AP Calculus AB units, explained

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

The AP Calculus AB course is organised into eight units. Here is what each one covers, in plain English, plus the two extra units in BC.

The eight AB units

1. Limits and Continuity. What happens to a function as the input gets close to a value, and what it means for a graph to have no breaks.
2. Differentiation: Definition and Fundamental Properties. The derivative as an instantaneous rate of change, and the basic rules for finding it.
3. Differentiation: Composite, Implicit, and Inverse Functions. The chain rule, implicit differentiation and derivatives of inverse functions.
4. Contextual Applications of Differentiation. Rates of change in real situations, related rates, motion and linear approximation.
5. Analytical Applications of Differentiation. Using derivatives to find maximums, minimums and the shape of a graph, plus optimisation problems.
6. Integration and Accumulation of Change. Riemann sums, definite integrals, the Fundamental Theorem of Calculus and techniques of integration.
7. Differential Equations. Slope fields, separation of variables and exponential models.
8. Applications of Integration. Average value, motion, area between curves and volumes of solids.

The two extra BC units

9. Parametric Equations, Polar Coordinates, and Vector-Valued Functions.
10. Infinite Sequences and Series.

BC also adds some extra topics inside the earlier units. See our guide to AP Calculus AB vs BC for the full comparison.

Which units carry the most weight?

The College Board publishes approximate exam weightings for each unit in the course and exam description. The units on applying derivatives (Unit 5) and on integration (Unit 6) are among the most heavily weighted in AB. Check the current figures on AP Central, as they can change.

How to study the units

• Units 1 to 3: build fluency. Practise derivative rules until they are automatic.
• Units 4 and 5: focus on reading the question. Most marks are lost on setting up the problem, not on the calculus.
• Units 6 to 8: connect integrals to accumulation. Draw the graph and say what the area means.
• All units: practise free-response questions and write clear justifications.

A worked example from every unit

Unit 1: a limit

Find lim_x→2 (x² − 4)/(x − 2).

• Factor: (x − 2)(x + 2)/(x − 2) = x + 2 for x ≠ 2.
• The limit is 2 + 2 = 4.

Unit 2: a derivative

f(x) = 3x⁴. Find f'(1).

• f'(x) = 12x³, so f'(1) = 12.

Unit 3: chain rule and implicit differentiation

Differentiate (2x + 1)⁵, and find dy/dx for x² + y² = 25.

• d/dx(2x + 1)⁵ = 5(2x + 1)⁴ × 2 = 10(2x + 1)⁴.
• 2x + 2y(dy/dx) = 0, so dy/dx = −x/y.

Unit 4: motion

A particle has position s(t) = t³ − 6t². When is it at rest?

• v(t) = 3t² − 12t = 3t(t − 4).
• At rest when v = 0: t = 0 and t = 4.

Unit 5: extreme values

Find and classify the stationary points of f(x) = x³ − 3x.

• f'(x) = 3x² − 3 = 0 gives x = ±1.
• f''(x) = 6x: f''(1) = 6 > 0, so a minimum at (1, −2); f''(−1) = −6 < 0, so a maximum at (−1, 2).

Unit 6: a definite integral

Evaluate ∫₁³ 2x dx.

• [x²] from 1 to 3 = 9 − 1 = 8.

Unit 7: a differential equation

Solve dy/dx = 2y with y(0) = 3.

• Separate: dy/y = 2 dx, so ln|y| = 2x + C, giving y = Ae^2x.
• y(0) = 3 gives A = 3, so y = 3e^2x.

Unit 8: area between curves

Find the area between y = x and y = x² from x = 0 to x = 1.

• On [0, 1], x ≥ x², so area = ∫₀¹ (x − x²) dx = ½ − ⅓ = 1/6.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Which unit should I review first?

Start with Units 2 and 3. Derivative skills support everything that follows.

Do the units appear in order on the exam?

No. Questions mix units, and free-response questions often combine several ideas.

Key takeaways

• AP Calculus AB has eight units, from limits to applications of integration.
• Derivative rules (Units 2 and 3) underpin the whole course.
• Units 5 and 6 are among the most heavily weighted.
• Free-response questions combine units in context.
• BC adds two more units: parametric/polar/vector and series.

How the units fit together

The course tells one story. Limits (Unit 1) make it possible to define the derivative. Units 2 and 3 build the rules for finding derivatives. Units 4 and 5 use derivatives to answer real questions about rates, motion and the shapes of graphs. Unit 6 introduces integrals as accumulated change and connects them to derivatives through the Fundamental Theorem. Units 7 and 8 use integrals to solve differential equations and find areas, volumes and averages.

Seeing this story helps with review: weak derivative skills cause trouble in every later unit, so fixing them first pays off everywhere.

More questions students ask

How long does each unit take?

It varies by school. Units 2, 5 and 6 usually take longest because they contain the most skills.

One more worked example combining units

A particle moves along a line with velocity v(t) = 6 − 2t for 0 ≤ t ≤ 5. Find its displacement and total distance travelled.

• Displacement = ∫₀⁵ (6 − 2t) dt = [6t − t²] from 0 to 5 = 30 − 25 = 5.
• The velocity changes sign at t = 3, so split the distance: ∫₀³ (6 − 2t) dt = 9 and the integral of the absolute value from 3 to 5 is 4.
• Total distance = 9 + 4 = 13. Displacement and distance differ because the particle turned around.

Study tip

After finishing each unit, write one question that combines it with an earlier unit, then solve it. Exam questions mix units, so practise that early.

Where to go next

With the course map in mind, focus next on the skills that carry the most weight. Derivative rules support every later unit, and a structured review plan keeps everything fresh before May.

• AP Calculus derivative rules
• Related rates, step by step
• AP Calculus AB review and study plan

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