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Parametric equations, polar coordinates and vector-valued functions (BC) practice problems AP Calculus, Unit 9

Written by STEM-Tutor's tutors · Updated October 2026 · 8 questions and 1 free-response question

BC only. Derivatives and arc length of parametric curves, motion with vectors, and slopes and areas in polar coordinates. Try each question before you look: use the hint if you are stuck, then reveal the worked solution one step at a time.

On this page
  1. Key equations
  2. Warm-up questions
  3. Exam-style questions
  4. Free-response practice
  5. Common mistakes
  6. Keep going

Key equations for parametric equations, polar coordinates and vector-valued functions (bc)

dy/dx = (dy/dt)/(dx/dt)parametric slope
Speed = √[(dx/dt)2 + (dy/dt)2]speed of a moving particle
L = ∫ √[(dx/dt)2 + (dy/dt)2] dtarc length
A = ½∫ r2 dθpolar area

Questions marked No calculator should be done by hand, as in Part A of the multiple-choice section. You can type fractions, such as 3/4. The full sheet is in our AP Calculus formula sheet.

Warm-up questions

Quick checks on the core ideas. If these feel easy, go straight to the exam-style questions.

Q1Warm-upNo calculator
x = t2, y = t3. Find dy/dx at t = 2.

Hint

Divide dy/dt by dx/dt.

Worked solution
  1. dy/dt = 3t2 and dx/dt = 2t, so dy/dx = 3t/2.
  2. At t = 2: 3.
Q2Warm-upNo calculator
A particle's velocity vector is ⟨3, 4⟩. What is its speed?

Hint

Speed is the length of the velocity vector.

Worked solution
  1. √(32 + 42) = 5.
Q3Warm-upCalculator
Find the area enclosed by the polar curve r = 3, as a decimal.

Hint

A = ½∫02π r2 dθ.

Worked solution
  1. A = ½ × 9 × 2π = 9π, the area of a circle of radius 3.
  2. ≈ 28.27.

Exam-style questions

Questions at the level of the AP Calculus exam. The ones marked Challenge combine more than one idea.

Q4Exam-styleNo calculator
x = 3cos t, y = 3sin t. Find dy/dx at t = π/4.

Hint

dy/dx = (3cos t)/(−3sin t).

Worked solution
  1. dy/dx = −cos t/sin t = −cot t.
  2. At π/4: −1.
Q5Exam-styleNo calculator
Find the arc length of x = t, y = (2/3)t3/2 for 0 ≤ t ≤ 3. Give a decimal.

Hint

dx/dt = 1 and dy/dt = t1/2.

Worked solution
  1. L = ∫03 √(1 + t) dt.
  2. = (2/3)[(1 + t)3/2] from 0 to 3 = (2/3)(8 − 1) = 14/3.
  3. ≈ 4.667.
Q6Exam-styleNo calculator
x = t2, y = t3. Find d2y/dx2 at t = 1.

Hint

d2y/dx2 = [d/dt(dy/dx)] ÷ (dx/dt).

Worked solution
  1. dy/dx = 3t/2, so d/dt(dy/dx) = 3/2.
  2. Divide by dx/dt = 2t = 2.
  3. 3/4 = 0.75.
Q7Exam-styleCalculator
Find the area enclosed by the cardioid r = 1 + cos θ, as a decimal.

Hint

A = ½∫02π (1 + cos θ)2 dθ.

Worked solution
  1. Expand: 1 + 2cos θ + cos2θ. Over a full turn, ∫cos θ = 0 and ∫cos2θ = π.
  2. A = ½(2π + 0 + π) = 3π/2.
  3. ≈ 4.712.
Q8ChallengeCalculator
Find the slope dy/dx of the polar curve r = θ at θ = π/2. Give a decimal.

Hint

dy/dx = (r' sinθ + r cosθ)/(r' cosθ − r sinθ).

Worked solution
  1. r = θ and r' = 1. At π/2: sin = 1 and cos = 0.
  2. dy/dx = (1 + 0)/(0 − π/2) = −2/π.
  3. ≈ −0.637.

Free-response practice

One original question in the style of the exam's free-response section. Answer it in full, with working, before you open the scoring guide.

Calculator-active free-response style · 5 points · about 15 minutes

A particle moves in the plane with x(t) = t2 − 3t and y(t) = 2t3/3, for t ≥ 0.

  1. (a) Find the velocity vector at t = 2.
  2. (b) Find the speed at t = 2.
  3. (c) At what time is the particle moving vertically?
  4. (d) Find the total distance travelled from t = 0 to t = 2.

Write your full answer on paper first, then open the scoring guide and tick each point you earned.

Scoring guide and model answer

Part (a)

x'(t) = 2t − 3 and y'(t) = 2t2, so v(2) = ⟨1, 8⟩.

Part (b)

√(1 + 64) = √65 ≈ 8.062.

Part (c)

x'(t) = 0 at t = 1.5, while y'(1.5) = 4.5 ≠ 0.

Part (d)

∫02 √[(2t − 3)2 + 4t4] dt ≈ 6.951, by calculator.

Your mark: 0 of 5

Common mistakes in parametric equations, polar coordinates and vector-valued functions (bc)

Keep going

Revise the ideas behind these questions:

Then try real exam questions: AP Calculus past papers, with official scoring guidelines. · All AP Calculus units