AP Calculus BC · Reference

AP Calculus BC formula sheet the formulas you have to memorize.

Q: Is a formula sheet provided on the AP Calculus BC exam?

No. AP Calculus BC does not provide a formula or reference sheet — you must memorize the derivatives, integrals, series, and theorems. Only a calculator is allowed, and only on the calculator sections.

Q: What is the difference between the AP Calculus BC formula sheet and equation sheet?

They are the same thing — the collected formulas you need to know. Because none is provided, it is a study reference you memorize.

Q: What formulas do I need for BC that are not on AB?

The BC-only additions: integration by parts and partial fractions, arc length, the parametric, polar, and vector formulas, and the sequences-and-series toolkit including Taylor series and convergence tests.

Q: Do I need to memorize the Maclaurin series for AP Calculus BC?

Yes — know the Maclaurin series for e^x, sin x, and cos x cold, since many series questions build directly from them.

Q: Is a calculator allowed on AP Calculus BC?

Yes, on Part B of the multiple choice and Part A of the free response. The other parts are no-calculator.

Here is a complete AP Calculus BC formula sheet — also called the equation sheet — covering everything from AB plus the BC-only series, parametric, and polar formulas. The catch up front: the AP Calculus BC exam provides no formula sheet, so treat this as a study reference to memorize, not something you carry into the test.

Updated June 2026Part of AP Calculus BC Review

Is a formula sheet provided on the AP Calculus BC exam?

No. Like AP Calculus AB, BC gives you no reference sheet at all — every derivative, antiderivative, series, and theorem has to be memorized. A graphing calculator (or the built-in Desmos) is allowed on Part B of the multiple choice and Part A of the free response, but the formulas themselves are on you.

That raises the stakes on the BC-only material, especially the sequences-and-series toolkit. The list below is grouped by topic, so each formula sits next to where you use it.

The AP Calculus BC formula sheet, by topic

Every formula you need on test day, with a plain-language note on when to use it.

Derivatives & integrals (the core)

Chain rule: d/dx[f(g(x))] = f′(g(x))·g′(x)

The rule BC leans on constantly.

Product & quotient rules

(fg)′ = f′g + fg′; (f/g)′ = (f′g − fg′)/g².

∫ xⁿ dx = xⁿ⁺¹/(n+1) + C · ∫ (1/x) dx = ln|x| + C

The reverse power rule and its exception.

∫ e^x dx = e^x+C · ∫ 1/(1+x²) dx = arctan x + C

From the derivative list, reversed.

Key theorems

Fundamental Theorem: ∫_a^b f(x) dx = F(b) − F(a)

Evaluate a definite integral with an antiderivative.

Average value: (1/(b−a)) ∫_a^b f(x) dx

The average height of a function on [a, b].

Mean Value Theorem: f′(c) = [f(b) − f(a)] / (b − a)

A guaranteed instant matching the average rate.

Applications of integration

Area between curves = ∫_a^b (top − bottom) dx

The setup for area problems.

Volume (disk): V = π ∫_a^b [R(x)]² dx

Rotating a region around an axis.

Arc length = ∫_a^b √(1 + [f′(x)]²) dx

BC-only — the length of a curve.

BC integration techniques

Integration by parts: ∫ u dv = uv − ∫ v du

BC-only — for products like x·e^x.

Partial fractions

Split a rational function into simpler fractions before integrating.

Improper integrals

Evaluate with a limit when a bound is infinite or the integrand blows up.

Parametric, polar & vector

Parametric slope: dy/dx = (dy/dt) / (dx/dt)

Differentiate a parametric curve.

Polar area: A = ½ ∫_α^β r² dθ

Area enclosed by a polar curve — do not drop the ½.

Vector motion: v(t) = ⟨x′(t), y′(t)⟩

Speed = √(x′² + y′²).

Sequences & series (the BC signature)

Geometric series: Σ arⁿ = a / (1 − r), for |r| < 1

Converges only when |r| < 1.

Taylor series about x = a: Σ [f⁽ⁿ⁾(a)/n!]·(x−a)ⁿ

Maclaurin series when a = 0.

Maclaurin series for e^x, sin x, cos x

Memorize these three cold — many questions build from them.

Convergence tests

nth-term, ratio, integral, comparison, and alternating-series — choosing the right one is the core BC skill.

Lagrange error bound

Bounds the error of a Taylor-polynomial approximation.

How to use this formula sheet

Because none of it is provided, the goal is recall under pressure — quiz yourself until you can reproduce each rule and each Maclaurin series from a blank page. Pair every formula with a worked problem so you know not just the formula but when it applies, and give the series section extra time, since it carries the most weight and the least familiar notation.

For a faster, concept-first version that pairs each unit with its key idea and the mistake to avoid, use our AP Calc BC cheat sheet, work full problems against the rubric with the FRQ guide, and the score calculator turns a practice raw score into a 1–5.

Frequently asked questions

Quick answers — written by humans, not a chatbot.

Is a formula sheet provided on the AP Calculus BC exam?

No. AP Calculus BC does not provide a formula or reference sheet — you must memorize the derivatives, integrals, series, and theorems. Only a calculator is allowed, and only on the calculator sections.

What is the difference between the AP Calculus BC formula sheet and equation sheet?

They are the same thing — the collected formulas you need to know. Because none is provided, it is a study reference you memorize.

What formulas do I need for BC that are not on AB?

The BC-only additions: integration by parts and partial fractions, arc length, the parametric, polar, and vector formulas, and the sequences-and-series toolkit including Taylor series and convergence tests.

Do I need to memorize the Maclaurin series for AP Calculus BC?

Yes — know the Maclaurin series for e^x, sin x, and cos x cold, since many series questions build directly from them.

Is a calculator allowed on AP Calculus BC?

Yes, on Part B of the multiple choice and Part A of the free response. The other parts are no-calculator.