AP Calculus AB covers limits through integration, totaling about 60% of what appears on Calculus BC. These notes hit every rule and theorem tested on the FRQ and multiple-choice sections.
Units 1–2: Limits and Continuity
Limit laws; squeeze theorem. L'Hôpital's rule for 0/0 or ∞/∞ forms. Continuity requires: limit exists, function defined, and they are equal. Intermediate Value Theorem: if f is continuous on [a,b] and k is between f(a) and f(b), then f(c) = k for some c. Asymptotes from limits at infinity or where denominator = 0.
Units 3–4: Differentiation Rules
Power rule: d/dx[xⁿ] = nxⁿ⁻¹. Product rule: (fg)' = f'g + fg'. Quotient rule: (f/g)' = (f'g − fg')/g². Chain rule: d/dx[f(g(x))] = f'(g(x))·g'(x). Derivatives of sin, cos, tan, ln, eˣ, arcsin, arctan. Implicit differentiation treats y as a function of x using chain rule. Related rates: differentiate an equation with respect to time t.
Units 5–6: Applications of Derivatives
Mean Value Theorem: f'(c) = (f(b)−f(a))/(b−a) for some c in (a,b). Critical points where f'(x) = 0 or undefined. First derivative test for local max/min; second derivative test (f'' > 0 = concave up = local min). Optimization: set up equation, differentiate, solve f'(x) = 0, check endpoints.
Units 7–8: Integration and the Fundamental Theorem
FTC Part 1: d/dx[∫aˣ f(t)dt] = f(x). FTC Part 2: ∫ab f(x)dx = F(b) − F(a). U-substitution mirrors chain rule in reverse. Integration by parts: ∫u dv = uv − ∫v du. Area between curves: ∫ab (top − bottom)dx. Average value of f on [a,b]: (1/(b−a))∫ab f(x)dx.
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