• 52 questions

AP Calculus Unit 4 Practice Test

Unofficial practice for Unit 4 of AP Calculus AB and BC: rates in context, motion, related rates, linearization, L'Hospital's rule.

W(t) is the mass of a puppy in kilograms, t weeks after birth. What are the units of W′(t)?

Pick an answer, then press Check answer. Then you see Correct or Not quite, with the reason. Keys: A to D pick, Enter checks.

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Unofficial practice questions, not affiliated with the College Board. Bank checked October 14, 2026.

Next step AP Calculus Unit 5: extrema and optimizationAP Calculus Unit 3: chain rule and implicit

All 52 questions

Topic 4.1: The derivative in context 5

  • C(x) is the cost in dollars of roasting x kilograms of coffee, and C′(200) = 3. What does this say? Wikipedia, "Derivative"; course topic 4.1
  • The depth of water in a tank is d(t) = 4√t centimeters after t minutes. What is d′(4), with its units? Wikipedia, "Derivative"; course topic 4.1
  • H(t) is the temperature of a cup of tea in °C, t minutes after it is poured, and H′(5) = −2. What does this say? Wikipedia, "Derivative"; course topic 4.1
  • The volume of a sphere is V = (4/3)πr³. What is the instantaneous rate of change of V with respect to r? Wikipedia, "Derivative"; course topic 4.1
  • W(t) is the mass of a puppy in kilograms, t weeks after birth. What are the units of W′(t)? Wikipedia, "Derivative"; course topic 4.1
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Topic 4.2: Straight-line motion 9

  • A ball thrown upward has height h(t) = 30t − 5t² meters after t seconds. When does it reach its highest point? Wikipedia, "Velocity"; course topic 4.2
  • A particle has position x(t) = 4 sin t. What is its acceleration at t = π/2? Wikipedia, "Acceleration"; course topic 4.2
  • A particle has position x(t) = t² − 6t + 10 for 0 ≤ t ≤ 5. On which interval is it moving in the negative direction? Wikipedia, "Velocity"; course topic 4.2
  • A particle has position x(t) = t³ − 9t² + 24t for t ≥ 0. At which times is it at rest? Wikipedia, "Velocity"; course topic 4.2
  • A particle has position x(t) = t³ − 9t² + 24t. What is its acceleration at t = 1? Wikipedia, "Acceleration"; course topic 4.2
  • A particle moves along the x-axis with position x(t) = t³ − 9t² + 24t. What is its velocity at t = 1? Wikipedia, "Velocity"; course topic 4.2
  • A particle moves on a line with velocity v(t) = t² − 4t − 5 for t ≥ 0. At what time does it change direction? Wikipedia, "Velocity"; course topic 4.2
  • A particle on a line has velocity v(3) = −7 meters per second. What is its speed at t = 3? Wikipedia, "Speed"; course topic 4.2
  • For the particle with x(t) = t³ − 9t² + 24t, is it speeding up or slowing down at t = 1? Wikipedia, "Speed"; course topic 4.2

Topic 4.3: Rates of change beyond motion 5

  • A spherical balloon has volume V = (4/3)πr³ cubic centimeters. What is dV/dr when r = 3 centimeters? Wikipedia, "Derivative"; course topic 4.3
  • A tank drains so that V(t) = 200(1 − t/20)² liters remain after t minutes, 0 ≤ t ≤ 20. What is the rate of change of volume at t = 10? Wikipedia, "Derivative"; course topic 4.3
  • A town has population P(t) = 1200e^(0.05t), with t in years. How fast is the population growing at t = 0? Wikipedia, "Derivative"; course topic 4.3
  • The charge that has passed a point in a wire is Q(t) = t³ − 2t² coulombs after t seconds. Current is dQ/dt. What is the current at t = 2? Wikipedia, "Derivative"; course topic 4.3
  • The cost of making x items is C(x) = 500 + 4x + 0.01x² dollars. What is the marginal cost C′(100)? Wikipedia, "Marginal cost"; course topic 4.3

Topics 4.4 and 4.5: Related rates 15

  • A 13-foot ladder leans on a wall. Its base slides away from the wall at 2 feet per second. How fast is the top sliding down when the base is 5 feet from the wall? Wikipedia, "Related rates"; course topic 4.5
  • A camera on the ground, 100 meters from a launch pad, follows a rocket rising straight up at 20 meters per second. How fast is the camera's angle of elevation changing when the rocket is 100 meters high? Wikipedia, "Related rates"; course topic 4.5
  • A cylindrical tank of radius 2 meters is filled at 6 cubic meters per minute. How fast is the water depth rising? Wikipedia, "Related rates"; course topic 4.5
  • A person 2 meters tall walks away from a lamp 6 meters above the ground at 1.5 meters per second. How fast is the length of the person's shadow growing? Wikipedia, "Related rates"; course topic 4.5
  • Air is pumped into a spherical balloon at 100 cubic centimeters per second. How fast is the radius growing when the radius is 5 centimeters? Wikipedia, "Related rates"; course topic 4.5
  • Each side of a square grows at 3 centimeters per second. How fast is the area growing when the side is 10 centimeters? Wikipedia, "Related rates"; course topic 4.5
  • The edge s of a cube changes with time t, and its volume is V = s³. What is dV/dt? Wikipedia, "Related rates"; course topic 4.4
  • In a ladder problem the height y of the top of the ladder has dy/dt = −5/6 foot per second. What does the negative sign say? Wikipedia, "Related rates"; course topic 4.5
  • In a related rates problem, when are the given values of the changing quantities substituted? Wikipedia, "Related rates"; course topic 4.4
  • Two cyclists leave the same corner at the same time, one riding north at 15 miles per hour and one riding east at 20 miles per hour. How fast is the distance between them growing 2 hours later? Wikipedia, "Related rates"; course topic 4.5
  • Two sides of a triangle have fixed lengths 6 and 8 centimeters. The angle between them grows at 0.1 radian per second. How fast is the area changing when the angle is π/3? Wikipedia, "Related rates"; course topic 4.5
  • Water runs into a cone-shaped tank, point down, that is 12 meters tall with a top radius of 4 meters, at 3 cubic meters per minute. How fast is the depth rising when the water is 6 meters deep? Wikipedia, "Related rates"; course topic 4.5
  • x and y are functions of t with x² + y² = 25. What does differentiating both sides with respect to t give? Wikipedia, "Related rates"; course topic 4.4
  • x and y are functions of time t, and y = x². What is dy/dt? Wikipedia, "Related rates"; course topic 4.4
  • z = xy, where x and y are functions of t. At one instant x = 3, y = 4, dx/dt = 2 and dy/dt = −1. What is dz/dt then? Wikipedia, "Related rates"; course topic 4.4

Topic 4.6: Linearization 8

  • A differentiable function has f(2) = 5 and f′(2) = −3. What does the tangent line at x = 2 give for f(2.1)? Wikipedia, "Linear approximation"; course topic 4.6
  • A function has f″(x) > 0 for all x. The tangent line at x = 3 is used to approximate f(3.2). How does the approximation compare with f(3.2)? Wikipedia, "Linear approximation"; course topic 4.6
  • The tangent line to f(x) = x³ at x = 2 is used to approximate (2.01)³. What value does it give? Wikipedia, "Linear approximation"; course topic 4.6
  • The tangent line to f(x) = √x at x = 16 is used to approximate √17. What value does it give? Wikipedia, "Linear approximation"; course topic 4.6
  • The tangent line to y = √x at x = 16 gives 4.125 for √17. Is that too large or too small, and why? Wikipedia, "Linear approximation"; course topic 4.6
  • What does the tangent line to y = e^x at x = 0 give as an approximation of e^0.1? Wikipedia, "Linear approximation"; course topic 4.6
  • What is the linearization of f(x) = sin x at x = 0? Wikipedia, "Linear approximation"; course topic 4.6
  • What is the linearization of f(x) = √x at x = 16? Wikipedia, "Linear approximation"; course topic 4.6

Topic 4.7: L'Hospital's rule 10

  • A student writes: the limit as x → 0 of (x² + 1)/(2x + 3) equals the limit of 2x/2, which is 0. What is wrong? Wikipedia, "L'Hôpital's rule"; course topic 4.7
  • For a limit of f(x)/g(x) with the form 0/0, what does L'Hospital's rule examine in its place? Wikipedia, "L'Hôpital's rule"; course topic 4.7
  • To which of these limits can L'Hospital's rule be applied directly? Wikipedia, "L'Hôpital's rule"; course topic 4.7
  • What is the limit as x → 0 of (e^(2x) − 1 − 2x)/x²? Wikipedia, "L'Hôpital's rule"; course topic 4.7
  • What is the limit as x → 0 of (e^(3x) − 1)/sin(2x)? Wikipedia, "L'Hôpital's rule"; course topic 4.7
  • What is the limit as x → 1 of (ln x)/(x² − 1)? Wikipedia, "L'Hôpital's rule"; course topic 4.7
  • What is the limit as x → ∞ of (5x + ln x)/(2x + 1)? Wikipedia, "L'Hôpital's rule"; course topic 4.7
  • What is the limit as x → ∞ of (ln x)/x? Wikipedia, "L'Hôpital's rule"; course topic 4.7
  • What is the limit as x → ∞ of x²/e^x? Wikipedia, "L'Hôpital's rule"; course topic 4.7
  • Which of these is an indeterminate form to which L'Hospital's rule applies? Wikipedia, "Indeterminate form"; course topic 4.7

How this list was compiled: Our own questions, written to Unit 4 (Contextual Applications of Differentiation; topics 4.1 to 4.7) of the College Board's AP Calculus AB and BC Course and Exam Description, https://apcentral.collegeboard.org/media/pdf/ap-calculus-ab-and-bc-course-and-exam-description.pdf, read 2026-10-14. Every answer is mathematics worked from what the question prints, and the computed ones were worked out a second time with computer algebra on 2026-10-14. The rule, theorem or definition behind each answer was read on 2026-10-14 in the Wikipedia article named under the question. Licence: Own work.

How it works

This AP Calculus Unit 4 practice test asks 52 questions of our own on Contextual Applications of Differentiation. It is unofficial, and this site is not affiliated with or endorsed by the College Board, which owns AP.

Read the full guide

What Unit 4 covers

The College Board's course description for AP Calculus AB and BC, read October 14, 2026, gives Unit 4 seven topics. Six sets follow them: the derivative in context (5 questions), straight-line motion (9), other rates of change (5), related rates (15), linearization (8) and L'Hospital's rule (10). The description weights no topic, so the spread is ours; related rates fills two of the seven topics and has the largest set.

Which limits are asked

The description says L'Hospital's rule is assessed on the forms 0/0 and ∞/∞ only. Other indeterminate forms, such as ∞ − ∞, are named as not assessed on either exam, so none appears here.

  • In a related rates problem, differentiate first and put the numbers in afterward. A value entered too early is treated as a constant.
  • Before L'Hospital's rule, check the form: both parts must approach 0, or both must become infinite.
  • A particle speeds up when velocity and acceleration share a sign, and slows down when their signs differ.

Questions and answers

What is Unit 4 in AP Calculus?

Contextual Applications of Differentiation. Its topics run from the meaning of a derivative in context (4.1) through motion, related rates and linearization to L'Hospital's rule (4.7).

How much of the AP Calculus exam is Unit 4?

From 10 to 15% of the multiple-choice section of the AB exam and from 5 to 10% of the BC exam, by the College Board's course pages.

Is Unit 4 the same in AP Calculus AB and BC?

Yes. Every topic belongs to both courses. The course description suggests 10 to 11 class periods for the unit in AB and 6 to 7 in BC.

Is this the AP Calculus Unit 4 progress check?

No. The Progress Check for Unit 4, about 15 multiple-choice and 3 free-response questions by the course description, belongs to the College Board and is assigned by teachers in AP Classroom. Every question here is our own.

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