Worksheets · AP Calculus AB

AP Calculus AB Unit 4 Worksheet: Derivatives in Context

20 exam-style multiple-choice questions on motion, related rates, linear approximation and L'Hospital's rule, free to use in class. Good practice before the Unit 4 Progress Check in AP Classroom. Free response for this unit: Unit 4 FRQ set.

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AP Calculus AB Unit 4: Derivatives in Context

Date Version A

Directions: Work each problem in the space below it, then choose the best of the four answer choices. Unless a question says otherwise, the domain of a function f is every real number x for which f(x) is a real number.

Part A: No calculator is allowed.

Questions 1–13
  1. Let J(t) be the temperature, in degrees Celsius, of a bottle of juice t minutes after it is placed in a room kept at 24 degrees Celsius to cool. The function J satisfies dJdt=−14(J−24), and J(0)=72. For all t≥0, J(t)>24. Using the tangent line at t=0, a student estimates J(1)≈60. Which of the following correctly describes this estimate?
    1. (A)An underestimate, because the graph of J is concave up for t≥0
    2. (B)An overestimate, because the graph of J is concave up for t≥0
    3. (C)An overestimate, because the graph of J is concave down for t≥0
    4. (D)An underestimate, because J is decreasing for t≥0
  2. The velocity of a particle on the x-axis is v(t)=2(t−1)2(t−7)(t−4) for t>0. At which times does the particle reverse its direction of motion?
    1. (A)t=1, t=4, and t=7
    2. (B)t=4 only
    3. (C)t=4 and t=7
    4. (D)t=1 only
  3. For 0≤t≤8 hours, hikers arrive at a campground at the rate E(t)=t+31 hikers per hour. The number of hikers who have departed from the campground during the first t hours is L(t)=2t2+13t. At t=5, at what rate is the number of hikers at the campground changing, in hikers per hour?
    1. (A)−79
    2. (B)−32
    3. (C)36
    4. (D)3
  4. A particle's position on a line is s(t)=t2−2t+4 for t≥0. At which times is the particle at rest?
    1. (A)t=−1 only
    2. (B)t=1 only
    3. (C)t=2 only
    4. (D)t=0 and t=1
  5. A conveyor drops road salt onto a storage yard, building a mound whose base is a circle of radius r meters. The mound holds V=r32 cubic meters. The base's circumference grows at a steady 6π meters per hour. At the moment the circumference is 8π meters, how fast is the volume of the mound increasing, in cubic meters per hour?
    1. (A)288
    2. (B)144π
    3. (C)24
    4. (D)72
  6. Evaluate lim⁡x→∞ln⁡ ⁣(1+e6x)3x.
    1. (A)nonexistent
    2. (B)2
    3. (C)13
    4. (D)12
  7. Two cyclists on straight perpendicular roads head toward the point where the paths cross. One is 15 kilometers west of the crossing and rides at 21 kilometers per hour; the other is 20 kilometers south of it and rides at 26 kilometers per hour. How fast, in kilometers per hour, is the straight-line distance between them changing at that moment?
    1. (A)−1675
    2. (B)1675
    3. (C)−1045
    4. (D)−47
  8. The position of a particle on the x-axis is x(t)=3sin⁡2t+3cos⁡2t, t≥0. Find the acceleration of the particle at the earliest time its velocity equals zero.
    1. (A)32
    2. (B)−32
    3. (C)π8
    4. (D)−122
  9. The path of a moving point is the curve y3+2xy=−12, and at one instant the point is at (1,−2) with dxdt=−2. What is dydt there?
    1. (A)27
    2. (B)47
    3. (C)−7
    4. (D)−47
  10. At time t, a particle on the x-axis is at s(t)=t3−6t2. When t=6, is it moving left or right, and is it speeding up or slowing down?
    1. (A)speeding up while moving left
    2. (B)slowing down while moving right
    3. (C)speeding up while moving right
    4. (D)slowing down while moving left
  11. A triangle is being reshaped: its base b grows at 4 inches per minute, and at the same time its height h shrinks at 3 inches per minute. Which of the following must be true about A, the area of the triangle?
    1. (A)The area A is decreasing only when b>43h.
    2. (B)The area A is always increasing.
    3. (C)The area A is decreasing only when b<43h.
    4. (D)The area A is always decreasing.
  12. The graph of a differentiable function f passes through (4,1), and f′(4)=−2. Where does the line tangent to the graph at that point meet the x-axis? (This value estimates a zero of f.)
    1. (A)0.5
    2. (B)9
    3. (C)6
    4. (D)4.5
  13. The point (−2,3) lies on the curve x3+2xy+y2=−11. A point of the curve near it has x-coordinate −1.7; its y-coordinate is estimated with the tangent line at (−2,3). What is the estimate?
    1. (A)8930
    2. (B)310
    3. (C)5710
    4. (D)−2710

Part B: A graphing calculator is allowed.

Questions 14–20
  1. A particle moves along the x-axis. For t≥0 its velocity is v(t)=5te−0.8t−2. How is the particle moving at t=2.5?
    1. (A)slowing down while moving right
    2. (B)speeding up while moving right
    3. (C)speeding up while moving left
    4. (D)slowing down while moving left
  2. V(t)=492+194cos⁡(0.6t) gives the rate, in visitors per hour, at which visitors enter a zoo; t is measured in hours. At t=6, is the rate at which visitors enter the zoo increasing or decreasing, and for what reason?
    1. (A)Increasing, because V′(6)=51.509>0
    2. (B)Decreasing, because V(6)−V(0)6=−61.329<0
    3. (C)Decreasing, because V′(6)=−0.128<0
    4. (D)Increasing, because V(6)=318.029>0
  3. A particle starts moving along the x-axis at t=0, and its position at time t is x(t)=3cos⁡ ⁣(1.5t). What is its velocity the first time it passes through the origin?
    1. (A)1.097
    2. (B)−2.149
    3. (C)−3.000
    4. (D)0.716
  4. A spherical ball of ice cream melts so that its volume decreases at a constant rate of 8 cubic centimeters per minute, and it stays spherical. At the instant its surface area is 300 square centimeters, at what rate is the surface area changing, in square centimeters per minute? (A sphere of radius r has volume V=43πr3 and surface area S=4πr2.)
    1. (A)−0.027
    2. (B)−0.670
    3. (C)−3.275
    4. (D)−1.637
  5. The line tangent to the graph of f(x)=ln⁡ ⁣(x2+3) at x=1 is used to estimate f(0.7). What value does the tangent line give?
    1. (A)−0.150
    2. (B)1.236
    3. (C)1.266
    4. (D)1.250
  6. A paper cup has the shape of a cone with its point at the bottom; it is 12 centimeters tall and the radius of its rim is 5 centimeters. Lemonade flows in at 6 cubic centimeters per second. How fast, in centimeters per second, is the level of the lemonade rising when the lemonade is 3 centimeters deep? (A cone with radius r and height h has volume V=13πr2h.)
    1. (A)1.222
    2. (B)3.667
    3. (C)0.037
    4. (D)0.818
  7. The rate, in bagels per hour, at which a bakery sells bagels is modeled by Q(t)=31t2e−0.9t+7 for 0≤t≤7 hours. Which of the following gives Q′(3), to three decimal places, together with what it means here?
    1. (A)Q′(3)=4.375: at t=3 hours, the rate at which the bakery sells bagels is increasing at a rate of 4.375 bagels per hour per hour
    2. (B)Q′(3)=25.750: at t=3 hours, the rate at which the bakery sells bagels is 25.750 bagels per hour
    3. (C)Q′(3)=−4.375: at t=3 hours, the rate at which the bakery sells bagels is decreasing at a rate of 4.375 bagels per hour per hour
    4. (D)Q′(3)=6.250: from t=0 to t=3, the rate at which the bakery sells bagels increased at an average rate of 6.250 bagels per hour per hour

    End of Version A

Answer key · Version A

12345678910
ACDBDBADDC
4.64.24.34.24.44.74.54.24.44.2
11121314151617181920
ADBCABCBAC
4.54.64.64.24.34.24.54.64.54.1

The second row is the correct letter; the third is the course topic each question tests.

Free AP Calculus AB worksheets at sigmaprep.io/worksheets/ap-calculus-abOriginal questions written by Sigma Prep. AP is a registered trademark of the College Board, which is not affiliated with this sheet.

What does the AP Calculus AB exam ask on Unit 4?

Less than a tenth of the multiple choice: Unit 4 was 28 of the 315 official multiple-choice questions on the seven 2012–2018 practice exams (8.9%), but it is 19.5% of the free-response points on the 2012–2026 exams.

This worksheet has 20 questions in that mix: Part A (13) without a calculator and Part B (7) with one; the exam put 42% of its official Unit 4 multiple choice in the calculator section.

Counted from the official AP Calculus AB exams: multiple choice from 17 exams, 1969–2018, free response 2012–2026. No official question is reproduced here.

AP Calculus AB Unit 4 question types: how often the exam asks each

Official AP Calculus AB multiple-choice questions by question type, Unit 4 (Contextual Applications of Differentiation), and how many are on this worksheet
Question typeOn this worksheetOfficial questions, 1969–2018Exams it was on (of 17)On the 2012–2018 practice exams
Interpret a derivative in contextTopic 4.11554
Velocity and acceleration from position or velocityTopic 4.221094
Particle at rest or changing directionTopic 4.22773
Speeding up or slowing downTopic 4.22222
Rate of change in an applied contextTopic 4.31222
Is the quantity increasing or decreasing at time t? (FRQ)Topic 4.31Asked on 15 free-response questions since 1998, most recently in 2024.
Related rates with a given equationTopic 4.42333
Related rates: circles and spheresTopic 4.51881
Related rates: ladders and right trianglesTopic 4.51441
Related rates: cones and similar trianglesTopic 4.51221
Related rates: other geometryTopic 4.51221
Tangent-line (linear) approximationTopic 4.63984
Tangent-line approximation: over- or underestimate? (FRQ)Topic 4.61Asked on 11 free-response questions since 1998, most recently in 2026.
L'Hospital's RuleTopic 4.71332

Counts are of official multiple-choice questions whose main topic is that kind, on 17 College Board exams 1969–2018 (7 released 1969–1998, 2003 and 2008 released, the 2008 practice exam and the seven 2012–2018 practice exams). No official multiple-choice questions after 2018 are public.

Free-response practice for this unit: 4 FRQs with a scoring guide.

Why these kinds of question: AP Calculus AB Unit 4 Review: The Most-Tested Motion and Related Rates Questions.

Questions about this worksheet

Unit 4, Contextual Applications of Differentiation, has 7 topics in the College Board's course description: interpreting a derivative in context, with units; straight-line motion (position, velocity and acceleration, when a particle is at rest or changes direction, and when it speeds up or slows down); rates of change in other applied contexts; related rates, from a given equation and from geometry such as circles, spheres, ladders and cones; approximating a value with the tangent line (linearization); and L'Hospital's rule for 0/0 and ∞/∞ limits. The course description weights the unit at 10–15% of the multiple-choice section.

Only in Part B, as on the exam. Questions 1–13 are Part A, with no calculator; questions 14–20 are Part B, where the exam allows a graphing calculator (on the digital exam, the Desmos calculator built into Bluebook is available there). Unlike Units 1 to 3, most of these Unit 4 Part B questions need one: the acceleration or speed of a particle from a model, related rates and tangent-line approximations worked on the calculator, and the value of a rate with its meaning in context. Their answers are given to three decimal places, as on the exam. From May 2027 the exam's multiple-choice section is 29 questions without a calculator and 13 with one.

We tagged 760 official multiple-choice questions from 17 College Board exams (1969–2018) and the free-response questions from 1969 to 2026, sorted the Unit 4 questions into 14 kinds, and weighted each kind by how often and how recently the exam asks it. Every kind the exam actually uses gets at least one question, and the most-tested get more (3 on tangent-line (linear) approximation, 2 on velocity and acceleration from position or velocity, 2 on particle at rest or changing direction, 2 on speeding up or slowing down, 2 on related rates with a given equation).

The progress check is College Board's own end-of-unit quiz in AP Classroom, assigned by your teacher. Its questions are not public, and we do not have them: Sigma Prep is not affiliated with College Board. This worksheet is exam-style practice on the same unit: 20 original multiple-choice questions, weighted by how often the official exams ask each kind of question, with a no-calculator Part A and a calculator Part B. Time yourself, check the answer key, then redo the kinds you missed. For the free-response part, use the Unit 4 FRQ set, which has a scoring guide.

No. Every question here is original, written by Sigma Prep in the style of the exam. We studied which kinds of questions the official exams ask and how often; no official question is copied or reworded. Each answer was checked by an independent computer-algebra check before the sheet was published.

Yes. It is shown with the sheet, and the PDF prints it on its own last page. There is also a PDF without the key to hand out. Worked solutions are not included.

AP is a registered trademark of the College Board, which was not involved in making this worksheet and does not endorse it. All questions are original to Sigma Prep. Frozen 2026-10-24.