FREE AP Calculus BC Study Guide 2026: All 10 Units, Series & Polar

This free AP Calculus BC study guide teaches every unit the College Board tests, organized in the official order.^[1] AP Calculus BC is a superset of AP Calculus AB: it covers all the AB material and adds the BC-only topics — parametric, polar, and vector-valued functions, and infinite sequences and series.^[3]

Because the BC-only units (9 and 10) are roughly 30% of the examand trip up the most students, this guide gives them extra weight — while still teaching the AB foundation cold. It’s interactive, not a wall of text: every unit has a built-in checkpoint quiz, hover-able glossary terms, worked examples, and concept questions, so you learn by doing.

Taking AB instead, or want the shared foundation on its own? See our AP Calculus AB study guide. Read this guide unit by unit, test yourself at each checkpoint, then round out your free prep with our practice questions and flashcards.

AP Calculus BC is one of the 17 AP exams — explore our AP study guides to compare and prep across the whole family.

AP Calculus BC Exam Snapshot

Section I (multiple choice) and Section II (free response) are each 50% of your score.^[2] Spend your study time across all ten units, but know that Integration (Unit 6) and the BC-only Infinite Series (Unit 10) are the two biggest units — together about a third of the exam:

College Board reports each unit’s exam weight as a range, so the exact mix shifts slightly each year.^[4] This guide teaches all ten units in order, with the BC-only Units 9 and 10 emphasized because they are both heavily weighted and the most common place BC students lose points.

1 · Limits and Continuity

About 4–7% of the exam. Limits are the foundation of the whole course — both derivatives and integrals are defined as limits.^[1]

Evaluating Limits

A limit is the value $f(x)$ approaches as $x$ approaches some number — and it can exist even where the function is undefined. Resolve an indeterminate $0/0$ form by factoring, multiplying by a conjugate, or applying L’Hospital’s Rule. For a rational function as $x \to \infty$, compare the degrees of numerator and denominator.

Continuity & the IVT

A function is continuous at $x = c$ when the limit exists there, $f(c)$ is defined, and they are equal. The Intermediate Value Theorem then guarantees a continuous function takes every value between $f(a)$ and $f(b)$ on $[a, b]$.

2 · Differentiation: Definition & Fundamental Properties

About 4–7% of the exam. What a derivative is, and the basic rules for computing one.^[1]

The Limit Definition

The derivative of $f$ at $a$ is $f'(a) = \lim_{h \to 0} \dfrac{f(a + h) - f(a)}{h}$ — the slope of the tangent line, the limit of secant-line slopes. It is differentiability that requires this limit to exist; a function must be continuous to be differentiable, but not vice versa.

Power, Product & Quotient Rules

Once you trust the definition, use the shortcuts:

3 · Composite, Implicit & Inverse Functions

About 4–7% of the exam.Differentiating the functions the basic rules can’t reach directly.^[1]

The Chain Rule

The chain rule differentiates a composite function: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$. Differentiate the outer function, keep the inner one, then multiply by the inner derivative — for example $\frac{d}{dx}\sin(5x^3) = \cos(5x^3)\cdot 15x^2$.

Implicit & Inverse Derivatives

Use implicit differentiationwhen an equation isn’t solved for $y$: differentiate both sides, attaching $\frac{dy}{dx}$ to every $y$-term, then solve. The inverse-trig derivatives follow the same logic — for instance $\frac{d}{dx}\arctan x = \frac{1}{1 + x^2}$.

4 · Contextual Applications of Differentiation

About 6–9% of the exam. The derivative as a real rate — motion, related rates, and linearization.^[1]

Motion & Rates

For a particle on a line, position $s(t)$ gives velocity $v(t) = s'(t)$ and acceleration $a(t) = v'(t)$. The particle speeds up when velocity and acceleration share a sign and slows down when they differ; speed is $|v(t)|$.

Related Rates & L’Hospital

In a related-rates problem, two quantities change with time and are tied by an equation. Differentiate that equation with respect to $t$, then substitute the known values and rates and solve. Set up the relationship before plugging in numbers.

5 · Analytical Applications of Differentiation

About 8–11% of the exam.Using derivatives to analyze a graph’s shape and to optimize.^[1]

Increasing, Decreasing & Extrema

The first derivative gives slope: $f' > 0$ means increasing, $f' < 0$ means decreasing, and a sign change at a critical point marks a relative extremum. The first-derivative test classifies it from how $f'$ changes.

Concavity & the MVT

The second derivative gives concavity: $f'' > 0$ is concave up, $f'' < 0$ is concave down, and a sign change is an inflection point. The Mean Value Theorem guarantees a point where the instantaneous rate equals the average rate over a closed interval.

6 · Integration and Accumulation of Change

About 17–20% of the exam — the single biggest unit. Antiderivatives, the Fundamental Theorem, and the BC integration techniques.^[1]

Antiderivatives & the FTC

A definite integral measures net accumulated change. The Fundamental Theorem of Calculus evaluates it as $\int_a^b f(x)\,dx = F(b) - F(a)$ and tells you that $\frac{d}{dx}\int_a^x f(t)\,dt = f(x)$.

Approximate integrals with Riemann sums (left, right, midpoint) or the trapezoidal rule when you only have data points. A left sum underestimates an increasing function; a right sum overestimates it.

Substitution, Parts & Partial Fractions (BC)

u-substitution reverses the chain rule. BC adds two techniques: integration by parts ($\int u\,dv = uv - \int v\,du$, useful for products like $\int x e^x\,dx$) and partial fractions, which splits a rational function into integrable pieces.

Improper Integrals (BC)

An improper integral has an infinite limit or an unbounded integrand. Evaluate it as a limit — for example $\int_1^\infty \frac{1}{x^2}\,dx = \lim_{b \to \infty}\left(1 - \frac{1}{b}\right) = 1$ converges, while $\int_1^\infty \frac{1}{x}\,dx$ diverges.

7 · Differential Equations

About 6–9% of the exam. Modeling change with equations that involve a derivative.^[1]

Slope Fields & Separable Equations

A slope field sketches the solutions of $\frac{dy}{dx}$ as little tangent segments. Solve a separable equation by getting all $y$-terms on one side and $x$-terms on the other, then integrating both sides and applying the initial condition for $C$.

Euler’s Method & Logistic Growth (BC)

Euler’s method approximates a solution numerically in steps: $y_{\text{new}} = y_{\text{old}} + \frac{dy}{dx}\cdot \Delta x$. BC also tests the logistic model $\frac{dP}{dt} = kP\left(1 - \frac{P}{M}\right)$, whose population grows fastest at half the carrying capacity $M$.

8 · Applications of Integration

About 6–9% of the exam. Using the integral to find areas, volumes, average values, and (in BC) arc length.^[1]

Area, Volume & Average Value

The area between two curves is $\int (\text{top} - \text{bottom})\,dx$ over the overlap. Volumes of revolution use the disk or washer method ($\pi r^2$ cross-sections) or shells. The average value of $f$ on $[a, b]$ is $\frac{1}{b - a}\int_a^b f(x)\,dx$.

Arc Length (BC)

Arc length is a BC topic: the length of $y = f(x)$ from $a$ to $b$ is $\int_a^b \sqrt{1 + (f'(x))^2}\,dx$. (The parametric and polar arc lengths come back in Unit 9.)

9 · Parametric, Polar & Vector-Valued Functions (BC-only)

About 11–12% of the exam, and BC-only. Curves described by a parameter, by a radius and angle, or by a position vector — and how to do calculus on them.^[3]

Parametric & Vector-Valued Functions

For a parametric curve $x(t),\ y(t)$, the slope is $\frac{dy}{dx} = \dfrac{dy/dt}{dx/dt}$ and the speed of a particle is $\sqrt{(dx/dt)^2 + (dy/dt)^2}$. A vector-valued function has position $(x(t), y(t))$, velocity $(x'(t), y'(t))$, and acceleration $(x''(t), y''(t))$.

Polar Curves & Polar Area

In polar coordinates, a curve is $r = f(\theta)$. The headline result is polar area:

10 · Infinite Sequences & Series (BC-only)

About 17–18% of the exam, and BC-only — the unit that decides most BC scores. When an infinite sum has a finite value, how to test for it, and how to build a function as a power series.^[3]

Convergence Tests

A series converges when its partial sums approach a finite limit. Two named series you must know cold: a geometric series converges to $\frac{a}{1 - r}$ when $|r| < 1$, and a p-series $\sum \frac{1}{n^p}$ converges when $p > 1$. Use a decision order for everything else:

The nth-Term Test only proves divergence; the Ratio Test is the workhorse for factorials and powers; and the Alternating Series Test handles sign-alternating series.

Power Series & Radius of Convergence

A power series $\sum c_n (x - a)^n$ is centered at $a$ and converges on an interval set by its radius of convergence— found with the Ratio Test. Always check the two endpoints separately, since the Ratio Test alone can’t decide them.

Taylor & Maclaurin Series

A Taylor series builds a function from its derivatives at a point: $\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x - a)^n$. Centered at $a = 0$ it’s a Maclaurin series. Memorize the standard ones:

How to Use This Study Guide

A study guide is a map, not the whole territory — use it alongside official College Board practice and our free tools. Work the ten units in order, because each builds on the last: limits ground derivatives, derivatives ground integrals, and integrals ground the BC-only series and polar work. Save extra time for Units 9 and 10— they’re BC-only, heavily weighted, and the most common place to lose points.

AP Calculus BC Glossary

Quick definitions for the terms you’ll see most across the ten AP Calculus BC units:

Alternating Series Test

A series whose terms alternate sign, decrease in size, and approach 0 converges.

Antiderivative

A function whose derivative is the given function; the indefinite integral, written with a constant of integration + C.

Arc length

The length of a curve y = f(x), found as the integral of √(1 + (dy/dx)²) dx — a BC topic.

Chain rule

The rule for differentiating a composite function: if y = f(g(x)), then dy/dx = f'(g(x)) · g'(x).

Continuity

A function is continuous at a point when the limit there exists, the function is defined there, and the two are equal — no holes, jumps, or breaks.

Critical point

A point where the first derivative is zero or undefined — a candidate for a relative maximum or minimum.

Definite integral

The net accumulated change of a quantity — the signed area under a rate curve from a to b — equal to F(b) − F(a) by the FTC.

Derivative

The instantaneous rate of change of a function — the slope of the tangent line — defined as the limit of [f(a + h) − f(a)] / h as h approaches 0.

Disk / washer method

Finding a volume of revolution by integrating circular cross-sections of area πr² (washer when there's a hole).

Euler's method

A BC step-by-step numerical method that approximates a differential equation's solution using tangent-line steps: y_new = y_old + (dy/dx) · Δx.

Fundamental Theorem of Calculus

The theorem connecting derivatives and integrals: integrating a rate gives net change (Part 1), and differentiating an accumulation function returns the integrand (Part 2).

Geometric series

A series a + ar + ar² + … ; it converges to a / (1 − r) when |r| < 1 and diverges otherwise.

Implicit differentiation

Differentiating an equation that is not solved for y by treating y as a function of x and applying the chain rule to y-terms.

Improper integral

A BC integral with an infinite limit of integration or an integrand that is unbounded; evaluated as a limit.

Inflection point

A point where a curve changes concavity, where the second derivative changes sign.

Integration by parts

A BC technique for integrating a product: the integral of u dv = uv − the integral of v du.

Intermediate Value Theorem

If a function is continuous on a closed interval [a, b], it takes every value between f(a) and f(b) at least once on that interval.

L'Hospital's Rule

A method for evaluating an indeterminate limit of the form 0/0 or ∞/∞ by taking the limit of the ratio of the derivatives instead.

Lagrange error bound

A BC bound on how far a Taylor polynomial approximation can be from the true function value.

Limit

The value a function approaches as its input approaches some number. The limit can exist even where the function itself is undefined.

Logistic model

A BC growth model dP/dt = kP(1 − P/M) with carrying capacity M; growth is fastest at half the carrying capacity.

Maclaurin series

A Taylor series centered at a = 0; the standard ones are for eˣ, sin x, cos x, and 1/(1 − x).

Mean Value Theorem

If f is continuous on [a, b] and differentiable on (a, b), there is a point c where the instantaneous rate f'(c) equals the average rate (f(b) − f(a)) / (b − a).

nth-Term Test

If the terms of a series do not approach 0, the series diverges. It can never prove convergence.

p-series

The series sum of 1/nᵖ; it converges when p > 1 and diverges when p ≤ 1.

Parametric equations

A BC way of defining a curve with x(t) and y(t); the slope is dy/dx = (dy/dt) / (dx/dt).

Partial fractions

A BC technique that splits a rational function into simpler fractions that can be integrated individually.

Polar area

The area swept by r = f(θ) from α to β: ½ times the integral from α to β of [r(θ)]² dθ — a BC formula.

Polar coordinates

A BC system locating points by radius r and angle θ; a polar curve is r = f(θ).

Power series

A BC series of the form the sum of cₙ(x − a)ⁿ; it converges on an interval set by its radius of convergence.

Radius of convergence

The distance from the center of a power series within which it converges, usually found with the Ratio Test.

Ratio Test

A test using the limit of |aₙ₊₁ / aₙ|: less than 1 converges, greater than 1 diverges, equal to 1 is inconclusive.

Related rates

Problems where two changing quantities are linked by an equation; you differentiate with respect to time to relate their rates.

Separable differential equation

An equation that can be rewritten so all y-terms are on one side and all x-terms on the other, then integrated.

Sequence

An ordered list of numbers aₙ; it converges if its terms approach a single limit as n → ∞.

Series

The sum of the terms of a sequence; it converges if its sequence of partial sums approaches a finite limit.

Slope field

A grid of small line segments showing the slope dy/dx of a differential equation's solutions at many points.

Taylor series

A BC representation of a function as an infinite power series built from its derivatives at a point a.

u-substitution

An integration technique that reverses the chain rule by substituting u for an inner function so the integral becomes a standard form.

Vector-valued function

A BC function giving position (x(t), y(t)); its derivative is the velocity vector (x'(t), y'(t)).

Free AP Calculus BC Study Materials & Resources

Everything you need to prepare for AP Calculus BC is free here — no paywall, no sign-up. This guide is the foundation; pair it with the rest of our free AP Calculus BC study materials for active recall, timed practice, and last-minute review:

• AP Calculus BC Practice Test — exam-style questions across all ten units, with explanations.
• AP Calculus BC Flashcards — active-recall decks for the formulas, convergence tests, and BC-only series and polar results.
• AP Calculus AB Study Guide — the shared AB foundation, if you’re taking AB or want to lock in the basics first.

AP Calculus BC Study Guide FAQ

References

1. 1.College Board. “AP Calculus BC — Course and Exam Description.” College Board. ↑
2. 2.College Board. “AP Calculus BC — Exam Format & Structure.” College Board. ↑
3. 3.College Board. “AP Calculus BC — AP Students.” College Board. ↑
4. 4.College Board. “AP Calculus BC — Assessment & Exam Weighting.” College Board. ↑
5. 5.College Board. “AP Calculus AB and BC Course and Exam Description (CED).” College Board. ↑

Sources for the concept answers

Every answer in the AP Calculus BC concept questions above is drawn from an official primary source:

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