• 55 questions

AP Calculus BC Unit 9 Practice Test

Unofficial practice for Unit 9 of AP Calculus BC: parametric equations, vector-valued functions, motion in the plane and polar curves.

A curve is given by x = f(t) and y = g(t). Where dx/dt is not 0, what is dy/dx?

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 BC Unit 10: seriesAP Calculus Unit 7: differential equations

All 55 questions

Topics 9.1 and 9.2: Parametric derivatives 12

  • A curve is given by x = f(t) and y = g(t). Where dx/dt is not 0, what is dy/dx? Wikipedia, "Parametric derivative"; course topic 9.1
  • For a parametric curve, how is d²y/dx² found? Wikipedia, "Parametric derivative"; course topic 9.2
  • For the circle x = cos t, y = sin t, what is d²y/dx² at t = π/2? Wikipedia, "Parametric derivative"; course topic 9.2
  • For the curve x = 4t, y = t² − 6t, what is d²y/dx²? Wikipedia, "Parametric derivative"; course topic 9.2
  • For the curve x = e^t, y = e^(3t), what is d²y/dx²? Wikipedia, "Parametric derivative"; course topic 9.2
  • For the curve x = t² + 1, y = t³ − 12t, at which values of t is the tangent line horizontal? Wikipedia, "Parametric derivative"; course topic 9.1
  • For the curve x = t², y = t³ with t > 0, what is d²y/dx²? Wikipedia, "Parametric derivative"; course topic 9.2
  • For the curve x = t², y = t³ − 3t, what is dy/dx at t = 2? Wikipedia, "Parametric derivative"; course topic 9.1
  • For the curve x = t³ − 3t, y = t², at which values of t is the tangent line vertical? Wikipedia, "Parametric derivative"; course topic 9.1
  • For the ellipse x = 2 cos t, y = 5 sin t, what is dy/dx at t = π/4? Wikipedia, "Parametric derivative"; course topic 9.1
  • Which equation in x and y describes the curve x = t − 1, y = t²? Wikipedia, "Parametric equation"; course topic 9.1
  • Which is an equation of the line tangent to the curve x = 2t + 1, y = t² at t = 3? Wikipedia, "Parametric derivative"; course topic 9.1
Show all 55 questions (43 more)

Topic 9.3: Arc length of parametric curves 6

  • What is the length of the curve x = 2 cos t, y = 2 sin t for 0 ≤ t ≤ π/2? Wikipedia, "Arc length"; course topic 9.3
  • What is the length of the curve x = 3t + 1, y = 4t − 2 for 0 ≤ t ≤ 2? Wikipedia, "Arc length"; course topic 9.3
  • What is the length of the curve x = t², y = (2/3)t³ for 0 ≤ t ≤ √3? Wikipedia, "Arc length"; course topic 9.3
  • What is the length of the spiral x = e^t cos t, y = e^t sin t for 0 ≤ t ≤ 1? Wikipedia, "Arc length"; course topic 9.3
  • Which integral gives the length of the curve x = f(t), y = g(t) for a ≤ t ≤ b? Wikipedia, "Arc length"; course topic 9.3
  • Which integral gives the length of the curve x = t³, y = t² for 0 ≤ t ≤ 1? Wikipedia, "Arc length"; course topic 9.3

Topics 9.4 and 9.5: Vector-valued functions 10

  • A ball has acceleration a(t) = ⟨0, −10⟩ and starting velocity v(0) = ⟨15, 20⟩. What is v(t)? Wikipedia, "Vector-valued function"; course topic 9.5
  • A path is given by r(t) = ⟨3t, t²⟩. What is the slope of the path at t = 2? Wikipedia, "Vector-valued function"; course topic 9.4
  • For r(t) = ⟨e^(2t), ln t⟩, what is r′(1)? Wikipedia, "Vector-valued function"; course topic 9.4
  • How is a vector-valued function r(t) = ⟨x(t), y(t)⟩ differentiated? Wikipedia, "Vector-valued function"; course topic 9.4
  • r′(t) = ⟨2t, cos t⟩ and r(0) = ⟨1, 3⟩. What is r(t)? Wikipedia, "Vector-valued function"; course topic 9.5
  • r′(t) = ⟨e^t, 2⟩ and r(0) = ⟨0, 0⟩. What is r(ln 3)? Wikipedia, "Vector-valued function"; course topic 9.5
  • What is the definite integral from 0 to 1 of ⟨6t², 4t⟩ dt? Wikipedia, "Vector-valued function"; course topic 9.5
  • What is the derivative of r(t) = ⟨t², sin t⟩? Wikipedia, "Vector-valued function"; course topic 9.4
  • What is the indefinite integral of ⟨2t, 3t²⟩ with respect to t? Wikipedia, "Vector-valued function"; course topic 9.5
  • What is the second derivative of r(t) = ⟨t³, 4t⟩? Wikipedia, "Vector-valued function"; course topic 9.4

Topic 9.6: Motion in the plane 9

  • A particle has position x = 3t, y = 2t². What is its speed at t = 1? Wikipedia, "Speed"; course topic 9.6
  • A particle has position ⟨cos 2t, sin 2t⟩. What is its acceleration vector at t = 0? Wikipedia, "Acceleration"; course topic 9.6
  • A particle has position ⟨t² − 4t, t³⟩ for t ≥ 0. At what time is it moving straight up? Wikipedia, "Velocity"; course topic 9.6
  • A particle has position ⟨t², t³⟩. What is its velocity vector at t = 2? Wikipedia, "Velocity"; course topic 9.6
  • A particle has velocity ⟨−3 sin t, 3 cos t⟩ for 0 ≤ t ≤ 2π. What are its total distance traveled and its displacement? Wikipedia, "Displacement (geometry)"; course topic 9.6
  • A particle is at the point (1, −1) at t = 0 and has velocity ⟨2t, 3⟩. Where is it at t = 2? Wikipedia, "Displacement (geometry)"; course topic 9.6
  • At one instant a particle moving in the plane has dx/dt = −5 and dy/dt = 12. What is its speed? Wikipedia, "Speed"; course topic 9.6
  • At t = 1 a particle in the plane has velocity ⟨−2, 3⟩. In which direction is it moving? Wikipedia, "Velocity"; course topic 9.6
  • What does the definite integral of a particle's speed over a time interval give? Wikipedia, "Speed"; course topic 9.6

Topic 9.7: Polar curves and their slopes 7

  • For a polar curve r = f(θ), what is dx/dθ? Wikipedia, "Polar coordinate system"; course topic 9.7
  • For the circle r = 4 cos θ, at which angle between 0 and π/2 is the tangent line horizontal? Wikipedia, "Polar coordinate system"; course topic 9.7
  • For the polar curve r = 3 + 2 sin θ, dr/dθ = −2 at θ = π. What does this say about the curve there? Wikipedia, "Polar coordinate system"; course topic 9.7
  • What are the rectangular coordinates of the point with polar coordinates r = 4, θ = π/3? Wikipedia, "Polar coordinate system"; course topic 9.7
  • What is the graph of the polar equation r = 5? Wikipedia, "Polar coordinate system"; course topic 9.7
  • What is the slope dy/dx of the cardioid r = 1 + cos θ at θ = π/2? Wikipedia, "Polar coordinate system"; course topic 9.7
  • What is the slope dy/dx of the spiral r = θ at θ = π/2? Wikipedia, "Polar coordinate system"; course topic 9.7

Topics 9.8 and 9.9: Polar areas 11

  • A region lies between an outer polar curve r = R(θ) and an inner curve r = r(θ) for α ≤ θ ≤ β. Which integral gives its area? Wikipedia, "Polar coordinate system"; course topic 9.9
  • At which angles in [0, 2π) do the polar curves r = 3 sin θ and r = 1 + sin θ meet with the same r? Wikipedia, "Polar coordinate system"; course topic 9.9
  • For 0 ≤ θ ≤ π, what is the area of the region inside the circle r = 3 and outside the circle r = 1? Wikipedia, "Polar coordinate system"; course topic 9.9
  • What is the area enclosed by the cardioid r = 1 + cos θ? Wikipedia, "Cardioid"; course topic 9.8
  • What is the area enclosed by the polar curve r = 2 sin θ, traced once for 0 ≤ θ ≤ π? Wikipedia, "Polar coordinate system"; course topic 9.8
  • What is the area of one petal of the rose r = cos 2θ? Wikipedia, "Rose (mathematics)"; course topic 9.8
  • What is the area of the region inside the circle r = 3 sin θ and outside the cardioid r = 1 + sin θ? Wikipedia, "Polar coordinate system"; course topic 9.9
  • What is the area of the region inside the circle r = 4 cos θ and outside the circle r = 2? Wikipedia, "Polar coordinate system"; course topic 9.9
  • What is the area swept out by the spiral r = θ from θ = 0 to θ = π? Wikipedia, "Polar coordinate system"; course topic 9.8
  • Which integral gives the area of one petal of the rose r = sin 3θ? Wikipedia, "Rose (mathematics)"; course topic 9.8
  • Which integral gives the area swept out by the polar curve r = f(θ) from θ = α to θ = β? Wikipedia, "Polar coordinate system"; course topic 9.8

How this list was compiled: Our own questions, written to Unit 9 (Parametric Equations, Polar Coordinates, and Vector-Valued Functions; topics 9.1 to 9.9, all BC only) 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 BC Unit 9 practice test has 55 questions of our own on Parametric Equations, Polar Coordinates, and Vector-Valued Functions. It is unofficial: this site is not affiliated with the College Board, which owns AP.

Read the full guide

What Unit 9 covers

The College Board's course description, read October 14, 2026, gives Unit 9 nine topics. Six sets follow them: parametric derivatives (12 questions), arc length (6), vector-valued functions (10), motion in the plane (9), polar curves and their slopes (7) and polar areas (11). No topic has a weight in the description, so the counts are ours.

Curves without pictures

The exam prints many of these curves as graphs. Here each curve is given by its equations, and a rose, a cardioid or a circle is named so that you can sketch it before you answer.

  • For a parametric second derivative, differentiate dy/dx with respect to t, then divide by dx/dt again.
  • Polar area is one half of the integral of r squared. Find the angles where r is 0, or where two curves meet, first.
  • Integrate velocity for displacement and speed for distance traveled. The two differ when the path turns back.

Questions and answers

What is Unit 9 in AP Calculus BC?

Parametric Equations, Polar Coordinates, and Vector-Valued Functions. Its topics run from differentiating parametric equations (9.1) to the area between two polar curves (9.9).

How much of the AP Calculus BC exam is Unit 9?

From 10 to 15% of the multiple-choice section, by the College Board's course page. The course description suggests 10 to 11 class periods for the unit.

Is Unit 9 on the AP Calculus AB exam?

No. Every topic of Unit 9 is marked BC only in the course description, as is every topic of Unit 10. The AB course has eight units.

Is this the AP Calculus BC Unit 9 progress check?

No. The College Board's Progress Check for Unit 9, about 25 multiple-choice and 3 free-response questions by the course description, is assigned in AP Classroom. The questions here are our own.

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