AP Calculus BC Study Guide

Series is the most distinctly BC topic and one of the heaviest-weighted units. Mastering convergence tests and Taylor series pays off enormously.

Convergence Tests

Test	When to Use	Conclusion
nth Term (Divergence)	Always try first	If lim aₙ ≠ 0, diverges. If = 0, inconclusive.
Geometric Series	Form: Σarⁿ	Converges if |r| < 1; sum = a/(1−r)
p-Series	Form: Σ1/nᵖ	Converges if p > 1
Integral Test	f(n) continuous, positive, decreasing	Series and integral share convergence
Comparison Test	Compare to known series	If 0 ≤ aₙ ≤ bₙ: bₙ converges → aₙ converges
Limit Comparison	Similar to known series	lim (aₙ/bₙ) = L > 0: same behavior
Ratio Test	Factorials, exponentials	L = lim|aₙ₊₁/aₙ|; L<1 converges, L>1 diverges
Alternating Series	Series alternates sign	Converges if bₙ decreasing → 0

Taylor and Maclaurin Series

A Taylor series centered at x = a, and the four Maclaurin series you must memorize:

Radius and Interval of Convergence

Apply the Ratio Test to a power series to find the radius of convergence R, then check both endpoints separately. State whether each is included or excluded.

Alternating Series Error Bound

For a convergent alternating series, the error using the first n terms satisfies:

The Fundamental Theorem of Calculus

Integration Techniques

u-Substitution: When you see a composite function, let u = inner function.

∫ 2x·cos(x²) dx   →   let u = x², du = 2x dx
= ∫ cos(u) du = sin(u) + C = sin(x²) + C

Integration by Parts (BC): Use when integrating a product. LIATE rule for choosing u: Logarithm, Inverse trig, Algebraic, Trig, Exponential.

Partial Fractions (BC): Decompose rational functions before integrating.

Improper Integrals (BC)

Replace the infinite bound with a limit. Example:

∫₁^∞ 1/x² dx = lim[b→∞] ∫₁ᵇ x⁻² dx = lim[b→∞] [−1/x]₁ᵇ = 1

If the limit exists and is finite, the integral converges. Otherwise it diverges.

Slope Fields

A slope field (or direction field) shows the slope dy/dx at each point (x, y) given a differential equation. A solution curve must be tangent to the slope marks at every point it passes through.

To sketch a slope field: plug in several (x, y) values into dy/dx, draw short line segments with the resulting slope.

Separation of Variables

If dy/dx = f(x)·g(y), separate variables and integrate both sides:

dy/g(y) = f(x) dx
∫ dy/g(y) = ∫ f(x) dx

Example: dy/dx = xy. Separate: dy/y = x dx. Integrate: ln|y| = x²/2 + C. Solve: y = Ae^(x²/2).

Exponential Growth and Decay

dP/dt = kP   →   P(t) = P₀eᵏᵗ

k > 0: growth. k < 0: decay. Common applications: population growth, radioactive decay, Newton's law of cooling.

Logistic Growth (BC)

dP/dt = kP(1 − P/L)

L is the carrying capacity (maximum population). Growth rate is fastest when P = L/2. The solution is an S-shaped logistic curve.

Parametric Equations

When x = f(t) and y = g(t), the derivative is:

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

Second derivative (concavity):

d²y/dx² = d(dy/dx)/dt ÷ dx/dt

Arc length of a parametric curve from t = a to t = b:

L = ∫ₐᵇ √[(dx/dt)² + (dy/dt)²] dt

Polar Coordinates

Area enclosed by a polar curve r = f(θ) from θ = α to θ = β:

A = (1/2) ∫ₐᵝ r² dθ

Area between two polar curves (r_outer − r_inner):

A = (1/2) ∫ₐᵝ (r_outer² − r_inner²) dθ

Vector-Valued Functions

Position vector: r(t) = ⟨x(t), y(t)⟩. Velocity and acceleration:

velocity:     v(t) = r′(t) = ⟨x′(t), y′(t)⟩
speed:        |v(t)| = √[x′(t)² + y′(t)²]
acceleration: a(t) = r″(t)