AP Calculus BC units, explained

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

AP Calculus BC has ten units: the eight AB units, with extra topics, plus two BC-only units. Here is the full map.

The ten units

1. Limits and Continuity
2. Differentiation: Definition and Fundamental Properties
3. Differentiation: Composite, Implicit, and Inverse Functions
4. Contextual Applications of Differentiation
5. Analytical Applications of Differentiation
6. Integration and Accumulation of Change, including integration by parts, partial fractions and improper integrals
7. Differential Equations, including Euler's method and logistic growth
8. Applications of Integration, including arc length
9. Parametric Equations, Polar Coordinates, and Vector-Valued Functions (BC only)
10. Infinite Sequences and Series (BC only)

Check unit weightings in the current course and exam description on AP Central.

The BC-only units

• Unit 9 extends calculus to curves described by parametric and polar equations, and to motion along vector paths.
• Unit 10 covers convergence tests, Taylor and Maclaurin series, and error bounds. Read our AP Calculus BC series guide.

How to prepare

1. Keep the AB core sharp; it is still most of the exam.
2. Start series early and revisit it often. It rewards repetition.
3. Practise parametric and polar questions with both calculator and non-calculator methods.
4. Use official free-response questions to learn how BC answers are marked.

Still deciding between the two? See AP Calculus AB vs BC.

Worked examples for the BC-only topics

Parametric speed

x = 3t and y = 4t. Find the speed.

• Speed = √((dx/dt)² + (dy/dt)²) = √(9 + 16) = 5.

Polar area

Find the area inside r = 2.

• A = ½∫₀^2π r² dθ = ½ × 4 × 2π = 4π, the area of a circle of radius 2.

Euler's method

dy/dx = x + y with y(0) = 1. Use two steps of size 0.1 to estimate y(0.2).

• Step 1: y(0.1) ≈ 1 + 0.1(0 + 1) = 1.1.
• Step 2: y(0.2) ≈ 1.1 + 0.1(0.1 + 1.1) = 1.22.

Logistic growth

dP/dt = 0.5P(1 − P/1000). Find the carrying capacity and the population when growth is fastest.

• Carrying capacity: 1000.
• Growth is fastest at half the carrying capacity: P = 500.

Integration by parts

Find ∫x cos x dx.

• Let u = x and dv = cos x dx, so du = dx and v = sin x.
• x sin x − ∫sin x dx = x sin x + cos x + C.

Partial fractions

Find ∫ 1/(x(x + 1)) dx.

• 1/(x(x + 1)) = 1/x − 1/(x + 1).
• ln|x| − ln|x + 1| + C.

An improper integral

Evaluate ∫₁^∞ 1/x² dx.

• lim_b→∞ [−1/x]₁^b = 0 + 1 = 1.

Arc length

Find the length of y = (2/3)x^3/2 from x = 0 to 3.

• y' = √x, so L = ∫₀³ √(1 + x) dx = (2/3)(1 + x)^3/2 from 0 to 3 = (2/3)(8 − 1) = 14/3.

Practice questions

Try each one before opening the answer.

Frequently asked questions

Which BC topic is hardest?

Most students say infinite series. Start practising it early and keep returning to it.

Are parametric and polar questions calculator-active?

They can appear in both calculator and non-calculator sections, so practise both ways.

Key takeaways

• BC adds parametric, polar and vector-valued functions, and infinite series.
• BC also extends AB units with by parts, partial fractions, improper integrals, Euler's method, logistic growth and arc length.
• The AB core is still most of the exam.
• Practise series steadily from the start of the unit.
• Use official free-response questions to see how BC answers are marked.

Understanding the BC-only units

Parametric, polar and vector-valued functions

So far, curves were described as y = f(x). Parametric equations describe both x and y using a third variable, usually time. This is natural for motion: x(t) and y(t) tell you where an object is at each moment. Vector-valued functions package the same idea as a position vector ⟨x(t), y(t)⟩. Velocity is the derivative of position, and speed is the magnitude of velocity.

Polar coordinates describe points by a distance r from the origin and an angle θ. Some shapes, such as spirals, roses and cardioids, are far simpler in polar form. The key BC skills are finding slopes of polar curves and areas enclosed by them, using A = ½∫r² dθ.

Infinite sequences and series

Series ask a surprising question: can you add infinitely many numbers and get a finite answer? Sometimes yes (1 + ½ + ¼ + … = 2), sometimes no (1 + ½ + ⅓ + … grows without bound). The unit builds tests to decide, then uses series to represent functions as infinite polynomials. Taylor polynomials let calculators approximate values of e^x, sin x and other functions.

How the BC topics appear in the exam

• Parametric or vector motion: velocity, speed, acceleration and total distance travelled.
• Polar: area enclosed by a curve, or between two curves, and the rate of change of r.
• Series: a Taylor polynomial, its interval of convergence and an error bound, often all in one question.

Example: total distance for parametric motion

A particle moves with dx/dt = 3 and dy/dt = 4 for 0 ≤ t ≤ 2. Find the distance travelled.

• Speed = √(9 + 16) = 5, constant.
• Distance = ∫₀² 5 dt = 10.

More practice

Try each one before opening the answer.

More questions students ask

How much of the BC exam is BC-only material?

A significant share, but most of the exam still tests the AB core. Check the current weightings on AP Central.

Can I learn BC topics over the summer?

Some students preview series or parametric equations, which helps, but the course itself will teach everything you need.

Study tip

BC moves quickly, so keep a running list of AB skills you feel shaky on and practise one of them every weekend. The AB core is still the largest part of the BC exam.

Where to go next

Series is usually the toughest BC unit, so give it its own focused practice. If you are still choosing between courses, the comparison guide will help.

• AP Calculus BC series
• AP Calculus AB vs BC
• What to memorise for AP Calculus