AP Calculus BC includes all Calculus AB content plus additional topics: parametric and polar calculus, infinite series, and numerical methods. About 40–45% of the BC exam is AB-only content. Master AB completely before tackling BC-only topics.
AB Foundations (Units 1–8)
All limits, derivatives, and integrals from AB apply here. Key additions: integration techniques — partial fractions for rational functions, integration by parts (∫u dv = uv − ∫v du). Improper integrals require limits: ∫a∞ f(x)dx = lim[t→∞] ∫at f(x)dx. L'Hôpital's rule applies to indeterminate forms beyond 0/0 (∞/∞, 0·∞, ∞−∞, 1∞, 0⁰).
Units 9: Parametric and Polar Equations
Parametric: dy/dx = (dy/dt)/(dx/dt); arc length = ∫√((dx/dt)² + (dy/dt)²)dt. Polar: area = ½∫r²dθ; arc length = ∫√(r² + (dr/dθ)²)dθ. Convert polar to rectangular: x = r cosθ, y = r sinθ, r² = x² + y².
Unit 10: Infinite Series
Convergence tests: nth term test (diverges if limit ≠ 0), integral test, p-series (converges if p > 1), comparison/limit comparison, ratio test (most useful for series with n! or rⁿ), alternating series test. Taylor series: f(x) = Σ f⁽ⁿ⁾(a)/n! (x−a)ⁿ. Maclaurin series to memorize: eˣ, sin x, cos x, 1/(1−x), ln(1+x).
Euler's Method and Differential Equations
Slope fields visualize dy/dx = f(x,y). Euler's method: yₙ₊₁ = yₙ + h·f(xₙ,yₙ) where h = step size. Logistic growth: dP/dt = kP(1 − P/M); carrying capacity M. Separation of variables: move all y terms to one side, all x terms to other, integrate both sides.
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