AP Calculus BC includes every AB topic plus BC-only content. If you are solid on AB material, your last session should focus entirely on BC-only topics — especially series, which is the largest and most heavily tested addition.
Series — The Most Important BC-Only Topic
Series questions appear in both MCQ and FRQ. Know these tests and when to use each:
- Geometric series: Σarⁿ converges when |r| < 1, sum = a/(1-r). First check whether a series is geometric before applying harder tests.
- p-series: Σ1/nᵖ converges when p > 1, diverges when p ≤ 1.
- Ratio test: use when terms contain factorials (n!) or exponentials (aⁿ). Compute lim|aₙ₊₁/aₙ| — if < 1, converges; if > 1, diverges; if = 1, inconclusive.
- Alternating Series Test: alternating series Σ(-1)ⁿbₙ converges if bₙ decreases monotonically to 0.
- Taylor/Maclaurin series: know the standard series for eˣ, sin x, cos x, and 1/(1-x). To construct a Taylor series for f(x) centered at a: Σ f⁽ⁿ⁾(a)/n! · (x-a)ⁿ.
- Interval of convergence: apply ratio test, solve for |x - a| < R, then test endpoints separately using simpler tests (p-series, alternating series).
- Lagrange error bound: |error| ≤ M/(n+1)! · |x-a|ⁿ⁺¹, where M is the max value of the (n+1)th derivative on the interval.
Parametric and Polar Formulas
- Parametric dy/dx: (dy/dt) / (dx/dt). For the second derivative d²y/dx²: [d/dt(dy/dx)] / (dx/dt).
- Parametric arc length: ∫√[(dx/dt)² + (dy/dt)²] dt over the given interval.
- Polar area: A = ½∫r² dθ over the given angular interval. If two curves intersect, find intersection angles first and subtract inner from outer: ½∫(r_outer² − r_inner²) dθ.
Logistic Differential Equation
dP/dt = kP(1 − P/M), where M is the carrying capacity. Key facts the exam tests: (1) the inflection point occurs at P = M/2, where the population grows fastest; (2) as t → ∞, P → M; (3) if P < M/2, the growth rate is increasing; if P > M/2, the growth rate is decreasing.
Calculator FRQ Strategy
On calculator-active FRQs: always write the full integrand or equation before writing the numerical answer. Graders require the setup for full credit — a naked decimal earns nothing. Store intermediate values in your calculator rather than rounding; only round at the final answer to three decimal places.
Use Cram Mode on AimFive to drill your weakest AP Calculus BC topics before the exam.
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