STEM-TutorLearn • Practice • ExcelBook Free DemoFree demo
AP CoursesGCSEInstructorsTestimonials
Guides
Past papers
ContactBook Free Demo

Contextual applications of differentiation practice problems AP Calculus, Unit 4

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

Rates of change in context, motion along a line, related rates, linear approximation and L'Hôpital's rule. 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 contextual applications of differentiation

v(t) = s'(t), a(t) = v'(t)motion along a line
L(x) = f(a) + f'(a)(x − a)linear approximation
lim f/g = lim f'/g'L'Hôpital's rule, for 0/0 or ∞/∞
Speeding upv and a have the same sign

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
A particle's position is s(t) = t3 − 6t2 + 9t. Find its velocity at t = 2.

Hint

Velocity is the derivative of position.

Worked solution
  1. v(t) = 3t2 − 12t + 9.
  2. v(2) = 12 − 24 + 9 = −3.
Q2Warm-upNo calculator
V(t) is the volume of water in a tank, in litres, t minutes after it starts filling. What are the units of V'(t)?

Hint

A derivative's units are the output's units divided by the input's units.

Worked solution
  1. V is in litres and t is in minutes.
  2. So V'(t) is in litres per minute. The answer is C.
Q3Warm-upCalculator
The radius of a circle grows at 2 cm/s. How fast is the area growing when the radius is 5 cm? Give a decimal.
cm²/s

Hint

A = πr2. Differentiate with respect to t.

Worked solution
  1. dA/dt = 2πr dr/dt.
  2. = 2π × 5 × 2 = 20π.
  3. 20π ≈ 62.8 cm²/s.

Exam-style questions

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

Q4Exam-styleNo calculator
A 10 m ladder leans against a wall. The bottom slides away at 1 m/s. How fast is the top sliding down when the bottom is 6 m from the wall?
m/s

Hint

x2 + y2 = 100. Differentiate with respect to t.

Worked solution
  1. When x = 6, y = 8.
  2. 2x dx/dt + 2y dy/dt = 0, so dy/dt = −x(dx/dt)/y = −6/8.
  3. dy/dt = −0.75 m/s: the top slides down at 0.75 m/s.
Q5Exam-styleNo calculator
Use a linear approximation of f(x) = √x at x = 4 to estimate √4.1.

Hint

Tangent line at x = 4: f(4) = 2 and f'(4) = 1/(2√4).

Worked solution
  1. f'(x) = 1/(2√x), so f'(4) = 1/4.
  2. L(x) = 2 + (1/4)(x − 4).
  3. L(4.1) = 2 + 0.025 = 2.025. (The true value is 2.0248.)
Q6Exam-styleNo calculator
Find limx→0 (ex − 1 − x)/x2.

Hint

It is 0/0. Apply L'Hôpital's rule, perhaps twice.

Worked solution
  1. 0/0, so differentiate top and bottom: (ex − 1)/(2x), still 0/0.
  2. Again: ex/2.
  3. At x = 0: 1/2.
Q7Exam-styleNo calculator
A particle has velocity v(t) = t2 − 4t + 3. Find its acceleration at t = 1.

Hint

Acceleration is the derivative of velocity.

Worked solution
  1. a(t) = 2t − 4.
  2. a(1) = −2.
Q8Exam-styleNo calculator
For the particle with v(t) = t2 − 4t + 3, at t = 0.5 the particle is:

Hint

Compare the signs of v(0.5) and a(0.5).

Worked solution
  1. v(0.5) = 0.25 − 2 + 3 = 1.25 > 0.
  2. a(0.5) = 1 − 4 = −3 < 0.
  3. Opposite signs mean it is slowing down: B.

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.

No-calculator free-response style · 6 points · about 15 minutes

A particle moves along the x-axis with position x(t) = t3 − 6t2 + 9t + 1 for 0 ≤ t ≤ 5.

  1. (a) Find the times when the particle is at rest.
  2. (b) Is the particle speeding up or slowing down at t = 4? Give a reason.
  3. (c) Find the total distance travelled from t = 0 to t = 3.

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)

v(t) = 3t2 − 12t + 9 = 3(t − 1)(t − 3) = 0, so t = 1 and t = 3.

Part (b)

v(4) = 9 > 0 and a(4) = 6(4) − 12 = 12 > 0. Same sign, so it is speeding up.

Part (c)

x(0) = 1, x(1) = 5, x(3) = 1. It turns at t = 1, so the distance is 4 + 4 = 8.

Your mark: 0 of 6

Common mistakes in contextual applications of differentiation

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