AP Calculus: how to study for AB and BC: the units, the free-response habit, where Grade 12 and first-year calculus overlap it, and a six-week plan

AP Calculus AB covers limits, derivatives, their applications, integrals and basic differential equations; BC adds parametric and polar functions, vectors, and infinite series, and moves faster. The exam has a multiple-choice section and a free-response section, each with a calculator part and a no-calculator part. The way to study it is the released free-response questions, done under time, graded with the College Board's own scoring guidelines, and logged by skill. Here are the units, the habit and a six-week plan.

AB or BC: what each covers (the AP Calculus units)

The College Board organizes both courses into units, and most teachers follow that order. Here is what each unit covers, in our words, not the College Board's names.

AB has eight units:

1. Limits and continuity. What a limit is, how to find one from a graph, a table or algebra, and what it means for a function to be continuous.
2. The derivative: definition and basic rules. The derivative as a limit, then the power, product and quotient rules.
3. Composite, implicit and inverse functions. The chain rule, implicit differentiation, and the derivatives of inverse functions, including the inverse trigonometric ones.
4. Contextual applications of derivatives. Rates of change in a situation: motion along a line, related rates, linear approximation, and L'Hôpital's rule.
5. Analytical applications of derivatives. The Mean Value Theorem, extrema, concavity, curve sketching and optimization.
6. Integration and accumulation. Riemann sums, the definite integral, the Fundamental Theorem of Calculus, antiderivatives and substitution.
7. Differential equations. Slope fields, separable equations and exponential models.
8. Applications of integration. Average value, motion from velocity, area between curves and volumes.

BC covers the eight AB units at a faster pace and adds two:

9. Parametric, polar and vector-valued functions. Derivatives and integrals when x and y each depend on a parameter, or when the curve is given in polar form.
10. Infinite sequences and series. Convergence tests, Taylor and Maclaurin series, and the error of a partial sum.

BC also carries a few extra integration techniques inside the shared units, such as integration by parts. The BC exam reports an AB subscore alongside the BC score, computed from the questions on AB material, so a BC student has two numbers to read. If your school teaches calculus under another name, the course page for first-year calculus has the same ideas in the order a university teaches them.

The exam in one table

Both exams have the same shape. The counts below are the College Board's as its exam pages state them at the time of writing; the course and exam description is the document to check before you sit it, because the format can change.

Section	Questions	Time	Calculator	Weight
I, multiple choice	42	1 h 40 min	Part A, 29 questions, no calculator; Part B, 13 questions, graphing calculator	50 %
II, free response	6	1 h 30 min	Part A, 2 questions, graphing calculator; Part B, 4 questions, no calculator	50 %

The whole exam runs about 3 hours 10 minutes and is scored from 1 to 5. The free-response questions are the half of the exam most students under-prepare for, because a school test usually gives marks for the number and the AP exam gives most of its marks for how you got there.

The free-response habit

The scoring guidelines for a free-response question award points for the set-up, the justification and the units, and usually one point for the final value. A right number with no work is worth less than a wrong number with the right set-up. So the habit is a way of writing, practised until it is automatic:

• Write the equation before you substitute. The integral with its limits, the derivative set equal to zero, the equation of the tangent line in general form. The set-up is a point on its own.
• Name the theorem you are using. "By the Mean Value Theorem, since f is continuous on [1, 4] and differentiable on (1, 4), there is a c …" The hypotheses are part of the name; a theorem used without them is not a justification.
• Justify with the sign of the derivative, not with the picture. "f is increasing because f′(x) > 0 on the interval" earns the point; "the graph goes up" does not.
• Include units where the problem has them, and say what the number is: metres, metres per second, metres per second squared.
• Answer the question asked. If it asks for an interval, give an interval. If it asks you to justify, write a sentence. If it asks for a value, give the value, and not the integral that would produce it.

Practise this with the released free-response questions on AP Central, under the real time, and then grade your own paper against the published scoring guideline for that question. The guideline tells you, point by point, what a reader would have given you and what you left out. That material is the College Board's and is not reproduced here; the example below is ours, written the free-response way to show the habit.

Worked example: a particle-motion question done the FRQ way

This is our own problem, not an AP item. A particle moves along a line with velocity v(t) = 3t² − 12 metres per second for 0 ≤ t ≤ 3 seconds.

(a) When is the particle at rest? The particle is at rest when v(t) = 0. Set-up: 3t² − 12 = 0, so t² = 4, so t = 2 or t = −2. Only t = 2 is in [0, 3]. The particle is at rest at t = 2 seconds. (Showing the rejected root is part of the answer: it tells the reader you checked the interval.)

(b) What is the displacement over [0, 3]? Displacement is the integral of velocity. Set-up first: ∫₀³ (3t² − 12) dt. An antiderivative is t³ − 12t. Evaluate: (27 − 36) − (0 − 0) = −9. The displacement is −9 metres: the particle ends 9 metres to the left of where it started.

(c) What is the total distance travelled over [0, 3]? Distance is the integral of speed, |v(t)|, so the interval has to be split where v changes sign, which part (a) found: t = 2. On [0, 2], v(0) = −12 < 0, so v is negative there; on [2, 3], v(3) = 15 > 0. The two pieces:

• ∫₀² v dt = (8 − 24) − 0 = −16, so the distance on [0, 2] is |−16| = 16 metres.
• ∫₂³ v dt = (27 − 36) − (8 − 24) = −9 − (−16) = 7, so the distance on [2, 3] is 7 metres.

Total distance: 16 + 7 = 23 metres. Check against (b): the displacement is −16 + 7 = −9, which matches. A common wrong answer is 9, from integrating v without the split; the set-up line "split at t = 2 where v changes sign" is the point that answer loses.

(d) Is the speed increasing or decreasing at t = 1? Justify. Speed increases when velocity and acceleration have the same sign. v(1) = 3 − 12 = −9 < 0. Acceleration is a(t) = v′(t) = 6t, so a(1) = 6 > 0. Velocity is negative and acceleration is positive, so they have opposite signs and the speed is decreasing at t = 1. The justification is the two signs stated; "it slows down" without them earns nothing.

Four parts, and every one opens with the set-up and closes with a sentence that answers the question asked, with units. That is the habit.

Where MCV4U and first-year calculus overlap it

For a Canadian reader: Ontario's Grade 12 calculus course covers the derivative half of AB and none of the integral half. A first-year university Calculus I course covers all of AB and most of BC's integration, without the series. The table reads our two course maps against the AB units; "covers" means the ideas are taught, not that one course is credit for another.

AB unit (in our words)	MCV4U	Calculus I
1 Limits and continuity	Yes: limits, limit laws, continuity	Yes, plus the formal definition
2 The derivative and basic rules	Yes: first principles, power, product, quotient	Yes
3 Composite, implicit and inverse functions	Chain rule and implicit yes; inverse trigonometric derivatives no	Yes
4 Contextual applications	Rates, motion, related rates yes; L'Hôpital no	Yes: related rates, linear approximation, L'Hôpital
5 Analytical applications	Extrema, concavity, sketching, optimization yes; Mean Value Theorem no	Yes, with Rolle's and the Mean Value Theorem
6 Integration and accumulation	No	Yes: Riemann sums, the Fundamental Theorem, substitution
7 Differential equations	No	Not by name on our map; antiderivatives yes
8 Applications of integration	No	Yes: area between curves, volumes
9 Parametric, polar, vectors (BC)	Vectors yes, as geometry, not as functions of t	No
10 Series (BC)	No	No

So the ideas new to an MCV4U student taking AB are accumulation, the Fundamental Theorem, substitution, differential equations and volumes, plus the Mean Value Theorem and L'Hôpital's rule. The MCV4U course page shows what the course does cover, in its order, so you can see the edge you are working from.

How to study for AP Calculus: the method

Calculus is procedures and the reasons behind them, and both are learned the same way: by producing them with the book closed.

1. Worked example, redo cold, sibling. Read one worked example slowly, cover it, redo it from the question alone, mark line by line, then do a sibling with different numbers. Sweller and Cooper (1985) found that students who studied worked examples before trying problems learned algebra procedures faster and with fewer errors than students who only solved problems. The cold redo is what stops the example from becoming reading. How to study math has the marking rule.
2. Mixed sets from week 2. Once two units are reviewed, every practice set shuffles problems from both. Rohrer and Taylor (2007) found that shuffled practice felt worse and tested better than practice blocked by type, and the AP exam is shuffled: no question tells you which unit it comes from. Why mixing beats blocking has the research.
3. The error log by skill. Every miss, with the unit's idea, the kind of error (did not know, slipped, ran out of time) and the fix, kept across the whole six weeks. The practice-test method is the hub's version of the log.
4. One released free-response set a week, under time, graded against the scoring guideline. Roediger and Karpicke (2006) found that recalling material produced better recall a week later than rereading it; a free-response set under time is recall with the format attached.
5. Recall cards for the theorems. The statement and its hypotheses on the back, the name on the front: the Intermediate Value Theorem, the Mean Value Theorem, both parts of the Fundamental Theorem. How to write a card that forces recall has the shape.

One honest sentence about score calculators: a calculator estimates a composite score from your two section scores. It is useful for understanding how the sections combine and useless as a study plan, because it cannot tell you which skills to fix. The error log can.

The six-week plan (AP Calculus AB review, week by week)

Each week has three parts: a recall task, a mixed set, and a free-response set.

Week	Recall	Mixed set	Free response
1	Diagnostic: one released full exam under time, then the error log; review units 1–2 by worked examples redone cold	Units 1–2 shuffled, two sets	The diagnostic's free-response section, graded by the guidelines
2	Units 3–4: chain rule, implicit, related rates, L'Hôpital	Units 1–4 shuffled, three sets	None; the week is for the log's top three skills
3	Units 5–6: the Mean Value Theorem, extrema, the Fundamental Theorem, substitution	Units 1–6 shuffled, three sets	One released set under time
4	Units 7–8 (BC: 9–10 as well): differential equations, volumes; series for BC	All units shuffled, three sets	One released set under time
5	A second released full exam under time; the log's top three skills, worked example, cold redo, sibling	The top three skills shuffled with everything else	The second exam's free-response section, graded
6	The compressed last week: the log's remaining skills on days 6 to 3, a timed section on day 2, a light day 1, and sleep the night before	Light: one short set a day	None after day 3

Week 6 is the one-week plan applied to this exam, and chapter 9 of the free playbook is the same seven days as a calendar. The last two days are light on purpose: the sleep stores the six weeks, and nerves fall when the format is familiar, which five released sets under time will have made it. The test-anxiety guide has the evidence on practice under conditions.

The plan, on one screen

• AB: eight units, limits through applications of integration. BC: the same plus parametric, polar and vector functions and series, at a faster pace, with an AB subscore.
• The exam: 42 multiple-choice questions in 1 h 40 min and 6 free-response questions in 1 h 30 min, half the marks each, calculator and no-calculator parts in both.
• The habit: set-up first, theorem named with its hypotheses, justification by the sign of the derivative, units, the question asked answered.
• The method: worked example, cold redo, sibling; mixed sets from week 2; the error log; one released free-response set a week; theorem cards.
• The plan: a diagnostic in week 1, two units a week, a second full exam in week 5, a light last week.