Calculus BC
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AP Calculus BC — Rules, Series & Theorems

6 sections · 36 entries · print it and keep it beside your practice sets

Everything on the AB sheet still applies — this page adds the BC-only material: advanced integration, parametric and polar calculus, and the series unit that is worth roughly a fifth of the exam. No formula sheet is supplied on exam day.

Practise with the sheet, not from memory. The College Board hands out its own version of this page on exam day, so nothing here is worth memorising for its own sake. What earns points is speed: knowing which section a quantity lives in, and reading off the right line without breaking your train of thought. Keep this open (or printed) for every practice set you do.

All of AB still counts

AB core

Limits, the differentiation rules, FTC, u-substitution, areas and volumes, separable differential equations — all of it is examinable on BC

See the AP Calculus AB reference sheet for those rules; only the BC additions are listed below.

Advanced integration techniques

By parts
∫ u dv = u v − ∫ v du

Integration by parts

Pick u so that du is simpler — the usual priority is logs, then inverse trig, then algebraic, then exponential/trig.

Partial fractions
1/((x−a)(x−b)) = A/(x−a) + B/(x−b)

Non-repeating linear factors

Improper integral
∫_a^∞ f dx = lim_(b→∞) ∫_a^b f dx

Rewrite as a limit

Also needed when the integrand blows up inside the interval. It converges only if the limit is finite.

Arc length
L = ∫_a^b √( 1 + [f′(x)]² ) dx

Length of y = f(x) from a to b

Differential equations (BC additions)

Euler’s method
y_(n+1) = y_n + h · f(x_n, y_n)

Numerical stepping with step size h

Logistic model
dy/dt = k y (1 − y/L)

Growth with carrying capacity L

Logistic solution & behaviour
y = L / (1 + C e^(−k t))

Solution and its key features

y → L as t → ∞, and the growth rate dy/dt is maximised at y = L/2 (the inflection point of the solution curve).

Parametric, vector & polar calculus

dy/dx
dy/dx = (dy/dt) / (dx/dt)

Slope of a parametric curve

d²y/dx²
d²y/dx² = ( d/dt (dy/dx) ) / (dx/dt)

Second derivative of a parametric curve

Speed
speed = √( (dx/dt)² + (dy/dt)² )

Magnitude of the velocity vector

Arc length (parametric)
L = ∫_a^b √( (dx/dt)² + (dy/dt)² ) dt

Distance travelled from t = a to t = b

Vector motion
x(t) = x(a) + ∫_a^t x′(τ) dτ, similarly for y

Position from velocity

Polar conversion
x = r cos θ, y = r sin θ, r² = x² + y², tan θ = y/x

Polar to rectangular

Polar area
A = ½ ∫_α^β [ r(θ) ]² dθ

Area swept by r(θ) from α to β

Polar slope
dy/dx = ( r′ sin θ + r cos θ ) / ( r′ cos θ − r sin θ )

Slope of a polar curve

Series — convergence tests

nth-term test
If lim_(n→∞) aₙ ≠ 0, then Σ aₙ diverges.

Test for DIVERGENCE only

If the limit IS zero the test tells you nothing.

Geometric series
Σ_(n=0)^∞ a rⁿ = a / (1 − r), |r| < 1

Converges only for |r| < 1

p-series
Σ 1/n^p converges ⇔ p > 1

The benchmark family

p = 1 is the harmonic series, which diverges.

Integral test
Σ aₙ and ∫_1^∞ f(x) dx both converge or both diverge

For positive, decreasing, continuous f with f(n) = aₙ

Comparison
If 0 ≤ aₙ ≤ bₙ and Σbₙ converges, Σaₙ converges

Direct comparison of positive series

Limit comparison
If lim aₙ/bₙ = L with 0 < L < ∞, the two series behave the same way

For positive series

Ratio test
L = lim_(n→∞) |a_(n+1)/aₙ|: L < 1 converges, L > 1 diverges, L = 1 inconclusive

Best for factorials and exponentials

Alternating series test
Converges if bₙ is decreasing and lim bₙ = 0

For Σ(−1)ⁿ bₙ with bₙ > 0

Absolute convergence
If Σ|aₙ| converges then Σaₙ converges

A stronger condition

Alternating error bound
|S − S_n| ≤ b_(n+1)

Truncation error of a convergent alternating series

The error is no bigger than the first omitted term.

Power, Taylor & Maclaurin series

Taylor series
f(x) = Σ_(n=0)^∞ f⁽ⁿ⁾(a) (x − a)ⁿ / n!

Expansion of f about x = a

Taylor polynomial
P_n(x) = f(a) + f′(a)(x−a) + f″(a)(x−a)²/2! + … + f⁽ⁿ⁾(a)(x−a)ⁿ/n!

Degree-n truncation

Lagrange error bound
|R_n(x)| ≤ M |x − a|^(n+1) / (n + 1)!

M bounds |f⁽ⁿ⁺¹⁾| on the interval between a and x

eˣ
eˣ = Σ xⁿ/n! = 1 + x + x²/2! + x³/3! + …

Maclaurin series, converges for all x

sin x
sin x = x − x³/3! + x⁵/5! − x⁷/7! + …

Maclaurin series, all x — odd powers only

cos x
cos x = 1 − x²/2! + x⁴/4! − x⁶/6! + …

Maclaurin series, all x — even powers only

1/(1 − x)
1/(1 − x) = Σ xⁿ = 1 + x + x² + x³ + …

Geometric series, |x| < 1

ln(1 + x)
ln(1 + x) = x − x²/2 + x³/3 − x⁴/4 + …

Converges for −1 < x ≤ 1

arctan x
arctan x = x − x³/3 + x⁵/5 − x⁷/7 + …

Converges for |x| ≤ 1

Radius of interval

Find R with the ratio test, then test each endpoint separately

Endpoints must always be checked by hand — the ratio test is inconclusive there.

Practise with the sheet openCheatsheet