AP Exam Playbook

The AP Calculus BC exam is on Monday, May 10, 2027, in the morning session. That's the same time as AP Calculus AB, so you can't take both in the same year. At most schools in the U.S. it starts at 8 a.m. local time; some places use a different time, so check with your AP coordinator.

The exam takes 3 hours and 10 minutes and has 48 questions: 42 multiple-choice and 6 free-response. The two sections count equally. Multiple choice is half your score, and free response is the other half.

That's a little over 2 minutes per question in multiple-choice Part A, a little under 3 minutes per question in Part B, and about 15 minutes per free-response question.

What's on it

BC covers everything in AB and more. About 60% of the exam is on the topics the two courses share, and the other 40% or so is BC only. In this course, Units 1 to 8 are the AB units with the BC-only topics added (integration by parts, partial fractions, improper integrals, Euler's method, logistic growth and arc length), and Units 9 and 10 are BC only: parametric, polar and vector functions, and infinite series. Those two units alone are 25% to 35% of the multiple-choice section, and series had a free-response question on every exam from 2023 to 2026.

Expect every kind of function you've studied, as formulas, graphs, tables and descriptions in words, plus the BC kinds: parametric and polar curves, vector-valued functions and power series. In multiple-choice Part B, not every question needs the calculator. Of the six free-response questions, at least two are set in a real-world situation, and about half of the work is carrying out procedures while the other half is explaining and reasoning.

The six free-response questions

Three of the six free-response questions are the same as on the AB exam, and three are BC only. On every exam from 2023 to 2026 they fell into the same slots:

That's a pattern, not a promise: the College Board says three questions are shared with AB, but the slots could change. The BC-only questions are worth 27 of the 54 free-response points, and they often open with a part that's plain AB calculus, so start every one of them.

Screen for multiple choice, paper for free response

The exam is hybrid digital. Here's what that means in practice:

• You answer the multiple-choice questions in the Bluebook app. Each one has four answer choices, A to D.
• The free-response questions appear in Bluebook, but you write your answers by hand in a paper booklet, in pencil or in black or dark-blue pen. Only what's in that booklet is scored. Scratch paper isn't.
• Each part has its own time limit. Within a part, you can move between its questions until time runs out.
• The on-screen clock turns red when 5 minutes are left. Nobody announces it, so glance at the clock now and then.
• There's no formula sheet. The derivatives, integrals and theorems you need have to come from memory.

What to bring

• Up to two graphing calculators from the College Board's approved list, with fresh batteries or a full charge. (Bluebook's built-in Desmos calculator is there too; see Calculator skills below.)
• Pencils, or black or dark-blue pens, for the free-response booklet.
• Anything else your AP coordinator asks for. They confirm the start time, the room and how Bluebook is set up at your school.

Before May, try a Bluebook test preview so the app and its built-in calculator feel familiar, including graphing a parametric and a polar curve.

No formula sheet: what to know by heart

With no formula sheet, BC has more to remember than AB. Know these cold, along with the series tests in the free-response section below:

Each of the 6 free-response questions is worth 9 points, 54 points in all, and together they're half your score. A question usually has 3 or 4 lettered parts worth 1 to 3 points each.

The graders (the College Board calls them readers) score each point on its own. Since 2025, the scoring guidelines number the points in every question, P1 to P9, and give each one a short name for what it rewards: a setup, an answer, a reason. That's good news for you. A question you can't finish can still earn most of its points, as long as you write down every piece you can.

The kinds of points

Points are often linked. A reason point usually needs the correct answer it supports, and some points are only available if an earlier one was earned. In a separation-of-variables part, for example, the points usually build on each other: if you never separate the variables, none of them can be earned, and leaving out the constant of integration limits how many you can get. One bad first step can cost a whole part, so slow down at the start.

Show the work: no bald answers

A bald answer is a value with nothing behind it. On a no-calculator part it usually earns the answer point at most, and sometimes nothing at all: an unsupported Riemann-sum value, or "yes, by the IVT" with no values, can score zero. If a part says justify, explain or show the work, the work is what's being scored.

You don't have to simplify. Once you've written a correct numerical expression you can usually stop, and a later arithmetic slip often won't cost the answer point. But a few parts do count errors made while simplifying, and an expression a reader can't read one way only (unmatched parentheses, for instance) earns nothing. Write the clean expression, then simplify only if it's easy.

Justify with calculus, and check the conditions

A justification is a reason built from the information you're given: the sign of a derivative, the values of a function, a theorem whose conditions you've checked. Name the function every time. "It", "the function", "the graph" or "the slope" with no name attached don't earn reason points. A sign chart on its own doesn't either: say in words or symbols what the sign is, on which interval, and what that tells you.

You don't have to name a theorem if its conditions and conclusion are clear. But if you name one, it has to be the right one. Mixing up the Intermediate Value Theorem (about values of f) and the Mean Value Theorem (about values of f′) is a classic way to lose a point.

Notation and linkage

• Use = only between things that are equal. A chain like f′(x) = 2x = 6 claims that 2x is always 6; write f′(3) = 2(3) = 6 instead. Errors like this come up every year.
• Write a definite integral in full: the integrand in parentheses, the limits, and dx (or dt). Never put ∫ in front of an antiderivative you've already found.
• Use ≈ for estimates, such as Riemann sums and tangent-line approximations.
• Keep dy/dx and dy/dt straight, and don't write a bare dy.
• Write lim in front of every expression in a chain of limits, including the one after you use L'Hospital's Rule.
• Use the names you're given. ∫₁⁴ (f(x) − g(x)) dx is enough, and it's safer than substituting long formulas, where parentheses get dropped.
• Label each part (a), (b) and so on, and label any function, graph or table you use.
• Copy given functions carefully. Re-read the formula before you differentiate it.
• Answer exactly what's asked. If the question asks for x-values, give x-values, not points. An extra wrong answer, like one inflection point too many, can cost both the answer and the reason.
• Extra work has to be right too. A wrong value you didn't need can still cost a point.

Decimals: three places, and no early rounding

• Give decimal answers correct to three places after the decimal point, unless the question asks for something else (like the nearest whole number). Rounding and truncating both work: 1.29583… can be written 1.296 or 1.295. 1.3 and 1.30 aren't accurate enough.
• Within one question you lose at most one point for rounding. It's still a point, so don't give it away.
• Don't round in the middle. Store intermediate values in your calculator and use the stored value in the next step.
• Any intermediate value you write down, like the place where two curves cross, has to be correct to three decimal places itself.

Units and "interpret" answers

When a question asks for units, they're a point of their own, and you can earn it even with a wrong number. A derivative's units are output units per input unit. An integral of a rate is an amount: rate units times time units. So if R(t) is in liters per minute and t is in minutes, R′(t) is in liters per minute per minute, and ∫₀¹⁰ R(t) dt is in liters.

An interpretation needs three things: what the value measures in the situation, its units, and when (at a specific time, or over a specific interval). Answer exactly what was asked: "the rate is increasing" and "the amount is increasing" are different claims.

The BC topics: where the extra rules are

Everything above holds on the BC exam. The BC-only questions (Q2, Q5 and Q6) add rules of their own, and they're where BC students lose the most points. Here they are, topic by topic.

Series: name the test and check its conditions

A series question is scored like any justification: the conclusion only counts with a reason, and the reason is a named test whose conditions you've checked in writing. Say "the series converges", not "the terms converge".

Intervals of convergence are where the most points slip away. On the 2025 series question, only 14% of BC students earned the point for checking both endpoints correctly and giving the final interval (College Board, 2025 Chief Reader Report).

• For a series in powers of x², like Σ x²ⁿ/9ⁿ, the ratio test gives x²/9 < 1, so |x| < 3: the radius is 3, not 9.
• Differentiating or integrating a power series term by term keeps the same radius of convergence. At an x outside the interval, the series doesn't converge, so it can't equal the function there.
• If a question asks whether a series converges at a point, give a clear yes or no, with the reason.

Error bounds: write an inequality chain

BC has two error bounds, and both are scored the same way: an inequality chain that starts with the actual error and ends at or below the target. Never write that the error equals the bound. You don't know the error, only that it's at most the bound.

• Alternating series error bound: if the series alternates and its terms shrink in size to 0, the error is at most the size of the first term you left out (the next nonzero term after the last one you used, not the last one you kept). Say why the bound applies: the series alternates, and its terms decrease in size to 0.
• Lagrange error bound: |f(x) − Pₙ(x)| ≤ (max of |f⁽ⁿ⁺¹⁾(z)| for z between a and x) · |x − a|ⁿ⁺¹/(n + 1)!. The derivative, the power and the factorial all use n + 1, and the word max (or a stated upper bound) has to be there.
• Use ≤ when the bound could equal the target, and < only when it's really smaller. "Bound < 0.01" is wrong when the bound is exactly 0.01.
• If the series alternates, use the alternating bound. Reaching for Lagrange, or for a geometric sum, out of habit is a common way to lose the point.

In 2025, 48% of BC students earned the point for writing the Lagrange bound in the right form, but only 28% earned the point for using it to reach the conclusion (College Board, 2025 Chief Reader Report).

Taylor polynomials

• Center it where you're asked: "about x = 1" means powers of (x − 1).
• A polynomial of degree n stops at the (x − a)ⁿ term. Ending with "+ …" or adding higher-degree terms has cost students the last point.
• The coefficient of (x − a)ᵏ is f⁽ᵏ⁾(a)/k!. You don't have to simplify it.
• Write a derivative's value at its point: f″(0) = 5, never f″(x) = 5 or f″(x) = f″(0).
• Derivatives found from a differential equation need the chain rule every time y appears (a dy/dx factor), and the product rule for products.
• "The first four nonzero terms" means skip any zero terms and write exactly four.
• A general term needs parentheses, the right starting index and (−1)ⁿ in the right place. Build it from a pattern of terms you've checked.
• Differentiate a series term by term with respect to x, not n.

Polar area: square r, and the outer curve goes first

• Area = ½ ∫ r² dθ. Square r, and keep the ½.
• Inside one curve and outside another: ½ ∫ (R² − r²) dθ, where R is the curve farther from the pole on that interval of θ. Check one value of θ in the interval, or look at the graph. It's R² − r², not (R − r)².
• The limits are the angles where the curves meet: solve R(θ) = r(θ), in radians. The setup has to be a definite integral with numbers as its limits.
• Symmetry is fine: twice the area of half the region.

On the 2025 polar question, only 16% of BC students earned the answer point for the area, the point that checks the ½ and the limits (College Board, 2025 Chief Reader Report).

Motion in the plane: speed, position and distance

• Velocity is the vector ⟨x′(t), y′(t)⟩, and acceleration is ⟨x″(t), y″(t)⟩. Give the one that's asked for, with the components in order.
• Speed is √((x′(t))² + (y′(t))²), never |dy/dx|. To find when the speed equals some value, write that equation and solve it.
• Position: x(b) = x(a) + ∫ₐᵇ x′(t) dt, a definite integral with dt. Working backward from a later time, subtract the integral instead.
• Total distance traveled is the integral of the speed over the interval, the same integral as the arc length of the path. It's not built from the two components' separate integrals.
• The slope of the path is dy/dx = (dy/dt)/(dx/dt) at the right time.
• Moving toward the x-axis means y(t) and y′(t) have opposite signs. Say both signs and the interval.

"Farthest", "closest" and other polar and motion claims

On the 2025 polar question, only 6% of BC students earned the point for a global argument about the farthest point (College Board, 2025 Chief Reader Report). A sign change at one value only shows a relative extremum.

Improper integrals: limit notation on every line

• Rewrite the integral as a limit, such as lim (b → ∞) of ∫₁ᵇ, and keep writing lim on every line until you actually take the limit. Plugging in ∞, or doing arithmetic with it, loses the point.
• An integrand that blows up at an endpoint, or anywhere inside the interval, makes the integral improper too: split it at the bad point and use a limit for each piece.
• If the limit is infinite or doesn't exist, write "the integral diverges".
• With a substitution, change the limits to the new variable correctly.

Integration by parts and partial fractions

• Show your u and dv. The tabular method counts too, as long as its columns start with the right functions.
• ∫ u dv = uv − ∫ v du: keep the minus sign, put parentheses around whatever you substitute, and end an indefinite integral with + C.
• Check your v by differentiating it. Coefficients flip easily: the antiderivative of e^(x/2) is 2e^(x/2), not ½e^(x/2).
• Partial fractions: factor the denominator, find the constants, and integrate to logarithms with absolute values, keeping the 1/a in front of ln|ax + b|. If the numerator's degree isn't lower than the denominator's, divide first.

Euler's method

• Take exactly the number of steps asked, with the step size given, starting from the given point.
• Each step: new y = old y + Δx · (the slope at the point you're at now).
• Write ≈ for the estimates, and show every step.

In 2025, 72% of BC students earned the point for the first Euler step, but only 44% earned the final answer (College Board, 2025 Chief Reader Report). A slip in the first step carries through to the end, so check it before you go on.

Logistic models

Logistic models and partial fractions haven't had a free-response part from 2023 to 2026, but they can still come up in multiple choice. Most logistic questions can be answered from the differential equation alone, without solving it:

• In dy/dt = ky(a − y), or dy/dt = ky(1 − y/a), the number a is the carrying capacity: the value y(t) approaches as t → ∞, when y starts above 0.
• If y starts above a, it decreases toward a.
• y grows fastest when y = a/2, halfway to the carrying capacity. Questions usually ask for that value of y, not a time.

When something goes wrong

• Use your earlier answers, even ones you doubt. If part (b) needs your value from part (a), use it: work that's consistent with a wrong earlier answer can still earn the later points.
• Stuck on a part? Move on. Later parts often stand on their own, and you can start fresh from the information in the question.
• Short on time? Write the setup. A correct integral or equation can earn its point even without the value.
• Cross out work you don't want graded with a single line instead of spending time erasing. Crossed-out work isn't scored.
• Commit to one answer. If two different answers to the same part are left standing, the reader won't pick the better one for you.

Before you leave a question

1. 1Did I answer what was asked, in the form it asked for: a value, an x-value, an interval, a sentence?
2. 2Is every number backed by work, and every calculator result by its setup?
3. 3Does every reason name its function, and check the conditions of any theorem I used?
4. 4Three decimal places? Units, if the question asked for them?
5. 5Is there a stray extra answer, or an = that isn't true?

And on the BC-only questions, also check:

1. 1Series: lim (n → ∞) written, absolute values in the ratio, both endpoints tested with named tests, and the brackets right?
2. 2Error bound: a chain that starts at the actual error and ends at the target, with no "error ="?
3. 3Taylor polynomial: the right center, the right degree, and no "+ …"?
4. 4Polar area: the ½, each r squared, the outer curve first, and θ-limits where the curves meet?
5. 5Motion: speed from both components, and each position as a definite integral with dt?
6. 6Improper integral: lim on every line until it's taken?
7. 7Euler: the right number of steps, each slope taken at the current point?

You need a graphing calculator in two parts only: multiple-choice Part B and free-response Part A. In those parts Bluebook has a built-in Desmos graphing calculator, and you can also bring up to two approved handheld graphing calculators. The rest of the exam is calculator-free.

Every approved calculator can do four things, and these four are the only calculator results you can use without showing the steps. Practice them until they're automatic:

1. 1Graph a function in a window you choose.
2. 2Solve an equation numerically, which includes finding where a function equals zero.
3. 3Find the value of a derivative at a point.
4. 4Find the value of a definite integral.

What to write when you use it

On a free-response question, write the setup in ordinary math notation (the equation you solved, the derivative you evaluated, or the integral you computed), then the result. Calculator commands aren't math notation and don't earn the setup point. Anything beyond the four skills, such as a built-in maximum finder, a regression, a table search or a program, doesn't count as a reason: show the calculus instead.

Desmos in Bluebook (multiple choice, Part B)

• Try the built-in Desmos in a Bluebook test preview before exam day. The Desmos website and app aren't allowed during the exam, but you can practice the same skills there.
• Define a function once, like f(x) = …, and reuse it: type f'(2) for the derivative's value at 2, and type int to start a definite integral.
• To solve an equation, graph each side as its own curve and tap the point where they cross. For zeros, tap where the graph crosses the x-axis.
• Set the window with the zoom buttons or the graph settings, then zoom out once to make sure you haven't missed a crossing.
• A point's label can show only a few decimal places. If two answer choices are close, zoom in on the point to see more digits.
• Desmos works in radians unless its settings are switched to degrees. Leave it in radians.
• Not every Part B question needs the calculator. If you can see the answer, take it and save the time.

Your handheld (free response, Part A)

In free-response Part A you can use the built-in Desmos or a handheld calculator. If you use a handheld:

• Set radian mode before the exam, and check it again when Part A starts.
• Enter each given function once, in the calculator's function list, and evaluate from there instead of retyping a long formula.
• Store intermediate results in a memory variable (A, B, and so on) instead of copying a rounded number, and use the stored value in the next step.
• Learn where your model keeps each of the four tools, and practice until you reach them without hunting through menus.
• Write the setup in the booklet first, then compute. If the calculator gives something odd, the setup shows what you meant.
• Bring fresh batteries or a full charge. A second approved calculator is allowed as a backup.

What BC adds

The calculator question on the BC side, Q2, has been about motion in the plane or a polar curve, and both lean on the calculator. Practice these on the calculator you'll bring and in Bluebook's Desmos:

• Graph a parametric curve. In Desmos, type the point (x(t), y(t)) and set the range of t; on a handheld, switch to parametric mode. The graph shows the path and which way it goes.
• Graph a polar curve. In Desmos, type r = …; on a handheld, switch to polar mode. Set the range of θ to the one in the question.
• Find where two polar curves meet by solving R(θ) = r(θ) for θ, in radians: those angles are your area limits. Curves can also meet at the pole, where r = 0, at different angles, so check the graph.
• Let numeric integrals do the heavy lifting: polar area, total distance, arc length, and a position found from a velocity. Enter each integrand as an ordinary function of one variable. The College Board's 2025 Chief Reader Report suggests evaluating a polar area integral in function (rectangular) mode, not polar mode.
• Use the numeric derivative for a velocity, an acceleration or a value of dr/dθ at one point.
• To find when the speed equals some value, solve the equation √((x′(t))² + (y′(t))²) = value with the solver, or by graphing both sides, and take the time that fits the question. TRACE isn't accurate enough.
• Store each value you'll reuse, like an angle where two curves meet, and use the stored value in the next step.
• Radian mode matters on every BC calculator question. In the 2023 BC scoring guidelines, a response worked in degree mode lost the first point it would otherwise have earned, and polar angles in degrees give the wrong area limits.

Which calculators are allowed

A handheld has to be on the College Board's list of approved graphing calculators for AP Calculus. The list marks the models that have the four required capabilities. Examples include the TI-84 Plus CE, TI-89, TI-Nspire CX and CX CAS, HP Prime and NumWorks; check your exact model on the official list.

These can't be used as your calculator: phones, watches, laptops and tablets, devices with a QWERTY keyboard or pen or stylus input, anything with a wireless, Bluetooth, cellular or internet connection, and touch-screen models that aren't on the list. You don't have to clear your calculator's memory, but you can't share a calculator with anyone.

A calculator with a computer algebra system (CAS) is allowed if its model is approved, but you don't need one: nothing in this course needs it.

AP scores run from 1 to 5. The College Board describes each score like this:

Each college sets its own policy. Many give credit or placement for a 3 or higher, and some require a 4 or 5, so check the policy of every college you're considering.

The AB subscore

Your BC score report also shows a Calculus AB subscore, from 1 to 5. It's based on the part of the BC exam that tests AB topics, about 60% of the exam. So the AB topics count twice: toward your BC score and toward the subscore.

The College Board asks colleges to treat the AB subscore the way they treat an AP Calculus AB score, because the shared topics are tested at the same level in both courses. Each college still sets its own policy, so look up both: what it gives for your BC score, and what it gives for the AB subscore. Exactly which questions count toward the subscore isn't published.

Recent results

Source: the College Board's AP Students website. The 2026 figures are listed in whole percents so far and may be updated. In 2025, 160,436 students took AP Calculus BC.

Source: the College Board's AB subscore distributions for each year. The 2026 subscore figures hadn't been posted when we checked in late October 2026. No subscores were reported in 2020, when the exam was shortened.

How the score is worked out

Your multiple-choice and free-response results are combined into one composite score, each section worth half. The College Board then decides how many composite points each AP score takes. Those cut points are set every year, they change from year to year, and the College Board doesn't publish them for recent exams. Old released exams printed conversion tables, but those exams had a different design.

The traditional estimate gave each multiple-choice question 1.2 points, so 45 questions made 54. With 42 questions, keeping the same 50/50 split means multiplying the number you got right by 54/42 (about 1.286) and adding your free-response points, for 108 points in all. It's an estimate, not an official formula.

That holds for BC too, and the College Board doesn't publish how the AB subscore is worked out either. Online score calculators make their own guesses. Here are three of them, out of the 108-point composite above:

Source: each site's AP Calculus BC score calculator, checked in October 2026. These are third-party guesses, not College Board numbers, and they disagree by several points.

The plan below counts down to the exam on Monday, May 10, 2027. It shares out the weeks you have left among the units you still need to study (all 10 of them to start with), in course order, and gives more time to the units that carry more weight on the exam. The last weeks are kept for review and a full practice exam.

When you're signed in, the plan skips the units you've already finished and gives their time to the rest. Treat it as a guide: if you're ahead, start the next unit; if you fall behind, keep going in order and protect the review weeks.

The 10 units, and what BC adds to each

The shares are for the multiple-choice section, from the College Board's course description; free-response questions draw on the whole course. Unit 10 is the longest: the College Board's suggested pacing gives it about 17 to 18 of roughly 115 class periods.

Where to spend extra time

The two lowest means from 2023 to 2025 were both BC only: the 2025 polar question (3.09) and the 2024 series question (3.38). So plan extra free-response practice for Units 9 and 10, and once you've finished Unit 10, keep doing a series question every week or two until May.

• Study a little on most days rather than a lot on one day.
• Finish each unit with its review and timed unit test, then go back to the topics still marked Practicing on the course page.
• Write your free-response practice by hand, on paper, the way you'll write it in May.
• Practice the four calculator skills on the calculator you'll bring, and try the built-in Desmos in a Bluebook test preview.
• In the review weeks, take the full practice exam in one sitting with real timing, then redo every question you missed.

• Took AB already? Use the Unit 1 to 5 reviews and timed tests to check what you remember, and move on quickly where you score well.
• There's no formula sheet, so make flash cards for the formulas in How the exam works, and say the series tests' conditions out loud until they're automatic.
• Practice graphing parametric and polar curves on the calculator you'll bring, and in the Bluebook Desmos.
• Write series justifications in full every time, even in practice: the limit, the test's name, its conditions, and both endpoints.