, 2 of the 4 tangent-line approximations, 1 of the 2 speeding-up questions, and 2 of the 4 geometric related-rates questions (a sphere and a cone). Every at-rest, given-equation related rates, L’Hospital and rate-in-context question was in the no-calculator part.

On free response, 19 of the 49 questions with a Unit 4 part allowed a calculator (counting 2020, when a calculator was allowed throughout): 7 of the 13 particle-motion questions and 11 of the 17 rate-in-context questions, but only 1 of the 11 tangent-line approximations and none of the 5 L’Hospital questions.

So know how to take a derivative at a point on your calculator, keep it in radian mode, and give answers to three decimal places. Do related rates, approximations and L’Hospital by hand.

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 4

Our free AP Calculus AB Unit 4 worksheet has 20 original multiple choice questions on motion, related rates, linear approximation and L’Hospital’s rule, a no-calculator Part A and a calculator Part B, weighted by how often the official exams ask each type. Most of the Part B questions 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 4 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 4 Progress Check in AP Classroom.

Common Questions

The CED gives Unit 4 10–15% of the multiple choice. On the 7 official practice exams from 2012–2018 it was 28 of 315 multiple choice questions (8.9%), 3 to 6 per exam. In free response it earned 19.5% of the points on the 2012–2026 exams, with a Unit 4 part on every exam: particle motion, rates in context, tangent-line approximations, related rates and L'Hospital's rule.

Mostly on the free response: a particle-motion part was on 13 of the 15 exams from 2012 to 2026. The usual parts are velocity or acceleration at a time, when the particle is at rest or changes direction (v changes sign), and whether it is speeding up (v and a have the same sign). Questions with a model too messy to differentiate by hand go in the calculator section.

Yes, and more than before. It has only 3 official multiple choice questions, but it was on the free response in 2018, 2019, 2021 and 2023. Check that the limit is 0/0 or ∞/∞ before applying it, differentiate the top and bottom separately, and on free response say why the rule applies.

Yes, but the kind has changed. Related rates with a given equation is rising: all 3 of its official multiple choice questions are on the 2012–2018 practice exams, and it was in 8 free-response questions from 2016 to 2025, often a particle moving along an implicit curve. The classic geometry problems (spheres, cones, ladders) are fading: circles and spheres have had no free-response part since 2009.

For some. Of the 28 Unit 4 multiple choice questions on the 2012–2018 practice exams, 12 were in the calculator part, including all 4 derivative interpretations and 3 of the 4 velocity and acceleration questions. Know how to take a derivative at a point on your calculator in radian mode, and give answers to three decimal places. Related rates from an equation, L'Hospital's rule and finding when a particle is at rest were all no-calculator.

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AP Calculus AB Unit 4 Review: The Most-Tested Motion and Related Rates Questions

The 10 most-tested Unit 4 question types

1. Tangent-line (linear) approximation

2. Particle at rest or changing direction

3. Velocity and acceleration from position or velocity

4. Related rates with a given equation

5. Speeding up or slowing down

6. L’Hospital’s rule

7. Rate of change in an applied context

8. Related rates: circles and spheres

9. Related rates: other geometry

10. Interpret a derivative in context

Four more Unit 4 types, lower priority

11. Related rates: ladders and right triangles

12. Is the quantity increasing or decreasing at time ttt? (free response)

13. Related rates: cones and similar triangles

14. Is the tangent-line approximation an over- or underestimate? (free response)

What’s rising and fading in Unit 4

In the CED, rarely asked in multiple choice

What students over- and under-practice in Unit 4

Calculator note

How we counted

Practice Unit 4

Common Questions

How much of the AP Calculus AB exam is Unit 4?+

How is particle motion tested on the AP Calculus AB exam?+

Is L'Hospital's rule on the AP Calculus AB exam?+

Are related rates still on the AP Calculus AB exam?+

Do I need a calculator for Unit 4 questions?+

Practice this unit: free AP Calculus AB Unit 4 worksheet

Keep reading

AP Calculus AB Review: The Most-Tested Questions, Unit by Unit

AP Calculus AB Unit 1 Review: The Most-Tested Limits and Continuity Questions

AP Calculus AB Unit 2 Review: The Most-Tested Derivative Questions

12. Is the quantity increasing or decreasing at time $t$ ? (free response)