AP Calculus AB review and study plan

By Ashraf Bhura, Co-Founder, STEM-Tutor · Updated October 2026

A strong AP Calculus AB review covers every unit, practises free-response questions properly and builds calculator confidence. Here is a plan.

What to review

• Limits and continuity: evaluating limits, continuity and the Intermediate Value Theorem.
• Derivatives: all the rules, implicit differentiation and inverse functions.
• Applications of derivatives: related rates, motion, optimisation and curve analysis.
• Integrals: Riemann sums, the Fundamental Theorem of Calculus and u-substitution.
• Differential equations: slope fields and separation of variables.
• Applications of integrals: area, volume, average value and accumulation.

See our guide to all eight AP Calculus AB units and the formulas to memorise.

Free-response skills

• Justify with theorems. Name the theorem and show its conditions are met.
• Use correct notation. Write the integral or derivative you are evaluating.
• Include units when the question gives a context.
• Answer every part. Later parts often score even if an earlier part went wrong.

Calculator skills

Make sure you can graph, find intersections, evaluate a derivative at a point and compute a definite integral on your calculator quickly. These save a lot of time in the calculator sections.

A plan for the final month

• Week 1: limits and derivatives, with daily mixed practice.
• Week 2: applications of derivatives and related rates.
• Week 3: integrals, accumulation and differential equations.
• Week 4: full timed practice exams and targeted review.

Use official past free-response questions and scoring guidelines from AP Central.

A free-response question, answered in full

Question: Water flows into a tank at r(t) = 20 + 5t litres per minute for 0 ≤ t ≤ 4. At t = 0 the tank holds 100 litres.

1. (a) How much water is in the tank at t = 4? 100 + ∫₀⁴ (20 + 5t) dt = 100 + [20t + 2.5t²]₀⁴ = 100 + 80 + 40 = 220 litres.
2. (b) Is the rate increasing at t = 2? Justify. r'(t) = 5 > 0, so yes: the rate is increasing because its derivative is positive.
3. (c) Find the average rate of flow over [0, 4]. (1/4)∫₀⁴ r(t) dt = 120/4 = 30 litres per minute.

Notice how each answer names the mathematics used, shows the setup, and includes units.

Writing justifications that earn marks

• Increasing or decreasing: "f is increasing on (a, b) because f'(x) > 0 there."
• Relative extremum: "f has a relative minimum at x = 2 because f' changes from negative to positive there."
• Intermediate Value Theorem: "f is continuous on [0, 3], f(0) = −1 < 0 and f(3) = 5 > 0, so there is a c in (0, 3) with f(c) = 0."
• Concavity: "The graph is concave up where f''(x) > 0."

Calculator skills to practise

• Graph a function and adjust the window to see its key features.
• Find intersections and zeros.
• Evaluate a derivative at a point numerically.
• Evaluate a definite integral numerically, such as total distance ∫|v(t)| dt.

Store intermediate answers in calculator memory and round only the final answer, usually to three decimal places.

Practice questions

Try each one before opening the answer.

Frequently asked questions

How many practice exams should I do?

Two or three full timed exams in the final month, with careful review of every mistake, is usually enough.

Where do I find past free-response questions?

AP Central publishes past free-response questions with scoring guidelines.

Key takeaways

• Review every unit, but weight practice toward derivatives, applications and integrals.
• Name theorems and show conditions in justifications.
• Write the integral or derivative you are evaluating before giving the number.
• Practise calculator skills until they are fast.
• Answer every part; later parts can still score.

How to review each big idea

Limits and continuity

Practise evaluating limits algebraically, from graphs and from tables, and know what makes a function continuous at a point. Review the Intermediate Value Theorem, because it appears in justification questions.

Derivatives

Drill the rules until they are automatic, then practise mixed questions where you must choose the rule. Include derivatives from tables and graphs, not just formulas.

Applications of derivatives

This is where many marks are. Practise related rates, motion (velocity, acceleration, speeding up and slowing down), linear approximation, and analysing graphs with f' and f''. Write the justification sentence every time.

Integrals and accumulation

Understand an integral as accumulated change: amount at the end = amount at the start + ∫ rate. Practise Riemann sums and trapezoidal approximations from tables, both parts of the Fundamental Theorem, and u-substitution.

Differential equations and applications of integration

Practise slope fields, separation of variables with initial conditions, areas between curves, volumes with discs, washers and cross-sections, and average value.

A sample review week

• Day 1: 20 mixed multiple-choice questions, no calculator, timed.
• Day 2: two free-response questions, marked with the official scoring guidelines.
• Day 3: rework every mistake from days 1 and 2.
• Day 4: calculator-section practice, focusing on speed.
• Day 5: one weak topic in depth.

More questions students ask

What calculator do I need?

A graphing calculator from the College Board's approved list. Practise with the same one you will use in the exam.

How do I improve my free-response scores?

Read the scoring guidelines for past questions. They show exactly which statements and steps earn each point.

Should I learn shortcuts?

Only after you understand the standard method. Shortcuts without understanding often break on unfamiliar questions.

One more worked example: reading a table

A tank's water level h(t), in metres, is recorded every 2 hours: h(0) = 1.0, h(2) = 1.6, h(4) = 2.0, h(6) = 2.3. Estimate h'(3) and explain what it means.

• Use the nearest values around t = 3: h'(3) ≈ (h(4) − h(2))/(4 − 2) = 0.4/2 = 0.2 m per hour.
• Meaning: at t = 3 hours, the water level is rising at about 0.2 metres per hour.

Study tip

Set a timer for 15 minutes per free-response question when practising. Learning to pace yourself is as important as knowing the content.

Where to go next

Use these guides to strengthen specific skills while you review. Each one has worked examples and practice questions with answers you can check straight away.

• What to memorise for AP Calculus
• Related rates, step by step
• AP Calculus derivative rules

Quick check