Unit 4 Summary & Key Theorems

AP Calculus · Axiom Academy

Summary Unit 4 Summary & Key Theorems Review of applications of derivatives Problems involving two or more changing quantities linked by an equation. Differentiate the relationship implicitly with respect to t Substitute known rates and solve for the unknown rate Always identify what you know and what you need before differentiating MVT: If f is continuous on [a,b] and differentiable on (a,b) , then with . Rolle's: Special case where f(a) = f(b) , guaranteeing f'(c) = 0 . Write objective function and constraint Reduce to one variable, differentiate, solve f'(x) = 0 Check endpoints and verify with Second Derivative Test Domain, intercepts, symmetry, asymptotes f' = 0 for critical points; sign chart for increase/decrease f'' = 0 for inflection points; sign chart for concavity Position s(t) , Velocity v(t) = s'(t) , Acceleration a(t) = v'(t) . Changes direction: v(t) changes sign Speeding up: v and a same sign Extreme Value Theorem: Continuous on [a,b] guarantees absolute max and min First Derivative Test: Sign change of f' classifies critical points Second Derivative Test: f''(c)>0 → min, f''(c)<0 → max

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